Top quark and Higgs boson masses: Interplay between infrared and ultraviolet physics

grog. Part h’ucl. Phys., Vol. 37, pp. l-90, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved

Pfrgamon

0146-6410/96

$32.00 + 0.00

SOl46-6410(96)00059-2

Top Quark and Higgs Boson Masses: Interplay Between Infrared and Ultraviolet Physics B. SCHREMPP’.2 and M. WIMMER’*

‘lnstitut fur Thcoretische Physik. UnivcrsitUt Kiel, D-24118 Kid. Germany 2Deutsches Elektronen-Synchrotron DESK D-22603 Hamburg. Germany

ABSTRACT We review

recent

top quark

efforts

The Standard in parallel

Model

(SM)

First.

the question

flow is independent

point

values

is addressed

ii) infrared

an infrared

iii) a systematical mass

the

mass

bounds

of heavy

matter

particles,

level in the renormalization

Supersymmetric

Tau-bottom-(top)

extent

Standard

mass,

Model

notably group

(MSSM)

of the

equations.

are considered

Yukawa

than

infrared coupling

masses fixed

attractive unification

than

for the experimental

top

tan /3 in the MSSM,

and

of attraction.

The mathematical lines,

hierarchies

ones being

The

the lower dimensional

as an ultraviolett

symmetry

aud on the

for all these

in the corresponding

i) infrared

nontrivial

an

triviality bound

backbone

surfaces,...

emerge:

SM, ii) generically,

,j

140 GeV)

strengths

points,

fixed

prominent

in the SM as well as the upper

are reviewed.

Interesting in the

attractive

mt = O(19OGeV)sin

the most

and the parameter

respective

top

attractive

transparent.

stronger

strongly

of their and

to rn~=O(

renormalization

are i) infrared

one being

parameters,

the top mass

in the MSSM

of infrared

is made

between

of the “top-down” issues

outstanding

in the SM, leading

Higgs

physics

The central

the most

between

assessment boson

the infrared

physics.

relations

relation

on the

Higgs

space,

are more

Higgs

fixed relation

is systematically

manifolds

the

attractive

rich structure

multiparameter MSSM

to which

analytical

stability

of the lightest

features,

on masses

at the quantum

of the ultraviolet

fixed top-Higgs and

vacuum

the information

as encoded

and the Minimal

for the top and

in the MSSM, infrared

boson,

throughout.

group

mass,

to explore

and the Higgs

higher

attraction

in the

dimensional

fixed

ones. property

of supersymmetric

renormalization group flow into the IH grand unified theories and its power to focus the “top-down” top mass fixed point and, more generally, onto the infrared fixed line in the mt-tan P-plane is reviewed. The program

of reduction

tions

couplings,

tary

between “bottom-up”

IR attractive

of parameters. guided

renormalization

fixed

manifolds

a systematic

by the requirement group

evolution,

are pointed

search

for renormalization

of asymptotically is summarized:

free couplings

group

invariant

rela-

in the comptemen-

its interrelations

with

the search

Supersymmetric

Standard

foi

out.

KEYWORDS Renormalization Infrared Infrared

group

evolution

/ Standard

Model

and

Minimal

attractive fixed point top and Higgs masses and mass relations fixed points, lines and surfaces / Unification of tau-bottom-(top)

symmetricgrand *supported

unification

and the infrared

by Deutsehe Forschungsgemeinschaft

top mass fixed point

/ Higgs and top mass Yukawa couplings

/ Program

of reduction

Model

/

bounds / in super-

of parameters.

B. Schrempp and M. Wimmer

2

Table of Content 1. Introduction 2. Theoretical

framework

2.1 Standard Model 2.2 Minimal Supersymmetric Standard Model 2.3 Grand Unification 2.4 Renormalization 2.5 Relations

Group Equations

between Pole Masses and MS Couplings

2.6 Effective Potential 3. Preview

of Infrared

and Vacuum Stability Fixed Manifolds and Bounds in the SM and MSSM

4. Infrared Fixed Points, Lines, Surfaces and Mass Bounds in Absence of Electroweak Gauge Couplings 4.1 The Pure Higgs Sector of the SM - Triviality and an Upper Bound on the Higgs Mass 4.2 The Higgs-Top Sector of the SM - a First IR Fixed Line and a First Vacuum Stability Bounc 4.3 The Top-gs Sector of the SM and MSSM - a Non-Trivial

IR Fixed Point

4.4 The Higgs-Top-gs

Approximation

Sector of the SM - a First Non-Trivial

4.5 The Top-Bottom-g3 IR fixed Property

Sector of the SM and MSSM - Top-Bottom

4.6 The Higgs-Top-Bottom-g3

Yukawa Unification

Sector of the SM - a First IR Fixed Surface

5. Infrared Fixed Points, Lines, Surfaces in Presence of All Gauge Couplings 5.1 The Top Sector of the SM and MSSM 5.2 The Top-Bottom-Sector 5.3 The Higgs-Top-Bottom 6. Infrared

Attractive

of the SM and MSSM Sector of the SM

Top and Higgs Masses, Mass Relations

and Mass Bounds

6.1 Top Mass and tan13 in the MSSM 6.2 Top and Higgs Masses and Top-Higgs Mass Relation in the SM 6.3 Lower Bound on the Higgs Mass in the SM 6.4 Upper Bound on the Lightest Higgs Mass in the MSSM 7. Supersymmetric 8. Program

Grand Unification

of Reduction

9. Conclusions

of Parameters

Including Yukawa Unification

as an

Top Quark and Higgs Boson Masses

1

Introduction

The

Standard

Model

(SM)

is highly

successful

at

describing

the

electromagnetic,

weak

and

strong

gauge interactions among the elementary particles up to presently accessible energies. It has, however. conceptual weakness: the masses of the matter particles, i.e. of the quarks, leptons and the theoretically predicted

Higgs boson,

enter

of the SM, i) its embedding interactions

in a single

to be instrumental Starting

point

ultraviolet upper

review

which The main

interest

physics

issues

on (at least

and lower bounds origin

top quark

and

the

mass

collider

value

LEP

the

GeV.

which

gauge

is considered

quantum

of the

eflects

over the last decade

in the framework

or so, with

potential

peak

of this

activities

of the quantum

effects

wide

during

the

for i) relating

(IR) phy sits and vice versa and in particular for ii) providing masses. In their mildest form these informations imply particle masses. Ultimatively, however, there even appears to open

one does not have to go beyond particle

boson

masses

but

that

to be specified

are the

dynamical

origin

in search

is provided

on the

below.

heaviest

directly

the SM and its extensions

this

matter

particles

of the

at the proton-antiproton

Standard

collider

Model.

Onl!

at FERMILAR

:

mt =

176 f 8 (stat.)

f- 10 (syst.)

Ge”.

(1)

DO collaboration[2]

:

m,

199 ‘iy

+ 22 (syst.)

Ge”.

(2)

and

The top quark

agreement

with

= the

present

(stat.) indirect,

evidence

from

the electron-positron

[3]

collaborations,

the

to a variation

t,han its partners

supersymmetry

extensions

the three

particle

- in a sense

is in good

value

in prominent unifying

collaboration(l]

at CERN

central

that

Higgs

LEP

correspond

persists (GUT),

of gravity.

on the inherent

has been observed

CDF

This

theory

to infrared

of (heavy)

effects

the top quark

focuses

for (heavy)

for the dynamical

recently

performed

the heavy)

possibility

level of the quantum

implementation

been

largely

unified

by the fermion-boson

are investigations

have

up the fascinating

The

grand

one, ii) its extension

for this

(UV)

informations

This deficiency

into an underlying

for an additional

class of theories, last few years.

as free parameters.

first

errors

quoted

of the central

is much

in the heaviest

heavier

value than

fermion

mt = 178 +11 +18

:

combined

_ll-

refer when

to a Higgs varying

all the other

generation,

mass

the Higgs

quarks

and

the bottom

of 300 mass

GeV,

between

leptons,

quark

I.11

19 Ge”;

with

the

second

errors

60 GeV and

even substantially

at p = mb as determined

[4] from

QCD

m, = 1.7771 which

will also play

For the Higgs

mass

sum rules.

T::i:“,t

1000

heavier

mass

mb = 4.25 f 0.15 GeV (the MS mass

a

i-1) and

the tau

lepton

with

mass

[5]

GeV.

a role in this review. there

exits

only

an experimental

lower bound

from

LEP

[.5]

mH > 58.4 GeV

(6,

at 9.5% confidence level. From the upgrade LEP200 of LEP and the future collider LHC one expects soon an extended experimental reach for the Higgs boson. In expectation of these future Higgs searches the activities for a precise determination of theoretical bounds on the Higgs mass have increased in the recent literature [6]-[2S], where Ref. [i’] has played the role of a primer in the field. particular to a lower (vacuum stability) bound within the SM and to an upper bound

This applies in for the lightr\t

B. Schrempp and M. Wimmer

4 Higgs

boson

within

developments Altogether,

one may

interaction

given

understand

expect

the

of the vacuum

the dynamical

disparity

with

origin

respect

emerged

analyses

from

In all the above couplings;

with

these

these describe

~1. They

have

allow to relate

to be very

physics

group and

roughly

The upshot

of these

of the

order

of the

weak

(UV)

scale

or IR behaviour

and

masses

leptons.

mass

respectively. “run”

equations

and

be of O(v) already,

Higgs

order

embedded

boson

challenge

to

and also for the important

masses,

in a grand and

of the Higgs

the Higgs

clues

which

have

theory

and

scales

p. Of interest

that

p N u and

and

and

differential couplings

in this review

are scales

weak

interaction

review

scale

The

RGE

generically

scale

the notions

p --) 0. The

is

as well as

in the momentum theory.

not to the limit

effects

equations.

of perturbation

this

t,o

the Higgs-

of the quantum

Yukawa

change

accessible

throughout

are related

scale p. The running

coupled

to a differential

in the framework

unified masses

signature

of a momentum

a set of nonlinear

of the presently

lepton

self interaction

A characteristic

(RGE),

momentum

to the scale

it is a great

As mentioned

as functions

couplings,

of the order

should

and the quark

of the theory,

gauge

Clearly,

over the last decade.

A. Let us emphasize

refer

Higgs

for the strength

in two-loop

scale

(7) field.

on the top quark

but

strong

at different (IR)

v of the Higgs

of the SM, possibly

interactions,

calculated

an infrared

value

of all couplings

weak

been

N 174GeV,

the Higgs

are not constant

the response

ultraviolet

region

of the SM. the MSSM.

v/h

effects

are a measure

Yukawa

couplings

the electromagnetic,

some

masses

quarks

frameworks

in the renormalization

between

Higgs

of informations

supersymmetry,

couplings

fermion-antifermion

They

extension

for why the top and

of the quantum

mentioned

endorsed

encoded

and

to the other

lie in the wealth

is that

top

expectation

for an answer

possibly

supersymmetric in this review.

scale

in terms

mass

the minimal

will be included

(7) and

IR. scale.

UV scale

may

IR

be as

large as A = MouT N O(1O16 GeV) in the framework of (supersymmetric) grand unification. in which the theory is supposed to continue to hold up to the scale M GUT where the three gauge couplings unify. or ultimatively become

as large

Let us anticipate symmetric and

and emphasize

extension

(MSSM),

the implications

the quantitative treat

as the Planck

scale

A = Mpianck ‘v 10” GeV:

them

In solving

already and

for particle

level.

This

in parallel,

masses

makes

as intended

the RGE,

which

here that

whichever

whichever

turn

out

to be similar

it a challenging

task

interactions

are a set of first order

from

strong

physical

tend to single out special

solutions are distinguished initial value conditions. The theoretical following.

motivations

motivations

solutions

by being

in principle,

to consider

differential

There

are, however,

SM or its minimal between

all these

super-

IR and

UV physics

even

though

different

cases

simultaneously

on and

in this review.

level:

l

graviational

the framework,

the size of A, the interplay

the same inherent deficiency as on the classical particle masses are still free parameters.

essence

where

important.

of the RGE.

determined

in the literature

different

in the first

values

sources

towards

boundary

instance

of the couplings

to be spelt

From the mathematical

by suitable

pointing

equations,

the initial

out below,

one faces thus

which

the

in

point of view these special

conditions

such special

and

in contradistinction

solutions

to

of the RGE are the

Consider the so-called “top-down” RG evolution, from the UV scale 12 to the IR scale. Determine the corresponding RG flow, which comprises all solutions of the RGE for any UV initial values for the Higgs self coupling and the Yukawa couplings which are admitted within the framework of perturbation theory. An important issue [29], [6], [7], [30]-[65], [11S], (1191 has recently been to determine the extent to which the IR physics is independent of the UV physics. i.e. independent of the UV initial

values.

This

happens

if the IR behaviour

is dominated

by special

solutions

of

5

Top Quark and Higgs Boson Masses the RGE -

which

correspond

in the space -

which

Indeed

a rich

structure

Higgs

IR fixed point

[119] leads

within

fixed lines,

for the whole

of such

top and

conspicuous

points,

IR attractive

mass and

values

line with

the MSSM

fixed

and

lines,

point

relations

Higgs

between

masses

within

SM e.g.

mass

the widest

coverage

value

value

in the literature

where

tan p is a ratio

rn,~, = O( 140 GeV) These

mass

equations

values

are clearly

a very appealing

Let us also anticipate RG flow which flow is roughly and finally manifolds

imply

of course

enhances

hierarchy MSSM

from

highly

non-trivial

particle

masses.

as to attract the IR fixed interest

into

independent Next,

Higgs

the

and bott,om

and tan 0. Within

the

(9 ) of the RG

is certainly close

the MSSM: than

in their relations

enhanced vicinity.

context.

even

upper

(triviality)

the

if the IR attraction

though

the

in 111~.

SM ones. correspond

to

and thus

between

is sufficiently

strong

RG flow comes

the UV to the IR. This

the IR fixed

pattern

interesting

manifolds

since they

couplings

the

a fixed line

This

A further

IR fixed

of

the RG

dimensional

couplings.

own right, between

from

along

the higher

in the corresponding

Of course path

i) Typically

this surface

out that

the involved

attractive

is the evolution

in this

within

IR fixed manifolds.

to be interesting

their

the longer

high UV scales

between

SM with

strongly

in the IR attraction

in this review.:

manifolds

the closer

to

explains

an

themselves

are

of the UV scale.

let us place

the

well-studied

and

lower

the Higgs mass and the top mass in context with IR attractive “top-bottom” flow towards them. The existence of these bounds strongly

and

IR attractive

into IR fixed manifolds

say, then

or to RG invariant

significance

RG flow into

manifold

are

the top,

hierarchies

dimensional

of the

more

for couplings

Their the

value,

results

with less

mt = 176 GeV.

Now, it turns

relations

of higher

may be considered

values

value

of the material

a fixed surface,

in the comparison

IR fixed manifolds

values

to

the research

the fixed point.

seem to be systematically

RG invariant

the experimental

the top mass

(phenomenological)

towards

the importance

emerges

and make

the presentation

this line towards always

leads

expectation

subject.

first attracted

along

[49]-(6.51.

value (8) is well compatible IR fixed point leads, though

between

between

for the experimental

two interesting

emerge

relation

interesting

most

[43]-[48],

of two vacuum

Furtherreaching

relation

mass

out

The

(S)

mass or. more generally,

top-Higgs

singling

masses.

-2OOGeV)sin,$

also of O(210GeV).

the top and Higgs

exists,

between

of

sector

the SM and an IR attractive

an IR attractive

to fixed manifolds

surfaces,...

relations

of the MSSM. The top mass the experimental value (2). Within the SM th e corresponding not too far from conspicuously, to a mass value 0(215GeV), IR fixed

for the Higgs

points,

IR attractive

to a top mass

to a fixed line in the tan a-mt-plane,

characteristic

in general

RG flow.

mt ‘v O(190 and

fixed surfaces,...,

of couplings,

are IR attractive

IR attractive [llg],

to fixed

of ratios

IR attractive

fixed manifold

and

the shape

of these

(vacuum

stability)

bounds

foi

fixed manifolds and the RGE may be traced back to the most

bounds

in the multiparameter

space

strongly reflects the position of this IR attractive manifold. In fact, since the evolution path from the UV to the IR is finite, t,here are IR images of UV initial values which fail to reach the most strongly IR attractive manifold: it is their boundaries which constitute the bounds. Clear11 the bounds will be the tighter the longer is the evolution path. i.e. the higher is the UV scale. The bounds are thus straight consequences of the perturhatively calculated quantum effects. In certain approximations they are supported by non-perturbative lattice calculations which will also be included in this review. The bounds to be discussed are the so-called triviality bound. a ton mass

dependent

upper

Higgs

mass

bound,

and the vacuum

stability

bound

a top-mass

dependent

6

B. Schrempp lower

Higgs

already Higgs

mass

LHC l

mass

been

bound

future

the headline One

the ongoing

is within

interest

supersymmetric

or even furtherreaching

property

it does not single

above.

furnishes

It rather

RG flow [49]-[65] point

and

focuses

line than

IR attractive

fixed

accompanying

in the

the

and

solution

relations

at LEP200

and

RG evolution

IR region

much

runs

line which

closely

ii) it appears

allows

very

couplings

couplings.

thus

In

reducing

out

i) this

the

IR attractive

to be the

the implementation

of

at the UV

in the sense proclaimed

values,

onto

appealing

a unification

of the gauge

As it turns

more

the

Yukawa

UV initial

RG flow.

one;

from

to provide

of the RGE

between

“top-down”

unconstrained

point

models

the unification

out a special

in the

It starts

unified

of the tau-bottom-top

symmetry

of free parameters

.4s has

on the lightest,

at FERMILAB.

of the “top-down”

unification.

of some grand

A=MGUT

the first instance number

bound.

for the Higgs

the top mass

in the context

grand

the tau-bottom

scale

to pin down

top mass

in the SM, resp.

unification;

symmetry

unification

upper

mass

in view of the search

efforts

and economical grand

dependent

on the Higgs

are of high actuality

and

Yulcawa coupling

access

a Higgs-mass

the bounds

issue [49]-[65] of high recent

A second

-

earlier,

in the MSSM,

in the near

under

or, conversely,

mentioned

and M. Wimmer

very

the

constrained fixed

existence

of the

of tau-bottom

Yukawa

unification. -

Another

access

implements is only . The

of interest

complementary

scale

theory)

of the RGE gauge

asymtotically a systematic

There

which

and

of the RGE,

it certainly

the special

. supersymmetric

between

solutions

approach grand

and

RGE.

from

the

IR

to all orders

subject

search

asympfof-

for special

self coupling

towards

this approach

in perturbation

simultaneously in such

zero. i.e. become

solutions

a way to the

simultaneousl! 011

has been the first to concentrate to the implementation

developments

in the systematic

of asymptotic search

for special

IR attractive.

the results

of the RGE

approach

that later

of the

become

and the Higgs

of the RGE,

to being

ii) they

to a systematic

simultaneously

has influenced

subject

which

from

the different

are singled

are solutions

which

approaches:

out as IR attractive

implement

asymptotic

(at the one-loop freedom

within

the

vice versa;

unified

fication or parameter reduction the IR fixed manifolds; l

decrease

evolution

(in principle

such that

coupling

be emphasized

interrelations

level) in the “top-down” “bottom-up”

they

which

also

to the SM amounts

search for special solutions

solutions

. among

that

It should

freedom;

are interesting

as applied

manifold

to p + oo), the direction of evolution of reduction of parameters. The central issues are

program

as possible

fixed

at all scales p, which again

unification

the MSSM.

invariant relations

link the top Yukawa

coupling free.

couplings

an IR attractive

“bottom-up”

mathematically

so-called group

as many

free. The program

strong

A (and

exists coupling

theory,

is the so-called

UV scale

renormalization

between

there

Yukawa

in the supersymmetric

by the interesting

to establish

that

top-bottom

approach

up to some

advocated

idly

is the observation

approximate

theories beyond

with

additional

the grand

features

unification

like the tau-bottom scale

MG”T drive

top-bottom Yukawa coupling unification may be viewed as an UV symmetry from grand unified theories, it also appears to be encoded in an IR attractive implies approximate top-bottom Yukawa unification ut all scales p.

Yukawa

uni-

the RG flow into

input, as motivated fixed manifold which

Thus, different aspects pointing towards special solutions of the RGE may be viewed as different facrts of some global regularities in the interplay between IR and UV physics. This is a strong incentive to review all these issues under the same headline as intended in this review.

7

Top Quark and Higgs Boson Masses Altogether, RGE

it is clearly

for couplings

which

extent

IR and

Let us next

a physical

which

new physics

where

Here

is where

a large

Mour

N 0(2 The

scale The

material

in the

masses

and to

are addressed of the SM it is the scale at

it is envisaged

that

the SM is embedded

scale A; accordingly

The

UV scale

at which

the large

0( 1015 GeV)

becomes range

is realized scales

the SM can be viewed

will presumably

gravity

possible

GeV)

be smaller

important.

than

In case

of

O( lo3 GeV) 5 A ,< bfnianck 2

e.g.

in a technicolor

A are possible,

the SM into a left-right

e.g.

scenario,

accounting

symmetric

gauge

for

theory.

(18) and revival,

appears

into the discussion.

Extension

of the Standard

boson-fermion properties

supersymmetry

allow

a grand since

to work

is reasonably

in this with

relating

Sect.

3 serves

parameter

space

naturally

unification

unification

out very

considered

the large

Model

two vastly

scale

of the

M~nr.

gauge

different

which

scales

Furthermore.

couplings

well quantitatively

only in the grand

(MSSM),

imple-

into the SM, has its merits.

First

in the the-

the MSSM

has

at a unification

in supersymmetric

unification

to finally

scale

grand

framework

with

culminate sector

couplings,

Sect.

the SM and the MSSM of the respective Higgs-top-bottom

5 treats

unifi-

a high

This

which from

I-V

strictly

of IR attraction masses

relations,

the detailed they

derivation

include

of t.he IR fixed

the IR attractive

relation

in developping

a

fixed surfaces....

pedagogically,

to include

sets

analyticall!

procedure

also

for the pure

Higgs

in absence

Sect.

allows and

a the

of the electroweak

couplings.

the IR attractive between

-4 and

the coupling

(for the Higgs selfcoupling)

also a (largely)

manifolds.

for the top and Higgs,

is spent

This

relevant

of the electroweak

in Sects.

in which

parameter

in the literature.

the

of all radiative

fixed lines,

space reduced

calculations

inclusion

in parallel;

effort

for

for grand

is also included.

in detail

the material

considered

lattice

4 provides

the non-trivial

5 much

fixed points,

to develop

have

A collection

developped

from a one-parameter allows

developments

results

fixed point

4 and

basis

background

pole masses

in form of a table,

IR attractive

is enlarged

the latest

are treated

mass

In Sects.

non-trivial

publications

of the SM. Sect.

for the material

fixed manifolds

by entry.

space.

with

strengths

IR attractive

guideline

which

non-perturbative

the theoretical

Some minimal

logical

entry

of couplings

2 summarizes

of Yukawa couplings is provided. in the MS scheme to the physical

into the highly

of pioneering

with

Higgs-fermion

MSSM bounds

and

Sect.

masses

of ail IR attractive

to a five parameter

body

as follows.

the SM as well as in the MSSM.

on unification

is enlarged

of ratios

comparison

resulting

is organized within

the running

insight

by step

gauge

emphasis

a summary

comprehensive in a space

review

as a preview

: it contains

and

scale

RGE evolution

unification

and

encoded

A = IMoor.

corrections

step

effects

and lepton

N O( lo*’ GeV) is appropriate in a grand unification scenario. Though whether this unification can work out on the quantit,ative level, we shall

a strong GeV)

quark

In the framework

Intermediate

Supersymmetric

interaction lOi

perturbative

5

particle.

renormatizability

MSSM

the scale

as A = 0(103

way the appealing

experienced

questions

momentum

or for embedding

UV scale

Minimal

of all its improved

cation.

as small

as A = Moor recent doubts

the

these

to consider

and quarks

the quantum

further

Generically

at a higher

motivations

of leptons

in a minimal

ory, the weak

which

as the UV cutoff.

is a composite

to include

recently

A acting

boson

A scale as large there are strong

ments

within

theory

.4n UV scale

compositeness

continue

with

are physical

extent

top and

are interlocked.

A_< MpranC~ N lO”GeV,

the Higgs

to which

the Higgs,

the SM is encountered.

underlying

mass

O( 10” GeV).

to trace

of the UV scale A is required.

beyond

theory

issues

the scope

interpretation

as an effective the SM there

about

specify

in a more complete Planck

task

information

UV physics

further

First,

the

a fascinating

yield

In both

analytical 6 then

summarizes

top-Higgs,

the top mass

sections

assessment

I he

top-bottom

and tan 9 in

I hr

on the level of the present state of the art. The dynamical origin for the triviality (upper) and vacuum stability (lower) bounds in the Higgs-top mass plane of the SM is developed step

by step in Sects. 4.1-4.4; Sect. Higgs mass from various sources

4.1 also contains (including lattice

an estimate calculations).

of an absolute upper bound on the S\I The most recent determinations of the

SM bounds as well as an upper bound for the lightest Higgs boson mass in the MSSM are presented in Sects. 6.3 and 6.4. Sect. 7 is devoted to the interrelated issues of implementing tau-bottom(-top) Yukawa unification into supersymmetric unification and the IR attractive top fixed point mass which

8

B. Schrempp

has received

so much

of parameters

2

attention

In order book

to render

material

physics

the review

addressed

Standard

among

selfcontained hand,

on the one hand

we shall

introduce

particle

theory

comprises

is broken

elementary

particles

spontaneously

The field content tions,

quark

symmetry

breakdown.

generation

of quarks

and

in detail

the theory

derive

to SU(3)

of the theory

the fermionic

the program

of reduction

for IR attractive

to avoid only

repetition

those

manifolds.

of too much

elements

the Glashow-Weinberg-Salam

chromodynamics,

SU(3) which

S summarizes with a search

pertinent

text

to the

in this review.

[66] and quantum

interactions

Sect.

Model

The SM of elementary teractions

Finally,

to the SM and its interrelation

Framework

on the other

issues

2.1

in the literature.

in its application

Theoretical

and M. Wimmer

is given

and

lepton

from

a local gauge

x N(2)

x U(1)

x U(l),, by means in terms

matter

For the purpose and leptons

of strong

of this

consisting

principle

fields,

the Higgs

review

with

gauge

in-

fundamental

group

which

mechanism mediate

field responsible

we confine

of the left-handed

of electroweak

[67]. These

(10)

of the Higgs

of the gauge

fields and

model

interactions

the discussion

SU(2)

the gauge

interac-

for the spontaneous to the third,

top-bottom

heaviest

and tau-neutrino-tau

doublets tL

4L

(bLL

=

(11) and the corresponding a(z)

with

U(1)

right-handed

hypercharge

SU(2)

a= where

the suffixes

The most

general

+,O characterize gauge

tR, bR, q.

singlets

invariant

the electric

The complex

SU(2)

doublet

Higgs

( $0>

charge

and renormalizable

(12)

9

+l,

0 of the components.

interaction

Lagrangian

is

t = -Cgaugt + LY"krev.¶ - V( @). L e=w contains

the gauge

9s. ~2, gr, with

91 normalized

interactions

in terms

as motivated

contains

the

Higgs

field

self interaction

of the respective

by grand

V(@) =

field

+

Y = 1 is

(1.7)

SU(3)

unification,

x SU(2)

g1 = $&/.

x U( 1) gauge

couplings

The potential

-r&t@ + X(&q2

in terms

is parametrized such that the Higgs field acquires spontaneous electroweak symmetry breakdown

of the

a priori

a vacuum

0 <@>=Jfi ( v >

(1J) unknown

expectation

Higgs

self coupling

value

responsible

A.

It

for the

(1.5)

9

Top Quark and Higgs Boson Masses with

(16) The numerical

value

of v is given

in terms

of the Fermi

G,D = 1.16639(2)

constant

10-s

GeV-2

(17)

to be l/2 = 246.218(Z)

v = ( fiGF)Of the four Higgs degrees

degrees

of freedom

The remaining

of freedom

three

for the massive

one corresponds

are Goldstone

weak gauge

degrees

bosons,

to the physical

thus

Higgs

GeV.

(18)

of freedom,

providing

boson

furnishing

the W boson

the longitudinal mass

mry = trgz/‘L.

field

h = v’$Recj” - U/I,‘?).

ICvut_._ describes

the interactions

of the doublet

~Y”kaWB

Cp’ = irz@* is the charge top,

bottom

and

All masses

tau

in the

zi. The weak (18).

The tree 2’ and

conjugate

Yukawa

boson

level top,

their

-tJ&@‘tR

-

field with

gbqL@bR

of (9, rs the second

-

the fermion

&@Tfl

Pauli

+

matrix,

matter

fields

h.c.

(‘0)

gt, gb, g7 are the a priori

unknown

couplings.

SM are induced

gauge

value

=

Higgs

(19)

masses bottom,

respective

by the spontaneous allow tau

to determine and

Higgs

symmetry

breakdown

the size of u which

masses

are given

and

are proportional

was already

in terms

introduced

of the vacuum

to in Eel.

expectat.ion

couplings (21)

and mH = 67~. SinCe

St,

gbr

QT

and

,! are

free parameters,

the

tree

(“2,

level

masses

mt,

mb. m,

and

mH

are

a priori

undetermined.

Minimal

2.2

A strong

reason

properties, without Within

Supersymmetric to implement

which running

allows into

the SM higher

Standard

supersymmetry

to retain

the

Higgs

into boson

the (interrelated)

problems

order

to the Higgs

corrections

the

Model SM is the

as elementary

of naturalness,

improvement particle

fine tuning

mass are quadratically

in renormalizability

up to a high and

divergent,

does not supply

any dynamical

mechanism

which

allows

naturally

.1

i.e. the “natural”

mass to O(v) only

size of the Higgs mass is the high UV cut-off A. Renormalization brings down this by means of an unnatural finetuning of parameters order by order in perturbation theory

UV scale

hierarchy.

theory.

the coexistence

Thus,

the

of two vastI>

different scales, the weak interaction scale and a very high UV scale, as is e.g. necessary in grand unified theories. This is the hierarchy problem. A dynamical mechanism could be supplied by an appropriate additional symmetry. This is indeed the case for supersymmetry. In a supersymmetric multiplets containing divergence partners

is naturally

textbook) particles are classified in supertheory (see Ref. (681 for an excellent bosons and fermions. In a supersymmetric extension of the SM the quadratic cancelled

of the SM particles

by the related

contributing

loop diagrams

to the divergent

involving

loops.

the fermionic

Supersymmetry

has,

supersymmetric however,

to he

10

B. Schrempp and M. Wimmer

broken in order to account for the fact that so far no supersymmetric partners for the SM particles have been found experimentally. A soft supersymmetry breaking at a scale Msusv close to the weak interaction

scale

The masses

of the supersymmetric

more

can

on this scale

In order

(MSSM)

this

way the

partners

naturalness

and

will be of the order

hierarchy

the

implications

of the

for the renormalization

important

ingredients

Following

the text

group

book

Supersymmetric relevant

For excellent

[70] the interactions superpotential

Minimal

equations

have to be introduced.

the supersymmetric

tensor

of the unknown

Here

fir

HI,* the respective

and

Yukawa

of Higgs

given

couplings

fiz are the two Higgs

tau-neutrino-tau

superfields,

reviews

bosons

extension

of the

see e.g.

and third

Refs.

masses

Standard only

a few

[69], [70].

generation

fermions

is obtained

by

ht, hb, h, and

way.

The

up type

top quark

forbidden

in Eq.

the anomalies scalar

invariant

They

the scalar

squark

two Higgs

doublet

+ ~t,,A,tEj,’

the parameter

and

(23)

/I; t,, is the antisymmetric

at tree

is given

V(H,,Hz)

of their

in terms

slepton

fermionic

quark

to Eq.

Hl =

singlet

scalar

top,

sector,

to i) provide

the appearance

and ii) provide

bottom,

masses of fi;

The dimension

four terms

involve

of the supersymmetric

mutual

nonvanishing

vacuum

expectation

( s) =

< with

nrr 2)~ positive

with

u given

ur ,

in this

supersymmetric

and gauge

HI.2 of the superfields

;g,Z~H;*H;~“+p2(H;‘H;

H1,2r as follows

+ H;‘H;)

parameter

(24)

couplings

for quark

(2,5) gr and g2 exclusively,

masses

requires

both

a characteristic

Higgs

fields to have

7~2 which may be different < Hz >= i

7

(26)

v5

and

?I; + 21; = v2 in Eq.

is of

( c;;) = q

gauge

The necessity

values

i?;

cancellation

(12)

the electroweak

theory.

tau

for the

and

Hz=($)= ($)d2 feature

compo-

top-bottom

fiz.

arising

field components

- H;‘H;)‘+

in analogy

in order

of A, and

level for the Higgs

S11(2)

(since

components)

components

the

their

weak doublet

lepton fields their respective superin a gauge The SU(2) m d’Ices are contracted

fields. bottom

besides

and

are necessary

type

= ;(;g;+g;)(H;“H;

field notation

T, B, i are

and

of the scalar

containing L are the SU(2)

the SM quark

superfields

by the fermionic

field potential

superfields,

partners-Q,

besides

as well as for the down

introduced

the H&s

doublet

respectively, contain

(23) on account

theory,

SU(2)

supersymmetric

superfields,

partners,

invariant

with

chiral

respectively.

symmetric

The

resolved.

in two dimensions.

nents and

remain

We shall elabomte

for the SM particle

W = Eij(htQ’fjzjrS’+ hbQ’Ei~~ + h,i’~j;i) in terms

problems

of this scale Msusv.

at the end of this subsection.

to understand

Model

from

be arranged;

(18).

in the MSSM

This defined

leads

to the sensible

(27)

introduction

of an angle

B as an additional

tan d = D~/v,

(‘S)

with 0 5 p 5 lr/2. In terms of B the tree level fermion masses become moment the effect of the soft SUSY breaking to be discussed below) v

mt = --hf fi

key

by

sin 0.

mb = Iha

Jz

cos d.

naively

mr = -h,cosD. d3

(disregarding

for the

(291

II

Top Quark and Higgs Boson Masses Of the eight the weak lightest

Higgs

gauge

degrees

bosons

one of them

two charged .4 soft

and

is the SUSY

a CP-even

supersymmetry

particles

must

breaking

is achieved

dimensional

The

would

and small

enough

In this

masses

(smaller

we are tau

all masses,

have masses than

concerned

masses

the MSSM

and RGE

the

lumped

into one scale,

the supersymmetry

parameters.

This

will be also adhered

in practical

typically

mt to several

TeV. at most

below

‘The transition usual

mz

soft mass

and

of MSSM

running (but

part,ners

lower experimental

In crude

applications

RGE

spectrum

responsible

at energies

are valid,

the SM RGE

high

with

at energies

hold.

or heavy

boson

region

approximation,

10 TeV. In Ref.

couplings

thr

region

is

mass threshold

is

can be approximatpl!

the size of &1sl,su

gt(Msusu-)

=

&(&usu+)

pointed

is expected

allowed

out that

to varh

even valuc,s

gb(&LJSY-)

=

~b(hl_EY+)c’=

=

h,(Ms~sY+)

case that

out the heavy

at p = A4srrsv is approximat.ed

for i = 1,2,3.

(Xl)) (31) IX)

9.

(33)

cos d.

the heavier Higgs

Higgs

bosons

field combinations

are sufficiently

much

at the scale /I = .I&rsv

ho=~((Re~~-111/~)cos9+(Re~~-~~/~)sin~), field ho to be identified to the SM Higgs.

Higgs

selfcoupling

While

x(MsusY-)

=

heavic>I leavcss (34)

with the SM Higgs the SM Higgs

X is subject

as

conditions

sin d.

g~(hfsuSY-)

not unlikely

couplings

matching

gi( nfSUSY+)

one, integrating

difference

to

well below

as a free paramet.er.

[73] it has been

to SM running

=

considered

level the MSSM

the

for respect

The intermediate

Higgs

this transition

it is treated

g*(n4USY-)

In the frequently

a crucial

and

bounds

The resulting

on the

scheme)

effectively

not differentiable)

than the lightest the combination

boson

Higgs

divergencies.

may be appropriate.

by the continuous

the light Higgs

for the

and slept,ons.

of quadratic

natural.

r!.

(higher

terms

the supersymmetric

Generically,

bosons

potential

for the squarks

with

S\I

[71],[51].

mass.

to in the following.

- 0( 1 TeV),

for Msus~

Refs.

of supersymmetry

that

Higgs

Higgs-squark-antisquark

that

Higgs Higgs

the

of the

supersymmet

scale hf. susy, absorbing the effect of the soft supprsymmet I.! of the complex situation has been widely used in the literature

idealization

to be of O(v) from

t.rilinear

effect

[72]-[74] h owever,

into

spoil the cancellation

in the RGE each time a superparticle

It has been argued

and

with

the (lightest)

partners soft

to keep the theory

see e.g.

to The

ones comprise

supersymmetric

not to be in conflict

in order

of masses,

and the heavy

by a change

passed. breaking

enough

particles.

the heavier

by supersymmetry),

by the two conditions.

large

to give mass

Higgs

.4n appropriate

soft mass terms

and

parameters

boson,

two terms

bosons,

leptons.

the

limits.

(in the MS renormalization

of the superpartners

characterized

and

l-10 TeV),

range

since

achieved

of the gauge

serving

to physical

boson.

dimension

the naturalness

of freedom.

SM Higgs

Higgs

experimental

None of these

over a whole

review

bottom.

of the physical

to be introduced

are monitored

bosons

degrees

five correspond

neutral

their

of the quarks

new free parameters

will be spread

are Goldstone

additional

destroy

couplings.

Higgs

has

beyond

superpartners

superpartners

the heavy

top,

masses

the fermionic

the scalar

analogon

by introducing

slepton-antislepton

three

and a CP-odd

breaking

have

terms

gauginos,

of freedom,

as in the SM, the remaining

field h below

selfcoupling

Msusv.

There

X is undetermined

to the tree level condition

is. however. at the tree

at p = fifsusu

;(~g:(nfs”,Y) +g:(‘~~susY))coszw,

( 1%

)

which leads to a tree level Higgs mass rn~* 5 m$ co? d 5 rni. The Higgs mass is lifted by radiative corrections as a function of the top mass and the size of the scale Msusv. which will be summarized in Sect. particle

2.6.

The

relation

in the MSSM.

(35) is the origin Since the Higgs

for the rather

selfcoupling

low upper

is fixed at Msusy

mass

bound

in terms

for the lightest

of the electroweak

Higgs gauge

12

B. Schrempp

from p = Msusy down to p = 1))~ to be rather small, it has only the SM RC 1 evolution to increase its value and correspondingly the value of the Higgs mass. In contradistinction to

couplings available

the SM, where

the upper

in the MSSM in Sect. This and

and M. Wimmer

depends

Higgs

mass

on Msusv.

bound

depends

How this works

on the UV scale

out in detail

A, the upper

in professional

Higgs

analyses

mass

bound

will be reviewed

6.4.

idealized

MSSM

the couplings

applications

framework,

relevant

- to involve

key parameter

the effective

Grand Unification

Grand

unification

is a magnificent

it is an appealing

scheme

between

to describe

top,

allows

UV initial

values

theory

[75] the

RG

evolution

of the

gauge

bottom

parameter

theoretical

which

the

and tau masses, may be viewed ht, hb, h,, X (the latter one below Msusy),

the free parameters

tan p and

2.3

relations

appropriate

for the Higgs,

Msusy

(varying

framework

to single

for couplings

between

in itself.

out

From

solutions

or in short,

which

bounds).

the point

of the

couplings

- for practical the new SUSY

RGE

constrains

of view of this review,

by providing

symmetry

the “top-down”

RG flow

considerably. In a grand group group

unified

mathematically respect

by the grand

to (irreducible)

responsible

for the

symmetry global

spontaneous

between

unification

of this review,

unification the three

gauge

gauge

At p = MoUr

p = Msusv,

thus,

and

below

will be given

This scheme

an underlying

=

gauge

theory

again

is successful

point

to discuss

=

breakdown

$72(p

scale

grand

with

a gauge

and lepton

Higgs

sector

symmetry

fields with

of the theory,

to the

SM gauge

framework,

endorsed

[76] with

and superstring

theories.

Excellent

unification

are e.g.

Refs.

[77].

a kind of minimal

framework popular in the literature. at the M our scale a symmetry r&tiOIl

to establish

=

MGUT)

p ,< Moor

to the RG evolution Sect.

gauge

unification

in a minimal =

&i (p

unifying below

to the “top-down” (two-loop) unification it is subject to the

in the next

essentially

unifying

for supergravity

of the grand

of the

of the quark

and the specific

The grand

is assumed

&UT)

subject grand

group

of the SM as provided

for values

explicitely

the classification

of the grand &four.

starting

it suffices

couplings

gr(p), gs(p). gs(p) are minimal supersymmetric form

scale

symmetry

the spontaneous

effective;

group.

and supersymmetric

91 (p

becomes

into

of the unifying

is the natural

on grand

For the purpose

gauge

breakdown

unification

supersymmetry,

The grand

unifying

representations

at the grand

textbooks

.IS embedded

SM

The minimal grand unifying gauge the SM gauge group SU(3) x ScI(2) x ci(1). further groups of interest are e.g. SO(10) and Es. The different scenarios are defined

containing is ScI(5);

=

SU(5)

theory (:36)

&VT).

gauge h4 our

group the

RG evolution RG evolution

of the SM; the correponding

to the SM gauge running

gauge

group

couplings

of the SM; in case of the MSSM down RGE

in their

of to

two-loop

2.4.

if i) the initial

value condition

(36) combined

with the high precision

data for (L and sinBw at p = mz leads to a (two-loop) value for gs(p = nz) compatible with data and ii) if MQJT turns out to be sufficiently large, in order not to run into conflict with the experimental limits on proton decay which is mediated by the exchange of heavy gauge bosons of the grand unifying gauge

theory.

The MSSM

has the advantage

over the SM of an additional

parameter,

the effective

scale

Msusv, which, however, for consistency reasons is strongly constrained as has been detailed in Sect. 2.2. (Threshold corrections and non-renormalizable operator corrections at the high scale as well as at the low scale are usually neglected; see Refs. (72]-(74) f or an estimate of these effects). Applying these criteria, recent reevaluations singled out supersymmetric grand unification 31016GeV.

and

the strong

gauge

coupling

of gauge coupling unification [78],[.55],[73],[59],[74] as successful with a grand unification scale bfo”r [74] gz(p

= mz)/(4*)

‘v 0.129 * 0.010

being

have z 2.

a bit on thr

13

Top Quark and Higgs Boson Masses high side, but within without

the errors

supersymmetry

This revival called

of interest

SIi(5)

grand

Yukawa

unified [49]-[53]

with

disfavoured

into supersymmetric

in the literature

unification

compatible

is strongly

grand

coupling

theory

that

the experimental

unification

unification

it implies

the

Yukawa

couplings unification

also of the

symmetry

more

generally

which

involve

appealing

fermion

(quark

is a symmetry fields

with

in grand Higgs

respect

unified

fields

and economic

symmetry

property

Implementing mass

and

2.4

the MSSM

theories

provided

order

that

within

their

explicit

of tau-bottom

Yukawa

coupling

=

in some

Yukawa

&UT)

=

fascinating

for Yukawa

hiit holds

couplings Even

at the scale

more

p = Jlorlr. (:Is)

&XJT)i

complex

tau-bottom-top

constrains

unification

repectively.

a single

or even

strongly

tan p. These

=

involving

of tau-bottom

unification

EC theories

unification

ht b

Yukawa

of the fermions

Yukawa

27 representations,

SO( 10) models

Higgs

Yukawa

the IR parameters

investigations

unification

such

will be reviewed

IO-plet

as the top

in Sect.

7.

Group Equations of an interacting transmutation);

any given

=

correspondingly

tau-bottom

SO( 10) and

and

fields;

on the classification

Thus,

as SfJ(5),

h6 (p

[:$7)

as well as Higgs

group.

5,lO

properties

the parameters

that

pendences

e.g.

parameter

(dimensional

ever,

such

MGUT)

and renormalization

to define

lepton)

of tau-bottom-top

grand

Renormalization

Quantization

the interest into what is feature of the minimal

= h* (/I = :W,,,).

to the gauge

is the option

supersymmetric

of a momentum

property

not only dependent

in the fundamental

the UV symmetry

into minimal

and

property

hr (p = a

SI!( 5) unification

at p = Mour

involve

Higgs

The grand

level.

has also renewed It .IS a well-known

[49]-[65].

h, (/I = MC”T)

coupiing

value.

on the quantitative

of the theory. order

of the renormalized

is cancelled parameters

introduces parameter

The important

of perturbation

p dependence

field theory

it is a hidden

a scale /1 with which

property

of RG invariance

measurable quantities

theory

by implicit

p dependences,

of the theory.

the dimension

has to be introduced

i.e. couplings

ensures.

in horv-

are ,u independeni.

introduced and masses,

through

i.f..

the p de

and of renormalized

wave functions. This

implies

in particular

probed;

vacuum

electric

coupling

that

polarization which

grows

of the SM to a differential the from order

differentially all those

virtual

differential

~1. Quite

resolution:

particle

emissions

theory.

This

vary

with

for example,

the

screen

generally,

momentum

the electric

the response

transfer charge,

with

resulting

which

of the set of renormalized

couplings

from

p to p + d p reflects according to the uncertainty corrections it allows to “see” higher order radiative and reabsorptions

response

is summarized

due to allowed in the RGE,

interactions a system

t,hey arc

in an effective principle resulting

in the considered of nonlinear

coupl~l

equations.

The

renormalization

mass

independent

in Refs.

with

scale change

increased

of perturbation

the couplings effects,

group

equations

MS renormalization

[79] and for the couplings

up to two scheme

loops

in perturbation

for the couplings

theory

gi, gi, gi, h:, hz, hz of the MSSM

in Refs.

may be found in Ref. [55]. The Yukawa couplings of the first and second are so small that they lead to negligible contributions to all quantities are assumed to vanish identically in the following. Then the two-loop renormalization group equations. form in terms of the common independent variable

were

calculated

gf, gi, g$, g:, gt. g? and [SO]; a compact

in the

X of the S.\I summar!

generation quarks and leptons relevant in this review. The!-

This assumption also precludes generation mixing. valid well above p = m,, mu, take the following

14

B. Schrempp and M. Wimmer

In the SM:

dg:=g, dt

4

41 iij

8x2

dgzZ=g24_19 dt 87r2 ( SC!& dt

87rZ

dg: -= dt

gt s

+&($gt+;g;+;g:-;g:

+ &(ig:

6

( -

7

+ &

+ ;g:

(kg:

+ 129,’ - ;g: - ;gt - kg:))

(40)

(41)

+ ;g: - 26g,’ - 2s: - 2g:))

(42)

2

- ;gf

(

+-&

- ;g; - 8g,’ + ;g: + fg: + g:

( Eg;

+ig:g;

- ig:g;

+ ;g:g:

_12gf - ;,;,: dgbZ=& dt

1 - -$

8n2 ( +--&(

-agf

- Tg:

+ zg;g:

+ Eg:g;

+ ;g:ga’

- ig: - igfg;

+ 9g;g: - 1089; + gg:g:

+ $g;g3

+ 36g;g: + 4g:g:

+ zga2& - ;g: - 12g;X - 4gfA + ,x2))

(43)

9 - ;g; - 8s: + ;g: + ;g; + g;

- ;g:

237 +xg:gf

- ;g:g;

+ ;,:gZ - ;gfgf

- $:

+ ;g;g:

+ $3’

+ 9g;g: - 108g; + ;g:g:

+ $g;g:

+ fg:g;+

- 12g,4+ zgfg; - ;g;g;

49329:+ 36g;g:

- ;g, - 4g,2X- 129,2X+ 6A’))

(44)

2 9 9 9, - ig: - ,g; + 3g: + 39: + gg: 8?r2 (

dg2 --I=dt

+&(Zg:

+;g;g: -Tgf dA dt

- ;g: - 591,)

=

+ ;g:g:

+ zg;g:

+ yg:g:

+ ;g:g:

+ Es:2

+ ;g:g:

+ 209329:+ 2og:gb2

+ igfga’ - Tg;

1 27 %g; 167r2 -(

- $g:

9 + ,og:g;

- ;g;g: 9 + $

- yg;gf

9 - ;g:x

- 3g; - 12g;X + 6h’))

- 9g;l - 69,“ - 6gb4 - 2gT4

+lZgfX + 12g;X + 4gjA + 24X2 1 3411 +16*2 ( - =g; 9

-,g;s:

63 + iyjg:g;g:

1677 - ~g:gzz

289 - og:g;

27 33 + iijg:g;gbz + 1og:s;gz

305 171 + 16926 - loog;g: 9 - ig;g:

+ $gfg:

9 3 - 49249: - jg:9:

(4.5)

Top Quark

and Higgs

+SOg;g;X + 8Og;g;X + ?,:A’

Masses

+ 108g;X’

- 6gt4g; - 6gtgb“ + 30gb6 + 10gT6 - 3gt4X + 6g:g;X - 3gb4X

+30gt6

-gr4x - 144g:X’

In the

Boson

- 144g;X’

- 48g;X’

- 312A3))

(46)

MSSM:

4

ds: dt

;

91

=

dgzZ= dt

924 8x2 (

dg,2= dt

A__

dh;

1

hZ

8x* (

13 -- 15g;

+&

+;g:hi d h;

dt=

(sg:

d hZ,

+ ;g:

- ;h;

- ;hf

- 392’ - ‘6 * + 6h; + h; 3 g3 + ;g;

+ 8g;g,’

g; + g;g; + $7:

+ ;g:g: + 8g;g: - fg,

(.50)

+ ;g;h:

+ ;g:hi

- 5h; - 5h;h,’ - 22h; - 3h;h: - 3h:))

- 3s; + 3h,; + 4h’,

+$-&g:

+ ig:g;

+ ;g:

tan $ is negligible

+ ig:h:

+ h; + 6h; + hf

h: - ;g; 8*2 (

of the parameter

- ;g!

- 22h; - 5h;h; - 5hf - hihf))

- 392”- ;g;

+ 6g;h; + 16g;h;

+ sg:g:

2 - ,g:G

+ ;g:h;

+ 6g;hZ, + 16g;h;

-9h,” - 9h;hf - lOh:)),

The running

- yh;))

+ 9s; + 149,’ - 4h; - 4h;))

(Jjg:

+ g:g;

hi - ;g; 8?rz (

+;g;h:

&

+ 6g;h; + 16g;h;

+&(E

-xi=

+ yg;

(ig: + 259; + 249: - 6h; - 6h; - 2hj))

+ &

3 +

8x2 ( t

dt=

+ --&;g;

879

(51)

- 3hfh;

(,52)

and ignored

as usual

16

B. Schrempp

In considering

the perturhative

in the generic

forms

well guaranteed scale O(v): anyway; run

g2/(4x),

of pertubation between

a grand

are valid in the given

(40))(52)

the

region

well above

where

gs increases

towards

the UV scale A, besides

non-asymptotically

free coupling

sections

theory.

physically toward

below

Relations

2.5

The

remains

SM tree

couplings,

level

Eqs.

within

the perturbative

relations

The physical

mass,

pole mass

pole

between

the

top

(bottom,

the top and Higgs and

leaves leads

pole and

which

is gauge

invariant,

[81]. It is defined of perturbation

tau)

to 1. This

is

mass

thresholds

one would

eventually

the

perturbative

into the region

thus

IR

leaves

region: where

the

the perturbative

g&,/(4*) resp. h$,/(47r) have to RG flow down to an IR scale O(t!)

infrared

finite

as the real part

order

$“le(l

and will henceforth

+ S,(p))

masses

be denoted

their under

scheme

pole position

subtraction which

and

theory

and renormalization

of the complex

the running couplings in the modified minimal the MS running couplings and the pole masses =

Higgs

in perturbation

(43)(46)

ml(g)

theory

and

to the order

in the considered between

that

between Pole Masses and &Tf$Couplings

the so-called

relation

as compared

sure

couplings,

region.

(21) and (22), have to be adapted

furnish

to make

all involved

this threshold,

region. The UV initial values for all the other couplings A/(4*), be chosen sufficiently much smaller than 1, then their “top-down” automatically

small

unmotivated,

its Landau

that

UV scale and the weak interaction

form only above

its Landau

being

gi increases

unification

variants

one has first

This requires

K) or X/(47r), have to be sufficiently

h*/(4

couplings

but even if one were to use the appropriate

into

going

of validity

for all gauge

the RGE

RG flow in future

“top-down”

one does not leave the region

and M. Wimmcr

respective discussion.

independent,

is

of the propagator

by mn”‘e. Since the RGE (MS) scheme, we need the

are as follows

with

1 ------g/(p); d2&GF

for f = t, b, r

and

respectively,

where

m(p)

mu

=

m~‘“(1

w(p)

=

&z

is the running

mass 2, =

given

in terms

of the Fermi

+SH(~))

(.5(i)

in terms

of the running

are used.

contributions

to correct

various

It seems

coupling

and

as a reminder

(JZGF)-"*.

j.57)

constant lo-’

GeV-*.

The radiative corrections b,(p) and 6~(p) h ave to be taken g,2 and A, if running couplings and correspondingly running (43)-(46)

(5.5)

=

GF = 1.16639(2)

their

with

worthwhile

are scattered typographical

*we are grateful to authors of Refs. the sign in front of the bracket in the only in the limit WLHB ml, ii) in Ref. the whole Eq. (12) holds only in the replaced by its negative as implemented

to collect

over a number errors

into account to order masses resulting from

here all relevant

of publications

j.58)

formulae

and in particular

gf, gi, gi, g,‘, yz. the two-loop RGE

for 6f(p)

and SH(p) since since this is an occasion

in the literature.*

[87],[15] for a g wing with us on the following typographical errors: i) last formula of Eq. (29) has to changed from - to + and the formula [15] the first term. &A, in the bracket of Eq. (12) has to be replaced limit ZRH >> mu, iii) the entry for the quantity o1 in the table of Ref. in Table 1 in t.his review.

in Ref. [a.>] (28) applies by &A and 1871 ha.5 fo

Top Quark and Higgs Boson Masses The correction

terms

6,(g)

and 6H(n)

[15] and [SS], (271, respectively. and

QCD

contribution

which

QCD

correction

be taken Ref.

are separately 6f(P)

The

may

Following

sFCD(p),

from

very

[ST] 6,(p)

IS d ecomposed and gauge independent

finite

= s;(P)

+ bFED(P)

only

for quarks,

applicable

the partially

17 recent

literature.

into a weak,

[S+[S7].

electromagnetic

(.59)

+ 6,QCD(/L). is numerically

the largest

one.

It has been

calculated to O(gg) in Refs. [82],[4] and to O(gi) in Refs. (831, [84]. It seems worthwhile to colect the relevant formulae defining 6TCD (p) implicitely (disregarding for the moment the electroweak contributions) pole mf mj(p = ml;ole)

4 os( my]‘) =

H

‘+3-

+

with quarks

as

= gi/(4*) with

in the MS scheme.

pole masses

rnpole < my’=.

The

second

From

Refs.

term

is an accurate

approximation

for n/-l

light

[83]

mt(p mt(~) = rnp,le) (61)

From

these

formulae

6fQCD(p) to O(os)

qyp)

=

becomes 2

;Y2y (-l+~ln>)+O(

: i

&/QED is obtained charge

from

of the fermion

S;(p)

was determined

other

particles

the authors

the leading

order

)

y

6yCD by substituting

(63)

)

$13 by Q;a.

where

Q,

is the electric

f = t. b. T. in Ref.

[87] in the limit

of the SM. Setting

the latter

where

masses

rnH and/or equal

m, are large

to zero and

omitting

as compared the subleading

to all the term\

obtain 312 arcoshfi

1

, (6i)

(fii i

with

mfi* 4m<*

(67)

18 Eq.

B. Schrempp

(64) is valid for r 2 1. For r < 1 one has to replace

Following leads

and M. Wimmer

Ref.

[87] one can expand

Eq.

(1 - l/r)3/2arcoshfi

(64) for mH >> 2mt

by (l/r

(r >> 1) and

mH <

- 1)3/2arccosfi.

2ml

(r <

l), which

to 3ln$+~-~ln$+~+0($ln~))],

(68)

(69) respectively.

In Ref.

6fQED including

these

[87] also the subleading

corrections

subleading

are very well approximated

corrections

d, = a, + b, In are added

on their

right

hand

sides.

6&)

has

identical

been apart

-6.90

calculated from

-5.82

b

1.52 x 1O-2

1.73 x 10-s

0

1.59 x 10-2

1.73 x 10-s

0

al,

in Ref.

bf and

cf for

[88] and negligible

a numerically

Si(m,)

+

if the terms

f = 1, b,

in Table

1. The correction

into Eq.

(70).

Cf x 1O-3

T

1: Coefficients

that

(64).(66)

+c%GX

1.73 x 1O-3

x 1O-3

It is found

by Eqs.

for f = t, b, T are listed

bf

Uf

t

Table

zizz

The coefficients

f

have been calculated.

T, to be inserted

more recently in Ref. [27]. The results turn redefinition of v. The results are summarized

out3 to be as follows

WI with

[ = rnklrni,

s2 = sin2 Bw, c2 = cos’ BW and 0~ being

fI((,p)

with

=

6ln!f+?ln(-~Z L rni

fo([,p) =

-6ln$

%e are grateful conclusion

=

+

to J.R.

c 0 F

2A arctan( Aln[(l

Espinosa

2

(E)

2[9

1

- A)]

for undertaking

(2 > t, (2 < f)

the effort

,

to compare

&+

(72)

0

+sln-$+2Z

i

3cr In cs + 7 + 12~’ In c2 - F

l/A)

+ A)/(1

angle:

2 -lnc’+! ?!!_

-Z

(E)

1+2c2-23

4csz

Z(z)

2

the Weinberg

(1 + 2~‘)

A=dm.

the results

and for providmg

(3)

(7.5)

us with th(z

Top Quark

so+100: - 150\

J

200

MS

1: Radiative

leading

to the corresponding

The following

applied

corrections,

asymptotic

physical

to the

pole mass

expressions

hold

250

MS Higgs-top pairs

mass

All radiative

corrections

representative squares

starting

indicate

where

are values these

pairs

region

of the absolute

glance

where

2.6

Effective Potential

For small tant.

the radiative

values

of the Higgs

2

become

l

These

in Sect.

sizeable

and Vacuum

selfcoupling [14].

1%~~In c’ + 24m, In m, m% 4

1

(77)

exposed

For a very nice and comprehensive

[12], supplemented by Ref. addressed in this review

squares.

[88]:

(to be discussed

corrections

by open

of masses

end

lower bound

denot~ed

1. More precisely, the squares represent in Fig. The arrows attached to the in tfrr,Cm, m.mt m-plane. We stop the presentation in the up in the mH -m, P”“-plane.

collectively for pairs

pairs,

at the tips of the arrows.

4 -

[GeV]

mt Pole

+

9

Figure

19

Boson Masses

and Higgs

X radiative review radiative

and

6.3).

This

in which

diagram

directions

allows they

to discern

at a

aim.

Stability corrections

on the subject corrections

to the Higgs and relevant become

potential references

relevant

become

impor-

we refer to Ref.

for two physics

issues

for the calculation of the vacuum stability bound [7].[11]-[17] within the SM. a lower bound on the Higgs mass. which increases for increasing top mass. resp. an upper bound for the top mass which increases for increasing Higgs mass;

20

B. Schrempp for the calculation MSSM as function

l

The

radiative

emission

corrections

to the Higgs

in Eq.

one-particle external

emerge

Th e contribution

(12)).

irreducible

graphs

legs and with

potential

on the interaction

of the SM, V(4) = -$m2c$* + $j4 fiRerjo

bound [18]-[28] for th e mass breaking scale Msusv.

of an upper of the SUSY

and reabsorption

with

zero external

and M. Wimmer

imply

in form

of a loop expansion

to a given

SM coupling,

number

corrections”

any number

of external

The loop expansion In (

4 the classical

d2 1 PO

legs with

boson

of virtual

to the scalar

(with

of loops

momentum.

Higgs

of the effect

The “quantum

( (Y is a generic

the calculation

energy.

a”+’ where

of the lightest

in the

particle potential

C#Jto be identified

results

from

the classical typically

summing scalar

contains

wit,h all the

field on the

terms

n

(78)

))

scalar

field, ~0 the arbitrary

momentum

scale and

n

the loop order. In order

for the loop expansion

the loop expansion

scale p. can be chosen In order

to keep potential.

improvement to all-loop

a In (4*/p:)

to make

the effect

for the investigation effective

to be reliable,

parameter

The

of the

relevant

on the two-loop in the effective

Including

the one-loop

gt as only

non-zero

logarithmic of vacuum state

as possible,

terms

small

instability,

of the art

but ~0 can only

over

one has

is to treat

level; in this case the leading

large

ranges

to take

take one value. of large

recourse

the one-loop

effective

and next-to-leading

4, as necessary

to the

RG improved

potential

logarithms

with

RG

are summed

potential.

correction

fermion

as small

In (4*/p:)

of the issue

it is not sufficient for the coupling (Y to be small, but also h as t o b e smaller than one. Of course the renormalization

in the ‘t Hoft

coupling),

Landau

the one-loop

gauge

effective

[14] (keeping

potential

the top Yukawa

coupling

of the SM reads

(79) with

H

=

-rn* + %#J~,

T

=

;gf42.

G

=

-m2+Xd2,

(80) The renormalization

group

improvement

consists At)

Since

the effective

potential

is independent

the explicit scale dependence in the effective and the field 4. For the resulting couplings, is given in terms of the two-loop RGE (46),

in introducing =

a variable

resealing

Poe’.

of the renormalization

(81)

scale,

the effect

of the resealing

on

potential has to be absorbed into changes of the couplings X(p), g*(p),..., th e d’ff I erential change at the two-loop level (43),... in terms of the variable t or p.

Next, the physical requirement of vacuum stability, i.e. of the stability electroweak vacuum, has to be fulfilled. This stability is only in danger

of the (radiatively corrected) at large values of the field 0,

The key point is now the following. The resealing allows to choose the scaling parameter t such that p(t) _ 4 at large 4. It has been shown [14] that for this choice the only term which is of importance in

Top Quark the potential

is the fJ+o’ term

X replaced a false,

by the two-loop

deep

question

minimum

of whether

A, the fact that the electroweak in fact large

scale p at which is to a good In Refs.

coupling importance [7].

the running

coupling

than

become

is strongly

of scale was refined

cut-off

A should

dependent.

be made.

to the effect

the vacuum

that

negative

well below

contibution

stability

was

from

the

Of course

no

Thus,

is p = A. Correspondingly,

for deriving

the

of the vacuum

a large

top mass

of

is simply

d rives the potential

of X to the stability

negative

condition

scale

Even for very small

fX(~h#~

for X(p) contains

bound

tree level coupling of the existence

at some

as t increases.

the term

the physical

could

the constant the question

minimum

of the evolution

the UV initial

[27], [16] the choice

that

stability

of the field larger

approximation

but now with

S ince the RGE

the vacuum

21

C orrespondingly

X(p) g oes negative

The

Masses

the electroweak

>> 1 means

in Ref.

coupling,

X(p).

destabilize

at d/rnz

recognized to values

coupling

could

the running

top Yukawa

extrapolation

which

minimum.

already

Boson

in the tree level potential,

running

this happens

and Higgs

the maximal X(p = ,I) = 0

bound

the scale dependence

in the IR.

of the one-loop

effective potential, RG improved at the two-loop level, becomes minimal. This leads the authors replace the role of X played in the argument led above by that of the slightly shifted variable 3

4

P2

3

_ &;

&-’

+ $2

ln (g,’ ;:g:)

3)

4

(

to

_ ;

i

(sa) !!

Preview of Infrared Fixed Manifolds and Bounds in the SM and MSSM

Since

IR fixed

manifolds

in the multiparameter

of the SM or of the MSSM them

quasi

contains

with

a brief

the space strength

of their

in in the following literature,

which

are omitted

in at a glance

the richness

and

most

it allows

picture

to emerge

Starting

point

4-6.

of the IR structures

towards

is to trade

the

to see each

section

is meant

of the art

including

masses

also

will contain

The purpose thus

and

as a first

predictions

hopefully

parameters to scrutinize

guideline

higher

order

the resulting the large

wetting

radiative

number

only:

in

masses

or

and tile

corrections.

bounds

are filled

of references

is twofold:

it

bounds

manifolds

mass

the appetite

chapters

for particle

of these

of this summary

of the following

between

as well as some one-loop

nor precise

between

sections

here altogether.

transparently only

These

This

relations

it is well worthwhile

from it. In fact all derivations

state

relations

interesting reality,

IR fixed manifolds

understanding

the latest

masses,

Sects.

glass.

may be inferred

IR attraction,

IR attractive

imply

on physical

one-loop

a thorough

bounds

space

bearing

magnifying

of the exact Neither

or mass

the resulting

a likely

a mathematical summary

of couplings.

mass relations

with

to the

it allows

to take

for more information:

in perspective

to the complete

the end. independent

variable

t for the

variable

gg, and

consider

the

ratios

of

couplings x/g;, This leads The most

to a set of RGE interesting

in one-loop

for the new dependent

IR fixed manifolds

approximation.

rev. h&/d.

s:a,&~

Therefore

variables

will appear this exploratory

(S3)

and &Is,‘. (83) and

the new independent

in this set of RGE chapter

variable

and will as a rule be exact

is strictly

based

on the one-loop

y:, onI> RGE

results. As has been mentioned in the introduction, bounds for masses flow which fail to be fully attracted onto the IR fixed manifolds.

are boundaries of IR points of the RG They lie the closer to the IR manifolds

the longer is the evolution path, i.e. the larger is the value of the UV cut-off A. Important this kind will be shown alongside the IR manifolds in this exploratory section. The most are the bounds stability It. is most

(lower)

on the Higgs bounds

instructive

selfcoupling

on the Higgs

to present

X responsible mass.

the material

cutting

for the triviality out a wedge-like

here and in the following

(upper) allowed sections

bound region

bounds of well known

and the vacuum

in the X-g,-plane.

by gradual/y

incrensrn
B. Schrempp and M. Wimmer

22

One is pedagogical: it allows to proceed in small steps from the parameter space for two reasons. warm-up exercises with simple examples to the complex structure, observing how with each step t,he physical implications become less and less trivial. The second one is that a number of crucial results in the literature as e.g. the non-perturbative lattice results have been obtained within such reduced parameter spaces and may be included most systematically this way. At each step in the gradual increase of the subset of considered parameters their evolution according to the RGE (either of the SM or of the MSSM) is evaluated by setting all the excluded parameters identical to zero. An a posteriori justification for this procedure emerges in the following sections. With the exception of the RG evolution of the Higgs selfcoupling, which is only treated in the SM, the MSSM results are displayed in parallel to the SM results. For simplicity the scale Msusv for the transition from REG of the MSSM to the RGE of the SM is chosen equal to mt = 176GeV in this exploratory section. Table 2 summarizes in four pages the RGE used (SM or MSSM), th e sets of nonzero couplings considered, the IR fixed points of the set of couplings or of their ratios (characterized by a diamond in the figures), the IR fixed lines (characterized by fat lines in the figures), the most attractive IR fixed surface and. finally, bounds for couplings in dependence on the UV cut-off scale A (characterized by thin lines in the figures). As representative values A = lo4 GeV, lo6 GeV, 10” GeV and 10” GeV have been chosen for the SM and A = MGIJT x 2 1016GeV for the MSSM. Each horizontal entry line needs a few commentaries which are given next. . Starting with a single parameter, the Higgs self coupling X of the SM, leads to the well studied four component e4 theory. The corresponding RGE for X exhibits the much discussed “trivial” IR fixed point A = 0, attracting the (perturbatively admissible) RG flow, and leads to the well-known A dependent upper bounds which are well confirmed also in the framework of non-perturbative lattice calculations. The details are spelt out in Section 4.1. . The enlarged system of RGE for two parameters, the Higgs self coupling X as well as the top Yukawa coupling gt, has still a common “trivial” IR fixed point at X = gt = 0, b,ut shows already the remarkable feature of an IR attractive (linear) line, which is more strongly attractive than the IR fixed point; mathematically it is the special solution of the RGE which fulfills the boundary condition of a finite ratio A/g: in the limit A,gt + co. The RG flow from arbitrary (perturbatively allowed) UV initial values is then first towards the fixed line and subsequently close to or along this line towards the fixed point. Representative upper and lower bounds are displayed. The details are worked out in Section 4.2. l

A next step includes the strong gauge coupling 93. It is instructive to first discuss the set of two couplings, gt and gs in the SM, resp. g3 and hl in the MSSM. It exhibits an IR attractive straight fixed line through the origin in the g:-gi plane resp. the hf-gjplane. More economically, this line (and similar ones in future variables) will be viewed as IR fixed points in the ratio variables g:/gi resp. h:/gi detailed in the table. The details are furnished in Sect. 4.3.

l

The set of three couplings X, gt and g3 is most economically reduced to the set of two ratios of couplings X/g,2 and g:/gi. This leads in the X/g:-g:/gi pl ane to a similar picture as in the X-g: plane in absence of gs. However, the IR fixed point is not trivial any more. Again there is an IR fixed line, which is now nonlinear. It is characterized by an analogous boundary condition (finiteness for the ratio X/g: in the limit A/g;, g:/gi + co) and has the same slope in this limit: this fixed line is more strongly IR attractive than the fixed point. Again the representative upper and lower bounds are displayed. The detailed discussion are found in Section 4.4.

l

The set of non-zero variables gt, gb and gs within the SM may be reduced to a discussion in the two ratios of variables g:/gi and gf/gi; similarly the ratios hi/g: and hi/g: become relevant in the MSSM. In both cases one finds qualitatively the same result: two IR fixed lines in the plane of

SM

MSSM

SM

SM

SM

model

1

x

couplin

considel

II

_

=

7

9

2

72

I/%-25

s32 -18

#ymbol 0 in figure

2

It

2

x

z

=

0

0,

point

= 2 %-_ _s32 9 h:_

gt

2

x =

x=0

IR fixed

’

16

fat line in figure

2

St

fat line in figure

&=’

IR fixed line

1

X 5 1.63,

0.37,

A = 104, lo6 lOlo, 0.22

resp.

1015GeV

figure of IR fixed line (fat), bounds at p = mt (thin) for

for

x

0,

02

0.3

04

OS

1015GeV

0

//’ 00

0.05

0. I

0. I5

0.2

0 25

A = 104, lOlo,

enlargement

SM

MSSM

model

A> !h,

gb,g3

ht, htz, .%

couplings

considered

symbol

z

sbz

s,2

2 Q3 2 gt

x

symbol

=

zz

T=3 Q3

1

1

5’

0 in figure

s

1

1 -, 6

24

LA%-9

0 in figure

_

-

g3’

h;

point

0 in figure

h;"

symbol

IR fixed

lines

attractive)

(weakly

IR

fat line 2 in figure

IR

in figure

attractive)

(strongly

fat line 1

(weakly IR attractive)

fat line 2 in figure

(strongly IR attractive)

fat line 1 in figure

f;;z

IR fixed

figure

g&gz;m

of IR fixed surface

9fl93

figure of IR fixed lines (fat), bounds at ~1 = mt (thin) for A = 104, 106, lOlo, 10”GeV

LA

“Z

"4

h:/g:

06

0.8

g:lg:~~

enlargement

1

A-_ 2

9:/g,'

model

h,, gi, gz, 93

considered couplings

initial

values

42. 43

for given

of Qj,

IR fixed surface

IR fixed surface

=

line in

=

line

i

fat line 1 in figurNe

for 1~ = 176 GeV

IR fixed

fat line in plot h:/gg versus l/g!

the surface

‘S

line ir I

93:

for 91, QZ, fat broken

value

initial

given

l/g:

versus

g:/gi

fat line in plot

the surface

fat broken

-

value s

2 QZ, g3:

initial

for 9

given

for 91, g2, g3

symbol

0 in figure

for 11 = 176 GeV

IR fixed point

values

IR fixed line for given initial

IR fixed point

iymbol

symbol

0 in figL

0 in figs

for p = 176Ge’

26

B. Schrempp and M. Wimmer

Top Quark and Higgs the two ratios,

the more attractive

fixed point

towards

line and then

the more

less attractive (in this

attractive

approximation

in Sect.

with

vanishing

The corresponding

and

this line towards

exa.ct top-bottom gauge

X, gtr gb and gs in the SM is discussed gt/gz

of couplings X/g:

versus

by exchanging

X/g:,

and gi/gi.

g:/g,’

and

g: for gt.

the

and gi/gi

couplings

and treating

them

first as free variables.

so far determined = 0. This

Since,

however

gi(p

fixed

interest

The details

are given

four-dimensional attractive

discuss

the resulting

attract

the

is followed

a The proposed

in the table

procedure

ht are considered

attractive

IR attractive

These

and

will be taken gi/gi

g:/g,”

and

again

Within

most

transparent.

which

allows

to isolate

surface

ratios

UV scale

a high

line is displayed

Note that

lies in a physicall)

I he

pick out

gs(p

= mt)

fixed manifolds increasing

and

which

the number

if first only the four parameters

the values values

A to this

space

of gi/gs

and gs/gs

for gi/gs

and

gs/gs

IR scale

m,=176

by means

of the ratios

of

l/g:. IR scales.

in a two-dimensional Also shown As expected

at p = mr the value

for gi = gs = 0, a measure

hf/‘gi

is then

along

this line much

it ends

for g:/g$

GeV(with

attracted

weakly

of this

resp.

hi/g:

is much

influence towards

towards

gf/g,‘,

RGE

If one feeds in

11 = 10” GeV within resp.

solution

IR fixed point higher

of gi and gs.

than The

of the t\vo

in the SM and

the surface

hf/gi,

For details

which

along

conveniently (small

for gf/gi

this

is III

as function

crosses)

beyond

= g,“/g:

= 0.

the IR fixed point

value

RG flow of gf/g,’

resp.

the IR fixed line (fat line) and then

the IR fixed point.

the most

g,“/g,” and g:/g:

the evolution

as a fat line, plotted

in the expected

of the significant

first very strongly more

figure

is the continuation

gi, ya, g3 and

of a figure

are unconstrained.

at p = mz,

A = MC,_,T = 2. 1016 GeV in the MSSM) traces a line of finite length denoted by a fat dashed-dotted line in the figure. The variable gf/g:,

the UV and

by

approximation.

to be p = mt = 176GeL..

after

and demonstrate

in the three-dimensional

the surface determined

of the variable

ratios

supplemented

and

the IR attractive

below,

account

the

in this parameter

of couplings,

= mt)

chosen

the

one has to look into bhe

of ratios

for gi(p

into

by adding

g,‘/gj,

In principle,

IR scale,

and the description

close to or along

IR fixed point

space

become

A and fixed

becomes

the experimentally from

manifold. UV scale

then

gi and gs in a crude

values

versus obtained

in steps.

gt resp.

h:/gi.

X/g:,

enough.

g:/gi

is trivially

to zero and thus the fixed point

feed in the known

lower-dimensional

surface,

They

gf/gi

for ignoring

in the five-dimensional

then

RG flow for increasing parameters

variables

is by far not good

surfaces

one and

in the

ignored.

X/g&

which

space containing

close to this fixed line towards

The common

justification

= mr) are unequal

the approximation

been

surface,

of the subspaces

g,‘/g,‘,

the attractive

of couplings

point

is an a posteriori

= mt) and gs(p

region,

procedure

gi and gs have

of ratios

is the

strongly

in the discussion

space

= g,‘/g,’

fixed

The

unification

is of eminent

IR attractive

one A/g,’ versus

The RG flow is first towards

gauge

space

resp.

coupling

in the three-dimensional

is a strongly

appeared

the corresponding

parameter

$‘/g,’

IR attractive

which

There

towards the more attractive fixed line 1 and finally unifying IR fixed point. For details see Sect,. 4.6.

So far, the electroweak by enlarging

considered

which

are also shown.

of ratios

inaccessible

The RG flow is first the fixed point.

Yukawa

couplings)

lower bounds

The set of four parameters

the surface top-bottom

This

imply

by 1 the less attractive

marked

of the two lines.

close to or along

electroweak

upper

all the fixed lines and fixed points

most

one being

4.5.

gi/gi,

gf/g,’

shaped

is at the intersection

line as well as the fixed point

in the MSSM.

l

quarter-circle

one by 2. The IR attractive

21

Boson Masses

see Sect.

close t,o or

ri.1.

. The set of parameters gi, gs, gs, g1 and gb leads to a t,hree-dimensional IR fixed surface in tllr four-dimensional space of ratios g:/gi. gl/gi, gi/gz and yi/gi. Proceeding as above leads to an IR attractive two-dimensional surface for g:/gz and g,‘/g,’ versus l/g:. The relevant curve in the g:/g:-gi/gz plane which replaces the fat IR attractive line of the case gi = gs = 0, is read off for p = mt. Sect.

.5.?.

The

corresponding

figures

for the

SISSM

are also shown.

The

details

are spelt

out

in

B. Schrempp and M. Wimmer

28

. Finally, including all parameters to be considered in this review, X, gt, gb, 91, gZ and gs within t,he SM, lead to a four-dimensional IR attractive surface in the corresponding five-dimensional space of ratios A/g;, gf/gi, gi/gi, gf/g,” and gi/gi. Feeding in as above the physical couplings 91 and ~2 and evaluating them at p = mt leads to the two-dimqnsional surface in the ~/g~-g:/g~-g~/g~-space. For details see Sect. 5.3. Boundary conditions singling out the various IR manifolds, discussed in Sects. 4 and 5.

4

as far as not mentioned

above, will be

Infrared Fixed Points, Lines, Surf&es and Mass Bounds in Absence of Electroweak Gauge Couplings

This section fills in the information throughout this section

into Table 2 in absence of the electroweak

gauge couplings,

91 = g2 = 0 for all scales p; the couplings considered and gi. This section l

(84)

at the end of this section will be X, gt, gb (and marginally

g7)

may be considered as a first warm-up exercise with exact IR attractive fixed manifolds in the (one-loop) RGE and their physical implications, leading already to a reasonable approximation to physical reality.

. It provides an excellent semi-quantitative vacuum stability bounds in the H&s-top UV cut-off scale A. l

i.?.

insight into the dynamical origin of the triviality and mass plane, which become the tighter the larger is the

Also it allows direct comparison with non-perturbative calculations on the lattice which have been performed in the pure Higgs and the H&s-top sector of the SM in absence of all gauge couplings.

As advocated in Sect. 3 the procedure of gradual increase of parameter space is followed, leading to less and less trivial IR structures and furnishing increasingly improving approximations to the SM resp. the MSSM . The inclusion of the electroweak couplings is deferred to Sect. 5. The final analysis, including two-loop RGE and radiative corrections to the relations between cquplings and masses, is presented in Sect. 6. The concept of an IR attractive fixed point, line, surface,... will be introduced step by step in conjunction with the applications. For mathematical background reading we refer to Ref. [S9]. For completeness let us also add that the notions IR (UV) attractive and repulsive used in this review are equivalent to the notions of IR (UV) stable and unstable, respectively. Furthermore, what physicists prefer to call fized lines, surfaces,..., is called by mathematicians [89], in fact more appropriately, invariant lines. surfaces,... Let us remind the reader of the definition of a fixed point which will allow most conveniently alization to fixed lines, surfaces,... . The differential equation for the function Y(I)

2 =f(Y)

a gener-

(S.5)

has a fixed point solution y = c for constant c. if it stays at y(l) = c for all values of I once its initial value yo = Y(Q) is chosen equal to c. In this case of a single differential equation (with a single dependent variable y) the fixed points are identical with the zeroes of f(y).

A system

of n coupled

differential

equations

for n dependent

E =f,(Yl>YZ..... variable

For future Consider

Ci

for

const.,

C, =

applications

it is important

to make

differential

the following

d YZ -

=

dx

hand

side of Eq.

(88,89) other

may

of the

hold

in turn

at some

value

equations

leads

function

for all values

procedure

(89)

left hand

is e.g.

q.41.

(‘30)

according

relation

of x, i.e.

(go),

at the point

yz in Eq.

the two equations

Now,

is not point

does not match (88,89)

equation

a solution

of the ratio

y1 = 0): Eq.

(90) is a non-constant

side d yz/d z which

to rewrite

to the differential however,

it is not a fixed

(except

of x; correspondingly

to a nonvanishing

side. A correct

example.

- CYI).

x = 20, signalling

of yz at x = ZO. The

not a fixed line in the yz-ll-plane,

is a non-constant

the aid of a simple

(89) has a zero at

derivative

of differential words

with

(SS)

bYz(Yz

Yz = equality

point

ad,

dx

vanishing

(87)

i = 1,2, .., n

equations

dyl -=

This

(86)

for y, = c,.

the set of two coupled

The right

yi(zr)

z has a fixed point yi =

if all fi vanish

functions

i = 1,2,...n,

y,),

of the independent

29

and Higgs Boson Masses

Top Quark

(89) a

of the set yz/y,,

(88) implies

or in that

y,

function

of I; this

the vanishing

right hand

as

YZ -zc+a b

(92)

Yl

is indeed

a fixed point

of the of Eq.

(91) for the ratio

yz/yI

. or equivalently,

Yz = cc + %)Yl a fixed

line in the Y1-yz-plane.

correct

solution

A fixed point “top-down”

(93),

Obviously,

the fake solution

(90) is only

a good

approximation

to the

if a/b < c.

is IR attractive

or - equivalently

RG Row, i.e. by any solution

IR stable

when evolved

- if it is approached

(asymptotically)

by the

from the UV to the IR. IR fixed Iines, surfaces,...

most easily within the applications to follow. Like the fixed points they will turn out to be special solutions, not determined by initial value conditions but by boundary conditions. These boundary conditions as a rule require a certain behaviour in a limit which is outside of the region of

will be introduced

validity of perturbation the physical perturbative In order

theory. Nevertheless the effect of IR attraction region mt ,< /.I,< A.

to keep the discussion

simple,

on the RG flow persists

within

let us fix the scafes.

the IR scale MSUSY

~1 = =

mt = 176GeV,

(94)

176GeV.

(9.5)

mt

=

30 For the numerical

calculations os(mz)

if the three-loop The

and M. Wimmer

8. Schrempp

discussion

absence

= 0.117 f 0.005

QCD

evolution

in this

section

of the electroweak

the general A partial

expressions

couplings

leading

to

[5]

gg(ml

= 176 GeV)

[90] f rom p = rn~ to p = mt = 176GeV will be exclusively

gauge

of this

set of ratios

divided

value

couplings

within

the one-loop

(96)

is used.

the framework contribution

= 1.34,

of the one-loop

to the RGE

RGE.

In the

may be read off from

(40)-( 52).

decoupling

the following

we use the experimental

coupled

system

of variables,

by the square

Higgs

g,’ of the strong

x PH

of differential

the

equations

self coupling gauge

may

be obtained

by introducing

as well as the squares

of the

Yukawa

coupling,

in the SM,

=

2 Pi

Pb

=

=

pt =

resp.

in the SM

and

treating

differential

the

ratio

equations

=

3

in the SM

resp.

pb = 2 in the MSSM, 93’

in the SM

resp.

pT = 2 in the MSSM, 93”

” 2

variables

in the MSSM,

93’

93’ pr

3

” Z

as functions

of the

independent

variable

(97) gi.

The

resulting

set of

is SM

MSSM

I g dt

-7g,ld=pt($t+

$b

+

pr

-

1)

-3!dd

=

ds,’

dPb

3

_39,2d

supplemented

within

3

_ - PT(~P* +

17

42

It will be the basis of the Subsects.

The

+b

+

7,

pb

-

3j

+

$7

+

Pbh’t

+

6pb

+

~7

-

;)

-3ddp’=p,(3pb+dp,+3) dd

7)

,

the SM by

_7g,2dPH = -PH*

4.1

=

ds:

d9s _7g,ldpr @

PA@,

dsi

-_r=Pb(~&+$b+/%-1)

-793’

= -3934. &r*

+ ~PHP~

+ 6P.vPa

+ ‘2Pffp,

+

7pH

-

3pt2 - 3pb* -

pIa.

(99)

4.3-4.6.

The Pure Higgs Sector of the SM - Triviality and an Upper Bound on the Higgs Mass first,

though

trivial,

IR fixed

point

is met

in the

scalar

sector

of the

SM, i.e.

the

pure

four-

component b4 theory in terms of a single coupling, the Higgs selfcoupling X. All the other SILI couplings are considered to be zero throughout this subsection. This model also allows already a semi-quantative

31

Top Quark and Higgs Boson Masses insight

into

allows

the origin

comparison

complement

of the so-called

with

a large

the perturbative

The RGE

in the pure

sector

bound

of lattice

results

Higgs

triviality

body

[6]-[lo]

calculations

in the non-perturbative is known

for the

for the

SM Higgs

four

mass.

component

Moreover,

44 theory

it

which

region.

even to three-loop

order.

In the MS scheme

it is [91]

(100) The coefficient

of the three-loop

The key observation relations

which

is that

term

is scheme

the RGE

exhibits

will shed light on different

SM. For pedagogical one loop order

purposes

dependent. an IR attractive

facets

this is done

fixed point4

of perturbative

in one-loop

order.

at X=0.

“triviality”

Let us first list the

and its implications

The general

solution

of the RGE

for the (100)

in

is 1 A(P) =

(101) -&j

in terms Landau

of the unknown pole.

UV initial

We are, however

X(p) is bounded

from

above

value

interested

X(A).

+ &In+ -A

For p increasing

in the evolution

beyond

towards

A, X(p) increases

towards

by 2*s J+(P) < -. 31n;

Inserting leads

the appropriate

lowest

to the implicitely

its

the IR, i.e. in p < A. Since X(A) > 0,

order

defined

mass

relation

perturbative

lowest

(102)

= mH) u into the upper

rnH = \/2X(p order

triviality

bound

for the Higgs

bound

(102)

mass

(103)

with

u = (\/ZGF)-‘/2.

Eqs.

(101)(

l

l

The

It is exhibited

103) may

be interpreted

perturbative

“top

solutions

of Eq.(2)

attracted

towards

The

IR value

the upper

decrease

In the fictitious to any physical RG flow is drawn non-interacting

arbitrary

the

The

large

UV initial

more

bound

increasing

p = A to the IR value

may

allowed

independent

of the value

comprising

X(A)/($s)

all

< 1. is

be interpreted

initial

value

1(A) (within

to approximately

and

the

closer

to

the framework

of

the IR images

of

collect

values.

value

into the IR fixed point (within

values,

X(A) is chosen

bound

(102) and the triviality

of the UV cut-off

value

the framework

X=0

(103) for the Higgs

toy model need not be subject and indeed the full perturbative

for the renormalized

of perturbation

bound

A.

case of the pure c$~ theory, which as a mathematical UV cut-off A, the limit A -+ 00 can be performed theory

at p = mH,

initial

X=0.

the UV initial

A(p) as well as its upper with

curve.

perturbatively

fixed point

the larger

theory).

a The IR coupling

RG flow from

from

becomes

(102),

2 as dotted

as follows

the IR attractive

X(p)

all sufficiently

mass

down”

starting

bound

perturbation

in Fig.

theory.

coupling,

discussed

4The twckmp ditTerential equation has formally a further fixed point at X/(4rr) = 4n/13 outside the range of validity of perturbation theory and is therefore physically meaningless.

.

leading

to a trivial

so far). z 0.97 which lies. however

B. Schrempp and M. Wimmer

32

s $ V

600 -

E’500400 -

Figure 2: Upper bound on the Higgs boson mass as a function of A/mH, where A is the scale of new physics. The dotted curve is obtained by identifying A with the Landau pole of X and is given by Eq. (103). The solid curve [93] is the renormalization group improved unitarity bound (107). The dashed curve is the result of a lattice calculation of Ref. [94]. The figure was taken from Ref. [93]. In the SM we do, however, expect a physical UV cut-off A to play the role of the scale above which new physics enters, as expanded on in the Introduction. So, as far as these simple-minded perturbative arguments go, we expect a A dependent upper bound (103) for the SM Higgs mass which decreases for increasing UV cut-off A. As was first pointed out in Ref. [8], this allows to determine an approximate absolute (perturbative) upper bound for the Higgs mass: on the one hand the triviality bound for the Higgs mass increases with decreasing A; on the other hand the Higgs mass is a physical quantity, which the SM is supposed to describe; so, for consistency, one has to require rnH < A. This implies an absolute upper bound for the Higgs mass close to the region where mH and A meet. A more subtle issue is to determine an absolute (perturbative) upper bound for the Higgs mass rn” sufficiently much smaller than A, such that the SM physics continues to hold even somewhat above rnH

l

without

being too close to violating unitarity

[92],[93],

l

without

running out of the region of validity of perturbation

l

without being significantly influenced by the nearby cutoff effects, i.e. by the new physics becoming relevant at energies O(A).

theory and

These three issues are of course not unrelated. In the following we shall summarize efforts in the recent literature to determine such an absolute bound for mH on a quantitative level for each of these itemized issues within the framework of perturbation theory. All of them apply to the full SM and not only to its Higgs sector, which does, however, not make a significant difference to the issue of an upper Higgs mass bound. They end up with very similar results. As we shall see, these results are also supported in a mellowed form by non-perturbative lattice results in the pure Higgs sector. Altogether rather convincing conclusions can be drawn. In order to implement the constraint of the unitarity bound, an interesting improvement on the perturbative triviality bound (103) was introduced in Ref. 1931. The UV cut-off :1 is identified with thr

.

33

Top Quark and Higgs Boson Masses momentum tightest

scale where condition

partial

perturbative

is obtained

wave amplidude

unitarity

bound

from

unitarity

is violated,

the (upper)

unitarity

of the isospin

]Reas(l

tightened

IReao(l in Ref.

[94]. In the limit

of center

of mass

one may

conclude

that

which

goes back

to Ref.

[92]. The

on ]as(Z = O)], where as is the zeroth -+ W,W, scattering. The well known

to

= O)l < l/2

energy

ao(l thus,

in W,W,

I = 0 channel

= O)] 5 1 has been

an ansatz bound

>

(104)

mH the tree level expression

for ~(1

= 0) is

= 0) = -5X/(16a);

the maximally

allowed

value

(10.5)

for X(A) is

X(A) I T. Feeding

this

triviality

inequality

the one-loop

with

u = (figs)the bound mass

‘I2 . This so-called (103);

bound,

applicability

(101)

leads

quantitative in Ref.

to be a factor

analysis [95].

to the

criterion

improved

perturbative

one-loop

scale p as well as on the scheme theory

calculated

breaks

cross

down

sections

The effects

of a nearby

mH/A

been

have

in the one-loop

cut-off

studied vacuum

in three order

diminish GeV)

[96]. The starting

is clearly

tighter

an absolute

upper

for mH

= 600GeV:

Higgs point

for a large

mass

Higgs

schemes which

mass

was

(the

MS scheme

are known

at two-loop

on the renormalization

in X. The conclusion

[95] is that

be mH ,< O(400 GeV)

for perturbativelh

of mass

energies

of 0(2 TeV).

in terms

of corrections

is the contribution

of weak gauge

fI$” = -ig’“(A,,

theory

the dependence

by order

up to center

diagrams

A, is reached

observables

is that

and mH must

A in case of a large

polarization

bound off that

renormalization

physical

should

in Ref.

cut-off

different

theory

to be trustworthy

the

of perturbation

of perturbation

for mH=G(700

unitarity

2. One can read

mH 5 530 GeV.

within

is investigated

for validity

improved

in Fig.

below

breakdown

p dependence

and the on mass shell scheme) The

two

a bound

of the

The

group

as solid curve

up to A = 2 TeV requires

A recent

bation

renormalization

it is exhibited

required

performed level.

relation

bound

than Higgs

into

(106)

of a virtual

pertur-

of the order Higgs

of

part,iclc

bosons

+ q’Fij(q*)) + q’q” terms,

(108)

i and j stand for W*, Z, y. At Q* 5 rni only A,, and the F,,(O) are retained, moreover so that the loop corrections are contained in = A+, = 0 as required from the Ward identities, renormalization three independent the six quantities Aww, Azz, FWW, FZZ, F7z, F^-,. Through combinations enter the definition of a, G F, mz and there remain three finite and scheme independent where

A,

parameters,

two of which c,

show

Azz

=

2-

a logarithmic

Aww mw

c3

where

indices

Weinberg

3 and

angle.

=

cotBWFsO

0 stand

These

on the Higgs SGFrn&

4 2X/S+

4 2$&s

_-+__

2=

mZ

dependence 3 G,c mt2

1GFmh = -6 “,i%?

The mH dependent

cut-off

(103)

by replacing

mH in the logarithm

hand

side to reach

its maximal

are chosen

scale A, at which

as probes

new physics

on the right

tan’@w

1 GFmZ, - -12 2fii$

log($)

for W, = cos 0w Z + sin 0~ y,

two quantities

mass

,&),

Iv0 = cos&

y - sin 0~ Z and

is extracted

side by u/&

(109)

mHz

for the sensitivity

is expected, hand

mZv In(m,2)7

and

to a cut-off

0w is the

‘2.

[96] from the inequalit!, assuming

mH on the left

value

(110)

B. Schrempp and M. Wimmer

34

El 5 10-4

s

A b) momentum c) exponential d) Pauli-Wars

O-

c3 8 S q

-5 104

a 10-4 -

a) infinite

r

2 8

d

-1 109

41044

CL

-

\ 2 10-4 -

?

-3 E

^I 5 -1.5 10-3 w’ -2 10-3

aio-4-

CJ

L

1 100

I

L

200

300

400

500

‘C

o-

Z

I

I

\

I

600

700

-2 10-4 L 100

a00

I

t

200

300

I 400

3: Higgs

value

boson

at infinite

The figure

When

mass

cut-off

was taken

a theory

dependence

and with from

becomes

Ref.

sensitive

different

integration,

with

the result

a Higgs mH/A

mass

(log),

In summary,

effects,

it also becomes

of new physics regularization

ii) by employing

cs. The four curves regularization

regularization

at the mH dependent

sources

the predictions within

as functions

part

of the proof

schemes,

scheme

i) by introducing

the exponential the

a cut-off

representation

relevant

of cr and

dependent.

the framework

of triviality

it into the region

down.

days

Higgs

and

mass

theory

are trustworthy

parameter

schemes,

Starting

the need to retain

with

Refs.

a finite

proofs

[99], [8], [loo],

UV cutoff

is

together

the four curves contributions

split

at

of order

come to the conclusion up to the upper

for a triviality

A(A)/(47r)z

of triviality

the triviality

A has meanwhile

that

bound (111)

1, where

bound

of the scalar

been established

for the SM Higgs

perturbation

of the one-component

the large N limit of the N-component theory [98] were given. Meanwhile, @4 theory with the inverse lattice spacing playing the role of the cut-off physics.

limit

propagators

cut-off

cs in the different

the

as an upper

of euclidean

dimensionful

of perturbation

its implications

of large couplings,

close to rigorous

The

A by calculating

of the SM any more.

of increasing

mass is to continue In the early

the

analyzed.

mH 2 G(500 GeV).

The harder

600

represent

schemes

cut-off

O(500GeV)

trust

different

cr and

in the three

for A -+ co, are shown in Fig. 3 [96]. In both cases scheme dependent d ue to regularization

than

and one cannot

the SM calculations

corrections

In all cases regularization. Results for the mH dependence valid

higher

cut-off

to cut-off

quantities

and ii) the Pauli-Villars replaced by Eq. (110).

4

700

[96].

the effects

for the momentum

of the two quantities

finite

idea [96] is now to simulate c1 and cs in three

600

mH [GeVl

mH NV]

Figure

I

500

the dominant A, i.e.

sector

theory

b4 theory of the

breaks

[97] and of

tool is lattice scale

of the SM and

of new therefore

for some time by analytical

and

numerical lattice calculations [loll, [94], [102]. Th e calculations are based on a representative class of lattice actions, all of which respect the property of reflection positivity, the property in this Euclidean formulation which corresponds to unitarity in Minkowski space. A lattice triviality bound [94] is shown in Fig. 2 [93]. It is in surprisingly good agreement with the renormalization group improved unitarity bound. A conservative conclusion [103] on an absolute triviality bound for the Higgs mass as obtained from a representative class of lattice actions is the following: the SM will describe physics to an accuracy of a few percent up to energies of the order 2 to 4 times the Higgs mass rnH only if mH 5 710 f 60 GeV.

Altogether, the non-perturbative tive framework and even relax

lattice calculations confirm the results obtained the absolute upper bound on mH somewhat.

(II’)

within

the perturba-

35

Top Quark and Higgs Boson Masses The

Higgs

This

dependence

mass

triviality

bound

is a weak

will be explicitely

function

discussed

In the next

step towards

the top Yukawa

the SM a reduction

coupling

reduced

system

is more strongly turns

out

to two coupling

with

is of interest

in this

lowest

order

top mass. in this

IR attractive two coupling

vacuum

Stability

parameters,

than

a non-trivial

the top mass.

Bound

the Higgs

origin

selfcoupling

becomes

= 0.

with

“trivial”

bound,

a weak

function

the lower bound find again

support

the upper

(43) and

(46) reduce

mass

to both

analytically. bound

of the top

on the Higgs

features

respect

couplings.

mass

again

by non-perturbative

which

This fixed line

of the Higgs

mass.

as well as ii) for the

being

a function

lattice

of the

calculations

X-g,-framework.

The corresponding

RG equations

in one-loop

order

to

ds:=

(11.1)

dt

dX

is a coupled

system

of differential IR attractive

equations. trivial

(Il.,)

+ 2x9: - gt4).

&(4P

dl= This

X and

(113)

of the two couplings

and it can still be solved

i) for the triviality

bound,

important

it is still

couplings

IR fixed line in the plane

the fixed point

framework

stability

Furthermore

all the other

since even though

as we shall see

to be the dynamical

which

resp.

91 is considered, ~#0,9~#0

X and 9$, it exhibits

coupling,

4.2, 4.4 and 6.2.

The Higgs-Top Sector of the SM - a first IR Fixed Line and a First Vacuum

4.2

This

of the top Yukawa

in Sects.

A first observation

fixed point

at

is that

it exhibits

a common

x = 0. gr = 0.

(116)

If the variable

f&

(117,

St is introduced, solutions

the system

gr = St(p)

and

of RG equations

(114),

( 115) may be rewritten

in a decoupled

form with

nestrrl

R = R(gt(p))

dg: dt

Ls' =

16~~

(IIS)

t

(Ii!,) The right

hand

side of the differential

equation

(119)

R=??=+%%-I), The positive

R=??=

+i%-

(120)

1).

zero 71 is [40] an IR attractive

An analogous

has two zeroes

IR fixed

fixed point point

to a fourth heavy fermion couplings R = X/9:. which

in the variable

R

at

R = R = $( v'%-

1) E 0.441.

(1’1)

had already been pointed out in an earlier publication [3’2] in application Here we meet for the first time a fixed point in the ratio of generation. has also been termed (in a context to he discussed in the next subsection J

B. Schrempp and M. Wimmer

36 a “quasi-fixed through

point”

[30].

In the

X-gr-plane

it corresponds

to a fixed

line,

which

is linear

and

goes

the origin x = $5-

It has the property chosen

that

the solution

on it. Its analytical

RGE

which

limit

A, g: +

the region

is defined

form

on the fixed line for all values

is independent

by the boundary

00; as announced

of validity

stays

of perturbation

this

theory;

to the RG flow in the perturbative

region.

Going

determine

beyond

approaches

Ref.

[40], one

the IR fixed

of differential

of initial

condition

earlier,

can

line and finally

(122)

l)gt’.

values

that

the ratio

boundary

condition

nevertheless

of p, once

A, gtc and X/g:

has a Jinite

refers

analytically

the

way

from

non-zero

to a limit

the IR attraction

the IR fixed point

its initial

value

As. It is the solution which

value leads

in the outside

of this IR fixed line applies

in which

the

the analytical

“top-down”

solution’

RG flow

to the system

(119)

equations

g:(t) =

do 1+

(123)

&$0 ln 4

R(g:)

for arbitrary

(perturbatively

(124)

allowed)

initial

values

R&=w.

gto=g*(A),

Evolving

from the UV to the IR, gt(p)

indeed

approaches

R = z, i.e. the IR fixed line. A measure

(12.5)

d(A)

St0

the value

is

of the

0 and correspondingly

for the strength

the RG flow approaches

of IR attraction

towards

the fixed

‘v 2.69. So. the RG flow is roughly as follows: line is the exponent in (g:/grs)’ &I3 which is large, m/3 first towards the IR fixed line and then close to it or along it towards the IR fixed point (116). The IR attractive for the vacuum RGE

can

evolution from

line X = (( 6 stability

bound.

cross

the IR fixed

from

an initial

an initial

value

below

gf

important

line.

value it.

IS the dynamical

- 1)/16) The

first

The

above

line is the lower it and

The

the upper

solutions

starting

below

the line will end up on the line.

The IR images

initial

value

X allowed

triviality values

of the Higgs

bound

for the

of X, which

self coupling IR values;

are &=O,

according

constitute

considerations. Given a finite evolution will be at some finite difference from

origin

observation

for all solutions

starting

for all solutions

starting

initial

values

by perturbation 2.6 the

theory

from

the top-down the evolution closely

the largest

will constitute

IR images

vacuum

sufficiently

st,arting

of the lowest

stability

bound

as well as

of the considered

bound

of the solutions

i.e.

bound

no solution

bound from

to Sect.

the lower,

for the triviality

is that

above

the upper, possible

in these

or

possible i.e.

initial

lowest

order

path from some initial scale p = A to the IR scale, these bounds the IR fixed line, their position strongly depending on A; they

will be the closer to the line the larger the value of A. For A -+ co, which is of mathematical interest only, the RG flow will first contract towards the IR fixed line and then towards the IR fixed point X=0. g:=O. These lowest order bounds may be translated into lowest order Higgs mass bounds in terms of the top mass

in the Higgs-top

ml = (ul&*(cl

= mt).

mass

plane,

The analytical

using

formula

‘This solution was first written down by F. Schrempp

the lowest

order

relations

for the gf dependence (unpublished)

mH = JZ$PZ$n

of the triviality

bound

and may

be

37

Top Quark and Higgs Boson Masses

gR

0

10

5

G‘W

G.G b)

4 Figure

4: a) lattice

of G&,

results

(corresponding

curve)

and

upper

bound;

The mass

A/m*

= 4 (dotted

the triangles

ratio

comparison

[105] for the upper

to gi/Z)

with

obtained

from

symbols

the 6 3. 12 lattice

on gR (corresponding estimates

denote

results,

for scale

the lower bound, the squares

to 6X) as a function

ratios

A/mH

= 3 (solid

the full symbols

the S3. 16 lattice

the

results.

b)

to mH/mt)

as a function of GR,J (corresponding to gl/fi) in curve), mRo = 1 (full curve) for mn,, = 0.75 (dotted

estimates

The figures

the Xo ---) co, i.e. &

the one-loop

The open

perturbative curve).

and lower bounds

with

(corresponding

one-loop

= 1.25 (dashed

curve).

represent

pR$/m&,

and mn.,

together

were taken

+ 00 limit

from

Ref.

of the general

[105].

solution

(124)

(1’6)

the one for the lowest i.e. &

order

vacuum

stability

bound

from

the solution

R fWvac.stab.

The

(124)

with

initial

value

X0 = 0.

= 0.

corresponding

(116) indicated and

the vacuum

thin

lines,

discussion

figure as diamond, stability

which including

cut out figures,

=

in Table

491) 2 9t

2 shows

= vacstab.

the

the IR attractive lower

bounds

wedgeformed more

proper reference to the literature coupling leads to a more realistic

(!+-I3

I _

54l

( l-

A-gf-plane

for four

values regions

(127)

,,$,,3

with

the

fixed line (122) drawn

allowed

1.

trivial

of A, A = 104, which

decrease

quantitative

information

about

will be given situation.

in Subsect.

4.4. where

IR attractive

fixed

as fat line and the triviality 106,

lOlo,

10” GeV,

for increasing

the degree

A.

of IR attraction

the inclusion

point upper

drawn

as

A thorough as well as

of the strong

gauge

Again it is of interest to go beyond perturbation theory and look for confirmation to non-perturbative results for the trivlattice calculations in the Higgs-fermion sector. In Refs. [104],[105] non-perturbative iality bound as well as the vacuum stability bound have been obtained. The investigations are based on

38

B. Schrempp and M. Wimmer

a lattice

action

which

is the sum of an o(4)

SU(2)r. x sCr(2)~ symmetry

chiral mented

by a corresponding

problem

mirror

fermion

performed

doublet

selfcoupling

The results

relevant

in order

coupling

A=4mH

are shown.

As in the pure

bounds;

the lattice

bounds

mass:

they

A of the order

of three

An interesting

result

the two couplings.

stability

bound

Higgs

case, the lattice

are “absolute

bounds”

the physics

to four

times

the Higgs

is the right-most It implies

however

point

a lattice

leaves

space.

The

rightmost

a safety

4b) (1051 the lattice

margin

in Fig.

resp.

tend

p&J/m& = 0.76 which corresponds determined

in Eq.

supported

4.3

(121)

.Thus

doubler

doublets

spectrum.

is arranged

The calculations

bound

Higgs selfcoupling

fermion

in Fig.

results

are

from

gn corresponds

as the a4 lattice

4a) [105] with

1ar g e error

bound

bare

to the with and

perturbative

bound

(112)

for

to be valid up to scales

point

(112)

bars

at largest

for the fermion

values

(12s)

up to a cut-off

the Euclidean

a ratio

since

R=X/g:, position

of unitarity

it allows

Eq.

for the fermion

of one-loop

A z 1.9 mr. The upper

analogue

interest,

in the ratio

of

mass

f SOGeV),

to the IR fixed point

lattice

sense

the one-loop

to continue

for the model

towards

confirm

sector

upper

positivity,

all features

by non-perturbative

fermion

the

triviality

calculations

4a) [105] is of further

the IR fixed

bounds

the notorious

mass.

“absolute”

of reflection

point

line in the A-g: plane, Fig.

only

by the requirement

supple-

X = lo-‘.

in the same

of the Higgs

m,_<0(600

provided

from

The

with

splitting,

between

the physical

phase.

part

mass

and the renormalized Yukawa coupling Gn$ is to be identified F or comparison the perturbative one-loop results for A=3mH

gt/&

allow

from

mixing

and a fermionic

without

to overcome

Mass

symmetry

part

doublet

4a) [105], w h ere the renormalized

in Fig.

6X in our nomenclature

the IR top Yukawa

the Higgs

broken

A = co, the vacuum

are shown

IR coupling

which

fermion

regularization on the lattice. decouple [106] th e mirror fermions

in the physically

scalar

x Su(2)R)

a heavy

accompanying

such as to exactly Higgs

(21 sum

involving

(121),

mass

to trace

theory

data.

In

Higgs mass

by the

R = X/g: = (&$

perturbation

the IR fixed

in the lattice

divided

limit is

in Minkowski

- I)/16 21 0.141,

in the Higgs-top

sector

are

calculations.

The Top-g3 Sector of the SM and MSSM - a Non-Trivial IR Fixed Point

As a first step

towards

including

gr, gs # 0, in the SM resp. is treated.

A first

corresponding will turn

out

pr=hf/gi

top mass mi

coupling

ht, gs # 0, in the MSSM

in the MSSM.

to be at the heart

to an IR attractive

gauge

IR fized point appears

non-triuiat

variable

the strong

of the quantum

with

in the

Let us point effects,

gs the subset

all the other

variable out

as encoded

of the two couplings

Pt=g:/gi

already

here

in the RGE,

couplings

= 0

in the SM and that

this

which

IR fixed will finally

(129) in the point lead

value pole

p& mt

z

215GeV

FZ O(l90

in the SM - 200) GeV sin p in the MSSM

after inclusion of further couplings, in particular of the electroweak gauge couplings, contributions in the RGE and of the radiative corrections leading to the pole mass.

(130) (131) of the two-loop

In case of the SM the fixed point was first pointed out in this form to be discussed next by Pendleton and Ross [30]; their paper may justly be considered to be the primer for all following investigations into IR fixed points, lines, surfaces in the space of ratios of variables. Earlier references to an IR fixed point were made in Refs. [29]. Subsequent important results concerning the IR fixed point were obtained in Ref. [31], also to be discussed below, and in Refs. [32]-[37], concentrating mainly on the issues

Top Quark and Higgs Boson Masses of new

heavy

rediscovered

fermion and

the framework

or of two Higgs

on in Refs.

of reduction

[38] and

of parameters,

doublets.

The

Pendleton-Ross

[39] as a renormalization

to be reviewed

in Sect.

IR fixed

group

8. Later

invariant

point

was

solution

developments

in

were made

in

[40]-[42], (481.

Refs. The

generations

expanded

39

corresponding

opment

IR fixed

(49]-[65]

focused

supersymmetric _

grand

The introduction

point

in the

MSSM

on the interrelated unification

of the ratio

was discussed

issues

and of an IR fixed top mass

of coupling

[43]-[48]. An explosive

in Refs.

of tau-bottom(-top)

Yukawa

value,

coupling

Eq. (131),

devel-

unification

to be reviewed

in

in Sect.

squares

2 pt = 2

in Refs.

[30]-[42]

and

in the SM

[43]-[48]

leads

and

pt = 5

in the MSSM.

to th e respective

RG equations

SM

(IX)

respectively.

in pt

MSSM

I (133)

in terms general

of the variable solutions

gj.

Following

of the RGE

Refs.

[31]-[39],[42]

(133) for it and

that

for the SM and

Ref.

[47] for the MSSM,

the

for ga, are

SM

MSSM 219

P&7,2) =

1 _ (I _ &-)

d(P)

(&-1"

PJS3

szl

=

=

I_

9:(p)

(1 _ g)

(13-r)

dl

=

1 --&g&In+ sir

(?g"

1 - &g&In

$

’

where

do = &A).

From first

Eqs.

(133)

discovered

[31] within

and

(134)

by Ref.[30]

the SM, may

the following within

be read

Pt(d(A)).

important

analytical

the SM and off.

(13.3)

Pt0 =

Further

ii) an effective references

(136)

results

i) about

IR fixed

were given

an exact

point,

already

first

IR fixed point.

discovered

by Ref.

in the introduction

of this

subsection.

l

The differential

equation

for pr(gj)

has an exact

fixed point

SM

MSSM

Pt = f

This fixed point that

pc approaches

corresponds a finite

to the special value

solution

(# 0) in the limit

at

i Pt = 18.

of the RGE gj +

(137)

defined

0. This

by the boundary

special

solution

condition

remains

in the

40

B. Schrempp fixed point, particular plane,

once it has started of the choice

this fixed point

(mathematicians’

value

in pt translates

A, gj,,

and

on this line, the solution

introduction

4, all further

IR fixed points,

of variables

it is more

The IR fixed point bative)

“top-down”

the SM this MSSM,

initial

of validity

Of high

interest

worked

large

respectively,

a triviality

bound

bound

in Ref. UV initial

allows

point theory

is the upper

out analytically

for sufficiently 7/18,

of perturbation

-l/7.

is -7/g, (137)

i.e.

only

at the IR scale,

and of mathematical

accessible,

its.

from

bound

Neglecting’in

(137).

upper

[31] went even further

(134)

evolution

In any

case,

of

however.

is outside

of the

only.

bound

l/p,,

bound,

with

which

pt.

As has been

for pt is approached respect

could

to 2/9 and

also be termed

7118

is roughly

upper

bound

(&-“”

the IR image

of all suficiently

large

IV

co or gs + co, which of course both are physically not, the Pendleton-Ross fixed point. Thus, as in the case of X, it is the finite

length

for the finite gap between decreases

and very intuitively

for increasing

reinterpreted

IR fized point for solutions

On the one hand, considering be rewritten approximately as

(13s)

< 1-

selfcoupling

is responsible

SM as an effective intermediate as follows. (138) may

large.

upper

Eqs.

F&7:)

bound

A + towards

tends

The

arbitrary

not the case [47] in the

[40] for the variable

upper

(&-“’

for the Higgs

the UV to the IR which

the IR fixed point Ref.

bound

from

MSSM

~to for pt. In the limits

the upper

the triviality

starting

gs -+ co which

interest

its independent

219

this

of

in the following.

pt

I -

values

in the

in spaces

the logarithmic

reasonably

[6], [7], [31], [32], [lo],

pt(932) <

Evaluated

that

the location

will appear

is, however,

in the limit

SM

initial

where

This

to read off the ~~0 independent

for the variable

line

implies

as anticipated

pt as new variable

like (gi/g&)-1/7,

[31], an almost values

or invariant

this line; again

all solutions

i.e. in

in the ht-gi

This

however,

lines and surfaces

to the ratio

exponent,

exponent

the fixed

terminology)

along

of gi

resp.

respectively.

Since,

i.e. it attracts

weak,

small

the corresponding

evolves

and g$.

plane,

g is, pro > 0 at p = A for arbitrary A, in short the (perturit. However, as has been pointed out in Ref. [31], in case of

is exceedingly

the full RG flow reaches range

values

by the prohibitively

where

to stick

RG flow, towards

attraction

gi is damped

economical

A, gf

(137) in pf is IR attractive,

allowed)

of the evolution

In the gf-gi

h: = 7/lSgz,

values

of this Sect.

irrespective

of 9:s.

into a fixed line (physicists’

[89]), g: = 2/9g:

is chosen

value.

and

of the initial

(perturbatively

l

values

of this line is independent ratios l

in this fixed point

of initial

terminology

once the initial

and M. Wimmer

$ ln(gi/g,‘,)

of the RG evolution the upper

bound

(138)

value of the UV cut-off

the upper

bound

path and A.

(138) in case of the

with large UV initial values ~~0 or gto

as small

expansion

.

parameter,

the bound

i.e.

(139) On the other

hand,

in going

back

to the RGE

for g:,

(1lOj

‘following Ref. [31] it suffices in fact to neglect

qptO [(-$-)“7-L].resp.

tp,,~[($-)“‘-I]

withrespecttol

41

Top Quark and Higgs Boson Masses Hill points

out [31] that

of g: is driven constant

average

IR region, p where

for large initial

by the term value

a transient

ig:

82.

In running

slowing

the bracket

values

down

which

with

Since

in the running d gf

and consequently

this slowing

down

justifies

p towards

is expected

of g: is expected

89: in the bracket upper

by some

bound

in the vicinity

in the

of the value of

zero for

82.

=z

to happen

of the UV scale the running

to replace

the ~~0 independent

/d t become 9 ,g;

identify

>> g& in the vicinity

gt

in the bracket,

(141)

in the vicinity

of the upper

bound

(138),

one may

[31] 7 9:(P)

&-

(142)

8+’ In the supersymmetric

case,

the analogon

of the transient

slowing

down

condition

(141)

is

161 6ht = 393r which

was pointed

IR fixed point

out

in Ref.

As the Hill intermediate (138),

treated

as synonymous strong

. A very

fixed

initial

Hill effective

liV

physics

in the reason

the upper

IR physics:

out

authors)

electroweak large and

ends

the

breaking

values

for the top mass,

we also shall repeatedly since it reflects bound”.

with

large

strong

attraction

effect

of effacing

theory at some

is present. scheme

of the upper been

frequently

prefer the notion more directly the

~~0 as well as its independence

of

towards

the upper

the memory

bound,

of the details

pairs

thus

under

scale

A a new

providing

and a composite

Higgs

in the A dependent

example the

will end up in

physics

boson. largely

of top sector

[lo’i’].

the force

for

for the spontaneous

The RG evolution the details

values

worked

condensation

provides

mechanism

effaces

top Yukawa

retained

is (as has also been

name

a dynamical

couplings

resp. of t.hc

high UV scale A, which supplies for some

A very good running

high

for the top Yukawa

to translate which

rq

=

J2/993(~

mt

=

\/78g3(~

considered

of the natural11 of the UV theor!

of the IR upper

bound,

i.c.

so far, already

460.

175sin

z 95 GeV

= q)v/&sin

mtmax x

300,

$GeV

position

pt into the corresponding

resp.

for A = bf~“~

in the SM.

(144) (145)

/3 x 126 GeV sin j? in the MSSM a comparatively

to the Hill effective 230GeV

in the variable

order

= ml)u/fi

corresponding

570,

the fixed point

is in lowest

on the top mass

z

have

fixed point.

The final aim is of course

mtmax

this

a strong

At some

up in the IR region

In the approximation

any

appealing

[108].

symmetry

the Hill effective

interpretation

fixed point

large UV initial values for the top Yukawa coupling,

of top-antitop

UV initial

is that

new physics

in Ref.

the condensation

“upper

A as well

~~0.

in the Hill eflective fixed point in the IR region; the only memory

A at which

reviewed

values

as an approximate

for solutions

implies

suficiently

is the scale e.g.

to realize point,

bound, resp.

by the

over the notion

intermediate

on the UV cut-off

Pto.

point fixed

initial

and of an effective

In the following

of the bound

value

important

the

dynamical

bounds

point”

[31], the effective

depends

for smaller

appears

bound

in the literature.

in Ref.

since its position

fixed point

of an upper

IR attraction

the (large)

values

effective

the notions

of “Hill effective very

fixed point,

value g& and it is not attractive

as on the initial

bound

A s was also stressed

[45].

is not a genuine

(143)

large

fixed point

top mass may

for A = 104, 106, lOlo,

results.

be estimated

Also the upper that

1015 GeV in the SM.

x 2. 1016 GeV in the MSSM

way (1.46)

j 1.t7i

42

B. Schrempp

Given

the approximation,

shows

that

of a heavy

top quark,

to get quantitatively

the introduction

results,

already

as encoded

in Eq.

much

step towards

in the RGE

heavier

than

one has i) to switch

tau couplings, to determine

of the electroweak

anticipated

as a very convincing

effects,

i.e. a top quark

reliable

as well as the Higgs, bottom and the radiative corrections allowing values

be considered

at the level of the quantum

possibility order

this may

and M. Wimmer

equations,

the other

physical resides

quarks

back on the electroweak

reality.

It

the intrinsic

and

leptons.

couplings

In

gr, gz

ii) use the full two-loop RGE and iii) take into account the pole mass @“‘. These “corrections” (in particular

couplings) will turn Th e corrections

out to significantly

(131).

increase

will be introduced

the top mass

to the

step by step in the following

sections.

4.4

The Higgs-Top-g3 Sector of the SM - a First Non-Trivial Approximation

Now the which

necessary

is non-trivial

physical

point

information

of view.

been

A dynamical

to the one for the a heavy The three

has

accumulated

from the mathematical

point

source

top quark

for a heavy

discussed

the

first

Higgs

boson

approximation

already

arises

which

to the

informative is intimately

SM

from the related

in the last section.

couplings A. gt,

are considered As above

to discuss

of view and qualitatively

to be the only

it is more

economical

non-vanishing

Q3

#

(148)

0

ones in the RGE

to consider

the ratios

of couplings

x

2

pt = 2

and

of the SM in this subsection.

pi = 1

(lJ9)

Q3

or even pt =

2

and

R =

f!

=

?_

Pt By rewriting tively

as in the last subsection

R as a function

gi as function

of pt, one ends

up with

(1.50)

d'

of t, pt as function

the following

system

of gi and finally

of decoupled

pi or alterna-

differential

equations

1421 (151) -14g+

ds3’

Pt (9pt - 2) 2 or alternatively

pt(9pt -2)

=

24~~~ + (12pt + 14)pH

(152) 6pt2

(1.53)

[42]

dd dt -14g;dp ds3’ (9pt - 2)g The system out in Ref.

=

of differential equations [30] and further analyzed Pt =

2

;,

=

_Qi!S7r2

(154)

=

p,(9p, - 2)

(1.X)

=

24R2 + (3 + E)R

- 6

for pt and PH resp. R, has a common in Refs. [31], [35]-[39], [41]. [42]. PH

=

&%772

25

resp. *

(156) IR fixed point

R = V%@ - 25 I6

first pointed

(157)

43

Top Quark and Higgs Boson Masses Thus, in presence subsection, trivial

IR fixed

physical

of the gauge

is supplemented point

reality,

since

it corresponds

rn~ .4gain,

the common origin

=

@gs(p

=

dMg3(p

IR fixed point for large

all the appropriate

As in the X-g: plane or alternatively [39],[41].

This

of the RGE the limit

Since slope

in the

ptY PH +

limit

out

(i.e.

214GeV 210GeV,

are applied

behaviour

(162)

in powers

2 Pt=9: pt+m; A precursor

discussed

Before

further

figures.

sections fixed line TH(P1) in the pt-pH-plane already contained in Refs. [35]a finite

of gs.

It is the solution

R = p~/p~

ratio

= X/q:

in

is

larger Indeed,

interpolating

as a possible values

(161)

the limit

than

we are back

gi,

this limit

in absence

of the strong

fixed

to the discussion

is consistently

also the trivial

the IR attractive

[42] by infinite

it = $ and

RPt)

=

Wt)

=

J&t)

=

in powers

; Pt + m-

point

(157)

led

identical

with

coupling

g3. Kow

gauge

fixed point

power

series

of l/pt

at R=O at pt=O,

at pl=2/9

$)Pt2- & pt3 - $ 25 + ,7307

.5 contains

expansions

in the limit

and

the

which

shows

d&3'" 1 17-G yj-- ~ 42

- llv%%

line solution

1

the

investigations

let us present

(163)

pt=O

(l&J )

(Pt- 1, + ...% 1

12789

the

of pt arountl

00

48.5+11~

pt+ around

in powers

pt +

pt4 + ,..,

1360

of the fixed

analytical Fig.

approximation lead to mass

for pt -+ co.

of the expansion

entering

( Ii!,)

to the one in absence

line in the X-gi-plane

to

to

(160)

solution of the program of reduction of parameters to be discussed expansion around pt + 00 had been put forward in a four generation

illuminating

theory

t ht

approximation

z 53 GeV.

of approaching

4.2.

as a crude

previous

replaces

in the SM

in the next

condition

in the point

(I,%)

is again an IR attractive [42)? with prior indications

on in Subsect.

of pt - f around

pt = 0 :

fixed

It will eventually

in contradistinction

UV attractive),

discussed

perturbation

in the presently

2

of the RGE

point, common

z 9.5 GeV

masses!

X and g: are much

reported

order

pole N mH

The fixed line %!(p,) can be represented pt=O.

pole

of gs, there

(122) of the IR attractive

asymptotic

Higgs

p,-R-plane

the couplings

This

= rn,)fi~~

by the boundary

the fixed line is the solution is now IR repulsive

this lowest

M or g:, x -+ co. In fact

(321, [40] and

t,op fixed point.

= m,)v/&

and

fixed line is nonlinear

singled

in this

in Refs.

in absence

fixed

of gs. It is also interesting

can be viewed

top quark

corrections

??(pt)

nontrivial

Higgs

within

mi

when

gs the

at X = gf = 0 in absence

mt

dynamical

coupling

by a nontrivial

2

is found

(16ji + “.’

[38],[39]

in a one-loop

in Sect. 8: a precursor model [3.5]. main

numerical

results

of

t tie

in several

the IR fixed

line in the &pH-plane (fat line); it is an update of it interpolates t,he IR repulsive the corresponding figure in the R-p,-plane in Ref. [42]. A s expected fixed point at pt=O, the IR attractive fixed point at pt=2/9 (diamond) and approaches a straight line with slope (6 - 1)/16 for pt + cm. The RG flow from the UV towards the IR is indicated by a set of solutions of the (one-loop) RGE starting at representative UV initial values at pt values above arid below the IR fixed point (thin lines). Clearly the solutions are much more strongly attracted by the IR fixed line than by the IR fixed point. They first move towards the fixed line and then proceed close to or along the line towards the fixed point.

44

B. Schrempp

Figure

5: The IR attractive

pt-plane;

solutions

the strong

(thin

An important This

follow

the solutions

A further

This

that

within

This

values

towards

figure

is an update

starting

from

above

a plane

of at, all solutions

negative

values.

that

the total

of the

range

i.e. within

pt indeed cluster

bound

question

is: if one starts

for IR values a square

with

accessible

0 <

in Ref.

above

fully (Even

demonstrat,e

[42].

the line can end up below the fixed

sectors.

line tend

The fixed line solution,

UV initial large

being

and

towards

singled

between

vice

If one were to infinit!

out by t,he

the two classes

values

at p = A = 10” GeV, say. randomly

0 5

pt C: 10, 0 < p,q 5 2.5, as presented

square

has shrunk

to within

onto

IR values at ~1 = mt = 176 GeV cluster hand side in Fig. 6 (1091: notice first

a square

pt 5 1, 0 < pi 5 0.25. Within

on or very closely

for A = 1O’5 GeV.

of a figure

0) in the PH-

which

for a fixed line.

the perturbatively

line fail to be drawn

(symbol

into two disjoint

on the left hand side in Fig. 6, how closely do the corresponding around the IR fixed line? The result is presented on the right square,

fixed point

RG flow are shown,

p~/p* be finite in this limit, is quasi the “watershed”

is characteristic

important

line.

any fixed line divides

large

tend

for the “top-down”

no solution

feature:

towards

below

condition

of solutions.

distributed

is that

is a general

and all solutions

representative

of the fixed

observation

versa.

boundary

fixed line (fat line) and the IR attractive

lines)

IR attraction

and M. Wimmer

of one tenth

this square

of the length

the points

to the fixed line (fat line); for lower values the line;

though

their

the allusion

upper

boundary

to a trivial

of

with

values

pt the points

above

is the pendant

IR fixed point

of the UV larger

of the

“triviality”

does not apply

any more

and the notion of an upper bound would be more appropriate we follow the usage in the literature and maintain the expression triviality bound). In comparison with the triviality bound for the Higgs mass, discussed

in the framework

into the discussion

of the d4 theory

has turned

the upper

The lower bound, the one-loop of all points starting from the the fixed line (from

below!).

in Subssect.

Higgs

mass

4.1, the inclusion

bound

into a top mass

of the top Yukawa dependent

bound

coupling [7], [lo].

vacuum stability bound (see Subsect. 2.6) [7], [ll]-(171, is the IR image UV initial values & = 0, i.e. XC, = 0; they all end up very closely to

Thus

for this large

value

A = 10” GeV for the UV scale

the lowest

order

vacuum stability bound is very close to the IR fixed line; the IR fixed line is clearly an upper bound for the (lowest order) vacuum stability bound. It becomes again clear from Fig. 6 that it is IR attraction of the fixed line rather than of the fixed point (diamond) which determines the “top-down” RG flow.

Top Quark

45

Boson Masses

and Higgs

uv

IR p = 1Ol5GeV

/.I = 176 GeV

PC= d/g: Figure

6: Randomly

are subject

[log]

chosen

to the

(figure

on the left hand

side).

The

strong

demonstrated.

UV initial

values

in the pH-pt-plane

RG evolution

down

to the

side) has scaled

down

by a factor

IR attraction

of the IR fixed

tip of the clustering

~~0 > 1, i.e.

of 90% of the randomly

is the absolute

upper

IR points

upper

initial

bound

as well in pt (which

well as in PH, both

calculated

for A = 1015 GeV;

of the upper

and also that

bound

So far we have discussed and

fixed point

bounds

(thin

restricted

lines)

0)

the triviality

and

of A. In the physically

drawn

into the Pendleton-Ross

Obviously,

from

Figs.

for the triviality

the points

which

UV scale

A down

The discussion

fail to reach

for couplings

treated in this section) attractive RG invariant here for the ratios for the Higgs

stability

inaccessible fixed point

interpreted clearly

Fig.

UV plane hand

IR fixed point,)

is

PH.

with

UV initial

demonstrates

values

[37]. [40], [42].

as Hill effective

the IR fixed

7 shows

to the A dependent the A dependent

fixed

This

point)

the attraction

bounds limit

(symbol that

line after

A = lo*,

are drawn

A -+ a

the IR fixed

triviality

bounds

106,

lOi’,

tip of the wedge closer

the upper

ns

polvrr

[ 1‘11

line (fat line)

and

vacuum

1015GeV

slides down

stabilit!

within

these

the IR line [:<‘;I.

to the IR fixed line for increasing bound,

i.e. the tip of the wedge.

is

0). the IR attraction

as well as for the vacuum

to the IR scale

on pt and

the

on the right

0 for the

of all the UV points

this figure

A = 1015 GeV.

6 and 7. we may conclude bound

symbol

that

(figure

[6], [7], [31], [32], [lo],

been

[42] for the UV scale

Clearly

vacuum

values

origin

UV scale,

[30] m relation

considerations.

p = A = 1015 Gr\;

notice

it lies on the IR fixed line [32], [37], [40], [42].

[6], [7], [lo]-(171,

one-loop

[42] and

a large

(symbol

bound

values had

UV scale

of 10 to the IR plane

is the IR image

chosen

at the

p = 176GeV;

line (fat line with

The tip of the line is the absolute

The outermost

IR scale

stability

evolution

of the fixed line is the dynamical bound.

with

They

the finite

are the boundaries

evolution

path

of

from

t ttc

176 GeV.

or rather

for ratios

of couplings

may be translated

(within

the lowest

order

into results for the top and Higgs masses. The IR fixed line corresponds to an IR top-Higgs mass relation. The triviality and vacuum stability bounds. formulated

~,q = X/g:

mass as functions

or R = X/g: of couplings, of the top mass

hy means

may

be translated

of the lowest

order

into corresponding relations

bounds

m, = y,(m,)r*/v?

and rn~~ = V-1.. The final

results

for the IR fixed line in the mH-mt-plane

and the corresponding

triviality

and

vaculiil,

B. Schrempp

and M. Wimmer

1.2

L’H = 1 A/!?,2 0.8

/

/ 0.6

Otf

Figure

7:

triviality

The

IR fixed

bounds

bounding

demonstrated.

stability

(fat

line)

allowed

insight differential

regions.

equation

The

higher

features

IR attraction upper

values order

&I = R(pto),

&

in relation

of the bounds

towards

bound

A dependent

lo6 10”

1O’5 GeV,

the IR fixed line is

on pt and PH slides

down

the IR fixed

of A. will be presented

[42] f rom the general

in Sect.

6.

solution

of the one-loop

??

R3)exd--~bt~ ho)1

(Ro -

&l) +

to the

A = 104,

of the IR fixed line solution

_ = R(pto)

0)

for UV scale

corrections

1 + f (hi, with

(symbol

lines)

may be obtained

(156) in terms

R(pt) =

point

(thin

the absolute

for increasing

all known

into all these

fixed

bounds

The tip of the wedge,

including

and

stability

the IR fixed point

bounds

Analytical

line

vacuum

wedge-formed

line towards

Riccati

and

(166)

Pto)l fi,,$%p,exp[-l%‘7 Pt’- f

and (167)

The difference term

of any general

on the right

in the numerator then

hand and

-

& - R,, sufficiently of approach -

the fixed line solution, which

small,

to the fixed

such

in turn

If the initial

i.e. the solution

ptabove and below the IR fixed point path; this reflects the IR attractiveness has to distinguish two cases: l

and

the denominator.

R(pt) = &),

clearly

solution

side of Eq. (166),

remains

is controlled value

&

R(pt)

- 7i(pt),

is given

by the exponential

happens

on the fixed

on the fixed line for all values

by the second

exp[-F(p,,

pto)]

line, i.e. &, = %, of pt. For values

of

it = f F(p,, ~to) is positive and increases with increasing evolution of the fixed line. For the discussion of the rate of attraction one

that

line E(pt)

the denominator is given

of Eq.

by exp[F(p,,p,o)]

(166)

remains

in the numerator

close to one; the rat.e with

a behaviour

for p,, pto close to 219:

exp[-~(phpt~)]

=

(_cJ

EE)““‘” =(AJi”‘” =(1_ _Lg~oln(.$)-‘2’, (168)

Top Quark and Higgs with

the high power

the exceedingly power

with

IR attraction -

a strong

approach

R(pt)

rate

is attracted

bound,

shrinks

for sufficiently

towards

the fixed line to within

a finite

WPt) +

~z(Pt) +

for R,a = 0, the analytical (166)

R(Pt1=

A final point

length

point

monitorrs fairly

large

for the strength

concerns

Wt) +

the hierarchy

of two IR attractive

pt0 -

m

of

(169)

for

dpt’

path

triviality

the

enhanced,

due to the solution

(170)

Ro -+ co,

ho)1

Pt’- ;j

of evolution

controlling

difference

---_Texp[-F(p,‘,

(i.e.

for increasing

value

of A). This

is tlie

bound.

of the lowest

order

vacuum

-R0exp[-JYhpto)l 1-_8W 3 0

is the rate

138) on the fixed line fi(p<).

Even for Rc -t co the general

$wd-J%t, pt0)l

form

This

the fixed line z(z(pt) is strongly

towards

for the A dependent

pts.

fixed point(

(166).

Likewise, Eq.

large

in solution

for increasing

form

+ 0 for

i.e. the Hill effective

of the denominator

from

l

which

s3'

s,qf

which

l/7

The resulting

measure

= ($)ia’3--$&

of IR attraction

the IR attraction

increase

analytical

2/9.

z 1.25, is the appropriate

x (!.$)a’3 pto - 9

to the upper

for large I& - s, substantial

low power

the IR fixed point

of the fixe9d” line R(pl).

exp[_~(p,,pto)]

l

m/21

47

the forbiddingly

of pt towards

to 4,

Masses

pt0L O(l):

for Pt,

with

x E.75 beating

m/3

slow approach

respect

Boson

pt0)l

Pt0dpt, exp[-F(pr’.

J

for

bound

is obtained

(171)

Ro = 0.

PiI - f

Pl

of IR attraction.

lines in the pH-pt-plane:

stability

The fixed

the trivial

point

(157)

one, pr=2/9.

is in fact

attracting

the intersection

exceedingly

weakI>

and the nontrivial one, isiJ(~r). discussed in this section. This follows a general rule like (sZ/s~o)“‘, [89]: A fixed point in a plane of two varables is the intersection point of two fixed lines in the plane. The

strength

equations

of attraction

the RG flow than that

the fixed

pe-pi-plane: more

4.5

of the two lines

for the two variables.

strongly

the other

point

one; the degenerate

attracts

the point attractive

The general

is regulated case

is that

than

the more

it is gratifying trivial

one of the fixed

case in which

the RG flow “radially”,

to make

by the coefficients

is that

pt=2/9

both

is, however, that fixed

in the coupled lines is more

are roughly also

equally

possible.

differential attractive

for

attractive,

Coming

back

such to the

the physically non-trivial fixed line E(pl)

is

line.

The Top-Bottom-g3 Sector of the SM and MSSM - Top-Bottom Yukawa Coupling Unification as an IR fixed Property

The experimental top mass is much larger than the experimental bottom mass. to see, which are the IR attractive fixed manifolds in the top-bottom sector. Early analyses Refs. [31]-[36] revealed sions were applied to fictitious heavy

already much of the basic higher fermion generations.

It is therefore

interestiug

structures, though most of the concluImplicitely the analyses [49]-[6.j] of

the consequences of tau-bottom Yukawa coupling unification in supersymmetric physics single out a narrow band of allowed values in the tan d-m,-plane which

unification for the 1R turns out to lie in the

B. Schrempp and M. Wimmer

48 vicinity

of the IR fixed line in the top-bottom

results

look like will be discussed

the discussion

to the IR structure

Let us begin and

by a very

gb in the SM and

off for the moment equations IR fixed resp.

that line,

kl=kb-plane. result

This

in the light

Such which

from

kind

of an “escape

route”

The

top-bottom

After

with

Yukawa

these

and

non-zero

bottom

Yukawa

couplings

to see from

the resulting

condition

that

g1

(switching one-loop

kb = 0, accompanied

ht,

by the boundary

couplings,

considered

RG by an

the ratio

g,/gb,

the line kt=kb in the

in the g,-gb-plane.

resp.

approximation

to exact

top-bottom

Yukawa

out an IR attractive property and a very intriguing

in top-bottom

Yukawa

at the UV scale in

unification

models. in the SM, since

the experimental

unification tan@

any disparity

largely

this is so and how the the issue by confining

it would

situation.

imply

in this approximation

The subsequent

discussion

mt=mb

will show some

out of this dilemma.

free parameter

tan /3 being

Why meet

gr, gb = 0, resp.

one-loop

interest

GUT

is of no interest

variance

with

in this

renewed some

next. [47],[48]

the top

It is easy

distinguished

line corresponds

is at strong

and

coupling).

00. It is the line gt’gb

of the recently

an IR unification

couplings

where

fixed point,

RGE

analyses

are the only

at all scales p. This is as it turns

as motivated

additional

gauge

of the

fixed

of a setting

common

to be discussed

recent

in the SM and the MSSM.

hb in the MSSM,

for gt, gb +

unification

the MSSM

discussion

is a trivial

the solution

7. More

of the RGE

also the strong

there

kt/k,, be finite

coupling

brief h, and

system,

in Sect.

is, however,

a fascinating

and

viable

according

to the lowest

order

Eqs.

allows

in the masses

since

option

in the

(29) to have

MSSM. equal

The

Yukawa

in this order

mf -=

&tan

mb

hb

(172)

13

a free parameter.

introductory

remarks,

let us discuss

the

much

more

non-trivial

case

of three

non-zero

couplings St, while

all the other

[47], which Again

couplings

was performed

it is economical

Qb,

are put

,!a

#f-4

rev.

to zero in the

ht,

RGE.

hb,

Q3 #

The

(173)

0,

discussion

follows

closely

the analysis

in the MSSM.

to consider

the following

ratios

SM

of couplings MSSM

I

(174)

which

lead to the following

-14g+

-14g$@

dd

partially

decoupled

form of the RGE

= pt(9p, + 3pb - 2)

= pb(9pb + 3p, - 2)

ds32

The obvious symmetry of the set of equations kb. will be reflected in all following results.

with

-39:dPt = p2(6p, + pb - ;) ds3’

(17.5)

-3ddPs=pa(6pa+pr;)

dd

respect

to the exchange

of gt and gb, resp.

of kt and

and Higgs Boson Masses

Top Quark Clearly, only

the system

IR attractive

of the two coupled

differential

one is the following’

equations

(175)

49 has a number

of fixed

points:

thr

[47]; SM

MSSM (1X)

pb

Pt = the other

fixed points

=

;

pt

=

Pb =

5

are MSSM

SM pt = 0 (IR repulsive),

pb = 0 (IR repulsive),

pt = 0 (IR repulsive),

pb = 0 (IR repulsive).

pt = $ (IR attractive),

pb = 0 (IR repulsive),

pt = 6 (IR attractive).

pb = 0 (IR repulsive)

pt = 0 (IR repulsive),

pt = 0 (IR repulsive),

pb = 8 (IR attractive),

pb = $

(IR attractive). (177)

There

are two IR fixed lines in the pt-pb-plane.

IR attractive

one, does not come

as a surprise; pt = pb

It signifies discussed

again

top-bottom

wh’ICh will turn out to be the less strongly

The one [47]

Yukawa

in the SM and the MSSM.

coupling

unification

(1X)

for all scales p with all the implications

already

above.

The other

one is the solution

distinguished

by the boundary

pt -+ 0. It is the solution

Pb + 0 as well as in the limit

condition

[47] which

that

pb

=

o),

(pt

=

0 in the limit

the three

fixed points

MSSM

SM (p1 = ;,

it be finitef

interpolates

pb

=

i,~

(pf

=

o,pb

=

5,

(pt

=

Pb =

6.

oh

(f’t

=

Pb =

i,.

(Pt

=

o,pb

=

$1 (l7Si

This

solution

point

(137)

has

at pt=O. The pt-pb-plane.

roughly

at pb=O and line can

the shape

be considered

Its end

symmetric

points

point of the

two IR attractive

Pb

2

fixed

pt

Pi

(f’b

=

(Pb

lines

-

fi

Pt 1’

ln

fi-

24p,“2pb3t2

intersect

each

$

other

1 fi

+

i?T’

+

ET’

Jir;

“$,

_t

in their

Pendleton-Ross fixed

fi

only

1

fixed

of pt with

the exchange

analytical

Pt)’ lnJir;-fi

-

i?4p,1/2pb3t2

Pt =

are the

under

Pendleton-Ross

In case of the SM this IR fixed line has even an implicit

For

The

circle.

mirror

as the generalization

For Pb5 pt

fixed point

of a quarter

the corresponding

point

into

ph

the

solutions

pb+Pt

Pb +Pt (

common

point,

the

IR attractive

(176).

Next we need some analytical as well as of the IR fixed point

insight

into the respective

(176).

In absence

the system (175) of differential equations point (I 76) and by solving it analytically pt0

=

strengths

of attraction

of an analytical

solution

for pt and pb in the neighbourhood for UV initial values pt(P

=

I\),

PbO

=

Pb(P

‘One of us (B.S.) is grateful to W. Zimmermann and F. Schrempp ‘this solution was first written down by F. Sehrempp (unpublished)

=

of the two IR fixed lines one proceeds

IR fixed

(1s’)

A).

for a communication

by linearizing

of the common

and discussion

on this pant

50

B. Schrempp and M. Wimmer

The solutions

are (47)

I

SM p&,2)

= 5 + fr(Pto +

PbO

-

4,

MSSM

P&7;)= $ + ;(Pto+ /‘bO

(&-’

-

$1

(&-’ 5 9

+

$tO

-

PbO)

($-)-”

+

$‘tO

-

Pbo)

(2)

(183) P&7;)

=

;

+

;(Pto

+

PbO

-

5,

(2)-j

Pb(g:)

=

;

+

$‘tO

+

pb0

-

5,

($)-” 5

-A -1

In this approximation read

off from

2(Pt~

-

ho)

the second

the approximate

2

(

&‘tO

)

IR attractive analytical

relevant

negative

power

of gz/gio

to 519 in the MSSM.

Thus

the fixed line proceeds this line towards &=Pb

ml, leading

to top-bottom

. In the SM both l/14 serious

dilemma

of the

that

“escape

however,

There

route”

is again

initial line.

values The

pbn=O.

from

value

Hill effective

fixed point attracts

couplings

in this

small,

it does

This

equality.

ii) it does not attract dependent

mathematical

interest

attractive

on the A and

the exponent

not

seem

This

is a practical

to be a

is the basis

argument

which

is.

are much larger than in the SM; the quarter the Pt=Pb fixed line with a strength 5/9.

in the

pl-pb-plane

under

discussed exchange This

upper

of the RGE

solutions

on the length only,

to start p between

in the MSSM i/9,

for pts=O.

RGE

the

of view.

which

collects

the

IR images

in Subsect.

4.3. is one point

of the variables boundary with

pt and

is an IR effective

sufficiently

large

on this

Pa* there

initial

of solutions

the upper

with

smaller

initial

of the evolution boundary,

values

path,

line,

with

fixed

line in the sense ~~0 or &,u and

~tu or PM). The Hill type effective

for

symmetric that

that

it

i) it depends on the I:V shaped fixed line, and

i.e. on A (in the limit

i.e. the Hill type

the point

is a mirror

values

is independent of those initial values. It is not a genuine IR fixed line since scale A, in fact it shrinks with increasing A towards the genuine quartercircle line is again

attractive,

in part,icular

that

are so far from

section.

happens

i.e. for all values

A to mt)

point

a strength

point,

all solutions

line is more

Ptu or Pbs. Since it is essentially independent of sufficiently large UV strongly attractive, we shall also call it a Hill type e$ective IR fixed

very

fixed

from

the mathematical

boundary

Due to the symmetry

it strongly

Yukawa earlier

One can immediately

in the SM and 7/9 as compared

if the solution

path,

are exceedingly

the evolution

exponents

with

pto or ,@u and

Hill effective

as well as l/14,

mentioned

UV initial

Only

at all scales.

negative

an upper

large

(176).

unification

the physical

fixed line attracts

Sufficiently

fixed point

(given

>?

RG flow from all UV initial values above and below the quarter-circle fixed line and then close to or along

Yukawa l/7

not satisfactory

. The corresponding circle

“top-down”

to l/14

evolution

small

2

(

= J, resp.;.

IR attractive

to be compared

on it for the whole

exponents,

is so ridiculously

l/7

first towards

the IR attractive

fixed line it remains

PbO)

(183)

the quarter-circle

being

the

roughly

line has the form Pt+Pb

solution

. As well in the SM as in the MSSM,

-

effective

A -+ co, which

line tends

towards

fixed is of

the more

IR fixed line).

Let us add two illustrative figures for the MSSM. In Figs. 8a) and Sb) we show the two IR fixed lines. the one denoted by 1 is the more attractive one, t,he one marked by 2 is the less attractive one, which incorporates top-bottom Yukawa unification at all scales p: their intersection is the IR attractive fixed point, marked by a symbol 0. For comparison in Fig. Sa) the upper boundary, playing the role of a Hill type effective fixed line, as calculated for an UV scale A = Afour z 2 1016 GeV is shown. In Fig.

Top Quark

and Higgs

Boson

Masses

51

Pt = ht”/g;

4 Figure and

8: The

more

attractive

the IR attractive

addition

fixed

a) The upper

is shown.

b) The

b) IR fixed

point

bound

“top-down”

RG flow, indicated

a selection

“watershed” tend

between

towards

SM. Finally, the MSSM realistic

of general

solutions

zero in the same Fig.

14 in Sect.

“top-down”

case where

An inclusion

(thin

lines). towards

limit.

Fig.

9 in Subsect.

into the RGE of the MSSM

attractive,

supplemented thus justfying

The Higgs-Top-Bottom-g3

and

two-loop

by the trivial a posteriori

effective

lines),

In

fixed line.

is shown

to be

fixed scale

is demonstrated

line appears

p and

as the

solutions

which

the RG flow in case of the values

at A=MC;ur%

subject

to

of the IR fixed lines for the more

RGE

equations

are used.

has been taken

fixed point

its omission

value

into account

p,=O.

in the present

which

in Ref. turns

out

discussion.

Sector of the SM

of A. St,

leads

line 2)

a First IR Fixed Surface the discussion

us to a reasonable

one-loop

of UV initial

into the close vicinity

coupling

strongly

line (fat

of the MSSM.

the IR fixed point

4.6 demonstrates

of the T Yukawa

is then

(thin

attractive

for increasing

are included

to be rather

Finally,

infinity

the contraction

RG evolution,

towards

the more

couplings

(176)

-

solutions

all gauge

[47]. The fixed point

4.6

Again

tend

6 will show

IR fixed

in the Pb-pt-plane

the role of a Hill type

by representative

which

two-loop

0)

fixed line line 1 and then close to it or along it towards top-bottom Yukawa unification at all scales p.

the IR fixed line 1 and then

solutions

less attractive

(symbol

line). playing

strongly attractive Line 2 implements

8b) the RG flow, first towards

line 1). the

intersection

for A=Mour(thin

first drawn towards the more the IR attractive fixed point.

with

line (fat

at their

RGE

approximation

for the three

( 175). supplemented

ratio

gb?

g3

of the SM and

variables

pH=X/gj,

#

(1SJi

0

to the first pt=gf/gj

IR attractive and

pb=gi/gi

fixed surface are given

[48]. The

hy the

Eqs.

hy

- 2dPH= -,g32 %Wz + 6pHpt + 6pHpb + 7pH _ ;3p,2 _ dg3

XPb2,

(1Si)

52

and M. Wimmer

B. Schrempp

Figure

9: The IR attractive

surface

line l), the less attractive “top-down”

RG flow is first drawn

IR fixed line 1, then along solutions

(thin

The common

lines).

in the PH-&-Pa-space

containing

towards

the surface

or close to this line towards

The figure

fixed point

was taken

from

[48] of the set of three

In the

space

of the

surface

in the PH-&-plane the exchange PH-pb-plane.

three

ratio

[48], shown as discussed

variables

in Fig.

IR attractive

fixed

surface

fulfils

and

4.4. and

Solutions

IR fixed line (fat intersection.

the surface

demonstrated

the boundary

their

evolution

on the fixed surface,

the

by representative

is

(1S6)

24 Pb there for pb=o

exists

a non-trivial,

by the IR attractive

- due to the mirror equally

symmetry

IR attractive

very fixed

strongly

condition

of the RGE

under in the

(lS7)

of finite

between solutions

evolve

IR

line =(pc)

line E(@)

ratios

pH/p~

and

pH/Pb

for pt

solutions above the surface, for which below the surface which tend towards

values. Again the IR fixed surface separates t.he pH-pt-pb-space in two distinct of the RGE can penetrate from above to below the surface or vice versa. starting

The

towards

$ - f +$f +...

or pb increasing towards infinity. It is the “watershed” the ratios pH/p1 or PH/Pb tend towards infinity, and negative solution

within

at their

m-9

of pt and Pb - for pt=O by the corresponding For large values of Pt=Pb=P it has the expansion

pH(p) =

The

equations

PH=-

9. It is bounded

in Subsect.

0)

[48].

differential

PH, it

then

the IR fixed point,

Ref.

6

(symbol

(not shown),

1 Pt=Pb=-,

attractive

the more attractive

one (fat line 2) and the IR fixed point

within

the surface,

independent

regions:

no

of the r\’

initial values PH,,, pto, gio and A. The full “top-down” RG flow (not shown in Fig. 9) is first strongI> attracted towards the IR fixed surface, from above and from below, and then proceeds on it or close to it as displayed in Fig. 9. Within the surface WC rediscover the generalizations of the two IR attractive fixed lines, the more attractive quarter circle line (fat line 1) and the less attractive line pt=pb (fat line 2), which were discussed in Subsect. 4.5. So the RG flow within the fixed surface (indicated by the

Top Quark thin

lines for representative

and

from

symbol

below,

solutions

then

along

in Fig.

or close

9) is towards

to it towards

the more

attractive

the common

fixed

fixed

point

line, from

above

denoted

by the

(lS6)

0.

A last

comment

surfaces above,

concerns

again

the hierarchy

in the PH-&&-space. and

the two more

respectively, between

in the surfaces

it is gratifying

The

trivial

of IR attraction.

by far most

ones,

pt-pb-plane.

the non-trivial

attractive

5

and

53

Boson Masses

and Higgs

rising

In fact

vertically

the

and the corresponding

is the most

attractive

to find the physically

strongly

are in fact

fixed

lines

1 and

more

trivial

IR attractive

surface,

line and

9 are

The criterion

in the specific

surface

most non-trivial

circle

2 in Fig.

ones.

three

non-trivial

over the quarter

one lies buried

Infrared Fixed Points,

There

IR attractive

each

to be the most

line,

an intersection

for which

coefficients

discussed

the pt=pb of the three

IR

of the RGE.

Again

IR attractive

one.

strongly

Lines, Surfaces in the Presence

of

All Gauge Couplings In this section entering last years

l

the analyses

the analytical

The top down

RGE

to collect

ztpwards

section.

Thus,

. Thus, there

fixed

point

couplings

are switched of large

is the fact

evolved

towards

also smallish IR value.

type from

coupling

large

since one expects

the

UV initial

values

ht, >

I

the phenomenon

in Subsect. the power

of an upper

4.3. also to be present of the Hill effective

fixed

values.

UV initial

values

A corresponding

[43]-[47],

of

in the MSSM.

at the IR scale p = m, = 176 GeV.

for the ratio figure

[49]-[65]:

[llS],

pto = h,i/y&

will be supplied

value as small as, say, hto z 0.4 is evolved

safe to conclude

91 and g2, Before development

towards

[119] that

are R(i

later

in this

h,( 176 GeV)

in the top Yukawa

z 0.X.

coupling

is a very strongly

directly

leads,

The near-at-hand

closely

couplings

important

from

on; it was precisely

that

with

all radiative

IR attractive

above,

ht(p)

into one point

UV initial

IR attractive which

gauge

the most

[31], as discussed

a larger

e.g. an initial

it is perfectly

coupling

essentially

the IR images

. More surprising

in words

is not all too surprising,

as a Hill effective

if the electroweak point

flow of the top Yukawa

zz 1.1. This

acting

the electroweak

the RG flow of the top Yukawa

A=MCUT is focused

at the UV scale h,( 176GeV)

to include

let us summarize

(1181, [119] concerning

[43]-[65],

bound.

are extended

discussion,

qualitative

is that

switching

fixed point

(137)

together, the following

. The effect

corrections

top mass

interpretation.

and the effective their

analytical

discussion. also present

(188)

to the

mt = 0( 190 - 200) GeV sin 9.

gauge

to a certain couplings

Hill fixed point

focusing

h, = O( 1)

at

applied.

implied

on the electroweak

combining

is qualitatively

fixed point

power

(13s)

for the

this is roughly

extent

effects both

also in references

the genuine

quoted

Pendleton-Ross

to move up and to move more

“top-down”

what

(1%~)

RG flow.

As we shall

see

happens.

in the SM. but the focusing

effect

for the RG flow is much

less pronounced.

The analysis mathematical

1481 described in the following traces the dynamical origin of these phenomena magnifying glass back to IR attractive fixed manifolds in the spaces of ratios

In case of the MSSM those known already

the analytical are somewhat

quasi with a of couplings.

insight is much improved. but the practical consequences limited. However, in the case of the SM the improved

beyond insight

54

B. Schrempp and M. Wimmer

allows the exact determination of an IR attractive in presence of the electroweak gauge couplings.

top mass, Higgs mass and top-Higgs

mass relation

The presentation still sticks to the one-loop RGE, since in that framework all discussed IR fixed manifolds are exact. The appropriate two-loop results as well as their translation into results for the Higgs and fermion masses, mass relations and mass bounds including radiative corrections are reviewed in Sect. 6. In the following it is again consistently introduced in Eqs. (97), supplemented

economical to consider the ratios of couplings PH, ptr Pb and P, by the ratios

(190) The one-loop RGE, rewritten

for these variables, are SM

MSSM

I

(191) where gs is treated

as before as a function of 2, and MSSM

SM

zdpl

3dg;

=

-7gF

=

Pt(iPt

Pa(%Pt

-P2

-

+

+

;Pb

+

- yp

d p2

19

2dP2 _

93s

-7g2*

41 2

= -P1

g3Q

+

;Pb

+

2 93d9,2

zP22

pr

pr

-

;p*

-

-

;pl

$2

-

$2

sdPt -3gsa

1)

-

-

1)

-3dd

~t(6pt

Pb(Pt

+

-P2

+

6pb +

Pb

-

-

pt +

2

YjPz

;/JI

p,

-

-

Ep,

3P, - 5)

-

3p2

-

f,

dd

ds3’

-7d* dd =,43p, + 3pb + ;/IT- $,, _ ;p2+ supplemented

=

=

1

=

+$G

7)

3

= PJ3P6 + 4p, - iPI - 3pz _t3),

(1921 * ,

within the SM

-7gzdPH dd

=

6pHpb $2p~p,

12PH2+~PHPL

•k

-3Pt2 - 3PbZ -

Pr2

In Eqs. (192) and (193) the ratios are treated

+

gp12

as functions

- ApHPl

$

iPIP

+

_ !fpHP2 +7pH

(193)

;p22.

of the independent

variable gj.

The general one-loop solutions for p1 and pz are SM

MSSM PI0 p1(g3 = 2

&+

I) _ yp,o (19-1)

Pzo PZ(d)

=

PZ(S3

* 2

(-ZPzo+

1) +

+$zo

=

* 2

P20

(5Pzo + 1) -

$Pzo’

Top Quark and Higgs Boson Masses Let us, for the course and

purpose

unphysical, (193).

In this

differential attractive

of mathematical

since

it leads

limit,

equations fixed point

the

which

combined region, only

with

the

knowledge This

Higgs

reason

attractive

which

to be strongly

persist

is most

in the next

leading

for the first

in Sect.

8.

Within

RG flow from

pt

=

zero.

Thus,

-+ co, which

the

is of

(192)

RGE

coupled

system

of the SM, has a single

gauge

gauge

defined

couplings

Already

p7

=

common

=

p2

=

of IR

0

supplemented

are

including

This

small

the electroweak

leading

gauge

circumstance. in the

couplings

averaging

IR

amounts couplings

procedure

to substantially

[42],[47]. higher

IR

[119] quoted

in

[31]-[37],[41],[43]-[65],

[118],

in more or less exact

frameworks.

effect of the electroweak

gauge

by boundary

for [urge values

conditions

~1, ps=O.

by

of the electroweak

numerically

elaborate

considerably,

analyses

Pl

0,

of the electroweak

when

even in the IR region

studied

f?

in the last section.

inclusion

is shifted

=

couplings,

performed

true.

many

pb

of the RGE in absence

[30] or by a more

the strong

couplings:

there of

exist

p, and pz.

p1 and pz are small as compared

where

in the case of the top sector

to success

are exact.

time this

known

Of course step and

starts

the space in Refs. enlarged

by treating

of ratio

in presence

of all gauge

initial

space

values

for

these

wi!l consist

in feeding = 0.2319

Such

of reduction

of parameters

fixed subspaces

IR manifolds

to

couplings

in the experimentally (MS); i g noring

level the resulting

initial

the errors

and

was put

and

the flill

for the moment IR fixed manifolds

IR fixed point

measured

pi, pt. 1)6

to be discussed

(disregarding

in the (unphysical)

f 0.0005

to

a procedure

are determined

is considered

p1 and ~2). On the one-loop terminate

in parallel

by two dimensions.

the IR attractive

all the IR fixed manifolds

sin*&(mz)

p1 and ps as free variables

parameters

[39] in the framework

the UV to the IR towards

experimentally

next

0

this statement

IR fixed manifolds

pl). This increases

l/127.9

=

electroweak

The

attractive

p2

the systematic

averages

[48] behind

=

coupling)

corroborate

transparently

forward

The

towards

gi

subsection.

The procedure (and

ps tend

discussion

is not quite

values.

of this section,

is an intricate

p1

fixed points

mass

very strongly This

of the perturbative

of the electroweak

the that

i.e. non-running,

the introduction

one.

that

one to expect

top and

0,

the consistent

of the IR attractive

attractive

=

of the T Yukawa

correction.

constant,

the position

There

in retrospect

lead

to a small

by their

the limit

(19.5) pr

’

24

in absence

might

pr and

to the one in absence

allows (and

variables

for a moment of applicability

a-9 PH=-

is identical

c.ouplings

trea.t

the framework

(192), supplemented by Eq. (193) in case (in the unphysical region of the RGE)

1 Pt=Pb=;’

This feature

insight,

outside

55

conditions evolving

(195).

[5] o(mz)

=

up to p = ml =

176 GeV , we find g:(ml

if the initial which known

= 176GeV)

=

0.215,

gi(rn,

= 176GeV)

= 0.418

pl(mt = 176GeV)

=

0.160,

pz(m, = 176GeV)

= 0.312,

value

(96),

will be called functions

gi(mt

henceforth

of the scale

= 176GeV)

= 1.34, is taken

“experimental p, or, more

relation

conveniently

into

between

account. p1

and

for our purposes,

leading

(196) This

So. finally

feeding

the n-dimensional

in the values

in Eq.

IR fixed manifold

relates

ps”.

Both

they

become

of l/g:, which in turn is a known function of the scale 11. Finally, for calculating Higgs mass values, one is interested in evaluating p1 and p2 at the IR scale, chosen in the following,

to

p,(p) to p2(p)

functions known

are then functions

IR attractive top and as p = mt = 176 Ge\.

(196). for free variables

p1

and ps shrinks

to an n-2 dimensional

manifold if the IR values for p1 and p2 are introduced. It is very important to realize that these submanifolds become IR fixed points, lines, surfaces,... in the following sense: the “top-down” RG ROU tends more and more closely towards

them from above and from below, if the UV scale

A increasrs.

B. Schrempp and M. Wimmer

56

while the IR scale (in our choice) p = mt = 176GeV) remains constant. In the limit A -+ 00 while keeping the IR scale fixed, which is of mathematical interest only, the full RG flow is drawn onto them. Let us also point out that the IR scale, chosen as 176 GeVin the following, is not really a free parameter. In determining the top mass r-n?” or the Higgs mass mu” from an IR fixed point or fixed line, the appropriate IR scale p is determined implicitely from the conditions Sg&

= my’=)

u&j/_

=

m~ie(l

+&(/J = mP)),

=

m$“O’“( 1 + 6~(p = mg”))

(197) in the SM,

(198)

in the MSSM. %,(p = mpOrc)sin B = mr”( 1 + &(/J = my”)) (199) Jz Since, however, the dependence on p in gt, Aand ht is only logarithmic, the correction to the result will turn out to be negligible. Again, for simplicity,

5.1

Msusv=mt=176

GeV is chosen throughout

this section.

The Top Sector of the SM and MSSM

The minimal and most instructive subsystem of ratios of couplings in the presence of all gauge couplings is pt, pr and ps. It provides also the basis for the IR fixed point in the top mass of the MSSM, mt = O( 190 - 200) GeV sin p [44]-[65]. To start with, there are several IR fixed surfaces to be found in the ~r-~i-~s space. The first surface to be put forward in the literature [39] t urns out to be not the most strongly IR attractive one. It was found in the framework of parameter reduction, where IR attraction is not a criterion; it will be discussed in Sect. 8. By far the most strongly IR attractive surface [48] is characterized by its boundary conditions for large values of pi and ps. These limits again lie outside of the region of validity of perturbation theory; however, as we had also experienced in other cases, the surface defined by these boundary conditions determines the properties in the perturbative region. The origin of this surface is most easily exposed, pi # 0 are discussed. For ps=O, pi # 0 a strongly

IR attractive

if first two limiting

line in the pt-pi-plane

cases ps=O, ps # 0 and ps=O,

appears

with the following properties

[481 SM

MSSM

I for pi + 00 it behaves asymptotically

as

I 11 Pt + GP’

Pt 4

with the general solution approaching

56

EPI

it in this limit as

WV

I

Clearly, this fixed line is strongly IR attractive, since p1 decreases in its evolution towards the IR; the attraction towards the fixed line is controlled by the large exponent 99/82, resp. 112/99 of p,.

51

Top Quark and Higgs Boson Masses Incidentally

the fixed line corresponds

to an IR fixed point P~PI = d/d,

Similarly

instructive

different

for the SM and

(201)

rev. hfld.

[48], w h ere pi is considered

is the limit

is even qualitatively

in the variable

MSSM

SM there

p,=O, which

of pz only with

as a function

the MSSM.

is an IR attractive

there

fixed point

is an IR attractive

fixed

line

in the pt - pz plane 42

Pt=E

p2=19t

227

The general for ~2, ~20 near

g

2

pi =

gp2

for

PZ

-+

00

solution

(202)

for large

is

values

of p2 is

I -227 pi~~_(zL&&-22”266

Clearly,

the fixed point

the IR and The

resulting

mentioned

The numerical

into pt=7/18,p,=pz=0 flow with

is much

more

strongly

pt-p1-p2

plane

in the pt-P1-p2-space

into the IR fixed point RG solutions

is bounded

IR attractive

is displayed

towards

by t,he above

than

in Fig.

pt=2/9. p~=pz=O

do. The important

point,

the IR fixed point.

for pt,p1 and pz is first attracted

values

can be represented

towards

10 for the

for the SM

however,

Thus

is that

the

the “top-down”

the surface

and then

and

RG

close to

analytically

in various

regions

by double

power

For

series expansions9

are [48]

11

=

in the

in its evolution

are large.

the fixed point.

for p1 -+ cm, around

Pi

merges

, as all L‘top-downn

it towards

the SM they

pz decreases

the attraction,

surface

of the IR fixed surface

The surface

free UV initial

The surface

IR attractive:

4, controlling

IR attractive

determination

IR fixed surface

resp.

lines and fixed point.

for the MSSM.

it or along

the fixed line is strongly 227/266

two-dimensional

IR fixed

SM and

resp.

the exponents,

Py_;;)(~)~

EPl+i$y--

p2 = 0 in powers

2150720

176

1

S616143

99

of l/pi

and p2

7656563200

p1 + 362”

1298880

1

+ 35869003309

2

1

___-

384780

1

~-

p, p2 - 8616143

+ S616143

p, "' + "' (203)

for p, + 00, around

Pt

=

%cidentally by a boundary relation”

between

3110427623

MSSM

1 -pl

even an

IR

oflIp,

in the

unphysical

p,

p?, but

not sufficiently

and

E

- p2

1270201090400

g(g-P2)+ _

1

1069676059.5497

2

- p.# +...

attractive

condition and

in powers

- Pz) - s;(;

;($

in the

pz = 42119

418655600

11 5423 lop'+-3439 + l;;;;;;;7

42

region,

fixed which

near

line exists even

to make

(204)

in the p1-p?

is not a point.

far

from

So. this

plane, the fixed

pl=(5/33)pz,

line line

representing will

again the

be ignored

characterized “experimental

subsequently

B. Schrempp and M. Wimmer

58

MSSM

SM

Figure

10: The strongly

be free variables. and

ps from

the surface. figure

With

a high

attractive

IR fixed surfaces

the input

of the experimental

UV scale

to the IR scale

At its IR tip is the IR fixed

was taken

from

Ref.

point

in the Pt-P1-,sz-space initial

p = 176GeV (symbol

values traces

0)

towards

with

for pi and

an IR fixed which

pi and

ps considered

pz, the evolution line (fat

broken

to of pi

line) on

the RG flow is drawn.

The

[48].

MSSM

SM

5

4

Pt = 3

Pt

=

31

&9,” 2

1

I00

I /

1

I CL= mt

2 I/932

31

4

p = 101.j GcV

Figure 11: pt as a function of ~1 or, more conveniently. as a function of l/g:. The IR attractive fixed line (fat line) in presence of all gauge couplings, identical with the fat broken in Fig. 10. is shown to attract the RG flow, represented by selected solutions (thin lines). The figure was taken from Ref. [#I.

Top Quark and Higgs Boson Masses

59 MSSM

SM

I

I

/J = 17GGeV Figure

fixed line (fat line 1) in presence

12: The IR attractive

the IR attractive is shown,

which

The figure

Feeding

fixed

in the

singles

point

(symbol

is favoured

was taken

within

from

initial

Ref.

values

0).

the pl values

11. Shown

length,

solutions

we find first of all the quantitative 5: the

similar.

mostly

is seen to be much

Next

let us discuss

in Fig.

10, fat

(This it from

even more

above zero).

The

At the IR scale,

11 with

line (thin

line 2)

to be discussed

in Sect.

8.

experimental

relation

pi and

pz

between

an UV scale

A down

to the IR scale

of l/g,”

results

the RG flow (thin

lines).

for the qualitative

discussion

similar

below is focused

figures

by a fat line mt=176GpV.

in the fat line in Fig.

led at the beginning

into a very

narrow

for the RG flow in the literature

h,. In case of the SM the focusing

effect,

though

while

of the RG

line cut

[44]-[16].

qualitative!!

evident

out

of the

IR attractive

fixed

surface,

(fat

again

as the

isolated

1 Fig.

12.

It clearly

flow for all solutions when

persuing

towards is stronger

starting

the solutions

it from infinity.

line from

towards

while those

in the MSSM,

as expected

above large below

and values

from

broken

below

from

like

the

line.

of p. where

the line are drawn the analytical

the IR scale

and from above.

the Pendleton-Ross second

line

acts

all the towards

discussion.

in the SM.

p = ml = 176 GeV.

keeping

of of IR

band

the following

conclusions

can be drawn.

The IR point (symbol 0) on the fixed line 1 in Fig. 12 plays the role of an IR attractive point in presence of the electroweak gauge couplings as well in the SM as in the MSSM. fixed point in the sense that for increasing UV scale .2(which is of mathematical interest from below

“the

are many

11) displayed”

IR attraction

less strong

from

support

coupling

the line are drawn

and somewhat

l

the effect

line on this

becomes

as in Fig.

less pronounced.

line in Fig.

an IR attractive solutions

There

for the Yukawa

resulting

above and from

RG flow from

in case of the MSSM.

[50]-[65],

couplings

of parameters

this line as a function

representing

Sect. values

i!‘&‘T

less IR attractive

pI-p2-pt space, which has been depicted

in the

running along

Here

“top-down”

of all gauge

a much

of reduction

gg. the

surface

of the surface

are also general

I Y=

[48].

for pi, pz and

10; the line has finite

Plotting

In comparison

the program

out a line in the IR attractive

in Fig.

I p = 17GGcV

/I = 10’z GvV

/J = ml = 176GeV This fixed point

fixed point

line 2 will be discussed

(137).

in Sect. 8

fixed,

replaces

the RG flow contracts in the presence

towards

of the electroweak

this

fixed It is a only). point

couplings

60

B. Schrempp and M. Wimmer The IR image of all solutions Hill fixed point

starting

in presence

from a high initial

of the electroweak

gauge

value of pt or in other

couplings,

represents

words

the upper

the effective bound

of all

IR points. This

Hill fixed

point

close to it since the introduction

to Sect.

In the perturbatively IR fixed point This

part

inaccessible

at pi=ps=O

responsible

for the strong

top mass

when

It is worthwhile

pr=ps=O

will be lifted the fixed

by radiative that

point

and

upper

of sin p. (This

The enhanced than

of the electroweak mathematical

in the SM, which

gauge

It is most

is extended

Let us repeat

(651 in the framework in Sect.

that

The

to be discussed towards

for the MSSM, fact

that

ps values

and

(symbol

in

below.

the genuine as expected.

the fixed

(196)

line rises

at the IR scale

correspondingly

is

for the IR fixed

0) and of the Hill upper rnl(p

bound

in the

= mr) = dmr~s(p

in Ref.

Therefore

appropriate

to see that with

been pointed

couplings

in the literatur

gratifying

compatible

covers the

=

out already

the range

IR fixed

the experimental

mass

in an approximate

between

value

lies at

within

errors

treatment

of

[47]). for the SM. There

the IR fixed point

the IR attraction

and the Hill effective

to use the IR fixed point

will be done in Sect.

to assign

is much fixed point

an IR attractive

value

6.

Sector of the SM and MSSM to the top-bottom early

results

grand

allowed

for the IR manifolds [47] for the MSSM,

where

with

an allowed

In this section

in the RGE the exact

we expect

in Refs.

unification

to trace

of the IR fixed line.

to the search

sector

were obtained

of supersymmetric

given

in Ref.

effect

very

already

couplings.

is very advantageous

7. This framework

out to lie in the vicinity

had

case.

The Top-Bottom

reviewed

fact

insight

different.

Next the analysis

12.

pr and

GeV sin /3 quoted

a top mass

in the supersymmetric

are distincly

IR scale.

17/18

One finds from

It is, however

admitting

for top mass

5.2

in Fig.

of the fixed point

(190-200)

even at larger

weaker

pt=

small

gauge

at this level.

bound.

range

the inclusion

has a sizeable

of the MSSM

was anticipated

180 GeV sin /3 and 190 GeV sin p, respectively. These values p& z 190 GeV sin P and my” z 200 GeV sin p, respectively. to m,

the lower end of that values

in case

This

of approximately of values

the

however,

IR fixed pt value

of the

already

corrections

the range

crosses

the still rather

on the electroweak

values

sin p th e values

point,

IR attractive.

for the SM resp.

by small

and

the positions

top mass

IR fixed

strongly

below p = mt the IR fixed line tends

pt=2/9

increase

switching

to translate

pt into

This shows

region

and

of the line is indicated between

m,)(v/fi)

is very

5. For the SM the distinction

so strongly

variable

above the

is dtstinctly

the IR fixed point

tau-bottom region

followed

fixed line at the

by the analyses[49]-

Yukawa

unification

in the tan P-m,plane

we restrict

of the SM and

treatment

an IR attractive

[31]-[36],

the discussion

MSSM;

an approximate

for the SM as well as the MSSM

which

again

to be turns

exclusively

treatment is found

was in Ref.

1481. The extension of this search to the pt-pb sector leads [48] t o a strongly IR attractive three-dimensional subspace in the four dimensional Pr-Pb-Pl-Ps space (the analogue of Fig. 10 in the pt-pIp2 case). The analytical treatment includes e.g. the boundaries for p1 + co, pz=O SM pt = gp1

MSSM

for pb=O

pt = $

for &=O

(20.5) Pt = ‘pi 8

’

Pb

=

4sp,

pi = g’p,,

pb

=

$1.

61

Top Quark and Higgs Boson Masses MSSM

,p = 1015GeV 3.57 3. 2.5~. 2.. 1.5.. 1..

:V

0.5.. 2.9

+z

‘2.5

13: IR attractive

Figure

of p or l/g:.

Shown

line 2). the fixed point solutions

The

latter

(thin

fixed surface are the more

0)

lines).

relation

dimensional flow from followed surface

and

circle

at substantially by the fact

that

for increasing

RG flow concentrates towards

more

the

in pt determined

IR fixed unaccessible

pi=ps=O

pt-pb plane discussed

and

in Sect.

for pi=ps=O A, while

closely (except

last

region

the genuine

Fig.

0.

in Fig.

p, the

IR

by selected

1.5.

point

p = nz,, draws

pt=O

down

couplings

The

running

the genuine

(the

the role but lies

is distingishrd fixed,

the line more

line at &=O.

is

the IR

shape.

p = mt = 176GeV or pb=O).

which

= mt) plane

gauge

along

the RC:

lines),

it. Running

the a two-

attracts

(thin

It has a similar

IR fixed finally

into

= m,)-pb(p

all solutions

lines and

Injecting

on it (Diamond)

the IR scale values

more. projection

13 which

onto

the line and shrinks lies on this

IR fixed

any

of the electroweak

IR fixed

for UV initial

below

fixed line (fat

is represented

the surface

line in the pt(p

keeping

subsection,

P!,P~ as functions

at all scales

to the

attracted

in Subsect

the

towards

shown

been

fixed

the line and

the

and more

IR fixed

point

(Evolution

into

in the IR attractive IR fixed

point

in

t tlr

4.5).

IR fixed line (fat line 1 in Fig.

by the symbol

evolution.

point

space,

leads

RG flow within

in the presence

UV scale

and more

for Pb=O in the

the perturbatively

The resulting

Again

graphically

of l/g,, *

or have

out an IR attractive

line substitutes

values.

are the

unification the surface

be represented

on the surface

in the space

[as].

in the pl-pa-l/g:

fixed line, determined

higher

towards

start

Ref.

ps as a function

Also displayed

which

13. This

from

cannot

fixed surface

/L = mt singles

of the quarter

denoted

pi and

below.

by solutions to scale

surface

subspace

IR attractive above

fat line 1 in Fig.

closely

fixed

Yukawa

The RG flow within

was taken

between

couplings

h,2/9;

=

line (fat line l), the less attractive

top-bottom

intersection.

The figure

of all gauge

IR fixed

approximate

at their

three-dimensional

in the presence

attractive

implementing

(symbol

experimental

RGE

= h:/& “.= o-o “.= Pt

k'b

Pb

To be more

14 also illustrates

realistic,

13) is plotted the figure

the dramatic

in Fig. contains

IR attraction

14 as fat line, the IR fixed point. is already of this

the result fixed

from

line and

a two-loop of the fixed

point on it. On the left hand side a dense lattice in the large plane 05 pt, Pb 5 25 is shown. The lattice points are taken as UV initial values at the scale Moue z 2 10’s GeV. The right hand side shows the IR image of the lattice at the scale p = ml = 176GeV. Notice first of all that the lattice has shrunk by a factor of 25 in each dimension to within a square Ospt,pb 2 1. Secondly one sees, how the large initial values pt or pb have shrunk towards the boundary. the Hill effective fixed line. which is close to the IR fixed line but distinct from it. The IR fixed line is independent of the UV scale :1=_\4our. The Hill effective line. however. reflects the choice .1 = .kfot!r x 2 1016GeV: in the mathematical limit

B. Schrempp and M. Wimmer

62

Pt = h:/g; Figure

14:

The

lattice

points

2. 1016GeV and subject scale

p = 176 GeV;

of 25 to the

line (fat line) and constitute

whole

that

(figure

keeping

close to the IR fixed

the UV plane

the

IR scale

fixed,

Let us also come

back

13 signalizes

(0)

very

was taken

the upper

the IR fixed

point

boundary

side)

strong Ref.

has scaled

and

down

E

by a factor

of the IR fixed

The upper

towards below

the exception

have moved

Mour

to the IR

bound

of all lines

[48].

tends

above

at A = couplings

IR attraction

apparent.

from

boundary (with

p*-pb-plane of all gauge

the

IR fixed

and then

of the UV initial

much

closer

line.

proceeds

together

The

along

values

or

hl=O

in the presence

couplings.

fat line 2 in Fig. of the electroweak

The

0) on it becomes

The figure

in the

in presence

on the left hand

side).

by the IR fixed line from

line towards

gauge

(figure

hand

(symbol

fixed line.

values

of the MSSM

The IR fixed line and the upper

of the electroweak

fixed point

as UV initial

on the right

RG flow first is attracted

and hb=O).

chosen

the IR fixed point

the Hill effective

A --t co, while

are

to the RG evolution

notice

IR plane

h?/g:

pt =

to the important

gauge in Figs.

that

couplings 13 and

issue of the top-bottom

top-bottom

Yukawa

as an approximate

14 implies

unification property.

approximate

Yukawa

unification

at all scales It is weakly

top-bottom

Yukawa

at all scales.

p survives

IR attractive. unification

The

the inclusion Also the IR at the IR scale

p = mt = 176 GeV. The translation in Sect.

5.3

of the results

in the Pt-Pb-plane

into the tan B-mpole -plane

of the MSSM will be performed

6.

The Higgs-Top-Bottom

Sector of the SM

The two-dimensional IR attractive surface, discussed in the absence of the electroweak gauge couplings, presumably turns into a four-dimensional IR attractive subspace in the five dimensional p~-p~-pb-pr-p~ space.

Injecting

the experimental

relation

between

pi and

pz and evolving

down

to ~1 = mt leads

to the

two-dimensional IR attractive fixed surface in the PH(~ = mt)-pt(p = rnl)-~b(p = mt)-space shown in Fig. 15, replacing in the presence of the electroweak gauge couplings the surface Fig. 9. The surface in Fig. 15 1431 is again distinguished by the fact that in the (unphysical) limit h --f oo. while keeping the IR scale fixed, the RG flow is drawn first towards it, then within the surface towards the IR fixed line (fat line) and finally

along

or close to the line towards

the fixed point

(0).

(Again

the IR fixed surface.

Top Quark

Figure

15:

The

the presence (symbol

0).

The

the surface figure Fig.

6

strongly

IR attractive

of all gauge

IR attraction

towards

was taken

couplings,

9, is approached

Ref.

in the PH-Pt-Pbmspace

the IR attractive

line (fat

line)

of the RG flow is first towards

the fixed

from

surface

line and

finally

along

63

Boson Masses

and Higgs

at the and

IR scale

the fixed surface,

or close

n = 176GeV

the IR attractive then

to the line towards

fixed

along

in

point

or close to

the fixed

point.

The

[48].

in the perturbatively

Infrared Attractive

unaccessible

region

for p below

mt).

Top and Higgs Masses, Mass Relations

and Mass Bounds The

previous

one-loop

sections

level, since

the results

served

to develop

it allows

to determine

on IR attractive

and top mass

bounds

top and

the

Higgs

on the professional

subject

step

by step

the IR fixed manifolds mass level.

values

and

pedagogically, exactly.

IR attractive

The present

state

still

keeping

to the

Next

we are going

to revie\\

mass

relations

on Higgs

of the art in most

and

publications

is the

following. . Two-loop

RGE

for the couplings

are mathematically allowed

region

not exact rn*_
involved

any more,

they

turn

are used.

At the two-loop

but numerically

out

to shifted

level the IR fixed manifolds

well-determined

by at most

within

10% with

the perturbativel!

respect

to the one-loop

ones. . In order

to determine

the running the Higgs

masses boson.

masses

from

in the MS scheme Then

in most

couplings.

the two-loop

according

publications

running

to the relations

the radiative

couplings

are first

(54) for the fermions

corrections,

relating

related

to

and (56) foi

the running

masses

to the physical pole masses (or at least the most important ones, the radiative QCD corrections to the quark masses), as detailed in Sect. 2.5 are applied. For convenience let us collect the relevant formulae again, including the matching conditions (33) for Yukawa couplings in the MSSM foi the transition

from

the RGE m,(p)

of the SM to the RGE =

r+‘+(l

+a,(/~))

of the MSSM with

at the scale

,Wsr:sy.

64

B. Schrempp

gt(hJsY-) gb(&USY

-)

The choice breaking

l

e.g.

scale

Higgs

The

bottom

known

masses

and

the value

6.1

mH(PL)

=

varies

and

with

IR attractive

mass

the more

where

mb(p

collected

of an upper

bound

are considered. from

use of the relations in Sect.

of the rather

cr, = gz/(4rr)).

can give rise to

the supersymmetry

2.5.

sizeable

Notice

(See.

~1 = mb, resp. between e.g.

dependence

that

the bottom

Ref.

[55] for a

of mb(p mass

from

the running = ml) on

quoted

in Eq.

= mb) in the MS scheme.

in the MSSM

conspicous

[43]-[65],

one makes

masses,

which

Typically

h ave to be evolved

(4,5),

also a figure

bounds.

1 TeV; for the determination

up to 1OTeV

Eqs.

pole

of mass

mt and values

masses,

including

with

p = rn~ to p = mt in the literature,

determination

For this purpose,

for a,(mz),

fixed point

from between

tau

1 + 6&))

in case of the MSSM

(206)

in the MSSM

Top Mass and tanp

Let us start

cos /3 in case of the MSSM

m~“(

p independent

(4) is the running

cos /3,

=

is varied

mass

presentation

the initial

= =

p = mr up to the IR scale. detailed

sin @? in case of the MSSM

hb(&usv+)

in the precise

Msusy

of the light

ht(Msusv+)

mH(P)

of the IR scale

deviations

=

h,(Msusv+)

g,(&usY-)

slight

and M. Wimmer

and

well known

results

in the MSSM,

the much-quoted

strongly

(1181, [119] rnpole = 0( 190 - 200) GeV sin B

and

a tendency

these

results.

for IR values

of tan fi to settle

In the top-bottom-tau sector at the IR scale 176GeV, there are four unknown parameters plings

to be my”

parameters and

an IR fixed

and

,.+,=hz/gi

corrections

IR fixed

and

of the experimental which

are applied IR fixed

parameter

tan /?.

Let us emphasize fixed

(for h,=O).

line implies

line in the &@-plane

again

that

line (thin

An approximate

a strongly

the IR fixed line)

dissection

line (fat

relation

line)

the upper

had

is independent IR bound

most

14, can then

of

leads

to

sensibly

be translated

the definitions pt=ht/g,2 are used and all radiative

also been

between

masses,

in the literature

at p = mt, Fig.

treatment

IR attractive

represents

the analytical

tau and bottom

have been chosen

line in the tan p-rnp0” -plane, Fig. 16 [48], by remembering and inserting g,‘(p = mt) = 1.34 , Eq. (96). Two-loop RGE

resulting

effective

at p = rnr = 176 GeV,

tan 4. The

into

tan 0 = O(60)

to be set for the purpose of the argument at p = ml = to be considered, the top, bottom and tau Yukawa cou

at p = rnr = 176 GeV and tan d. The input

two unknown

type

around

(207)

given

in Ref.

the top mass

of the scale

for A = &UT

and

[47].

A, while z 2

The

the MSShI

the Hill

. 1016GeV:

in

the mathematical limit A + 00, while keeping the IR scale fixed, the upper bound along with all other solutions tends towards the fixed line. The IR fixed point, implementing approximate top-bottom Yukawa unification, is denoted by a symbol 0 in Fig. 16. The whole RG flow is attracted from above and below very strongly first towards the IR fixed line and then along or close to it towards point (with the exception of the solutions starting from initial values ht=O or hb=O). The results

t.o be read

off Fig.

16 are the following

[48]

the IR fixed

Top Quark

and Higgs Boson Masses

65

60

IR fixed line

pole mt

Figure 16: The very strongly and the IR attractive bottom

Yukawa

line, i.e. figure

the

fixed

unification.

IR image

was taken

from

IR attractive

point

The upper

of all large Ref.

fixed line (fat line) in the tan &mp”“-plane

(symbol

0)

are shown;

boundary

initial

with

values

the fixed point

implements

the interpretation

of the

of a Hill type

for ht or hb at A=Mour,

is shown

MSSM

approximate

top-

effective

as thin

fixed

line.

The

[48].

. In the large tan 9 interval l_
to top mass values 150 GeV_
well compatible approximation

with

the experimental

top

The

IR fixed point

(for h,=O)

p&

implementing

approximate

the experimental

a The upper bound, parametrized by

pole

(XJ!J)

190 GeV

(2) within values

t,he experimental

bounds.

A good

of tan b is

E 192 GeV sin B.

(210)

lies at mt

with

mass

for the IR fixed line for not too large ml

l

(‘OS)

top-bottom

x

182 GeV.

tan D x 60.

unification.

This

top

mass

(211) value

agrees

amazingly

well

value. i.e. the Hill type

effective

fixed line. is very nearby

my’= zz ‘202 GeV sin 3.

and may

be approximatel!

(21”)

Obviously the result (207) quoted in the literature [43]-[65]. [llS], [119] covers the band of rnyle values between the IR fixed line and the upper bound; so, Fig. 16 contains quasi a dissection of this band into

8. Schrempp and hf. Wimmer

66

a genuine genuine with

IR fixed line, attracting

all solutions,

and an upper

the experimental

Let us anticipate rn~“-tan

top mass

already

P-plane

of tau-bottom

that

roughly

Yukawa

this issue we devote

the shape

coincides

of the band

with

unification

Sect.

7, where

[50]-[65]

in the framework why tau-bottom

we come

back

rather

to the knowledge

the IR scale

to the result This

Since,

the

resulting

from

of supersymmetric

grand

Yukawa

focuses

unification

bound

in the

the requirement unification,

To

the IR physics

r$”

from

line or

an IR fixed

the dependence

on p in hl is only logarithmic,

with

M susy=mt=176GeVand

the correction

are varied. top mass

Such

value

a,(mz)

a variation

emerging that

within

scenario

and

Of course

in Sect.

the framework

its IR attraction

of the supersymmetric

= 0.117.

will be reviewed

of the MSSM

is so strong.

the result

7 as well, is very close

Still, one has to await

a measurement

of tan p before

an

drawing

conclusions.

Higgs Masses and Top-Higgs

In the SM the input the IR scale

which

= ml),

of the known we shall

bottom

mass

set, in order

this can be turned

to the two-loop

analogue

leads

into a knowledge

of the IR fixed

an IR fixed line in the pH_pt-plane than

the fixed

inclusion

point.

of all radiative

with

Mass relation in the SM

to a determination

to be definite,

of the ratio

a fixed point

corrections

variable

couplings

With

pb=g,‘/g,’

at p = mt.

of pb at p = mt cuts

in it, the fixed line being

much

gb at

the known

at p = mt in the Higgs-top-bottom

The insertion

Translating

of the Yukawa

at p = m; = 176GeV.

surface

9, i.e. in the pH=X/g~-pt=g~/g~-pa=g~/g~-space.

after

IR values

the top mass

to the top mass

value and it is gratifying

Top and

attractive

that

comparison

to be 176 GeV is in fact not a free parameter: implicitely m, Pole which in turn is determined

however,

performed

confirmation

any further

turn

to be identical

the IR attractive

experimental

of gi(p

so far chosen

two parameters

to the experimental

6.2

in determining

had been

if these

Altogether

to notice

favourable

is negligible.

analysis

varies

that

(199).

a more

$-plane.

of the RG evolution, has

the condition

It is gratifying

allows

the IR fixed line and the upper

of allowed

we review

Here

the IR endpoint

between

the band

or close to the IR fixed line in the tnr”-tan

point,

bound.

and thus

value.

onto

from

values

IR fixed line lies at the lower end of my’=

input

Now we

sector,

Fig.

out of the surface more

strongly

IR

pH and pt at p = mt into X and gt at p = m, and these pole into mH and my’=, one ends up with the following result in

the m~le-m~‘e-plane. l

A strongly

IR attractive

top-Higgs l

mass

fixed

and on the fixed line a weakly IR fixed

point

fixed point The strongly

line in the

mg’e-mr’e

-plane,

implying

a strongly

IR attractive

relation

top and

(157)

Higgs

IR attractive mass

in the presence

IR attractive

top-Higgs

(Below

a top mass of 150 GeVthe

It may

be considered

fixed point This

in the m~‘c-m~‘e-plane,

fixed point

of the electroweak

gauge

plays

corresponding

to

the role of the Pendleton-Ross

couplings.

[48] is shown in Fig. 17 (fat line) above m,p”“=150 GeV. determination of the fixed line starts to become unreliable).

mass relation

theoretical

as the non-trivial

[32], [40], [37]. The weakly

values.

attractive

update

of the corresponding

IR fixed point,

characterized

figure

in Ref.

by a the symbol

[42]; see also Refs. 0 in Fig.

17 is at

1481 IR attractive IR attractive The IR attractive experimental top

top fixed point Higgs

fixed point

mass mass

pole m, x pole mH z

214GeV, 210GeV.

top mass value is clearly outside the combined one standard deviation mass; still it is impressive that the SM, which is not endorsed with

(213) errors of the an additional

Top Quark and Higgs Boson Masses

L

:

160

180

200

220

240

260

2

pole *t

Figure

17: The

weakly

IR attractive

lines)

strongly

IR attractive

fixed point

for A = 107, lo”,

free parameter

lOIs,

like tanp

experimental

value.

contradistinction

more

to the top mass

say, leads pole mH

more

IR attractive

This is a very interesting further

conclusions

As we had pointed and

then

point

Higgs

below.

out already This

17) for four

is within

“wedges”.

the

to Subsects.

the

keeping stability

A = 10” GeV.

than

value

stability Ref.

the

bounds

(thin

[48].

so relatively

close to the

is very weakly

attractive.

in

the IR fixed point.

So it seems

top mass

value,

justified

to

m, P0”=176GeV.

[4S] mass

but again,

sections,

relation,

evaluated

experimental

at rnpole = 176 GeV.

confirmation

reflected

The

IR scale

(For a thorough

point.

values

discussion

fixed).

discussion

triviality

of A = 10 ‘, and

tip of the wedge

increases

The attraction

in the combined

of the UV scale

bound

the RG flow is first drawn

the IR fixed

(For the theoretical

.2 4

vacuum

value

value,

from

this IR fixed point

the line at the experimental

in previous

2.6, 4.1-4.3). value

co. while

was taken

top mass

of the SM and

vacuum

in the MSSM.

top - Higgs

representative

fixed line for increasing A the

that

IR attractive

it towards

is again

lines in Fig. refer

point

mass

mass

The figure

and

is needed,

(211)

before

an!

can be drawn.

close to it or along

than

however

strongly Higgs

in the m~‘e-m~‘e-plane

has an IR attractive

to it. Evaluating

to the corresponding zz 141 GeV,

fixed

line)

as well as the triviality

10” GeV are shown.

in the MSSM,

significance

line (fat

0).

It has to be stressed,

The IR fixed line is much attach

fixed

(symbol

Also clearly the

10”.

lOi’,

vacuum

visible

of the lower bounds

becomes line and

above

The

of the relevant

bound,

the IR fixed point

IR fixed

the IR fixed line

stability

10”GeV.

the quotation

is the upper

A (towards

towards

and

towards is stronger

which the fact

(thin

allowed

region

literature down

in the mathematical

is already

see Subsect.

slides

the fixed

bounds

we

the

IR

limit

that

for increasing

very

close

to it for

6.3).

Notice again that the IR scale, set throughout this analysis to p = 176GeV. is not a free parameter. Rather the IR scale has to be identical to mpO1’, resp. rnr”, which in turn is determined implicitel! from the condition (197), resp. (198). Since, however, the dependence on p in the couplings is onl!, logarithmic.

the corrections

to the results

are very small.

68

B. Schrempp

and M. Wimmer b)

4

b)

pole Figure 18: a) Vacuum stability bounds for the SM Higgs mass mH as a function of the top mass m, PdC for different values of the UV scale A. b) Limits on the Higgs mass rn~” as a function of the UV scale A for various

6.3

values

of the top mass

The figures

were taken

from

Ref.

[15].

Lower Bound on the Higgs Mass in the SM

In order

to be prepared

for future

and at the LHC collider, stability There

bound

and Ref.

[17] for a refinement) their

In both

the vacuum

analyses UV scale the

scale,

Sect.

2.6.

include

Ref.

that

given

in Ref.

bound

their

refined

= 0.124.

for the two extremal

values

in Sect.

Higgs

mass.

LEPBOO upgrade bound,

of LEP

i.e. the vacuum

appear

having

stability

A; both

bound

[15] the bound

assume

significant

the

influence

for A = lo3 Ge\

was determined

from

X(p) reach zero at p = A, as justified in to a nearly scale independent one-loop effective

leading the

In Ref.

UV scales

without

selfcoupling

running

order, mentioned MS couplings

2.5 in this context).

In Ref.

at the end of Sect. to the physical [15] a,(mz)

[15] an analytical approximation for the lower for A which allows to vary mrleand dmz)

mpHOle >

135 + Z.l(mpo’C - 174) - 4.5

mu’=

72 + 0.9(mpo’=

>

for various

possibly

In this sense the vacuum

method.

in relating

In Ref.

at the

of the lower

as a function of the top mass and the UV scale .\. [15] and [16] (see also Refs. [27] for providing the such a lower bound and which agree to within a few

to the next-to-leading

corrections

boson

analysis

is determined could

on the Higgs

running

(see the two footnotes

[16] a,(mz)

provide

bound

new physics

the two-loop

SM Higgs

of 3-5 GeV.

stability

is RG improved

the radiative

pole masses

lower

[16] used

which

which

errors

to be A = lo3 GeV.

“absolute”

the requirement potential

quoted

A at which

at the electroweak provides

for the

to have a precise

on the SM Higgs mass professional analyses in Refs.

GeV well within

minimal

searches

it is important

[7], [Ill-[17],

are two recent

basis

Ref.

mpo”.

Higgs

Both

and

top

= 0.118 is used, Higgs

mass

forA = 10” GeV,

- 174) - 1.0

2.6.

forA = lo3 GeV,

in

bound

(215)

(216)

(161 for comparison mP,o

> 127.9 + 1.92(mp””

- 174) - 4.25

cx,(mz)

- 0.124

0.006 The results

of the two analyses

One important conclusion at LEP200 would imply

are exhibited

in Figs.

18 and

for A = 10”GeV.

is

(217)

19, respectively.

is that for a top mass larger than 150 GeV the discovery of a SM Higgs boson that the SM b reaks down at a scale A much smaller than a grand unifying

69

Top Quark and Higgs Boson Masses

160

80

60

130

140

150

160

170

180

190

200

Mt (GeV) Figure

19: Vacuum

for different

stability

values

scale of 0( ICP) GeV. (upper)

bound

bounds

for the SM Higgs

of the UV scale

Actually,

for the Higgs

A. The figure

as can be inferred mass

in Fig.

mass

mgle

was taken

with

as a function

from

Ref.

the additional

17, for rr~pa’~--176

GeVand

of the top

mass

rnpolr

[16].

information

about

AL 10” GeV.

only

the triviality a mass

range

130 GeV 5 mpHDre ,< 190 GeV is allowed.

6.4

Upper Bound on the Lightest

The conclusions fact that gauge

for the lightest

in the MSSM

couplings,

remaining

task

Let us start

Eq.

with

its initial

value,

Since in Fig. mass. Msosy.

the initial

(35), which

discussion.

according

Eq.

(35),

reaches

the higher

a qualitative

selfcoupling

is fixed,

at p = Msusv

17, one expects

the evolution

The

upper

maximally order The

to the scale

There is a vast literature on the cannot all be reviewed here. We In Ref. [27] the supersymmetry Msusy = 1 TEV and Msusv = the one-loop

threshold

mass

in dependence

IR attractive

the Higgs

mass,

This

is due to the

in terms

of the electroweak

mass

for cos’ B = 1. The

are mainly

of the SM from

p = mu.

the longer

top-Higgs

corrections

RGE

bound

different.

is given

corrections.

radiative

to increase

boson

the value of the Z boson

is well below the

for the Higgs

are completely

Higgs

radiative

will be the stronger,

or at best on the IR attractive

(‘271 include

in the MSSM

to the (two-loop)

value

The increase

boson

the tree level mass of the lightest

is to calculate

the Higgs

Higgs

Higgs Mass in the MSSM

is the evolution

as a function

and

path,

p = Msusy,

of

where

on the size of the top mass.

Higgs-top

selfcoupling

due to the running

the scale mass

relation,

correspondingly

i.e. the larger

of the top mass,

will have

displayed the Higgs

is the value to settle

for

below

relation.

determination of the higher order radiative corrections [18]-[28] which shall concentrate on the results of a recent, very careful analysis [27]. breaking scale is assumed to be well above the lightest Higgs mass: 10TEV are the values for which results are presented. The authors contribution

due to a possible

stop

mixing

at &&yin

the following

B. Schrempp

70

and M. Wimmer

x-2 t

(219)

12Ms”syz

This

term

has to be added

the right values

hand

for Xt, X:

They

Higgs from

(35).

= GMsusv*

use the framework

at the two-loop corrections

p = Msusv

of the stop

for maximal

down

depends

threshold

to p = rn”

approximation

gives

positive

with

of 176GeV

7

Supersymmetric

p&

mH

masses

corrections.

conservative

Fig.

bound

bound

of mixing.

potential,

and

mr’e.

Since

Fig.

20 shows

RG improved the full radiative the tree

level

17, the RG evolution

the result

of the analysis.

is [27] a,(mz)

- 174) - 0.85

the lower bounds

the upper

effective

on

two extreme

Msusy ,Xt and cos* 2p and of course on my]‘.

< 126.1 + 0.75(m4”‘”

of order

at p = Msnsv consider

2.6 and 6.3, and include

fixed line in the m~‘e-m~‘e-plane,

for the most

mass

one-loop

in Sects. pole

selfcoupling

The authors

and Xt = 0, for absence

independent

already

the IR attractive

level Higgs mixing.

effects

to the physical

on the parameters

can be compared

scale

has been quoted

rng”

which

for the tree

the MS masses

mass lies well below

An analytic

value

X t IS a measure

[27] of an almost

level, which

relating

The bound

to the initial

side of Eq.

on the SM Higgs

on the Higgs

- 0.124 0.006

mass

mass

discussed

is of the order

in Sect.

6.3.

For a top

of 130GeV.

Grand Unification Including Yukawa Uni-

ficat ion As has

been

symmetry

developed

relations

supersymmetric unification. Yukawa

grand Recent

coupling

tau-bottom a realistic structure

l

points only

They Yukawa

l

More

have

particle

interestingly

gauge

two global

unification

angles

a crucial

[78],[55],[73],[58],[74]

in Sect.

and investigation

at the grand electroweak

terms

Yukawa and including

top mass,

and cover in addition

of the Higgs

iv) radiative

of minimal

tau-bottom-top

of the quantitative

unification sector,

scale

MC"= with

iii) implementation

symmetry

or vi) investigation

to lead the discussion

breaking,

program

coupling coupling

involves

constant constant

several

unification unification

of

v) search

of the fixed

for the heaviest

point

fermion

2.

test

of supersymmetric

context

grand

unification

including

tau-bottom

(-top)

level.

of this

review,

they

demonstrate

that

tau-bottom

unification

requires the IR values for the top mass to be close to its IR fixed point or, more generally, values in the tan B-rnrle -plane to be close to the IR fixed line discussed in Subsect. 6.1.

The

[49]-

issues like i)

aspects.

at the quantitative in the

matrix

mass

RG flow by

framework

the issue of an IR attractive

ii) exploration

outlined

sector

into the analysis

We are going

“top-bottom”

in the

or even

in this review

mass

breaking

theory.

the

is provided

tau-bottom with

considered

spectrum,

the framework

represent

combined

implemented,

to constrain

in the gauge

for the fermion

in the supersymmetry

within

analyses

recently

and mixing

unification

of the underlying

generation These

masses

tool UV scale

in addition

the framework

(textures)

supersymmetric

for IR fixed

implying

more

go beyond

Yukawa

at the

have been performed

unification,

of ansatze

an interesting

values

unification

of all fermion

implications

2.3,

initial

analyses

[65]. Most of them inclusion

in Sect.

between

the IR

steps. A first step is the exploration [XX], [55],[73],[.58],[74] of gauge (36). The preferred way to perform this analysis is to assume gauge (36) at a scale n/lGur which at this stage is not specified: so there are two

Top Quark and Higgs Boson Masses

MS=1 130

-

110

-

TeV

/

z s

go-

r' 70

,

50

,

/

/

/

/

,

/

/

/

/

,

,

/

/_

/

90

70

-

50,

/' 30(' 120

' 140

/ /

"

"

"

160

160

220

/

/

* 140

(GeV)

Mt

/

/

/'

30." 120

' 200

,

/

/

,

/'

.

-

'I 160 MI

4

"I' 160

200

220

(GeV)

b) 200

j,,

I1

I,

170 _ lso -

160 -

M.=l

TeV

Xl'=6

M,'

140 -

90 -

70 ,

, r

30’ 120

TeV

/-

130 -

50-

M.=lO

,

/

/

/

/

/

/

,

/

80 -

,'

’

60-

’

’

140

160

’

’

190

’

’

’

’

200

-, ,

40 120

220

/ /

, s

/'

’

140

”

MI (Ge'd

Figure 20: The upper of the k&y figure

of the

top

supersymmetry

*

I 180

s

" 200

220

d)

bound

mass

160

Mt (Gev)

c)

function

/

on the mass

rnyo”

scale

for cos’p

Msusy

= 10 TeV, .Yt = 0, c) k&y was taken from Ref. [37].

and

rngle = the

of the

1 (solid mixing

= 1 TeV. X:

lightest

line)

and

parameter

Higgs

mass

boson

in the MSSM

as a

cos “20 = 0 (dashed line) and values X,: a) Msusy = 1 TeV, .Yt = 0. hi

= 6A&~sy~~ d) A4s~~sy = lOTeV,

St = 6Msusy’.

The

12

B. Schrempp and M. Wimmer

unknown parameters, the unified gauge coupling at fl = .&fouT and the unification scale hfot,r itself: in addititon there is a mild dependence on the supersymmetry breaking scale Msnsy which regulates the transition from the RGEs of the MSSM to those of the SM. Th ese parameters are adjusted such that the two-loop RG evolution of the MSSM gauge couplings, including among others the two-loop contribution of the large top Yukawa coupling, leads to the experimental values for o(mz) and sin* On: this procedure results in a value for Mocr and for o,(mz), both depending mildly on n/r,usv. As has been stressed in Ref. [73],(58],[74] I‘t is important to estimate the theoretical error for o,(mz), allowing for a variation of Msnsv within reasonable bounds and for threshold corrections and nonrenormalizable operator corrections at the low and high scales. A most recent analysis [74] along these lines determines Mo”T to be &UT N 3 . lOI GeV and a,(mz) with the appropriate theoretical errors to be o,(mz) which is larger than the experimental errors.

2: 0.129 f 0.010,

value, however,

(221)

still within the experimental

and theoretical

The analysis so far fixes the gauge sector and MoUT, having used the experimental value of the top quark mass already for the two-loop contribution of the top Yukawa couplings to the RGE of the gauge couplings. Next, the two-loop RGEs for the heavy fermion generation, the tau, bottom and top quark are considered. There are to start with four completely free parameters, the initial values for the top, bottom and tau Yukawa couplings at the unification scale MouT, say, and the parameter tan 4. characterizing the ratio of the two different Higgs vacuum expectation values in the supersymmetric theory: a further rather constrained parameter is the value of Msus~, Now, the tau-bottom

Yukawa unification

(37)

&(P = is introduced

MGUT)

("L2)

= hb(P = MGUT),

as well as the known tau and bottom

masses within their errors, Eqs. (4.5).

What happens then is rather subtle (see e.g. Ref. [55] for a comprehensive presentation) and requires some preparatory remarks. In Subsect. 2.5 and the introduction to Sect. 6 we have seen. how to translate the given input of the top and bottom masses into the MSSM bottom and T Yukawa couplings at some higher scale p. For definiteness, let us fix Ms”sy=mt=176 GeV. Then the MSSM bottom and r Yukawa couplings at this scale are given as follows mb(P

=

mt)

=

m,(p = mt) =

-$

hb(P

=

mt)

cos

13,

3

h&l

= mt) cos p.

(323)

(22-L)

in terms of the unknown parameter cosp. Now, one realizes that the input of tau-bottom Yukawa unification on the one hand and of the tau and bottom masses on the other hand leads to fixing the ratio (22.5)

at two scales Rb/r(p==cu~)

=

1

and

The input of two initial values for the same quantity Ra,,(p) at two different scales p can only be accommodated, by tuning the contribution of the top Yukawa coupling hl(,a), which enters the RGE of

Top Quark hb on the one-loop analytically

level and is absent

[56],[55]

by looking

and Higgs

Boson

from the one-loop

at the one-loop

RGE of h,.

RGE

13

Masses This can be made

most

transparent

for Rbjr (227)

the solution

of which

tau

couplings

Yukawa

fixed

has to accommodate

at p = ml and

clearly

do not

play

p = Mour

the contribution

the two initial

a significant

and

thus

not available

in the fine tuning unification

coupling first

h,(p)

order

initial

ht(p

Now comes

the

admits

altering

the IR value

As it turns

out,

UV initial

&,/,(ml)

respectively.

and

couplings

of the initial

the

to their

values

values

prescribed

Now, the running

it has to satisfy

its own RGE

product

the appropriate

which

(u/&)h,(p

radiative

the

value

increasing

beyond

safe region

values

There

on the value the

are (226).

by tau-

of the top Yukawa

in the system

of coupled

= mt)sinD

involving

its

is required

to match

the

corrections.

theory.

point

viz.

This

which

casts

Yukawa

should

be an IR starting

resp.

in its translation

UV starting

Rt+

of tau-bottom

coupling

the

depends

coupling

which

implies

led for small

with

besides there

values

in Sects.

and

without

large

values

value

for h,( :2f,, 1)

for the

(via the evolution a,(mz)

and

so large

that

of mh

for n,(l,,z)

I IIP

it leaves

the top Yukawa is a whole

over for larger

of tan/3

expect

circumstance

unification.

relevant

5.2 and 6.1 one would

unification

rather

on a,(mz)

take

well

R~,,(,~IGuT)

respectively,

the necessary

Fortunately, may

strongI>

tan 8.

h,( MGUT) becomes

becomes

(227).

This

ratios

and tau masses.

(226)

the

is, as we know,

h,(MGUT).

on the perturbative

point

Yukawa

value

in the analysis:

doubt

exhibits

point

= mt) onto or close to the fixed

the free parameter

in turn

equation

As announced

ht(p

as to accommodate

on or close to the half-circle into the tan /3-rnrle -plane, which similarly

for the ratio

for h,(p) fixed

for h,(M GUT) increases

in the ht-hb plane,

in Sects.

RGE this

the IR value

two conditions corner

value

in the argument

tl~, ,liscussion

there

(226)

of mb(mt) required

the

3-6 and

and the bottom fixing

of the

side of the differential

IR fixed

the consequences

such

unification

is a tight

of tan a the bottom

hand

From

large

MGUT)

the value 0.12 the top Yukawa

in the ,?-pb plane,

role of the

=

accommodation

of perturbation

on the right

Setting

= mr) and without

ht(MGUT).

sensitively

h,(p

a way out: in Sects.

any sufficiently

by tau-bottom hr(p

provides

at length value for mt.

UV value

p = mb to /I = mt):

happens.

and

masses,

furthermore

with

discussed

practically

the

as prescribed

values

adorned

point,

&,,,(m,)

available

and

key observation

to adjust

For larger

tau

not free to choose,

with the experimental

value,

depends

Rblr(&ur)

and

equations

= mt),

IR fixed

compatible

from

for the fine tuning

gauge

top mass.

attractive

allows

tan ,/? the bottom

d ln P h?(p)

J mi

1679

of the ratios

the bottom

is in principle

experimental

point

and

differential

value

For small

As the running

&UT

(-

exp

is crucial

(226j.

RGE.

[52],[53],[54],(55]

1

bottom

values

role in this

that

above.

values This

coupling

IR fixed

is exactly

for any value

line

I he

of tan p

tan dL.50

what -60

type IR fixed line in the h,-hb plane. allows to accommodate the two initial

4.5, 5.2 and

for the IR physics

6.1. the exploitation isolates

narrow

[49]-[65] allowed

of

bands

of IR values

in the tan/3-mrle which turn out to be close to the IR attractive fixed line as -plane, So in this very implicit way the IR attractive fixed line in the translated into the tan p-mpole -plane. tan &rnpole -plane had been isolated without explicitely having been recognized as such. Let us present typical the tan ,!3-rnrle -plane

Th ey are represented results from Refs. [55],[63],[58]. in form of allowed regions in at a suitable IR scale. The analyses are based on the two-loop RGE of the MSS\I.

the grand unification scale is determined by the unification of the electroweak gauge couplings and o(mz) the is treated as a free parameter. In Refs. [55], [63] the IR scale was chosen to be mt=l.50GeVand analysis was performed for two values of the supersymmetry breaking scale. 1lfsusv=m,=I50 GeVanrl

B. Schrempp and M. Wimmer

74 h4snsv

= 1 TeV,

was chosen requirement a narrow Msnsv shifted these

and

two values

according

to Eq.

of tau-bottom dark

unification

admits

is not mpO”but

rather

= 0.12 is shown

mt(p

updated

in Fig.

was made,

where

parameter

p-l.

to Mour;

a theoretical

for o(mz)

= 0.12 is shown

= mt),

22. In Ref.

rrzb” is the prediction This correction

A comparison

e.g.

the IR result

of the

it is rather

implementing A final

issue context

of hr and

of Fig.

hb, required

mass

values

the

shown

as

Yukawa

requires

In a setting, is allowed

one at small Additional

where

interesting

which

that

indeed

fixed

translate

to it

GeVand

(229) corrections

value.

incorporated

in the running The

tau-bottom

line,

which

(38) within

= Monr)

implements

in the

from mz up

resulting

Yukawa

allowed

region

unification

line.

requires

A closer

look reveals

from

the analyses

grand

unification.

is determined

supersymmetric

= hb(p = Mour) Yukawa Yukawa

[63], Fig.

In Sects.

top-bottom unification

22, shows

at the UV scale is at large

for the UV initial

unification.

approximate

the top-bottom

the analysis

only

from

couplings

recent

bottom

values

requires

analysis

couplings

values

that

Yukawa input

values

4.5 and

ij.2

unification

at p = MonT

the IR image

to

of combined

of tan p, i.e. in the vicinity

the

mass

the strong

gauge

errors.

are too large

0.11

the

allowed

value

and figures

coupling

Taking

of the

and

and

0.12,

Yukawa

sector.

unification

settling

can be learnt

tan$

gauge

function

from

values

at the grand

at tanp

the whole

to be rather

value

tau-bottom

couplings

tau-bottom of the

cr,(mz)

Yukawa

large. = 0.129

unification

reliable

is required

Yukawa

become

of o,(mz). Taking Fig. 22 as a guideline errors, at face value, two narrow windows

top-bottom

o,(mz)

the central

IR parameters

a weak

shown

to lead to perturbatively

of the electroweak

values

I,< tan p,< 4 and one at large

lessons

described

of this kind at face value,

which

unification

between

the analyses

and theoretical

As a consequence

generation (211).

in order

for Msusv=mr=l50

all uncertainties

to the tau-bottom

emerge

of gauge

to vary of the

the theoretical

unification h,(p

in addition

features

Msusv and the value (2L m,=176GeVwith

quark point

Yukawa constraint

we expect

the most

implemented.

to be added

< 4.45 GeV,

= 0.85 as reasonable

16 shows

the experimental

from

GeV)

Hill effective

Indeed,

UV top Yukawa

function

to p-l

the

unification

unification

emerging

5% have

16.

the following

Grand

= p-‘mb0(5

Fig.

bound,

to hold

in Fig.

so about

unification.

to the IR scale.

but still within

l

bottom

mass

in the tang-mt-plane

[63] in the tan p-rr~p”‘~ -plane

is to incorporate

fixed line was isolated

tau-bottom-top

Several

bands

The input

of bottom

RG flow to lie on or close to the IR fixed

CL; correspondingly

down

Altogether

22 with

“top-down”

it is a symmetry

IR fixed point

l

band

23.

is the tau-bottom-top

at all scales

l

allowed

for mb without

leads

in Fig.

the upper

an IR attractive

l

= 0.12.

[5S] the ansatz

parameter

estimate

tau-bottom

In this

persist

cr(mz)

For this narrow

a narrow

figure

mb(5 GeV)

that

= 0.11 and

in Fig.

into mtpo’e. The corresponding cr(mz)

a(mz)

21 for a) Msusv=mt, os(mz) = 0.11, b) Msnsy=mt, a,(mz) = 0.12, c) = 0.12. Obviously, the allowed region is = lTeV, a,(mz) = 0.11, d) M susy = 1 TeV, os(mz) towards larger top masses for increasing scale Msnsvand for inreasing o(mz). The top mass in

figures

region

of o(mz),

(4) as 4.25 f 0.15GeV.

and

unification

tightly

constrained

supersymmetry

breaking

results. a,(mz) may

be as a

scale

and the experimental top mass for solutions for tan,!? remain,

42 ,< tan p ,< 66. unification

= @SO),

i.e.

close

scale

totally

to the value

fixes the heavy of the IR fixed

discussion.

The very existence of the IR fixed point, in ht and, more generally, of the IR fixed line in the /L-/I~ plane, which are inherently to a large extent independent of the UV initial values at p = &four of

Top Quark and Higgs Boson Masses

)MsIr~~=rn,;

a.dMd=O.lI

= 1 TeV;

“loo

no

140

c~JM~)=0.11

160

180

m,tGav)

Figure

21: Contours

subject

to the

a,(mz)

= 0.11,

a,(mz)

= 0.12.

of constant

tau-bottom b) Ms,,sy=mt,

The top mass

higher values if plotted the IR fixed line shown

m,, in the tan .&ml-plane

Yukawa

against in Fig.

(I, shown

unification

const.raint

obtained

from the RG evolution

at ~Ifou~ for the parameters

of the MSS\I a) Afscrsy=m(.

= 0.12, c) M susy = 1 TeV, a,(mz) = 0.11, d) Msosv = 1 TeV. is the MS mass mt; the curves experience a shift of 5% t,owards

rr~po’~. With this shift the contours appear 16. The figure was taken from Ref. [55].

to lie in the close vicinity

of

B. Schrempp and M. Wimmer

70 60 54 40 Lo 20 10 0 Is0

170 pole 180

160

m,

Figure

22: An update

of Fig.

21 for Msusy=mt

shown

is the line which

results

from tau-bottom-top

to lie in the close vicinity the close vicinity

of the IR fixed

of the fixed point

60

190

and

o,(mz)

in Fig.

I

50 _

210

= 0.12 with

Yukawa

unification

line, the IR result

shown

200

wv)

of tau-bottom-top

16. The figure

I

mt replaced

at Mour.

I

was taken

I

by n~po’~. Also

The contours Yukawa

from

Ref.

appear

unification

[63].

U

allowed region

a, (m,) = 0.12

20 _ 10 _

100

120

140

160

180

200

mtPoie[GeV] Figure 23: The region and with the constraint 15%

in the tan P-my” -plane which is consistent with tau-bottom (229) on the bottom mass for a,(mz) = 0.12. This figure

Yukawa is adapted

unification from Ref.

in

h.l, resp. of ht and hb, provide

the basis for being

the physical

masses

tau

is twofold: initial

and

they

implement

value allows

Yukawa

unification.

. As we have

Seen from and

seen,

cation

enforces

initial

values

for solutions

RGE

solutions

&UT

to mt.

scale

values

Thus.

Yukawa

seen from

independent

scenario

are drawn

matters

of the MSSM.

and

their

independence

and

function of the I’\-

a\ ivell as tau-bottom

nl \-lew. the

tau-bottom

presence

of

is again

of the theory

evolution

path

that path

Yukawa

unification

and above

of the RGE.

for the

values

the from

unification focllscas

circumstance.

grand around

form

initial

the

for small

RG flow which

a very gratifying

of a supersymmetric

point

the long evolution

“top-down”

uniti-

are exactly

the IR fixed

the tau-bottom

of the

\I’ukawa

These

It is for small

of view,

is the specific

the long

onto

or line. even given

point

of the details

for the IR physics

with

safely

a portion

This

sector

their

and top masses

uniilt
unification

at nf~“~

in the top-bottom

high degree

and

unification

pointedly,

for h,(hf GUT) or for hb(,tf~u~).

the IR physics

principle

more

circumstance.

which

in the IR fixed manifolds.

What

values

bottom

the IR fixed line in the tan I;l-mp”‘e -plane.

IR physics

to be those

and

grand

initial

of the RGE

by a symmetry

hfc~~.

top mass

the experimental the gauge

tau-bottom

Formulated

fail to end up in the IR fixed point

at the IR scale

rather

acceptable

the supersymmetric large

able to accommodate

simultaneously.

line is a very fortunate

rather

hb(MGUT), resp.

preselects

highly

to accommodate

the IR fixed point

. The

bottom

II

and Higgs Boson Masses

Top Quark

RGE

scenario

is to n

the unification

which

had

flow from

scale

been chosc,n

a very

high

I.\’

i1=:Zf~,-r.

In summary,

there

is an amazing

conspiracy

between

the ultraviolet

physics

issues

and IR physics

issllrs

at work. As mentioned metric

already

grand

there

unification.

is a large

with

grand

unifying

groups

explore

larger

. which

include

all quarks Higgs

. a realistic . radiative l

search

spectrum

discussion

point

for mt, resp.

.4 very scale.

interesting This

symmetry points

goes

beyond

mixing

sector

the scope

the dependence

the framework

of supersylll-

unification

parameters

of the MSSM

into the numrriral

analysi\

(,I

or

of the ordinary

particles

or

or

in the supersymmetry structure

[50]-[65]in

Yukawa

or

partners

breaking

papers

tau-bottom

and their

in the two-Higgs

of the IR fixed point

Their

the quantitative

leptons

of supersymmetric

electroweak

. investigation

and

bosons

for IR fixed

of interesting

adorned

. which

. all physical

body

usually

breaking

mass

of the underlying

of this review.

of the top mass

terms grand

The qualitative

on tan $, persist.

or unifying

gauge

theor)

conclusions

about

the IR fixcarl

There

are of course

changes

011

level. question

can be “measured”

is:

which

is the extent

by varying

the physics

of the

loss of memory

at the UV scale

of UV physics

and evaluating

at the

the effect

IR

of this

variation in the top-bottom sector of the MSSM at the IR scale as e.g. proposed and analysed in R(~f. in the framework of reduction of parameters to (651. Similar investigations [118],[119] were performed be discussed in the next section. The top mass IR fixed point turns out to be remarkably stable towards such variations.

78

B. Schrempp

8

Program of Reduction

The

program

of reduction

field theories

with

the full content

we shall

MSSM.

restriction

This

approach.

What

[llO]-[112]

already

model

than

mainly

the discussion

will, unfortunately,

is that

search

for IR attractive

to a certain

Starting

of the reduction

program

X,(p) for i = 1, . . . . n and g(p);

generically

point

these

couplings

group

equations,

the

program,

to the SM and occasionally

for special

to the scope

starting

solutions

with

of the

reported

is a renormalizable

the

RGE,

and

very

has

on in previous

parameter

non-linear

program

field theory

g is an asymptotically

of the scale

n + 1 coupled reduction

for renormalizahlc

first

had

to the

beauty

of the

publications

an effect

sections,

with

as role which

it

extent.

as functions

The complete of these

[IlO]-[I121

to do full justice

fixed manifolds,

is interrelated

generally

to its application

not allow

however,

as a systematic

quite

It goes beyond the framework of this review to summarize including all the recent developments [118],[119]; after some

[llO]-[119] confine

will emerge,

for the search

was formulated

one coupling.

of the program

introduction

of Parameters

of parameters

more

and M. Wimmer

p is controlled

differential

amounts

with

free (gauge)

The running

by the perturbative

equations

for these

of

renormalization

couplings.

[119] a s y s t ematic

to [llO]-[112],

a set of n + 1 couplings coupling.

search

for special

solutions

RGE

. which

establish

renormalization

X;(g); this requirement and

ps = dg/d

invariant relations

group

is shown

to imply

the relations

which

. rather plings

the /? functions

PA, = d X,/d

X, = In(p)

In(p) a$$

. and

between all couplings in the form

between

are not determined

by the theoretically Xi vanish.

= @A,

by initial appealing

value

conditions

boundary

asymptotically

i.e. become

form,

the dependent

but

condition

that

with

free in the UV limit for

x; + 0

In the most restricted

i = l,...,n

for

couplings

the coupling

g also the cou

p --) co

g + 0.

X,(g) emerge

(231)

as expansions

in powers

of g, typically

in the form X:(g)

=

with constants pi;, pz,, . . . . less restricted logarithms of g. The reduction is called (232)

trivial if they

are positive,

are also mixed recent In any a single

modes

discussion case, one.

perturbation

see Ref.

complete g.

It is instructive

9’

+

+

P3i g6

+

and

(TE)

...

also to contain

if all the first order

are zero or, more

generally,

partly

trivial

coefficients

if XT/g* + expansions

fractional

powers

0 for g --+ 0; of course possible;

or

pli in the expansion there

for a comprehensive

[119]. amounts

a reduction

to reducing

is complete.

and

in practice

to the order

to relate

the result

of this program

as advocated

PZ* g4

allow the expansions

non-trivial,

nontrivial

reduction

Provided

theory

of the RGE

of partly

Pli

forms

in earlier

sections.

the number it allows

to which

the RGE

to a general

A complete

n + 1 of independent

in principle

have been

search

non-trivial

a reduction

to in

calculated.

for IR fixed points, reduction

couplings to all orders

lines, surfaces

of the type

(232)

with

positive constants pri amounts at the one-loop level to determining [ill] an IR attractive i.e. IR stable fised point XT/g2 = ~1, for i = 1, . . . . n of the one-loop RGE in the space of the ratios of couplings X,/y. In the framework of reduction of parameters, this point has the interpretation as an UV repulsive i.e. UV unstable fixed point solution, it exists as such in principle to ail orders of perturbation theory, more precisely

the existence

of a renormalization

group

trajectory

that

asymptotically

approaches

the point

79

Top Quark and Higgs Boson Masses in the level,

UV limit

is ensured.

the important

solutions

which

couplings

in such

Despite

difference

are IR attractive a way that

In the application

with the strong

that

couplings

as many

gs such as to share fails in the sector within

a finite

A partial gram

of gauge

reduction

couplings and

one and

Let us start jlll]

with

the

UV sraie

in the UV limit

to the SM [35],[39] (231)

then

special

out which p +

link all

03.

the “driving”

coupling

the physical

condition

implies group

in the UV limit

A, introduced

invariant

manner

p + 00. A complete

to

reduction to

step

in form bottom

in the introduction.

[38],[39]

in a two-step

in a second

at the

already

The analysis

proceeds

as corrections

process:

of the partial

line of the

power

interactions

are systemat-

series

analyses

pro-

the electroweak

the electroweak

of a double

reduction

in a first step expansion

[38],[39]

are

in the ratios

two

solutions.

a

one.

non-trivial

meaningful

are singled

in a renormalization

freedom

nevertheless.

As results

a trivial

solutions

at the one-loop

of selecting

91, gs and gs. The rJ( 1) gauge coupling gi being not asymptotically limit /I -+ m, but rather limits the final analysis

off, gi=gs=O;

included

manifolds instead

of the formal

a physical

are switched

are linked

of asymptotic

of the SM then

pz=g;/gi.

non-trivial

of parameters gs; the condition

can be performed sector

consistently

pi=gf/g!and

physically

below

criterion:

to zero simultaneously

coupling

couplings

for IR fixed

RG flow, special

of reduction

gauge

the performance

in the matter

ically

decrease

to the search

lies in the selection

of the SM as possible

interval

gauge

they

its property

free also prohibits

vicinity

search

for the whole

of the program

g is identified

this

to this

solution.

non-trivial

with the IR attractive

In the

solution

fixed point

absence

is found

of electroweak

in the top-Higgs-gs

by Pendleton

couplings

sector:

and Ross 1301, Eq.

a consistent

and

at one loop it, is identical

(15i).

in the variables

p,=yf,‘!li

and p"=X/gj

at the two-loop

level it is given

r\t the one-loop

[113],[114]

31359 +41&W 15552

912

=

2 9932+

x

=

&-B-2sg2+ 72

level the electroweak

gs4

147015115

of gi

+ .. .

s

3 couplings

in powers

-.535843&B 3856896

gs4 m

(1’:: Lj

+

[39] as perturbations

are included

for small

\,alues

of

of a double power series expansion in powers of p, ant1 ,j2 into the ansatz for a special solution of the RGE f or sufficiently small values for p, and p2, subject to solution (233) is recovered; the result is the boundary condition that for pr,pz + 0 the unperturbed the ratios

pl=gf/g,' and

by the expansion

pz=g,2/g,2in form

2 Pt

=

9:19,’

=

x/9,2

=

799

17

9 -

540

Pl

-

m-25 PH

=

It is interesting

to translate

of convergence

result

fixed surface

(35)

gjgjjjP12+~P’P2-&P22+‘.’

1295

7&% Pl

66960

the one-loop

an IR attractive

P2 +

t&B-

72

This expansion has a finite radius be extended beyond it numerically.

it constitutes

_

&

in pi and

(235)

into

-

- 163

pI=p2=0. but

pz around

the language

in the three-dimensional

(mi)

pz +

1488

the solution

of IR attractive

pr-pi-pz-

space

which

mai

manifolds”: attracts

the

“top-down” RG flow very weakly. This can be seen by solving [117] the linearized version of the RGI(192) and around the common IR attractive fixed point pt=2/9 and pl=pz=O, for pb=&=O: it has the IR fixed solution (235) with only the linear terms in p1 and pz present. The general solution to this linearized problem can also be obtained [1X]: it shows that the IR attractive fixed surface (235) attract\ the “top-down” ‘lone letter.

of us (B.S.)

RG flow in the vicinity is grateful

to 14.

of the IR fixed

Zimmermann

for stimulating

point

pt=2/9. p,=p2=0

the following

discussion

with

and

the small

investigation

po\~ei

[I 171 h> a

80

B. Schrempp

and M. Wimmer

i.e. exceedingly weakly. This surface had not been discussed (&“r, competition with the much stronger IR attractive fixed surface. given in Eqs. Ref.

(203),

(204)

[39] and Sect.

allows

to express

the pt-l/gz

RG flow may

numerically

5.1, i.e. by introducing The resulting

be read

in Sect.

of the solution

off from

(235)

The conclusions

Fig.

11; clearly

from

which

is dictated

by the boundary

has

exhibited

yet

of pi and

pz into

initial

most

values

than

reduction

for gi(mz)

by following

and gz(mz).

This

of the two surfaces

in Fig.

12. The

the more

attractive

into

“top-downfixed line

the projection

program.

little

for the search weight

of asymptotic

interrelation

it loses thr expansions

transparently

the thin line 2, representing

are important

condition

5.1, since of analytic

projection

the fat line 1, representing attractive

while it carries

another

be seen

fixed lines are displayed

the one-loop

counts,

may

and plot the resulting

at in the last paragraph

the rate of IR attraction

This

the experimental of l/g&

more strongly

emerging

arrived

10.

two IR attractive

5.1, is much

where

discussion

in Fig.

pi and pi as functions

plane.

discussed

and

in Sect. in form

freedom

between

for IR attractive

for the parameter

in the UV limit.

IR motivated

and

manifolds

reduction

approach

In any case the

UV motivated

physics

issues. The

inclusion

powers

of pi and

eventually

ps cannot

become

the parameter

large

and

reduction

IR attractive

fixed

the mathematical

the

solutions

be extended,

run out of the region

scheme

manifolds: UV limit

faces the

p d

increase

between

the IR scale p ‘v m, and an UV scale

top and

Higgs

mass

are of the order

two-loop

approach

by experiment, on a higher

the discussion

At the two-loop

from

of gi for decreasing

Thus,

search

from

p prevents

pi will

in a way.

as the

the former

have to make

the one-loop

results

mH N 64.4 GeV.

a welcome

for

reaching

the latter

do with

a finite

deeper

[39] by evaluating

More

the resulting

b esi‘d es an expansion

pb=g,'/g," is included.

17

precise

values

from

interval

pt and emerge

top mass value is excluded

insight

~~~~~5P13P2

93’ 31359 +41m

+4*2t 25

83&i%-

+.022843

PI’

2

into

the parameter

pH at

from

the

meanwhile

reduction

-

of pi and ps also an expansion (236) are then

1 PIP2

1

-

+

0.8262~~

283

method

PI’PZ

-

0.02158~ip~~

replaced

9 p2’

+

400

in powers

by

54 PlPb

-

175

Pb2

- o.oo9p12p* - 0.001p,pzp*

9.90905pip23 + 0.1690pi2

163

1488

-

-

o.ooo14p24+ ...

+ 0.1824pip2

- 0.0664p22 + ,,,) + ,,

pz - 0.037161 pb

0.0531783plp~ + 0.1092913~~~ + 0.0362~~~~

- 0.032795

+ 0.00885ps3

+ 0.024~~~~ - o.0160pi~pb

+ 0.2725~~2 + 0.0215p,p2pb

p~'pb PI4

+

o.0226P13P2

+

numbers

O.O152p1~p,* + 0.00588p,p,3 + 0.00923p24 + ... - O.O3088p, - 0.1205 pz

15427584

+o.os3.5pIz - 0.0115p1p2 where the decimal

9360

7v@i?PI

14701515 -535843a

$

+

o.oooo8p12p22

1295

66960 +

-6.01722~~3 +O.O124

-

- 0.2231 PI

62208

i2

+0.09348

and

pi3 - 62400 p12p2 + 56160 plpzz - mpz3

+".oo2gp14 +

m-

pl’

17

323

972000

=

(235)

119

f&300

5593

in powers

Eqs.

799

g-&P’-;P2-;Pb+-

--

PH

theory.

in

series

00, since

problem

p prevents

approaches

Even though

provides

level [113],[114]

parameter

2 =

determined next.

SM a similar

power

p d

A 5 10” GeV.

of mt z 91.3 GeV and

to be presented

of perturbation

to the

Both

of a double

loop level.

of the small

Pi

values

in form

to the UV limit

of validity

the increase

p -+ 0 [115].

the mathematical

p = mz

(236)

of pi for increasing

co, while

IR limit

and

in application

reaching

The

(235)

even not in principle,

-

0.00083 pz2 + ,._) + ...

are numerical

approximations,

('37)

Top Quark Including

a physical

for the top quark following

value and

for the bottom

the Higgs

pole masses

boson

and Higgs

mass

[113]

top mass

It is interesting framework gauge

and

is too low to be compatible

to ask whether

of the MSSM.

couplings

the one-loop

[117] which

Introducing

the experimental

The

12, which

attractive

manifolds

pz leads

for p1 and

less strongly

attractive

result

leads

to a success

in absence

solution

(195)

a la Pendleton fixed

as the one (Fig

10) discussed

pz, leads

than

and

surface

to the solution

the one drawn

in the

of the electroweak

to an IR attractive

IR attractive

conditions

analysis

one-loop

fixed point

of p1 and

value.

Ross.

in the p-

in Sect.

drawn

5.1.

by the thiu

by a fat line 1 representing

5.1.

of the

masses.

reduction

The

(‘23s)

= 64.6 * 0.9 GeV.

the experimental

parameter

is no [117].

less strongly

is clearly

rnrle with

the IR attractive

initial

in Sect.

solution

Higgs

with

is again

discussed

“trivial”

top and

answer

of pt in powers

P1-pz-space

t.he solution

the analogous

The

is identical

expansion

line 2 in Fig.

81

Masses

(5 GeV), radiative corrections relating the pole masses MS masses at p = rnz the two-loop results lead to the

to their

my”‘e = 99.2 f 5.7 GeV The predicted

Boson

parameter

There

are,

reduction

however,

at the one-loop

level.

program

again

very

applied

to the

interesting

The one-loop

trivial

cross

SM leads

relations

solution

t,o even

smaller

to the search

[39] in absence

for IR

of electroweak

couplings 1 1 p.q = j pt2 + iij pt3 + . . . establishes

a relation

expansion

(163)

between

the ratios

of the IR attractive

R = p~/p~). This solution corrections

PH

extends

up to the IR fixed point in Fig.

out

to be identical

with

the

pt=p~=O (remembering t,hat at pt=2/9, i e . is historically a precursor of around

5. The corresponding

one-loop

expansion

including

tht.

[39] is

;

=

pt. It turns

pH and

fixed line in the pt-pH-plane

the low pt end of the fixed line shown electroweak

of couplings

(239)

Pt2

+

+j

Pt3

i&PlV

+Pt)+

&P,P2(1

+pt)

_

&P2?1

+

+pt)

351 -a .4gain

PI3

this is the expansion

apphed

of recent

in Sect.

interesting

to supersymmetric

MGUT. The advantage

is that

go clearly

The

invariant

RG

p 2 MGUT are summarized physically

appealing

all dimensionless

coupling

is sufficiently

-

theories

large;

can be singled

relations

between

P23

the

program

out which

theory, bottom

6

m*

+

(‘10)

....

fixed

of reduction

allow a complete

some

gauge

Generically

-

IR attractive

the

however, the

discussed cases

&

manifold

in

the

of parameters

p 2 MGUT above the grand

and

the

unification

reduction.

of the conclusions

coupling(s)

“gauge-Yukawa

parameters.

in many

+

pt=p~=~l=pz=O.

for scales

this review,

to each

around

[118],[119]

theories

the headline

PlP22

three-dimensional

beyond

coupling

&

4.6 (for gb=O),

unified

under

solutions

between

P12P2

publications

grand

of the applications resulting

ggj

of the corresponding

PI-pH-PI-p@pace,discussed In a number

-

The details

are very Yukawa

is scale

pertinent.

couplings

for

unification”. each the

Yukawa

one

There are typically several providing RG invariant relations

UV initial coupling

value

for the

is of the

order

top

Yukawa

of the

top

coupling. The authors [118],[119] th en explore the corresponding strongly constrained “top-down” RC: flows from MGUT to the IR scale mt according to the RGE of the MSSM with MSUS~ varying within reasonable bounds. Here their various solutions for various theories are caught to a certain extent in the trap of the strongly the top mass discussed to be rather the different

IR attractive fixed manifolds and and in particular in Sects. 5 and 6. Thus, in zeroth approximation

of the IR fixed point for one expects the IR reslllts

independent of the UV input. The authors work out to which extent the results vary fol theories and for the different solutions within a given theory. Their top masses lie aroulltl

B. Schrempp

82 190GeV,

as expected,

the values These

with

gauge-Yukawa

unification

with

and

. The dynamics

9

have

unification

two

features

discussed

one model

or solution

see Sects.

in common

in Sect.

is a rather

Mour

singles

also for the bottom

RG flow which plane.

from

unification,

to another

5.2 and

with

one:

6.

supersymmetric

gmnd

7.

insensitive

testing

ground

for the details

of the

/I = MouT.

at the UV scale

line in the h,-hb

10GeV

top-bottom

of the IR region

above

top and generically “top-down”

scenarios

Yukawa

sector

around

of at most

63, signalling

reduction

tau-bottom

. The top-bottom theory

a variation

for tan /? are around

and M. Wimmer

out

Yukawa

is contracted

largish

fully onto

This enhances

strongly

values

coupling.

for the UV initial

This

in turn

the IR fixed point

the significance

selects

that

in ht, resp.

of these

values

for the

portion

of the

on the IR fixed

IR fixed manifolds.

Conclusions

The efforts effects,

to trace

as encoded

a possible

dynamical

in the RGE,

origin

have

been

for the top and Higgs

reviewed.

The SM and

masses

at the level of the quantum

the MSSM

have

been

considered

in

parallel. The most

important

approached

by the

l

answers

to this question

“top-down”

in the tan @-n~f)o’~-plane -

for small

bottom

lie in IR attractive

RG flow from

an UV scale

fixed lines and fixed points

A to the IR scale

which

are

O(v)

of the MSSM: Yukawa

couplings

the much

quoted

very strongly

IR attractive

fixed point

for the top mass rnpole = 0( 190 - 200) GeV sin /? which

is resolved

into a genuine

and

an upper

UV initial -

bound,

values bottom

tan /3-mt P”‘e-plane,

Fig.

the IR image

Yukawa

16, with mi

approximate

. in the m~‘e-m~le-plane -

a weakly

IR attractive

on a strongly

a very

m, strongly

fixed point

% 182 GeV, Yukawa

z

190 GeV sin/3,

(242)

pole

x

200 GeV sin 8.

(243)

IR attractive

fixed

line in the

at

tan p NN60,

(24‘4)

unification.

of the SM: fixed

point

mt lying

coupling

couplings

pole

pole mt

at

of large

an IR attractive

bottom-top

p&

-

IR fixed point

for the top Yukawa

for unconstrained

implying

(211)

attractive which

implies

at z 214 GeV,

pole mH x 210 GeV,

IR fixed line in the m~‘e-m~‘e-plane, POlC mH =z 141 GeV

for

Fig.

my”‘e = 176GeV.

17, (246)

83

Top Quark and Higgs Boson Masses As it turns scale,

out,

v/v’?,

resulting

the IR attractive

much

larger

Of course, reality,

as we knew

for the supersymmetry

An intricate attention

content

detection

interrelation

unification

values

are roughly

matter

UV physics

onto

in the MSSM

the experimental is altogether

the significance

The MSSM

the

value,

but

a very striking

of these

results

needs an experimental

for

support

value for tan 8, the SM as well as the MSSM

and

await

IR physics

in the MSSM,

which

has received

much

as follows.

of tau-bottom

precisely

with it. This

to judge

of the electroweak

In particular

boson.

be summarized

constraint

of the order

particles.

information.

the RG flow towards

more

in order

and an experimental

may

focuses

value,

experimental

between

. The UV symmetry

the outset,

of the Higgs

in the literature,

mass

from

one needs further

the experimental

mass

of all other

SM value mr z 215 GeV is not very far from

also the corresponding physical

the masses

rnpole = 0( 190 - 200) GeV sin /3 is well compatible

top mass

result.

top and Higgs

than

Yukawa

coupling

the IR scale much

unification

in supersymmetric

more strongly

grand

into the IR fixed point

the IR fixed line in the tan $mpo’e -plane,

than

top

the unconstrained

RG flow. It appears

l

in the scale

to be the very presence

grand

Much

effort

surfaces,...) space

second

to implement

a symmetry

property

appear

couplings.

gf/gi,

as far as possible

gt/gi,

at the

1%’

scale in (supersymmetric)

From

gz/gi,

this

(fixed

absolutely

of approach

in table

towards

the

for

X/g~;. These and lines in

2 in Sect.

the various

investigation

the

in parallel

into the IR fixed points form

points,

are included,

by the variable

in a compact

collective

couplings

treated

increased

for an insight rates

IR fixed manifolds

all gauge

is further

are summarized

the respective

analytically.

When

gt/g&

to be essential

issue was to assess

at the unification

of parameters.

The results

of the IR fixed line

unification

property.

in case of the SM the space

manifolds

resp.

Yukawa

masses.

to work out the underlying

of ratios

of variables

in the top mass, tau-bottom

of the tau and bottom

in this review

space

the MSSM;

important

manifolds

spent

is the

of all gauge

values

unification,

in the space

dimensional

of the IR fixed point allows

is also an IR attractive

has been

the SM and higher

Yukawa

unification,

lines,

presence

which

as well as the physical

. Bottom-top

relevant

-plane,

tan/3-m~‘e

:3. ,A

IR attract

following

i VP

hierarchies

emerge.

l

The SM and the MSSM for the “top-down”

. Generically than

This

nontrivial

the lower

last statement

higher

dimensional

needs

generically

of attraction

is equal

lines in turn

are intersections

is that

dimensional

most of these

to one variable held constant. out to be the most attractive

stronger

structure.

in the MSSM

IR attractive

manifolds

In the high dimensional

lies at the intersection for both

IR fixed manifold

However, than

the IR attraction

in the SM.

are more strongly

IR attractive

ones.

qualification:

fixed point

The point

have a very similar

RG flow is systematically

lines,

multiparameter

of two IR attractive

it is always

of two IR attractive lines and surfaces

lines.

Unless

one of the two lines which surfaces, are trivial

It turns out that typically ones, leading to non-trivial

one of which ones,

e.g.

space

an IR attractive

accidentally

attracts

more

will be more

the strength strongly.

The

attractive,

etc.

in the sense that, they

the most non-trivial IR attractive relations

correspond

lines, surfaces, etc. turn between the considered

parameters. The triviality and vacuum stability of the boundaries of the “top-down”

bounds on the SM Higgs mass provide a measure for the distance RG flow from the IR fixed point and fixed line in the m~‘e-m~‘p-

plane as a function of the UV scale .A. These bounds are also of relevance for future Higgs searches. Therefore. the most recent precise determinations of the vacuum stability bound in the SM as well as

B. Schrempp and M. Wimmer

84 of the upper

bound

on the mass

of the lightest

Higgs

boson

in the MSSM

points

of view.

have

been

included

in this

review. Let us close with One

could

an outlook

entertain

UV physics.

the

This

A, at which

(i.e.

be achieved IR stable)

new physics

the IR fixed manifolds, of the RGEs

necessary

lepton step

sector

towards

Very interesting

From

further

our present

First

A much

the IR fixed

and/or lished

structure

weakly

towards

is the specific structures

form in Eqs.

in the remaining

IR attractive)

[SS], (1181, [119], which

and the known

would

be a

in supersymmetric

of the top Yukawa

evolution

Yukawa

rely very much

IR physics.

new physics

values

the RGE

will lead into the

coupling

is large

grand

couplings

in addition,

are fixed

“trap”

of the

one ends

up

how “stable” this memory [65], [118], [119] t o investigate, sector is with respect to “variations” in the new physics

[119] indicate

of the theory

of the symmetries

of approach

then

IR fixed manifolds

whichever

that

issue [65] is the question

to determine

and

lying and

to which

extent

breaking

mass

stable. the quark

terms

and lepton

are determined

masses

and

in terms

the SM. First

answers

content

of the theory

beyond

the SM is largely

results

in the low energy

to lead to very

promising

[65] indicate

of

beyond

multiplet

the IR structure

it is remarkably

that

of the theory.

One of us (B.S.)

I. Montvay, enlightning results,

is grateful

that

O(1).

The issue is then in the top-bottom

point

Acknowledgements: Kniehl,

appear

counts

the IR fixed point

into

UV scale

if the bottom

[65], Ills],

In this case the UV scale

which

to match

of view.

as long as the UV initial than

of the RGEs.

to be only

in the literature

at a high

it may

out

of

independent

the parameters

this angle,

as well as the soft supersymmetry

the knowledge sector

been raised

to be larger

answers

angles

sufficient

of this point

new physics

furtherreaching

mixing

to turn

be largely

as well as the rate

An investigation

support

for the top mass;

with tanP=0(60). loss of UV physics

l

point. were

knowledge

by the UV dynamics IR fixed point

Seen under

should

by arranging

The only input

if they

one formulates:

IR physics theory

or fixed lines etc.

irrelevant.

starting (even

that

of the new physics

of the IR scale.

between

unification

sector.

largely

issues have recently

on an interrelation

l

the details

become

are a promising

and

expectation in a renormalizable

fixed points

arises,

in the vicinity

(241)-(246) quark

two complementary

“conservative”

may

the IR attractive

from

F. Schrempp, discussions.

obtained

to the DESY

partly

is grateful

F. Wagner, We also thank

by himself,

theory

group

to J.R.

P. Weisz,

Espinosa,

F. Schrempp

partly

G. Isidori,

W. Zimmermann for granting

in collaboration

for the continuous

with

F. Jegerlehner,

for valuable

B.

communications

the inclusion

of some unpub-

one of us (B.S).

One of us (B.S.)

hospitality.

References [I] CDF

Collaboration,

[2] DO Collaboration,

F. Abe et al., Phys. S. Abachi

et al., Phys.

[3] D. Schaile, in Proc. of the 1994 Knowles, London 1994, p. 27. [4] J. Gasser [5] Particle [6] L.Maiani.

and H. Leutwyler, Data

Group,

G. Parisi

Int.

Phys.

L. Montanet and

R. Petronzio,

Rev.

74 (1995)

Rev. Lett.

Conf.

Rep.

Lett.

74 (1995)

2632.

Energy

Physics,

on High

87 (1982)

77.

et al., Phys.

Rev. D50

Nucl.

B136

Phys.

2626.

(1994) (1978)

1173. 11.5.

ed.

P.J.Bussey

and

I.G.

Top Quark [7] N. Cabbibo,

L.Maiani,

[8] R. Dashen [9] M.A.B. [lo]

and

H. Neuberger,

M. Lindner,

Z. f. Physik and

B. Grzadkowski, M. Lindner,

C31

M. Sher

[13] M. Sher

, Phys.

Lett.

[14] C. Ford,

D.R.T.

Jones,

B31’7

and

G. Isidori,

[16] J.A.

Casas,

J.R.

Espinosa

[17] J.R.

Espinosa

and W.J.

[19] Y. Okada, J. Ellis,

M. Yamaguchi G. Rodolfi

R. Barbieri

and

D. Pierce,

and

A. Brignole,

Phys.

Chankowski,

(211 Y. Okada,

and

and

Casas, (1995)

[28] M. Carena, [29] C.D.

Lett.

T. Hugo,

[26] R. Hempfling

and and

and J.R.

and

A.H.

1987) 64.

(1989)

39.

B353

(1995)

Lett.

B262

Lett.

Lett.

B258

Phys.

Rev.

(1991)

233;

(1992)

284;

Phys.

B283

Lett.

(1992) and

Ii. Sasaki,

Phys.

B395

(1995)

171

Phys.

(1991)

85 (1991)

83; ibid.

Lett.

66 (1991)

(1991)

477;

68 (1992)

3678.

1815;

Lett.

B274

(1992)

191.

Phys.

Lett.

B262

(1991)

54;

Lett.

B258

(1991)

167;

Lett.

B301

(1993)

(1994)

7035.

(1993)

4280.

(1991)

389.

287. H. Nakano, Phys.

Phys.

Rev. D50

Phys.

Lett.

B331

(1994)

Espinosa,

M. Quiros

and

A. Riotto,

Espinosa,

M. Quiros

and

C.E.M.

83.

99.

Nucl.

Phys.

B436

(1994)

3; Erratum

466. J.R. and

H.B.

Nielsen,

[30] B. Pendleton

and

G.G.

Ross,

Hill, Phys.

Rev. D24

Nucl. Phys.

(1981)

Phys. Lett.

691.

17

1;

B262

Phys.

B266

(1993)

395; Rev.

Phys.

Rev. D48

Phys.

Nucl.

257.

Theor.

Phys. Lett.

448

54.

B257 (1991)

S. Johnson,

B324

(1991)

Prog.

Phys.

(1994)

141.

Phys.

Lett.

331

Einhorn,

(1994)

T. Yanagida,

Frogatt

[Bl] C.T.

M.B.

B337

F. Caravaglios.

Hoang,

81; B199

ibid.

Lett.

N. Maekawa

Y. Yasui

and

and J. Rosiek,

M. Quiros,

Phys.

, J. Iiodaira,

B439

B281

R. Hempfling,

Espinosa

[24] M. Bando,

[27] J.A.

Lett.

B228

Addendum

T. Yanagida,

S. Pokorski

M. Frigeni

[23] B. Kastening,

(251

B263

Lett

Phys.

and

M. Yamaguchi

R. Barbieri,

Lett

Phys.

Lett.

883

Lett.

M. Quiros,

R. Hempfling,

Phys.

Phys.

F. Zwirner.

A. Papadopoulos

Haber

H. Haber

and

and

Rev. Let t. 52 (1984)

1897.

Phys.

Phys.

193)159;

Phys.

M Frigeni,

A. Yamada,

[22] J.R.

Stirling,

295.

50 (1983)

B158

273.

Phys.

M. Quiros,

(1979)

B178 (1986)

Lett.

S. Theisen,

Stephenson

and

Phys.

295;

(1993)

P.W.

[15] G. Altarelli

[18] Z. Kunszt

and

Lett.

85

Masses

Nucl.

A. Sirlin,

Phys.

179 (1989),

Rep.

and

and

and H. Zaglauer,

Phys.

P.H.

Rev.

(1986)

M. Lindner

Boson

R. Petronzio,

Phys.

M. Lindner,

[12] M. Sher,

[20] H.E.

and

Beg, C. Panagiotakopoulos

B. Grzadkowski

[ll]

G. Parisi

and Higgs

B147

B98

Wagner, (1979)

(1981)

291.

Phys. 277.

Lett.

B355

(1995 209.

ibid.

86

B. Schrempp

[32] C. Wetterich,

Phys.

Lett.

104B

(1981)

269.

[33] E. Paschos, Z. Phys. C26 (1984) 235; J.W. Halley, E. Paschos and H. Usler, [34] J. Bagger,

S. Dimopoulos

[35] S. Dimopoulos [36] J. Bagger, J. Bagger, 1371 C.T.

and

and

S. Dimopoulos S. Dimopoulos

Hill, C.N.

Leung

and and

and

Phys.

E. Masse,

S. Theodorakis,

Lett.

Nucl.

Phys.

Nucl.

Bl55

Phys.

Lett.

E. Masso, E. Masse,

S. Rao,

and M. Wimmer

(1985)

B253

154B

Phys. Phys.

Lett. Rev.

Phys.

B262

107.

(1985)

(1985)

397.

153.

B156 (1985) 357; Lett. 55 (1985) 1450. (1985)

517.

Nucl. Phys. B259 (1985) 1381 J. Kubo, K. Sibold and W. Zimmermann, K. Sibold and W. Zimmermann, Phys. Lett. B191 (1987) 427. J. Kubo,

K. Sibold

and

W. Zimmermann,

Phys.

Lett

1391J. Kubo,

K. Sibold

and

W. Zimmermann,

Phys.

Lett.

(411 C.D. Frogatt,

R.G.

Moorhouse,

Phys.

Lett.

B249

(1990)

273.

F. Schrempp,

Phys.

Lett.

B299

(1993)

321.

Gaume,

I441 J. Bagger,

S. Dimopoulos

I451 M. Carena,

T.E.

J. Polchinski and

Clark,

and

M.B.

Wise,

p.403,

Nucl.

E. Masso,

Phys.

Rev.

Wagner,

W.A.

Bardeen

C.E.M.

191.

(1989)

Trieste

[431 L. Alvarez

(1987)

B220

Proc.

and

Workshop

(1989)

1401 C. Wetterich,

1421B. Schrempp

HEP

B226

331;

185.

and preprint

Phys.

Lett.

B221

DESY-87-154

(1983)

55 (1985)

and

(1987).

495.

920.

K. Sasaki,

Nucl.

Phys.

B369

(1992)

33.

I461 C.D. Frogatt,

LG. Knowles

I471 B. Schrempp,

Phys.

1481B. Schrempp

and

1491MS. Chanowitz, [501 L.E. Ibanez

J. Ellis and

and

C. Lopez,

T. Kugo,

Gaillard,

Nucl.

N. Maekawa

and

H. Nakano,

S. Mikaelian, D.J.

M.K.

Q. Shafi,

and

B298

(1993)

356.

to be published.

and

Hall

Lett.

193.

Lett

Keszthelyi,

Phys.

(1995)

Phys.

Phys.

L.J.

H. Arason,

Moorhouse,

G. Lazarides

A. Giveon, 2933 and

344

R.G.

M. Wimmer,

B. Ananthanarayan, M. Bando,

Lett.

and

U. Sarid, E.J.

B126

Phys.

Piard,

Rev. D46

(1992)

Castelano,

E.J.

Phys.

(1983)

Lett.

P. Ramond

B128

54; Nucl.

Phys.

and

Phys.

Rev. D44

Mod. B271

(1977)

Phys. B.D.

B233

(1991)

Lett.

(1991)

506. (1984)

A7 (1992)

138H. Wright,

Arason, Phys.

Giudice,

Mod.

Phys.

Lett.

[511 G.G. Ross and R.G. Roberts,

Rev.

and

P. Ramond,

A7 Nucl.

(1992)

Phys.

Rev. D47

(1993)

H.

67 (1991)

Phys.

B377 Rev.

(1992),571.

S. Raby.

Phys.

I531 S. Dimopoulos.

L. Hall and

S. Raby

, Phys. Rev. D45

T. Han and

232; Phys.

2429.

L. Hall and

M.S. Berger,

Castelano, Lett.

3945; Piard

1521S. Dimopoulos,

I541 V. Barger.

3379; D.J.

S. Kelley, J.L. Lopez and D.V. Nanopoulos. Phys. Lett. B274 (1992) 387 and (1992) 140; J. Ellis, S. Kelley and D.V. Nanopoulos, Nucl. Phys. B373 (1992) 55; G.F.

511;

1613;

M. Zralek.

Lett.

Phys.

68 (1992) (1992) Rev.

1984.

4192.

Lett.

68 (1992)

3394.

Lett.

B278

Top Quark and Higgs Boson Masses

[55] V. Barger,

M.S. Berger

[56] G. Anderson,

and

P. Ohmann,

S. Dimopoulos,

[57] M. Carena,

S. Pokorski

M. Olechowski W. Bardeen,

and

and

Hall and

C.E.M.

S. Pokorski,

M. Carena,

M. Carena,

L.J.

Rev. D4’7 (1993)

S. Raby,

Wagner,

Nucl.

Nucl.

Phys.

S. Pokorski

L. Clavelli,

Phys.

and

D. Matalliotakis,

B404

C.E.M.

Phys.

1093.

Rev. D47

Phys.

B406

(1990)

590;

Wagner,

(1993)

59;

Lett.

B320

Phys.

H.P. Nilles and C.E.M.

(1993)

Wagner,

3072.

(1994)

110;

Lett.

B31’7

Phys.

(1993)

346. [58] P. Langacker

and

N. Polonsky,

[59] V. Barger,

M.S. Berger,

V. Barger,

M.S. Berger

[60] B.C.

Allanach

and

[61] P. Langacker

and

Phys.

Rev. D49

P. Ohmann and

and

R.J.N.

P. Ohmann,

S.F. King,

Phys.

N. Polonsky,

M.S. Berger

and

P. Ohmann,

[63] V. Barger,

M.S. Berger

and P. Ohmann,

Lett.

Phys.

B314

(1994)

360.

D50

(1994)

2199.

Phys. Proc.

Rev D49 High

Lett.

(1993)

B328

Rev.

[62] V. Barger,

1454.

Phillips,

Phys.

Lett.

Phys.

(1994)

(1994)

Energy

B314

(1993)

351;

351.

4908.

Physics

Conf.,

Glasgow

Phys.

B419

(1994)

Vol.2.

563. [64] M. Carena, Nucl.

M. Olechowski,

Phys.

BB426

M. Carena

and

S. Pokorski

(1994)

C.E.M.

and

C.E.M.

Wagner,

Wagner,

Nucl.

Phys.

B452

(1995)

and G.G.

Ross,

Phys.

Lett.

B349

(1995)

M. Lanzagorta

and G.G.

Ross,

Phys.

Lett.

B364

(1995)163:

Ross,

Phys.

[66] S.L. Glashow,

Nucl.

S. Weinberg, A. Salam, (1969)

Lett.

B364

Phys.

Phys.

Rev.

(1994)

213 and

45.

[65] M. Lanzagorta G.G.

Nucl.

269;

(1995)216.

B22 Lett.

in Elementary

319;

(1961)

579;

19 (1967)

Particle

1264;

Theory,

N. Svartholm,

eds. Amquist

and

Wiksells.

Stockholni

p.376.

[67] H. Fritzsch, H.D.

M. Gell-Mann,

Politzer,

D. Gross,

Phys.

F. Wilczek,

S. Weinberg, [68] J. Bagger

Phys. and

H. Leutwyler,

Rev.

Lett.

Phys.

Rev.

Rev. Lett.

J. Wess,

Phys

30 (1973) Lett.

Lett.

30 (1973)

31 (1973)

Introduction

47 (1973)

365:

1346: 1343;

494.

to Supersymmetry,

Princeton

Univ.

Press,

Princeton

(1983). [69] H.P.

Nilles,

[70] J.F.

Gunion,

Publishing [71] S. Bertolini,

Phys.

Rep.

H.E.

110

Haber,

Company

(1984) G. Kane

1. and

The Higgs Hunter’s

S. Dawson,

Guide,

Addison-Wesle!

B353

(1991)

591.

Phys.

B406

1990.

F. Borzumati,

A. Masiero

and

G. Ridolfi,

Nucl.

Phys.

1721 P. Langacker an M. Luo, Phys. Rev. D44 (1991) 817; R. Barbieri and L.J. Hall, Phys. Rev. Lett. 68 (1992) 752; K. Hagiwara and Y. Yamada, Phys. Rev. Lett. 70 (1993) 709; P. Langacker M. Carena. [73] P. Langacker

and

N. Polonsky,

M. Olechowski, and

Phys.

S. Pokorski

N. Polonsky,

Phys.

Rev. D47 and

(1993)

C.E.M.

Rev. D47

4028;

Wagner,

(1993)

-1028

Nucl.

(1993)

59

X.1

B. Schrempp and M. Wimmer

88 [74] P. Langacker [75] H. Georgi

and

and S. Glashow,

H. Georgi,

H. Quinn

[76] S. Dimopoulos,

and

N. Sakai,

Zeitschr.

E. Witten,

Nucl.

Marciano

L.E. Ibanez M.B.

and

J. Ellis,

W. de Boer

S. Kelley and

F. Anselmo,

Machacek

(1985)

and

I. Jack, I. Jack I. Jack

and

B196

3092;

439: (1982) 439.

Benjamin/Cummings,

Rev.

D44

Lett.

B222

1984; 1992.

B260

(1991) (1990)

447; 441;

817;

A. Zichichi,

Phys

New York, New York,

B249

Lett.

(1991)

and

Nucl.

Sringer,

Phys. Phys.

A. Peterman

J. Oliensis, Al6

J. Phys. and

I. Jack,

Nuovo

(1983)

Cimento

83; ibid.

A104

B236

(1991)

(1984)

1817.

221;

ibid.

B249

H. Osborn,

Phys. and

Lett.

ibid.

B147

H. Osborn, and

C. Ford,

D.R.T.

Jones,

D.R.T. P. West,

Phys.

A. Parkes D.R.T.

and

and

M.E.Machacek

and

S.P. Martin [81] S. Willenbrock R.G. Stuart, 3193; A. Sirlin, [82] R. Tarrach, Tarasov,

Lett.

M.T.

Nucl.

2027;

Prog.

Theor.

Lett.

B136

B138

Phys.

(1984)

Lett

67 (1982)

(1984)

221;

Phys.

B237

(1984)

307;

and M. Quiros, (1986)

247. J.E. 73; B160

Vaughn,

Phys.

Rev.

D50

Vladimirov

Kataev,

Phys.

Lett.

(1991)

67 (1991)

B183

(1993)

1889; ibid.

17.

68 (1982)

92i:

293;

Nucl.

B262

B395

99;

Savoy, Cl0

Phys.

242;

(1984)

Lett.

A.L.

Phys.

B236

(1985)

3291; S.G. Gorishny,

Nucl.

(1984)

B138

Phys.

A.L. Kataev

(1992);

Einhorn,

Phys.

B156

S.G. Gorishny,

472;

Nucl.

Lett.

Phys.

(1985)

Parkes,

Lett

A.A.

(1983)

371;

Lett.

G. Valencia,

Rev.

Rev. D28

Vaughn,

C.A.

A.J.

and

Phys.

(1984)

f. Phys.

Phys.

385; Phys.

No. NSF-ITP-92-21 and M.B.

Phys.

J. Perez-Mercader

and and

B249

Stephenson

Phys.

M.T.

Phys.

Jones

(1982)

1101;

Rep.

S. Takeshita,

B137

Zeitschr.

(1985) 533; A.J. Parkes, D.R.T.

and

and

J. Leon,

N.K. Falk,

Phys.

L. Mezinescu,

Derendinger

B. Gato,

(1983)

Jones,

P.W.

P. West,

Jones

B119

405;

L.Mezincescu,

Lett.

and

Al6

D.R.T.

A. Kakuto

Jones

Lett.

(1984)

Nucl.

I. Jack

[80] K. Inoue,

Phys.

(1983) 1083;

C. Ford,

[83] O.V.

(1982)

(1982)

Phys.

H. Fiirstenau,

Vaughn,

1681;

150;

70;

M. Fischler

J.P.

Nucl.

M. Luo, Phys.

451

(1981)

(1981)

Rev. D24 B105

D.V. Nanopoulos,

L. Cifarelli,

33 (1974)

Rev. D24

B193

and Supersymmetry,

and

and

438;

Lett.

153;

Lett.

Jones,

3081.

513; Phys.

Phys.

Unification

Phys.

(1981)

Unified Theories,

Grand

P. Langacker

[79] M.E.

Ross,

Rev.

Phys.

(1981)

(1995)

32 (1974)

Phys.

Nucl.

Cl1

D52

Lett.

F. Wilczek,

B188

D.R.T.

Mohapatra,

[78] U. Amaldi,

Rev.

and G. Senjanovic,

Einhorn

R.N.

and

f. Phys.

G.G.

Rev.

S. Weinberg,

H. Georgi,

Phys.

and

[77] G.G. Ross,

Phys.

Phys.

and

S. Raby

S. Dimopoulos,

W.J.

N. Polonsky,

2127;

(1981) and

B259

113; ibid. Phys.

Nucl.

Phys.

Bjijrkman

B253

and D.R.T.

285; Jones,

Nucl.

Phys.

B259

(1985)267; (1994)

2282.

(1991) B272 Lett.

373; (1991)

B267

353; (1991)

Phys.

Rev.

Lett.

70 (1993)

240.

384. A.Y Zarkov,

and S.A. Larin, S.A. Larin

and

Phys.Lett.

93B

Yad. Fiz. 40 (1984) L.R. Surguladze.

(1980) 429;

517 [Sov. J. Nucl. Phys. Mod.

Phys.

Lett.

40 (1984)

A5 (1990)

2703.

89

Top Quark and Higgs Boson Masses [84] N. Gray,

D.J.

Broadhurst,

[85] R. Barbieri,

W. Grafe

M.Beccaria,

[86] A.I. Bochkarev

and

P. Ciafaloni,

R.S. Willey,

[87] R. Hempfling

and

[88] A. Sirlin

R. Zucchini,

and

B.A.

Kniehl,

and

[91] A.A.

Vladimirov,

I. Jack

and

[92] B.W.

Lee, C. Quigg

[93] W. Marciano, [94) M. Liischer [95] U. Nierste

P. Weisz,

and

Tarasov,

(1983)

(1993)

105.

1386

Oscillations,

Systems

Dynamical Verlag

and Bifur-

(1983).

Sov. Phys.

Rev. Dl6

B212

(1977)

Phys. (1988)

Technical

JETP

50 (1979)

521;

1101.

Phys.

Lett.

K. Riesselmann,

B409

2049.

Vol 42, Springer

S. Willenbrock,

Phys.

Phys.

389.

Non-linear

O.V.

Al6

and

Nucl.

673

580.

H. Thacker,

G. Valencia and

and

(1986)

Sciences,

(1984)

J. Phys.

and

B266

Math.

D.I. Kazakov

H. Osborn,

Phys.

C48 (1990)

Phys.

.4. Vicere.

Rev D.51 (1995)

P. Holmes,

Rev. D29

and

Zeitschr.

D51 (1995)

Rev.

Phys.

cations of Vector Fields, Applg. Phys.

I<. Schilcher,

G. Curci

Phys.

Nucl.

[89] see e.g. J. Guckenheimer

[90] W. Marciano,

and

Univ.

1519.

Rev. D40

(1989)

1725

472.

Munich

preprint

TUM-T31-90/95

(1995).

hep-

ph/9511407. [96] S. Cortese.

E. Pallante

[97] J. Frijhlich.

Nucl.

M. Aizenmann, J. Glimm (981 W.A.

and

[loll

D. Callaway

Rev. Lett.

A. Jaffe,

Bardeen

and

R. Petronzio,

Ann.

Lett.

B301

Henri

Phys.

and

R. Petronzio.

Poincare

Rev.

D28

D Weingarten, Nucl.

22 (1975) (1983) Phys.

b240 (1984)

Phys.

and

P. Weisz,

Nucl.

Phys.

290

(1987)

25;

and

P. Weisz,

Nucl.

Phys.

295

(1988)

65;

M. Liischer

and

P. Weisz,

Nucl.

Phys.

300

(1988)

353:

M. Liischer

and

P. Weisz.

Nucl.

Phys.

B318

L. Lin and Y. Shen,

(1989)

K. Jansen,

Rev.

Lett.

C.B. Lang,

B113

(1982)

481.

577.

705.

61 (1988)

678;

T. Neuhaus

and H. Yoneyama.

Nucl.

Phys.

B317

81;

G. Bhanot

and

G. Bhanot,

K. Bitar,

Phys.

Phys.

J. Jersak,

(1989)

97.

Lett.

M. Liischer

A. Hasenfratz,

203.

1372.

M. Liischer

(1021 J. Kuti,

(1993)

47 (1981)l;

Inst.

M. Moshe,

P. Smolensky and

Phys.

BlOO (1982) 281;

Phys. Phys.

1991 B. Freedman, [loo]

and

B375

U.M Heller,

K. Bitar,

(1992)

Phys.

U.M. Heller

Rev.

Lett.

61 (1988)

and H. Neuberger,

798; Nucl.

Phys.

B353

(1991)

551 (erratum

Nucl.

503;

H. Neuberger

and P. Vranas.

Nucl.

Phys.

B399

(1993)

271; Phys.

Lett.

B238

(1992)

335; M. Giickeler, U.M. Heller. [103] for a recent

H.A. Kastrup, M. Klomfass, review

T. Neuhaus H. Neuberger

see e.g. U.M.

Heller,

and F. Zimmermann, and P. Vranas, Nucl. Nucl.

Phys.

B (Proc.

Nucl. Phys.

Phys. B405

Suppl.)

B404 (1993) (1993) 555.

(1994)

101.

517;

90

B. Schrempp

(1041 L. Lin, I. Montvay, C. Frick, B397

(1993)

1. Lin and [105] [106]

431; Nucl.

H. Wittig,

L. Lin, I. Montvay, A. Borrelli,

Goltermann

[107]

W.A.

Bardeen,

[108]

M. Lindner,

[log]

B. Schrempp

[llO]

W. Zimmermann, R. Oehme R. Oehme,

[113] J. Kubo,

M. Plagge

G.C.

Rossi, Petcher,

Hill and

of Mod.

F. Schrempp Comm.

K. Sibold Lett.

B317

(1993)

143.

R. Sisto and

M. Testa,

Phys.

Lett.

B221

(1989)

360;

Phys.

Lett.

Phys.

Suppl.

Math.

(1990)

1647. therein.

(1991)

211; Phys.

86 (1985)

97 (1985)

569.

215.

Phys.

Lett.

B153

(1985)

142

472 249.

[115]

W. Zimmermann,

Phys.

Lett.

B308

(1993)

117.

[116] W. Zimmermann,

Lett.

Math.

Phys.

30 (1994)

61.

unpublished.

D. Kapetanakis,

J. Kubo,

159.

2176 and references

97 (1985)

Comm.

(1993)

[119]

(1989)

Rev. D41

A8 (1993)

B311

J. Kubo,M.

B225

Phys.

Lett.

M. Mondragon

Nucl.

Lett.

Phys.

[118]

H. Wittig,

647;

Phys.

[114] W. Zimmermann,

(1171 B. Schrempp

30 (1993)

165;

and

H. Wittig,

W. Zimmermann,

B262

(1985)

(unpublished).

Phys.

and

B355

and

Phys.

Math.

Theor.

Phys.

T. Trappenberg

331.

M. Lindner,

W. Zimmermann,

Phys.

Suppl.)

G. Miinster,

Int. Jounal

Progr.

(1121 R. Oehme,

B (Proc.

D.N.

Nucl.

M. Plagge,

C 54 (1992)

C.T.

and

H. Wittig,

Z. Phys.

and

and

and

G. Miinster,

Phys.

L. Maiani,

M.F.

[ill]

G. Miinster

L. Lin, I. Montvay,

and M. Wimmer. .

M. Mondragon and

and G. Zoupanos,

G. Zoupanos,

Mondragon, M. Mondragon

Nucl.

N.D. Tracas and

Phys.

and

G. Zoupanos,

Zeitschr.

B (Proc.

G. Zoupanos, Nucl.

Phys.

f. Phys.

Suppl.)

37C

Phys.

Lett.

B424

(1994)

C60

(1993)

(1995) B342 291.

181;

98; (1995)

155.

Phys