Hadroproduction of t anti-t pair in association with an isolated photon at NLO accuracy matched with parton shower
PPreprint typeset in JHEP style - PAPER VERSION
Hadroproduction of t anti-t pair in association with anisolated photon at NLO accuracy matched withparton shower
Adam Kardos
MTA-DE Particle Physics Research Group,University of Debrecen, H-4010 Debrecen P.O.Box 105, HungaryE-mail: [email protected]
Zolt´an Tr´ocs´anyi
Institute of Physics and MTA-DE Particle Physics Research Group,University of Debrecen, H-4010 Debrecen P.O.Box 105, HungaryE-mail: [email protected]
Abstract:
We simulate the hadroproduction of a t ¯t-pair in association with a hard photonat LHC using the
PowHel package. These events are almost fully inclusive with respectto the photon, allowing for any physically relevant isolation of the photon. We use thegenerated events, stored according to the Les-Houches event format, to make predictionsfor differential distributions formally at the next-to-leading order (NLO) accuracy andwe compare these to existing predictions accurate at NLO using the smooth isolationprescription of Frixione. Our fixed-order predictions include the direct-photon contributiononly. We also make predictions for distributions after full parton shower and hadronizationusing the standard experimental cone-isolation of the photon.
Keywords:
QCD, Top physics, Photon production, Hadronic Colliders, LHC. a r X i v : . [ h e p - ph ] M a y ontents
1. Introduction 12. Details of the implementation 33. NLO-LHE comparison 54. Photon isolation as a generation cut 75. Independence of the generation isolation 96. Estimation of the non-pertrubative contribution 117. Effect of the parton shower 128. Predictions 139. Conclusions 17
1. Introduction
Isolated hard photons are important experimental tools for a variety of processes at theLHC. Most notably, one of the cleanest channels to identify the Standard Model (SM) Higgsparticle is its decay into a pair of hard photons. Although this channel has a small (about0.2 %) branching ratio as compared to the hadronic and leptonic channels, the spectacularresolution of the electromagnetic calorimeters of the ATLAS and CMS detectors and therelatively low background made this as one of the prime discovery channels [1, 2].From the theoretical point of view isolated hard photons are rather cumbersome ob-jects. Unlike leptons, the photons couple directly to quarks. If the quark that emits thephoton is a light quark, treated massless in perturbative QCD, then the emission is en-hanced at small angles and in fact, becomes singular for strictly collinear emission. Theusual experimental definition of an isolated photon allows for small hadronic activity eveninside the isolation cone. Due to the divergence of the collinear emission, this isolation can-not be implemented directly in a perturbative computation at leading-order (LO) accuracybecause even small hadronic activity inside the cone leads to infinite results.Of course, one can approximate the experimental definition with complete isolationof the photon from the coloured particles inside a fixed cone and obtain a perturbativeprediction at LO. The problem however, comes back with a different face if we want todefine the isolated photon in a computation at the next-to-leading order (NLO) accuracy.– 1 –t NLO there are two kinds of radiative corrections: (i) the virtual one with the samefinal state as the Born contribution, but including a loop and (ii) the real one that involvesthe emission of a real parton in the final state. These two contributions are separatelydivergent, but their sum is finite for infrared (IR) safe observables according to the KLNtheorem [3, 4]. The IR-safe observables are represented by a jet function J m , where m isthe number of partons in the final state: for an n -jet measure m = n at LO and for thevirtual corrections, while m = n + 1 in the real correction.There exist general methods (see e.g. ref. [5]) to combine the real and virtual correctionsfor infrared (IR) safe observables J m , for which J n +1 tends to J n smoothly in kinematicallydegenerate regions of the phase space, namely when two final-state partons become collinearor a final-state gluon becomes soft. The problem with the isolated-photon cross section inperturbative QCD is that the cone-photon isolation is not IR safe beyond LO. The extragluon in the real radiation may be radiated within the isolation cone in which case theevent will be cut even if the gluon energy tends to zero.There are ways to make predictions for photon production in perturbation theory, butall have drawbacks. In a pioneering work [6] the measurement of the inclusive photoncross section was advocated, but that is not very useful from the experimental point ofview. In ref. [7] an isolation procedure was proposed that is similar in spirit to the caseof inclusive cross section, yet provides a smooth isolation prescription that is IR safe atall orders in perturbation theory. However, the implementation of the smooth prescriptionexperimentally is very cumbersome as it requires very fine granularity of the detector, soit has never become popular among experimenters.There is a precise way of defining the isolated photon theoretically, but that requiresthe inclusion of the photon fragmentation component as well (see e.g. [8]). The drawbackof this approach is the need for non-perturbative input and the extra computational effortfor a contribution that is mostly discarded when the experimental isolation is used (conewith small hadronic activity inside that is described by the fragmentation contribution).Thus one would be tempted to neglect the fragmentation contribution, which is however,uncontrolled from the theoretical point of view and thus is not a viable option in a fixed-order computation.In the last decade new approaches were proposed to make predictions that are formallyaccurate to NLO but include the advantage of event simulations of the shower Monte Carlo(SMC) programs [9, 10, 11]. By now many processes have been included in the genericframeworks of these NLO+PS approaches, the aMCatNLO [12] and the POWHEG-BOX [13]codes. In a series of papers we combined the
POWHEG-BOX with the
HELAC-NLO package [14]into
PowHel to make predictions for the hadroproduction of a t ¯t-pair in association with ahard boson (scalar [15], pseudoscalar [16], vector [17] or jet [18]). The only missing bosonof the SM in this list is the hard photon. In view of the above, the reason is clear: thephoton has to be isolated, which makes this computation more involved than for the othercases.In this paper we use the
PowHel framework to simulate events containing direct photonsonly, that is we neglect the fragmentation contribution. We generate the events with looseisolation cut, resulting in an almost inclusive event sample. We argue that with sufficiently– 2 –oose generation isolation the fragmentation contribution should be indeed small. Theoutput of the
POWHEG-BOX consists of pre-showered events stored according to the LesHouches accord (LHEs) [19]. The LHEs when fed into a SMC, result in showered eventson which the usual experimental cone isolation can be applied. We discuss the validity ofthis approach on the example of
W γ hadroproduction for which predictions including amodelling of fragmentation as well as experimental results exist. Using events generatedwith loose isolation cuts, we make predictions to t ¯t γ hadroproduction, but the approachis general and can be used to make predictions for any other process that involves isolatedhard photons in the final state at NLO accuracy matched with PS.
2. Details of the implementation
PowHel is a computational framework composed of the
POWHEG-BOX [13] and the
HELAC-NLO [14] packages to provide predictions at the hadron level with NLO QCD accuracy in thehard process. The essential ingredients needed for a particular process are the matrixelements for the Born, virtual and real-emission contributions, spin- and colour-correlatedmatrix elements and a suitable phase space for the Born process.The matrix elements are provided by
HELAC-NLO while the Born phase space is con-structed by us using the relatively simple kinematics at the Born level. The Born phasespace is generated with the help of one kinematic invariant and three angles. An overallazimuth is kept fixed and randomly reinstated at the end of the calculation as a commonpractice in
POWHEG-BOX . Matrix elements are generated for the following subprocesses:q ¯q → γ t ¯t, g g → γ t ¯t (tree-level for the Born process and at one-loop for the virtual) andq ¯q → γ t ¯t g , g g → γ t ¯t g for the real emission (q ∈ { u , d , c , s , b } ). The ordering amongparticles follows the convention of POWHEG-BOX : non-QCD particles, massive quarks, mass-less partons. Matrix elements for all other subprocesses are obtained from these by meansof crossing.All matrix elements, including the crossed ones, are compared to the stand-alone ver-sion of
HELAC-NLO in several, randomly chosen phase-space points. The internal consistencybetween the Born, spin-, colour-correlated and real-emission matrix elements is checked bycomparing the limit of the real-emission part and the corresponding counter terms in allkinematically degenerate regions of the phase space.In order to check the whole implementation we compare differential distributions tothose in ref. [20] using the LHC setup in the published paper: the calculation was performedfor LHC at centre-of-mass energy √ s = 14 TeV with a CTEQ6L1 and
CTEQ6.6M
PDF at LOand NLO accuracy and a one- and two-loop running α s , respectively. The mass of thet-quark was m t = 172 GeV, the fine-structure constant, was set to α EM = 1 / m t . In the analysis aphoton was required to be hard, p ⊥ ,γ >
20 GeV and the smooth isolation of Frixione [7]was employed with isolation parameters δ = 0 . (cid:15) γ = n = 1. The cross sectionsobtained with PowHel are enlisted on Tab. 1. We found complete agreement with thepredictions of [20] both for the cross sections and for the available distributions as well.Two out of these are depicted in Fig. 1. – 3 – σ PHLO [pb] σ PHNLO [pb]2 m t . ± .
004 2 . ± . m t . ± .
005 2 . ± . m t / . ± .
006 3 . ± . Table 1:
Cross sections obtained with
PowHel at LO and NLO accuracy using the setup and cutsof [20]. The renormalization and factorization scales are made equal to µ . − − − d σ / d p ⊥ , γ [ pb / G e V ] PowHel -NLOMSS NLO14 TeV, µ = m t p ⊥ ,γ >
20 GeV , δ = 0 . M SS / PH p ⊥ ,γ [GeV] − − − d σ / d p ⊥ , ¯ t [ pb / G e V ] PowHel -NLOMSS NLO14 TeV, µ = m t p ⊥ ,γ >
20 GeV , δ = 0 . M SS / PH p ⊥ , ¯t [GeV] Figure 1:
Comparison between
PowHel and [20] at the central scale with NLO accuracy for thedifferential cross section as a function of the transverse momentum of the photon and anti-t quark.Lower panels depict the ratio of predictions in [20] (MSS) to ours. The uncertainties appearing onthe lower panels only take into account the statistical uncertainty of our calculation.
Having checked the implementation of the NLO computation, we generated eventswith the
POWHEG-BOX . The final state in the Born contribution, t ¯t γ , is composed of twomassive and one massless particles. The cross section when the photon is emitted from amassless (anti)quark can become singular. This can happen when the photon is emitted byone massless (anti)quark from the initial state, or from a final state one in the real-emissioncontribution. These configurations have to be avoided such that the physical cross sectionsfor isolated photon production do not depend on the actual implementation.Let us first focus only on the singular radiation present at the Born level. In this casethere are two simple solutions to avoid infinite contributions to the cross section. Thefirst is a generation cut [18], which if applied on the transverse momentum of the photon,can avoid the singularity. This cut has to be sufficiently small so that when physical cutsare applied, the prediction becomes independent of this generation cut. Although thismethod offers an easy way to avoid the singularity, yet we end up generating events mostlywith photons having small transverse momentum. Hence the majority of events will begenerated in a region of phase space which has no physical importance and the efficiencyof the event generation is small.The other solution is the inclusion of a suppression factor [21] which can be usedto enhance event generation in certain regions of the phase space. The distributions are– 4 –lways independent of the suppression used as events are generated according to a crosssection obtained with a suppression factor, but the weight of each event is multiplied bythe inverse of the suppression factor (for details see [22]). In our calculation the analyticalform of suppression was chosen to be F supp = 11 + (cid:18) p ⊥ , supp p ⊥ ,γ (cid:19) i , (2.1)and we found i = 2 a suitable choice and p ⊥ , supp = 100 GeV was set throughout the wholecalculation. It is not necessary, yet we included also the generation cut on the transversemomentum of the photon, by requiring the transverse momentum of the photon in theunderlying phase space to be larger than 15 GeV. We checked that this cut does not affectour predictions with physical cuts larger than 15 GeV. Our strategy to handle singularitiescoming from collinear photon-emission from final state massless (anti)quarks will be coveredin the next section.In order to speed up the event generation the real emission part can be decomposedinto a singular and finite contribution such that the former contains all the kinematicallydegenerate regions of phase space, while the latter is finite over the whole phase space.When this decomposition is implemented the POWHEG Sudakov factor is evaluated withonly the singular contribution. For the decomposition we used the original suggestion of[13] which became standard in all calculations done with the help of POWHEG-BOX . Beside ofthis decomposition and the generation isolation nothing is taken into account which couldalter the shape of the POWHEG Sudakov, that is the matching systematics. In particular,we have not used the hfact option which is only used in [23] and in all the other cases theseparation of real emission contribution, mentioned above, was considered only.
3. NLO-LHE comparison
In this and all the upcoming sections predictions are made for proton-proton collisions at √ s = 8 TeV with the following parameters: CT10nlo
PDF using
LHAPDF [24] with a 2-looprunning α s considering 5 massless quark-flavours, m t = 172 . α EM = 1 / µ = 12 ˆ H ⊥ = 12 (cid:0) m ⊥ , t + m ⊥ , ¯t + p ⊥ ,γ (cid:1) , (3.1)where the hat reminds us that underlying-Born kinematics was used to evaluate the sum.For the NLO-LHE comparison the following set of cuts was employed: • The photon had to be hard enough, p ⊥ ,γ >
30 GeV. • The photon was constrained into the central region, | y γ | < . • To avoid the quark-photon singularity a Frixione-isolation was used with δ = 0 . (cid:15) γ = n = 1. – 5 – .10.20.30.40.50.60.70.80.91.0 σ [ pb ] PowHel -NLOLO8 TeV, µ = ˆ H ⊥ / δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < . K - f a c t o r − ξ Figure 2:
Cross section with cuts listed in the text and also shown in the figure at LO (blue dotted)and at NLO (red solid) accuracy as a function of the equal renormalization and factorization scalenormalized to the default scale µ . The lower panel shows the NLO K-factor. The cross section at LO and NLO accuracy as a function of the equal renormalization andfactorization scale normalized to the default scale µ is shown in Fig. 2. We find significantreduction of the scale dependence and an NLO K-factor K = 1 .
21 at our default scalechoice.Next we turn to comparisons of predictions at NLO accuracy with those obtained fromthe pre-showered events. With this comparison our only aim is to demonstrate that ourframework can generate meaningful pre-showered events using the Frixione isolation (thestandard in fixed-ordered calculations). On Figs. 3–5 six sample distributions are depictedto illustrate the effect of the POWHEG Sudakov factor. In general we find agreementbetween the corresponding predictions except for the transverse-momentum distributionfor the extra parton (left plot of Fig. 5). The effect of the POWHEG Sudakov suppressionis clearly visible in the low p ⊥ region where the radiation activity is highly limited, asexpected. The presence of the extra cut in the real-emission part (the Frixione isolation)causes small distortion in the Sudakov shape as seen at about 0.75. in the left plot ofFig. 5. These comparisons show good agreement between the fixed-order predictions andthose from the pre-showered events. The visible differences can be accounted for the effectof the POWHEG Sudakov factor. It is worth mentioning that the formal accuracy is stillNLO, the difference is due to higher order terms.– 6 – − − − − d σ / d p ⊥ , γ [ pb / G e V ] PowHel -LHE
PowHel -NLO8 TeV, µ = ˆ H ⊥ / δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < . N L O L H E p ⊥ ,γ [GeV] − − d σ / d p ⊥ , t [ pb / G e V ] PowHel -LHE
PowHel -NLO8 TeV, µ = ˆ H ⊥ / δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < . N L O L H E p ⊥ , t [GeV] Figure 3:
Comparison between predictions from LHEs (solid red) and at NLO (blue dashed) usingFrixione isolation for the transverse momentum of the photon and t-quark. On the lower panel theratio of the two predictions is shown. d σ / d y γ [ pb ] PowHel -LHE
PowHel -NLO δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < .
58 TeV, µ = ˆ H ⊥ / N L O L H E -3 -2 -1 0 1 2 3 y γ d σ / d y t [ pb ] PowHel -LHE
PowHel -NLO δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < .
58 TeV, µ = ˆ H ⊥ / N L O L H E -3 -2 -1 0 1 2 3 y t Figure 4:
The same as Fig. 3 for the rapidities of the photon and the t-quark.
4. Photon isolation as a generation cut
When photons are produced with massless partons in the final state the usual soft/collineardivergences coming from parton-parton splittings are accompanied by a new type of collinearsplitting, namely the quark-photon one. The singularity produced by a collinear photonemission off a massless (anti)quark can be absorbed into the photon fragmentation func-tion, decomposing the cross section into direct photon production and a fragmentationcontribution.The only known solution that leads to an IR-safe cross section at all orders in pertur-bation theory that avoids the fragmentation contribution is offered in ref. [7] where QCDactivity is considered in a continuously shrinking cone around the photon such that theallowed activity decreases with decreasing cone size.While in a theoretical calculation the shrinking cone size can be easily implemented,in an experiment the finite resolution of the detector does not allow for taking the smoothlimit. As a result most of the experiments adopt a different isolation criterion: reduced– 7 – − − − d σ / d l og p ⊥ , j [ pb ] PowHel -LHE
PowHel -NLO δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < .
58 TeV, µ = ˆ H ⊥ / N L O L H E log ( p ⊥ ,j /GeV) d σ / d ∆ R ( γ , t) [ pb ] PowHel -LHE
PowHel -NLO δ = 0 . p ⊥ ,γ >
30 GeV , | y γ | < .
58 TeV, µ = ˆ H ⊥ / N L O L H E ∆ R ( γ, t) Figure 5:
The same as Fig. 3 but the differential cross section is depicted as a function of thelogarithm of the extra-parton transverse momentum and the separation of the photon and thet-quark. The separation is defined in the rapidity–azimuthal angle plane. hadronic activity is allowed around the photon in a cone with finite size such that for thetotal hadronic transverse energy inside the cone E ⊥ , had = (cid:88) i ∈ tracks E ⊥ ,i Θ ( R γ − R ( p γ , p i )) < E max ⊥ , had . (4.1)In Eq. (4.1) E ⊥ ,i is the transverse energy of the i th track, R γ is the isolation cone size, R ( p γ , p i ) is the separation between the photon and the i th track measured in rapidity–azimuthal angle plane, while E maxhad is the maximal hadronic energy allowed to be depositedin the cone of R γ around the photon. In the following we call this quantity hadronicor partonic leakage depending on whether the process is considered on the hadron orthe parton level. In a fixed-order calculation an isolation of the form of Eq. (4.1) does notcompletely remove the singularity of collinear quark-photon emission and therefore, cannotbe applied. Setting E max ⊥ , had = 0 removes this singularity, but cuts into the phase space ofsoft gluon emission in the real correction, hence it is not IR-safe.As shown in the previous section, there is one photon isolation which is free fromperturbative singularities and can be used to generate meaningful pre-showered events.When generating LHEs according to the POWHEG formula, we can generate events usingthe smooth isolation prescription of the photons according to the formula (Frixione-typeisolation with (cid:15) γ = n = 1) E ⊥ , had = (cid:88) i ∈ partons E ⊥ ,i Θ ( δ − R ( p γ , p i )) ≤ E ⊥ ,γ (cid:18) − cos δ − cos δ (cid:19) , (4.2)for all δ ≤ δ , where δ is a sufficiently small, pre-defined number. This isolation canbe considered as a generation isolation, Θ genisol ( δ ). Then the inclusive cross section can bedecomposed as σ incl = σ incl Θ genisol + σ incl (cid:0) − Θ genisol (cid:1) , (4.3)where the first term on the right hand side is the perturbatively computable cross sectionwith smooth photon isolation of Eq. (4.2). The second one contains non-perturbative con-– 8 –ribution to, therefore, cannot be computed in perturbation theory. Thus Eq. (4.3) canbe considered as (an unconventional) factorization of the quark-photon singularity intonon-perturbative contribution. Applying a physical isolation on Eq. (4.3), we obtain theexperimentally measurable cross section for isolated photon production, σ expisol = σ incl Θ expisol = σ incl Θ genisol ( δ ) Θ expisol + σ incl (cid:0) − Θ genisol ( δ ) (cid:1) Θ expisol . (4.4)If the experimental isolation is simply a tighter version of the smooth isolation of Eq. (4.2),then the non-perturbative contribution trivially vanishes, as (cid:0) − Θ genisol ( δ ) (cid:1) Θ expisol = 0. Thusthe events generated with smooth isolation can be used to make such physical prediction.If the physical isolation is the cone-type isolation of Eq. (4.1), then the non-perturbativecontribution is non-zero. Nevertheless, we shall argue that if the generation isolation issufficiently loose and the photon is sufficiently hard, then for cone-type isolation with valuesof parameters used in the experiments, the non-perturbative contribution is negligiblewithin the accuracy of the perturbative one, thus the first term still gives a sufficientlygood description of data.First let us note that left hand side of Eq. (4.4) is independent of δ , so must be the righthand side, too. Below we shall demonstrate that for sufficiently loose generation isolation,in the range δ gen0 ∈ [0 . , . σ incl Θ genisol Θ expisol obtained with usual experimentalcone-type isolation of Eq. (4.1), depends on δ gen0 very little. As a result, the second term onthe right hand side of Eq. (4.4) has to be almost independent of δ gen0 , too. Although thissecond term is not computable in perturbation theory, making Θ genisol looser, it decreases, andwe expect it becomes negligible within the accuracy of the calculation, when δ gen0 ≤ . σ incl Θ genisol Θ expisol approximates the experimentally isolated hard photon cross sectionwithin the accuracy of the prediction.With such a generation isolation we can generate a sufficiently inclusive sample of pre-showered events. On the events prepared this way it is easy to apply a close-to-experimenttype of cut such as Eq. (4.1), the quark-photon singularity is appropriately screened henceallowing for a small hadronic (or partonic) activity in the cone around the photon andcannot lead to infinite predictions. This procedure of making theoretical predictions ismade possible by the generation of LHEs as opposed to producing differential distributionsdirectly, as in the case of computing cross sections at fixed order in perturbation theorybeyond LO accuracy.
5. Independence of the generation isolation
We generate an almost inclusive event sample with a loose photon isolation. The generationisolation parameter δ gen0 should be chosen such that the distributions with experimentalphoton isolation obtained at various stages of event simulation (from LHEs, after partonshower and after full SMC) should be independent of it. In order to see this independence,we generated events with three different generation isolation values: δ gen0 ∈ { . , . , . } .– 9 – .00.050.10.150.20.250.30.350.4 σ e x p i s o l ( R γ ) SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV, µ = ˆ H ⊥ / r a t i o R γ σ e x p i s o l ( E m a x ⊥ , h a d ) [ pb ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV, µ = ˆ H ⊥ / r a t i o E max ⊥ , had [GeV] Figure 6:
Isoloted photon cross sections obtained after full SMC with different generation isolationsusing cuts listed in the text, as a function of a) the radius of experimental photon isolation cone,b) the hadronic leakage inside the photon isolation cone.
These event generations are done with parameters listed in Sec. 3. Then we compare thepredictions made with different values of δ gen0 at various stages of the event simulation.Although the particle content can be different at different stages of event evolution, wekept the set of cuts applied to the events the same: • There is a cut on the transverse momentum of the hardest photon: p ⊥ ,γ >
30 GeV. • The hardest photon should be central: | y γ | < . • A jet algorithm is applied using the anti- k ⊥ algorithm [25] provided by FastJet [26, 27] with p j ⊥ >
30 GeV and R = 0 . • The hardest photon should be well-isolated from the jets: ∆ R ( γ, j ) > . • A hadronic (or partonic) leakage is allowed in an R γ = 0 . E max ⊥ , had = 3 GeV.We have checked that for δ gen0 ∈ [0 . , .
1] the physical predictions depend marginally on δ gen0 at all stages of the event evolution, but show here only for predictions obtained at thehadronic stage, i.e. after SMC. The cross section values after full SMC and given selectioncuts are presented as a function of the radius of experimental photon isolation cone, and ofthe hadronic leakage inside the photon isolation cone in Fig. 6. We see that independentlyof these parameters (within the ranges shown here), the physical cross section depends onthe generation isolation weakly.For kinematic distributions we find even smaller dependence on δ gen0 . Six sampledistributions are presented on Figs. 7–9. – 10 – − − − − − − d σ / d p ⊥ , γ [ pb / G e V ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV, µ = ˆ H ⊥ / r a t i o p ⊥ ,γ [GeV] − − d σ / d p ⊥ , t [ pb / G e V ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV, µ = ˆ H ⊥ / r a t i o p ⊥ , t [GeV] Figure 7:
Transverse-momentum distribution for the hardest photon and the t-quark after partonshower and hadronization with
PYTHIA for smooth generation isolation with δ gen0 ∈ [0 . , . d σ / d y γ [ pb ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV µ = ˆ H ⊥ / r a t i o -3 -2 -1 0 1 2 3 y γ d σ / d y t [ pb ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV µ = ˆ H ⊥ / r a t i o -3 -2 -1 0 1 2 3 y t Figure 8:
The same as Fig. 7 but for the rapidities of the hardest-photon and the t-quark.
6. Estimation of the non-pertrubative contribution
In Eq. (4.4) we decomposed the isolated photon cross section into a perturbatively com-putable part (first term) and a non-perturbative contribution (second term). Furthermore,we argued that provided δ gen0 sufficiently small, we expect the non-perturbative contribu-tion to be small. This statement can only be verified by explicit comparison to experimentaldata, which is presently not possible for isolated photon production in association with at ¯t pair. It is possible however, for the case of massive vector boson + isolated photonproduction for which the ATLAS collaboration published results for both isolated photon+ 0 jet (exclusive) and isolated photon + N ( ≥
0) jets (inclusive) in the final state [28]. Thisfinal state has also been considered recently at NLO accuracy interfaced to a shower gen-erator according to the POWHEG prescription supplemented with the MiNLO procedure[29]. In this work the fixed order result is matched to an interleaved QCD+QED partonshower, in such a way that the contribution arising from hadron fragmentation into photonsis fully modeled. Thus for this process the comparison is possible not only for experimental– 11 – .050.10.150.20.25 d σ / d ∆ R ( γ , t) [ pb ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV, µ = ˆ H ⊥ / r a t i o ∆ R ( γ, t) d σ / d ∆ R (t , ¯ t) [ pb ] SMC, PY , δ = 0 . PY , δ = 0 . PY , δ = 0 .
18 TeV, µ = ˆ H ⊥ / r a t i o ∆ R (t , ¯t) Figure 9:
The same as Fig. 7 but for separations in the rapidity–azimuthal angle plane. results, but also with a theoretical prediction where the fragmentation is included througha shower model.Within
PowHel the
W γ process can be implemented straightforwardly. We generatedevents for
W γ production with the same three values of δ gen0 as in the case of t ¯t γ productionand checked that the predictions from the pre-showered events agree with those at NLOaccuracy, just as in the case of t ¯t γ production in Sec. 3. Next, we checked the dependenceon the generation isolation parameter, similarly as in Sec. 5 and found that the perturbativeprediction depends weakly (below 10 % for the exclusive and below 5 % for the inclusivecase) on the choice of δ gen0 in the range δ gen0 ∈ [0 . , .
1] if a similar physical isolation is usedas in Sec. 5. Thus we have implemented the event selection of ref. [28] and made predictionsfor the inclusive case and for the exclusive case using events obtained with δ gen0 = 0 .
7. Effect of the parton shower
In the previous sections we estimated the effect of the neglected non-perturbative contribu-– 12 – − − d σ d p ⊥ , γ [ f b / G e V ] ATLAS data
W γ
POWHEG
W γ
PowHel / ≤ ξ R , ξ F ≤ √ s = 7 TeV N jet ≥ t h e o r y d a t a POWHEG P o w H e l p ⊥ , γ [GeV] − − d σ d p ⊥ , γ [ f b / G e V ] ATLAS data
W γ
POWHEG
W γ
PowHel / ≤ ξ R , ξ F ≤ √ s = 7 TeV N jet = 0 t h e o r y d a t a POWHEG P o w H e l p ⊥ , γ [GeV] Figure 10:
Transverse momentum distribution of the isolated photon for the a) W + γ + N ≥ W + γ + 0 jet final states. The lower panels show the ratio of the predictions to thedata. tion, as well as demonstrated that predictions for isolated photon cross section made withfull SMC do not depend upon the sufficiently small generation-isolation. In fact, we alsochecked that the latter is true at various stages of event simulation (LHE, PS and SMC).To quantify the effect of the parton shower and in the next section to present physicalpredictions after full SMC we decided to use δ gen0 = 0 .
01 in our generation isolation. Forthis comparison we used the setup of Sec. 5. Our standard distributions can be found onFigs. 11–13. While for rapidities and separations the difference between the LHE and PSstages only manifest in an overall change in normalization, for the transverse-momentumdistributions the change is not only a constant factor in normalization, but there is even achange in the shape. As we expect, the shower softens the spectra. This softening added tothe difference between the predictions of LHEs and at NLO suggests very small PS effect athigh transverse momenta. (We cannot compare Figs. 3 and 11 directly as photon isolationsare different.) In the case of the photon p ⊥ the change remains small, around 5%, whilefor the transverse momentum of the t-quark it reaches even 12% when the p ⊥ approaches500 GeV. If our default, rather tight, criterion on the allowed hadronic leakage is loosen up(going from 3 GeV to 10 GeV) the difference observed in the photon transverse-momentumdistribution remains more-or-less the same, but in the case of the transverse momentumof the t-quark the difference drops below 10% in the high- p ⊥ region. The relaxation inthe hadronic leakage condition results in a smaller difference, ∼
8. Predictions
We conclude with a simple phenomenological study at the hadron level. To this end
PYTHIA-6.4.25 was chosen to decay, shower and finally hadronize the events. The eventsample with δ gen0 = 0 .
01 at 8 TeV was selected,
PYTHIA was run with the 2010 Perugia tune– 13 – − − − − d σ / d p ⊥ , γ [ pb / G e V ] LHEPS, PY µ = ˆ H ⊥ / δ gen0 = 0 . r a t i o p ⊥ ,γ [GeV] − − d σ / d p ⊥ , t [ pb / G e V ] LHEPS, PY µ = ˆ H ⊥ / δ gen0 = 0 . r a t i o p ⊥ , t [GeV] Figure 11:
Transverse-momentum distribution for the photon and t-quark at the LHE stage andafter parton shower. The lower panel shows the LHE/PS ratio. d σ / d y γ [ pb ] LHEPS, PY µ = ˆ H ⊥ / δ gen0 = 0 . r a t i o -2 -1 0 1 2 y γ d σ / d y t [ pb ] LHEPS, PY µ = ˆ H ⊥ / δ gen0 = 0 . r a t i o -3 -2 -1 0 1 2 3 y t Figure 12:
The same as Fig. 11 but for the rapidities of the photon and t-quark. d σ / d ∆ R ( γ , t) [ pb ] LHEPS, PY µ = ˆ H ⊥ / δ gen0 = 0 . r a t i o ∆ R ( γ, t) d σ / d ∆ R (t , ¯ t) [ pb ] LHEPS, PY µ = ˆ H ⊥ / δ gen0 = 0 . r a t i o ∆ R (t , ¯t) Figure 13:
The same as Fig. 11 but for the separation of the photon-t and t ¯t systems. [30], omitting photon showers, making τ ± and π stable and we turned off multi-particleinteractions. The cuts employed in this analysis were the following: • The analysis was done in the semileptonic decay-channel by requesting exactly one– 14 – − − − d σ / d p ⊥ , γ [ pb / G e V ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o
50 100 150 200 250 300 350 400 450 500 p ⊥ ,γ [GeV] − − − d σ / d p ⊥ , t [ pb / G e V ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o
50 100 150 200 250 300 350 400 450 500 p ⊥ , t [GeV] Figure 14:
Transverse momentum distribution for the photon and the t-quark at the hadronicstage. On the lower panel the ratio of predictions to that obtained with our default scale choice isshown. hard lepton or antilepton in the final state with p ⊥ ,(cid:96) >
30 GeV, the (anti)lepton hadto be isolated from all the jets with ∆ R ( (cid:96), j ) > . • The final state had to contain one hard photon in the central region, | y γ | < . p ⊥ ,γ >
30 GeV, isolated from all the jets by ∆ R ( γ, j ) > .
4. A minimal hadronicleakage was allowed in a R γ = 0 . E max ⊥ , had = 3 GeVaccording to Eq. (4.1). • The (anti)lepton and photon had to be separated from each other, ∆ R ( γ, (cid:96) ) > . • Jets were reconstructed with the anti- k ⊥ algorithm [25] with R = 0 . p j ⊥ >
30 GeV. • The event had to have significant missing transverse momentum, /p ⊥ >
30 GeV.In our calculation, throughout, a different scale choice was used than that in theliterature [20] for t ¯t γ production. Our default scale choice, the half the sum of transversemasses ˆ H ⊥ / µ R = ξ R µ and µ F = ξ F µ , respectively, and the band is formed as the upper-and lower-bounding envelopes of distributions taken with( ξ R , ξ F ) ∈ (cid:26)(cid:18) , (cid:19) , (cid:18) , (cid:19) , (cid:18) , (cid:19) , (1 , , (1 , , (2 , , (2 , (cid:27) . (8.1)The antipodal choices ((1 / ,
2) and (2 , / MCTRUTH .– 15 – − − − − − d σ / d p ⊥ , ‘ [ pb / G e V ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o
50 100 150 200 250 300 350 400 450 500 p ⊥ ,‘ [GeV] − − − − d σ / d / p ⊥ [ pb / G e V ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o
50 100 150 200 250 300 350 400 450 500 /p ⊥ [GeV] Figure 15:
The same as Fig. 14 but for the spectra of the transverse momentum of the charged(anti)lepton and the missing momentum. d σ / d ∆ R ( γ , t) [ pb ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o ∆ R ( γ, t) d σ / d ∆ R ( γ , ‘ ) [ pb ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o ∆ R ( γ, ‘ ) Figure 16:
The same as Fig. 14 but for separations measured in the rapidity–azimuthal angleplane.
Taking a look at the transverse momentum of the photon the static scale results in anarrower band with a shrinking width. This hints a cross-over point at a higher p ⊥ value,while in the case of the dynamical scale the band, although wider, keeps the same width allacross the whole plotted transverse momentum spectrum. While for the p ⊥ -distribution ofthe photon the presence of a cross-over point is only hinted by the narrowing uncertaintyband, for the transverse momentum of the t-quark it is indeed visible around 350 GeV.Until this point the uncertainty band taken with the static scale decreases in width thanafter opens up. This is somehow expected since a highly boosted t-quark with a heavycompanion anti-t and a photon correspond to a system with a large summed transversemass hence lying far away from the central scale m t .In Fig. 15 the spectrum of the transverse momentum of the charged lepton and that ofthe missing momentum are shown. For both distributions a cross-over can be seen around250 GeV when static scale is used. The dynamical scale choice appears to give reliable scaledependence over the whole plotted range for these observables.If we turn our attention to the separations between the photon and the t-quark, as– 16 – .0050.010.0150.020.0250.030.035 d σ / d y γ [ pb ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o -2 -1 0 1 2 y γ d σ / d y ‘ [ pb ] SMC, PY , µ = ˆ H ⊥ / PY , µ = m t δ gen0 = 0 . r a t i o -3 -2 -1 0 1 2 3 y ‘ Figure 17:
The same as Fig. 14 but for the rapidity distributions of the photon and (anti)lepton. well as between the photon and the charged lepton, measured in the rapidity–azimuthalangle plane, we do not find significant difference between the two scale choices, as seenin Fig. 16. The static scale gives somewhat higher cross section and a slightly narroweruncertainty band below ∆ R = π and larger scale dependence above. Similar conclusionscan be drawn from the rapidity distributions for the photon and the (anti)lepton shownin Fig. 17. In general, the scale dependence is moderate, below 20 % for both scale choicesand all observables, except for the predictions at large transverse momenta with the staticscale.
9. Conclusions
In this paper we presented a new way to make predictions for the hadroproduction ofisolated photons which uses event samples that emerge in simulations aimed at matchingpredictions at NLO accuracy with PS. Our approach uses only the direct-photon contribu-tion, i.e. we neglect the fragmentation. We demonstrated that the presence of a sufficientlysmall smooth isolation of the direct photons, applied during generation of the events,does not affect the physical predictions, within the numerical accuracy of the calculation.Hence it can be used to generate sufficiently inclusive pre-showered event samples. Thepre-showered events obtained this way can be further showered and hadronized to obtaindifferential distributions at the hadronic stage, which include NLO QCD corrections in thehard process, and either smooth or standard experimental photon isolation can be applied.Using the POWHEG method one can make predictions at various stages of the eventsimulation. In particular, for most of the phenomenologically interesting distributions weestimate fairly small (about 10 %, or less) corrections for the t ¯t γ final state due to theparton shower. We also studied the dependence of our predictions on the renormalizationand factorization scales and found small and rather uniform scale dependence for the defaultscale ˆ H ⊥ / W boson in association with a hard isolated photon by comparing our predictions tomeasured data if the photon is harder than the accompanying jets. Our method is com-pletely general and can be used to any process with isolated hard photons in the final state,in particular also for t ¯t production in association with hard isolated photons. Acknowledgments
This research was supported by the Hungarian Scientific Research Fund grant K-101482,the European Union and the European Social Fund through Supercomputer, the nationalvirtual lab TAMOP-4.2.2.C-11/1/KONV-2012-0010 and the LHCPhenoNet network PITN-GA-2010-264564 projects.
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