Reduction of couplings in the two Higgs doublet extension of the electroweak standard model

Nuclear Physics B347 (1990) 184—202 North-Holland

REDUCTION OF COUPLINGS IN THE TWO HIGGS DOUBLET EXTENSION OF THE ELECTROWEAK STANDARD MODEL Ansgar DENNER* Max-Planck-Institut für Physik und Astrophysik, Werner-Heisenberg-Institut für Physik, P.O. Box 40 12 12, Munich, ERG Received 17 April 1990 (Revised 29 June 1990)

The method of reduction of couplings is applied to the two Higgs doublet extension of the standard electroweak model. From the solutions of the reduction equations predictions for the top quark and Higgs boson masses are obtained.

1. Introduction In recent papers [1,21 the masses of the top quark and the Higgs boson were predicted by applying the reduction method [3] to the Minimal Standard Model (MSM). This method can be considered as an appropriate procedure to solve the Renormalization Group Equations (RGE) taking into account their singularity structure and the hierarchy of gauge couplings in the standard model. The RGE are integrated in two steps. First the electroweak couplings are turned off. By introducing the effective strong coupling as the new variable instead of the scaling parameter in the RGE one obtains the reduction equations. These are solved under the additional restriction that all couplings vanish together with the strong gauge coupling in the weak coupling limit. This is equivalent to extending the requirement of (ultra-violet) asymptotic freedom from the strong gauge coupling to the Yukawa and Higgs couplings. Special solutions of the reduction equations are constructed corresponding to perturbative solutions (i.e. power series solutions in the strong gauge coupling constant) and bounds on the general solutions are derived. To one-loop order those special solutions are just the constant solutions of the reduction equations. Next the stability properties of the special solutions are studied. From this one obtains the asymptotic behaviour of the actual general solutions in the weak coupling limit. In the second step electroweak corrections to the reduction solutions are computed by constructing solutions to the full set of RGE as power series *

Present address: Physikalisches Institut der Universität Wiirzburg, Am Hubland, 8700 Wbrzburg, FRG.

0550-3213/90/$03.50 © 1990

—

Elsevier Science Publishers B.V. (North-Holland)

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expansions in the electroweak couplings. They are fixed by the requirement that their asymptotic behaviour in the infrared equals that of the undisturbed solutions (i.e. solutions of the corresponding reduction equations with vanishing electroweak couplings). In this paper this program is carried through for the simplest extension of MSM, namely the Standard Model with two Higgs doublets. From the solutions of the corresponding reduction equations and using the Fermi constant and the mass of the bottom quark as input we obtain predictions for the top quark mass and for the masses of the four Higgs particles of the two doublet model. These predictions turn out to be in conflict with recent experimental data [41.Thus assuming the reduction principle the two Higgs doublet extension of the electroweak standard model is ruled out. The paper is organized as follows: In sect. 2 the model is described and the RGE are given. The solutions of the corresponding reduction equations with vanishing electroweak couplings are investigated in sect. 3. In sect. 4 electroweak corrections to the solutions obtained in sect. 3 are calculated. In sect. 5 we summarize and discuss our results. —

—

2. The model We consider the gauge theory of the group SU(3) x SU(2) x U(1) with two Higgs doublets and three generations of fermions [5]. The most general renormalizable couplings between the fermions and the two Higgs doublets I~and 12 are given by ~Yuk

=

—

—

~

~

~

+

~

+

+

h.c.)

a~2)G~dR~ + h.c.)

+ h.c.).

(2.1) dR LL left-handedcharged quark and leptonupdoublets, respectively, and ~ and and UR QL are are the the right-handed leptons, quarks and down quarks. f= u, d, denote the (for simplicity) diagonal Yukawa couplings. To avoid flavour changing neutral currents we demand that no fermionic charge species couples to more than one Higgs doublet [6], i.e. the parameters aj~and a~which can take the values 0 or 1 fulfil a 1~~ 0. This can be enforced by imposing suitable discrete symmetries, namely +

—

1R’

~,

=

~1~~I’

~‘~2’

~2’

together with appropriate transformations for the fermion fields.

(2.2)

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Reduction of couplings

The most general, renormalizable, SU(2) x U(1) invariant Higgs potential, which is also invariant under the discrete symmetry (2.2) reads =

—1x~’P~1 _bL2~2t~b2 + ~ [~1(~~i1)~ +

+

A3(~ 1)(~4’~2) + A4(Il1~P2)(D2cP1) (2.3)

The requirement that the vacuum has to conserve electric charge is equivalent to A4 <0, the requirement that the potential is bounded from below implies the conditions A1 >0,

~

A2>0,

—A3

—

A4

+

1A51.

(2.4)

A5 can be chosen real and negative without loss of generality (i.e. the phase contained in A5 can be absorbed in the fields). Then the vacuum expectation values of ~D1and ‘2 have the form

~

(2.5)

K~2)_w(v2),

with real i~ and 1)2. There are four physical Higgs particles, two neutral scalars H12, a neutral pseudoscalar H~and a charged scalar H~.In the tree approximation their masses are 2 m~2

=

~(AiL~

+ A2L’~±~(A~v~

—

A2vfl

+

4(A 3

+

A4 + A5)2v~v~),

m~=—A5(v~+vfl, m~= —~(A4+A5)(D~+vfl.

(2.6)

In addition there are two mixing angles in the Higgs sector, tan/3=—,

tan2a=

2( A3 + A4 + A5) 2 2 A1v1 —A2v2

(2.7)

The fermion masses originate from the Yukawa couplings m7~ (1/V~)(a~v1+ a~v2)G~”~, =

(2.8)

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187

and the vacuum expectation values are related to the Fermi constant by 1 (2.9)

~GF

Denoting the SU(3), SU(2) and U(1) gauge couplings by g3, g2 and g1, the renormalization group equations read in the one-loop approximation [71 2~x----— 7g~’, dg1 d~

(2.10)

=

16~-

dg 2~x— 2 = 161T

d,u.

—

3g~,

(2.11)

dg 2~x—= 3 —7g~,

(2.12)

16ir 16~2~ dG~

=

G~(

—

(~g~+ ~g~)

=

G~( (8g~+ —

+

+

~(a~+ a~)G~’~2 + ~ {(a~+a~)G~~~2

+ 3(a~a~ + a~a~)G~2}),

+3(a~a~ +

16~2~ dG~ d~

+

~

+ ~g~)

+

+

(2.13)

a~)G~2

a~a~ 3a~a~3a~a~)G~2 + ~ {3(a~+ a~)G~2 —

—

I +

16~2~ dG~~ d~x =

(a~a~+ a~a~ )G~’~2 + 3(a?a~+

—

+

(8g~+ ~

+ ~g~)

+

~(a~

+

a~)G~2

~-(a~a~+ a~a~ 3a~’a~’3a~a~)G~2 + ~ {3(a~+ a~)G —

dA 2~t— 1

=

16ir

d,u

2

—

J

+

(2.14)

a~a~)G~2}),

(a~a~+ a~a~ )G~’~2 + 3(a~a~ + a~a~)G~2}),

12A~+ 4A~+ 4A 3A4 + 2A~+ 2A~ 3A1(3g~+g~) —

3~’~ (2.15)

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Reduction of couplings

2 + 3a~G~~1)2} +

+ 4A

1~

+

+

+ 3a~G~

—4 ~ {a~’G~’~ + 3a~G~4 + 3a~G~t04}, I

dA 2~t 2 16’~-

=

(2.16)

12A~+ 4A~+ 4A 3A4 + 2A~+ 2A~ 3A2(3g~+ g~) —

dp.

2 + 3a~G~2 + 3a~GJ~2} +

—4 ~ {a~GJ~4 + 3a~G~4 + 3a~G

—

16ir

=

+

(2.17)

4}, 3~1)

I

dA 2~x 3

+

4A2 ~ {a~G~

+

(A 1

d1x

+

A2)(6A3 + 2A4)

+ 4A~+

2A~+ 2A~ 3A3(3g~+ g~) —

2 + 3(a~+ a~)G~2 + 3(a~+ a~)G~2} +2A3E I {(a(+ a~)G~ +

+

—

—

12(a~a~ + a~a~) ~ {G~2G~2},

(2.18)

1

dA 2jx-—-— 25 —3A 4’~ 13g~+g~) 4 2(A 16i~- d~ 1 +A2)A4+ 4(2A3+A4)A4+ 8A =

+2A

2+ 3(a~+

+

3(a~+ a~)G~2}

4E )G~ j {(a(+ a~ +3g~g~+ 12(a~a~ + a~a~) ~ {G~1)2G~’)2},

(2.19)

I

dA

2/x~ã_~~_ 5 = A42(A ‘ IL 16ir 1 + A2) + 8A3 + 12A4 + 3(a~+

2

—

I 3(3g~+g~)+ 2~{(a~’+ a~)GJ”~

a~)G~2 + 3( a~+ a~)GJ~2}).

(2.20)

For convenience we introduce the following variables: u=g~/g2 3, =

G~’~/g~, p,

=

A1/g~,

2

2

v=g1/g3, (2.21)

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189

and use x as our independent variable instead of the scale IL Thus we arrive at the reduction equations du

—

14x— dx

=

14u

—

—

14x—

dv dx

=

14v

+ 141)2,

14x

dpV~ =

dx

(2.22)

6u2,

—

p~(14 —

(2.23)

—

~v

+

3(a(+ a~)p~ + 2 ~ {(a~+a~)~~/)

+3(a~a~ + a~a~)p~ + 3(a~a~ + a~a~)p~})~

(2.24)

d ~(d) —

14x~

(d)L 2

dx

—

—

°~

~v + 3(a~+ a~)Pt(d)

+ (a~a~ + a~a~ — 3a’~’a~ — 3a~a~)p~ + 2~{3(a~+ a~)p~ I

+ (a~a~+ a~a~)p~~ + 3(a~a~ + a~a~)p~}),

(2.25)

dp(h1) —

14x—~—dx

(u)l

2—

—

-~v+ 3(a~+ a~)p~~

du + a~a~ — 3a1a1

—

du 3a2a2)p~+ 2~{3(a~+ I

+ (a~a~+ a~a~ )p~ + 3(a~a~ +

—

dp1 dx

14x—

=

12p~+ 4p~+

4~O3~O4 +

(2.26)

2p~+ 2p~+ l4,o1

3p —

1(3u + v) +

~ {a~’p~ + 3a~’p~ + 3a~’p~} 3

2+ + ~u

+

~1)2

—

4~ {a~’p~”)2 + 3a~p~2 + 3a~p~2}, I

(2.27)

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—

14x~

=

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Reduction of couplings

12p~+ 4p~+ 4~3~4+ 2p~+ 2p~+ 14P2 3p 2(3u + v) + 4p2E {a~p + 3a~p~ + 3a~p~°} 2 +

—

14x~

=

—

4~ {a~p~2+ 3a~p~2 + 3a~p)~2}, (2.28)

+ ~uv + ~v

(P1 + p~)(6p~+ 2p~)+ 4p~+ 2p~+ 2p~+ l4p

3

+2p3~ {(a~’+aflp 2—~uv +

—

3p3(3u + v)

+ 3(a~+ a~)p~+ 3(a~’+ a~)p~}

~

~

(2.29)

+~u

dp 2(p~+p

4 —

14x—

dx

8f)52 + 14/34

—

2)p4 + 4(2p3 +p4) /74 +

=

E ((a(+

3p 4(3u + v)

a~’)p~ + 3(a~+ a~)p~+ 3( a~+ a~)p~}

+3uv + 12(a~a~ + a~a~) ~ {p~p~},

(2.30)

d p~ —14x——— =p~ 2(p1 +P2) + 8/33 + 12/74 + 14— 3(3u + u) dx +2~ {(a~+a~)p~ + 3(a~+ a~)p~+ 3(a~+ a~)~~}). (2.31)

The gauge couplings are fixed by their experimental values at the scale of the W-mass, 2 O~(M~) 0.233 ±0.005, (2.32) =

a~(Mw) 0.12 ±0.01, =

a(Mw)

=

sin

~g,

corresponding to x a U

2

a~s1n ~

=

=0.2794...,

=

0.12, a V

2

a~cos 0~

=0.08488....

(2.33)

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191

Using further GF= 11.66 TeV2,

m~,=4.S GeV

(2.34)

to fix the vacuum expectation values, all masses are determined once the couplings are given in terms of x, u and v.

3. Solution of the reduction equations for vanishing electroweak couplings In order to find solutions of the reduction equations (2.24)—(2.31) we first set the electroweak couplings U and v equal to zero. (The solutions of the corresponding simplified reduction equations are denoted by w.) In the weak coupling limit all other couplings are required to vanish together with the strong gauge coupling constant, G 1,A1—~0 forx=a.,—*0,

(3.1)

or equivalently all the w, are finite in this limit, wJ
forx—’O.

(3.2)

We determine all possible constant solutions of the reduction equations consistent with the positivity properties of the w1. Using the fact that all physical solutions respecting (3.2) have to approach these constant solutions in the limit x 0, we derive bounds on the general solutions. A stability analysis of these special constant solutions provides us with asymptotic expansions for the general solutions in the weak coupling limit and leads to predictions for the particle masses. —~

3.1. SPECIAL SOLUTIONS AND BOUNDS ON GENERAL SOLUTIONS

The reduction equations for the Yukawa couplings can be treated separately. Eq. (2.24) for the Higgs—lepton couplings leads to

—

dw~ 14x dx

=

w~(14 + 3(a~+a~)w~+ 2~{(a~+a~)~V)

+3(a~a~ + a~a~)w3c~ + 3(a~a~ + a~a~)wJ~}).

(3.3)

The positivity of the w~ and the af allows only one constant solution for the leptonic Yukawa couplings, namely w~’~0.

(3.4)

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Reduction of couplings

All general (non-constant) solutions of eq. (3.3) are either negative or diverge for x 0. Thus the reduction equations together with the asymptotic requirement (3.2) and the positivity of the w~1~uniquely determines the leptonic Yukawa couplings. Inserting ~ 0, the a’ drop out of the reduction equations and we are left with only two interesting choices for the a7 (q u, d), —~

=

=

case I:

a~=a~=1,

a~=a~=0,

casell:

a~=a~=1,

a~=a~=0.

(3.5)

All other relevant cases can be obtained by interchanging I~and ~ In case I ~2 decouples from the fermions and the resulting fixed point equations for the quark—Higgs couplings are identical to the ones obtained in the MSM. In ref. [1] it was shown that they have only two constant solutions which yield satisfactory agreement with phenomenology, (Ia):

w~

(Ib):

w~=0

W~h1)=

2

w~’~ = 0,

or

for all i,q.

(3.6)

The general solutions are bounded by (3.7) where the upper limit can only be reached for one of the w~ if all others vanish. In case lIthe resulting reduction equations read

—14x

—

14x

dx

=w~3w+w+6E{~’~}—2 I

dw”’~

dx

=

/

3w~+ w~+ 6 ~ {wJ~}—2

W~11)(

I

.

(3.8)

There are 64 constant solutions. The nonvanishing values for the w~ range between 1/23 and 2/9. These would lead to quark mass ratios less than 5 for quarks of equal charge. To achieve gross agreement with phenomenology we must therefore require w~ 0 for the first two generations. Thus we are left with 4 =

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193

constant solutions, (ha):

Wb~W~Wt~,

(lIb):

Wb=O,

w1=~,

(IIc):

Wb=~,

w~=0,

(lid):

Wh=Wt=O.

(3.9)

The general solutions are bounded by 0<

O
~WJL1)<+,

0< Lw~<~.

(3.10)

It was shown in ref. [1] that the influence of nonzero couplings of the light quarks on the top quark and Higgs couplings is negligible in the MSM. Since this is also expected here, we consider in particular the case where these couplings are set equal to zero. We then find in addition 0
(3.11)

We now turn to the Higgs self-couplings. In case I we start examining the reduction equation for w2 originating from eq. (2.28), 2 + 2w~+ 14w

dw2

—

14x—~-----= 12w~+ 2w~+ 2(w3 + w4)

2.

(3.12)

It is independent of the Yukawa couplings. Like in the case of the leptonic Yukawa couplings, the condition w2 ~ 0 [from eq. (2.4)] implies that w2~0

(3.13)

is the only (constant and general) solution of (3.12) which is finite for x addition eq. (3.12) uniquely determines the further Higgs couplings w3=w4=w5~0.

—~

0. In

(3.14)

Taking this into account eq. (2.27) gives 2+ W~}, —

14x~

=

12w~+ (14

+

12E (w~+ w~))w1

—

12~{w~

(3.15)

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which coincides with the reduction equation for the Higgs self-coupling in the MSM [1]. Only one of its two constant solutions is positive and thus physically acceptable. With the Yukawa couplings of case Ta we obtain W~•=

—36+36\/~—0.034689... .

(3.16)

This constant solution provides an upper bound to the general solutions for the Higgs coupling w1 in case h. Inserting eqs. (3.13) and (3.14) into the reduction equations (2.29) and (2.30), we find an additional restriction for the Yukawa couplings, Ewi1~wi~=0,

(3.17)

not present in the MSM but fulfilled for the solutions Ia and lb. Case II is more involved. We study only the constant solutions of the reduction equations. They fulfill the following relations: 0

=

12w~+ 4w~+ 4w3w4 + 2w~+ 2w~+ 14w1

E

12w~

+

0

=

{WJ1~}

—

E {wI~’~}

12

l2w~+ 4w~+ 4w3w4 + 2w~+ 2w~+ 14w7 2}, 12w2E{wJ~} 12~{w~©

+

0

=

—

(w 1 +

0

=

+

2w~+ 2w~+ 14w3

—

+

6w4

+ 4w~+

E {wJ~+ w1~°~}12 E {w~wj°~}

6w3

2(w1 +

w2)(6w3 + 2w4)

w2)w4 + 4(2w3

+

w4)w4 + 8w~+ 14w4

E {w~+ wJ°~}+ 12 E {w)w~ht)},

0=ws[2(wi

+w2)

+8w3+ 12w4+ 14+6E(wJ~+w~}].

(3.18)

From the bounds on the Yukawa couplings (3.10) and w~,w2>0, the following

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195

properties of their solutions can be derived: 12~.w~’~~2 2 <—=0.0740...

0
14 + 12~1wJt1)

27

2 2 00740”’ 12~.w~~ 2< 14+12L3wj<27

0
12~.wc~wc~ 14

+

2 <—=0.0740...,

‘~

6~

1{wJ~ + w~}

27

—~w4
(3.19)

w5~0.

In particular w4 <0 implies that the solutions of the reduction equations lead to vacua which conserve the electric charge. Neglecting the Yukawa couplings of the quarks of the first two generations and using (3.11), stronger bounds follow: w1,w2<~4=0.0355...

w4~<~~=0.0292...

,

.

(3.20)

One can show that there exists only one constant solution of the equations (3.18) respecting (3.19). It can easily be found iteratively from 12~1{wI~} [12w~ + 2w~+ 2(w3 14 + 12~j{wI~}

+ w4)2]

—

1

=

2}

—

[12w~ + 2w~+ 2(w

12E1{wJ~ 2

—

w4=

+

w4)2j

14 + 12~1{w~}

—

—

3

3

12E1{wJ’~w~°}[(w1 + w2)(6w3 + 2w4) 14 + ~ + wI~} —

12~1{w~wj~} + 2w4[w1

+

w2

+

4w3

+

+

4w~+ 2w~]

2w4]

—

(3.21)

.

14

+

6~1(w~ + w~}

For the special Yukawa couplings of case lIa we thus obtain w1=w2=w3=0.028571...

,

w4= —0.028868...,

w5=0.

(3.22)

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Reduction of couplings

Case lib is equivalent to case Ia and case lIc follows from lIb/la by exchanging w1 and w2. Finally for case lid which is equivalent to case lb all the w1 turn out to be zero. This last constant solution of the reduction equations is called trivial reduction, whereas the solutions with a least one nonzero w~are called nontrivial reductions.

3.2. GENERAL ASYMPTOTIC SOLUTIONS AND MASS PREDICTIONS

In ref. [1] it has been demonstrated how asymptotic expansions solving the reduction equations can be obtained from the constant solutions. One has to calculate the stability matrix S,3 of these solutions. It’s elements are given by the derivatives of the right-hand sides of the ith reduction equation (2.24)—(2.31) with respect to the jth coupling divided by 14 (left-hand side of reduction equations). From the nonnegative eigenvalues of S.1 the general asymptotic solutions can be constructed as power series in xEj. Case la/lIb resembles closely the MSM. However, in contrast to the constant solutions the stability matrix depends on the a( and a~.For definiteness we take a(= I and a~=0 and case I. Different choices would lead to different eigenvalues however, with the same signs. Thus we get —

~,

~,,

We

W1,

W~

W,j

W~

W6

W~

W~

W~

Wj

W2

W4

W3

—c ~/w1

—d

W5

23

21 23 2! 23

21

0 0

s,)=

2

2

2

2

63

63

63

21

/w~

-/w~

-/W~

—5~W1

2 2!

-~w1

1 21

-/~w,

21 2 21

2 21

—~w~

—~W1

7

—a

—b

0

(3.23) with 41

—

‘/~

42

‘

21’

c=

251

+

252

d= ‘

67

+

84

.

(3.24)

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197

From the form of the stability matrix 5,3 we can directly read off the eigenvalues. The negative ones correspond to unstable, the positive ones to stable solutions. Hence the lepton, top and Higgs couplings are unstable; no general solutions are approaching their fixed points for x 0. Thus these couplings are uniquely determined. The other quark couplings are stable and there are general solutions —~

(3.25)

Wq~~CqX~’!

with arbitrary coefficients Cq approaching their corresponding fixed points Wq 0 for x 0. From the uniquely determined couplings we can deduce the corresponding masses and mixing angles using the tree level formulae (2.6)—(2.9), i.e. neglecting radiative corrections. As in the MSM the leptons are massless in the limit of vanishing electroweak couplings. For the masses of the top quark and the Higgs particles we find using a~from eq. (2.32) and GF from eq. (2.34) =

—*

m

~i

101 GeV,

~1

m1

56 GeV,

m2

=

m

=

m

+ =

0.

(3.26) The upper limits (reached for ~2 0) equal the MSM predictions. Case JIb yields different eigenvalues ~, of the stability matrix but the same mass predictions. In case lIc the top coupling is arbitrary whereas the bottom coupling is uniquely determined. Using mb 4.5 GeV as input we find the Higgs masses =

=

ml~0.56mh~2.5GeV,

m2=m~=m+=0.

(3.27)

The corresponding analysis for case ha reveals uniquely determined couplings for the leptons, the bottom and top quark, and the Higgs particles. Using again the bottom quark mass to fix ~2 we obtain m~=96GeV, m1 =51 GeV,

m2=2.4 GeV,

tan/3=0.047,

m~=0GeV, m~=36GeV

tan2a= —0.001.

(3.28)

For the trivial reduction lb/lid and also for the general solutions of the quarkonic Yukawa couplings in cases I and II, the leptonic Yukawa couplings and the Higgs self-couplings are uniquely fixed. Since the only physical constant solution [obtained from eq. (3.21)] turns out to be unstable, there is no other general solution which is finite for x 0 and respects the positivity properties of the w,. The quark couplings can be adjusted arbitrarily within the domain of attraction of the stable fixed point ~ 0. From the upper limits on the couplings —*

=

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Reduction of couplings

obtained above, 0
w1,w2<0.0355...

,

w41 <0.0292...,

(3.29)

one can derive upper limits on the top quark and Higgs masses,

m1 < {2~GF

~

0.99~/~ 101 GeV,

m~<~‘~G1)

~

~0.36~/~

m1 <

~

m2<~[~

~37GeV,

o.s~y’~57GeV, y~0.40~/~~40GeV,

m0=0.

(3.30)

The limits for m1 and m1 agree with the corresponding limits in the MSM. In the evaluation of the limits for the Higgs masses we have used the stability conditions (2.4) and neglected the Yukawa couplings of the light quarks. Furthermore these limits do not include the effects of the electroweak couplings.

4. Electroweak corrections Since the contributions of the electroweak couplings are small they can be treated as a perturbation of the solutions found in sect. 3. For their determination we use the method of partial differential equations described in ref. [2].We neglect the Yukawa couplings of the light fermions and consider only corrections to the special constant solutions. Since these special solutions are close to the upper bounds of the general solutions we expect that also the corresponding corrected solutions will essentially give the upper bounds of the general corrected solutions. For the constant solutions the method works as follows: The p, are considered as functionals of u and v, which are solutions of the corresponding reduction equations eqs. (2.22) and (2.23). Evaluating the derivatives dp~/dx according to the chain rule the reductions equations turn into partial differential equations for

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199

the functionals p,(U, v) of the following structure: (I4u—6U 2 )—+(14v+14v

2

)—

01)

i9u

=

~ ~

2

+ D,p

1

k,l

—

B11p~u— B,2p,v + E~1u

+

2E

2

,

(4.1)

112uv + E122v

where the indices i, j, k run over all Yukawa and Higgs couplings. The coefficients C~k/ Clik, D~,B, and E~are determined by the corresponding reduction equations. Eqs. (4.1) are solved by a power series ansatz =

P~ ~a~,qu’~’v”, leading to the following relations between the coefficients a~q:

(4.2)

~Mj 1a~,q

=

=

~

(

—2~C~~1a0~0 + (14( p

E CIk,

k,1

~

+ q)

—

Di)~ii)a~,q

a~q.a~p.q_q

p,q=0

p’+qO,p±q

[6(p —1) —Bji]a~_iq [14(p —1) +B,2]a~q_i 2Ej 3q + Eiii~p2~qO+ 12~p1 1+ Ej22~p0~q2 +

—

,

(4.3)

which is a system of linear equations for the a~t,qonce the a~.q. with p’ + q’

f,—W1

foru,v—*0.

(4.4)

This condition is required for all solutions and consequently has to be checked also if all a~qare already determined from the partial differential equations (4.1). Considering first case Ila the corresponding matrix M,, turns out to be non-singular for all p + q > 0 and the coefficients a~q are uniquely determined. The

200

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Reduction of couplings TABLE

p + ~\Mass

[GeV]

0 1 2 3 4 5

I

m,

m

1

m2

m±

mp

tan /7

tan 2a

95.5

51.1 41.1 60.4 59.9 60.1 60.1

2.41 2.26 2.73 2.73 2.73 2.73

36.3 30.3 26.3 26.8 26.7 26.7

0.0 0.0 0.0 0.0 0.0 0.0

0.047 0.050 0.050 0.050 0.050 0.050

—0.001 0.002 0.049 0.049 0.050 0.050

89.4 89.5 89.5 89.5 89.5

asymptotic requirement (4.4) is satisfied. From these solutions together with the central values of the input parameters given in eqs. (2.32) and (2.34), we obtain the predictions for the masses to order p + q given in table I. The large corrections in the second order originate from the terms in eqs. (4.1) and (4.3) containing the coefficients E,. These only appear equations for the Higgs 2 O,~byin±the 0.005 corresponds in the fifthself-couplings. order to the Varying a~by ± 0.01 and sin following range: for the top quark mass m, 85.0 93.7 GeV, for the Higgs =

masses

=

59.2

—

—

61.1 GeV, m

2 2.70 2.76 GeV, m 24.0 29.1 GeV, and for the mixing angles tan f3 0.042 0.059 and tan2a 0.048 0.053. In case lib/la M1, becomes singular for p + q 1. The coefficients a~~1 and a~11 of the Higgs coupling /32 have to be fixed from the asymptotic condition (4.4). They turn out to be zero. Having determined these coefficients, all others follow from the recursion formula (4.3). Starting from the order p + q 2, p4 becomes positive. This would lead to a vacuum which violates charge conservation. Consequently, this case must be ruled out. The same reasoning holds for case hIc. For comparison the corresponding results of the reduction method in the MSM are2 given table 2. into account the numerical in a~and O~,weinobtain theTaking mass ranges m, 89.7 98.9 GeVuncertainties and mH 63.0 65.5 sin GeV. =

=

—

~=

—

=

—

—

=

=

=

TABLE

p+qNj~Mass{GeV] 0 1 2 3 4 5

—

=

2 mt

mH

100.8 94.3 94.4 94.4 94.4 94.4

56.3 46.0 64.4 64.0 64.2 64.2

—

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201

5. Summary and conclusions We have studied the reduction of couplings in the standard model with two Higgs doublets and three fermion generations. The Yukawa couplings were assumed to be diagonal. We found definite values for the Yukawa and Higgs self-couplings in terms of the gauge couplings for the nontrivial reduction solutions as well as upper limits for the trivial (general) reduction solutions. Using the tree level expressions for the particle masses we obtained predictions and bounds for the latter. The results are as follows: (i) The masses of the leptons are zero. (ii) The quarks must be lighter than approximately 100 GeV. This concerns in particular the top quark. (iii) For the two scalar Higgses there are upper limits m1 ~ 65 GeV, m2 ~ 46 GeV. The mass of the lighter scalar is probably less than 5 GeV. (iv) The mass of the charged Higgs particle is bounded by about 27 GeV. (v) The pseudoscalar is massless. (vi) For given quark masses also all the Higgs masses are uniquely determined. Nonzero masses for the leptons and the pseudoscalar Higgs are made possible by the free integration constants in the electroweak infrared regime. This implies, however, to give up the asymptotic freedom of the respective couplings in the ultraviolet. The limits for the top quark and the Higgs mass m~ m11 are also valid within the MSM. The systematic error of these predictions originates mainly from two-loop effects and is estimated to be 5—10%. Recent LEP results [4] contradict the predictions above. This means that assuming the reduction principle — the SM containing two Higgs doublets and three fermion generations is experimentally ruled out. Concerning the MSM, the experimental lower bounds on the top quark mass are approaching the upper limits derived from the reduction principle. A top quark heavier than about 100 GeV would be in conflict with our results. Consequently, either the reduction method would not be valid or the MSM would have to be modified. =

—

The author would like to thank K. Sibold and W. Zimmermann for suggesting the problem and for many helpful discussions. References [1] J. Kubo, K. Sibold and W. Zimmermann, Nucl. Phys. B259 (1985) 331 [2] J. Kubo, K. Sibold and W. Zimmermann, Phys. Lett B220 (1989) 185, 191 [3] W. Zimmermann, Commun. Math. Phys. 97 (1985) 211; R. Oehme and W. Zimmermann, Commun. Math. Phys. 97 (1985) 569 [4] OPAL Collab., Phys. Lett B236 (1990) 224; ALEPH Collab., Phys. Lett. B236 (1990) 233; B237 (1990) 291; B241 (1990) 141; DELPHI Collab., Phys. Lett. B241 (1990) 449;

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[5]

[6] [7] [8]

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ALEPH Collab., Phys. Lett. B241 (1990) 623; OPAL Collab., Phys. Lett. B242 (1990) 299 D. Toussaint, Phys. Rev. D18 (1978) 1626; N.G. Deshpande and E. Ma, Phys. Rev. D18 (1978) 2574; R.A. Flores and M. Sher, Ann. Phys. (N.Y.) 148 (1983) 95 S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958 CT. Hill, C.N. Leung and S. Rao, NucI. Phys. B262 (1985) 517 W. Zimmermann, in preparation