All-orders resummation for diphoton production at hadron colliders

We present a QCD calculation of the transverse momentum distribution of photon pairs produced at hadron colliders, including all-orders soft-gluon resummation valid at next-to-next-to-leading logarithmic accuracy. We specify the region of phase space in which the calculation is most reliable, compare our results with data from the Fermilab Tevatron, and make predictions for the Large Hadron Collider. The uncertainty of predictions for production of diphotons from fragmentation of final-state quarks is examined.

hadron colliders through its decay into a pair of energetic photons, a challenging prospect at the Large Hadron Collider (LHC) in view of the intense background from hadronic production of non-resonant photon pairs [1]. Theoretical predictions of these background processes may be of substantial value in aiding search strategies. Moreover, the perturbative quantum chromodynamics (QCD) calculation of photon-pair production is of theoretical interest in its own right, and data from the Tevatron collider offer an opportunity to compare and test results against experiment.
In this paper, we present a new calculation of the diphoton cross section in perturbative QCD. We include contributions from all perturbative subprocesses (quark-antiquark, quarkgluon, antiquark-gluon, and gluon-gluon) to next-to-leading (NLO) accuracy. In addition, to describe properly the behavior of the transverse momentum Q T distribution of the pairs in the region in which Q T < Q, where Q is the invariant mass of the photon pair, we include the all-orders resummation of soft and collinear logarithmic contributions up to next-to-nextto-leading log (NNLL) accuracy. This calculation goes beyond the previous resummation treatments of diphoton production [2,3]. Its components are summarized briefly below, and a more complete discussion is presented elsewhere [4].
A full treatment of photon pair production requires that we address the contributions from non-perturbative processes, such as π and η meson decays, and the quasi-collinear fragmentation of quarks and gluons into photons. Elaborate isolation procedures are applied by the experiments to reduce these long-distance contributions, procedures that are only approximately reproducible theoretically. Some final-state fragmentation contributions invariably survive the isolation, especially at the LHC, where the efficiency of isolation is reduced by event pile-up and the large number of energetic hadronic fragments in each event. A new feature of diphoton production, with respect to single photon production, is the prospect that both photons may be produced from fragmentation of the same final-state parton. This fragmentation contribution is expected to be most influential in the region in which both the diphoton invariant mass and the separation ∆ϕ between the azimuthal angles of the two photons are relatively small, Q < Q T and ∆ϕ < π/4. Diphoton production is characterized by large radiative corrections, distributed in a complex pattern over the accessible phase space. The influence of initial-state gluon radiation on the predicted transverse momentum distributions can be evaluated to all orders with the Collins-Soper-Sterman (CSS) resummation procedure [5], the method that we follow. Our results are implemented in a Monte-Carlo integration program RESBOS. We use a simple, efficient approximation for the fragmentation contributions. We compare our results with data from the Collider Detector at Fermilab (CDF) collaboration at pp collision energy √ S = 1.96 TeV and integrated luminosity 207 pb −1 [6], and we observe good agreement. We make several suggestions for a further more differential analysis of the data that would allow refined tests of our calculation. In view of theoretical uncertainties associated with the fragmentation component of the cross section, and the presence of other large radiative corrections, we question the conclusion in Ref. [6] that the inclusion of single-photon fragmentation contributions within the NLO calculation of Ref. [7] uniquely explains the observed kinematic distributions of the diphotons at the Tevatron. We also include predictions for diphoton production at the LHC.
Analytical Calculation. The CSS resummation method is used in Refs. [2,3] to treat the direct production of photon pairs from qq, q (−) g, and gg scattering. The NLO perturbative cross sections (i.e., cross sections of O(α s ) in the qq and qg channels [8,9,10], and O(α 3 s ) in the gg channel [3,11,12,13]) are included as a part of the resummed cross section. Singular logarithms arise in the NLO cross sections when the transverse momentum Q T of the γγ pair is much smaller than its invariant mass Q. These logarithms are resummed into a Sudakov exponent (composed of two anomalous-dimension functions A(µ) and B(µ)) and convolutions of the conventional parton densities f a (x, µ F ) with Wilson coefficient functions C. In Refs. [2,3], the functions A(µ), B(µ) and C are evaluated up to order α 2 s , α s , and α s , respectively. An approximate expression is used there for the C-function of order α s in the gg subprocess (borrowed from the gg → Higgs resummed cross section). In this work, we include the exact C-function of order α s for gg → γγX [14] and O(α 2 s ) expressions for A(µ) and B(µ) in all subprocesses [14,15,16]. These enhancements elevate the accuracy of the resummed prediction to the NNLL level. We use an improved model for the non-perturbative contributions at large impact parameter [17]. When expanded in a series in α s , the resummed predictions for the total rate, γγ invariant mass, and γγ rapidity (y) distributions are equal to the fixed-order QCD cross sections, augmented by higher-order contributions from the integrated Q T logs. The resummed Q T distribution is well-behaved as Q T → 0, unlike its fixed-order counterpart which is singular in this limit. As Q T grows, our resummed cross section crosses the perturbative NLO cross section at Q T ∼ Q, and, for each Q and y, we switch from the resummed to the NLO cross section for values of Q T above this point.
A fragmentation singularity arises in the matrix element when the momentum of a photon is collinear with that of an outgoing quark or gluon. The fragmentation singularities do not appear in the resummed terms since those correspond to initial-state radiation. At the lowest order, the fragmentation singularity appears in the qg → γγq channel and is proportional to P γ←q (z)/(n − 4) in n-dimensional regularization, where P γ←q (z) is the q → γ splitting function, and z is the fraction of the fragmenting quark's light-cone momentum carried by the photon. The fragmentation singularity is subtracted from the direct contribution. It is resummed in the photon fragmentation function D γ (z) through the introduction of a "one- production of a photon from a quark. For a wide class of two-to-two partonic processes, such as qq → qq, etc., there is a second type of "one-fragmentation" contribution that arises in low-mass photon-pair production (Q < Q T ). In this case, a final-state parton may fragment into a low-mass pair of photons, a process described by a different fragmentation function D γγ (z 1 , z 2 ). This new contribution is not included in the existing calculations. "Two-fragmentation" contributions arise in processes like involve convolutions with two functions D γ (z) (one per photon).
Isolation constraints must be imposed on the inclusive photon cross sections before the comparison with data. Isolation can be applied to the cross sections at each order of α s [18,19,20].
The magnitude of the fragmentation contribution is controlled by the isolation procedure chosen and can be strongly affected by tuning the quasi-experimental isolation model. An isolation condition in a typical measurement requires the hadronic activity to be minimal (e.g., comparable to the underlying event) in the immediate neighborhood of each candidate photon. Candidate photons may be rejected because of energy deposit nearby in the hadronic calorimeter, which introduces dependence on the calorimeter cell geometry, or because hadronic tracks are present near the photons. A theory calculation may approximate the experimental isolation by requiring the full energy of the hadronic remnants to be less than a threshold "isolation energy" E iso T in the cone ∆R = √ ∆η 2 + ∆ϕ 2 around each photon, with ∆η and ∆ϕ being the separations of the hadronic remnant(s) from the photon in the plane of pseudo-rapidity η and azimuthal angle ϕ. The two photons must also be separated in the η − ϕ plane by an amount exceeding the approximate angular size ∆R γγ of one calorimeter cell. The values of E iso T , ∆R, and ∆R γγ serve as crude characteristics of the actual measurement. The size of the fragmentation contributions depends tangibly on the assumed values of E iso T , ∆R, and ∆R γγ , as is shown below.
We find it sufficient in our work to use a simplified fragmentation model to represent the isolated cross section. We regularize the fragmentation region by imposing a combination of a sharp cutoff E iso T on the transverse energy E T of the final-state quark or gluon and smooth cone isolation [21]. We impose quasi-experimental isolation by rejecting an event if (a) the separation ∆r = (η − η γ ) 2 + (ϕ − ϕ γ ) 2 between the final-state parton and one of the photons is less than ∆R, and (b) E T of the parton is larger than E iso T . This condition excludes the singular fragmentation contributions in the finite-order qg cross section at ∆r < ∆R and The fragmentation contributions at ∆r < ∆R and E T < E iso T are suppressed by rejecting events in the ∆R cone that satisfy E T < χ(∆r), where χ(∆r) is a smooth function satisfying χ(0) = 0, χ(∆R) = E iso T . This "smooth cone isolation" [21] transforms the nonintegrable fragmentation singularity associated with D γ (z) into an integrable singularity of a magnitude dependent on the functional form of χ(∆r). Infrared safety of the cross sections is preserved as a result of smoothness of χ(∆r). The cross section for direct contributions is rendered finite by this prescription without the explicit introduction of fragmentation functions fications to the function χ(∆r) lead to only mild variations of our predicted Q T distribution In our calculation, we use the electroweak parameters [22] 1882 GeV, and m W = 80.419 GeV. We use two-loop expressions for the running electromagnetic and strong couplings α(µ) and α S (µ), as well as the NLO parton distribution function set CTEQ6M [23] and set 1 of the NLO photon fragmentation functions from Ref. [24]. Our choices of the renormalization and factorization scales are the same as in Ref. [2]; in particular, we set µ R = µ F = Q in the finite-order perturbative expressions.
In impact parameter (b) space, used in the CSS resummation procedure, we must integrate into the non-perturbative region of large b. Contributions from this region are known to be suppressed at high energies [25], but it is important nevertheless to evaluate the expected residual uncertainties. We use a model for the non-perturbative contributions ("revised b * model") based on the analysis of Drell-Yan pair and Z boson production in Ref. [17]. A nonperturbative Sudakov function for the factorization constant C 3 = 2e −γ E ≈ 1.123 is used here to describe the non-perturbative terms in the leading qq → γγ channel [17]. We neglect possible corrections to the non-perturbative contributions arising from the final-state soft radiation in the qg channel, as well as additional √ S dependence affecting Drell-Yan-like processes at x 10 −2 [26], as those exceed the accuracy of the present measurements at the Tevatron. The non-perturbative function in the gg → γγ channel is approximated by multiplying the non-perturbative function for the qq channel by the ratio C A /C F = 9/4 of the color factors C A = 3 and C F = 4/3 for the leading soft contributions in the gg and qq channels. Comparing our results based on the "revised b * model" with those obtained with the original b * approach, we find at most 10% differences in our predicted dσ/dQ T at the lowest values of Q T at the Tevatron collider energy, and smaller differences at larger values of Q T , all well within the experimental uncertainties. The differences are even smaller at the LHC energy [4].
Comparison with Tevatron Data. Our analysis provides a calculation of the triple-differential cross section dσ/dQdQ T d∆ϕ. Its relevance is especially pertinent for the transverse momentum Q T distribution in the region Q T ≤ Q, for fixed values of diphoton mass Q. It would be best to compare our multi-differential distribution with experiment, but the published collider data tend to be presented in the form of single-differential distributions in Q, Q T , and ∆ϕ, after integration over the other variables. We follow suit in order to make comparisons with the Tevatron collider data, but we comment on the features that can be explored if more differential studies are made. In accord with CDF, we impose the cuts |y γ | < 0.9 on the rapidity of each photon, and p γ T > p γ T min = 14 (13) GeV on the transverse momentum of the harder (softer) photon in each γγ pair. We choose E iso T = 1 GeV, ∆R = 0.4, and ∆R γγ = 0.3, unless stated otherwise. Fig. 1a. It exhibits a characteristic lower kinematic cutoff at Q ≈ 2 p γ 1 T min p γ 2 T min ≈ 27 GeV. Our calculation (RESBOS) agrees well with the data. In this figure we also show the perturbative QCD contributions evaluated at finite order, represented by the DIPHOX code [7]. Unless specified otherwise, the scales µ R = µ F = Q are used to obtain the DIPHOX results presented here. The overall agreement between the two calculations is anticipated, since both evaluate the inclusive rates at NLO accuracy. The differences are due to different isolation prescriptions, resummation of higherorder logarithms as well as NLO contributions to the gg channel in our calculation, and single-photon one-and two-fragmentation contributions in DIPHOX.

The invariant mass (Q) distribution is shown in
The transverse momentum (Q T ) distribution of diphotons is shown in Fig. 1b. The finiteorder calculation, represented here by DIPHOX, displays an unphysical logarithmic singularity as Q T → 0. In our work, the initial-state small-Q T singularities are resummed in the CSS formalism, resulting in a reasonable overall shape of the cross section at any Q T . The fragmentation contributions exhibit a double-logarithmic singularity when Q T approaches E iso T from below [7], as it is evident in the DIPHOX Q T distribution for E iso T = 4 GeV. No such singularity is present in our Q T distribution, which instead has a mild discontinuity at the point Q T = E iso T where we switch from the quasi-experimental to smooth-cone isolation.  In the shoulder region, the increase in E iso T to 4 GeV strongly enhances the DIPHOX cross section to the value shown in the CDF publication. The magnitude of the one-fragmentation cross section associated with D γ (z) is increased on average by 400% when E iso T is increased from 1 to 4 GeV.
Our calculations show that most of the shoulder events populate a limited volume of phase space characterized by ∆ϕ 1 rad, Q < 27 GeV, and Q T 25 GeV. The location of the shoulder in the Q T distribution is sensitive to the value of the cut on the minimum transverse momentum, p γ T , of the individual photons, moving to larger Q T if these cuts are raised. It has also been noted [27] that non-zero values of p γ 1 T and p γ 2 T disallow contributions with small Q T if the azimuthal angle separation between the two photons is small, ∆ϕ < π/2. The excess of the experimental rate over our prediction in the region ∆ϕ < 0.6 radian (cf. Fig. 2a) contributes the bulk of the excess seen in the shoulder in the Q T distribution in Fig. 1b. We note, in addition, that the excess at small ∆ϕ and large Q T is characterized by Q T Q.
From a theoretical point of view, when Q T > Q, as in the shoulder region, the calculation must be organized in a different way [28,29] in order to resum contributions arising from the fragmentation of partons into a pair of photons with small invariant mass. In addition, a small azimuthal separation ∆ϕ often implies that the photons are produced at polar angles θ * ≈ 0 or π in the Collins-Soper diphoton rest frame [30]. The matrix element for the Born scattering process qq → γγ diverges as |cos θ * | → 1. Large QCD corrections are known to exist when The theoretical ambiguities arise in a small part of phase space, where the cross section is also small. Our theoretical treatment is most reliable in the region in which Q T < Q. When the Q T < Q selection is made, the contributions from ∆ϕ < π/2 are efficiently suppressed, and dependence on tunable isolation parameters and factorization scales is reduced (cf. Fig. 2b). The fixed-order predictions agree well between our calculation and DIPHOX, while our resummmed cross section also provides an accurate description of the rate at small values of Q T . After the selection Q T < Q, we expect that the large Q T shoulder will disappear in the experimental Q T distribution.
An important prediction of the resummation formalism is a logarithmic dependence on the diphoton invariant mass Q. In Fig. 3a  Results for the LHC. To obtain predictions for pp collisions at √ S = 14 TeV, we employ the following cuts on the kinematics of the individual photons. For each photon, we require transverse momentum p γ T > 25 GeV and rapidity |y γ | < 2.5. We impose a somewhat looser isolation restriction than for the Tevatron study, requiring less than E iso T = 10 GeV of extra transverse energy inside a cone with ∆R = 0.7 around each photon. Higgs boson, e.g., 115 to 130 GeV, the diphoton background that we consider in this paper has Q T ∼ 27 GeV, to be compared with the expectation for the signal of ∼ 40 GeV [25].
The harder transverse momentum distribution for the signal arises because their is more soft gluon radiation in the dominant gluon-fusion Higgs boson production process [25]. Additional predictions for the LHC are presented in Ref. [4].
Summary. We present a new QCD calculation of the transverse momentum distribution of photon pairs produced at hadron colliders, including all-orders resummation of initial-state soft-gluon radiation valid at next-to-next-to-leading logarithmic accuracy. This calculation is most appropriate for values of γγ transverse momentum Q T not in excess of the γγ invariant mass Q. Resummation changes both the shape and normalization of the Q T distribution, with respect to a finite-order calculation, in the range of values of Q T where the cross section is largest. Comparison of our results with data from the Fermilab Tevatron shows good agreement, and we offer suggestions for a more differential analysis of the Tevatron data. We also include predictions for the Large Hadron Collider.
Our calculation accounts for the effects of soft gluon radiation on transverse momentum distributions through all orders of α s . The NLO calculation with inclusion of single-photon fragmentation [7] is another important approach to γγ production. However, theoretical uncertainties are present in the rate of fragmentation contributions associated with the kinematic approximations and tunable parameters in the quasi-experimental isolation condition.
For Q T > Q (∆ϕ < π/2), new types of higher-order contributions are expected to enhance the rate above our predictions. The interpretation of the region of small ∆ϕ remains ambiguous, as several distinct processes may contribute to the enhanced rate. This interesting region warrants further theoretical investigation. With the contributions from the Q T > Q region removed, our calculation describes the leading contributions in the qq + qg and gg diphoton production channels at NNLL accuracy.