The calculation of the matrix element for proton decay

Volume 105B, number 1

PHYSICS LETTERS

24 September 1981

THE CALCULATION OF THE MATRIX ELEMENT FOR PROTON DECAY V.S. BEREZINSKY a, B.L. IOFFE b and Ya.I. KOGAN b a Institute for Nuclear Research, Academy o f Sciences o f the USSR, Moscow, USSR b Institute o f Theoretical and Experimental Physics, Moscow, USSR

Received 17 June 1981

Matrix elements and proton decay probabilities are calculated in the SU(5) model of grand unification with the help of the matrix element of the proton transition into three quarks, found previously by the QCD sum rules. The results show that an increase of the existing experimental lower limit for the proton lifetime by a factor of 3-4 would rule out the standard SU(5) model.

1. I n t r o d u c t i o n . At present the theoretical predictions for the proton lifetime in unified models of strong and electroweak interactions cover a very wide interval of rp. Even in the framework of the simplest SU(5) model [1 ] with three generations of fermions, 5- and 24-plets of Higgs fields the uncertainty in the proton lifetime is estimated (see reviews [2-4] ) by a factor of 100-500. This uncertainty is caused basically by two reasons: (i) The uncertainty in the superheavy gauge boson mass rn x which arises mainly due to the uncertainty in the constant % in QCD (or equivalently, the IR cutoff parameter A). The values of A differing from each other by a factor of 2 - 3 cannot be excluded nowadays and since rp ~ m 4 and m x ~ A, this circumstance yields the uncertainty in rp of the order of 10-100. (ii) Uncertainties in the model calculations of the nucleon matrix element of the given quark lagrangian for the transition of three quarks into a lepton. These uncertainties, depending on the model, result in the values of rp differing by a factor of about 40 (see refs. [2,4] ). In this paper we calculate the nucleon matrix element without any nucleon model using the matrix element of the nucleon transition into three quarks obtained in ref. [5] by QCD sum rules. This procedure will allow one to decrease substantially uncertainty (ii) in the proton lifetime. But at first a few words are in order about the previous calculations of this matrix element.

The nucleon matrix element has been calculated by several methods: in the SU(6) quark model [6], in the quark bag model [7,8] and by the use of PCAC and some additional assumptions. The essence of the quark model method is the following. Two quarks interacting at one point produce an antiquark and a lepton. The antiquark together with the third quark-spectator produces a meson. The three-quark annihilation into a lepton (with the coherent emission of the final meson by three quarks) is neglected because the probability of finding three quarks at one point is small. To see the restrictions of this method, let us consider the hypothetical proton decay into the isovector photon 7v and a fictitious r-lepton, such that m r < mp and mp - m r ,~ mchar where mchar is the characteristic mass of strong interactions. The preceding consideration is not changed and the decay p -~ r + + 7v should occur via the two-quark collision at one point (fig. la). But in fact, according to the Low theorem, at small photon momenta k ~ mchar the matrix element corresponding to the graph in fig. I a vanishes and only the coherent emission of the isovector photon by three quarks survives (fig. Ib). This consideration is directly related to the process p ~ e+p 0, but refers also to any process p ~ e%r0, p ~/a+K 0, etc. [9]. Under the real conditions k ~ mchar, therefore the contribution of the graph in fig. 1b cannot be neglected, and rather vice versa, it may be that it will exceed

0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company

33

Volume 105B, number 1

PHYSICS LETTERS

24 September 1981

the anomalous dimensions of the three-quark currents. For the calculation of the matrix element two different, although related, methods will be used. M (a)

(b) Fig. 1.

the contribution of the graph in fig. la. In ref. [5] from the QCD sum rules for the polarization operator of the three-quark current

rl(x) = eelk [ui(x)CT ul.(x)] 75~u dk(X),

(1)

the matrix element = XVp

(2)

was found. Here u(x) and d(x) are the field operators of the u and d quarks, i, L k are colour indices, C is the charge conjugation matrix and Vp is the proton wavefunction. For the constant X2 the numerical value was found to be X2 = 1.2 X 10 -3 GeV 6 (-+50%). We restrict our consideration by the SU(5) model for which the low-energy lagrangian is

£ = 4(GGuM/X/2)Aei/k

-¢ UL/ (URi3'~

X [(1 * coS20c)e~u dLk * sin 0 c cos 0 c ~ ' ~ dLk + (1 + s i n 2 0 c ) ~ ' dRk

SLk + sin 0 c cos 0 c ~ '

SLk

ktR')'~SRk ]

+ ~fiLV/~dRk)} ,

where GGUM/X/-~=4UM/8m2x -gGUM/8my _ 2 2

(3)

is the effective four-fermion constant, 0 c is the Cabibbo angle, A is the renormalization factor arising due to 34

1 = Ima(p2'k2'

a(p2'k2'O)=~ fo

p2

2

P~)dp2"

(4)

The integral on the rhs of eq. (4) rapidly converges. The saturation of the rhs of eq. (4) by the one-baryon pole is a good approximation due to the suppression of non-pole terms by a s, the vanishing of the/x (3,3) resonance contribution and the dp2[p 4 convergence of the integral. The examples of less convergent sum rules of Adler and Weisberger and Cabibbo and Radicarl demonstrate that the contribution of heavy intermediate states does not exceed 10-50%. As follows from eq. (3), for the decays p ~ e+zr0, p ~ e+p 0 2 will be proportional to and p -+ e + w, Im a(p 2 , k 2 , Pe) Im a(p 2, k 2, P e2) ~ e-+ ( 3 v s - 1)5 X 6(p 2 - m2),

+ fi~i"f~(dLj cos 0 c + SLy sin Oc) X (ffel'I'/a dRk

2. The pole approximation method. Consider the proton two-particle decay p ~ ~M where ~ is a lepton = e+, /a+, ~e, ~u and M the M-meson, M = 7r, p, co, K. We shall study the dependence of any form factor of this process a(p 2, k 2, p2) on the lepton masssquared p2 at fixed momentum-squared of the proton, p2, and the meson k 2. Using the quark counting rule [10] it can be readily shown that at p2 _+ oo, a(p2, k 2, p2) decreases as lip 2 or faster. (The decrease of a ~ 1/pQ stems from dimension considerations, an additional supression ~ 1/p~ is a result of chirality conservation.) Thereby, for a(p 2) the subtractionless dispersion relation in p2 takes place Because p2 = m 2 , m 2 ,~ m 2 , we can put p 2 = 0 and for a(p 2, k 2 , 0) we write

(5)

where JM is the current of the corresponding meson and 0 c = 0 has been used. Thus, the relations (4), (5) allow us to express the decay amplitudes p ~ e+rr0, p -* e+p 0 , p ~ e+co through the known matrix element (2) and the coupling constants of the rr0, p0 and co-mesons with nucleons. As a result the matrix elements and the decay widths of p ~ e+0r 0 , p0, co) are

Volume 105B, number 1

M(p -~ e+Tr0) =

PHYSICS LETTERS

(GGuM/X/~)AX(g~r/mp)~+(75--3)Vp,

F(p -> e+Tr0) = (5/167r)G2uM A2X2-2~m 2 g ~ r /p'

written as (0 c = 0):

(6a)

M = V~ GGUM A ( 2 ~ - ~ ) T h ,

(6b)

where

M(p -~ e+(p 0 , co)) = (GGuM/X/'2) AX(gp,to/2mp) X g+(1 - 33,5) [3,u +

24 September 1981

(8)

T h = i f d 4 x eikX([] +/a2)(01T (tp(3)(x),

i~Va'S/mp)°xkx ] V ,

(7a)

F(p -~ e+(p 0, co))

r/(0)} IP),

(9)

~o(3)(x) is the pion operator. At arbitrary k 2 and (pk) Th(P, k, Pc) is expressed via two form factors Th = g l ( p 2, k 2, pk)75 lip

5 2 . 2 . 2 2 (m2p-- mo,to 2 )2 327r GGUM 2't A g,o,to m5 p

+ g2(.p 2, k 2, pk) (fC/mp)75 Vp.

v,s - --v,s~2 2m2 + m2 to"

X 1 +P +3//a a-~/a ) 2m2,to

--2 m2p

(10)

Using the PCAC relation aujsu = (ts2f,JV~)~0(3), we find from (9) at k 2 = 0:

.

/ (7b)

Here g~r o to are the cou[~ling constants of the 7r, p, ~-mesons with nucleons, g~/4rr = 14, g2/4rr = 2.5, 7r p g2co/4rr = 18, ~uav'sare the isovector and lsoscalar anomalous magnetic moments of the nucleons, ~uv = (pp 1 - Un)/2, USa= ~ p - 1 + Un)/2. When deriving formula (7a) we have used the expression for the matrix element (plfo,to IP) followed from the VDM. The widths of other two-particle proton decays can be calculated similarly. The ratios of these widths to the p -* e+rr0 decay width are presented in table 1. (The values F(p -~/a+K 0) and F(p ~ e+~7) are unreliably determined because of the poor knowledge of the constants gpKo ~÷ and gprm ")

3. The PCAC method. Using the PCAC method, the form factors of the decay p ~ e+Tr0 can be calculated at ku ~ 0 where kg is the pion momentum. Note that owing to kinematical relations, the transition to the limit k , -+ 0 in the matrix element is impossible. However, the form factor in this method can be calculated in the nonphysical region p2 _~ mp2 and (pk) -+ O. The matrix element of the decay p -+ e+lr0 can be

Th=~(k~M~+fd4xeikx(o[[J5,0(x),~7(O)]xo=O (11) where Mu is the matrix element of the process p -+ e + + axial current. The equal-time commutator can be calculated using the explicit form of r/(x) andJs,0(x ) = (fliT0 75ui -- d/TOTS di)/2. [/'5,0(x), r/(0)] Xo=0 = 2-1/23, 5 r/(0) ~ 3 (x). Consider now T h at k u -> 0. The nonvanishing contribution to k u M . comes only from the pole graph (fig. 2) and thus

Th -~ (XIvr2frr)((gA -- 1)3,5 Vp +gA [rap I¢/(pk)] 3,5 V } . (12) Assume now that the form factors gl and g2 are smooth functions of (pk) within the interval 0 < pk < m2/2. From (8) and (12)it then follows:

Fig. 2.

Table 1 The ratio of the proton two-particle decay widths to the p --*e%° decay width. Final state f

e÷p°

e÷to

Velr÷

g+K o

e÷rl

tt+lro

rf/r (p -~ e÷Tr°)

0.11

0.10

2/5

0.03-0.10

0.01-0.08

1 sin20e

35

Volume 105B, number 1

PHYSICS LETTERS

M = GGUM AX[(gA + 1)/2frr] ~+(75 - 3)Vp.

(13)

Comparing the eqs. (6a) and (13) we find that the ratio of these two expressions is v~g~fJ(gA+l)mp and close to unity (~1.15) due to the GoldbergerTreiman relation• Thus, the PCAC method agrees within 15% uncertainty with the pole approximation method ,1 The calculations for the decays p ~ p+lr 0 , p ~ e r r 0, p ~ p+K 0 and p ~ ~uK 0 are analogous to the above. For the last two decays significant uncertainties appear due to the use of PCAC for K-mesons.

4. The proton lifetime. At a fixed value of GGUM (i.e. mx) the uncertainty in the calculation of the partial widths of the proton decay is determined in our pole approximation method mainly by the contribution to eq. (4) of other, besides the one-nucleon, intermediate states, and in the PCAC method - by the extrapolation of PCAC relations up to momenta mp/2. According to the discussion in sections 2 and 3 we cautiously estimate this uncertainty (including the 50% error in X2) to be a factor of 2 -+ 1. To compare our results with those of other authors we use m x = 2 × 1014 GeV, g2UM/41r = 0.024 and A 2 ~ 10. Then we have r(p ~ e+lr0) = 5.3 × 1028 yr, rp,to t = r(p ~ e+TrO)/B = 2.6 × 1028 yr, B = Ptot/F(p ~ e+zr0) ~ 2. •

(14) +

0

Our expectatmn of r(p ~ e rr ) and rp. tot are smaller by a factor of 2 - 5 than those ma~e in the SU(6) quark model [2,3,6]. But at the same time there is a strong disagreement with the calculations made in the quark bag model [8], which give values 2 0 - 1 0 0 as large for r(p ~ e+Tr0) and rp,to t. Since we exclude the possibility of such large errors in our resuits, we have to consider the results of ref. [8] to be wrong• To find a more confident value of rp,to t we have repeated the calculation of m x (see refs. [ 2 - 4 ] ). In ,l The PCAC method was used also by Tomozawa [11] for the calculation of the p ~ e+lr° amplitude. In his approach the constant h was calculated with the help of the BetheSalpeter equation for the quark wavefunction in the relativistic oscillator potential, i.e. in a quite different way than in ref. [5]• Nevertheless, our results are in surprisingly good agreement. 36

24 September 1981

our calculations we started from the value of as(Q 2) at fixed 002 but not from A and we took as(Q2 ) from directly measured physical parameters. For any regularization scheme, for which the higher-order corrections to as(Q02) are small, the dependence of a s on Q2 is determined by the renormalization group equation and is independent on the regularization procedure• Using the charmonium data we took as(10 GeV 2) = 0.20 which corresponds in the leading logarithmic approximation to ALLA = 95 MeV (or in the twoloop approximation to A 2 loop = 220 MeV). This value of as(10 GeV 2) agrees with the a s measurements at production of gluonic jets in e+e - annihilation [12] as well as with the recent data on deep inelastic eN and vN scattering [ 13 ]. Using this value of a s we have obtained m x = 1.8 × 1014 GeV in the standard SU(5) model taking into account two-loop approximation and threshold effects and taking a~-n~(4M2 ) = 128.5. This value of m x agrees with the calculations presented in refs. [ 2 - 4 ] which now seems to be most reliable. Therefore, we consider the estimate (14) as a sufficiently confident prediction for the proton lifetime in the standard SU(5) model. Since (14) contradicts the experimental lower limit rp,ex p > 1.2 × 1030 yr [14,15], possible errors which could increase the ~-p,theor are to be estimated. It seems unlikely for as(10 GeV 2) to exceed the value 0.25. At as(10 GeV2) = 0.25, (ALLA = 190 MeV, A 2 loop = 400 MeV) and a -em l ( 4 M 2w ) = 129 "0, m x equals to 4.4 × 1014 GeV • (The increase o f a -e lm( 4 M w2 ) from 128.5 to 129 enlarges m x by a factor of 1.2.) Taking as a reasonable theoretical bound m x = 5 × 1014 GeV and accounting for the uncertainty in the nucleon matrix element squared (factor 2) we have rp,theor ~ 2 X 1030 yr. Thus, it follows that increasing of the experimental lower limit for rp by a factor of 3 - 4 would safely rule out the standard SU(5)model. We are grateful to M.V. Terentyev, M.A. Shifman and M.B. Voloshin for useful discussions and remarks•

References [1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [2] J. Ellis, M.K. Gafllard, D.V. Nanopoulos and S. Rudaz, Nucl. Phys. B176 (1980) 61.

Volume 105B, number 1

PHYSICS LETTERS

[3] T. Goldman and D.A. Ross, Nucl. Phys. B171 (1980) 273. [4] P. Langacker, preprint SLAC-PUB-2554 (1980). [5] B.L. Ioffe, preprint ITEP-10 (1981). [6] A.J. Buras, J. Ellis, M.K. GaiUard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66; C. Jarlskog and F. Yndurain, Nucl. Phys. B149 (1979) 19; M. Machacek, Nucl. Phys. B159 (1979) 37; G. Kane and G. Karl, Phys. Rev. D22 (1980) 1808: M.B. Gavela, A. Le Yaouanc, L. Oliver, O. Pene and J.C. Raynal, Phys. Rev. D23 (1981) 1580. [7] A. Din, G. Girardi and P. Sorba, Phys. Lett. 91B (1980) 77. [8] J.F. Donoghue, Phys. Lett. 92B (1980) 99; E. Golowich, Phys. Rev. D22 (1980) 1148.

24 September 1981

[9] A. Hurlbert and F. Wflczek, Phys. Lett. 92B (1980) 95; M.B. Wise, R. Blankenbecker and L.F. Abbot, Phys. Rev. D23 (1981) 1591. [10] V.A. Matveev, R.M. Muradyan and A.N. Tavkhelidze, Lett. Nuovo Cimento 7 (1973) 719; S.J. Brodsky and G.R. Farra.r, Phys. Rev. Lett. 31 (1973) 1153. [11] Y. Tomozawa, Phys. Rev. Lett. 46 (1981) 463. [12] B.H. Wiik, preprint DESY 80/129 (1980). [ 13] H. Wahl, report XVI Recontre de Moriond Elementary particle physics Meeting (1981). [14] J. Learned, F. Reines and A. Soni, Phys. Rev. Lett. 43 (1979) 907. [15] E.N. Alekseev et aJ., report XVIIth Intern. Cosmic Ray Conf. (1981).

37