Fixed-angle elastic hadron scattering

Transcription

1 Fixed-angle elatic hadron cattering R. Fiore,, * L. L. Jenkovzky, 2, V. K. Maga, 3, and F. Paccanoni 4, Dipartimento di Fiica, Univerità della Calabria, Itituto Nazionale di Fiica Nucleare, Gruppo collegato di Coenza, Arcavacata di Rende, I Coenza, Italy 2 Bogoliubov Intitute for Theoretical Phyic, Academy of Science of the Ukraine, Kiev, Ukraine 3 Section of Theoretical Phyic (SENTEF), Department of Phyic, Univerity of Bergen, Allegaten 55, 5007 Bergen, Norway 4 Dipartimento di Fiica, Univerità di Padova, Itituto Nazionale di Fiica Nucleare, Sezione di Padova, via F. Marzolo 8, I-353 Padova, Italy Received 3 April 999; publihed 3 November 999 The cattering amplitude in the dual model with Mandeltam analyticity and trajectory () 0 ln ( 0 )/( 0 ) i tudied in the limit, t, /t cont. By uing the addle point method, a erie decompoition for the cattering amplitude i obtained, with the leading and two ubleading term calculated explicitly. S PACS number :.55.Jy, 3.85.Dz, Cm PHYSICAL REVIEW D, VOLUME 60, 6003 I. INTRODUCTION: WIDE-ANGLE SCATTERING IN QCD A,t 0 4 i dx i i x i T x i,,t, 2 Although attempt to apply perturbative QCD to wideangle elatic hadron cattering have been undertaken in a number of paper 0, explicit prediction have been available only for elatic procee involving external photon, uch a hadron, Compton cattering of hadron, etc. Prediction baed on perturbative QCD ret on three premie: hadronic interaction become weak at mall invariant eparation r QCD, 2 the perturbative expanion in (Q) i well defined, and 3 factorization, implying that all effect of collinear ingularitie, confinement, nonperturbative interaction, and bound tate dynamic can be iolated at large momentum tranfer in term of the proce independent tructure function G i/h (x,q), fragmentation function D H/i (z,q), or, in the cae of excluive procee, ditribution amplitude H ( i,q). Conequently the hadronic cattering amplitude take the form A H H x i,q T x i,p H ;Q dx i, (x i,q) i a univeral ditribution amplitude which give the probability amplitude for finding the valence qq or qqq in the hadronic wave function collinear up to the cale Q /2, and T i the hard cattering amplitude for valence quark colliion. The technical complication which ha made it particularly difficult to compute the behavior of hadron-hadron amplitude i the poibility of multiple cattering. The tandard factorized form for the elatic cattering of hadron i i * addre: FIORE@CS.INFN.IT addre: JENK@GLUK.APC.ORG addre: VLADIMIR@SENTEF2.FI.UIB.NO addre: PACCANONI@PADOVA.INFN.IT x i repreent collectively the fractional momenta of hadron i carried by it valence parton. According to thi concept, all of the parton collide in a mall region of the pace-time of typical dimenion Q. The relevant contribution to the amplitude behave according to dimenional counting,2, i.e., A,t 2 n/2 2 f /t, 3 for n parton participating in hard cattering, repreenting hadronic ma cale, which make the amplitude dimenionle. An extenion of thi ingle-cattering cenario i the double independent-cattering picture, due to Landhoff 8, in which two pair of parton catter independently off two cattering center. According to thi picture, the lowet order diagram contribute with A m,t 2 (n m )/2 2 f m /t, 4 m i the number of independent cattering. If o, the multiple cattering hould dominate in the cae of wideangle cattering. A olution to thi problem wa pointed out in Ref. 5 and, it wa hown that the Sudakov logarithm aociated with the recattering diagram do not cancel. In the leading logarithmic approximation they exponentiate to uppre the typical double-cattering contribution by a factor exp cont ln Q 2 ln ln Q 2, characteritic of the Sudakov uppreion in QCD. More quantitatively 2, /99/60 / /$ The American Phyical Society

2 FIORE, JENKOVSZKY, MAGAS, AND PACCANONI PHYSICAL REVIEW D A 2 2c ln(/r) Q Q 4 QCD, 5 r 2c/ 2c, c 32/ 33 2n f, and n f i the number of flavor. Interetingly, for n f 3 the power turn out to be Q 3.8, nearly the ame a the dimenional counting power Q 4 in the ingle-cattering cenario. Higher order diagram were calculated, e.g., in Ref. 9 ; however, oon it became evident that even the firt order QCD correction involve an immene number of Feynman diagram, o further attempt to go beyond the imple quark counting rule were abandoned. It may be that perturbative QCD i not the relevant or not the only phyically intereting expanion of the wide-angle cattering amplitude. Recent development in M-brane ee, e.g., Ref. 22 may open new propect in the realization of a hypothetical duality between mall and large ditance or, equivalently, large- and mall-angle cattering. The earch for a relevant expanion parameter i of crucial importance in thi matter. In thi paper we olve an invere problem : we ue the known explicit expreion of the dual amplitude with Mandeltam analyticity DAMA, which ha correct wide-angle caling behavior. By identifying it with that reulting from the quark counting rule, we then calculate two ubleading term in the expanion of the known full dual amplitude and tudy the behavior of the reulting erie. II. WIDE-ANGLE BEHAVIOR OF THE DUAL AMPLITUDE WITH MANDELSTAM ANALYTICITY Wide-angle caling behavior within the S-matrix approach wa dicued in Ref. 3, by mean of a logarithmic Regge trajectory an interpolation from the oft Regge behavior to the hard caling regime wa uggeted. The motivation of the logarithmic trajectory came from earlier paper 4, a cla of dual model requiring a logarithmic trajectory wa uggeted. The logarithmic aymptotic behavior of the trajectory and the large-angle caling behavior are uniquely related alo in a different cla of dual model, called dual amplitude with Mandeltam analytic 5,6. The link between thi cla of model in the caling limit and the parton model in the infinite momentum frame wa tudied in Ref. 7. In all thoe paper only the leading aymptotic (, t, /t cont) term wa treated. The reult of different approache vary in uch detail a the form of the caling violation normally, logarithmic, the form of the angular dependence f ( ), and the way active quark are counted. In thi paper we calculate the ubleading term in the preaymptotic large and t ) behavior of DAMA. Since the model i realitic enough in the ene that it atifie the general requirement of the theory ee Ref. 5,6, we believe that our reult i univeral and thu it may be ued a a guide, e.g., in QCD calculation. Apart from the leading term, we have explicitly calculated two more ubleading term. Our technique allow further calculation of till higher order, but the obtained firt three term of the erie already how a regular trend that may be interpreted a the expanion in the running coupling contant g() /ln, valid at large and t. Thi ituation take place for any one trajectory with the logarithmic aymptotic. The aim of the preent paper i twofold. Firt, by identifying the leading term of the aymptotic wide-angle expanion of DAMA with that derived from perturbative QCD 5 we tentatively aume that DAMA in the wide-angle aymptotic region i equivalent to the aymptotically free regime in QCD 8. With thi identification in mind, we calculate within DAMA correction to the leading term in the hope that their form may give ome inight into the relevant correction in perturbative QCD that are known to be very complicated. Clearly, the above identity ha the chance to be true only in the vicinity of the wide-angle region mall ditance, perturbative calculation are aumed to be till valid. The econd apect i purely phenomenological. Since, however, the experimental ituation in the wide-angle region did not change for almot two decade, we are left with the earlier fit to the data. Let u now calculate the perturbative expanion of DAMA. We write the elatic cattering amplitude for pinle particle in the following ymmetric form 6 : A,t,u C u D,t D u,t, C i a contant and D,t 0 dx x g ( ) x g 6 (t ). 7 Here ( x), t tx, and g i a dimenionle parameter, g. Only one leading trajectory wa included and it wa choen in a imple, but repreentative form 0 ln 0 0, 8 which account both for the threhold and the aymptotic behavior and i nearly linear for very mall, 0. For implicity we have included only the leading trajectorie in both channel: the Pomeron trajectory in the t channel and the exotic trajectory in the channel. While the parameter of the Pomeron trajectory are well known, only a little i known about the exotic trajectory. Fortunately, thi ha no ubtantial effect on our reult, ince our goal i the functional form of the erie and it individual term rather than fit to the data. Given the carcity of the data and the freedom available in the model, the wide-angle behavior of DAMA cannot be determined completely. Let u conider the aymptotic behavior of Eq. 7 in the limit, t, /t cont. For the Regge trajectorie we have 0 2 ln 2 i 2 ln a, 2 9

3 FIXED-ANGLE ELASTIC HADRON SCATTERING PHYSICAL REVIEW D with t 0 2 ln 2 2 ln t 2 ln b, 0 a 0 2 ln 2 i 2, b 0 2 ln 2 2 ln t, 0, ln. 0 2 From here on,,t,u will be dimenionle variable, meaured in unit of 0. In thi domain the addle point method can be ued to calculate the integral in Eq To do thi we can rewrite Eq. 7 in the following form: D,t 2g a b 2 g ln 2 2 g u e f (u) du, 2 we have changed the variable x to u, x ( u)/2, and introduced new function g u u ã u exp b ln u u ln 2 2, 3 f u ln u 2, 4 ã a 2 ln g, b b ln g. 5 2 We ee now that f (u) ha a harp maximum at the addle point u 0 0. We quote the explicit expreion for the addle point expanion in Appendix A. Uing formula from thi appendix we obtain the power erie for D(,t) in Appendix B. It read D,t A ln 2g ln t ( /2)ln 2g h ã,b ln h 2 ã,b 2, ln 6 A, h, h 2 are given by the expreion B5, B8, B9. The expreion for D(u,t) can be calculated in a imilar way ee Eq. B0 in Appendix B. In the kinematical region, t, t/ cont we can ue the ubtitution t in 2 /2, u co 2 /2. 7 Subtituting the reult for D(,t) and D(u,t) into Eq. 6 and changing the variable we get the expreion for the full amplitude a a function of the and variable ee Eq. B4 in Appendix B : A, CA N ln f I,, 8 A, N, f ( ), I(, ) are given by the expreion B3, B5 B7. To ummarize, we have expanded the wide-angle cattering amplitude in a power erie of /ln and have evaluated explicitly the coefficient of the firt two term beyond the leading one. III. COMPARISON WITH THE DATA AND DISCUSSION OF THE RESULTS New experimental data on wide-angle cattering are not likely to appear any more becaue of the imple reaon that a energy increae more particle tend to fly in the forward direction and there i no chance to detect, e.g., the protonproton differential cro ection at 90 for, ay, 0 GeV. Wide angle, of coure, extend beyond 90. Still the complication due to the huge number of Born diagram contributing to large-angle excluive reaction 5, overwhelming the contribution due to the Landhoff pinch ingularity 8, will remain for long topical in thi field. We ue the data given in the compilation of 20 to fix the cale. The error, quoted in the original paper ee Ref. 20 and reference therein, are typically about 0%. Thi cale i the overall normalization factor, the quark counting power in the cro ection being et equal to N 4 in the cae of proton-proton cro ection, in agreement with the data 20,5 ee Fig.. Our main goal i the behavior of the caling-violating correction to the leading term obeying quark counting rule. Figure 2 how the relative contribution of thee term. We draw the correction power erie: J, I, 2 2 Re f /Z 2 Re f 2 /Z ln ln 2 f /Z 2 ln 2 O 3, 9 f ( ), f 2 ( ), Z( ) are given by expreion B8 B20. We can ee that the correction are quite large for mall, epecially for angle cloe to 90. That i not a urprie, ince the lowet order of our expanion i valid for large ( ln /2 ). In the experimental energy interval the correction give a factor of 4 6 to the cro ection and hould not be neglected. Thi wa mied in Ref. 5,6. Moreover, we find that the correction are very enitive to variation of and

4 FIORE, JENKOVSZKY, MAGAS, AND PACCANONI PHYSICAL REVIEW D APPENDIX A: COEFFICIENTS IN THE SADDLE POINT METHOD In thi appendix we preent the explicit expreion for the addle point expanion from Ref. 9. Here f (k) (u 0 ) f k, g (k) (u 0 ) g k : g u e f (u) du e f (u 0 ) a 0 a a 2 O 2 3, A a 0 g 0, a 4 g 2 3 3g 2 g 0 3, FIG.. Cro ection d /dt for pp pp cattering at variou center of ma cattering angle. Both axe are in logarithmic cale. Star denote the experimental point from Ref. 20. The traight line correpond to a falloff of / 0. They are calculated according to the power erie for the cattering amplitude, dicued above d /dt 4 /( 0 ) 2 A(, ) 2, with the following et of parameter: 0, N 4, 2.84 (g 2.9), 0.05 GeV, C GeV 2, and 0 4m 2. ACKNOWLEDGMENTS V.K.M. i thankful for the hopitality extended to him by the Bogolyubov Intitute for Theoretical Phyic in Kiev part of thi work wa done. Thi work wa upported in part by the Reearch Council of Norway program for nuclear and particle phyic, upercomputing, and free project, in part by the Minitero italiano dell Univerità e della Ricerca Scientifica e Tecnologica, and in part by IN- TAS grant extenion. a 2 32 g 4 5 0g g g g 4 0g 3 2 g 0 5, f 2, 2 3 f 3f 2 2, 3 4 f 4f f 3 2 f 2 2 3, 5 f 5f 2 f 4 f 3 f f 3 3 f 2 3 4, 6 f 6f f 5f 3 f f f 2 A f 3 4 f f 4f 3 2 f A3 APPENDIX B: CALCULATIONS OF THE SCATTERING AMPLITUDE Uing the definition of function g(u), f (u), Eq. 3, 4, we obtain f 2 2, g 0 e ln22, f 3 0, f 4 2, f 5 0, f 6 240, B g 2 g 0 ã b 2 ã b 2 ln 2, FIG. 2. The correction J(, ), given by Eq. 9, to the differential cro ection d /dt for pp pp cattering. We have ued the ame value of parameter a in Fig. : 0, N 4, 2.84 (g 2.9), 0.05 GeV, and 0 4m 2, coming from the comparion with the data. g 4 g 0 ã ã ã 2 ã 3 4ã ã ã 2 b 6ã ã b b 4ãb b b 2 b b b 2 b 3 2 ã b 2 ã b ln ln ln2 5. From Eq. A3, B we get B

5 FIXED-ANGLE ELASTIC HADRON SCATTERING PHYSICAL REVIEW D , 2 0, 3 3 2, 4 0, B3 Finally we get D,t A N t /2 ln 2g I ã,b,, ln B4 Subtituting Eq. B4 and B0 into Eq. 6 we get the expreion for the full amplitude: A,t,u C A 0 N 0 i /2 ln u 2g t ( /2)ln 2g I ã,b, tu 2 ( /2)ln 2g I c,b,, B2 A 2g 2 0 ln 2 ln 2 i /2 2, B5 A 2g 2 (0) ln 2 ln B3 N ln 2g, I ã,b, h ã,b ln h 2 ã,b 2. ln Coefficient h (ã,b ), h 2 (ã,b ) A2, B2, B3 : h ã,b 3 4 g 2 2g 0, h 2 ã,b g 4 5g 2 8g 0 8g 0 B6 B7 are calculated from Eq.. B8 B9 The expreion for D(u,t) can be calculated in a imilar way. It turn out to be D u,t A 2 N ln ut 2 ( /2)ln 2g A 2 2g 2 0 ln 2 ln 2 2, I c,b,, c c 2 ln g 0 2 ln 2 2 ln u 2 B0 ln g. B In the kinematical region, t, t/ cont we can ue the ubtitution 7. So the expreion for the cattering amplitude a a function of and appear to be A, C A N ln f I,, N N ln 2g, ln 2g Z, f co 2 2 in 2 I, f Z ln f 2 Z ln, 2 B4 B5 B6 B7 f h ã,b 2g i /2 h c,b 2g ln co( /2), B8 f 2 h 2 ã,b 2g i /2 h 2 c,b 2g ln co( /2), B9 Z 2g i /2 2g ln co( /2), b 0 2 ln 2 2 ln in c 0 2 ln 2 2 ln co B20 ln g, B2 ln g. B22 S. J. Brodky and G. R. Farrar, Phy. Rev. Lett. 3, V. A. Matveev, A. N. Tavkhelidze, and R. M. Muradyan, Lett. Nuovo Cimento 7, S. J. Brodky and G. R. Farrar, Phy. Rev. D, A. V. Efremov and S. J. Brodky, Theor. Math. Phy. 42, G. P. Lepage and S. J. Brodky, Phy. Rev. D 22, ; J. C. Collin, D. E. Soper, and G. Sterman, in Perturbative Quantum Chromodynamic, edited by A. H. Muller World Scientific, Singapore, A. Duncan and A. H. Mueller, Phy. Rev. D 2, N. Igur and C. H. Llewellyn Smith, Nucl. Phy. B37, P. V. Landhoff, Phy. Rev. D 0, G. R. Farrar, G. Sterman, and H. Zhang, Phy. Rev. Lett. 62, J. Bott and G. Sterman, Nucl. Phy. B Proc. Suppl. 2, P. V. Landhoff and D. J. Pritchard, Z. Phy. C 6, A. H. Mueller, Phy. Rep. 73, D. D. Coon et al., Phy. Rev. D 8, M. Arik, Phy. Rev. D, L. L. Jenkovzky and Z. E. Chikovani, Yad. Fiz. 30,

6 FIORE, JENKOVSZKY, MAGAS, AND PACCANONI PHYSICAL REVIEW D Sov. J. Nucl. Phy. 30, A. I. Bugrij, Z. E. Chikovani, and L. L. Jenkovzky, Z. Phy. C 4, M. J. Schmidt, Phy. Lett. 43B, H. Kluberg and J. Zuber, Phy. Rev. D 2, ; 2, ; 2, M. V. Fedoryuk, Aimptotika: ryady i integraly Nauka, Mocow, P. V. Landhoff and J. C. Polkinghorne, Phy. Lett. 44B, V. Maga, Ukr. Fiz. Zh. 44, J. Polchinki, Rev. Mod. Phy. 68,