Volume 117B, number 3,4 PHYSICS LETTERS 11 November 1982

ON THE CONSISTENCY OF CHIRAL SYMMETRY AND THE QUARK MODEL IN PROTON DECAY

Nathan ISGUR 1 Department of Physics, University of Toronto, Toronto, Ontario, Canada, MSS 1A 7

and

Mark B. WISE 2 L yman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 29 June 1982 Revised manuscript received 13 September 1982

We have examined an apparent discrepancy between the predictions of chiral perturbation theory and the quark model in proton decay and found that the disagreement is resolved by taking into account a neglected diagram in the quark model. Our results increase the predicted rate for p -~ ,rOe+; in minimal SU(5) this increase leads to a discrepancy with experiment i f A ~ = 150 MeV.

1. Introduction. Models which unify the strong, weak, and electromagnetic interactions typically contain new interactions which violate baryon number. Aside from their intrinsic interest, such interactions are now recog- nized as offering an explanation of the present baryon number asymmetry of the universe in the standard cos- mology [1,2] and in the new inflationary cosmology [3]. In a large class of grand unified models, renormalization group analyses [4,5] of the SU(3), SU(2) and U(1) gauge couplings have indicated that the scale of grand unifica- tion is of order 1015 GeV so that baryon-number-violating nucleon decay should be observed in the present gener- ation of proton decay experiments if these models are correct.

I f the unification scale is of order 1015 GeV, and there are no exotic particles much lighter than this mass scale, then only baryon-number-violating operators of dimension six contribute to nucleon decay at an observable rate: there are no lower dimension baryon-number-violating operators consistent with SU(3) × SU(2) X U(1) symme- try, and higher dimension operators are suppressed by additional powers of the unification mass. The dimension- six baryon-number-violating operators are [6,7] (in the two-component notation of ref. [7]):

(1) _ Oabcd - ( d~aR U3bR)(qi3,cL ~/dL) e~vei / , O(2) d = (qiaaL qj3bL) (UycR £dR) ea~ eij , (la, b)

O(3)cd = (qieaL q]3bL)(qkTcL I~ldL) ealyreile]k , O(4) d = (deaR U3bR)(U-rcR g.dR) eo<3.r . (lc, d)

Here a,/3, 3' are SU(3) color indices, i, j, k, 1 are SU(2) indices, a, b, c, d are generation indices and L and R denote left- and right-handed fields.

The nonrelativistic quark model [8], the bag model [9], and chiral perturbation theory [10] have been used to compute the hadronic matrix elements of the operators in (1) for various nucleon decay modes. In chiral perturba- tion theory, relative rates for decays to a pseudoscalar meson plus an antilepton were calculated and it was found

1 Work supported in part by the Natural Sciences and Engineering Research Council of Canada. 2 This research is supported in part by the National Science Foundation under Contract No. PHY77-22864, and the Harvard Soci-

ety of Fellows.

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Volume 117B, number 3,4 PHYSICS LETTERS 11 November 1982

that both the non-pole diagram in fig. la and the pole diagram in fig. lb play an important role. For example, in the decay p -77r0e + the two diagrams are about the same magnitude and interfere constructively. Previous non- relativistic quark model and bag model calculations of nucleon decay rates and branching ratios have neglected the pole diagram [8,9] (fig. lb) and therefore reached conclusions which differ from those of chiral perturbation the- ory. This paper resolves this discrepancy by performing a nonrelativistic quark model calculation of both the pole and non-pole contributions.

In section 2 decay rates for p -+ rr°e + and p -* rTe + are computed in the nonrelativistic quark model assuming that vector exchange dominates the baryon-number-violating interactions. We find that the pole diagram does play a significant role and that our results are consistent with those of chiral perturbation theory. In addition, we are able to understand why lowest order chiral perturbation theory should work for nucleon decay despite its failure in the superficially similar nonleptonic hyperon decays. In section 3 we re-examine nucleon decay rates in the minimal SU(5) model [11 ] and find a discrepancy with experiment of two orders of magnitude if Agg = 150 MeV. We also discuss the significance of this result in section 3 and conclude that a modest improvement in the experimental limits would rule out this model. Finally, section 4 contains a summary of our work.

2. Baryon decay in the quark model. The amplitude for fig. la has been calculated in the nonrelativistic quark model on many occasions [8]. We calculate it here to set the stage for our calculation of fig. lb. For the baryons and pseudoscalar mesons we need in our calculation we take the states

ec~tyr ~ d ~i ] I B(.p, s)> = (.EB/a~B) 1/2 3 -3/4 fd3po f d3ph - ~ YdaijkPB;abcX2;s 1g2S3 ~k(Po, Ph)

X Iqaa(lp + 6-1/2px + 2-1/2po; Sl) q~b(~P + 6-1/2px -- 2-1/2pp; s2) q'rc(~P -- T . V~p~,, s3)>, (2)

[ P(q)> = (2°3)I/2 fd3p V;,b ffeCo) I q~(½ q + p; s) #~b(½ q -- P; ~>, (3)

where (/~B/MB) 1/2 and (26o) 1/2 are normalization factors for these nonrelativistic states *~ and in which we have taken the approximation mu =md = ms. Here abc are flavor indices, while the ask are SU(6) X 0(3) coupling coefficients to a configuration d which appears in the baryon B with strength/d; 4, × and #7 are the corresponding flavor, spin, and momentum wave functions [12].

Assuming gauge boson exchange dominance, the relevant operators for both sub-processes are (in four-compo-

,1 In this approximation the mass M of one of these states is just the sum of the masses of its constituent quarks.

\ \

II

N

(o)

x x I \ \ \ \

N

(b) Fig. 1. Feynman diagrams contributing to the amplitude for nucleon decay to a meson (dashed line) and antilepton.

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nent notat ion)

Q_+ = eag 7 [dc ½(1 ¥ 75) up] [u~ ½(1 ± 3'5) el, (4)

and the effective hamiltonian for baryon-number-violating interactions is

H = C+Q+ + C _ Q _ . (5)

Assuming the nucleon is pure [56,0 +] ,2, taking the static limit of all the Dirac spinors and using harmonic oscil- lator wavefunctions [12]

fP(P) = (1/7r3/4/33/2) exp (--p2/2/32), ~N(Pp, P?,) = (1/n3/2°~3) exp [ - (P 2 + P~)]2~2] , (6,7)

we find that the helicity ampli tude from fig. 1 a is [A + = A (a) + A +Co) is the amplitude for a proton with Sz = + 1/2 to decay to a pion moving in the +z direction].

A_~a)(p -+ e +n0) = -C+_(3wEe/8me) 1/2 [2V:J-n/(3/32 + 2a2)] 3/2~p(0) f iN(0 ,0 )F(q2) • (8)

ttere ffp(r) and ~N(P, k) are the normalized spatial wavefunctions conjugate to (6) and (7) and F ( q 2) is a form factor ,3. Since the matr ix element of the operators Q+ vanish for ud -7 e+g, we find tha tA a(p _+ e+rT) = - 2 - 1 / 2 A a(p -+ e+n 0) for the "perfect ly mixed" [15] ~ - 2-1 /2 [2-1/2(uff + dd~ - sg].

We next turn to the pole term of fig. lb , which gives a contribution (in old-fashioned time.ordered perturbat ion

theory)

A (_+b)(p __> e+Tr O) = ~ C+_@[)en/(EBn + co - MN), n

where we have dominated the intermediate states by a sum over all baryon resonances Bn, and where

(9)

a (+-) ~- (2zr)3(e+(--q, +--)lQ+_ I Bn(-q, +-)), (lO)

/ f i f f en ~ (2n)3(Bn(-q, +-) nO(q)l stronglp(0, -+)) ~- -gBnNnH(q 2) q/2MN ( l l a , b)

in which H(q 2) is the BNn form factor ,4. Note that eq. (9) neglects the "z-graph" time ordering of fig. lb . Since in the static limit an is proport ional to ffk(0,0), we find that in this approximation only the nucleon and its radial excitations will contribute to (9). Since, in addition, both the Un'S and the Cn'S for radial excitations decrease [12] and their energy denominators increase with excitation number, we find that A ~) is almost completely dominated by the nucleon pole term. This observation serves as a justification for the use of lowest order chiral perturbat ion theory in this problem and tfighlights the feature which distinguishes it from the problem of weak nonleptonic

,2 The nucleon is believed to have a mixture of other SU(6) × 0(3) configurations. See ref. [13] for a discussion of some of the effects of such mixing on proton decay.

,3 We find F(q 2) = exp[-3q2/(16a 2 + 2482)] and so conclude that, with the values of c~ and a used here F(q 2) is near unity. Our formula for F(q 2) does not appear to agree with those in Kane and Karl, Donoghue and Karl, or Gavela et al. in ref. [8]. It actually does agree with the first of these papers since their meson harmonic oscillator parameter, ~, is ~ times ours, corre- sponding to their using (1/x/~) (rq - r~q) as theft meson coordinate (Karl, [ 14]). In the second paper this "discrepancy" is fur- ther compounded by a misprint. We do not, however, agree with some of the quantitative conclusions of these first two papers. Our apparent disagreement with the third paper, on the other hand, disappears when theft R N and R M are interpreted not as

I 1/2 1 the rms nucleon and meson radii, but as wavefunction parameters corresponding to a - and 2 - ~3- in our calculation (Pbne [141).

,a We find, for example, that the NNn form factor is H(q 2) = exp[-qZ/6c~2].

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hyperon decays where orbitally excited states would contribute. For the pole diagram we therefore obtain the am- plitude

A~b)(p -+ e+rr 0) = -C+(x/~Eelrne) 112 [qgNN~rH(q2)I4MN(EN + co - MN)] qJN(0,0) • (12)

For p ~ e+~ one obtains the analogous formula with gNNrr replaced by gNN~ = (3X/~ 10) gNNTr- The parameters appearing in these helicity amplitudes are reasonably well known from other applications of

the quark model. The parameters ~ and/3 are both known from studies of baryon and meson spectroscopy and decays to be about 0.32 GeV and 0.22 GeV respectively ,s . The parameter c5 is, as mentioned above, the energy of the system of two non-relativistic constituent quarks. Note, for example, that in the same approximation

fn = (2k/~/rnn 1/2) t~p(0), (13)

which shows that r~r should indeed be related to the constituent quark mass, not to the physical value of the pion mass.

The quark model results are now constituent with the results of chiral perturbation theory. In particular we find that A (b)(p _+ e+rrO)/A (+a)(p _+ e+rr0) = 0.8, while in chiral perturbation [101 theory tttis ratio is gA ,6. The con- stmctive interference between these two contributions enhances the predicted e+~r ° rate by about a factor of four relative to previous quark model calculations. For p -+ e+r/the analogous ratio is -0 .31 so the rate for this process is suppressed, once again in agreement with the results of chiral perturbation theory.

We conclude this section by noting that the absolute rates for these decays may be obtained via

V(p ~ e+P) = (qme/4rrMN)([A+ 12 + IA_ 12). (14)

3. Predictions for minimal SU(5). In minimal SU(5), the coefficients of the operators Q._ are [7,17]

2 2/13~ 3) 9/4~ 2) 51 + 18/60~ 1) C._ = [(6 +- 2) 7rc~5/Mx ] [c~3(MN)/C~5 ] [c~2(MW)/C~5] [5cq(Mw)/3c~5] - . (15)

if mixing angles are neglected. In (15)

/3~ 1 ) = l l - - ~ N f , /3~2)=~ _ ~ N f 1 , / 3 ~ 3 ) = - ~ N f - ~ , (16a, b,c)

where Nf is the number of quark flavors, a5 = 0.024 and [5] MX = 1.5 × 1015 A m . Using (15) we predict that

1/F(p ~ e+Tr 0) = 6 X 1029(A~1~/0.15 GeV) 4 y r , 1/F(p ~ e+rT) = 10 × 1030(A~/0 .15 GeV) 4 y r .

In minimal SU(5) V(p ~ ~eTr +) = ~F(p ~ e+n 0) so since about 1/3 of the primary 7r+'s will survive to produce a/J+, we also have the result that 1/P(p ~/~ + X) ~< !~ 1/I ' (p ~ e+Tr0). The present limit on the rate for p - ' e+Tr 0 comes from the Kolar Gold Fields Experiment. They find [18] 1/F(p ~ e+lr 0) > 2 × 1031 yr. Other ongoing experiments ,7 are sensitive to decays leading to a muon and give the limit 1/F(p -,/~ + X) > 3 X 1030 yr. The experimental limit on the p ~ e+Tr 0 mode is above our prediction, with A~-g = 150 MeV, by a factor of about thirty.

,s These values are the ones we believe are most appropriate to the static limit being used here. See reL [ 16]. 2 :1-6 Note that this ratio in the quark model has, inserting (13) into (8) and going to the chiral limit mrr--, 0, q --' 0, the value

x/~[(3t32 + 22)/12~32 ]3/2 (gNNTrf~r/x/~MN) ~_ x/~[(3~32 + 2~2)/12132 ]3]2 gA

by the Goldberger-Treiman relation, so this agreement is not just a numerical coincidence. ,7 The experimental situation is reviewed, for example, in ref. [ 19].

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The significance of this discrepancy can, however, only be assessed by considering the uncertainties in our cal- culation. The uncertainties in the quark model calculation arise mainly from the parameters a and/3, determined from hadron spectroscopy and decays, and give rise to an uncertainty in our predicted rate for p ~ e+n 0 which we estimate to be about a factor of 3. The agreement with the results of chiral perturbat ion theory probably indicate that the error is not much worse than this. A more significant source of error is the value of Afi~. For example, if A~-S = 300 MeV, then the discrepancy is reduced by a factor of 16. There is an additional uncertainty in our predicted rate for p ~ n°e + of about a factor of 5 due to the possibility of superheavy scalars having masses which differ from MX by up to several orders of magnitude. In view of these assessments, we cannot exclude minimal SU(5) on the basis of these calculations. Nevertheless, with our predicted increase in the p -~ e+n 0 rate, a modest improvement in the present experimental limits would now create a discrepancy which could not, in our opinion, be accomodated.

4. Summary. In tiffs paper we have resolved the discrepancy between the predictions of chiral perturbat ion the- ory and the nonrelativistic quark model for baryon-number-violating nucleon decay. This was done by performing a quark model calculation of the rates for p ~ 7r°e + and p -~ r~e + that includes both the effects of fig. 1 a and fig. 1 b. We found that form factor suppressions are not significant and that the nucleon intermediate state dominates fig. lb . These results support the validity of lowest order chiral per turbat ion theory for nucleon decay processes. In minimal SU(5) we predict, for Afi~ = 150 MeV, a rate for p -~ n°e + that is now a factor of about thir ty above file experimental limit. Although the uncertainties are large enough to accomodate this discrepancy, a modest im- provement in the experimental l imit would, in our opinion, rule out minimal SU(5).

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