Electroweak one-loop corrections to the decay of the neutral vector boson

ELECTROWEAK

ONE-LOOP

CORRECTIONS

OF THE NEUTRAL

TO THE DECAY

BOSON

VECTOR

1. Introduction In the near provide

future.

an opportunity

theory

to an accuracy

is the creation detailed

study

of e +em accelerators.

a new generation of testing previously

of fermion

SLC

the Glssho\v-Weinberg-Salam unattainable.

pairs around

of the most frequent

Among

the Z-boson

decay branches

and LEP.

(GWS)

or standard

the most interesting pole. This

\vill

reactions

process allo~vs the

of the \\.eak neutral

gauge boson

z: Z+e’e

If the t-quark

Z + dd. ss.bb.

(4)

mass HI, is not

better

determination

too

large (01, -c IA{,).

study

of the coupling so this theory

(1)

(3)

Z + tt. The experimental theory,

.r’r_.

z +uu.cc.

euista: standard

./I&/I

of decays-( l)-(4) constants

may be verified

yet another will

of the \veak including

decay

channel

allow

a lo-30

times

neutral

current

in the

one-loop

effects.

We will

F-ig. I

Born and one-loop electroaeak contributions dGl\

not discuss

here the more fundamental

in the unitary gauge to the partial \vidth fy z -

of the

f, f,.

problems.

such as those connected

with the

number of fermion generations or the existence of nonstandard contributions from supersymmetric particles or compositeness, and others on which Z-decays may shed some light too. Radiative corrections been

studied

to the partial

in the standard

tunately. the calculational scheme Trieste Conference on electroweak malization satisfying

widths of the leptonic

theory

decays (1) and (2) have

some time ago by Consoli

et al. [l]. Unfor-

they used is somewhat complicated. After the radiative corrections in 1983 [2], Sirlin’s renor-

scheme [3] became accepted by the physical community in many respects. It is characterized by renormalization

as the most on-mass-shell

(renormalization with the use of (Y.Mz. Mw, M,. m, as parameters) and a certain choice of parameters in actual calculations: (Y.G, (the Fermi constant in muon decay). Mz. M,, m,. As has been show in refs. [4-71, the complete electroweak one-loop

corrections

to a variety

may be taken into account ~(4’). ~(4’) if the condition

of neutrino-induced

through ~;/9’<

processes

the introduction 1 is valid (q2

(ve-, vN-scattering)

of only two form factors is the squared momentum

transfer). Here p and K have a simple physical interpretation: PC,, = Gkfff(ql) is the effective Fermi constant, and K sin%, = sir&$( 9’) is the effective mixing parameter for the given process with neutral current exchange. Here and in the following \ve use

An immediate consequence of the developed framework is the possibility ing in terms of a “corrected Born amplitude”. Furthermore. it became

of thinkrelatively

easy to compare in detail the results of different authors, which is often no simpie task (e.g. for the above-mentioned Ye-scattering of [6] and [7] this has been done successfully). In this article. the approach developed in [4-71 will be used to discuss the electroweak one-loop corrections of fig. 1 to the partial decay widths of the Z-boson for the processes (l)-(4) in the framework of the standard theory*. For each of the

decay

channels

the corresponding

are determined. calculational t-quark

The article scheme

and

mass dependence

discussed expressions

electroweak

is organized general

formulae

is analyzed.

as a function

p( - M.j) and IC(- Mf )

form factors

as follows:

sect. 2 contains

definitions.

used in the following.

Numerical

of the Higgs boson

results are presented

mass

h,,

the

In sect. 3 the in sect. 4 and

and of w,.

Some explicit

are given in the appendix.

2. Amplitudes, partial widths. renormalization As has been Z ---) f,f,

may

introduction

shown

in [9]. the one-loop

be Lvritten

in strong

of two constant

corrected

resemblance

form factors

3,,.

matrix to the

sJ,

element

Born

for the decay

approximation

by

(eqs. (B.9) and (B.lO) of ref.

]91):

(6)

and Q, is the charge of fermion f, (Q, = - 1). The Born amplitude .+,, =.F?, = 1. The partial decay width derived from (6) is

corresponds

to

where K, = Re( .F?!/$t,), and c, is the color factor: c, = l(3) for leptons(quarks). In the calculational scheme of Sirlin [3]. as independent input parameters of the GWS theory, one uses (Y. Mz. M,.. M, and the fermion masses PPZ~(on-mass-shell renormalization).

So, the partial

ividths

(8) are well-defined

terms of (5) and (7). But. a further ingredient of the approach actual calculations of the Fermi constant from muon decay. GU=(1.16634f0.00002)~10~‘GeV~~~, instead of the W-boson mass M,. One immediate advantage the knouledge of G, to a high precision. Furthermore. using

---E-

R?

sA4;.

expressions chosen

if read in is the use in

(9) of this choice lies in

-iTa

2s3.M;.

= ,/fGJl

+ O( (Y)] ,

(10)

the Born amplitudes are scaled by r f G, instead of the normalization in (8). This in fact is a finite renormalization of the Born term of the order of one-loop corrections.

4. 4. .-Iklllrml~wL’,‘Il. ;/ t:leaw~wlX otlt’~1i,o[~ c,W)‘l?~l,~,,li

-l

has been extensively

As

discussed

the constant

electroweak

consideration:

among

tion

in the literature

corrections

cited, this takes away that part of

not directly

them large contributions

connected

with the process

of fermions

to the vacuum

under

polariza-

- IQ f ln( 01f/M;!, ).

The necessary formulae

calculation

of 121, has been done

iteratively

using

the following

[lo]*:

(11)

A = s\vM,

=

4, (1 - J#’

(14

’

= (37.2810 _+0.0003)

Here.

Jr

is a calculable

GeV.

(13)

from the muon decay expression:

Jr=

5X( M,.

n4z. M,.

17Zf),

04)

where X may be taken from eq. (E.8) of [9]**. Correction terms arising from a nonzero t-quark mass (vanishing for ttz, = 0) may be found in [7]: see also this appendix. Inserting (12) and (13) into (10). the above-mentioned finite renormalization of the original Born widths of (8), the partial widths r to be used in the following are obtained:

(15) where

the electroweak

corrections

of fig. 1 are contained

in (16) (17)

l

* Concerning the calculation of M,. from a. .%I,. M, and wt. we would like to remark that we got an impressive agreement of our result5 already uwd in [ll] with those of [12] Lvhere in table ?. the .&I, ho.\ hccn given with a four-digits precision. * The hadronic vacuum polarization influence> the results of thib article only through the calculation of into hadrons .II,, Differing from [Y]. here we take it from the cros\ section of c + e annihilation [3,13,1’].

-I. 4. -lXlrlrtl‘lor’ er trl. / I~kv l,.lllwlrl\ 011d1~01) U,)‘)‘6’C’l,,,,,Y

The .F,,?,

differ from S, ?, of [9] by the exclusion

together

with

analogue

real and virtual

partial

where

lvidths

bremsstrahiung containing

0.170;, whereas

bremsstrahlung

all one-loop

D, = 0( 1) for lepton(quark)

exceed

of some pure QED-terms

lead to the well-known

gluon

production.

correction

factor

Lvhich

[14]. The

also add up [14]. thus yielding

corrections

the QCD-correction

5

of the standard

The QED-correction may be estimated

the

theory.

in (1X) does not

to be about

4Y [15].

Of course. one could include both of them in the definition of p,: being interested in the pure electroweak corrections of fig. 1. we \vill not do so. The explicit expressions for p, and K, are given in the appendix.

3. Influence of the t-quark mass In the standard model there are two mass parameters not yet fixed - ))I, and M,, - so they should be varied in estimating one-loop effects. The influence of the

Higgs mass may easily be derived

from ref. [9] so we do not discuss here any details.

The existing experimental findings on the t-quark mass. /vt = (40 f 10) GeV [8], are rather preliminary. Nevertheless. they show the necessity of taking into account ITI, in precision calculations. In the follo\ving. we will take for granted 111t 1 30 GeV. The t-quark mass sho\vs up in several ways: (i) The calculation of hl, from the measured parameters (Y,GP. Mz through ( 11) and ( 14) depends on the fermion masses including WI,. (ii) The definitions of X. S,,. SF?, or. after the finite renormalization. of the form factors p,. ti, explicitly depend on 111,. From fermion integrals. decays

the technical masses which into

point

of view, all the quantities

through self-energy are easily calculated

down-type

quarks.

M,:. X..

. . , K,

depend

on

diagrams (via the counterterm). or l-fold for nonvanishing /)z,. In case of Z-boson

171, leads to additional

corrections

resulting

from

charged current loops in the diagrams of fig. 1. They necessitate the calculation of I-fold integrals tvith three different nonvanishing masses Mz. M,v, I?!~. These vertex corrections are proportional to 1V,q1’. Lvhere Liq is the Kobayashi-Maskalva matrix element

for t-q

transitions.

The t-quark

has small

mixing

with the light quarks:

0 d 1P;,( G 0.024. 0.036 < 1C;>l < 0.069. 0.997 < 1I,‘,hl G 0.999 [16]. Thus. one ma) expect numerical nz,-dependent vertex corrections only for the decay Z -hb. Formulae for flavor-changing Z-boson decays have been given in [17] and in the references cited therein. Those results. being obtained in the ‘t Hooft-Feynman gauge. are applicable in the present context too. since the piece containing the 111, dependence. say P’(ttz,) - I’(0). is gauge-in\,ariant. Nevertheless. we independently recalculated these terms. including the corresponding counterterms [ 1X] in the unitary gauge used here. and got excellent numerical agreement Lvith the analytical

result leading

of [17]. The exact expressions

for p, and

K, are given in the appendix.

The

terms in the limit nzf > Mi? are: apt = 6p, + d, * 6pY .

i?K:

=

8K,

+

iw

d, . 8Kj:.

(20)

(21)

where d, = 1 for down-type quarks and d, = 0 for all other fermions. (19)--(22) remain true for a fourth-generation sequential quark doublet top-bottom

Formulae with large

mass splitting. 4. Results

For the Z-boson and K, have been four-digits partial

decays (1) and (3)-(4). the one-loop exhibited in tables 1. 2 as functions

precision.

widths

In table

3. the percentage

electroweak form factors p, of Mz, M, and t7zt with

corrections

are presented

~,,, = Y” - 116 I p 10 arhere r,Sp = ly(

neutral

to the

$T.

neutrinos

p, =

K, =

(23)

1) is the Born approximation.

For decays into electrically

K,,IQ~,\ = 0 so that p,, = 1 + 8,y may be taken from table 3.

The form factor p, contains the difference between the coupling constant acting in the decay Z + f,f, and the Fermi constant GP. whereas K, measures the deviation of from the mixing parameter sin%,, which is fixed in our scheme sin%$, = K,sin%, by the gauge boson masses. The existence of and only of p and K has a far-reaching consequence - experiments may be analyzed in terms of the usual coupling conbeing corrected for electroweak stants. e.g. vector and axial vector couplings. one-loop effects in a simple manner: 1', =p

;,,“(l - ~s;~~Q,~K,)s,,

a, = p),’ ‘s,

.

(24)

s, = +( -) for up(down)-type fermions. This explicitly shows that data on Z-boson decays may be conveniently described in terms of “Born-like” expressions forgetting the complicated details of one-loop calculations.

where

~0.3Xl O.?Y.l 0,407 1.183 1.164 1.137 1.130 1.114

- O.lflK ~0.16R ~ ij.167 - 0.165 - 0.161 1.375 1.37-l 1.373 1.377 1.381

Yl1

O.‘dS

~ O.lOh

Y2 Y4 Yti YS ‘)t)

~ 0.301 ~~0.31?. ~ 0.373 0.331

0.350 ~ 0.366

.?I 1

-.‘30

1.30

0.103 - 0.0’)‘) - O.OY4 o.oxx 1..I42

9’

1.135

1.444

44 Yh YS

1.710 1.706 1.192

144s

I .-I53 1.45Y

0.7Y3 ~ O.‘M ~ 0.175 ~ 0.355 ~~0.35 1.245 1.251 1.759 1.267 1.276

0.123 0.13Y 0.155 0.171 (I.IXX 7.673 5.3Sl Y.376 10.715 11.153

~0.14) I).136 m~o.131 1).12k ~ 0.1’3 7.163 8.037 s.x.w Y.6YY 10.5Y7

-

MIX1 0.096 0.1 11 0.127 0.143 7.631 x.437 Y.31 10.171) 11.107

~0.1s~ ~0.171) ~0.176 -0 177 - 0.167 7770 7.YY3 S.XCl3 Y.654 10.551

~ O.YO7 0 Y-x O.YS5 ~~1.W.: ~ 1 .(rho

O.Shj

- O.YO3

-O.YJl - O.Y7N - 1.015 6.330 h.YlS 7.637 S.3Yl Y.lX?

h.lS7

6.874 7 5Y7 S.316 Y 137

We would like to begin the discussion of numerical results \vith a rough cstimats of the anticipated experimental accuracy. Based on the large statistics of Z-boson physics at resonances of about 1 million Z-decays per experiment, one may expect kvidth measurements with an accuracy reaching or exceeding the 1Y level. For &b-production. e.g. the corresponding tagging efficiency is about 10’ &b-pairs [ 151. So. it seems worthwhile trying a more detailed discussion of the tables. We remark that for the leptonic channels (1.2) our results coincide aith those of [l] within the precision of their figures. To start with. for small ~1, ( - 30 GeV) we get for all channels: Ip - 11 -< 0.5%.

Ih.- 11 < l.lT,,

16’” 1 < 0.5%.

implying that a simple interpretation of data in terms of a Born approximation using GL, as coupling constant and sin%, from (5) is a rather good approximation. Of course. this may be traced back to the calculational scheme chosen. as explained in hect. 2.

The t-quark mass (or. analogously. a fourth-generation hlrt with large mass splitting) has only a small influence leads to 0 Q i -\p, 5 1.5%; the minus

sign to be taken

sequential fermion douon p. A JM, of 200 GeV for b-quark

production

as a

consequenct3 of the dominating vertex corrections in accordance with eq. (22). Stated another \vay. 1.1% < (p - 11 < 1.6? for w, = 230 GeV in all channels. For the Lvidth corrections ~3,“’ the change Lvith /lzr is more channel-dependent: iSy’(

(0.8Y.

(AS,:w( 5 1.5?,

jJ8;;L,I 5 2.5%..

I-\S’,” 1 5 3.45.

The moht sensitive lvith respect to the 171, quantity is K. In the large IPI, limit con>idered. K also becomes large: 62 < (K - 1) -< 12.4”;! As a consequence of 1~ ,. a difference arizrb between the b-quark and other down-quark channels (see eq. (22)) \\hich already influenced the discussion. For /H, = 230 GeV the follo\ving values may

be

quoted: (PI,-p,,)-

-?.5$,.

(K,,-K~.J)-

1.27.

(6yiy)-

-3%.

111,

\I,,

= 10

\I,,

31

=

=

III0

\I,,

IlKIll

=

\I,,

III

=

Ii.04 Ib.(iS oti7 -I)06 0.05 -I).!7 0.22

-0.11 l).Oh O.ill 0 07

I

0.1

t1.13 tr.16 I1.b II.OY O.(lY 0.10 Il.1 I 0.1 I ~I,01 0.01 ~ Il.01 0 III) 0 (UJ

tabulated

All

value5

show

some

dependence

AI,-drpendencr

will

not be so important

bince then

will

be known

more

interesting

estimate detail

51,

in [l I]

the variations 100 GeV

Tlwse most wzm~

since

of the Higgs

in a related bring

to 1000

numbers sensiti\v

one boson

found

GeV

with

could

context

hope

to

(combined

and

accuracy. derive

The

from

fit For

to AI,

‘44,.

cariation

radiative

detailed are done,

b.ith

M,,

corrections

and discussed

and sin%,v

111, = 30 GeV.

The

experiments

is an

in more

). Unfortunately.

a change of ,W,,

from

in:

bhow that it would

to he rsachable.

41,

has been proposed

here arc small.

results

parameter)

excellent

This

II~S.

on

by the time Z-decay

which

be necessary

is clearly

below

to reach a sensitivity the percentage

level.

to K (as the This

hardly

From

our numerical

analysis

one may conclude

that perhaps

is (for all channels) the co-factor quantity out of p, K. 6” percentage corrections and sensitivity to the Higgs and fermion which scales the effective accuracy lvidths

weak mixing

if one is interested

parameter.

in a precise

the most interesting with

K

one has to determine

determination

its largest

masses. Since it is k

of sin%,.

it with high

from the partial

of the Z-boson.

To conclude.

we determined

the electroweak

one-loop

corrections

Lvidths I’(Z + f,f,) to be of the order of at most a few percent

to the partial

in the scheme

precision experiments, and in dependence of M,, Mu. ~TI,. For the anticipated should be taken into account numerically in the analysis of data. We would for interest

like to thank

Profs. G.B. Abdullaev,

F. Kaschluhn.

used they

and D.V. Shirkov

in our work and support.

Appendix Using the expressions of ref. [9] for X. T,,. and (17) the following expressions for p, and a

“=‘+

477(1-R)\

,F?., one gets according to eqs. (16) in the approximation wf +z 12fG:

K,

1 ‘Z(-l)+Z,(-l)-Ct’(O)+~R(l+R)-5

_

9R 4(1 -R)

K’=‘-4vr(l-R)\(1-R)

~

In R + II, I I

(A.11

[ CV( - 1) - Z( - l,] - M( - 1) + $I,

_ (1 -R)’ R

Q,? [ v, ( - AI;. h;

\ j + +] :, 1

(.4.2)

11,=;X[I-6,~),1(1-~)+1Zg”(l-~)~][I:(-M~.Ml)+i]

+[1-2R-21Q,1(l-R)][C.~,(-MZ..iz~~~)+t] +3R[&(

-M;,

M;..

M;,)

+ t].

(A.3)

Further definitions used are taken from refs. [9,7]. Corrections due to nonvanishing ttt, deri1.e from (A.I)-(.4.3) by the replacement of If,‘(O) through [C+‘(O)+ t+~‘(O)]

7

C$V’

t+J) Inr;.i=6&[I,(q’.

(I-

U

M,‘,‘.(-1) Z[(-I)=

z;-( -1)

ir-r’+(l

=Inr-

-r’)lnll

-$8(1-R)+ x [/J

- .v;,

+ irl,,(

-iv;.

= - ir[l

X

-lM;;y(

!In(lR)+rR+ [-

I?( y2.0.0)]+ 3//,( ‘71.mf.0). --

(A.4) (A.5)

‘I.

‘;(l-R)y ttzf. rn;) - 1J -nr;.o.o)] tt,;. ttt;, . tt~i.tlli)]+jljj[?-x(l-R)+‘;-(I-x)’]

+I;.

i

(‘4.6)

-&

+ _Ir - r?R

!

nf;;f(

-,w;.

ttz;. tttf

) . (A.7) I (.4.X)

‘~~~‘(-1)=[-4+~(1-R)][I,(-d~~,t,2~.t~*f)-l~(-M~.O.O)].

Here. R = M&/M; and r = tnf/.kd~,. Whereas Z. I$‘. M functions stem from the counterterms. functions C’, and C’, are deri\,ed from the vertex diagrams of fig. 1. Thus. when they contribute to S,’ they depend on ttz, or on tT,. the corresponding Lveak-isospin partner masses. Consequently. through b’,. 1,; the ttz, may influence only decays into down-type quarks. has to include the weight 1l,;,l’: q

4;.

mainly

in the b-quark

channel.

In that case. one

hqv: mf)

= $s’d>+

- 3r(l

-!.)]lnlr,I

+ 2.Rrlnlr,I

- Rt

+(2+R)[F,(+F,(O)]-(i+R)[F,(r)-F?(O)] +!Rr(Z+R)F,(r)-2R

1-R ‘G[l

+ Inlr,l

-4?,(Y)

1 =-I R o

Id,%

{

++R)[F,(+F30)]

+r[2RInlr,I

+ i(l-2R)(lnlr,I xF,(r)+

-l)+

i(r-2Rr-4R-4)

:(I -r+7R+IRr)F,(r)]).

(.4.10)

Here. rR r, =

-

J(l

-v)

tj=_r+r(l

F,(r) =/,(l.

f‘,(a_h)=

u -

r,=r-~(1

1.

-J)R~‘.

/j = 1 - .I*(1 -

-.I,).

r).

_I’

)R

F,(r)=f,(r.l).

l h - ,,R

’

i= 1.2.

u-~‘(1

-~a)R-'

CL’+

h( 1 -_r.)

In

’.

.

1 -j’ f,(u.

h) =

u-h-

\qR ’

-(h+>.‘R

‘)fl(o.b).

As explained in refs. [17.18], the vertex counterterm contains a flavour-changing piece whose nl, dependence yields one further correction. with weight n/(2astv): r-2

a~:,_,=

The

-2&L,_,=

complete

integration.

vertex

-,!(l

+2R)&[j(5r-

correction

Il)+3~rIn~~~~
in analytical

form.

not leaving

(A.11)

one numerical

may be taken from ref. [17] to replace (A.9), (A.lO) and (A.ll):

where I’( IIZ~) is the function as defined the same meaning as in eq. (24).

in eq. (3.3) of [17] for 4’ = - Mg and s, has

-I. 1. -~~\/IwI&J~~et II/. ,i/ l:le rrswctr~ 011t~-ii~o~~ uwrccr~or~s [I I]

.A..A. Akhundcw. (IYX6)

[I?)

13.b.

D.Yu.

Bxdin

Riemann.

JINR-preprint

EI-X5-454

(19X5):

Phyh. Lctt.

lh6B

111 L>nn

and R.<;. Stuart.

I ?] E..4 Pa>chcw. Nucl.Phvs. 131 I). Alhcrt. 1151 .I M

and T

I?

u’...

Mar&o

Nucl.

Ph>s. B753 (IYXS)

B15Y (1970)

and D. Wylcr.

Nucl.

Dc>rfan.

Iccturc

prtxnted

at the Thwretical

Ann

Arbor.

Michigan.

lYX4:

ph\>io. ‘161 Rc\ic\\ [17]


1IS]

I> 1.~1.

of Particle

Mann

Properties.

and T. Rirmann.

Bardin.

P.Ch

Chribtnu

da

Ph!>. Blh6

Ph\z.

t IWO) -IhO

Advanced

SLAC-Pub-3407

Rev. hlod.

Ann.

‘16

2X5 Stud!

Institute

on Elcmrntay

(lYX4)

Phyh. 56 (19X4) 40 (19X3) 334

and 0 h,i. Fcdnrenko.

JINR-preprint

P7-X7-X40

(19831

particls