Volume 113B, number 3 PHYSICS LETTERS 17 June 1982

OPERATOR ANALYSIS OF PROTON DECAY IN AN SU(7) GUT ~

Ann NELSON 1 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 17 March 1982

An SU(7) model, with a surviving low-energy SU(4) × SU(3) × U(1) gauge symmetry, may reduce the numbei of low- energy effective four-fermion baryon number violating interactions from six to two. A simple argument, applicable to gen- eIal SU(N), shows that the terms allowed by SU(4) × SU(3) × U(1) are the same as those allowed by SU(7)..

Introduction. The observation o f baryon-number or lepton-number violating interactions would imply that the standard model is not complete, and that there is new physics at energies >>M~¢. All renormal- izable interactions in the standard model conserve B - L as an automatic consequence of the SU(3) × SU(2) × U(1) gauge symmetry. However, for some time theorists have been busily embedding this model into larger gauge groups, which are broken to SU(3) × SU(2) X U(1) in one or several stages at mass scales >M w. This new physics may be described at current- ly accessible energies by an effective lagrangian con- taining local non-renormalizable SU(3) X SU(2) × U(1) invariant terms [ 1 -4 ] , which are constructed from fields in the standard model. Such terms would be suppressed by coupling constants of magnitude 1/n (d - 4), where d is the dimension of the operator, and M is the associated heavy mass scale. Specific GUTS provide constaints on the relative coefficients of the various operators, just as experimental mea- surement of the coefficients would constrain GUTS.

An SU(7) model exemplifies the complementary relationship between the constraints of general opera- tor analysis and GUTS. In this model, the maximal subgroup SU(4) × SU(3) × U(1) survives down to en- ergies ~ M w. Using a simple method applicable to all SU(N) models, we f'md this subgroup provides some

* This research is supported in part by the National Science Foundation under Grant No. PHY77-22864.

1 Supported in part by an NSF Pre-doctoral Fellowship.

interesting constraints on baryon-number violating operators.

SU(7]. This model is the first of a general class o f SU(N) models found by Dawson and Georgi [5] which share the successful prediction of the popular Georgi Glashow SU(5) for sin20 w andMGu M [6,7]. Under SU(5)GG, the fundamental 7 contains a 5 and two singlets. At MGU M , SU(7) ~ SU(4) × SU(3) X U(1), where color is entirely contained in SU(4), SU(2)L , in SU(3), and the hypercharge generator is a sum of generators from all three. The renormalization group equations for the SU(3) X SU(2) × U(1) cou- pling constants are [5,7] :

[gi(Mw)] 2 = [gGu(MGuM) ] -2

+ 2b i ~n (MGuM/Mw) + const In (MGUM/#),

where the constant depends on the matter and Higgs representation. Note that # as determined from this equation is arbitrary, as its only effect is on gGU •

There are several possible anomaly free combina- tions of the fundamental reps of SU(7) which could contain the observed families. Each family can be in either a 21 ¢ 3 7's (case 1) or a 35 ¢ 2 7"s (case 2). The extra fermions are in representations which are real with respect to SU(3) × SU(2) × U(1) and get masses of order/a. For the second case, we must be careful how we break SU(4) X SU(3) × U(1) to SU(3) × SU(2) × U(1), or the renormalizable interactions may mediate baryon-number violation via the diagram infig. 1.

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Volume 113B, number 3 PHYSICS LETTERS 17 June 1982

QL "UL qL ~ ~

i eL ~ ~ ~ qL

UL

Fig. 1. q, Q are quark doublets. W' is a SU(3) gauge boson with SU(3) c X SU(2) X U(1) quantum numbers (1, 2, 1/2). G' is an SU(4) gauge boson with SU(3) c × SU(2) X U(1) quantum numbers (3, 1, - 1/3). QL, UL, and either qL and

+ - - t ' e L or u L and qL are heavy (mass -~t) and carry unconven- tional baryon and lepton number.

The symmetry must be carefully broken so that only two of the quarks participating in this diagram correspond to observed families. The others must combine with heavy particles to get mass ~, and are assigned unconventional baryon numbers. Then it will be possible to assign global quantum numbers to all fermions and light Higgs, which prevent proton decay by interactions below the grand unified scale.

Proton decay. Some convenient group-theoretical properties of SU(N) will help us classify the varieties of proton decay.

(1) N-ality is additively conserved in tensor p r o - ducts of representations.

(2) If a representation o f SU(N) contains a sing, let under a regular maximal subgroup G, the rep hasN- ality 0.

(3) A tensor product of fundamental SU(N) reps with total N-altiy o f 0 contains at least one SU(N) singlet.

From these facts we can deduce that if a term in an effective lagrangian, which is constructed from fields in fundamental reps of SU(N), is allowed by G, it is also allowed by SU(N). SU(N) only restricts the relative coefficients of some terms above the GUM. For example, all terms allowed by SU(3) X SU(2) X U(1) are also allowed by SU(5).

Table 1 Lowest dimension operators relevant for proton decay [1-4].

Case Operator Type

1 (u T C3AxqL) (e T C3,/aqL ) vector 2 (u T Cy#qL ) (~T CT~tdR ) vector 3 (f~T rC qL ) (qT rC q'L) scalar 4 (~T C qL ) (qT C qL ) scalar 5 (e T CUR) (u T CdR) scalai 6 (er CdR) C ui0 scalar

A number of authors [ 1 -4 ] have tabulated the lowest dimension operators relevant for proton de- cay. These are shown in table 1.

If the low-energy families are in 21's and 7's (case 1 in table 1), all of these terms are also allowed by SU(4) X SU(3) × U(1). However, if the low energy families are in 35's and 7's (case 2 in table 1), the four operators generated only by the exchange of heavy scalars have 7-ality v~ 0, and so are not allowed. Thus, we can reduce the number of parameters for proton decay from 6 to 2. The same reasoning may be transplanted to larger SU(N) groups with low- energy maximal subgroups with the same results in many cases.

I wish to thank Howard Georgi for many useful conservations.

References

[1] S. Weinberg, Phys. Rev. D42 (1980) 1694. [21 S. Weinberg, Phys. Rev. D43 (1980) 1566. [3] H. Weldon and A. Zee, Nucl. Phys. B173 (1980) 1571. [4] F. Wilczek and A. Zee, Phys. Rev. D43 (1980) 1571. [5] S. Dawson and H. Georgi, Phys. Rev. Lett. 43 (1979)

821. [6] H. Georgi and S.L. Glashow, Phys. Rev. D32 (1974)

438. [7] H. Georgi, H. Quinn and S. Weinberg, Phys. Rev. D33

(1974) 451.

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