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In the Weinberg--Salam model for weak interactions, the masses of all the gauge bosons, quarks and leptons are taken to arise from the Higgs mechanism. At the tree approximation, the couplings of the Higgs scalar field to itself determine the effective potential , which in turn determines the symmetry of the ``vacuum''. In this approximation is independent of the couplings (which determine the masses attained after spontaneous symmetry breakdown) of fermions and gauge bosons to . If, however, one-loop corrections to are included, then the gauge bosons and fermions will influence . The requirement that this influence should not serve to prevent the possibility of spontaneous symmetry breakdown places several constraints on the couplings in the theory, and hence on the ratios of masses of various particles. Linde and Weinberg [1] have derived a lower bound on the mass of the Higgs particle H by demanding that the energy density of the ``vacuum'' after spontaneous symmetry breakdown should not exceed its value when . In this note, we apply the more complete requirement that the conventional ``vacuum'' in which corresponds to the absolute, rather than only a local, minimum of , at least in the domain where may be obtained reliably from perturbation theory. If all fermion and gauge boson masses are generated from the vacuum expectation value of a single field, then this constraint allows one to place an upper bound on the fermion masses. The exact form of the bound involves and other parameters, but typically the mass of the heaviest fermion must satisfy . While this range is not immediately accessible to experimental investigation, the very existence of such a bound, coming solely from considerations of self -consistency, places constraints on models for weak interactions. Our bound is equivalent to an upper limit on the dimensionless fermion--Higgs Yukawa coupling, , and it ensures that is perturbatively small; corresponds to .

In a theory with more than one coupling constant, one-loop graphs can dominate over tree graphs, while perturbation theory remains reliable because all couplings are small. For example, with a gauge coupling and self-coupling (both small), but such that is of order , a gauge boson loop can compete with tree graphs, while yet higher-order corrections remain unimportant. However, even if the couplings are small, the perturbation expansion breaks down when logarithms of field strengths become large (1). In the following discussion, we shall simply require that the theory be consistent over the range of that can be explored perturbatively.

The complete formula for in the one-loop approximation is [2]

where

and the are the couplings of the gauge bosons (fermions) to the Higgs particles. Note that, because of Fermi statistics, the fermion contribution to is negative. The parameter in eq. (1a) is a renormalization mass. In the Weinberg--Salam model, is given by

In our numerical estimates, we use , so that . We have dropped the contributions of Higgs scalar loops to , since, as discussed below, these must be negligible if perturbation theory is to be valid (2).

For spontaneous symmetry breakdown to occur it is necessary that should have a non-trivial local minimum at such that

To investigate the consistency of a theory based on the ``vacuum'' , we shall assume such a theory and then find under what circumstances inconsistencies appear. In that case, the parameters and appearing in the effective potential may be eliminated in favor of and . It is convenient to introduce

in terms of which

[ Figure 1 ] The reduced effective potential as a function of for various choices of the combination of couplings defined in eq. (4). For usual spontaneous symmetry breakdown to occur, must correspond to an absolute minimum of , at least within the range of accessible to perturbation theory.

The requirement [1] necessary to allow spontaneous symmetry breakdown becomes

so that .

In fig. 1 we plot as a function of for various values of . As the Yukawa couplings increase, decreases, as does . For negative , a new phenomenon occurs: eventually turns over and goes to as . However, since our expression for is obtained from perturbation theory, we have no estimate of it for values of so large that . We therefore do not consider its behavior as , but rather, require that for all values of within the range over which is reliably calculated. If this is not satisfied, then the theory is inevitably inconsistent. Fig. 2 shows the values of for which becomes less than , as a function of . (We also show the values of corresponding to the second local maximum of .) For large values of , one finds

If the theory is to allow a stable ``vacuum'' in perturbation theory then must lie outside the range of validity of perturbative approximations. In practice, our final results are rather insensitive to the precise value of which is deemed unacceptable. Combining the Linde--Weinberg condition [1] with our requirements on one obtains

where is presumably much less than 1 and perhaps as small as .

[ Figure 2 ] The values of at which drops below and at which the second local maximum of occurs, as a function of . These values of must be so large that our perturbative methods fail if the ``vacuum'' is to be stable.

For any particular set of fields and couplings, one can translate these bounds on into bounds on ratios of particle masses. Consider first the case of the Weinberg--Salam model with its one complex doublet of Higgs fields and with a single heavy fermion. In this case ( and are defined in eq. (2))

The first inequality in eq. (7) then becomes

which reduces to the Linde--Weinberg bound [1] when . Note that this bound places no constraint on when . A constraint is, however, provided by the second inequality in eq. (7), which yields

The regions in and allowed by the bounds (9) and (10) are illustrated in fig. 3 for various choices of . If there are many fermions, then the in eqs. (9) and (10) is obviously replaced by ; for quarks each color is counted separately.

Our bound on does not come from the requirement that the Yukawa couplings of the Higgs bosons to the fermions should not be large; in fact, so long as is not enormous it is much more stringent. However, for a perturbative investigation of the theory to be at all meaningful, it is necessary that higher and higher orders in the perturbation series should give systematically smaller contributions. Experiments have shown that and satisfy this condition, and our bounds on ensure that it will hold for the . The quartic self-couplings of the Higgs bosons must also obey the condition, so that (3)

or

All predictions of the theory are obtained by perturbative methods, and, if the bound (11) were not satisfied then no predictions could be made (4).

[ Figure 3 ] The domains in the mass of the Higgs particle and of the heaviest fermion for which the Weinberg--Salam model is consistent. The value of depends on the region of validity of perturbation theory; is probably very small. The forbidden region in the lower left-hand corner represents the Linde--Weinberg bound.

We have given bounds on the Higgs particle mass (eqs. (9), (10) and (11)) which result from demanding consistency of the theory. However, by making the specific assumption that the term in vanishes (5) , one may obtain a definite prediction for [2]:

If the fermion term can be ignored, then this gives ---close to the range of present experiments.

In this paper, we have concentrated on the simplest workable model for weak interactions, since there is so far no compelling experimental evidence for a more complicated structure. In more complicated models our bounds may be strengthened, weakened or may even disappear entirely. For example, if one introduces an extra Higgs field which couples only to certain fermions, then our bounds (7) cannot be used, because they involve the vacuum expectation value of the new Higgs field, which would only be determined directly from the mass of a gauge boson coupled to it.

To conclude, we have investigated the Weinberg--Salam model for weak interactions, and find that unless ratios of particle masses obey certain bounds, no meaningful predictions based on the model may be obtained by perturbative methods.

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