

 11 Jul 94

Higgs and Top Quark Masses in the Standard Model without

Elementary Higgs Boson

V. N. Gribov Institut f"ur Theoretische Kernphysik, Universit"at Bonn, Nussallee 14-16, 53115 Bonn, Germany

and Research Institute for Particle and Nuclear Physics,

Budapest, Hungary

and L. D. Landau Institute for Theoretical Physics,

Moscow, Russia

(June 6, 1994)

Abstract In this short note I present a simple calculation of the top quark and Higgs masses, based on the idea that in the standard model without elementary Higgs the fact that the U (1)Y coupling becomes of the order of unity at the Landau scale * leads to spontaneous symmetry breaking and generation of masses.

Typeset using REVTEX 1

I. INTRODUCTION The basis of this paper is the idea that in the standard model without elementary Higgs particle the U (1)Y coupling becomes of the order of unity at the Landau scale *, thus resulting in a spontaneous symmetry breaking and generation of masses. In this case the longitudinal components of the W , Z0 and the Higgs have to be bound states similar to the deuteron in the zero-radius limit of nuclear forces. The usual SU (2) and SU (3) interactions would act like the electromagnetic interaction between nucleons in the deuteron, and the situation could be treated perturbatively. The existence of the longitudinal components of the W , Z0 is connected to the fact that the generation of quark and lepton masses leads to non-conservation of the two components of the SU (2) left-handed current and the left-handed component of the U (1) current. The existence of the Higgs is due to the degeneracy of states with positive and negative parity at short distances. In a more detailed paper we will show that there is a good chance for such a theory to exist.

II. THE W AND Z0 MASSES We start with the calculation of the W mass. The fermionic part of the polarization operator of the SU (2) bosons is given by

\Pi ab_* = \Gamma g

22

4 Xj Z

d4q (2ss)4i tr(o/

aflL

_ Gj (q)fl

L * o/

bGj (q \Gamma p)) (1)

=

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p ;

where flL_ = fl_ \Delta 12(1 \Gamma fl5) and the sum extends over all fermionic generations. In the absence of quark and lepton masses

\Pi ab_* = *\Pi ab_* = (g_* p2 \Gamma p_p* ) \Pi (p2) ffiab (2) due to current conservation. For massive fermions

Gj = 1m

j\Gamma 6 q : (3)

Here mj is a function of o/3 and can be written as

mj = m"j \Delta 12 (1 + o/3) + m#j \Delta 12 (1 \Gamma o/3) : (4) Because of (3), the form (2) is no longer valid. Instead, in second order in m, \Pi _* can be written for q AE m and p2 ! 0 as

\Pi ab_* = *\Pi ab_* \Gamma g

22

2 Xj Zm

jo/qo/*

d4q (2ss)4i tr(fl

L _ 6 q m2j fl

L * 6 q) 1q6 ffiab (5)

= *\Pi ab_* + \Delta \Pi ab_* : (6)

2

Applying (4),

\Pi ab_* = *\Pi ab_* + g_* ffiab 3g

22

2 \Delta 16ss2 Xq;l Z

m2

dq2

q2 (m

2 "q + m2#q + 13 m2l ) : (7)

The lower limit in (7) is defined by the physical masses of quarks and leptons. At first sight this integral looks divergent. But if we take into account the dependence of m on q2 the integral turns out to be convergent for the main contribution, given by the top quark mass, if the upper limit is the Landau scale *. The dependence of the masses on q2 is well-known in the standard model (see for example [1]). It is defined by the SU (3) and U (1) interactions only and reads (for three generations)

m"q(q2) = m"q " ffs(q

2)

ffs(m2"q) #

4 7 " ff0(m2"q)

ff0(q2) #

1 20 (8)

m#q(q2) = m#q " ffs(q

2)

ffs(m2#q) #

4 7 " ff0(m2#q)

ff0(q2) #

\Gamma 140

(9)

ml(q2) = ml " ff

0(m2

l )

ff0(q2) #

9 40 : (10)

The dependence on the U (1) coupling ff0 is very slow and becomes essential only near the Landau scale. We can therefore neglect it when integrating up to the Landau scale. Taking into account only the contribution of the heaviest quark - the top quark - and using the ffs dependence on q2,

ff\Gamma 1s (q2) = ff\Gamma 1s (m2) + 74ss ln q

2

m2 ; (11) and (8) we obtain

\Pi ab_* = *\Pi ab_* + g22 g_* ffiab 32g2

s m

2 t 8!:1 \Gamma " ff

s(*2)

ffs(m2t ) #

1 7 9=; ; (12)

where g2s = 4ssffs(m2t ). After replacing ffiab by (ffiab \Gamma ffia3ffib3) we find

m2W = g22 v

2

4 (13) v2

4 =

3 2g2s m

2 t 8!:1 \Gamma " ff

s(*2)

ffs(m2t ) #

1 7 9=; : (14)

In order to calculate the Z0 mass, we have to consider not only the transition between the zero-components of W , given by \Delta \Pi _*(2; 2). We also have to consider the transitions between W0 and the U (1) boson - \Delta \Pi _*(1; 2) and \Delta \Pi _*(2; 1) - and between U (1) bosons, \Delta \Pi _*(1; 1). We can do this very easily when noticing that the U (1) current has the structure

flL_ YL2 + flR_ YR2 = Qfl_ \Gamma flL_ o/32 ; (15)

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where Q is the operator of electric charge and Qfl_ is the conserved electric current. Therefore, because the non-conserved part \Delta \Pi _*(2; 2) is equal

\Delta \Pi _*(2; 2) = g22 v

2

4 (16) then

\Delta \Pi _*(1; 2) = \Delta \Pi _*(2; 1) = \Gamma g2 g0 v

2

4 (17)

\Delta \Pi _*(1; 1) = g02 v

2

4 (18) and this will reproduce the usual features of Z0 and fl with v=2 = 123:11 GeV.

Up to now we did not discuss Goldstone particles. As a result our \Delta \Pi _* is not transverse. But if mass generation is due to spontaneous symmetry breaking, currents have to be conserved via the existence of Goldstone particles and so \Delta \Pi _* has to be transverse. In order to find the correct expression for \Delta \Pi _*, we have to add to the left-handed SU (2) current the Goldstone contribution:

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p_

p2 : (20)

Here gb denotes the Goldstone-fermion coupling and fab is the current-Goldstone transition amplitude. These couplings are defined by the Ward identity for the new current

p_\Gamma a_ = 12 (1 + fl5) o/

a

2 G

\Gamma 1(q1) \Gamma G\Gamma 1(q2) \Delta 1

2 (1 \Gamma fl5)

o/ a

2 : (21)

For G having the form (3) the Ward identity for p = 0 gives us

1 4 [o/

a; m] + 1

4 fl5fo/

a; mg = \Gamma fab gb : (22)

For fab being a transition amplitude and because we suppose the Goldstone to be a fermionic bound state

fabp_ = flL_ o/

a

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Combining (22) and (24) and calculating the integral (24) in the same way as we did in the calculation of \Delta \Pi _*, one can derive

4

ifab = f ffia3ffib3 + f ?ab (25) f ?acf ?cb = f 2(ffiab \Gamma ffia3ffib3) (26) with f 2 = v2=4, as defined in (14) from (7) and (12). Further we can simplify ga by choosing an appropriate basis in the ab -12-plane,

ga = i ([o/ a; m] + fl5 fo/ a; mg) 12f ; (27)

g3 = ifl5 o/3 m2f : (28) Now it is easy to write down the correct expression for \Delta \Pi _*. We have to replace both vertices in the diagram of formula (1) by the two terms in (19) or (20). As a result \Delta \Pi _* would be represented by four diagrams:

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first two terms give the correct expression for \Delta \Pi _*

\Delta \Pi _* = f 2 g_* \Gamma p_p*p2 ! : (30) The last two diagrams have to cancel each other. The cancellation condition requires that the new loop in the fourth term of (29) fulfills

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which is a natural condition for the self-energy \Sigma G of a Goldstone type bound state. It is this cancellation condition that will enable us to calculate the Higgs mass.

III. THE HIGGS MASS The existence of a Higgs particle is suggested by the fact that at short distances, where all particle masses are zero, bound states have to be degenerate in parity. Therefore, if a pseudoscalar Goldstone particle exists, according to (28), there also has to be a scalar bound state which we will call Higgs particle. Due to large distance effects the mass of this state would not be zero. Because of the degeneracy at short distances, the self-energies of these two states would have the same structure:

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with

gHiggs = g j m2f : (34)

The mass of the scalar state is defined by the condition

\Sigma S (p2 = m2H) = 0 : (35) This condition can be written in another form, if we know that the pseudoscalar Goldstone state satisfies (31):

\Sigma S (p2) \Gamma \Sigma P S (p2) + p2 = 0 (36)

\Sigma S(p2) = \Gamma X

j Z

d4q (2ss)4i tr(g G

j g Gj ) (37)

\Sigma P S (p2) = \Gamma X

j Z

d4q (2ss)4i tr(ifl5g G

j ifl5g Gj ) (38)

so that

p2 = \Sigma P S (p2) \Gamma \Sigma S(p2)

= X

j Z

d4q (2ss)4i tr(gffl5; G

jgfl5g Gj) : (39)

The integral on the right-hand side is convergent in the same sense as we discussed before and we find

m2H = 2 \Delta 3f 2 \Delta 16ss2 X

q;l Z

dq2

q2 (m

4 "q + m4#q + 13 m4l ) : (40)

Using the expression for f 2 (see (14), (7) and (12)) we can write (40) in the form

m2H = 4 P

q;l R

dq2

q2 (m

4 "q + m4#q + 13m4l )P

q;l R

dq2

q2 (m2"q + m2#q +

1 3m2l )

: (41)

Neglecting the contributions of all quarks and leptons except the top quark we have

m2H = 4

*2R m2t

dq2

q2 m

4 t (q2)

*2R m2t

dq2

q2 m

2t (q2) (42)

or

m2H = 32v2ss2

*2Z

m2t

dq2

q2 m

4 t (q2) : (43)

6

Using (8) and (11) and neglecting the ff0 dependence we obtain

m2H = 8m

4 t

3g2s v2 8!:1 \Gamma "

ffs(*2) ffs(m2t ) #

9 7 9=; : (44)

Formulas (14) and (44) are the predictions we mentioned in the beginning of the paper. The quantity ffs(*2) can be expressed through the electromagnetic coupling ffe(m2t ). From

(ff0(*2))\Gamma 1 = (ff0(m2t ))\Gamma 1 \Gamma 53ss ln *

2

m2t ; (45)

in order to have ff0(*2) , 1

ln *

2

m2t ss

3ss

5

1 ff0(m2t ) : (46)

Using (11) and the relation ffe(m2t ) = ff0(m2t ) cos2 `W (`W - Weinberg angle), we have

ffs(m2t )

ffs(*2) = 1 +

21 20 cos

2 `W ffs(m2t )

ffe(m2t ) : (47)

The concrete values for mt and mH depend essentially on the value of the strong coupling at the top quark scale, which is not very well known. For example, for ffs(m2t ) = 0:11

mt = 215 GeV (48) mH = 255 GeV (49)

in contradiction to Fermilab data, and for ffs(m2t ) = 0:09

mt = 201 GeV (50) mH = 252 GeV : (51)

Accepting the Fermilab result mt = 174 GeV [2], we find mH = 167 GeV from (44) for ffs(m2t ) = 0:11.

However, the fact that the masses calculated for the Higgs and top quark very roughly, but essentially from first principles, turn out to be in the correct region seems to me very encouraging. At the same time, there may be an indication that some finite contribution to v2=4 from the region where q2 is of order of *2 could survive. In this respect the prediction of the Higgs mass becomes very important, because it is less dependent on the large momentum region of the theory than the top quark mass.

IV. CONCLUSION The idea of a non-elementary nature of the Higgs particle was widely discussed in connection with the so-called top quark condensate [3]. The essential difference to these approaches is that in my discussion the top quark has no special interaction. It becomes important only

7

because it is the heaviest fermion. My Goldstone and Higgs particles consist of all existing quarks and leptons on equal footing. The ratio of the contributions of the different fermions is defined by their masses and the U (1) interaction at short distances.

It is interesting to notice that the Golstone and the Higgs interact weakly with fermions at the scale mt. The largest of the Goldstone couplings - the one to the top - is, according to (14) and (34),

g2t 4ss =

m2t 16ss f 2 =

1 6

ffs(m2t )

1 \Gamma h ffs(*

2)

ffs(m2t )i

1 7 : (52)

If we accept this formula for an arbitrary scale mt ! _, we would find that g2t decreases with the momentum scale, similar to the QCD coupling ffs. After passing the "grand unification" point ffs , ff0, the behavior of g2t changes. It starts to increase with momentum and asymptotically becomes proportional to the U (1) coupling

g2t (_2)

4ss =

10

9 ff

0(_2) : (53)

The Higgs self-interaction has the usual properties of the Glashow-Weinberg-Salam model, but here there is no question of fine-tuning the Higgs mass, because it is fixed by the symmetry condition (36).

These and other details of the present theory, such as the influence of the Goldstone couplings on the quark mass dependence, will be discussed in the next paper [4].

Acknowledgments: I would like to express my deep gratitude to Yu. Dokshitzer for numerous discussions of various aspects of the theory without an elementary Higgs particle. I am also grateful to C. Ewerz for discussions and his help in preparing the manuscript. I wish to thank all the members of the Institute for Theoretical Nuclear Physics for the kind and inspiring atmosphere I experienced during my stay in Bonn.

References

[1] T.-P. Cheng, L.-F. Li, "Gauge theory of elementary particle physics", Clarendon Press,

Oxford, 1984; chapter 14.5

[2] CDF Collaboration, FERMILAB-PUB-94/097-E [3] Y. Nambu, in New Theories in Physics, Proceedings XI Int. Symposium on Elementary

Particle Physics, World Scientific, Singapore, 1989; A. Miransky, M. Tanabashi, K. Yamawaki, Mod. Phys. Lett. A 4 (1989), 1043 and Phys. Lett. B 221 (1989), 177. For recent developments see for example M. Lindner, Int. Journ. of Mod. Phys. A 8,3 (1993), 2167 and references therein.

[4] V. Gribov, Yu. Dokshitzer, under preparation.

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