Vacuum fine tuning and masses of t-quark and Higgs boson

19 January 1995

PHYSICS El SEVIFX

LETTERS

B

Physics LettersB 343 (1995) 295-298

Vacuum fine tuning and masses of t-quark and Higgs boson A.A. Andrianov a,1,N.V. Romanenko b ’ Department of Theoretical Physics, University of Saint Petersburg, 198904 Saint Petersburg, Russian Federation b Department of Neutron Research, Petersburg Nuclear Physics Institute, 188350 Gatchina, Russian Federation

Received 16 June 1994; revised manuscriptmzeived 4 November 1994 Editor: FX Landshoff

Abstract

The fine-tuning principles are exploited to predict the top-quark and Higgs-boson masses. A modification of the Veltman condition based on the compensation of vacuum energies within the Standard Model is proposed. It is supplemented with the requirement of stability under resealing and of finiteness of the e+e- H-vertex. The top-quark and Higgs-boson couplings are fitted for the ultraviolet scale A x 1015GeV. It yields the low-energy values mr = 175 f 5 GeV, mH = 210 f 10 GeV.

1. The Standard Model (SM) describes with good accuracy the strong and electroweak particle interactions for a whole range of energies available in experiments [ 11. Still, there are a number of well-known open problems within the framework of the SM that have to be resolved to justify all principles on which the Standard Model is based and to eventually determine all its phenomenological parameters. In particular, the detection of the top-quark is expected [ 21 and the discovery of the scalar Higgs particle is wanted [ 1,3]. In this connection, much effort has been made to estimate their masses for the purpose of understanding the possible extensions of the SM. At the same time there exist a few phenomenological principles within the minimal SM which make it possible to determine relations between top-quark and Higgs-boson masses having little dependence on the details of a fundamental theory underlying the SM. These principles are based on the assumption that the SM is actually an effective theory applicable consistently for low energies. Consequently, its cou-

’ E-mail: andrianovl @phim.niif.spb.su. Elsevier Science B.V. SSD10370-2693(94)01473-6

pling constants and dimensional parameters absorb all influence of high-energy degrees of freedom and of new heavy particles as well. Of course, the form of an effective action of Green-Wilson type [4] generally depends on the preparation procedure, but the consistent effective action coinciding with the SM action is supposedly minimally sensitive to very high energies. Still, the influence of the high-energy dynamics responsible for the parameter formation exists and is manifested in the relations between the dimensional parameters and certain coupling constants. The latter statement allows one to formulate the following phenomenological principles which could be realized in the quantum SM enabling one to predict the masses of heavy particles. - The strong fine tuning for the Higgs field parameters (v.e.v. and its mass) that consists in the cancellation of large radiative contributions quadratic in the ultraviolet scales which bound the particle spectra in the effective theory (Veltman condition [ 5-81) . - We propose also the strong fine tuning for vacuum energies [9] that envisages the cancellation of large di-

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vergencies quartic in the ultraviolet scales which might affect drastically the formation of the cosmological constant. Of course, the disbalance in vacuum energies for an effective theory may prove to be successfully compensated by those from virtual high-energy components. However, we suppose that the consistent preparation of an effective model is to provide the essential decoupling of the low-energy world from very high energies, and therefore for an appropriate choice of ultraviolet scales the huge vacuum energies should not naturally appear [ 91. This requirement of vacuum adaptation leads as we shall see to a modification of the Veltman condition. - The weak fine tuning that presupposes the cancellation of the logarithmic cutoff dependence in certain coupling constants thus causing the quasi-fixed ultraviolet behavior [ 10-121. - The stability of the renormalization-group (RG) flow for the Higgs self-coupling and/or for its coupling to the r-quark (quasi-fixed infrared point [ 131) at relatively low energies. In our paper we examine the compatibility of the abovementioned principles and find the corresponding estimates for r-quark and Higgs-boson masses. The vacuum-energy fine tuning when combined with others leads to predictions for the top-quark and Higgsboson masses within the range of validity of the Standard Model and in fair correspondence with the recent experiments [ 21. 2. It is well known that the scalar sector in the Weinberg-Salam theory contains quadratic divergences in the tadpole diagrams and in the scalar particle self-energy. In the early 80s the rule of cancellation for quadratic divergences [ 51 was proposed in the electroweak sector of the SM. This cancellation occurs if the fermion and boson loops are tuned due to specific values of coupling constants. At one loop level it was found that the condition

(1) flavors, colors

removes quadratic divergences both from the Higgsfield v.e.v. and the Higgs boson self-energy if the universal momentum cutoff is implemented for all fields. From ( 1) it is easy to see that if the r-quark is the only heavy fermion, the m, 2 70 GeV bound should be fulfilled.

The original Veltman condition is, however, not stable under resealing since its renorm-derivative cannot vanish simultaneously for any choice of the cutoff A below the Plank scale [ 71, i.e. the related cancellation of quadratic divergences can be provided only at a selected scale. The weak fine tuning (the cancellation of logarithmic divergences) has been applied to the e+e-Hvertex in Ref. [ 1I]. This vertex does not contain quadratic divergences and therefore it is safe in the weak fine tuning. The corresponding equation reads m2=5M2 f 2z

-M2

(2)

W.

It is compatible with ( l), and they together yield the mass predictions m, = 120 GeV, MH = 190 GeV. Still this t-quark mass is ruled out by a recent experiment

121. A useful tool to obtain the values of the Higgsboson self-coupling has been found while computing the RG-flow for coupling A,

B=Gg

1

14

1212 +ggg

32 fE&?.

(3)

It happens [ 131 that A(r) tends to the Hill quasi-fixed point for A (aA/& = 0) in the wide intermediate energy range for any boundary conditions at high energies. 3. Let us consider the SM as a low-energy limit of a more fundamental theory and suppose that only one heavy fermion, the r-quark, is involved in its dynamics within the selected energy range. Consequently, we neglect the masses of all lighter fermions. When there is no expected supersymmetry (below the Grand Unification scale) we apply different scales for the design of the SM-effective action for bosons, As < Acomp, and for fermions, AF < Acomp. Among bosons the universal scale is introduced in order not to induce the explicit breaking of a Grand Unification symmetry below a scale of compositeness A,. In its turn, the universal scale for fermions confirms the horizontal symmetry in the ultraviolet region. We require for the SM the suppression of very large contributions (leading divergences) to dimensional physical parameters, which is equivalent to the

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L..etters B 343 (199.5) 295-298

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absence of their strong scale dependence. The latter means that, in addition to the strong fine-tuning, the cancellation of contributions to the vacuum energy should occur, i.e., first of all the contributions which are quartic in cutoffs,

The explicit form of the second stability condition

TP N g&x

Df = gg2( 9g2 _ gg2 _ “g2 _ cg’2) f 21 3 4 12

= 0,

f=4g;-2a(A+A)

=O,

Df=

162:

=O,

r =ln(A/uc).

(7) is

-24a[A2+(g;-A)A+B-g;]

4NF + 2N, A; = (~NB + Ns) A;,

- $r(-19g4+

Yg’4),

(8)

(4) where NF = 21 is the number of flavor and color degrees of freedom for three generations of massive fermions, N, is the number of (Weyl or Majorana) neutrinos, NB = 12, Ns = 4 are the numbers of flavor and color degrees of freedom for vector and scalar bosons, respectively. If neutrinos are massive particles then one should replace N, --f 2N,, which yields cr2 = y. Since the problem of the neutrino masses is not solved yet, we shall keep in mind both possibilities. The strong fine-tuning condition in this case at the one-loop level reads

or

1.852.

(5)

Taking into account the effects of all loops one obtains the integral Veltman condition,

I 2 + ig’2

+

2g k2+M;

A E ig2 + tg12,

are borrowed

B = >4 + ig2g12 + &g2.

After elimination of A the equation Yukawa coupling constant reads

for the t-quark

k,g; + kzg: + k3 = 0, where the coefficients k, = 12(-2a

(9)

(10) are

+ 1 + 8/a)

M 22.5

(19.4),

k2=64g;+yg’2-24A(a+5),

4m;=a(2&+M;+m;), LYN 1.793

where in Eqs.( 7)) (8) the denotations from Ref. [13],

.g+

A k2+Mf,

>’

(6)

where the conventional denotations for the electroweak g,g’, Higgs-quartic, A and the Yukawa t-quark, gr coupling constants are used. When integrating by parts one can conclude that the leading contribution is the modified Veltman condition at the scale A. The latter one is supplemented in the nextto-one-loop approximation with its renorm-derivative (having a small combinatorial factor) and so on. At the one-loop level of accuracy, all the renormderivatives except the first one are zero. The requirement of the weak dependence of A brings about the modified Veltman condition and its renorm-derivative equal to zero,

k3 = ;a( 352A2 + 76Ag’2 + 61g’4).

(11)

Numerically the existence of a solution is very sensitive to both the values of cr N 1.79 (1.85) and the value of the strong coupling constant (~3 = gg/47r. In what follows the averaged value of cy3 is taken from Ref. [ 141 as as(Mz) = 0.118 5 0.007. For the case of Dirac neutrinos it can be found that the solution for gf exists only for the effective A > 1014 GeV. At the scale A = lOI GeV one obtains two solutions. The low-energy mass values, m, = 176,191 GeV, 171~= 209,225 GeV are obtained by the RG-flow for the Yukawa and Higgs self-couplings. The latter one reaches at low energies the infra-red quasi-fixed point [ 131. On the other hand, the joint solution of the modified Veltman condition at low energies and the Hill quasi-fixed point relation DA = 0 yields m, = 170 GeV, rn~ = 203 GeV. Hence the first solution only provides reasonable precision in the RG invariance of the modified Veltman condition. For the case of the Weyl or Majorana neutrinos one obtains numerically a similar solution but for the higher scale A N 10*6-10’7 GeV where the unification of coupling constants g2 = gs holds.

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The lack of precision in the as-value diminishes the accuracy of the scale A predictions. So one may consider (following Ref. [ 1 l] ) the finiteness of the e+e-H-vertex as an additional weak fine-tuning principle. This yields Eq. (2) independent of the Higgs mass. Luckily it proves to be compatible with the modified Veltman conditions. One can find the overlapping for A N 10’4-10’5 GeV for Dirac neutrinos, A N 10’6-10’7 GeV for Weyl or Majorana neutrinos, m, = 175 f5GeV,

mH = 210 f IOGeV.

(12)

A theoretical error can be estimated by the evaluation of the two-loop contributions [ 71 to p-functions. One can check that for these mass values the modified Veltman condition holds with good accuracy f 3 GeV both for the effective bound of the Standard Model A and for the vector boson mass scale p zz 100 GeV. This means that the modified Veltman equation does not depend on the resealing for a wide range of energies. The predictions are within the accepted range of different experimental and theoretical bounds [ l-31. We have shown that the modification of the Veltman condition due to the strong fine-tuning of vacuum energies makes it possible to define the effective SM with a finite cutoff and to determine the related particle masses which are essentially less than the cutoff. The relation between the t-quark and Higgs-boson masses is consistent within the wide energy range from the fundamental EW scale up to the cutoff (due to the vanishing renorm-derivative). It has not been available in the original formulation (for tr = I there is no solution). We are grateful to Professor R. Rodenberg for valuable discussions. This work is partially supported by the Russian GRACENAS.

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