1. Introduction
Direct photon production in hadronic collisions provides an excellent opportunity to test Standard Model (SM) predictions and explore potential new physics effects, due to its very clean experimental signature. Photons are also very relevant to study the internal structure of colliding particles [
1,
2] because they can shed light on the parton level kinematics, leading to new interesting paths to unveil the hidden symmetries of nature.
In this phenomenological context, one process of special relevance is the production of a pair of prompt photons (i.e., diphoton production), which was one of the key channels for the discovery of a new particle compatible with the SM-Higgs boson [
3,
4]. This indicates that direct photon production might also play an important role in the precision physics program for future colliders [
5,
6,
7,
8,
9,
10,
11,
12,
13].
Due to its relevance, the direct diphoton production has been extensively studied and its perturbative corrections were computed in several models. At the lowest order (LO), only the
partonic channel is available. The first quantitative correction appears at the next-to-leading order (NLO) in the strong coupling constant
[
14,
15,
16,
17], which includes the
channel and provides a noticeable enhancement due to the gluonic luminosity in the low-
x region. At the next-to-next-to-leading order (NNLO) in
(i.e.,
), the
channel becomes available. Its contribution is comparable to the Born cross-section, but the total NNLO correction is still dominated by the
channel. The NNLO QCD corrections were first calculated in Ref. [
18] and later in Ref. [
19]. The QCD resummed corrections have been recently obtained up to NNLL accuracy [
20], thus improving the phenomenological description by including the effects due to low-energy gluon radiation. Recently, articles on diphoton production at NNLO were published, studying in detail the isolation criteria [
21,
22,
23,
24,
25,
26,
27,
28], the weight of the fragmentation component [
21,
22,
24,
26], the reliability of the prediction under symmetric cuts on single final photons [
21], the scale sensitivity of the process and the hybrid isolation criteria [
29,
30], among other features of this important process. In addition, the diphoton background could be potentially interesting as an indirect measurement of the top-quark mass [
31,
32].
Besides the direct photon production from the hard subprocess, photons can be generated through the fragmentation mechanism, which involves non-perturbative transitions from QCD partons. The complete NLO single- and double-fragmentation contributions are implemented in
DIPHOX [
14]. The measured photon cross-sections use isolation prescriptions in order to reduce the large reducible background (associated to photons that are faked by jets or produced by hadron decays). Two of these prescriptions are the standard cone isolation and the smooth cone isolation [
33] prescriptions. While measured cross-sections rely on the standard cone criterion, theoretical calculations can be greatly simplified with the smooth cone prescription. It is worth noticing that both algorithms produce similar results [
22,
34] for those isolation parameters implemented by the experiment.
Other important effect to be considered in the context of photon production is the presence of electroweak (EW) corrections. Leaving aside the large gluon luminosity at the LHC, a simple power counting shows that the NLO QED corrections could compete with the NNLO QCD contributions because
. For this reason, the higher-order QED and EW effects deserve a serious treatment. Several studies about the EW corrections to gauge boson production [
35,
36,
37,
38,
39,
40] and diphoton production [
41,
42,
43,
44] showed small but still non-negligible contributions, which might play a crucial role in the context of the precision physics program. Moreover, we have explored the impact of pure QED and mixed QCD-QED corrections in the DGLAP equations [
45,
46,
47] and we found that they could provide non-negligible effects to the PDF evolution. In particular, the NLO QED corrections to the diphoton production process are sensitive to the photon PDF through the
channel, which might propagate the information related to this distribution to the final cross-section result.
From the point of view of new physics searches, the high-energy region of the diphoton spectrum could be used to impose constraints on many models. Since no new particles are predicted by the SM beyond the TeV scale, the inclusion of higher-order EW and QCD effects could only lead to an enhancement/decrease of the distributions without introducing any peak. Thus, a proper understanding of all the SM effects for this region might allow to impose tighter constraints to BSM models involving a continuum spectrum at high-energies (for instance, large extra-dimensions or Randall–Sundrum with composite fermions [
48]).
The purpose of this article is to describe interesting phenomenological features of the NLO QED corrections to diphoton production, which might be useful for deeper studies of this process including higher-order mixed QCD-EW corrections. We start in
Section 2 describing the theoretical framework used to perform the computation, centering the discussion in the different partonic channels contributing to the process. Special emphasis is put on the treatment of the electromagnetic coupling, because it leads to noticeable corrections to the theoretical predictions. Then, in
Section 2.1, we explain the experimental cuts applied, focusing on the photon isolation and reconstruction algorithms. In
Section 3, we present our predictions for LHC, highlighting the effects due to the ordering algorithms used to deal with the radiated photons. Besides that, we briefly comment about the implementation of a clustering (i.e., merging) algorithm for reconstructing the photons and the dependence of the results on the use of different PDF sets, and the photon PDF in particular. Finally, in
Section 4 we present our conclusions and a brief discussion about the importance of higher-order QED corrections in future experiments.
2. Computational Details
In our calculations we focus on the differential cross-section for the production of a prompt-diphoton system in hadron-hadron collisions. By virtue of the factorization theorem, the differential NLO QED cross-section for the process
is given by
where
denotes the PDF for partons of flavor
to be found inside the hadron
h,
are the longitudinal momentum fractions and
is the factorization scale. Additionally, we introduced the supraindices
to denote the perturbative expansion in QCD-QED; explicitly,
corresponds to the
correction to the Born process.
is the Born cross-section,
contains the QED one-loop finite correction to the LO and
are the real corrections to the Born subprocess due to QED extra-radiation, where the symbol
denotes the regularized partonic cross-section, i.e., without the infrared (IR) and ultraviolet (UV) singularities. In Equation (
1),
is the measurement function that defines the IR-safe observable at parton level for the Born (
) and real-radiation (
) kinematics. At Born-level, the only available subprocess is
, whilst the real-radiation contributions at NLO are associated to the subprocesses
therefore the sum performed over the
c parameter in Equation (
1) refers to the precedent two channels. Notice that, at
, the
channel starts to contribute and the cross-section becomes sensitive to the photon PDF.
In order to explicitly implement this computation numerically, we must properly define the regularized cross-sections, which implies canceling the UV divergences through renormalization and the IR singularities with a proper real-virtual combination. To tackle the IR regularization, we will rely on the
-subtraction formalism [
49,
50], accordingly modified to deal with QED corrections in a fully consistent way. Within this formalism, the cross-section associated to the production of any neutral final state
F with transverse momentum
is separated into a regular and a singular contribution in the limit
. Whilst the regular part is finite and process-dependent, the singular contribution possesses a universal structure, given by [
50]
where
is the Sudakov form factor,
is the hard-collinear factor and
is related to the Born level cross-section. The Sudakov factor embodies all the information related with the emission of low-energy photons from the parton
c entering in the hard-process
. It is obtained from the exponentiation of photon emissions, and is given by
which depends on the flavor of the parent parton
c and the resummation coefficients
and
. In Equation (
4),
is evaluated at scales typically present in the hard process,
b is the impact parameter and
(where
is the Euler–Mascheroni constant). The resummation coefficients
and
can be computed in perturbation theory,
It is worth noticing that these ideas can be applied for dealing with a mixed QCD-QED expansion, which justifies the inclusion of a double-superscript to keep track of the perturbative order in each theory [
40]. In the case
, the first terms of the expansions are
with
the electromagnetic (EM) charge of the quark
q;
for
. For the purpose of computing NLO QED corrections, we only need
: the remaining higher-order coefficients are available in Refs. [
40,
51].
On the other hand, the symbolic factor
in Equation (
3) for the
annihilation channel is given by
where the functions
are universal and the renormalised hard function
contains process-dependent contributions related to the virtual amplitudes. In both cases, they can be expanded in perturbative series, i.e.,
where we are using the double-superscript notation mentioned before. The only non-vanishing
coefficients are
with
the number of colors. It is worth emphasizing that the structure of Equation (
7) is different when dealing with vector particles (i.e.,
or
) due to the presence of spin correlations. More details about the mixed QCD-QED
-subtraction/resummation formalism can be found in Refs. [
40,
51].
For the particular case of the diphoton production, we can show that the differential cross-section can be written as
where the finite process-dependent pieces coming from the virtual amplitudes (and the Born cross-section) are contained in the hard coefficient function
. In this equation, the symbol ⊗ is understood as a convolution of momentum fractions and sum over flavour indices of the partons [
49,
50,
51].
includes the QED real-radiation contributions and
denotes the subtraction counter-terms, which are obtained through a perturbative expansion of Equation (
3) and collecting the corresponding
terms.
While in the QCD formalism the undetected final state
X can contain any number of QCD partons (i.e., quarks and/or gluons), in the QED case
X contains any number of photons. The two terms in the r.h.s. of Equation (
11) are finite, which implies that they can be numerically computed. However, both
and
diverge in the limit where the transverse-momentum of the system
vanishes (i.e.,
). The hard coefficient
is extracted in a process-independent way from the
process-dependent corrections to the scattering amplitudes. In this case, the only contribution to this hard coefficient comes from the
channel. Thus, the regularized hard coefficient is given by
with
the ratio of the partonic Mandelstam variables. This expression is obtained from the hard factor provided in Ref. [
50] after applying the Abelianization algorithm described in Refs. [
45,
46].
Finally, it is important to mention that we are using
massless quarks as active flavors and we include all the SM charged-fermions inside any closed fermionic loops. To be more precise, the QED beta function reads
where
q (
l) denotes all the possible flavors of quarks (leptons) in the SM with their corresponding EM charges,
(
).
2.1. Event Selection, Isolation Prescription and Setup of the Calculation
The present NLO QED computation is implemented by modifying the numerical program
2NNLO [
18], which originally includes up to NNLO QCD corrections to prompt-diphoton production. At this perturbative order in QED, the Born + virtual component only contains two photons, whilst the real-emission part introduces contributions with up to three observable photons in the final state (i.e., through the subprocess
). Since we are interested in computing
, we order the transverse-momenta of the triphoton system, thus reproducing the selection criteria applied by the experiment in direct diphoton measurements. It is interesting to emphasize that the momentum ordering of the photons introduces a dynamical cut in the available real-emission phase space, as we will explain in
Section 3.
As it was stated in the Introduction, we apply the smooth cone isolation prescription [
33]. This criterion differs from the one applied by the experiment (standard cone) but, for commonly used isolation parameters by the LHC, both criteria lead to similar results [
22,
34]. Explicitly, given a final state photon, we build a cone around it, whose radius in the
plane is
where we are using the well-known notation
and
for the pseudo-rapidity and azimuthal angle, respectively (see Ref. [
21] for more details). Then, we require that the partonic energy deposited inside the cone fulfils [
33]
with
the radius of the fixed outer cone used to initialize the isolation algorithm,
the maximum allowed deposited transverse energy and
n an arbitrary isolation parameter. In this work, we choose
.
We show results for two different center-of-mass energies using the guidelines applied by the ATLAS and CMS experiments. Explicitly, for
, we choose those events where
and
, restricting both photon rapidities to satisfy
and
[
52]. The isolation parameter was fixed to
. On the other hand, for
we required
and
whilst the rapidity range was slightly modified to
and
, and we imposed
in the isolation criterion [
53]. For all these setups, the separation between the hardest photons in the
,
plane must to be greater than
.
In order to reproduce the experimental measurement procedure, we consider the implementation of a
photon-clustering algorithm. From the experimental point of view, it is not possible to distinguish events with quasi-collinear photons due to the finite granularity of the detectors. Thus, if they are produced within a cone of radius
, they are identified as an unique particle. For instance, for the ATLAS detector [
54], this value ranges from 0.05 to 0.075, although a conservative estimate of the minimal detectable separation could be set to
. From the theoretical side, it is possible to implement such a procedure by working with the parton-level kinematics. In particular, notice that this algorithm will be activated only for the
channel, and that only one pair of quasi-collinear photons is allowed due to the other kinematical cuts (i.e., angular separation between
resolved particles). Schematically, we consider two variants of the clustering/merging procedure, which are defined in the following way:
Compute the distance among the final-state photons, where at NLO.
If for all pairs, all the photons can be resolved and the process is described with a full kinematics as normal.
If the photon
k is isolated but
, then the photons
i and
j are detected inside the same bin. Hence, we define the merged momenta as
which corresponds to applying the
and
E-schemes for QCD jets, respectively, [
55,
56]. In both cases, the definition of the original algorithm was slightly modified to deal with photons instead of jets.
Order the final state particles according to their new transverse momentum, and impose the corresponding cuts to select the events.
Since the matrix element is finite when two photons become collinear, there is some freedom in the implementation of the algorithm. In the -scheme, we have and the energy measured in the detector agrees with the total sum of the diphoton system. Moreover, the direction of motion of the merged particle corresponds to the sum of momenta, with a proper re-scaling. In this way, the reconstructed photon is identified as an on-shell particle with a well defined three-momentum. However, this approach does neither fulfill momentum conservation nor Lorentz invariance. On the other hand, the E-scheme is compatible with the last two properties, but the merged momenta is not massless (i.e., the on-shell condition is not fulfilled).
Finally, we comment on the treatment of the electromagnetic coupling and its running. In the first place, it is a well-known fact that the running effects arise from the renormalization of the interaction vertex. If the Born contribution only contains on-shell final-state photons, when collecting all the possible unrenormalized one-loop contributions we realize that there is a partial cancellation of UV singularities between self-energies and vertex corrections. This cancellation avoids the presence of UV-logarithmic terms within the renormalized one-loop amplitudes; including a running coupling would add corrections proportional to (with the renormalization scale) that might alter the high-energy behavior of the physical predictions.
On the other hand, when the LO process involves initial-state photons, the NLO QED corrections should include the one-loop running for
evaluated at the factorization scale
[
57]. Otherwise, some terms proportional to
would remain un-canceled and enhance artificially the scale-dependence of the hadronic cross-section. As pointed out in Ref. [
57], this behaviour is due the presence of the photon PDF, which introduces a dependence on
, even if photons entering the partonic processes are kinematically on-shell. This dependence is controlled by the splitting kernel
, that at the lowest order is given by [
46]
Notice that this expression involves the QED beta function at one-loop, since this splitting function is directly extracted from the one-loop photon self-energy. In consequence, there is an interplay between the renormalization and factorization corrections that require the proper inclusion of the running effects to achieve a consistent result.
As a consequence of the precedent observations applied to the particular case of diphoton production, we claim that we should fix
in order to provide a conservative estimate of the NLO QED corrections. However, it is worth emphasizing that we can introduce a
partial running for
to estimate effects due to missing higher-order terms. Explicitly, since the LO is
, we propose to modify the NLO behavior of the coupling according to
thus including the running effects without overestimating the correction by introducing additional logarithms. So, we consider the frozen coupling scenario as the default one, and we also take into account the full hadronic evolution given by Refs. [
58,
59,
60] with the purpose of studying the propagation of uncertainties due to the choice of
.
3. Results and Discussion
In this section, we present the numerical results for diphoton production comparing the strength of the NLO QED corrections with the well-known NLO QCD ones. In the following, we denote diphoton system as the system composed of the two hardest photons. We show results for the LHC at two different energies, TeV and TeV, making use of two different sets of cuts for the transverse momentum of the diphoton pair. For TeV we use nearly symmetrical cuts, whilst for TeV we apply asymmetrical ones as detailed in the previous section.
Concerning the PDF sets, we perform the QED computations with
LUXqed [
61,
62] unless otherwise specified. Since
LUXqed uses
PDF4LHC [
63] to describe the QCD parton distributions, we consistently use it when computing the pure QCD contributions. Another set containing the photon PDF is
NNPDF3.0QED [
64,
65]. We provide also results computed with the later set in order to compare the effects of different implementation of photon PDFs.
The default renormalization (
) and factorization (
) scales are set to the value of the transverse invariant mass of the diphoton system, i.e.,
As explained in
Section 2, the default value for the QED coupling is fixed to
and we consider a
partial running to estimate the effects due to missing
corrections. On the other hand, we also study the uncertainties originated by missing QCD-QED higher-order contributions. Starting from the formal additive result for the NLO QED corrections,
we define an estimate of the mixed QCD-QED corrections with the multiplicative approach, i.e.,
The multiplicative ansatz is described in Reference [
66] and is motivated by the fact that the IR-divergent structure of the mixed QCD-QED terms factorizes as in Equation (
23). However, this approach fails to describe the finite pieces of the total contribution because it does not take into account non-factorizable QCD-QED terms present in the full computation.
After describing the details of the implementation, we start the analysis of the results. In first place, we consider the invariant mass distribution of the two hardest photons for
and
in
Figure 1 and
Figure 2, respectively. We compare the impact of the NLO QED corrections (
) with the NLO QCD (
) and the corresponding Born-level contribution.
The two different channels that are part of the total NLO QED correction, i.e.,
and
, are separately shown in the plots. The NLO QED corrections are moderate to tiny in the whole invariant-mass range. To quantitatively describe them, we define the ratio
that reaches up to
for
TeV.
is dominated by the
channel in almost the entire
range. The only exception is the low-mass region, which is forbidden at the LO due to kinematical constraints. The kinematics of the LO subprocess is determined by the two final-state photons. Since the only kinematical configuration allowed by the Born contribution is such that the two photons are
back-to-back,
(i.e.,
) and therefore, only the value of
is effective as cut at LO,
Specifically,
GeV for photon-pair production at
TeV (and its associated LHC cuts) and
GeV for diphoton production at
TeV. Perturbative calculations in regions around unphysical fixed-order thresholds are known [
67] to be generally affected by perturbative instabilities at higher orders. The invariant mass distribution is very steep in the kinematical region around the LO threshold and even the effect of little instabilities is amplified by the large slope of
. Therefore, in order to obtain reliable predictions around the LO threshold in
it is enough to consider a bin size of roughly 2 GeV since we are dealing with integrable singularities [
21,
67].
The suppression of the
channel in the low-mass region is an artifact of the ordering in the transverse momentum of the final state photons. This ordering procedure reproduces the selection algorithm implemented by the LHC. It is worth noticing that we can describe quantitatively the interplay among the kinematical cuts, the ordering and the position of the threshold in the
channel. Let us consider a system with three massless on-shell particles in the center-of-mass frame. After ordering, we have
with
and
. In fact, the transverse momentum conservation implies
, and the ordering condition leads to
where
is the angular separation in the transverse plane. Since
by definition, we conclude that
. So, we explicitly find a constraint in the angular separation of the transverse momenta of the hardest particles by imposing an ordering condition. Of course, such a restriction is not present when selecting randomly two photons and using their momenta to classify the event, as we did for the QCD corrections to
. Moreover, from Equation (
27) we can appraise that
, which constitutes another constraint.
On the other hand, it is crucial to notice that imposing both the transverse momenta ordering and the
cuts detailed in
Section 2, we end up with a cut in
. To find the minimum allowed value for
distributions measured at ATLAS with
, we choose
and
. Then, inserting these values in Equation (
27), we find
that increases the minimal angular separation between the transverse momenta of the hardest photons. Besides this, the invariant mass of the hardest subsystem is given by
where
is the angle between
and
. This angle is measured in the plane containing both three-vectors, which is not equivalent to the transverse plane to the colliding axis: thus, in general,
. However, we impose
in order to find the minimal invariant mass configuration. This choice is equivalent to settle the production of the triphoton system in the transverse plane, which leads to
,
and
Below these limit, the
channel is not compatible with the experimental cuts. Analogously, when dealing with ATLAS measurements at
, we find
These results explain the behavior of the distributions shown in
Figure 1 and
Figure 2, in particular, the steeply suppression of the cross-section in the region
for
and
for
.
In
Figure 3, we present the results for the transverse momentum distribution of the two hardest photons for
TeV. We compare the NLO QCD and NLO QED corrections, as well as the individual channels contributing to the last one. Notice that in the whole
range (with the exception of
GeV), these are
effective LO contributions. As we appraised from
Figure 1 and
Figure 2, the
channel for the invariant mass distribution was sub-dominant (and almost negligible) in all kinematical regions except the low-mass region.
This can be regarded as a hint of what we are obtaining after the shoulder ( GeV) in the distribution for the QED corrections: dominates the NLO QED cross section.
We present also two different ratios between the NLO QCD corrections and possible estimates of the higher-order QED corrections in
Figure 4. Besides
, we define
where
denotes the NLO QED correction including running effects (as described in
Section 2.1). Both
and
are similar for
. It is a well-known fact that QCD transverse momentum resummation is requested to recover the reliability of the calculation and improve the description of data in the low
region [
20]. The same consideration applies here for the NLO QED corrections and the resummation QED program, which will be presented in a forthcoming paper.
In order to explore additional measurements that could be more sensitive to QED corrections, we define a jet-veto algorithm to partially suppress the QCD components. In the first place, we consider only jets produced by QCD partons since we assume that photons can be unambiguously identified. Of course, from the experimental point of view, this is not completely true: highly energetic photons might decay into hadrons and they could be reconstructed as a jet. However, we claim that these effects are related to higher-order corrections beyond NLO QED, and that the jet identification is well-defined within the theoretical computation (at least at this perturbative order). So, after generating the event, we request the jet to be generated by a quark or gluon (but not by a photon) and to be observable within the detector. Then, we reject all the events with and . It is worth emphasizing that this definition is IR safe at this perturbative order, since all the IR-singular configurations automatically pass the cuts. (This jet-veto algorithm properly works for computing NLO corrections, since only one extra-parton is present in the final state. At NNLO and beyond, some subtleties could arise and the algorithm should be carefully redefined).
After this discussion, we consider the diphoton production for
TeV with an additional jet-veto in
Figure 5. In the left panel, we show the effects on the different contributions of the total cross section. The NLO QCD correction is noticeably reduced beyond the veto threshold (i.e.,
), as well as the
channel contribution to the NLO QED correction. We can appraise a discontinuity in the distribution at
because of the modification of the event selection criteria imposed by the jet-veto. Of course, this is something artificial and not related to any underlying physical phenomena. Furthermore, as expected, since the
corrections to the
channel are associated to the process
, this process is not affected by the jet-veto and is not suppressed. In order to quantify these effects, we extend the definition of the ratios given in Equations (
24) and (
32) introducing
where the additional superscript denotes that we are taking into account the jet-veto. The results are shown in the right panel of
Figure 5. While the low
region is similar to the one without imposing the jet-veto (see
Figure 3), at large values of transverse momentum (in particular for
GeV) the
channel dominates the total cross section, and the ratios reach
.
As next, we present the transverse momentum distributions of the hardest (
) and second hardest (
) photons in
Figure 6. The
distribution is strongly affected neither by the NLO QED corrections nor by the estimated mixed QCD-QED contributions. Even if a jet-veto is applied, the NLO QED corrections for this observable are still negligible, because the LO and NLO QCD terms are much more relevant.
On the contrary, we notice strong effects due to the jet-veto in the
distribution: an important suppression of the total NLO QCD corrections and the
channel are shown in
Figure 6 (right panel). Since the NLO QED cross section is dominated by the
channel, which is not affected by the jet-veto, an enhancement of the QED-to-QCD ratios takes place. The estimate of the mixed QCD-QED corrections is similar to the one obtained when considering the running of
. The exception to this behavior is found in the low transverse momentum region (i.e.,
GeV), since it is populated with real-radiation events appearing for the first time in the NLO contributions. It is worth emphasizing that the higher-order QED terms introduce corrections of
of the total NLO QCD contributions for
GeV.
In
Figure 7, we present our results for the transverse momentum distribution of the two hardest photons but using the distributions of the set
NNPDF3.0QED. We compare the NLO QCD cross section with the NLO QED contribution, considering also in particular its two channels:
and
. The main differences with the results using the
LUXqed PDFs, detailed in
Figure 3, are due to the
channel. Even if there is a disagreement among them, due to an enhancement of the
channel in the middle and high-energy region, we should consider the predictions obtained with
NNPDF3.0QED as a qualitative estimate in that region.
We would like to emphasize here that the purpose of the previous comparison is to highlight that different methodologies on the extraction of PDFs could have a noticeable impact of the phenomenological analysis. In particular, a purely data-driven extraction of PDFs suffers from the lack of good-quality data-points in some regions of the parameter space, which directly impact in the error propagation. The methodology applied for extracting
NNPDF3.0QED is more sensible to the lack of experimental points for very high energies, which leads to huge errors in the
region. The approach followed by
LUXqed uses all-order theoretical predictions to constrain the photon PDF with data-points in the low-energy region, thus reducing the fitting errors. It is worth noticing that the updated versions of
NNPDF combine the methodology developed by
LUXqed in order to improve the predictions for the photon PDF [
68,
69,
70], leading to an impressive increase on the precision achieved.
Finally, we comment about two other aspects of this computation. In the first place, we explore the effects due to the merging algorithm for photons, as described in
Section 2.1. For
, both the
and
E-schemes present significant deviations neither between these nor compared with the distributions obtained without any clustering algorithm. The precedent consideration is valid for the total cross section, as well as for all the differential distributions presented in this work. Thus, we conclude that the implementation of a clustering algorithms with the current experimental conditions has a negligible impact in the measurements, which is compatible with the experimental and theoretical uncertainties.
4. Conclusions
A discussion about the computation of NLO QED corrections to diphoton production was presented, putting special emphasis in the phenomenological aspects for hadron colliders. Besides the size of these contributions and the (partial) estimate of the mixed QCD-QED terms, we emphasize that keeping under control all the possible uncertainties in this process is crucial since it is the main background for many new physics searches. Taking under control the QED corrections at the LHC allows to potentially decide about the origin of future discrepancies between data and theory. Moreover, the importance of high-precision predictions becomes completely indispensable when considering the next generation of high-energy and high-luminosity experiments [
6,
7,
8,
9,
10,
11,
12,
13].
In this article, NLO QED corrections were obtained through the application of the QED version of the -subtraction method, after a proper Abelianization of the NLO QCD implementation available in the program 2NNLO. It is important to notice that we carefully studied the cancellation of IR singularities in the limit, in order to guarantee the theoretical consistency of the approach.
In general, we found small corrections for the distribution, whilst the ratio reaches for the spectrum. This is because only the real-emission terms contribute to the cross-section for : a simple power counting shows that the QED-to-QCD ratio behaves like . Besides that, we studied the effects due to the treatment of the EM running coupling, which could also introduce percent-level effects in all the distributions considered.
An interesting observation is that the
channel is not always negligible, specially in some kinematical regions for distributions involving the transverse momentum. For instance, it dominates the NLO QED corrections to the
distribution for moderate values of transverse momentum, i.e.,
GeV, as we showed in
Figure 3. From this results, we conclude that it is always recommended to include the photon-initiated processes when computing differential distributions.
Besides that, using the approach described in Ref. [
66], we provided an estimate of the mixed QCD-QED corrections. Other interesting effects were found when considering the dependence on
(the transverse momentum of the second hardest photon), exhibiting
corrections relative to the NLO QCD contributions. These effects are enhanced when including a jet-veto to suppress the QCD component, although the
distribution was slightly modified.
Finally, we would like to highlight that QED corrections cannot be neglected in the context of precision high-energy physics. Moreover, such computations must be done within a fully consistent framework, which carefully includes all the ingredients without introducing additional sources of uncertainties.