1 Introduction Asymtotic freedom has led to imortant redictions for hard inclusive hadronic rocesses in erturbative QCD. The basic tools are factoriza

Transcription

1 UPRF July 1997 One article cross section in the target fragmentation region: an exlicit calculation in ( 3 )6 M. Grazzini y Diartimento di Fisica, Universita di Parma and INFN, Gruo Collegato di Parma Abstract One article inclusive cross sections in the target fragmentation region are considered and an exlicit calculation is erformed in ( 3 ) 6 model eld theory. The collinear divergences can be correctly absorbed into a arton density and a fragmentation function but the renormalized cross section gets a large logarithmic correction as exected in a two scale regime. We nd that the coecient of such a correction is recisely the scalar DGLAP kernel. Furthermore the consistency of this result with an extended factorization hyothesis is investigated. PACS Ni y grazzini@arma.infn.it

2 1 Introduction Asymtotic freedom has led to imortant redictions for hard inclusive hadronic rocesses in erturbative QCD. The basic tools are factorization theorems [1] together with erturbative evaluation of scaling roerties. Nevertheless erturbative QCD has been successfully alied also to various semi-inclusive rocesses, for which a comlete factorization roof is still lacking. In a dee inelastic semi-inclusive reaction a hadron with momentum is scattered by a far o shell hoton with momentum (Q =? QCD) and from the inclusive nal state, a hadron with momentum 0 is detected. Until some years ago the main interest has been devoted in hadron roduction in the current fragmentation region, that is in the region in which the momentum transfer t =?(? 0 ) is of order Q []. In the last few years, after the introduction of fracture functions [3], a novel attention has been ayed to the target fragmentation region (t Q ) [3, 4, 5]. With the observation of diractive dee inelastic events at HERA art of this attention has been focused on the limit in which the longitudinal momentum fraction of the observed hadron is close to unity and t is of order 1 GeV or less. In this aer we want to deal with semi-inclusive dee inelastic scattering (DIS) in a rather simlied context, that is we will work in ( 3 ) 6. The scalar ( 3 ) 6 model share with QCD some imortant asect. It is an asymtotically free theory and the Feynman diagrams have the same toology as in QCD. Moreover there are no soft but only collinear divergences and so factorization become simler to deal with. For this reasons it has been customary, in the ast, to face a QCD roblem by studying it rst in ( 3 ) 6 [7, 8, 9, 10, 11]. In ( 3 ) 6 one can dene as in QCD a scalar arton density and a fragmentation function which obey DGLAP evolution euations [8, 11]. It is in this framework that we are going to calculate here the one article dee inelastic cross section in the limit t Q. The aim of such a calculation is twofold. On one hand we want to verify that the renormalized one article cross section gets large log Q =t corrections, as one naturally exects in the two 1

3 scale regime t Q. On the other hand we want to check the consistence of our result with the factorization ut forward in Ref.[1]. This aer is organized as follows: in Sect. we will recall the result one gets in ( 3 ) 6 for inclusive DIS. In Sect. 3 we will erform the calculation of the semi-inclusive cross section. In Sect. 4 we will show how an interretation in terms of cut vertices can be given. In Sect. 5 we will sketch our conclusions. DIS As a rst ste we will focus on the dee inelastic inclusive cross section. We consider the rocess + J()! X where J = 1 is a scalar oerator which lays the role of the electromagnetic current. We dene as usual Q =? x = Q : (.1) The structure function is W (x; Q ) = Q Z d 6 ye iy <jj(y)j(0)j> : (.) We want to calculate the arton-current cross section w(x; Q ) at = 0 in dimensional Figure 1: Lowest order contribution to the dee inelastic cross section

4 regularization (D = 6? ), by retaining only divergent contributions. At Born level we have (see Fig. 1) w 0 (x; Q ) = Q (( + ) ) = (1? x): (.3) The rst order corrections are shown in Fig. and give 4 w 1a (x; Q ) = (4) 3 x(1? x)? 1 Q (.4) w 1b (x; Q ) = (1? x) (.5) (4) 3 Q w 1c (x; Q ) = 1? 1 4 (1? x): (.6) 1 (4) 3 Q The factor in e. (.5) takes into account the contribution from the symmetric diagram. The diagram in Fig.d gives only a nite correction. External self energies have not be taken into account 1 since we work at = 0. Usually one has not to worry about current renormalization, because the electromagnetic current is not renormalized by strong interactions. However in our simlied model the oerator J 0 = 1 0 is renormalized by the interaction. The renormalization constant Z J is dened by J = Z?1 J J 0: (.7) We get, in the MS scheme Z J = (4) 3 : (.8) As a matter of fact, being the renormalization scale deendent, the couling to the current becomes scale deendent, so it turns out more convenient to dene a Q deendent renormalization Z J (Q ) = (.9) 1 (4) 3 Q 1 Actually one should also include self energy corrections on the current line, which are of the same order in and cause the mixing of the oerator with the oerator [13]. Nevertheless these diagrams are vacuum olarization of the external robe, and in the real world are suressed by one ower of em and so are neglected. We decide here to follow the literature [9, 10, 11] and not to include them in our denition of the structure function. 3

5 k (a) (b) (c) (d) Figure : One loo corrections to the dee inelastic cross section which takes into account this eect. Therefore the cross section is obtained summing u all the contributions and multilying by Z? J (Q ). U to nite corrections we get w(x; Q ) = (1? x) +? (4) P (x) 1 4 (.10) 3 Q where P (x) = x(1? x)? 1 (1? x) (.11) 1 is the DGLAP kernel for this model [8, 14]. We see that this result has the same structure one gets in QCD. The contribution to the structure function is obtained as a convolution with the bare arton density f 0 (x) W (x; Q ) = x du u f 0(u)w(x=u; Q ): (.1) 4

6 The collinear divergence in w(x; Q ) can be absorbed as usual by dening a Q deendent scalar arton density f(x; Q ) by means of the euation du f 0 (x) = (1? u) + 4 u (4) P (u)1 3 Q x f(x=u; Q ): (.13) This renormalized scalar arton density obeys the DGLAP evolution f(x; Q ) = x du u P (u)f(x=u; Q ): (.14) On the same footing one can dene for the rocess J()! + X with timelike a fragmentation function d(x; Q ) which obeys the same DGLAP evolution euation. Thanks to the Gribov-Liatov recirocity relation [15] at one loo level the timelike DGLAP kernel is the same as in the sacelike case, but this relation is broken at two loo, as exlicitly veried for ( 3 ) 6 in Ref.[9]. 3 Semi-inclusive DIS We are now ready to discuss the rocess + J()! 0 + X. In this case a semi-inclusive structure function can be dened as We dene W (; 0 ; ) = Q XZ X d 6 xe ix <jj(x)j 0 X ><X 0 jj(0)j> : (3.1) z = 0 : (3.) We will calculate the artonic cross section in the limit t Q (3.3) at leading ower, by keeing only divergent terms and ossible log Q =t contributions. It turns out that the aroximation (3.3) selects a secial class of diagrams, those in which the roduced article is radiated by the incoming one. As a matter of fact at the lowest 5

7 Figure 3: Leading order contribution to the one article dee inelastic cross section in the region t Q order in there is no contribution in the region t Q. The rst diagram is the one in Fig. 3 which gives w 1 (x; z; t; Q ) = 0 x(1? x? z) (3.4) t where 0 is the bare couling constant. Notice that, at this order it is necessary to distinguish between and 0. The other contributions come from the one loo corrections listed in Figs. 4,5. It aears that the toology of the diagrams is much richer than the one for the inclusive scattering. The diagram in Fig. 4(a) gives w a (x; z; t; Q ) = Q Z d D k () D 1 k 4 1 (k? 0 ) 4 4 +? (? k) +? (k? 0 + ) : We choose as a air of lightlike vectors and 0 + x and set (3.5) k = + + k? (3.6) 0 = ? (3.7) s = : (3.8) 6

8 (a) (b) (c) (d) (e) (f) (g) Figure 4: One loo leading contributions to the one article dee inelastic cross section 7

9 (h) (l) (m) (n) Figure 5: One loo nite corrections to the one article dee inelastic cross section The delta functions can be used to integrate over and. It turns out that the integral over k? gets the leading contribution in the region of small k?. In this limit we have w a (x; z; t; Q ) ' 1 x(1? x? z)3 4 Z d D? k? 1 1 () D?1 k 4? ((1? z)k? (1? x?? z)0 k (3.9)?? + (1? x? z)(x + z)t) so there is a collinear divergence at k? = 0 which corresonds to the conguration in which k becomes arallel to. The result is w a (x; z; t; Q ) = (4) 3 t x1? x? z? 1 4 : (3.10) (x + z) t In the same way the diagram in Fig. 4(b) has a collinear ole when k gets arallel to 0 and we get w b (x; z; t; Q ) = (4) 3 t x1? x? z? 1 4 : (3.11) (1? x) t 8

10 The other calculations are straightforward and give w c (x; z; t; Q ) = x (4) 3 t 1? z (1? x Q ) log 1? z t (3.1) w d (x; z; t; Q ) = 1? 1 (4) 3 t x 1 4 (1? x? z) (3.13) Q w e (x; z; t; Q ) = (4) 3 t x (1? x? z) (3.14) t w f (x; z; t; Q ) = (4) 3 t x (1? x? z) (3.15) Q w g (x; z; t; Q ) = 1? 1 (4) 3 t x 1 4 (1? x? z): (3.16) t Again the factor in es. (3.14)-(3.16) takes into account the contribution of the symmetric diagrams. The diagrams in Fig. 5 are nite and don't give log Q =t contributions so they will be neglected. There are of course many other diagrams which have not been considered here. They are either diagrams in which the roduced article comes from the fragmentation of the current and diagrams which reresent interference between target and current fragmentation. However one can verify immediately that these diagrams are suressed by owers of t=q, and so they are not relevant here. For examle, consider the diagram deicted in Fig.6(a), in which the observed nal state article comes from current fragmentation. The exlicit calculation shows that this diagram is suressed by two owers of t=q. This fact can be better understood if we redraw it as in Fig. 6(b): in the large Q limit there are actually four lines going into a hard vertex and the diagram is suressed by ower counting [1]. Summing u all the contributions, multilying by Z? J (Q ), introducing the running couling constant (t) = log t=4 '? 3 1 t 4 (4) 3 log 4 9 (3.17)

11 (a) (b) Figure 6: (a) A current fragmentation contribution to the semi-inclusive cross section (b) The same diagram deicted in a dierent form and using 0 = 1? (4) 3 = (t) 1? (4) 3 t (3.18) we nally get w(x; z; t; Q ) = (t) x (1? x? z) (1? x? z) t 6 (4) 3 + 1? x? z (4) 3 (x + z) + 1? x? z? 1 4 (1? x) t + 1 x P x 1? z Q log 3 (4) t 4 t : (3.19) We have now to verify that the collinear divergences in e. (3.19) can be correctly absorbed in the redenition of arton density and fragmentation function. We have W (x; z; t; Q ) = = 0 0 du u du u 0 0 dv v f 0(u)w(u; 0 =v; )d 4 0 (v) dv v f 0(u)w(x=u; z=uv; tu=v; Q )d 4 0 (v): (3.0) Here the integration measure dv=v 4 element of the detected article. takes into account the scaling of the hase sace 10

12 By using the denition of scale deendent arton density (.13) and the corresonding denition for fragmentation function, eventually we nd Z du 1 dv W (x; z; t; Q ) = x+z u z v f(u; (t) v h 4 t) t u u?x + x=u (4) P log Q 3 1? z=uv t 1? x=u 1? z=uv i d(v; t) (3.1) where again only leading log Q =t terms have been considered and the integration limits are derived using momentum conservation. E. (3.1) is our nal result. We see that the renormalized hard cross section gets, as exected, a large log Q =t corrections whose coecient is recisely the DGLAP kernel. Such correction, if not roerly resummed, can soil erturbative calculations in the region t Q. Moreover we nd that in the limit t! 0 a new singularity aears in the cross section, which corresonds to the conguration in which becomes arallel to 0. As ointed out in Ref.[4], when integrating over t, such singularity can't be absorbed in ordinary arton densities and fragmentation functions and the introduction of a new henomenological distribution, the fracture function [3] becomes necessary. E. (3.1) can also be rewritten in the following form (see Fig. 7): W (x; z; t; Q ) = (t) zt?z x + (4) 3 P x r dr r z+r log Q t du u(u? r) ^P i d z u? r ; t r u f(u; t) h 1? x r (3.) where we have dened the A-P real scalar vertex ^P (x) = x(1? x). The function E (1) (x; Q ; Q 0) = (1? x) + Q P (x) log 3 (4) Q 0 (3.3) is the rst order aroximation of the evolution kernel E(x; Q ; Q 0) which resum the leading logarithmic series [14]. This fact suggests that an interretation of e. (3.) can be given in terms of Jet Calculus [14]. 11

13 Q t d f Figure 7: Grahical interretation of e. (3.) 4 Interretation in terms of cut vertices The cut vertex exansion is a generalization of the Wilson exansion originally roosed by Mueller in Ref.[16] and alied to a variety of hard rocesses in Ref.[17]. We will briey recall it for DIS in ( 3 ) 6. Let us go back to Sect. and set < 0 with = ( + ; 0;? ). If we choose a frame in which +? we can write for the arton-current cross section [16] Z du w(; ) = u v( ; u)c(x=u; Q ) (4.1) where v( ; x) = Z V (; k)x x? k + d6 k (4.) + () 6 is a sacelike cut vertex, which fully contains the mass singularities of the cross section, and C(x; Q ) is the corresonding coecient function. In e. (4.) V (; k) is dened as the discontinuity of the four oint amlitude in the channel (? k) and the integration over k is roerly renormalized. 1

14 The results of the revious sections can be easily recast in the form of cut vertex exansion. If we dene v(x; ) = (1? x) +? (4) P (x) 1 3 (4.3) C(x; Q ) = (1? x) + we can write e. (.10) in the form w(x; Q ) = x Q P (x) log (4.4) 3 (4) 4 du u v(u; )C(x=u; Q ): (4.5) Here v(x; ) is a sacelike cut vertex dened at = 0 whose mass divergence is regularized dimensionally. ~ * Figure 8: Factorization of the semi-inclusive structure function and It turns out that a similar interretation can be given of e. (3.19). We dene x = x 1? z v(x; z; t; ) = (t) t h (1? x) + P (x) 4 log (4) t (4) (1? x)? (1? z) x(1? x) (x(1? z) + z)? (1? z) x(1? x) i (1? x(1? z)) t (4.6) (4.7) 13

15 as a generalized cut vertex [1] which contains all the leading mass singularities of the cross section. We can write u to O(t=Q ) corrections (see Fig. 8) w(x; z; t; Q ; ) = x du u v(u; z; t; )C(x=u; Q ) (4.8) where the coecient function is the same which occurs in inclusive DIS. It is well known that in dimensional regularization there is a mixing between collinear and ultraviolet divergences. In order to avoid it one should distinguish between and 0 to regulate UV and collinear divergences, resectively. Moreover external self energies should be taken into account since they are zero on shell for a cancelation of the two kind of divergences. In this framework one can show that factorization (4.8) actually holds grah by grah [19]. 5 Conclusions In this aer we have studied the dee inelastic semi-inclusive cross section in the target fragmentation region and we have erformed an exlicit calculation in ( 3 ) 6 model eld theory. We have shown that the renormalized hard cross section gets a large log Q =t correction as exected in a two scale regime. Furthermore we have found that the coecient driving this logarithmic correction is recisely the scalar DGLAP kernel. This result suggests that the Q deendence of the cross section in such rocesses at xed z and t is driven by the same anomalous dimension which controls the inclusive DIS, as roosed in Ref. [1], and in in the context of diraction in Ref.[6]. We have then examined our result from the oint of view of extended factorization and we have found that it is consistent with such an hyothesis. In this framework the artonic semi-inclusive cross section factorizes into a convolution of a new object, a generalized cut vertex v(; 0 ; x) [1], with four rather than two external legs, and a coecient function C(x; Q ). The former is of long distance nature and embodies the leading mass singularities of the cross section, while the latter is of erturbative nature and, what is imortant, it is the same as in inclusive DIS. 14

16 Therefore these results verify the validity of the aroach roosed in Ref. [1]. Of course the calculation erformed here is only a one loo calculation in a scalar model. Nevertheless we believe that the results obtained maintain the same structure in QCD, by assuming in ( 3 ) 6 a articularly simle and aealing form. Acknowledgments The author would like to thank G. Camici, S. Catani, D.E. Soer and G. Veneziano for useful discussions, and articularly L. Trentadue for his advice and encouragement during the course of this work. References [1] D. Amati, R. Petronzio and G. Veneziano, Nucl. Phys. B140 (1978) 54, B146 (1978) 9; R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G. Ross, Nucl. Phys. B15 (1979) 85. [] G. Altarelli, R.K. Ellis, G. Martinelli, S.Y. Pi, Nucl. Phys. B160 (1979) 301. [3] L. Trentadue and G. Veneziano, Phys. Lett. B33 (1994) 01. [4] D. Graudenz, Nucl. Phys. B43 (1994) 351. [5] D. De Florian and R. Sassot, Phys. Rev. D56 (1997) 46. [6] A. Berera and D.E. Soer, Phys. Rev. D53 (1996) 616; Z. Kunszt and W.J. Stirling, he-h/ [7] J.C. Taylor, Phys. Lett. B73 (1978) 85. [8] Y. Kazama and Y.P. Yao, Phys. Rev. Lett. 41 (1978) 611; Phys. Rev. D19 (1979) [9] T. Kubota, Nucl. Phys. B165 (1980)

17 [10] L. Baulieu, E.G. Floratos and C. Kounnas, Phys. Rev. D3 464 (1981). [11] J.Collins, D.E. Soer and G. Sterman in Perturbative QCD ed. by A.H. Mueller (198) 1. [1] M. Grazzini, L. Trentadue and G. Veneziano, to aear. [13] J. Collins, Renormalization, Cambridge University Press (1984). [14] K. Konishi, A. Ukawa and G. Veneziano, Nucl. Phys. B157 (1979) 45. [15] V.N. Gribov and L.N. Liatov, Phys. Lett. B37 (1971) 78; Sov. J. Nucl. Phys. 15 (197). [16] A.H. Mueller, Phys. Rev. D18 (1978) 3705; Phys. Re. 73 (1981) 37. [17] S. Guta and A.H. Mueller, Phys. Rev. D0 (1979) 118. [18] L. Baulieu, E.G. Floratos and C. Kounnas, Nucl. Phys. B166 (1980) 31. [19] M. Grazzini, PHD thesis, in rearation. 16