SUM RULES AND PION FORM FACTOR IN QCD

Transcription

1 SUM RULES AND PION FORM FACTOR IN QCD V. Nesterenko, A. Radyushkin, A. Efremov To cite this version: V. Nesterenko, A. Radyushkin, A. Efremov. SUM RULES AND PION FORM FACTOR IN QCD. Journal de Physique Colloques, 1982, 43 (C3), pp.c3-242-c < /jphyscol: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1982 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE CoZZoque C3, suppze'ment m no 12, Tome 43, de'cembre 2982 page C3-242 SUM RULES AND PION FORM FACTOR IN QCD V.A. Nesterenko and A.V. Radyushkin Presented by A.V. Efremov Laboratory of Z'heoretieaZ Physics, JINR, Eubna, USSR The discovery that perturbative QCD methods can be applied to study the asymptotic behaviour of the pion electromagnetic form factor (see, e.g., refs./l,2/ and the reviews /3/) has been an important step in the development of the'theory of hard processes.the pion form factor for all orders and for all logarithms in perturbation theory was shown to have the factorized form: where Q2 =-q2, p is a renormalization parameter, 6 is the pion wave function and E is the amplitude of the short-distance (SD) sub-?recess q&* + q'q' (Fig. 1 ) a)(x I %- "PC AA YP, p, "2 Fig. 1 As it is now generally accepted /4-6/, the best choice for p is in the region where the higher order PT corrections are small. The 1- l~op~yorrection to E have been calculated by Dittes and Radyushkin /7/. Its magnitude depends on the wave function +(x, p). It was shown /9/, that asymptotically +(x, p) + 6fTx(l-x) as p - +. However, for ph I GeV the wave function should be very broad for nearly-massless strongly interacting quarks. Thus; if one parametrizes +(x) simply as +(x) - xr(l-x): then one should expect that for pion r<< I (say, r z ). As a result, for the naive choice p =Q and small r the correction is huge: X, A sixilar calculation was performed by Field et al. /3/, but their results for some diagrams are incorrect. Article published online by EDP Sciences and available at

3 A.V. Efremov C3-243 To reduce it, one should take p2 = p Ept tq2exp(-3/2r) ( Q2 ). This means that for Q2 ( 10 Gev2 the exchanged gluon (Fig. la) has virtuality much smaller than 1 G~v'. In such a situation it is, of course, misleading to rely on the asymptotic freedom. First attempt to find the way out was a QCD inspired model /10,4/ with the result that for moderately large Q2 the?ion EM form factor is dominated by the Feynman mechanism /11/. The aim of this report is a brief description of a new approach /12,13,14/ that enables one to analyse the behaviour of 3,( J2) for moderately large ~ 2. Technically our analysis is based on the QCD sum rule approach /15/. We consider the three-point amplitude (see for-notation Fig. Ib) where Jp is the electromagnetic current and ja= ~Y~Y,U is the axial one: < 0 1 fa (0)l P > = ifspa, where IP> is the I-pion state and f,= 133 MeV the pion decay constant. The invariant amplitudes Ti (corresponding to various structures present in T,B! depend on 3 variables: p;, p q2 = (P -p2) 2. Owing to the asymptotic freedom one may calculate T,&I\, pg, q2 ) in the Euclidean region P y, P$, q ( -1 G ~v~. To extract hhe desired information about the physical states, we use the dispersion relation and apply the Bore1 procedure /15/ in p: and p; : Ti (p:,pg,612)-@(~f,~~~,~~). For the pion form factor analysis the most important is the inva- riant amplitude Q, related to the structure Pa PBP, where P = g lcp2. Neglecting quark masses ( 210 MeV) we calculated the contribution of the diagram Ib into the p density: In the real world p(sl, s2.q2) differs, of course, from po (sl, s2, q2) especially for small sl, s,. In particular, it contains the pion term p,(sl, s2, q 2 ) = n 2 f ~ ~ n ( ~ 2 ) ~ m:) ( s 16(sz - - m:).. The basic idea of the QCD sum rule approach /15/ is that the -ao is mainly due to power corrections*) ( 1 / ~ generated ~ ) ~ by nonzero vacuum expectation values of quark and gluon operators. Taking into account the contributions proportional to <C;,G;> and a,<qq> (cf. ref. /I 5/) we arrive at the following representat~on for the pion form factor - 6 where (as/n)<gg> = G ~ v a,<\p2=1 ~, GeV2 /15/ and so is the "effective threshold" characterizing the region where one can substi- X, To treat initial and final states on equal footing we take henceforth M1 = M2 = FA.

4 C3-244 JOURNAL DE PHYSIQUE tute p(sl, s2, q2) by its f ree-quark value P~(s~,s~, q2). It is natural to define that the "true" so is that for which the region of weak sensitivity to variations of unphysical parameter M is the most broad. For Q ' = 1-3 Gev2 this gives so= Gev2 respectively (Fig, 2) which is in good agreement with the value s o=0.7 G ~ ex- V ~ tracted from the analykis /15/. To get a reliable result for Yn(Q2)one should choose M~ SO as to reduce to a minimum both the power corrections (blowing up as M~+o) and the contribution of states with sl, s2 > so. In our case, however, the situation IS more complicated than in /15/, mainly because the free -quark term in eq. (2) behaves like l/c14 for large~~while the power corrections grow with CI2. In particular to reduce the power corrections to 30 % for Q2 = 2 Gev2 one should take ~ ~ Gev The contribution of higher states (estimated by use of the free-quark density) 0 2 L 6 is then =I00. However, it is the difference M'(G~v') between the true higher state contribution and its free-quark approximation that matters, Fiq. 2 which may well be even zero for an appropriate choice of so. As is seen from Fig.3, in the region 1<Q2<4 Gev2 our theoretical curve (broken line) agrees with experimental data /16/ within 10-20% accuracy. Taking M' sufficiently large one can control the power corrections for arbirarily high Q' values. The most radical way out is to take M2 =-. Then the power corrections are absent altogether, and if one assumes that the "effective threshold" s, is Q2-independent, one obtains the finite energy sum rule L.0 q2 (Gev2) (3) is nothing else but a duality rela- :?on between the pion and free-quark contri- Fig. 3 butions (cf. /17/ and references bherein). Using ( 1 ) one can reduce eq. (3) to For so = 4n2f 2 = 0.7 Gev2 this formula is in excellent agreement with experimental data (fig. 3, solid line). The agreement indicates that the local duality between the pion and free-quark contributions is indeed a very good approximation. In principle, apart from power corrections one must take into account also higher perturbative corrections due to diagrams like that shown in Fig. Ic. However, their contribution is damped by thea /n=0.1 factor with respect to that of Fig. Ib. Our estimates show that for Q G ~ these V ~ diagrams produce only 10% (positive) correction to 3,(Q2). Thus, the celebrated one-gluon-exchange contribution is of little importance for available Q2. However, the situation changes drastically in the asymptotic Q2+ - region, because Fig. Ic gives in this region 1/Q2 -contribution (corresponding to the asymptotic QCD analysis /I-3/) exceeding the 1/Q4 -contributions of Fig. Ib for sufficiently large Q2.

5 A.V. Efremov References 1. A.V.Radyushkin, JINR preprint P (June, 1977); A.V.Efremov and A.V.Radyushkin. Phys-Lett. 94B, 2451 (1980) and references therein. 2. S;J.~rodsk~ and G.P.Lepage. Phys-Rev., E, 157 (1980). 3. V.L.Chernyak. Proc. XV LINP Winter School, Leningrad (1980),1, p. 65; A.H.Mueller. Phys.Reports, 2, 237 (1951). 4. A.V.Efremov and A.V.Radyushkin, JINX preprint E (1980j, submitted to Madison Conference. 5. P.M.Stevenson. Phys-Rev., GI 2916 (1981). 6. W-Celmaster and D.Sivers. Phys.Rev. G, 227 (1981). 7. F.-11.Dittes and A.V.Radyushkin. Yad-Fiz., 34, 529 (1951). 8. R.D.Fheld et. al. Nucl.Phys., e, 429 (1-1). 9. A.V.Efremov and A.V.Radyushkin, JINR preprint E , (October, 1978). 10. A.V.Radyushkin. JINR report P (1990); see also Proc. of the Symp. on Particle Physics, Visegrad, Hungary (September, 1981), p R.P.Feynman. Photon-Hadron Interactions. Benjamin (1972). 12. V.A.Nesterenko and A.V.Radyushkin. Pis ma v JETF, 2, 395 (1932). 13. V.A.Nesterenko and A-V-Radyushkin. JINR preprint E (1932), to be published in Phys.Lett. B. 14. B.L.Ioffe and A.V.Smilga. Preprint ITEP-27 (1982). 15. M.A.Shifman, A.I.Vainshtein and V.I.Zakharov, Nucl-Phys., m, 447 (1 979). 16. C.Bebek et al. Phys.Rev. m, 1693 (1978). 17. N.V.Krasnikov and A.A.Pivovarov. Report NBI-HZ (Kopenhagen