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QCD Corrections to Photon Production in

arXiv:hep-ph/9207232v1 10 Jul 1992ETH-TH/92-26June 1992QCD Corrections to Photon Production inAssociation with Hadrons in e+e−Annihilation †Zoltan Kunszt and Zolt´an Tr´ocs´anyi ‡Theoretical Physics, ETH,Zurich, SwitzerlandAbstractA detailed investigation of the theoretical ambiguities present in the QCD de-scription of photon production in e+e−annihilation is given. It is pointed out thatin a well-defined perturbative analysis it is necessary to subtract the quark-photoncollinear singularities.

This subtraction requires the introduction of an unphysicalparameter in the perturbative part of the cross section.The subtracted term isfactored into non-perturbative fragmentation function. The dependence on the un-physical parameter cancels in the sum of non-perturbative and perturbative parts.It is pointed out that for Eγ ≤√s/(2(1+ǫc)) the non-perturbative contributions aresuppressed.

Using a general purpose next-to-leading order Monte Carlo program,we calculate various physical quantities that were measured in LEP experimentsrecently.†Work supported in part by the Schweizerischer Nationalfonds‡On leave from Kossuth University, Debrecen, Hungary

1IntroductionThe production of a photon (or an isolated photon) in association with hadrons in e+e−annihilation is a useful process to learn about the differences in the properties of q¯qγand q¯qg final states, to measure the parton-photon fragmentation function and to testQCD predictions in a channel crossed to photon-photon annihilation. The correspondingtheoretical problems are well understood in the case of prompt photon production athadron colliders, photo-production of jets and heavy flavor and photon-photon scattering.It is an important development that experiments at LEP give us high statistics data andopen ground to study even photon plus multijet final states [1, 2].

The better data call fora quantitative QCD description.The QCD description of inclusive photon production has a simple, but important fea-ture: the photon has hadronic component.In the perturbative treament this fact isreflected by the appearance of collinear photon-quark singularities. In order to obtainwell defined cross sections in perturbative QCD in all orders of the running coupling αs,these singularities are to be subtracted and absorbed into the photon fragmentation func-tions (factorization theorem) [3, 4].

The fragmentation functions of the photon satisfyinhomogeneous evolution equation; it grows with Q2 therefore, it is called “anomalous”[5, 6, 7].It is also interesting to study the case of isolated photon. Physical isolation meansthat we isolate the photons from hadrons and so we cannot make distinction betweenquarks and gluons.Gluons, however, cannot be isolated completely from the photonwithout destroying the cancellation of soft gluon singularities between the virtual and realgluon corrections.

Therefore, a physical isolation cannot eliminate completely the collinearphoton-quark singularities, and so, even in the case of isolated photon production the crosssection contains “anomalous” (non-perturbative) piece. This problem has been recognizedclearly in the next-to-leading order QCD study of isolated photon production at hadroncolliders [8, 9].The theoretical subtleties of defining isolated photon cross section inperturbative QCD, however, have not been clearly formulated in previous studies in thecase of e+e−annihilation [10, 11].In section 2 we review the next-to-leading order description of the inclusive (non-isolated) photon production.

In section 3 we outline the change in the formalism due tothe introduction of isolation cuts for the photon production. We point out that isolationcannot completely eliminate the non-perturbative fragmentation contribution, although itcan reduce its size.

In section 4 a detailed perturbative study is given for the cross sectionof isolated photon plus jet production up to order O(ααs). We review the mechanismsof the cancellation of the infrared singularities and point out that in perturbation theoryfor processes containing a photon in the final state the definition of a finite hard scatter-ing cross requires a counter term which necessarily introduces an unphysical parameter.Section 5 contains our numerical results for the isolated photon plus n-jet production atLEP.

To demonstrate the flexibility of our numerical program to calculate any jet shapeparameters, we calculate the distribution of the photon transverse momentum with respect1

to the thrust axis as well. The last section contains our conclusions.2Inclusive photon production in e+e−annihilationAccording to the factorization theorem, the physical cross section of inclusive photonproduction is obtained by folding the fragmentation functions Dγ/a(x, µf) with the finitehard-scattering cross sections dˆσa:dσγdEγ=XaZ √s/20dEaZ 10 dx Dγ/a(x, µf) dˆσadEa(Ea, µ, µf, αs(µ))δ(Eγ −xEa),(1)where αs(µ) is the strong coupling constant at the ultraviolet renormalization scale µ andµf is the factorization scale.It is instructive to investigate the decomposition of this generally valid expression upto next-to-leading order.

First we remark thatDγ/γ(x) = δ(1 −x) + O(α2),(2)therefore, to leading order in the electromagnetic coupling, the term in eq. (1) given bya = γ is a purely perturbative contribution.

We use this equation to eliminate Dγ/γ(x)from eq. (1).

The hard scattering cross section dˆσγ/dEγ is of order α in comparison to theleading order annihilation cross section σ0.1 The leading non-perturbative part given bythe fragmentation function, however, is of order α/αs. This contribution is the “anoma-lous” photon component.

Its enhanced order is due to the fact that the scale dependenceof the fragmentation functions Dγ/a(x, µf), a = q, ¯q, g are given by the inhomogeneousrenormalization group equations [12, 14]:µ ∂∂µDγ/a(x, µ) = απ Pγ/a(x) + αsπXbZ dyy Dγ/b xy , µ!Pb/a(y),(3)where Pb/a(x) denote the Altarelli-Parisi splitting functions. To order ααs the inhomoge-neous terms have the expressions [4]Pγ/a(x) = P (0)γ/a(x) + αs2πP (1)γ/a(x),(4)whereP (0)γ/q(x) = e2q1 + (1 −x)2x,P (0)γ/g(x) = 0,(5)and after trivial replacement of the color factors in eq.

(12) of ref. [4], we haveP (1)γ/q(x) =(6)e2qCF−12 + 92x +−8 + 12xlog x + 2x log (1 −x) +1 −12xlog2 x+log2 (1 −x) + 4 log x log (1 −x) + 8 Li2(1 −x) −43π2P (0)γ/q(¯q)(x),1 In the following analysis, when the order of a contribution is given, it is always understood incomparison to the leading order annihilation cross section σ0.2

P (1)γ/g(x)=⟨e2q⟩TR−4 + 12x −1649 x2 + 929 x−1+10 + 14x + 163 x2 + 163 x−1log x + 2(1 + x) log2 x.In the last equation,⟨e2q⟩≡NFXq=1e2q,(7)where NF is the number of flavors.2The unique solution of these inhomogeneous equations requires non-perturbative input3at a certain initial scale µ. At asymptotically large values of µ, however, the solutions areindependent of the initial values and one obtainslimµ→∞Dγ/q(x, µ)=α2π log µ2Λ2aγ/q(x),(8)limµ→∞Dγ/g(x, µ)=α2π log µ2Λ2aγ/g(x).

(9)Exact analytic expressions for the Mellin transforms of the aa/γ(x) functions have beenfound in refs. [5, 6].

These are related to the aγ/a functions via crossing. It is useful,however, to have a parametrization in x-space.

Formulas which accurately reproduce theexact leading logarithmic solutions were given in ref. [15]:aγ/q(x)=e2q1x"2.21 −1.28x + 1.29x21 −1.63 log(1 −x) x0.049 + 0.002(1 −x)2x−1.54#,(10)aγ/g(x)=1x[0.0243(1 −x)1.03x−0.97].

(11)A new parametrization of the photon fragmentation functions is described in ref. [20].

Themost striking feature of these solution is that they increase as 1/αs with increasing theevolution scale. Therefore, at high energy the contribution from the quark fragmentationinto a photon gives the leading order (α/αs) termdσ(0)γdEγ= σ04√sXqe2q⟨e2q⟩Dγ/q 2Eγ√s , µ!+ O(α),(12)In next-to-leading order, the µ dependence of Dγ/q has to be calculated with the next-to-leading order evolution equation and we should also add the order α hard scattering crosssectiondσγdEγ= σ04√sXqe2q⟨e2q⟩Dγ/q 2Eγ√s , µ!2We assume e+e−annihilation via virtual photon.

In order to obtain formulas valid at the Z0 peak,trivial modifications of charge factors are required.3In the literature it is usually called Vector Meson Dominance (VMD) contribution [13, 14, 3].3

+XaZ √s/20dEaZ 10 dx Dγ/a(x, µf)dˆσ(1)adEa(Ea, µ, µf, αs(µ))δ(Eγ −xEa)+ dˆσ(0)γdEγ(Eγ, µf, αs(µ)) + O(ααs),(13)where dˆσ(1)a /dEa denotes the order αs cross section of quark and gluon production.The O(α) hard-scattering cross section dˆσ(0)γ /dEγ is defined by subtracting the photon-quark collinear singularity in the MS schemedˆσ(0)γdEγ= limε→0d˜σ(0)dEγ+ dσ(0)CTdEγ,(14)where the first term on the right hand side is the partonic cross section in 4−2ε dimensionsas defined by Feynman diagramsd˜σ(0)dEγ= σ0Xqe4q⟨e2q⟩α2π2√sH 4πµ2s!ε1Γ(1 −ε)Zdy12 dy13 dy23 θ(1−y13 −y23)(y12y13y23)−ε(15)×"(1 −ε) y23y13+ y13y23!+ 2y12 −εy13y23y13y23#δ(1 −y12 −y13 −y23)δ 1 −y12 −2Eγ√s!,where H = 1 + O(ε), while the second term is the MS counter-termdσ(0)CTdEγ= α2π(4π)εεΓ(1 −ε)XqZ √s/20dEqZ 10 dx P (0)γ/q(x)dˆσ(0)qdEq(Eq)δ(Eγ −xEq). (16)The integrations in eqs.

(15), (16) are easily performed. The collinear poles cancel in theirsum.

Setting ε = 0, one obtainsdˆσ(0)γdEγ= σ0α2π2√s2NFXq=1e2q⟨e2q⟩P (0)γ/q(xγ) log s(1 −xγ)x2γµ2!,(17)where xγ = 2Eγ/√s.The O(αs) corrections to the dˆσq,g hard-scattering cross sections are defined by theFeynman diagrams of fig. 1.First we note that dˆσ(1)gcan be obtained from dˆσ(0)γbymodifying the charge factors:dˆσ(1)gdEg= CFσ0αs2π4√sNFP (0)g/q(xg) log s(1 −xg)x2gµ2!.

(18)The cross sections dˆσ(1)qreceives both real and virtual corrections. The loop correction canbe written asdσloopdEq= CFσ0αs2π2√sH µ2s!ε(4π)εεΓ(1 −ε)−23 −3 + (π2 −8)εδ √s2 −Eq!.

(19)4

The Bremsstrahlung contribution has an expression similar to d˜σ(0)/dEγ (15):dσrealdEq= CFσ0e2q⟨e2q⟩αs2π2√sH 4πµ2s!ε1Γ(1 −ε)Zdy12 dy13 dy23 θ(1 −y13 −y23)(y12y13y23)−ε(20)×"(1 −ε) y23y13+ y13y23!+ 2y12 −εy13y23y13y23#δ(1 −y12 −y13 −y23)δ 1 −y23 −2Eγ√s!,The sum of the loop and Bremsstrahlung contributions has the simple expressiond˜σ(1)qdEq= CFσ0αs2π2√sH(4π)εεΓ(1 −ε)(21)×(−P (0)q/q(xq) + ε"P (0)q/q(xq) log sµ2!+23π2 −92δ(1 −xq) + 2 log xq1 + x2q1 −xq+(1 + x2q) log(1 −xq)1 −xq!+−32 11 −xq!+−32xq + 52#),where the index + denotes the usual “+ prescription” of regularizing singular behavior atxq = 1. The remaining single pole is cancelled when one adds the MS counterterm dσ(0)CTwhich is defined asdσ(1)CTdEq= CFσ0αs2π2√s(4π)εεΓ(1 −ε)P (0)q/q 2Eq√s!.

(22)The final result is obtained after setting ε = 0:dˆσ(1)qdEq= CFσ0αs2π2√s(23)×(P (0)q/q(xq) log sµ2!+23π2 −92δ(1 −xq) + 2 log xq1 + x2q1 −xq+(1 + x2q) log(1 −xq)1 −xq!+−32 11 −xq!+−32xq + 52),where xq = 2Eq/√s. This result can also be deduced after replacing trivial color factorsfrom the coefficient functions of inclusive single hadron production calculated in ref.

[3].The theoretical input described in this section is sufficient to extract the photon frag-mentation functions from experimental data in next-to-leading order accuracy. A completeanalysis requires the measurement of the inclusive photon production cross section at var-ious energies.

The recent LEP data give information at the Z-pole. Unfortunately, thedata obtained at PETRA, LEP and TRISTAN suffer from low statistics.

Needless to saythat such an experimental study would give very important complementary informationon the fragmentation functions of the photon obtained at hadron colliders.5

3Inclusive isolated photon production in e+e−anni-hilationLet us now consider the inclusive photon cross section with photon isolation. One can arguethat due to isolation cuts the fragmentation contribution is suppressed.

As a consequence,isolation changes the relative importance of the different contributions. It is reasonable toconsider the effect of isolation typically as an order αs effect.

After imposing the isolationcuts, the fragmentation contribution will be of order α, i. e., the same order as the order ofthe pointlike perturbative cross section dˆσ(0)γ /dEγ. Isolation in practice can only be madewith finite energy resolution.

Therefore, we require that in a cone of half angle δc aroundthe photon three momentum the deposited energy be less than a fraction ǫc of the photonenergy. In experiments this parameter ǫc has a value typically about 0.1.

Calculatingdˆσ(0)γ, iso/dEγ, we should insert a combination of θ functions in the phase space integrals asfollowsS(ǫc, δc)=θ(ϑqγ −δc)θ(ϑ¯qγ −δc)+θ(ϑqγ −δc)θ(δc −ϑ¯qγ)θ(ǫcEγ −E¯q)(24)+θ(ϑ¯qγ −δc)θ(δc −ϑqγ)θ(ǫcEγ −Eq).Let us require thatǫc < 12andsin2 δc2 < 12and choose integration variablesxγ = 2Eγ√s ,y = y13xγ.We define the hard scattering cross section again with a collinear counter-termdˆσ(0)γ, isodxγ= limε→0d˜σ(0)isodxγ+ dσ(0)CT, isodxγ,(25)where the first term in the right hand side is calculated as given by Feynman diagrams in4 −2ε dimensions and for the counter-term, we use the MS-type expressiondσ(0)CT,isodxγ= 2σ0α2πXqe2q⟨e2q⟩(4π)εεΓ(1 −ε)P (0)γ/q(xγ)θ(xγ −11 + ǫc). (26)After performing the integration over y and setting ε = 0, one obtainsdˆσ(0)γ, isodxγ= 2σ0α2πXqe2q⟨e2q⟩("P (0)γ/q(xγ) log s(1 −xγ)x2γymµ2(1 −ym)+ e2qxγ(1 −2ym)#θxγ −11 + ǫc(27)6

+ P (0)γ/q(xγ) log 1 −ycyc−e2qxγ(1 −2yc)),where yc and ym are defined as followsyc =1 −xγ1 −xγ sin2 δc2sin2 δc2 ,ym = min(yc, 1 + ǫc −1xγ).One can make several comments on this result.• The unisolated case can be recovered in the limit ǫc →∞(cf. eq.

(17)).• Imperfect isolation allows for a contribution from the fragmentation: the photonlooks isolated since the relatively soft fragments surrounding it are not counted.• Assuming perfect energy resolution (ǫc = 0) we obtain vanishing counter term. Inhigher order, however, we can not isolate the photon from the soft gluons completely(we shall discuss this point in great detail in the next section), therefore, one cannot set the value of ǫc to zero.• In the leading logarithmic approximation one can define a fragmentation functionwith isolation satisfying a modified inhomogeneous evolution equationµ ∂∂µDγ/a(x, µ, ǫc) =(28)απ Pγ/a(x)θx −11 + ǫc+ αsπXbZ dyy Dγ/b xy , µ, ǫc!Pb/a(y).Clearly, if Dγ/a(x, µ) is a solution of the evolution equation without isolation thenDγ/a(x, µ, ǫc) = Dγ/a(x, µ)θx −11 + ǫc(29)will be the solution of the evolution equation with isolation.In next-to-leadinglogarithmic approximation and/or choosing a different counter-term (for examplecompletely subtracting the contribution of the singular region as defined by the thirdterm of eq.

(24)), the isolated fragmentation can also be dependent on δc therefore, ingeneral, one cannot simply identify the isolated fragmentation with the non-isolatedfragmentation in the high-x region.In next-to-leading order, the physical cross section of isolated photon production is givenby the terms as followsdσisoγdEγ(ǫc, δc) = dˆσ(0)isodEγ+ dˆσ(1)isodEγ+ 2σ0α2πDisoγ/q(2Eγ√s , µ)θ(2Eγ√s −11 + ǫc)+XaZ √s/20dEaZ 111+ǫcdx Disoγ/a(x, µf) dˆσadEa(Ea, µ, µf, αs(µ))δ(Eγ −xEa). (30)7

This decomposition is scheme dependent. The first term on the right hand side of thisequation has been calculated in the MS scheme (see eq.

(27). It also appears useful tocalculate the next-to-leading order perturbative cross section dˆσ(1)/dEγ in the MS scheme.This requires the calculation of the next-to-leading order splitting function P (1)γ/a in thepresence of isolation cuts and a corresponding local subtraction term has to be found.This is a complex but feasible calculation.

Since such a result is not yet available, inthe next section we carry out the calculation of dˆσ(1)γin a different subtraction schemewhere the photon is completely isolated from the quarks but not from soft gluons (“conesubtraction”). In this scheme, in leading order, the counter-term is vanishing and the crosssection becomes independent of ǫc:dˆσ(0)dEγ= 2σ0α2πXqe2q⟨e2q⟩(P (0)γ/q(xγ) ln 1 −ycyc−xγ(1 −2yc)).

(31)We note that the logarithmic divergence at xγ = 1 is the usual soft singularity. Contrary tothe case of the MS scheme, with cone subtraction the cross section is continuous and alwayspositive (see fig.

2 for comparison). One may argue that in this scheme the perturbativepart is separated more efficiently, consequently the contributions of the non-perturbativeterms (proportional to Disoγ/a) become relatively smaller.In general we find that the non-perturbative terms contribute mainly in the regionxγ > 1/(1 + ǫc) thus we conclude that the perturbative predictions appear to be reliablefor Eγ < √s/(2(1 + ǫc)).In the next section we present the results of our next-to-leading order perturbativecalculation of dˆσ(1)iso for isolated photon plus n-jet production.

We conjecture that a jetalgorithm applied to the isolated photon hard scattering cross section (eq. (30)) providesan infrared safe isolated photon plus n-jet cross section.

This is supported by the fact thatour isolation prescription does not influence the soft-gluon structure of the cross section.If we can define a jet algorithm,dˆσisoγdEγ(δc, ǫc) = dˆσisoγ+1 jetdEγ(δc, ǫc) + dˆσisoγ+2 jetsdEγ(δc, ǫc) + dˆσisoγ+3 jetsdEγ(δc, ǫc) + . .

. ,(32)such that every term on the right hand side is finite and we count every particle only once,then isolated photon plus n-jet cross section appears to be infrared safe.We shall see that the non-perturbative (“anomalous”) contributions are important onlyin the case of photon plus 1 jet production when the cross section is dominated by thexγ > 1/(1 + ǫc) region.4Isolated photon plus n-jet productionIn QCD, the differential cross section at O(α2s) is a sum of the real and virtual corrections:dσ = |M4|2dS(4) + |M3|2dS(3),(33)8

where dS(n) is the n-body phase space element with the flux factor included. For infraredsafe quantities both terms on the right hand side are separately divergent, but the sumis finite.

It is very difficult to handle numerically this cancellation. Fortunately, at leastat one loop, the divergencies can be cancelled analytically.

There are two commonly usedalgorithms to achieve such a cancellation — the subtraction method [16, 17] and the phasespace slicing method [18, 19]. They both rely on the fact that after partial decomposition|M4|2 can be written as a sum of terms with single pole singularity.

Focusing our attentionto the case of q¯qγg final state, we find four such terms:|M4|2 = CFααs Mgqygq+ Mg¯qyg¯q+ Mγqyγq+ Mγ¯qyγ¯q!,(34)whereyij = (pi + pj)2/s,(s = M2Z)(35)The pole part of each term is defined asPijyij,wherePij = limyij→0 Mij. (36)It can be integrated analytically over either the whole or a part of the phase space.

In thisway, in general, we obtain analytical expressions for the regularized divergencies of dσ(4)which cancel against the divergencies of the virtual corrections, dσ(3) (KLN theorem).When a photon in the final state is observed, the cancellation mechanism described abovedoes not apply to the yγq(¯q) poles. The reason for this is that the process is exclusive inthe photon and the virtual corrections with the photon in the loop cannot contribute forkinematical reasons.To make the discussion more transparent, let us consider contributions from the regionwhere only yqg is small.

The virtual corrections can also be split into three terms|M3|2 = CFααsM(3)gq + M(3)g¯q + M(3)f,(37)such that M(3)gq contains one half of the singularities, the second term contains the otherhalf and the third is the finite part.4 (Notice that there are no M(3)γq , M(3)γ¯q terms.) Thenwe shall concentrate onMijyijdS(4) + M(3)ij dS(3)(38)parts of the cross section.In the subtraction method, one considers the combinationMgqygqdS(4) −PgqygqdS(4→3) + Z PgqygqdS(g)!dS(3) + M(3)gq dS(3),(39)4For the reader’s convenience, we give the explicit expressions for Mij, M (3)kl and M (3)fin the appendix.9

where the integration over dS(g) is meant to be an integral over the gluon variables. dS(4→3)means the factorized four-body phase space element in the limit when the gluon is soft orcollinear to the quark: dS(4→3) = dS(g)dS(3).In the phase space slicing method, formula (38) is written asMgqygqθ(ygq −y0)dS(4) + Mgqygqθ(y0 −ygq)dS(4) + M(3)gq dS(3).

(40)If y0 is chosen small enough (y0 ≤10−4), thenMgqygqθ(ygq −y0)dS(4) + Pgqygqθ(y0 −ygq)dS(4→3) + M(3)gq dS(3)(41)is a good approximation. The first and second terms depend on y0 strongly, but their sumis independent of this unphysical parameter.

The strong y0 dependence originates mainlyfrom the slicing of the soft gluon region.If one wishes to calculate isolated photon production, one has to make sure that therestriction of the phase space does not disturb the cancellation mechanism of soft andcollinear gluons.At hadron level the meaning of photon isolation is well-defined.Atparton level however, one has to be careful because the isolation prescription is differentfor states with different number of partons.5 If the photon is isolated form the partonswith yγ, we should include isolation cuts with respect to all partons:θ(yγq −yγ)θ(yγ¯q −yγ)θ(yγg −yγ). (42)The isolation from the gluon can be implemented only in the first term of formula (39).However, if we cut the soft gluons in the first term, then the cancellation of singularitiesbetween the first and second terms breaks down.One can maintain the cancellationintroducing an energy resolution parameter ǫ such that a gluon is isolated from the photononly if its energy is greater then ǫEγ.Accordingly, isolation for the first term meansmultiplication withθ(yγq −yγ)θ(yγ¯q −yγ)(1 −θ(yγ −yγg)θ(Eg −ǫEγ)).

(43)Clearly, this criterium is “not physical” in the sense that one cannot implement it athadron level since we apply different cuts to quarks and gluons.If we introduce photon isolation in the slicing method, from formula (41) we obtainMgqygqθ(ygq −y0)dS(4)θ(yγq −yγ)θ(yγ¯q −yγ)θ(yγg −yγ)+(44)5One may object isolation at parton level arguing that the fragmentation process inevitably scattershadronic matter into the isolation cone. For a purely perturbative analysis, this objection is not valid.To understand the reason for this, let us consider the same process at higher energies, say √s = 10 T eV ,in which energy region perturbation theory is expected to give even better description.

Clearly, at thisenergy, fragmentation does not alter the energy flow, therefore isolation at parton level corresponds toisolation at hadron level.10

Pgqygqθ(y0 −ygq)dS(4→3) + M(3)gq dS(3)!θ(yγq −yγ)θ(yγ¯q −yγ).Usually, yγ ≫y0. This means that when changing y0 at fixed yγ, the contribution from thesoft gluons will be cut independently of y0 and consequently, the y0 dependence is dampedin the first term.

On the other hand, in the second term the y0 dependence is not dampedby the gluon-photon isolation condition. The conclusion is that the y0 dependence will bedifferent in the two terms and, therefore, the cross section will depend on the unphysicalparameter y0.

It is important to notice that if ygq > y0, then there exist an ǫ′ such that ifǫ < ǫ′ then1 −θ(yγ −yγg)θ(Eg −ǫEγ) = θ(yγg −yγ),(45)therefore (44) defines a finite cross section, but y0 plays in a sense the role of ǫ used informula (43).To demonstrate the y0 dependence of the isolated photon cross section explicitly, wecalculated the isolated photon plus 1- and 2-jet cross sections using the isolation criteriumgiven by formula (44). First we make two technical remarks about the slicing method.Choosing large y0, the pole approximation is not precise enough in the singular region;one has to take into account the non-singular terms in the same region, i. e., one shouldadd the MgqygqdS(4) −PgqygqdS(4→3)!θ(y0 −ygq)(46)correction term.

The calculation becomes analogous to the subtraction method and onehas to introduce the ǫ energy resolution parameter.It is a practical question to establish at what value of y0 the correction term (46)becomes important. The most straightforward way to calculate the finite terms in (44) isto perform the integration by a Monte Carlo method, which leaves sufficient flexibility tocalculate any jet shape parameter one wishes to obtain.

The Monte Carlo calculation has afinite statistical error which of course, can be reduced by generating more points. Then thecriterium which determines the importance of the correction term (46) is to require thatthe systematic error introduced by neglecting (46) has to be smaller then the statisticalone.

Clearly, this critical value of y0 depends on the jet resolution parameter yJ as wellas on yγ. For the case of 3-jet production without photon in the final state, an analysiswas carried out in ref.

[19] to determine the critical value of y0 above which the systematicerror dominates. They found that choosing y0/yJ ≤0.01 removes the systematic error.The number of isolated photon plus n-jet events can be conveniently parametrized inthe form1σ0σγ+n jets(yJ, y0)=α2πXqe4q⟨e2q⟩g(0)n (yJ, y0) + αs2πg(1)n (yJ, y0)(47)≡α2πXqe4q⟨e2q⟩g(0)n (yJ, y0)(1 + αsRn(yJ, y0)),11

where σ0 is the leading order cross section of the reaction e+e−→hadrons and theRn(yJ, y0) functions are defined by the equation. In figs.

3 and 4, we show the y0 de-pendence of the O(αs) QCD corrections, Rn(yJ, y0), to the isolated photon plus 1-jetand the isolated photon plus 2-jet cross sections. To obtain the corrections, we used thefollowing algorithm:1. select isolated γ + n-jet events by requiring the invariant mass of the photon withany particle in the event to be larger than yγ (see formula (44);2. apply E0 cluster algorithm to the hadronic part of the event;3. separate γ+ 1-, 2-, and 3-jet events by the number of remaining clusters of hadrons.We used yγ = yJ.

As expected, the y0 dependence in Rn(yJ, y0) is strong up to y0 = yγ.As explained before, in formula (44) the y0 cut plays the role of the ǫ parameter of formula(43). Therefore, the (apparently) physical cut, (44) is in fact unphysical because y0 is nolonger a dummy variable of the cross section.In order to demonstrate that we control the numerical evaluation of the integrals atsmall y0 values, we calculated Rn(yJ, y0) with θ(yγg −yγ) in (44) removed.

We denote thecorresponding quantity with ˜Rn(yJ, y0). According to the discussion after formula (44),this alteration should remove the y0 dependence.

The explicit calculation shows that thisindeed happens.6 We see that in order to obtain a y0 independent result, we have to usean unphysical cut: different cuts are applied to quarks and gluons.We conclude from this discussion that if we want to define a finite isolated photonplus n-jet cross section we have to make a subtraction which depends on some unphysicalparameter no matter which algorithm — the subtraction or slicing one — is used (seeformulas (43), (44) and (45)). In other words, the physical isolated photon plus n-jetcross section always contains some non-perturbative (“anomalous”) contribution whichis expected to give contributions comparable or somewhat smaller than the O(αs) QCDcorrections.

For the separation of perturbative and non-perturbative parts of the crosssection, one must introduce an unphysical (non-zero) parameter. Of course, the sum ofthe perturbative and non-perturbative pieces is independent of this parameter.

In theprevious section we pointed out that the non-perturbative contribution is expected to besmall for xγ < 1/(1 + ǫc).5Numerical resultsAs advocated in section 4, we carry out our calculation with the subtraction method. Theevent definition is the following:6In fact, we can see weak y0 dependence in ˜Rn(yJ, y0).

The origin of this dependence is the use of thepole approximation. It can be observed for values y0 > 10−3 (this value depends on yJ) in accordancewith the observation made in ref.

[19]. The results are shown for yJ = 0.06.

For other values of yJ thedependence is similar.12

1. isolate the photon;2. apply E0 cluster algorithm to the hadronic part of the event;3. separate γ+ 1-, 2-, and 3-jet events by the number of remaining clusters of hadrons.Photon isolation can be achieved either by isolating the photon in a cone (cone isolation)or by requiring the invariant mass of the photon with any particle in the event to belarger then an invariant mass cut yγ. From experimental point of view the cone isolationis more natural.

Unfortunately, the results by OPAL [2] are corrected experimental valuesin order to compare the measured rates with the matrix element calculation of [10] whereinvariant mass isolation was used (with yγ = yJ). We give results for both.

Since the QCDcorrections are very sensitive to the event definition we give explicitly how formula (39) ismodified in the case of cone isolation:θ(ϑqγ −δc)θ(ϑ¯qγ −δc)×((1 −θ(δc −ϑgγ)θ(Eg −ǫcEγ))MgqygqdS(4)(48)−PgqygqdS(4→3) + Z PgqygqdS(g)!dS(3) + M(3)gq);and in the case of invariant mass isolation:θ(yqγ −yJ)θ(y¯qγ −yJ)×((1 −θ(yJ −ygγ)θ(Eg −ǫcEγ))MgqygqdS(4)(49)−PgqygqdS(4→3) + Z PgqygqdS(g)!dS(3) + M(3)gq).To obtain the isolated photon plus n-jet rates, the formulas above are multiplied withθ functions as follows:• One photon plus 3-jet:θ(yqg −yJ)θ(y¯qg −yJ)θ(yq¯q −yJ)θ(yqγ −yJ)θ(y¯qγ −yJ)θ(ygγ −yJ). (50)• One photon plus 2-jet:Denote i and j the partons which when combined have the smallest invariant mass inthe hadronic part of the event, so they are combined into pseudoparticle c. Denotek the third parton.

In the three-body phase space the momentum of the j particleis identically zero. Then we useθ(yJ −yij)θ(yck −yJ)θ(ycγ −yJ)θ(ykγ −yJ).

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• One photon plus 1-jet:θ(yJ −yqg)θ(yJ −y¯qg)θ(yJ −yq¯q)θ(yJ −yck). (52)In the case of cone isolation, we also required that the energy of the photon has to belarger than 7.5 GeV.

The half-opening angle of the cone is 15◦.We shall give the results of our calculation for the partial widths Γ(Z →γ + n jets) asratios to the hadronic width:Γ(Z →γ + n jets)Γ(Z →hadrons) =89cu + 13cd α2π(2cu + 3cd)1 + αsπ + 1.42αsπ2gn(yJ),(53)wherecf = v2f + a2f(54)and vf and af are the weak vector and axial vector couplings:vf = 2I3,f −4ef sin2 θW(55)af = 2I3,f,(56)so with sin2 θW = 0.23, vu = 0.39, vd = −0.69, au = +1 and ad = −1. The gn(yJ) functionscan be expanded in αs:gn(yJ) = g(0)n (yJ) + αs2πg(0)n (yJ) ≡g(0)n (yJ)(1 + αsRn(yJ)),(57)and our aim is to compute the g(i)n functions (of course, g(0)3 (yJ) = 0.

)It is interesting to study the photon energy spectrum of the jet cross sectionsdσisoγ+1jetdEγ(δc, ǫc),dσisoγ+2jetdEγ(δc, ǫc)(58)at some realistic values of the isolation parameters δc, ǫc.In fig. 5 the photon energydistibutions of one jet and two jet production are shown in the Born approximation, whilein fig.

6 the same curves are plotted but including the next-to-leading order corrections.Due to obvious kinematical reasons one jet production is completely dominated by the hardphoton region xγ > 1/(1 + ǫc). In this region as we pointed out one may get substantial(but not overwhelmingly large) “anomalous” photon contribution.

It is difficult to estimatethe “anomalous” contribution since we do not have yet enough phenomenological input.Certainly a combined study of the hadron collider and LEP data would help to understandits size better. We note that in the high x region the application of perturbative QCD byitself requires some care due to the appearance of large logarithms of type log(1−x).

Indeedthe QCD corrections are larger for one jet than for two jet production. It is interestingto compare the one jet data to the perturbative QCD prediction, but one should not14

be surprised if one does not find perfect agreement.The non-perturbative correctionsappear, however, negligible in the case of 2-jet production since it is dominated by thecomplementary region xγ < 1/(1 + ǫc). Requiring that xγ < 1/(1 + ǫc), the 2-jet resultsremain practically unaffected, while this cut largely eliminates the 1-jet production.

Thisis illustrated by the numbers given in Table 1. There is a tendency that if we shrink theisolation region the perturbative contribution increases.

From figs. 5, 6 and, we can seealso that the total one jet and two jet rates should depend weekly on ǫc.

The reason forthis is that the photon energy distribution changes weakly if we change ǫc in the physicallyinteresting region of 0.06–0.2 .In addition to the ambiguities due to “anomalous” photon production there are also theusual scale ambiguities. In fig.

7 we present the predicted values of the Γ(Z →γ + n jets)(n = 1, 2) partial widths for the cone isolation with ǫc = 0.1. The bands between thedashed lines represent the scale dependence between the scales MZ/2 and 2MZ.

We usedαs(MZ) = 0.12 and α = 1/137. The ǫc dependence is so weak for experimentally feasiblevalues that the uncertainty introduced by the ǫc dependendce is much smaller than thescale dependence and therefore we did not show it.

The scale dependence of the 1-jet rateis rather large. This is a reflection of the fact that the QCD corrections are large.

In figs.8 and 9 the same curves as in fig. 7 are depicted in the case of invariant mass isolationwith yγ = yJ for the 1-jet and 2-jet rates, respectively.

In the same figures, we show theenhancement induced by the choice of smaller isolation region. In accordance with ourprevious discussion, the enhancement is larger for the 1-jet rate than for the 2-jet rate.We note, however, that when comparison is made to the data at a given isolation it isimportant to use exactly the same isolation and event definition both in the experimentaland theoretical analysis.

Therefore one can not just change the value of yγ such that theprediction fits better the data. In particular one is not allowed to use different values ofyγ in case of one jet and two jet production.

In a given subtraction scheme with welldefined experimental isolation cuts all the parameters of the perturbative part are fixed.In particular the discrepancy between the measured γ+ 1-jet rate and the perturbativeprediction at yγ = yJ may indicate non-negligible anomalous contributions.As mentioned in the section 4, the Monte Carlo approach is useful because it leavessufficient flexibility to calculate any jet shape parameter. To demonstrate this feature ofour work we present the result of matrix element calculation for the distribution of thephoton transverse momentum with respect to the thrust axis (fig.

10). The thrust axis hasbeen calculated all particles taken into account, including the photon.

We used invariantmass isolation (with yγ = 0.005 and 0.06) to isolate the photon from the partons. Wealso required the photon to be more energetic than 7.5 GeV.

For small pT, configurationswith thrust value close to one may occur. The histogram is normalized to one, thereforethe uncertainty in the small pT region influences the behaviour in the large pT region.

Wenote, however, that requiring xγ < 1/(1 + ǫc) the small pT region will be suppressed.Finally, in fig. 11, we present the predicted values of the Γ(Z →γ + n jets) (n = 1, 2)partial widths for the cone isolation with ǫc = 0.1 when Durham clustering algorithm is15

used [21]. In this algorithm, two jets are combined in to a single jet ifyDij = 2min(E2i , E2j )(1 −cos θij)s(59)is smaller than the jet resolution parameter yJ.

For pure QCD events, this algorithmtends to emphasize 2-jet events as compared to other algorithms and suited better forresummation purposes [23]. When a photon in the final state is observed, we find higher1-jet rate and lower 2-jet rate and the QCD corrections are smaller as compared to the E0cluster algorithm.6ConclusionsPhoton production in association with hadrons in e+e−annihilation provides us interestinginformation on the non-perturbative component of the photon and new possibilities to testthe underlying structure of perturbative QCD.In this paper we paid special attention to the importance of the correct treatment ofthe collinear photon-quark region.

It was shown that next-to-leading and higher ordersthe perturbative part can only be defined using some non-physical parameter, no matterwhether non-isolated or isolated photon production is considered.The physical crosssection defined as the sum of the perturbative and non-perturbative part is, of course,independent of such a parameter.We briefly reviewed the theoretical description of the inclusive non-isolated photonproduction in e+e−annihilation.It was pointed out that the LEP data can be usedto constrain the parametrization of the fragmentation functions of the photon, Dγ/q(x, µ),Dγ,g(x, µ). The measurement of these fragmentation functions would give important inputinformation for the other inclusive photon production measurements at hadron collidersand at HERA.

Furthermore, one could test the anomalous µ-dependence at asymptoticallylarge µ values predicted by perturbative QCD.The case of isolated photon production was studied as well. Under well defined cir-cumstances, isolation can suppress the numerical contribution of the non-perturbativecontributions.

We pointed out that the non-perturbative (“anomalous”) contribution canbe sizable only for Eγ > √s/2/(1 + ǫc), where ǫc is the energy fraction in the isolationcone with respect to the photon energy. When a jet algorithm is used, then the non-perturbative contribution is expected to be further suppressed for isolated photon plusn-jet cross section for n > 1, but not for n = 1.We demonstrated the difficulty due to the quark photon collinear singularity withcareful calculation of the next-to-leading order QCD corrections to isolated photon plusone or two jets.

We argued that in the case of isolated photon plus 2-jet production indeed,as suggested by Kramer and Lampe, the perturbative contribution dominates the physicalcross section. The next-to-leading order corrections are calculated by developing a MonteCarlo program which can be used to calculate the perturbative corrections to any physicalquantity.16

AcknowledgementWe thank P. M¨attig and C. Markus for helpful correspondence.One of us (Z.K.) is greatful to R. K. Ellis and G. Sterman for illuminating discussions.Appendix AIn this appendix we give the explicit expressions for the Mij and M(3)ijexpressions usedin formulas (34) and (37) respectively.

We shall make the following renaming:q→particle 1¯q→particle 2(60)g→particle 3γ→particle 4.Then the following relations are valid:M23 = M13(1 ↔2),M14 = M13(3 ↔4),M24 = M13(1 ↔2,3 ↔4),(61)therefore, it is sufficient to spell out M13. The corresponding expression can be obtainedfrom the four-parton matrix element given in Appendix B of ref.

[16] by setting NC = 0and TR = 0. After performing partial fractioning one obtainsM13=24π2" 2y12y123(1 + y34)y134y234(y13 + y23) + 2y14(1 −y24)y234(y13 + y24) + 2(1 −y13)y23y134(y13 + y24)+ 1y2134(y24y34 + y12y34 + y13y24 −y14y23 + y12y13) +y34y13 + y24+y12y24y34 + y12y14y34 −y13y224 + y13y14y24 + 2y12y14y24y134(y13 + y14 + y23) 1y13 + y23+1y13 + y14!+y12y24y34 + y12y14y34 −y214y23 + y14y23y24 + 2y12y14y24y234(y13 + y23 + y24) 1y13 + y23+1y13 + y24!+y12y23y34 + y12y13y34 −y14y223 + y13y14y23 + 2y12y13y23y134(y13 + y14 + y24) 1y13 + y14+1y13 + y24!+2y12y123y124y13 + y23 + y14 + y24 1y13 + y23 + y24 1y13 + y24+1y13 + y23!+1y13 + y14 + y24 1y13 + y14+1y13 + y24!+1y13 + y14 + y23 1y13 + y14+1y13 + y23!

!+1y134y234(y13 + y24)(y12y234 −y13y24y34 + y14y23y34 + 3y12y23y3417

+3y12y14y34 + 4y212y34 −y13y23y24 + 2y12y23y24−y13y14y24 −2y12y13y24 + 2y212y24 + y14y223+2y12y223 + y214y23 + 4y12y14y23 + 4y212y23+2y12y214 + 2y12y13y14 + 4y212y14 + 2y212y13 + 2y312)−1y134(y13 + y14)(y14y24 + 2y14y23 + 2y12y14 + y213+y13y23 + 2y13y24 + 2y12y13 + y214)Due to partial fractioning, this expression is finite if a single yij →0 (and for the samereason the expression is lengthy. )The virtual corrections can also be obtained easily from eq.

(2.20) of ref. [16] by settingNC = 0 and TR = 0.

In our decompositionM(3)gq = M(3)g¯q =14π2−1ε2 −12ε(3 −2 log y12),(62)and the finite part isM(3)f=14π2"y12y12 + y14+y12y12 + y24+ y12 + y24y14+ y12 + y14y24(63)+ log y14"4y212 + 2y12y14 + 4y12y24 + y14y24(y12 + y24)2#+ log y24"4y212 + 2y12y24 + 4y12y14 + y14y24(y12 + y14)2#−2"y212 + (y12 + y14)2y14y24R(y12, y24) + y212 + (y12 + y24)2y14y24R(y12, y14)+y214 + y224y14y24(y14 + y24) −2 log y12 y212(y14 + y24)2 +2y12y14 + y24!#+ y24y14+ y14y24+ 2y12y14y24! 23π2 −log2 y12 −8#,whereR(x, y) =(64)log x log y −log x log(1 −y) −log y log(1 −x) + 16π2 −Li2(x) −Li2(y)andLi2(x) = −Z x0 dz log(1 −z)z.

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Figure captionsFigure 1 Typical Feynman diagrams contributing to the calculation of the O(αs) correc-tions to the inclusive quark production in e+e−annihilation.Figure 2 Leading order hard-scattering cross section for inclusive isolated photon produc-tion calculated in the MS subtraction scheme (eq. 27) and in the cone subtractionscheme (eq.

31). δc = 15◦and ǫc = 0.1 isolation parameters were used.Figure 3 The dependence of the QCD corrections to the isolated photon plus 1-jet pro-duction on the unphysical parameter y0 when physical cuts are applied (see formula(44)) — solid curves — and with unphysical cuts (only quarks are cut) — dashedcurves.

The slicing method in the pole approximation was used with yγ = yJ.Figure 4 The dependence of the QCD corrections to the isolated photon plus 2-jet pro-duction on the unphysical parameter y0 when physical cuts are applied (see formula(44)) — solid curves — and with unphysical cuts (only quarks are cut) — dashedcurves. The slicing method in the pole approximation was used with yγ = yJ.Figure 5 Leading order photon energy spectrum of the partial widths Γ(Z →γ + n jets),(n = 1, 2) normalized to 10−3Γ(Z →hadrons) at yJ = 0.1, δc = 15◦and ǫc = 0.1.Figure 6 Next-to-leading order photon energy spectrum of the partial widths Γ(Z →γ + n jets), (n = 1, 2) normalized to 10−3Γ(Z →hadrons) at yJ = 0.1, δc = 15◦andǫc = 0.1.Figure 7 Partial widths Γ(Z →γ + n jets), (n = 1, 2) as a function of yJ normalizedto 10−3Γ(Z →hadrons) when cone isolation of the photon is used (solid lines) withαs(MZ) = 0.12, α = 1/137.The dashed curves represent the scale dependencebetween scales µ = MZ/2 and µ = 2MZ.Figure 8 Partial width Γ(Z →γ + 1 jet), as a function of yJ normalized to 10−3Γ(Z →hadrons) (solid line) when invariant mass isolation is used with yγ = yJ, ǫc = 0.1,αs(MZ) = 0.12, α = 1/137.The dashed curves represent the scale dependencebetween scales µ = MZ/2 and µ = 2MZ.

The dashed dotted curve is the partialwidth calculated with yγ = 0.005 and the long-dashed short-dashed curve is thatwith yγ = 0.001.Figure 9 Partial width Γ(Z →γ + 2 jets), as a function of yJ normalized to 10−3Γ(Z →hadrons) (solid line) when invariant mass isolation is used with yγ = yJ, ǫc = 0.1,αs(MZ) = 0.12, α = 1/137. The dashed curves represent the scale dependencebetween scales µ = MZ/2 and µ = 2MZ.

The long-dashed short-dashed curve is thepartial width calculated with yγ = 0.001.21

Figure 10 Distribution of the photon transverse momentum with respect to the thrustaxis. The photon was isolated using invariant mass isolation with yγ = 0.005 and0.06.

The dotted histograms show the scale dependence between scales µ = MZ/2and µ = 2MZ. The width of one bin is Mz/100.Figure 11 Partial widths Γ(Z →γ + n jets), (n = 1, 2) as a function of yD normalizedto 10−3Γ(Z →hadrons) when Durham clustering algorithm and cone isolation ofthe photon is used (solid lines) with αs(MZ) = 0.12, α = 1/137.

The dashed curvesrepresent the scale dependence between scales µ = MZ/2 and µ = 2MZ.22

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