We discuss resummation of large logarithmic terms that appear in the
cross-section of semi-inclusive DIS in the case when the final-state hadron
follows the direction of the incoming electroweak vector boson in the c.m.
frame of the vector boson and the initial-state proton.Comment: Presented at the 8th International Workshop on Deep Inelastic
  Scattering (DIS2000), Liverpool, U.K., April 2000; 4 pages, 2 fig

Nadolsky, P.

Stump, D. R.

Yuan, C. -P.

English

arXiv

We discuss resummation of large logarithmic terms that appear in the cross-section of semi-inclusive DIS in the case when the final-state hadron follows the direction of the incoming electroweak vector boson in the c.m. frame of the vector boson and the initial-state proton

Nadolsky, P M

Stump, D R

Yuan, C P

CERN Document Server

MSUHEP-00615hep-ph/0006176SOFT PARTON RESUMMATION IN THE CURRENT REGIONOF SEMI-INCLUSIVE DEEP INELASTIC SCATTERINGyP. NADOLSKY(a), D. R. STUMP(a), C.-P. YUAN(b)z(a) Department of Physics & Astronomy, Michigan State University, E.Lansing,MI 48824, U.S.A.(b) Theoretical Physics Division, CERN, CH-1211 Geneva 23, SwitzerlandWe discuss resummation of large logarithmic terms that appear in the cross-sectionof semi-inclusive DIS in the case when the nal-state hadron follows the directionof the incoming electroweak vector boson in the c.m. frame of the vector bosonand the initial-state proton.During the past years, the H1 and ZEUS Collaborations at DESY-HERAhave put a signicant eort 1 into the experimental study of semi-inclusivedeep inelastic scattering (sDIS) ep ! eBX . In this process, in addition to thescattered electron (or positron) e, some specic nal-state hadron B, or theflow of the hadronic energy into a given region of phase space, is observed. Theproduced data set includes events with a large momentum transfer from theelectron to the hadronic system, Q2 = −q2  2QCD. The theoretical descrip-tion of such events can be attained with the help of methods of perturbativeQCD, under the assumption of a single-photon exchange between the scatteredelectron and the proton. If the renormalization and factorization scales in thecalculation are chosen to be of the order of the large photon virtuality Q, thenthe cross-section can be calculated perturbatively as a series in the small QCDcoupling s. However, the smallness of s does not guarantee fast convergenceof this series. Even when there is a large momentum scale in the collision,†Presented at the 8th Intl. Workshop on Deep Inelastic Scattering (DIS2000), Liverpool,U.K., April 2000‡On leave of absence from Michigan State University.1a)d)b)e) f)c)Figure 1: The diagrams contributing to sDIS at the next-to-leading orderit is not hard to imagine a situation in which the sDIS observable receivesimportant contributions from all orders of the perturbative expansion.To illustrate this point, consider semi-inclusive DIS in the c.m. frame of thephoton and the initial proton. The z-axis in this frame is dened to follow thedirection of the photon’s momentum. We assume all particles to be massless.At the order O(0s) (Fig. 1a), the photon interacts with an initial-statequark a whose momentum is~pa = x~PA = −x~q ; 0  x  1: (1)The nal-state quark b, which has momentum equal to ~pa+~q = (1−x)~q, escapesin the direction of motion of the incoming photon, i.e. along the positive z-axis.At the orderO(s), one needs to account for QCD loop corrections (Figs. 1b-d),as well as for diagrams with the emission of an additional parton (Figs. 1e,f).The individual diagrams may contain soft singularities, but these cancel inthe coherent sum of all contributions. The remaining collinear singularitiesare absorbed into the parton distribution functions (PDFs), or fragmentationfunctions (FFs).While the above computational procedure yields a nite result, it failsto describe accurately the rate of sDIS in the limit when the direction of theresolved nal-state parton is close to the one predicted in the leading order sub-process, that is in the direction of the initial electromagnetic current. Math-ematically, the higher order corrections in this kinematic region (the currentregion) of sDIS are dominated by large logarithmic terms logm qT =Q. Here qTis a variable that is related to the nal hadron’s pseudorapidity  in the γp2c.m. frame:qT = We−η; with W 2 = Q2(1x− 1): (2)The leading order process has  = +1, i.e. qT = 0. The singular logarithmicterms in the limit qT ! 0 are therefore given by1q2T1∑n=1ns2n−1∑m=1vnm logm qTQ; (3)where vnm denote coecients of order 1. As can be seen from (3), due to thepresence of large logarithms, the higher order contributions are not negligiblewhen qT is small. Thus, they all must be accounted for in order to predictreliably the observables in this kinematical regime.It is interesting to note that a structure of the singular terms that is similarto the all-order sum (3) also dominates the cross-section for the production ofback-to-back hadronic jets at e+e− colliders 3, and the cross-section for small-pT vector boson production at hadron colliders 4. For those two processes, ithas been proven that the analog of (3) can be computed in a closed form, byconsidering a Fourier transform of the cross-section from momentum space tothe space of the conjugate variable (the impact parameter) ~b 2,3,4. The resultis called the resummed cross-section. Crossing relations between sDIS, e+e−hadroproduction and vector boson production suggest that the same resum-mation technique may be successful in improving the theoretical description ofthe sDIS observables. Indeed, it was shown 5 that the b-space resummation for-malism provides a description of the energy flow in the small-qT region of sDIS.This approach was generalized 6 to describe particle production cross-sectionsand multiplicities, and a comparison with experimental data was presented.In the b-space formalism 4, the resummed cross-section in the limit qT  Qis written in the form(eA ! eBX)  N 0∫d2b(2)2ei~qT ~b ~WBA(b; Q) + pertBA − singBA ; (4)where~WBA(b; Q) =∑a,b,je−S(b,Q)[DB/b  Coutb/j ][Cinj/a  fa/A]: (5)In this formula, the C-functions, which are convoluted with the PDFs fa/A,or FFs DB/b, arise because of collinear radiation along the direction of theinitial or nal LO quark. The contributions from the sea of soft partons thatare not associated with either of the two jet-like structures are absorbed into3the Sudakov factor S(b; Q). Another part of the Sudakov factor is used toparameterize the contribution from non-perturbative physics, which cannotpresently be calculated from rst principles. The integral over the parameter~b is combined with the known xed-order perturbative cross-section pertBA . Toavoid double counting, one must subtract the singular part of the perturbativecross-section singBA , which was already included in the b-space integral.In Fig. 2 we compare experimental results of the H1 Collaboration 7 to thepredictions of the resummation formalism for the z-flow (rescaled transverseenergy flow in the γp c.m. frame). The precise denition of the z-flow isdzdx dQ2 dq2T=∑B∫ 1zminzd(eA ! eBX)dx dz dQ2 dq2Tdz; (6)where z = (pa  pb)=(pa  q) is a variable describing the fragmentation of thenal parton into the observed hadrons, and the cross-sections in the limitqT ! 0 are calculated according to (4). Using (2), the experimental valuesfor dΣzdxdQ2dq2Tcan be derived from the published pseudorapidity distributionsof the transverse energy flow hET i in the γp c.m. frame.As can be seen in Fig. 2, the results ofthe resummation calculation 6 are in ac-ceptable agreement with the data. Notethat the study of the small qT resum-mation formalism at HERA has severaladvantages, including the possibility totest the resummation formula at verysmall values of Bjorken x. It will beinteresting to compare the predictionsof the resummation formalism for vari-ous types of hadronic nal states, sincethis will allow a test of the universalityof the non-perturbative Sudakov fac-tor. To summarize, the study of semi-inclusive DIS in the current region ofthe γp c.m. frame may provide impor-tant insights on the nature of the softand collinear parton radiation in thehadronic interactions at high energies.Figure 2: Comparison of the resummed z-flowwith the experimental data from the H1 Col-laboration 74AcknowledgmentsP. N. thanks the organizers of the DIS2000 Workshop for their warm hospital-ity. This work was supported in part by the NSF under grant PHY-9802564.5References1. H1 Collaboration, Eur. Phys. J. C, 12, 595 (2000), and Refs. 2, 3, 9-11,19 therein.2. G. Parisi and R. Petronzio, Nucl. Phys. B 154, 427 (1979).3. J. Collins and D. Soper, Phys. Rev. D 16, 2219 (1977).4. J. Collins, D. Soper, G. Sterman, Nucl. Phys. B 250, 199 (1985).5. R. Meng, F. Olness, D. Soper, Phys. Rev. D 54, 1919 (1996).6. P. Nadolsky, D. R. Stump, C.-P. Yuan, Phys. Rev. D 61, 014003 (2000).7. H1 Collaboration, Phys. Lett. B 356, 118 (1995).6

Soft parton resummation in the current region of semi-inclusive deep inelastic scattering

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