Goldberger Miyazawa Oehme sum rule and a similar sum rule for τ decay

Nuclear Physics B (Proc. Suppl.) 12 (1990) 90-94 North-Holland

90

GOLDBERGER MIYAZAWA SUM RULE FOR ~ DECAY

OEHME

SUM

RULE

AND

A

SIMILAR

Tran N. TRUONG Centre de Physique Theorique de l'Ecole Polytechnique, 91128 Palaiseau, France

The Goldberger Miyazawa Oehme Sum Rules for lion Nucleon and for Nucleon Antinucleon scattering are reviewed in the light of the recent measurement of the real part of the forward proton-antiproton amplitude. A similar Sum Rule, but with stronger assumptions, relates the asymmetry between the axial and vector spectral functions to the Q.C.D. dimension 6 condensates.

At the last Blois Workshop held at the Rockfeller University in 1987, in the light of the recent measurement of a large ratio of the real to imaginary part of the forward protonantiproton amplitude, questions were raised on what would happen to the well-known Goldberger-Miyasawa-Oehme (GMO)sum rule1 if the difference of particle-antiparticle total cross-sections did not vanish asymptotically. The same question was raised almost 20 years ago by Lam and myself and a proper answer was provided2. In the following is a brief summary of this work and the more recent phemonological analyses of this sum rule for the difference of the proton proton and proton-antiproton forward amplitude. Since we discuss in this workshop the interface between soft and hard processes in QCD, it is appropriate to point out a new but s~mil~r sere-rule relating the difference between the axial and vector spectral sum-rule to the QCD short distance expansion; it can be used to analyse the hadronic ~ decay with some stronger assumptions. We were disturbed almost 20 years ago by typical text-book statement on the validity of the Pomeranchuck theorem:': Because the GMO sum rule is well satisfied by experimental data, the Pomeranchuck theorem for the difference of,he particle-antiparticle total cross section must be valid; if not, ~le sum rule integral would diverge and meaningless result would be

0920-5632/90/803.50(~ Elsevier (North-Holland)

Science Publishers B.V,.

T.N. Truong/Goldberger Miyasawa Oehme sum rule and a similar sum rule for r decay obtained". Our main motivation at that time was simply our belief that the low energy phenomena should not be influenced very much by the law of physics at very high energy, in particular the Planck's mass scale physics. In addition, there are sum rules in physics such as the Thomas-Reich-Kuhn sum rule which has little chance of converging, but are approximately verified so reasons for ~eir validity must be found3. It is not difficult to realize that the GMO and similar sum rules are consequences of analyticity, unitarity and the use of the Cauchy theorem. Provided the contour integration in the complex plane is closed, it is always possible to write an u n s u b u ~ ~b'persion relation. The essential point is the contribution of the circle at infinity must ~ways be taken into account, although with enough subtractions their contribution may vanish. Although the asymptotic behavi(~ur of the physical amplitude is usually established on the real axis, i.e.in the physical region; with the help of the Phragment-Lindeioff theorem, this behaviour can also be extended into the complex direction for the application of the Cauchy theorem. Returning now to the GMO sum rule, our point is that the usual text-book statement as quoted above is not correct. This is so because we can always write an unsubtracted dispersion relation provided that the high energy behaviour of the amplitude is known so that the contribution of the large circle can be taken into account. As a consequence of this remark, it is simple to show that the divergence of the integration of the real axis is cancelled by a similar divergence due to the contribution of the large circle at infinity; it is only when the contribution from the large circle at infinity is forgotten that the divergence would appear:. To show this let us consider a point s inside the circle Clof radius OA where O is the origin and A is a point on the real axis. By applying the Cauchy's Theorem for a real analytic function f(s) along the contour C1 and along the line AB above and below the real axis, where B is also on the real axis, and along the circle C2 with a radius OB with OB>OA. Using the Cauchy theorem, we have: •

1 | f(s')ds' 1 | 2zi ] C s ')- s S =l~

rSB

,

[

Imf(s')ds 1 ).C f(s')ds' A --"S"~S--+2~i 2 S'-S

(1)

Let us hold A fixed and let B tend to infinity.Suppose the high energy behaviour of f(s) is such that the first integral on the R.H.S. of Eq.(1) diverges.Because the integral on the L.H.S. is finite, the divergence of the integral on the cicle C2 is such that it cancels out the divergence of the integral on the real axis to yield a finite result. The usual GMO sum rule can now be rewritten as: ~ ( a l - a 3 1+

=f2 + 2 fNdE . 8z 2 ]Ix q (a.- O+) + 4~i

-~!dE' E';

(2)

where we set f=f'~-~ with f" and f" as, respectively, the forward x"p and x+p forward amplitudes and a's are the scattering lengths, f is the pion nucleon coupling constant and o's

91

92

T.N. Truong/Goldberger Miyazawa Oehme sum rule and a similar sum ru/e for 1" decay

are the total cross-sections. Because of the energy denominator in the sum rule and because If" -f+l/If-+f~'l ~¢1 at high energy, the contibution of the last term to the R.H.S of Eq.(2) to the sum rule is small, irrespectively whether there exists a Pomeranchuck-violating amplitude or not. Assuming some power law or some logarithm asymptotic behaviour for the scattering amplitude, given the imginary part of the forward amplitude as can be observed from the experimental data, one can deduce the behaviour for the real part,from which the asymptotic forward amplitude can be deduced in any complex s direction. An exception to this rule is in the situation when f'-f+ behaves asymptotically as a linear function of the laboratory energy E; in this case, the magnitude of the real part cannot be deduced from the experimental knowledge of the cross section. One must look'at the experimental data of the real part to estimate the magnitude of its contribution to the sum rule. From the experimental data of the re~ pq~rtof the gorward pion nucleon amplitude at even as low as 30 Gev. the contribution of this p~ssible term to the G.M.O. sum rule is completely negligible and the sum rule is well : ; ~ . . d by the experimental data. Let us now return to the m ~ t.h.e~e of ~,is conference, w,uneiy p-p scattering. A similar sum rule can be written for the difference of the forward pp and p~ amplitudes.About a year ago there were some controversy about the use of this sum rule by Igi and Kroll to put the limit on the Pomoranchuck-violating amplitude which is now called "Odderon". These authors claimed that the"Odderon" which was used to fit the high energy data by Bernard et al and others4 was in conflict with the GMO sum rule.There was,unfortunately, an error in the early version of this paper. In the published version5 the error was corrected and Igi and Kroll found that the low energy parameter scattering length is unaffected by the existence of the Oddemn,as it should be. Igi and Kroll5 obtained a remarkable agreement between the values of the scattering length which is obtained from one hand by the direct measurement, and on the other hand by the sum rule using the measured total cross sections.They obtain: ReF(m) (sum rule) = -45.5:!: 3 Gev "1 ReF(m) (exper.) = -46.8 i'0.5 Gev "1 (3) which is indeed in excellent e ~ e m e n t with each other. Sum rules written with the crossing-odd amplitude are usually much more reliable than those written with the crossing even amplitude.The reasons for this is, the high energy contibution of the absorptive part of the crossing even amplitude is not suppressed by the energy denominato~ a.-.d hence the calculated low energy parameters such as the scattering lengths are extremely ser~sidve to the high energy assumptions which make the related sum rules unreliable. Because this Conference is the interface between soft and hard processes in Q.C.D., it is appropriate to discuss a related sum rule which relates the short distance expansion coefficient

T.1V. Truong/ Goldberger Miyazaws Oehme sum ru/e and a similar sum ru/e [or r decry

of the di~¥erence of the the vector -axial correlation function Under some assumption on the rapid convergence of this sum rule, it says something about the difference of the contribution of the even -odd number of pions in the ~-decay and also to the well-known mi~ing one prong event in this decay. Let us define the Vector and the Axial Vector Conelation Functions as6: x~vV(q2) = (.gttVq2+qltqV)~)(q2) (4)

Z~AV(q2) = (-~vq2+q~qV)x2)(q2).~ ~i~qV~+f)(q2 ) Let s=q2 and let us define the function g(s)=H(I)v(S)-I'[(I)A(s)-r[(0)A(s ). In the chiral limit when the quark masses are neglected, the deep Euclidean behaviour for s --) -~ is given by the dimension 6 condensates and terms of higher dimension than 8. More precisely, the deep Euclidean behaviour of g(s) is = ..I_

g(s)

S--.)~

I,LS-ICa < 0 6

2Z2 17 I

(. S)3

•

(5)

where C a =-896~c3/81, 06 = a(q-q) 2 • Notice that although the deep Euclidean for g(s) is given by the Eq.(5) we cannot, in principle, say definitely its asymptotic time like behaviour. If we assumed that the time like g(s) behaviour was non oscillatory then we could use Eq(5) to continue its asymptotic behaviour throughout the complex s plane and onto the positive real s axis.Because the next leading order for g(s) decreases faster than s - 4 it is simple to use the Cauchy theorem to show that:

.....

ds ( l - s / M2) (I + 2s / M~2) (v(')(s) - a(1)(s)) - a(°)(s

_ _ {162/7)C6<06>

(6)

where v(l), a(t) and a(°) are the imaginary part of the corresponding spectral functions. Now the integrand has a double zero at s=M~2, it is a good approximation to set the upper limit of the integration at Mc 2, instead at ~, in Eq.(6) .In this approximation Eq.(6) b~comes: I" (~ --+ Vector + v) - I" (~_---yAxial v) _- . (162/7) C~ < O6 • cos2 ec

(7)

vv) Using the existing experimental data on ~ - de~ay7 the L.H.S.of Eq.(7) is approximately 0.25, which anplies C6= -0.35(Oev) 6 It is much larger than the estimate $hifman et al 8 but is in rough agreement with other calculations 9. It does not, however, solve ~he missing one prong problem in ~ -decay which would req-ire C6=0 10.

93

94

T.N. Truong/Goldberger M ~ m

Oehme sum rule and a similar sum ride for ~"decay

REFERENCES 1) M.L. Goldberger, H. Miyazawa and R. Oehme, Phys. Rev. 99 (1955) 986. 2) W.S. Lain and T.N. Truong, Phys. Lett. B31 (1970) 307. 3) T.N. Pham and T.N. Truong, Phys. Lett. B64 (1976) 51. D. Bernard, P. Gaumn and B. Nicolescu, Phys. Lett. B199 (1987) 125. K. K~-ngand B. Nicolescu, Phys. Rev. D l l (1975) 2461. 5) K. Igi and P. Kroll, Phys. Lett. 13218 (1989) 95. 6) See for ex#mple E. Br~!en, Phys. Rev. Lett. 60 (1988) !,y2)6. S. Narison and A. Pich, Phys. Lc.n..B211 (1988) 183. 7) For a review see K.K. Gan and M.L. Perl, .Int. J. Mod. Phys. A3 (1988) 531, B.C. Barish and R. Stmynowski, Phys. Rep. 157 (!988) 1. 8) M.A. Shifman, A.I. Vainshtein and V.I. Zakhamv, Nucl. Phys. B 147 (1979) 385, 448, 519. 9) R.A.Be~tlmann et.al. Z.Phys.C 39 (1988) 231 and references cited therein. 10) T.N.Tmong, Phys. Rev. D30 (1984) 1509.