The validity of the equivalent-photon approximation for two-photon processes
in electron--positron collisions is critically examined. Commonly used forms to
describe hadronic two-photon production are shown to lead to sizeable errors.
An improved two-photon luminosity function is presented, which includes
beyond-leading-logarithmic effects and scalar-photon contributions. Comparisons
of various approximate expressions with the exact calculation in the case of
the total hadronic cross section are given. Furthermore, effects of the poorly
known low-Q2 behaviour of the virtual hadronic cross sections are discussed.Comment: 10 pages, Latex, uses 12pt.sty, no figur

Schuler, Gerhard A.

English

arXiv

The validity of the equivalent-photon approximation for two-photon processes in electron--positron collisions is critically examined. Commonly used forms to describe hadronic two-photon production are shown to lead to sizeable errors. An improved two-photon luminosity function is presented, which includes beyond-leading-logarithmic effects and scalar-photon contributions. Comparisons of various approximate expressions with the exact calculation in the case of the total hadronic cross section are given. Furthermore, effects of the poorly known low-Q2 behaviour of the virtual hadronic cross sections are discussed.The validity of the equivalent-photon approximation for two-photon processes in electron--positron collisions is critically examined. Commonly used forms to describe hadronic two-photon production are shown to lead to sizeable errors. An improved two-photon luminosity function is presented, which includes beyond-leading-logarithmic effects and scalar-photon contributions. Comparisons of various approximate expressions with the exact calculation in the case of the total hadronic cross section are given. Furthermore, effects of the poorly known low-Q2 behaviour of the virtual hadronic cross sections are discussed

CERN Document Server

CERN{TH/96{297Improving the equivalent-photonapproximation in electron{positron collisionsGerhard A. SchuleraTheory Division, CERN,CH-1211 Geneva 23, SwitzerlandE-mail: Gerhard.Schuler@cern.chAbstractThe validity of the equivalent-photon approximation for two-photon pro-cesses in electron{positron collisions is critically examined. Commonly usedforms to describe hadronic two-photon production are shown to lead tosizeable errors. An improved two-photon luminosity function is presented,which includes beyond-leading-logarithmic eects and scalar-photon con-tributions. Comparisons of various approximate expressions with the exactcalculation in the case of the total hadronic cross section are given. Further-more, eects of the poorly known low-Q2behaviour of the virtual hadroniccross sections are discussed.aHeisenberg Fellow.CERN{TH/96{297October 1996With the advent of LEP2, measurements of two-photon processes in a new domain of cm energiesW will soon become feasible [1]. Together with HERA measurements [2] ofp collisions, detailed insights into the photon structure at the highest energies are aheadof us. At HERA, measurements of the scattered lepton in the luminosity system restrictsthe photon virtuality Q2in tagged p collisions to values below about 0:02GeV2. Thisnumber is well below the typical hadronic scale Q0 mand, hence, the extrapolationto the real photon{proton cross section is under good control. In contrast, two-photonprocesses at LEP cannot be measured at such low virtualities Q2iof the two photons.Either antitagging conditions are imposed or the scattered leptons are measured at ratherlarge angles (single- or double-tag events). In the rst case, photon virtualities up toseveral GeV2are included in the data sample, while Q2values well above 1GeV2areselected in the second case. Even the advent of the so-called very small angle tagger willselect events with an average Q2of still about 0:5GeV2[1] (see also table 1).In any case, the extrapolation of hadronic two-photon processes to zero Q2is highlynon-trivial, in particular in view of the recent very low-Q2data from HERA [3], whichshow a signicant change in the W -dependence of the virtual p cross section as Q2exceeds m2. We hasten to add that measurements of hadronic two-photon cross sectionsat non-zero photon virtualities might give a glimpse on the elusive QCD Pomeron [1,4]. In this letter we estimate the uncertainties associated with the extraction of thetheoretically interesting hadronic two-photon cross sections from the measured e+e onesand propose an improved equivalent-photon approximation (EPA). Let us recall that theEPA is implemented, in one way or another, in practically all programmes [5{10] togenerate hadronic two-photon interactions.The concept of the two-photon luminosity function L() arises when one relates thecross section for the scattering of two real photons (W2) to the measured e+e crosssection (s)(s) =Z1thd (W2= s)L() : (1)Hereps = 2E denotes the e+e c.m. energy and W the  c.m. (\hadronic") energy.Usually [11], L() is calculated as the product (or, more precisely, the convolution) of twoequivalent-photon approximations (EPA):LEPA() =1ZN(x1)N(x2= =x1)dx1x1N(xi) =2(h1 + (1  xi)2ilnQ2imaxQimin  2m2ex2i"1Q2imin 1Q2imax#)Q2imin= Q2i0m2ex2i1  xi(imin= 0)= (1  xi) E22imin(imin6= 0)Q2imax= minnm2; (1  xi) E22imaxomaxfx1min; =x2maxg  x1 minfx1max; =x2min; 1g : (2)The limits on the photon virtualities Q2iand the scaled photon energies xi= !i=E aredetermined by the experimental (anti-)tagging cuts on the angles iand energies Eiofthe scattered electrons ximin= 1  Emaxi=E, ximax= 1  Emini=E.It is important to realize the three approximations that lead to (2) [11]. First, the Q2idependence of TT, the cross section for transverse photons, is neglected. Rather, for a1hadronic cross section the Q2iintegrations are cut o at Q2i= m2in (2). For Q2i<m2the uncertainty is of the order of Q2i=m2. Clearly, the approximation will break down fortagged events where Q2>m2.Second, scalar-photon contributions are neglected. Again, forQ2i<m2the uncertaintyis bound by Q2i=m2, but scalar-photon contributions can potentially be large in taggedevents.Third, the kinematics is not treated exactly. Rather it is based on the approximationQ2i W2. For Q2i<W2, the uncertainty is of order Q2i=W2and, hence, presumably smallfor the measurement of cross sections at large W (W > 5GeV, say,) since the dynamicswill strongly suppress Q2values larger than a few GeV2.We now investigate the importance of the various contributions and propose an im-proved luminosity function. Consider the reaction e+(p1)+ e (p2)! e+(k1)+ e (k2)+Xproceeding through the two-photon process (q1) + (q2) ! X, qi= pi  ki, Q2i=  q2i,W2= (q1+ q2)2. In general, any two-photon process is described by ve non-trivial struc-ture functions (two more for polarized initial electrons). Three of these can be expressedthrough the cross sections abfor scalar (a; b = S) and transverse photons (a; b = T )(ST= TS(q1$ q2)). The other two structure functions TTand TScorrespond totransitions with spin-ip for each of the photons (with total helicity conservation, ofcourse). We emphasize that the hadronic physics is fully encoded in these structure func-tions while the connection with the measured e+e cross section is a pure matter of QED.This connection is most transparent1if we introduce~, the angle between the scatteringplanes of the colliding e+and e in the photon c.m.s.:d =216 4Q21Q22vuut(W2+Q21+Q22)2  4Q21Q22s (s  4m2e)d3k1E1d3k2E2n4 ++1++2TT+ 2 ++1002TS+ 2 001++2ST+ 001002SS+2 j+ 1+ 2j TTcos 2~  8 j+01+02j TScos~o: (3)In exact treatments of the phase space, the latter is most often [11{14] expressed interms of the virtualities Qi(or the polar angles i) and energies !i= qi (p1+ p2)=ps ofthe photons, and of the angle  between the planes of the two scattered electrons denedin the laboratory c.m.s.:d3k1E1d3k2E2=2 s  4m2edQ21dQ22d!1d!2d : (4)Note that, in general,~ 6=  and the hadronic energy W depends non-trivially on theintegration variablesW2=W2A+vuutW4B4 (E2 m2e)2cos ; (5)whereW2A= 4!1!2 Q22(E   !1)E Q12(E   !2)E+Q22Q122E21See, for example, [11] where also explicit expressions for the density matrices abiof virtual photonscan be found.2+2me2!1!2E2 me2+me22EQ22!1+ 2EQ12!2+Q22Q122E2(E2 me2)(6)W4B=  Q22Q124E2  4E !2 Q22  4E2+ 4E !1+Q12 4Q22me2!124E2  4E !2 Q22  4Q12!22me24E2  4E !1 Q12+4Q22Q12me2Q12+Q22  8E2+ 4E !2+ 4E !1+16!22!12me4+ 16me4Q12!22+Q22Q12+Q22!12: (7)It is only for small Q1(and/or small Q2) and small methat~ coincides with  and Wbecomes independent of  and, hence, the azimuthal integration becomes trivial.In order to obtain the most general e+e cross section at xedW , equivalently at xed , one needs to transform from one of the integration variables of (4) to W [15{18], forexample from  to W [15]d3k1E1d3k2E2=4 psqs  4m2edQ21dQ22dW2d!1d!2p 164; (8)where the Gram determinant 4is a quadratic function in each !i. Since the virtual-photon cross sections abdo not depend on xi, the e+e cross section at xed  = W2=scan be written asd(s; )d L(; s;m2e; cuts) (W2= s)=Xa;b=T;SZdQ21dQ22Jab(; Q21; Q22; s;m2e; cuts) ab(W2= s; Q21; Q22) ; (9)whereJab=1d!1d!2p 16422 2Q21Q22spXs  4m2ab1ab2: (10)Here we have neglected the~-dependent terms, which were signicant only when ex-perimental situations with asymmetric cuts on the two scattered leptons would not beaveraged. If one is interested in only the total cross section at xed W , analytic resultscan be obtained for the functions Jab. Indeed, an explicit expression for JTThas beenderived in [15]. However, in experimental measurements one always applies cuts on theangles and energies of scattered electrons, in which case exact results for Jabcannot beobtained analytically. On the other hand, accurate numerical results (better than 1%,say) are not easily obtained since (9) involves four non-trivial integrations.For applications at LEP, where the experimental interest is focused on two-photonphysics at high W , one can proceed further analytically by expanding in Q2i=W2. Westart from the observation that, for m2e=s 1, m2e=W2 1, and when at least one Qiissmall compared with W , the ! integration measure can be replaced by the approximateexpression with a Dirac delta distribution:1d!1d!2p 164=1s(d!1d!24!1!2 W2=4+O(m2e=s;m2e=W2; Q2i=W2)): (11)The approximation (11) is justied forW  m, since hadronic two-photon cross sectionsabvanish quickly for Q2i>m2. This yieldsx1dLdx1=ZdQ21ZdQ22Xa;b=T;Sfa(x1; Q21) fb(x2; Q22)ab(W2; Q21; Q22)(W2); (12)3wherefT(x;Q2) =2 (1 + (1  x)2Q2 2 m2ex2Q4) fLLA+ fNLfS(x;Q2) =1  xQ2: (13)The Q2i-integration limits are determined by the experimental (anti-)tagging cuts on theangles of the scattered electrons imin;max. For me !iwe ndQ2imin=Q2i0+ 4E2(1  xi) sin212imin Q2i0+ E2(1  xi)2iminQ2imax=Q2i0+ 4E2(1  xi) sin212imax Q2i0+ E2(1  xi)2imax: (14)Note the inclusion of the usually neglected term Q2i0= m2ex2i=(1 xi) in the upper limitof Q2. Its presence improves the behaviour for xi! 1. In the limit of small maximumscattering angle 1maxand 1min= 0, the Q21-limits (14) coincide with the results quotedin [19] found from an analysis of ep collisions.In order to further proceed analytically we make use of a second observation. AtQ2i W2one can, to a very good approximation, assume factorization of the W and Qidependences:ab(W2; Q2i) = ha(Q21) hb(Q22) (W2) : (15)Then (9) can be reduced to the single product of a \true" two-photon luminosity functionand the real  cross section, the former given byL =1Zdx1x12Yi=1ZdQ2ihfT(xi; Q2i) hT(Q2i) + fS(xi; Q2i) hS(Q2i)i: (16)Equation (16) denes the improved EPA. We emphasize that, in general, the luminosityfunction does not factorize into the product of two equivalent-photon approximations; theapproximation Q2iW2is essential.We now present our results for four typical experimental situations, antitag events(both electrons in the antitag region), low-Q2and high-Q2single-tag events (one electronin the antitag region and the other in the tag region), and low-Q2double-tag events. Wechooseps = 130GeV, W = 10GeV and dene the regions in table 1.Tag region E1minE1max1min1maxQ21minQ21maxAntitag 0 65 0 1:43 0:572 10 125:26High-Q2tag 30 65 1:55 3:67 2:85 34:7Low-Q2tag 30 65 0:28 0:68 0:0931 1:19Table 1: The cuts on the scattered electrons used in this paper. For convenience, thecorresponding Q2-ranges are also given (ps = 130GeV, W = 10GeV; energies in GeV,angles in degrees, and squared momentum transfers in GeV2).4In order to appreciate the uncertainty associated with the poorly known Q2fallo of the hadronic cross sections, we consider three models. The rst follows from aparametrization [20] of the p cross section calculated in a model of generalized vector-meson dominance (GVMD):hT(Q2) = r P 21(Q2) + (1  r)P 12(Q2)hS(Q2) = (rQ2m21P 21(Q2) + (1  r)"m22Q2lnP2(Q2)  P 12(Q2)#)Pi(Q2) = 1 +Q2m2i; (17)where we take  = 1=4, r = 3=4, m21= 0:54GeV2and m22= 1:8GeV2.The second model [21] adds a continuum contribution to simple (diagonal, three-mesons only) vector{meson dominance (VMDc):hT(Q2) =XV=;!;rV m2Vm2V+Q2!2+ rcm20m20+Q2hS(Q2) =XV=;!; Q2m2VrV m2Vm2V+Q2!2; (18)where r= 0:65, r!= 0:08, r= 0:05, and rc= 1 PVrV. Since photon-virtuality eectsare often estimated by using a simple -pole only, we consider also the model dened by(-pole):hT(Q2) = m2m2+Q2!2hS(Q2) = Q2m2 m2m2+Q2!2: (19)In all cases, the Q2iintegrations can be performed analytically so that one is left withonly a single one-dimensional numerical integration. A Fortran program is available onrequest from the author.Table 2 gives the results for the three models obtained from the improved EPA (16)in comparison with the exact results obtained2from (9). Table 2 contains also the resultsof four approximations, two of which are frequently used EPAs: (i) the -pole modelwithout scalar-photon contributions, i.e. hS(Q2i) = 0; (ii) the -pole model in the leading-logarithmic approximation, i.e. neglecting, besides scalar-photon contributions, also them2e=Q4term in fT(fNL= 0 in (13)); (iii) the EPA dened in (2) [11]; and (iv) the EPAobtained by integrating (2) with logarithmic accuracy [11]:LPDG=1224 lnQ2m m2e  1!2f(; m) 13ln31m35f(x; y) =1 +x22ln1y  2sxy1 +x2(1  y) +x2y1  y2Qm=min fm; maxEg ; m= max(4!2minW2; ): (20)2Details of the exact integration (9) will be described elsewhere [23].5Tag region Antitag [10 2] Single low-Q2tag [10 4]Model exact approx. exact approx.GVMD 105 105 608 610VMDc 105 105 622 624-pole 102 102 541 543-pole, hS= 0 99:5 99:5 493 495-pole, hS= fNL= 0 110 522EPA (2) 112 974PDG (20) 114 |Tag region Single high-Q2tag [10 5] Double low-Q2tag [10 5]Model exact approx. exact approx.GVMD 789 851 355 357VMDc 696 749 370 373-pole 267 286 290 292-pole, hS= 0 94:9 101 246 247-pole, hS= fNL= 0 106 247EPA (2) 0 862Table 2: The two-photon luminosity L atps = 130GeV and W = 10GeV for variousmodels of the two-photon cross sections calculated exactly by integrating (9), in theimproved EPA (16), and in the two commonly used EPA (2) and (20).6For W2 4E!min, (20) reduces to the expression quoted by the PDG [22].The fastest (in terms of CPU time) estimate is, of course, obtained with the closedexpression (20) and is accurate to about 10% for the antitag case. Clearly, (20) cannot beused for single-tag cases, but also the double-tag case is beyond its validity. The evaluationof the improved EPA (16) or the standard EPA (2) is still very fast and, denitely, muchshorter than the evaluation of the exact result (9). The result of the standard EPA(2) is slightly better than that of (20) for the antitag case. More importantly, (2) canalso be applied to the low-Q2-tag cases, at least for an order-of-magnitude estimate: theresults for the single and double low-Q2-tag cases are overestimated by factors of 1:6 and2:4, respectively. Results for high-Q2tags cannot be obtained since the lower-Q2limitexceeds m2. An additional drawback of the standard EPA is the fact that one cannotinvestigate uncertainties in the two-photon luminosity arising from the poorly knownlow-Q2behaviour of the virtual hadronic cross sections.As can be seen from table 2, the improved two-photon luminosity function (16) worksextremely well in the single-tag and antitag cases. The accuracy is better than 1% andthe dierences compared to the exact results are smaller than uncertainties associatedwith the low-Q2extrapolation of the hadronic cross sections. Even for the high-Q2case,the accuracy stays better than 10%.The eect of neglecting the non-logarithmic term proportional to m2e=Q2iin fTis aslarge as about 10% for the antitag case. The importance of this term clearly diminisheswith increasing average Q2i. In the case of double low-Q2tag it becomes essentiallynegligible.The dierences between the two models GVMD and VMDc is rather small: in thelow-Q2-tag cases less than 2% and in the high-Q2-tag case about 6%. This result is notsurprising as the two models are quite similar. Yet, the uncertainty in the extraction ofreal  cross section, particularly in tagged events, can well be much bigger. Dierencesof about 7% (12%) are found between the VMDc and -pole models for the single (double)low-Q2-tag cases. The high-Q2-tag results of these two models even dier by more than afactor of 2! These dierences can be traced back to the extra monopole factor (continuumterm) in the transverse-photon part of the VMDc model. On top of that, there is asimilarly large eect of scalar photons.A correct understanding of the hadronic cross sections at low photon virtualities ishence indispensable for a precision measurement of the total hadronic  cross section.The argument can, of course, be turned around: given the large sensitivity to the low-Q2behaviour, and provided the total cross section can be measured in various modes (antitag,low- and high-Q2single- and double-tag events) the transition region towards zero Q2can be investigated at LEP. It goes without saying that with only minor modications,the above-proposed improved luminosity function can also be used for other hadronicreactions, for example high-pTjet production.AcknowledgementsI thank F. Berends, M. Seymour, and D. Summers for useful discussions.References[1] Report on ` Physics', conveners P. Aurenche and G.A. Schuler, in Proc. Physicsat LEP2, eds. G. Altarelli, T. Sjostrand and F. Zwirner, CERN 96-01, Vol. 1, p. 2917[2] H1 collab., S. Aid et al., Z. Phys. C69 (1995) 27;ZEUS collab., M. Derrick et al., Z. Phys. C63 (1994) 391;A. Levy, DESY-95-204, to appear in Proc. EPS HEP 95, Brussels, Belgium, 1995,and references therein;Proc. Workshop on HERA Physics, Durham, 1995, J. Phys. G22 (1996);Proc. Photon '95, Sheeld, 1995, eds. D.J. Miller, S.L. Cartwright and V. Khoze(World Scientic, Singapore, 1995)[3] H1 collab., paper pa02-070 submitted to the 28th Int. Conf. on High Energy Physics,Warsaw, 1996;ZEUS collab., ibid., paper pa02-025[4] J. Bartels, A. DeRoeck and H. Lotter, DESY-96-168 , [hep-ph/9608401];S.J. Brodsky, F. Hautmann and D.E. Soper, SLAC-PUB-7218-T-E, [hep-ph/9610260][5] \HERWIG", G. Marchesini et al., Comput. Phys. Commun. 67 (1992) 465[6] \PYTHIA", T. Sjostrand, Comput. Phys. Commun. 82 (1994) 74; Lund Universityreport LU-TP-95-20 (1995)[7] \PHOJET", R. Engel and J. Ranft, Phys. Rev. D54 (1996) 4244;R. Engel, Z. Phys. C66 (1995) 203[8] \MINIJET", A. Miyamoto and H. Hayashii, Comput. Phys. Commun. 96 (1996) 87[9] \GGPS1/2", T. Munehisa, K. Kato and D. Perret{Gallix, in Proc. Physics at LEP2,eds. G. Altarelli, T. Sjostrand and F. Zwirner, CERN 96-01, Vol. 2, p. 211;T. Munehisa, P. Aurenche, M. Fontannaz and Y. Shimizu, preprint KEK CP 032(1995), [hep-ph/9507339][10] \GGHV01", M. Kramer, P. Zerwas, J. Zunft and A. Finch, in Proc. Physics at LEP2,eds. G. Altarelli, T. Sjostrand and F. Zwirner, CERN 96-01, Vol. 2, p. 210[11] V.M. Budnev et al., Phys. Rep. C15 (1975) 181, and references therein[12] J. Field, Nucl. Phys. B168 (1980) 477[13] \TWOGAM", S. Nova, A. Olshevski and T. Todorov, DELPHI Note 90-35 (1990)[14] \TWOGEN", A. Buijs, W.G.J. Langeveld, M.H. Lehto and D.J. Miller, Comput.Phys. Commun. 79 (1994) 523[15] G. Bonneau, M. Gourdin and F. Martin, Nucl. Phys. B54 (1973) 573[16] J.A.M. Vermaseren, Nucl. Phys. B229 (1983) 347[17] \DIAG36", F.A. Berends, P.H. Daverveldt and R. Kleiss, Nucl. Phys. B253 (1985)421; Comput. Phys. Commun. 40 (1986) 271, 285, and 309[18] \FERMISV", F. Le Diberder, J. Hilgert and R. Kleiss, Comput. Phys. Commun. 75(1993) 191[19] S. Frixione, M.L. Mangano, P. Nason and G. Ridol, Phys. Lett. B319 (1993) 3398[20] L.B. Bezrukov and E.V. Bugaev, Sov. J. Nucl. Phys. 32 (1980) 847[21] J.J. Sakurai and D. Schildknecht, Phys. Lett. B40 (1972) 121[22] Particle Data Group, L. Montanet et al., Phys. Rev. D50 (1994) 1173[23] G.A. Schuler, \Galuga: an event generator for two-photon processes in e+e colli-sions", CERN preprint in preparation9

Improving the equivalent-photon approximation in electron-positron collisions

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.348.2525

Improving the equivalent-photon approximation in electron--positron collisions

Abstract

Similar works

Full text

Available Versions

CERN Document Server