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A bag model calculation of the nucleon lifetime in grand unified theories
Volume 91B, number 1
PHYSICS LETTERS
24 March 1980
A BAG MODEL CALCULATION OF THE NUCLEON LIFETIME IN GRAND UNIFIED THEORIES A.M. DIN
CERN, Geneva, Switzerland and G. GIRARDI and P. SORBA i
LAPP, Anneey-le- Vieux, France Received 5 December 1979
The nucleon lifetime is evaluated using bag model wave functions for the quark in the effective baryon-number violating four-fermion interaction of the SU(5) grand unified theory. We obtain rp ~ 1030 years if m X - 4.2 X 1014 GeV.
Several proposals have recently been put foward with the purpose of improving the present experimental limit [1] on the proton lifetime 7p ~ 2 X 1030 yr. The great interest in such an endeavour stems from the fact that grand unified theories of electroweak and strong interactions [2] predict baryon-number violating nucleon decays. The theoretical proton decay rate estimates depend on the chosen unification scheme and a few key parameters. It seems, however, that rough estimates are so close to the experimental limit that an improvement o f the latter of, say, two orders o f magnitude might very well confirm the existence of proton decay, or in case of a negative result, at least exclude certain models. Before new experimental results appear it is important to eliminate as far as possible uncertainties in the involved parameters. In the specific case of the standard SU(5) unification scheme, which we will consider below, this has already been done to some extent. The absolute decay rate depends crucially on three different factors, the first one being the masses mx, Y o f the exchanged superheavy vector bosons which give rise to the effective four-fermion interaction
1 On leave from Centre de Physique Th6orique, CNRS, Marseilte, France.
£eff
-C
2-1/2GGu [ei]k UkL'),uU]L -+
-+
-+
(1) -+
X (2eL)'UdiL -- eR'),~tdiR + btL~,~zSiL -- pR'),uSiR) -c -c ")'~S/R)] + h.c. + CijkUkL~/pdjL (PCR')'~ZdiL + VpR where GGU = x/2g2/8m2,y, c stands for charge conjugation and i, j, k are colour indices. For simplicity we have here neglected Cabibbo mixing. The mass m x (assumed to be equal to m y ) thus enters in the lifetime as ~ x , Y ' The value of m X has been evaluated carefully taking account of thresholds, Higgs bosons and the electromagnetic coupling constant at the relevant scale [3]. It depends on the strong interaction scale parameter A. Secondly the lifetime also depends essentially on enhancement factors due to coupling constant evolution from the grand unification scale down to the energy scale 1 GeV. This renormalization of the effective lagrangian by SU(3) gluon exchange as well as by SU(2) and U(1) vector boson exchange has been studied in detail [4]. Apart from the inclusion of some other less important decay mechanisms [5] the third major uncertainty in the lifetime is the effect of the quark wave function which in the standard calculation enters as a factor I ~ (0) 12, i.e., the square of the wave function at the origin for the annihilating two-quark system. The 77
Volume 91B, number 1
PttYSICS LETTERS
same wave function factor appears in the calculation of nonleptonic hyperon decays and can be estimated to lie between 4 × 10 - 3 GeV 3 and 14 × 10 . 3 GeV 3 [6]. In view of this uncertainty and the superficial treatment o f the proton structure it would be desirable to have an alternative approach. We will here propose to treat the quark wave function effects in a standard bag model [7] with noninteracting quarks. The wellknown definitions and properties of the bag model will not be restated but we will simply use the lowest excitation wave functions with parameters such as the radius and the bag constant fixed from static hadronic properties. The physical mechanism behind proton decay, say p -+ n°e + to be explicit, will therefore be the following: two quarks with proton bag wave functions 6~ and ¢~ annihilate in the point r 1 and create there [because of the local interaction (1)] a positron with the wave function ~e +, taken to be a plane wave ~ e ik .r, and a antiquark with wave function 6~. For simplicity (i.e., to avoid use o f quark propagators in the bag) this antiquark is assumed to be created immediately in a pion bag. The third "spectator" quark q3 in the proton with wave function ~ is at some point r 2 transformed into a pion bag quark q5 with wave function ~ r . The two points r l and r 2 are constrained to lie in the overlap Vp r3 Vn o f the proton and pion bags. They could a priori be arbitrarily located in sPace but we will as a first approximation assume that their centres coincide. The bag radii R p and Rn are different but since RTr <~ Rp the bag overlap is thus effectively the pion bag V~r itself. The approximative picture which emerges is that of a kind o f " r a d i a t i v e " transition where a static proton bag is transformed into a static pion bag under the emission o f a positron. The rate for such a process [8] will be P = 2hiM2 [, where the transition matrix element is given by M~
f
d3r 1 ~L(rl)Tla~L(rl)~e+Leik'rxT"~b~L(rl)
v~
(2)
X f d3r 2 ~ ( r 2 ) f f s ( r 2 ) . V. One might argue that the fixed static bag approximation is more likely to be valid for proton decay inio a 78
24 March 1980
heavier particle than the n, say the p. A possibility is thus to go from the p rate to the n rate by includin~ a relativistic phase space factor (m p2 - m2)/(m p2 - m~) p • In a proper relativistic treatment one should also think of the pion bag as being Lorentz contracted with a factor (1 + k2/m2) 1/2. We will take this effect approximatively into account by substituting the integration in eq. (2) over the spherical volume V~ by integration over a sphere of volume V~r(1 + k2/m2) -112. We will not describe explicitly the evaluation o f the decay rate taking account of the proper SU(6) structure of the proton and meson wave functions. For the considered two-body decays p ~ n°e +, p0e+, co0e +, r/e+, n+Ue , P+Ue, K0/a+ ( in the latter case using the appropriate effective SU(5) interaction [5]) this is fairly straightforward. If one would like to get an idea of the importance of a three-body decay like p nOnOe+ one is confronted with the unpleasant difficulties of working with three bags. A simple way out is to use PCAC to reduce out a (soft) pion and subsequently use a chirally transformed version o f the previous two-bag calculation. Of course, the available nucleon rest mass energy is so big that neither pion might actually be soft, but assuming this not to be the case (which might be likely for a 2n 0 S-state) one finds in fact (provided one can extrapolate smoothly from the soft n point) that the direct 2n 0 decay is suppressed by a factor ~ 7 relative to the one n 0 decay. By isospin considerations one may argue that the 7r+n- rate is equally small (to within a factor 2). The results for the absolute lifetime are displayed in table 1, with account taken as mentioned above o f a relativistic phase space factor, and in table 2, without this factor, for a number of values of the mass m X and the generation number ng. The branching ratios are displayed in table 3. In tables 1 and 2 we have considered three different cases for the quark masses used in the bag wave functions: (1) mu, d = 0, m s = 280 MeV, (2) mu, d = 33 MeV, m s = 300 MeV [9], (3) mu, a = 110 MeV, m s = 350 M e V . It is seen that these different choices correspond to a variation of the lifetime with a factor 2. For a typical case o f r n X = 4.2 X 1014 GeV and ng = 3 we obtain rp 1030 yr which agrees roughly with the standard calculation. To be more specific about the relation between our treatment o f the quark wave functions with the standard approach one might evaluate the ratio
Volume 91B, n u m b e r 1
PHYSICS LETTERS
24 March 1980
Table 1 The proton lifetime ~-p in 103o yr versus the generation n u m b e r ng and the mass m X in GeV, with phase space correction, rp is displayed for three different choices o f quark masses. ng
mX 2.5 X 1014
3.3 X 1014
4.2 × 1014
5 X 1014
5.8 × 1014
3
0.1 0.19 0.15
0.27 0.56 0.45
0.65 1.3 1.0
1.2 2.5 1.9
2.0 4.0 3.5
4
0.08 0.15 0.12
0.23 0.45 0.34
0.53 1.1 0.85
1.05 2.2 1.7
1.8 3.6 2.9
5
0.045 0.09 0.07
0.13 0.28 0.22
0.32 0.65 0.51
0.63 1.3 1.0
1.1 2.3 1.7
6
0.015 0.032 0.025
0.05 0.1 0.08
0.12 0.24 0.18
0.23 0.47 0.36
0.4 0.8 0.63
Table 2 The proton lifetime r p as in table 1 b u t w i t h o u t phase space correction. ng
mx 2.5 × 1014
3.3 × 1014
4.2 × 1014
5 × 1014
5.8 × 1014
3
0.13 0.3 0.2
0.4 0.8 0.6
1.0 1.8 1.4
1.8 3.5 2.7
3.0 6.0 4.5
4
0.11 0.22 0.17
0.33 0.7 0.5
0.8 1.5 1.1
1.6 3.1 2.3
2.7 5.3 4.0
5
0.07 0.13 0.1
0.2 0.4 0.3
0.5 1.0 0.7
1.0 1.9 1.4
1.7 3.2 2.9
6
0.025 0.045 0.030
0.075 0.14 0.11
0.17 0.33 0.26
0.33 0.70 0.51
0.6 1.2 0.9
Table 3 Branching ratios in % for proton decay with (case a) and w i t h o u t (case b) space phase corrections. _
M
p~
Case a Case b
~rOe+
noTrOe+
pOe+
u)Oe +
rTe+
n+ff
p+~
K°ta +
31 15
4 2
21 32
19 29
5 4
11 5
8 12
0.5 ~0
79
Volume 91B, number 1
PHYSICS LETTERS
a = Z p (4y r )- / m_4J,-( G e V )- in various cases. In the work of Jarlskog and Yndurain [5] for example, a ~ 2 X 10.29 for ng = 3 whereas from our table 1 we find, taking ng = 3 and rnu,d = 0, that a = 2.5 X 10-29 for m x = 2.5 X 1014 and a = 2 X 10 . 2 9 for m X = 4.2 X 1014 . Here one has to remember, however, that we have included an SU(2) X U(1) enhancement factor of 3 [4] so that the conclusion is that our a effectively is a factor ~ 3 bigger. For the case of the branching ratios the results depend to some extent on the inclusion of the phase space factor, however, one might assume that a correct relativistic calculation will give results somewhere in between the stated ones. It thus appears that the branching ratios for p ~ 7r0e+, poe+ and woe + should be comparable while the branching ratios for 7r07r0e+ and K0/a+ are of the order of 1%. These results do not depend on rn x. In our bag model calculation above the approximations may actually have somewhat overestimated the wave function overlaps and it is likely that a more refined calculation might tend to increase the lifetime further. Our result roughly confirms the standard lifetime results but it is fair to say that lumping together all calculational uncertainties theory can only predict ~-o = 1030-+2 yr. In the case of neutron decay one may get an idea about the decay rates without reproducing the above calculations for proton decay by using isospin arguments [5]. Thus, for example, one may find F ( n e+Tr- ) = 2P(p ~ e+Tr0) etc. The corresponding neutron lifetime will be of the same order of magnitude as the proton lifetime.
80
24 March 1980
We thank V. Baluni for discussions during the initial stage of the work, and C. Meyers for interesting remarks. We are very grateful to J. Ellis and M.K. Gaillard for their interest and support all along this work.
References [1] F. Reines and M.F. Crouch, Phys. Rev. Lett. 32 (1974) 483. [2] J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240; tI. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1979) 438. [3] T.Goldman and D.A. Ross, Phys. Lett. 84B (1979) 208; W.J. Marciano, Phys. Rev. D20 (1979) 274. [4] A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, LAPP-TH06/TH.2749-CERN (1979); F.A. Wilezek and A. Zee, Princeton Univ. preprint (1979). [5] C. Jarlskog and F.J. Yndurain, Nucl. Phys. B149 (1979) 29; M. Machacek, Harvard Univ. preprint HUTP-79-A021 (1979). [6] C. Schmid, Phys. Lett. 66B (1977) 353; A. Le Yaouanc, L. Olivier, O. P6ne and J.C. Ragnal, Phys. Lett. 72B (1977) 53. [7] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471; T.A. de Grand, R.L. Jaffe, R. Johnson and I. Kiskis, Phys. Rev. D12 (1975) 2060; J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Rev. D12 (1975) 2875. [8] M. Schepkin, CERN preprint Th.2503-CERN (1978). [9] J.F. Donoghue and K. Johnson, MIT preprint CPT No. 802 (July 1979).
PHYSICS LETTERS
24 March 1980
A BAG MODEL CALCULATION OF THE NUCLEON LIFETIME IN GRAND UNIFIED THEORIES A.M. DIN
CERN, Geneva, Switzerland and G. GIRARDI and P. SORBA i
LAPP, Anneey-le- Vieux, France Received 5 December 1979
The nucleon lifetime is evaluated using bag model wave functions for the quark in the effective baryon-number violating four-fermion interaction of the SU(5) grand unified theory. We obtain rp ~ 1030 years if m X - 4.2 X 1014 GeV.
Several proposals have recently been put foward with the purpose of improving the present experimental limit [1] on the proton lifetime 7p ~ 2 X 1030 yr. The great interest in such an endeavour stems from the fact that grand unified theories of electroweak and strong interactions [2] predict baryon-number violating nucleon decays. The theoretical proton decay rate estimates depend on the chosen unification scheme and a few key parameters. It seems, however, that rough estimates are so close to the experimental limit that an improvement o f the latter of, say, two orders o f magnitude might very well confirm the existence of proton decay, or in case of a negative result, at least exclude certain models. Before new experimental results appear it is important to eliminate as far as possible uncertainties in the involved parameters. In the specific case of the standard SU(5) unification scheme, which we will consider below, this has already been done to some extent. The absolute decay rate depends crucially on three different factors, the first one being the masses mx, Y o f the exchanged superheavy vector bosons which give rise to the effective four-fermion interaction
1 On leave from Centre de Physique Th6orique, CNRS, Marseilte, France.
£eff
-C
2-1/2GGu [ei]k UkL'),uU]L -+
-+
-+
(1) -+
X (2eL)'UdiL -- eR'),~tdiR + btL~,~zSiL -- pR'),uSiR) -c -c ")'~S/R)] + h.c. + CijkUkL~/pdjL (PCR')'~ZdiL + VpR where GGU = x/2g2/8m2,y, c stands for charge conjugation and i, j, k are colour indices. For simplicity we have here neglected Cabibbo mixing. The mass m x (assumed to be equal to m y ) thus enters in the lifetime as ~ x , Y ' The value of m X has been evaluated carefully taking account of thresholds, Higgs bosons and the electromagnetic coupling constant at the relevant scale [3]. It depends on the strong interaction scale parameter A. Secondly the lifetime also depends essentially on enhancement factors due to coupling constant evolution from the grand unification scale down to the energy scale 1 GeV. This renormalization of the effective lagrangian by SU(3) gluon exchange as well as by SU(2) and U(1) vector boson exchange has been studied in detail [4]. Apart from the inclusion of some other less important decay mechanisms [5] the third major uncertainty in the lifetime is the effect of the quark wave function which in the standard calculation enters as a factor I ~ (0) 12, i.e., the square of the wave function at the origin for the annihilating two-quark system. The 77
Volume 91B, number 1
PttYSICS LETTERS
same wave function factor appears in the calculation of nonleptonic hyperon decays and can be estimated to lie between 4 × 10 - 3 GeV 3 and 14 × 10 . 3 GeV 3 [6]. In view of this uncertainty and the superficial treatment o f the proton structure it would be desirable to have an alternative approach. We will here propose to treat the quark wave function effects in a standard bag model [7] with noninteracting quarks. The wellknown definitions and properties of the bag model will not be restated but we will simply use the lowest excitation wave functions with parameters such as the radius and the bag constant fixed from static hadronic properties. The physical mechanism behind proton decay, say p -+ n°e + to be explicit, will therefore be the following: two quarks with proton bag wave functions 6~ and ¢~ annihilate in the point r 1 and create there [because of the local interaction (1)] a positron with the wave function ~e +, taken to be a plane wave ~ e ik .r, and a antiquark with wave function 6~. For simplicity (i.e., to avoid use o f quark propagators in the bag) this antiquark is assumed to be created immediately in a pion bag. The third "spectator" quark q3 in the proton with wave function ~ is at some point r 2 transformed into a pion bag quark q5 with wave function ~ r . The two points r l and r 2 are constrained to lie in the overlap Vp r3 Vn o f the proton and pion bags. They could a priori be arbitrarily located in sPace but we will as a first approximation assume that their centres coincide. The bag radii R p and Rn are different but since RTr <~ Rp the bag overlap is thus effectively the pion bag V~r itself. The approximative picture which emerges is that of a kind o f " r a d i a t i v e " transition where a static proton bag is transformed into a static pion bag under the emission o f a positron. The rate for such a process [8] will be P = 2hiM2 [, where the transition matrix element is given by M~
f
d3r 1 ~L(rl)Tla~L(rl)~e+Leik'rxT"~b~L(rl)
v~
(2)
X f d3r 2 ~ ( r 2 ) f f s ( r 2 ) . V. One might argue that the fixed static bag approximation is more likely to be valid for proton decay inio a 78
24 March 1980
heavier particle than the n, say the p. A possibility is thus to go from the p rate to the n rate by includin~ a relativistic phase space factor (m p2 - m2)/(m p2 - m~) p • In a proper relativistic treatment one should also think of the pion bag as being Lorentz contracted with a factor (1 + k2/m2) 1/2. We will take this effect approximatively into account by substituting the integration in eq. (2) over the spherical volume V~ by integration over a sphere of volume V~r(1 + k2/m2) -112. We will not describe explicitly the evaluation o f the decay rate taking account of the proper SU(6) structure of the proton and meson wave functions. For the considered two-body decays p ~ n°e +, p0e+, co0e +, r/e+, n+Ue , P+Ue, K0/a+ ( in the latter case using the appropriate effective SU(5) interaction [5]) this is fairly straightforward. If one would like to get an idea of the importance of a three-body decay like p nOnOe+ one is confronted with the unpleasant difficulties of working with three bags. A simple way out is to use PCAC to reduce out a (soft) pion and subsequently use a chirally transformed version o f the previous two-bag calculation. Of course, the available nucleon rest mass energy is so big that neither pion might actually be soft, but assuming this not to be the case (which might be likely for a 2n 0 S-state) one finds in fact (provided one can extrapolate smoothly from the soft n point) that the direct 2n 0 decay is suppressed by a factor ~ 7 relative to the one n 0 decay. By isospin considerations one may argue that the 7r+n- rate is equally small (to within a factor 2). The results for the absolute lifetime are displayed in table 1, with account taken as mentioned above o f a relativistic phase space factor, and in table 2, without this factor, for a number of values of the mass m X and the generation number ng. The branching ratios are displayed in table 3. In tables 1 and 2 we have considered three different cases for the quark masses used in the bag wave functions: (1) mu, d = 0, m s = 280 MeV, (2) mu, d = 33 MeV, m s = 300 MeV [9], (3) mu, a = 110 MeV, m s = 350 M e V . It is seen that these different choices correspond to a variation of the lifetime with a factor 2. For a typical case o f r n X = 4.2 X 1014 GeV and ng = 3 we obtain rp 1030 yr which agrees roughly with the standard calculation. To be more specific about the relation between our treatment o f the quark wave functions with the standard approach one might evaluate the ratio
Volume 91B, n u m b e r 1
PHYSICS LETTERS
24 March 1980
Table 1 The proton lifetime ~-p in 103o yr versus the generation n u m b e r ng and the mass m X in GeV, with phase space correction, rp is displayed for three different choices o f quark masses. ng
mX 2.5 X 1014
3.3 X 1014
4.2 × 1014
5 X 1014
5.8 × 1014
3
0.1 0.19 0.15
0.27 0.56 0.45
0.65 1.3 1.0
1.2 2.5 1.9
2.0 4.0 3.5
4
0.08 0.15 0.12
0.23 0.45 0.34
0.53 1.1 0.85
1.05 2.2 1.7
1.8 3.6 2.9
5
0.045 0.09 0.07
0.13 0.28 0.22
0.32 0.65 0.51
0.63 1.3 1.0
1.1 2.3 1.7
6
0.015 0.032 0.025
0.05 0.1 0.08
0.12 0.24 0.18
0.23 0.47 0.36
0.4 0.8 0.63
Table 2 The proton lifetime r p as in table 1 b u t w i t h o u t phase space correction. ng
mx 2.5 × 1014
3.3 × 1014
4.2 × 1014
5 × 1014
5.8 × 1014
3
0.13 0.3 0.2
0.4 0.8 0.6
1.0 1.8 1.4
1.8 3.5 2.7
3.0 6.0 4.5
4
0.11 0.22 0.17
0.33 0.7 0.5
0.8 1.5 1.1
1.6 3.1 2.3
2.7 5.3 4.0
5
0.07 0.13 0.1
0.2 0.4 0.3
0.5 1.0 0.7
1.0 1.9 1.4
1.7 3.2 2.9
6
0.025 0.045 0.030
0.075 0.14 0.11
0.17 0.33 0.26
0.33 0.70 0.51
0.6 1.2 0.9
Table 3 Branching ratios in % for proton decay with (case a) and w i t h o u t (case b) space phase corrections. _
M
p~
Case a Case b
~rOe+
noTrOe+
pOe+
u)Oe +
rTe+
n+ff
p+~
K°ta +
31 15
4 2
21 32
19 29
5 4
11 5
8 12
0.5 ~0
79
Volume 91B, number 1
PHYSICS LETTERS
a = Z p (4y r )- / m_4J,-( G e V )- in various cases. In the work of Jarlskog and Yndurain [5] for example, a ~ 2 X 10.29 for ng = 3 whereas from our table 1 we find, taking ng = 3 and rnu,d = 0, that a = 2.5 X 10-29 for m x = 2.5 X 1014 and a = 2 X 10 . 2 9 for m X = 4.2 X 1014 . Here one has to remember, however, that we have included an SU(2) X U(1) enhancement factor of 3 [4] so that the conclusion is that our a effectively is a factor ~ 3 bigger. For the case of the branching ratios the results depend to some extent on the inclusion of the phase space factor, however, one might assume that a correct relativistic calculation will give results somewhere in between the stated ones. It thus appears that the branching ratios for p ~ 7r0e+, poe+ and woe + should be comparable while the branching ratios for 7r07r0e+ and K0/a+ are of the order of 1%. These results do not depend on rn x. In our bag model calculation above the approximations may actually have somewhat overestimated the wave function overlaps and it is likely that a more refined calculation might tend to increase the lifetime further. Our result roughly confirms the standard lifetime results but it is fair to say that lumping together all calculational uncertainties theory can only predict ~-o = 1030-+2 yr. In the case of neutron decay one may get an idea about the decay rates without reproducing the above calculations for proton decay by using isospin arguments [5]. Thus, for example, one may find F ( n e+Tr- ) = 2P(p ~ e+Tr0) etc. The corresponding neutron lifetime will be of the same order of magnitude as the proton lifetime.
80
24 March 1980
We thank V. Baluni for discussions during the initial stage of the work, and C. Meyers for interesting remarks. We are very grateful to J. Ellis and M.K. Gaillard for their interest and support all along this work.
References [1] F. Reines and M.F. Crouch, Phys. Rev. Lett. 32 (1974) 483. [2] J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240; tI. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1979) 438. [3] T.Goldman and D.A. Ross, Phys. Lett. 84B (1979) 208; W.J. Marciano, Phys. Rev. D20 (1979) 274. [4] A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, LAPP-TH06/TH.2749-CERN (1979); F.A. Wilezek and A. Zee, Princeton Univ. preprint (1979). [5] C. Jarlskog and F.J. Yndurain, Nucl. Phys. B149 (1979) 29; M. Machacek, Harvard Univ. preprint HUTP-79-A021 (1979). [6] C. Schmid, Phys. Lett. 66B (1977) 353; A. Le Yaouanc, L. Olivier, O. P6ne and J.C. Ragnal, Phys. Lett. 72B (1977) 53. [7] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471; T.A. de Grand, R.L. Jaffe, R. Johnson and I. Kiskis, Phys. Rev. D12 (1975) 2060; J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Rev. D12 (1975) 2875. [8] M. Schepkin, CERN preprint Th.2503-CERN (1978). [9] J.F. Donoghue and K. Johnson, MIT preprint CPT No. 802 (July 1979).