Extended survival hypothesis and fermion masses

Volume 140B, n u m b e r 1,2

PHYSICS LETTERS

31 May 1984

EXTENDED SURVIVAL HYPOTHESIS AND FERMION MASSES S. DIMOPOULOS 1,z Department o f Physics, Stanford University, Stanford, CA 94305, USA and H.M. GEORGI 2 Department o f Physics, Harvard University, Cambridge, MA 02138, USA Received 5 March 1984

In this paper two thIngs are done. First we discuss the extended survival hypothesis which states that at each energy scale the only scalars in existence are those that are goIng to develop a vacuum expectation at a lower energy. Second we give some examples which show how the e x t e n d e d survival hypothesis can greatly enhance the predictive power of a theory and lead to several relations a m o n g fermion masses.

1. Natural mass relations. There are few mysteries in particle physics as puzzling as flavor. We do not know why there are flavors. We do not understand the flavor masses and mixing angles. We are not even sure what form such understanding might take. The situation is ripe for numerology. Though we do not really expect the mass matrix to be given exactly in terms o f a string of integers, a set of approximate numerological relations might be a clue to the underlying flavor dynamics. But approximate relations are subtle in quantum field theory. Exact relations often follow from symmetries. When symmetries are broken, the quantities which were related are sometimes renormalized differently, and so become independent unrelated parameters. So far, most serious* 1 attempts at understanding the flavor masses have been based on the concept o f 1 Alfred P. Sloan F o u n d a t i o n Fellow. 2 Work supported by the National Science Foundation Grants NSF-PHY 83-10654 and NSF-PHY-82-15249 and b y the A.P. Sloan F o u n d a t i o n . , l By "serious", we m e a n based on dynamical assumptions w h i c h have a reasonable chance o f being consistent. For example, we do n o t include " p r e e n " models which ass u m e peculiar u n b r o k e n chiral symmetries.

a "natural" relation [1 ]. Natural relations do not suffer arbitrary renormalizations. Rather, the corrections to such relations induced by quantum effects are finite and calculable. A typical example o f a natural mass relation occurs when the masses of a multiplet of fermions arise through their Yukawa couplings to a Higgs multiplet which develops a vacuum expectation value. If several Higgs multiptets with different symmetry properties can couple to the fermions, but only one is present in the theory, then the fermion masses produced may not be the most general possible, but can instead satisfy natural relations. In this case they are imposed by the group theoretical structure o f the Yukawa couplings. A familiar example occurs in the minimal SU(5) model where the charge-I/3 quark mass at the unification scale is equal to the corresponding charged lepton mass. The Finite corrections to this natural relation give the successful minimal SU(5) prediction for the b/r mass ratio [2].

2. Hierarchy. In this note, we discuss some consequences o f a slightly more general type o f relation which can appear in models with several very different mass scales. Such models are apparently necessary to describe physics at very short distances, although, 67

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at present, we have no compelling idea about the source of the large ratios between different scales. In our ignorance, we must impose each such hierarchy by hand in the form of a fine tuning of parameters. There is an important physical reason why such fine tunings are necessary. Suppose that there is a large gap between two scales and that physics at the lower scales involves only fermions (which are light because of global chiral symmetries) and gauge bosons (light because of gauge invariance). Then all of the renormalizable interactions are gauge interactions which preserve the global chiral symmetries. Explicit chiral symmetry breaking, which is almost always necessary on physical grounds, can come only from nonrenormalizable interactions which are suppressed by inverse powers o f the large scale * 2. If the ratio of scales is large, this symmetry breaking is usually not big enough. The problem, therefore, is the lack of communication of the symmetry breaking from the large scale to the small scale. This communication problem can be solved if there are scalar mesons in the effective theory below the large scale. Scalar mesons can have renormalizable Yukawa couplings to fermions which break the chiral symmetries without suppression. But why are these scalars light? This is a hierarchy puzzle. One approach to the hierarchy puzzle is supersymmetry. As long as supersymmetry remains unbroken, it is easy to keep scalars light. Indeed, there are lots of scalars. There is a light scalar multiplet for each light chiral fermion multiplet. Unfortunately, in supersymmetric theories, there is another hierarchy puzzle. Why should the scale of supersymmetry breaking be much smaller than the large scales in the theory? So far, no satisfactory answer has been suggested, although supersymmetry breaking can be tuned to give models which are phenomenologically acceptable. 3. The extended survival hypothesis. If supersymmetry is not relevant (either absent completely or broken at the largest scale), we can still keep scalars light, but we need one fine tuning of parameters for each light scalar multiplet. Under these circumstances, we may want to keep the number of light scalars and therefore the number of independent Fine tunings to the absolute minimum necessary. In particular, we

,2 Or from small fermion mass terms, which are finely tuned. 68

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need only those multiplets which contain scalar fields which get important vacuum expectation values (VEVs). The assumption that at any scale, the only scalar multiplets present are those that develop VEVs at smaller scales is sometimes called the extended survival hypothesis (ESH) [3]. As we will see, the ESH has important consequences for mass relations. In the effective field theory language, the essential point is extraordinarily simple. In general, in the effective theory between two scales,M 1 and M2(M 1 > M2) , the scalar multiplets present will not include all of the multiplets which could possibly couple to the fermions in the effective theory. Then the renormalizable couplings in the effective theory have a form which is restricted by the absence of some scalar multiplets. Often, this leads to nontrivial mass relations. In this case, however, there are corrections to these relations which come from the nonrenormalizable interactions in the effective theory (or equivalently from the effects of particles with mass/> M 1 in the full theory). These corrections receive divergent contributions in perturbation theory because they involve independent parameters in the full theory. But because they are absent in the effective theory, all these divergent contributions, once renormalized, are suppressed by powers of 1/M 1 . The same Free tunings which keep the light scalars light also suppress the corrections to the mass relation. It should be clear that we have simply generalized the notion of the natural mass relation to the effective field theory language. One of us has already shown that this idea can have interesting consequences for the flavor problem [4]. Below we explore these further. 4. A 4-family 0(10) model with t = 6b. In this section we illustrate the extended survival hypothesis (ESH) via an example introduced in ref. [4]. Consider an O(10) theory whose particle content includes: three scalars, one 210, one 10 and one 120; three fermion 16s and one 16. The complete four family theory contains additional multiplets which we shall not discuss here. The Yukawa coiuplings of the theory are defined in the graphs of figs. 1 and 2. All the quarks and leptons of the fourth family get degenerate (at the GUT scale) masses of order of the weak scale via the graph of fig. 1. The diagram of fig. 2 generates a Cabibbo mixing of the 4th and 3rd families.

Volume 140B, number 1,2

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31 May 1984 m

t64

164

16H

t63

16H

164

I I

21024

10

12045

Fig. 1. Mass generation for the fourth family (16) via its coupling to the weak Higgs (10). At this point in spite o f the fact that only two graphs give us all the quark and lepton 2 X 2 mass matrices o f this model no interesting predictions for the top quark mass carl be made. The reason for this is as follows. The VEV o f the 210 can point in any sU3 X SU 2 X U 1 direction. There are three basic directions in 210 which conserve SU(3) X SU(2) X U(1). They can be simply characterized b y the SU 5 representation to which they belong. Thus in an obvious notation they are denoted by 2101,21024 and 21075 depending on whether they belong to the singlet, 24or 75-dimensional representation o f SU(5). A general SU(3) × SU(2) X U(1) conserving VEV points along a linear combination o f these three directions and thus has the form a2101 + ~21024 + 721075. a,/3, and 7 c a n n o t be determined from O(10) group theory. They depend on parameters o f the Higgs potential o f the theory. Thus their presence greatly diminishes the predictive power o fthe theory and makes it impossible to obtain interesting relations involving the top quark. To see how the ESH ameliorates this situation consider a scenario in which O(10) breaks down to SU(5) at a large mass scale M 1 which can for example be

Fig. 2. Cabibbo mixing of the 4th and 3rd families. 16 and 16 get superheavy (~MGUT) masses via their coupling to the scalar 210. The subscripts denote the SU(5) representation to which the VEVs belong.

~Mplanck. SU(5) in turn breaks down to SU(3) × SU(2) × U(1) at a much smaller mass scale M 2 MGU T. At the large M 1 let 2101 and 21075 acquire super large masses ~ M 1 but keep 21024 massless. Then the only relevant SU 5 representation which is in existence at energies as low a s M 2 ~ M G u T is 21024. Thus only 21024 can get a significant VEV in this theory. The remaining 2101 and 21075 can only get VEVs highly suppressed by powers o f M 2 / M 1 . Note, o f course, that as always in implementing ESH b y keeping 21024 light we had to conduct a fine tuning o f the usual sort. Now that the VEVs o f the 210 point almost completely in the 21024 direction the couplings o f the 210 VEV to ups, downs and lepton is proportional to a simple group theoretic Clebsch that can be found in table 1. The 120 o f O(10) contains two types o f weak doublets denoted by 1205 and 12045 depending on whether they belong to the 5 or 45 o f SU 5 . The 1205 is undesirable because it does not contribute to the up quark mass matrix (this simply follows because 5 couples symmetrically to up quarks whereas 120

Table 1 Couplings of the various VEVs to the particle states in the 16 of O(10). In the last row we present the particle states appropriate to the "anti-SU(5)" case. The subscripts denote the SUs representation to which each VEV belongs. VEV direction

State (SUs case) = d

1

451 4524 2101 21024 21075

d

u

6

e÷

e-

v

Y

1

1

1

1

1

1

1

1

-3 2 1 -6 0

1 1 -1 1 -1

1 1 -1 1 -1

1 -4 -1 -4 +1

1 6 -1 6 +3

-3 -3 +1 9 0

-3 -3 +1 9 0

+5 0 +5 0 0

u

d

d

V

v

e-

e+

State (anfi-SU5) = ~

69

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PHYSICS LETTERS 4524

21024 I I t I

t6 T6 H

21o~

t I I t6 H

16~

lzo45

~'~-I

21o~5

Fig. 3. Mass of the 3rd family in the model of section 5. couples antisymmetricaUy). Thus again we use the ESH to give a large mass to the 1205 and keep only the 12045 down to the weak scale. The couplings o f 12045 to up, down and charged leptons are proportional to - 2 , 1, 3, respectively. From these and from the couplings o f 21024 (see table 1) we can easily obtain b = r and t = 6b. 5. A 3-family 0(10] model with t = 6b. In the previous model the r-family was light in comparison to the weak scale simply because it got a mass via Cabibbo feed down from a heavy fourth family degenerate at the weak scale. The model o f this section illustrates another class theory with just three families in which the lightness o f the r-family in comparison to the weak scale is explained in another way. The theory contains one 120, one 45 and three 210 scalars as well as 3 fermion 16s and two 16s. The Yukawa couplings o f the theory are defined by the graph o f fig. 3. This graph gives us no relation between t, b and ~"masses unless we invoke the ESH because of, as we em-

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phasized be fore, the uncalculable continuous variety o f directions in which each VEV o f fig. 2 can point. We use the ESH in a way completely analogous to the previous section;we imagine O(10) breaking down to SU(5) at a scale M 1 ~ Mplanck. The only SU(5) multiplets that do not obtain masses ~ M 1 are (in an obvious notation): 12045; 4524; 2101 ; 210~4; 210~5. These multiplets are therefore the only ones available to get VEVs at lower energies ~ M G u T and MWEAK. It is easy to see from fig. 3 and table 1 that we again obtain r = b and t = 6b. 6. Conclusion. The extended survival hypothesis can greatly enhance the predictive power of a theory. The examples o f this paper show how ESH can lead to predictions for the top quark mass. In ref. [3] more examples are worked out in which all o f the quark and lepton masses and mixing angles (a total o f 20 parameters) are obtained in terms o f just 4 input parameters. In those examples too the preferred value for t was 6b. References [1] See, e.g.S. Weinberg, Phys. Rev. Lett. 29 (1972) 388; H. Georgi and S i . Glashow, Phys. Rev. D7 (1973) 2457; H. Georgi and A. Pats, Phys. Rev. D10 (1974) 539. [2] M. Chanowitz, J. Ellis and M.K. Galliard, Nucl. Phys. 128 (1977) 506; A. Buras et al., Nucl. Phys. B135 (1978) 66. [3] F. del Aguila and L. Ibafiez, Nucl. Phys. B177 (1981) 60. [4] S. Dimopoulos, Phys. Lett. B129 (1983) 417.