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Eur. Phys. J. C (2013) 73:2625DOI 10.1140/epjc/s10052-013-2625-1

Special Article - Tools for Experiment and Theory

QED bremsstrahlung in decays of electroweak bosons

Andrej B. Arbuzov1,2,a, Renat R. Sadykov1,b, Zbigniew Wa̧s3,4,c1Joint Institute for Nuclear Research, Joliot-Curie str. 6, Dubna 141980, Russia2Dep. of Higher Mathematics, Dubna University, Universitetskaya str. 19, Dubna 141980, Russia3Institute of Nuclear Physics, PAN, ul. Radzikowskiego 152, Kraków, Poland4CERN PH-TH, 1211 Geneva 23, Switzerland

Received: 30 May 2013 / Revised: 13 September 2013 / Published online: 5 November 2013© The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract Isolated lepton momenta, in particular their direc-tions are the most precisely measured quantities in pp col-lisions at LHC. This offers opportunities for multitude ofprecision measurements.

It is of practical importance to verify if precision mea-surements with leptons in the final state require all theo-retical effects evaluated simultaneously or if QED brems-strahlung in the final state can be separated without un-wanted precision loss.

Results for final-state bremsstrahlung in the decays ofnarrow resonances are obtained from the Feynman rules ofQED in an unambiguous way and can be controlled witha very high precision. Also for resonances of non-negligiblewidth, if calculations are appropriately performed, such sep-aration from the remaining electroweak effects can be ex-pected.

Our paper is devoted to validation that final-state QEDbremsstrahlung can indeed be separated from the rest ofQCD and electroweak effects, in the production and decayof Z and W bosons, and to estimation of the resulting sys-tematic error. The quantitative discussion is based on MonteCarlo programs PHOTOS and SANC, as well as on KKMCwhich is used for benchmark results. We show that for alarge class of W and Z boson observables as used at LHC,the theoretical error on photonic bremsstrahlung is 0.1 or0.2 %, depending on the program options used. An overalltheoretical error on the QED final-state radiation, i.e. takinginto account missing corrections due to pair emission andinterference with initial state radiation is estimated respec-tively at 0.2 % or 0.3 % again depending on the programoption used.

a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

1 Introduction

Several of the most important measurements at LHC exper-iments, such as Higgs boson searches [1, 2], precision mea-surements of the W boson mass at LHC and Tevatron [3–6]or measurements of Drell–Yan (DY) processes [7] and elec-troweak (EW) boson pair production [5, 8] rely on a precisereconstruction of momenta for the final-state leptons [9, 10].A substantial effort of the experimental community was de-voted to optimize detector design and understand detectorresponses. Precision of 0.1 % (even 0.01 % for lepton direc-tions) is of no exception. For more details, see e.g. Refs. [9–12].

The QED effects of the final-state radiation (FSR) playan important role in such experimental studies. FSR is in-cluded in all simulation chains and indeed should be studiedtogether with the detector response to leptons. It can not beseparated, because of infrared singularities of QED. Differ-ent approaches based on various theoretical simplificationsare in use at present. At the level of the collinear approx-imation, expressions for higher order FSR corrections arenot only well defined but are in fact process independent. Ingeneral, QED calculations are process dependent, but meth-ods for obtaining results with O(α2) corrections and resum-mation of higher order effects are well established as wellas techniques for evaluating theoretical errors. There is noneed for introducing effective models. Situation is differentif intermediate or outgoing particles, such as Z or W , areunstable, an effort like documented in [13] is then needed.In case of Z and W decays if the narrow width approxi-mation is used, the theoretical framework for QED FSR ef-fects is unambiguous. In case of the Z decay (in fact forhard processes mediated by s-channel Z/γ ∗ exchange) theframework in which the QED final-state radiation is sepa-rated from other contributions is defined unambiguously aswell. This was explored in e+e− collisions at LEP and highprecision solutions were proposed.

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of 18 Eur. Phys. J. C (2013) 73:2625

For LEP I experiments Monte Carlo simulation programsbased on exclusive exponentiation and featuring second or-der matrix elements were developed [14, 15]. Considerabletheoretical effort was invested in the reordering of the per-turbative expansion. This opened a scheme for proper re-summation of vacuum polarization diagrams into terms con-tributing to Z width and remaining vacuum polarization cor-rections [16]. The definition of final-state bremsstrahlungwas affected in a minimal way. The by-product of this ef-fort was separation of the electroweak corrections into QEDparts, corrections to the Z boson propagator (which have tobe resummed to all orders) and remaining weak correctionswhich, with the proper choice of the calculation schemes,were small.

In the case of LEP II this theoretical approach had tobe reconsidered, because of the new W+W− pair produc-tion processes. The gauge cancellation between diagramsof electroweak bosons exchanged in s and t channels, hadto be carefully respected. The resummation of the domi-nant contribution of the vacuum polarization had to be re-stricted to the case of the constant width. It was not neces-sary however to reopen discussion on details of scheme forfinal-state bremsstrahlung calculations because of the rela-tively small available statistics of W -pair samples [17, 18]and thus limited interest in high precision calculations forthe QED bremsstrahlung.1

It is important to stress that the QED final-state effectsin processes where leptons are produced through decays ofW or Z/γ ∗ can be calculated and simulated with an essen-tially arbitrary high precision. They form a separate classof Feynman diagrams and already developed techniquesshould be explored at LHC, especially as the attractivenessof such an approach was confirmed [21] in the context ofW mass measurement of CDF and D0 collaborations. Withthe ever increasing precision, effects beyond photonic final-state bremsstrahlung have to be of course considered as partof FSR effects as well, in particular those due to emission ofextra lepton pairs. Also interference effects, such as initial-final-state bremsstrahlung interference, has to be taken intoaccount.

The interference becomes an issue for separating QEDFSR from the rest of the electroweak corrections, at thelevel of cross section. That is why, the interference of pho-ton emission from the initial state quarks and the final-state

1Consequences of resummation of parts of the electroweak effects arecomplex but will not be covered in this paper. Let us point to anothertheoretical constraint restricting naive resummation. It could be ob-served that even in the case of initial state QED bremsstrahlung for theprocess e+e− → νeν̄e , previously performed step of resummation hadto be partially revisited, because of amplitude featuring t -channel Wexchange [19]. For the complete second order matrix element of thetwo hard photon emission the diagrams of charged pseudo-Goldstoneboson exchange had to be taken into account [20].

lepton has to be discussed carefully. It is of practical im-portance to verify if the suppression of the interference dueto boson’s lifetime survives experimental cuts. If it is thecase, then separation of effects due to the QED final-statebremsstrahlung and remaining parts of the electroweak andstrong interaction corrections can be conveniently exploredin the experimental studies.

For this paper we assume, that in practical applications,all other corrections than QED final-state interactions, thatis remaining electroweak, and initial state hadronic inter-actions are expected to be measured with the properly de-fined observables. Thanks to this approach we will be able toachieve very competitive precision of theoretical predictionson the class of corrections directly affecting lepton momentameasurements. This represents a complementary approachto the one used in [22]. There, emphasis was put on the useof electroweak calculations together with all hadronic ini-tial state interactions necessary for complete predictions forobservables.

Our paper is organized as follows. In Sect. 2 we de-scribe two programs PHOTOS and SANC, and their theoreti-cal base. Section 3 is devoted to tests of the first order QEDcalculations. This is of importance in itself but also representconsistency checks of definitions of this part of electroweakcorrections which will be considered at the amplitude levelas QED final-state radiation. Definition of observables andcalculation schemes used all over the paper are also givenin this section. Section 4 is devoted to discussion of resultsfor multiphoton emission and theoretical uncertainties in φ∗ηmeasurements. Section 5 is devoted to discussion for otherthan photonic bremsstrahlung effects which contribute to thetheoretical error of QED FSR simulated using PHOTOS orSANC. In particular, comments on emissions of pairs and in-terferences between photon emission from the final state andother sources are given here. Section 6, Summary, closes thepaper.

Our discussion of theoretical error is limited to system-atic error of QED FSR only, but it is performed in thecontext of full event generation. Other effects, like orien-tation of spin state for the intermediate W or Z bosons,affecting input for calculation of QED FSR spin ampli-tudes are addressed in Sect. 5.3. Precision tag for the QEDFSR calculation implemented in PHOTOS or SANC is finallygiven for a broad class of observables: first for the photonicbremsstrahlung and then for complete FSR corrections.2

In our work we concentrate on the applications for LHCexperiments of techniques, calculations and programs devel-oped earlier: PHOTOS [23–29], SANC [30–39] and KKMC

2In this paper we use the name photonic bremsstrahlung whenever wewant to stress that only diagrams resulting from supplementing Bornlevel amplitudes with photon lines are considered. The name final-stateradiation is used when we stress presence of additional pairs and final-state interaction when we want to discuss separation with remainingparts of electroweak interactions.

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[14, 15] including systematic error estimation for these newapplications. Substantial and well known effort documentedin these references and which form theoretical basis of ourpresent work, will not be recalled. This would require sub-stantial increase of our paper size with the repetition ofpublished material. Our naming conventions originate fromthose papers as well. Aim of our present work is establish-ing physics precision of the new results of importance forLHC applications. Whenever necessary, (essentially for val-idating compatibility of our calculation schemes), we willaddress issues of technical correctness as well.

2 Description of the programs

Usually when discussing phenomenological processes atLHC in the context of the detector response one concentrateson description of the hard process and initial state QCD ef-fects embedded e.g. in general purpose Monte Carlo gener-ators featuring parton shower, models of underlying events,and finally QCD NLO or NNLO corrections to the hard scat-tering process itself are taken into account, see e.g. [40] fora review.

In this paper we concentrate on the QED FSR effects.This determines the choice of programs which will be usedby us for simulations. The final-state interactions consist ofQED bremsstrahlung in decays of electroweak bosons (in-cluding cases of substantial virtualities) and to some degreeon the other parts of weak corrections as well. Let us re-call the massive effort of years 1980–2000 for establishingdefinition of calculation scheme at LEP [41–43] where the-oretically sophisticated and numerically essential separationfrom electroweak corrections of the initial-state, final-state,vacuum polarization (including definition of the width) andinterferences was established.

In this context discussion of theoretical errors of all partsbeing necessary for predictions is important, since it is notstraightforward to disentangle effects of new physics andtheir genuine weak backgrounds. This represents howeverfurther separate work on weak corrections. In the presentpaper we concentrate on final-state radiation, especially onthe final-state photonic one.

For the purpose of these studies, two programs featur-ing QED final-state radiation for LHC applications will befirst described and later used. We will start with presen-tation of the SANC system [30], because it features com-plete electroweak corrections as well. Description of PHO-TOS [23–25, 29] implementing the QED final-state photonicbremsstrahlung only, will follow.

It might be useful to note that abbreviations LO andNLO in SANC and PHOTOS have somewhat differentmeaning. In the case of SANC, “LO” (the Leading Or-der) means just the tree-level Born cross section, while an

exclusive-exponentiation-like notation is adopted in PHO-TOS, where “LO” supposes exponentiation/resummation ofthe terms responsible for leading logarithmic terms of pho-tonic bremsstrahlung. Full coverage of multiphoton phase-space is assured. The same concerns “NLO”: in SANC itmeans the one-loop approximation (Born plus O(α) EWcorrections), while in PHOTOS exponentiation of the O(α)result is assumed.

2.1 SANC

SANC is a computer system for Support of Analytic andNumeric calculations for experiments at Colliders [30]. Itcan be accessed through the Internet at http://sanc.jinr.ru/and at http://pcphsanc.cern.ch/. The SANC system is suitedfor calculations of one-loop QED, EW, and QCD radiativecorrections (RC) to various Standard Model processes. Au-tomatized analytic calculations in SANC provide FORM andFORTRAN modules [35], which can be used as buildingblocks in computer codes for particular applications.

For Drell–Yan-like processes within the SANC projectthere are implemented:

– complete one-loop EW RC in the charged current [31]and neutral current [32] processes;

– photon induced Drell–Yan processes [33];– higher order photonic FSR in the collinear leading loga-

rithmic approximation;– higher order photonic and pair FSR in the QED leading

logarithmic approximation [38, 44, 45];– complete next-to-leading QCD corrections [34, 36];– Monte Carlo integrators [37] and event generators;– interface to parton showers in PYTHIA and HERWIG

based on the standard Les Houches Accord format.

Tuned comparisons with results of HORACE [46] andZ(W)GRADE [47] for EW RC to charged current (CC) andneutral current (NC) Drell–Yan (DY) were performed withinthe scope of Les Houches ’05 [48], ’07 [49] and TEV4LHC’06 [50] workshops. A good agreement achieved in thesecomparisons confirms correctness of the implementation ofthe complete one-loop EW corrections in all these programs.

An important feature of the SANC approach is the pos-sibility to control and directly access different contributionsto the observables being under consideration. In particular,SANC code allows to separate effects due to the final-stateradiation, the interference of initial and final radiation, theso called pure weak contributions, etc.

Separation of the FSR contribution in the case of neutralcurrent DY processes is straightforward, it naturally appearsat the level of Feynman diagrams. But the correspondingseparation in the case of the charged current DY processesis not so trivial. In general it is even not gauge invariant.

http://sanc.jinr.ru/http://pcphsanc.cern.ch/

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For the sake of tuned comparison with PHOTOS, a specialprescription for this separation3 was introduced into SANC.

Let us consider a formal separation of the pure weak(PW) and QED contributions δPW and δQED to the totalW+ → u + d̄ decay widthΓ

PW+QEDW = Γ LO

(δPW + δQED). (1)

This process at the O(α) is described by 6 QED-like dia-grams with virtual photon line and 3 other ones with realphoton emission which together lead to the formula

δQED = απ

[Q2W

(11

6− π

2

3

)

+ (Q2u + Q2d)(

11

8− 3

4log

M2W

μ2PW

)],

where parameter μPW is the ’t Hooft scale introducedfor separation of QED and PW contributions. In order toseparate the FSR QED contribution, we choose μPW =MW exp(− 1112 ). This value of the ’t Hooft scale makes thetotal QED contribution to the W boson decay being equalto zero. This is in agreement with the corresponding treat-ment in PHOTOS, where by construction the effect of FSRto a process does not change the normalization of the crosssection.

2.2 PHOTOS

Already in the era of data analysis of LEP experiments sim-ulation of bremsstrahlung in decays of resonances and par-ticles required specialized tools. In parallel, two programsoriented toward highest possible overall precision for thewhole processes in e+e− collisions such as KKMC [14] orKORALZ [51], programs dealing with decays only, graduallybecame of a broad use. The PHOTOS Monte Carlo was oneof such applications [23, 24]. Naturally comparisons withthese high-precision generators became parts of test-beds forPHOTOS package.

The principle of PHOTOS algorithm is to replace, on thebasis of well defined rules, the decay vertex embedded in theevent record such as HEPEVT [52] or HepMC [53] with thenew one, where additional photons are added. Such solution,initially not aimed for high precision simulations, turned outto be very effective and precise as well. Phase space pa-rameterization was carefully documented in [26]. Graduallyfor selected decays [26–28, 54], also exact matrix elementswere implemented and could be activated in place of uni-versal kernels.4 Originally [23], only single photon radia-tion was possible and approximations in the universal kernel

3This prescription should be respected also in electroweak, non-QEDFSR calculations used together with PHOTOS in practical applications.4Prior to introduction of the C++ interface matrix element kernels wereavailable for our test only. They require more detailed information from

were present even in the soft photon region. With time, mul-tiphoton radiation was introduced [25] and then installationof exact first order matrix elements in W and Z decays be-came available with C++ implementation of PHOTOS [29].The algorithm of PHOTOS is constructed in such a way, thatthe same function, but with different input kinematical vari-ables, is used if the single photon emission or full multi-photon emission is requested. Such an arrangement enablestests in a rigorous first order emission environment. For mul-tiphoton emission, the same kernel is used iteratively, thanksto the factorization properties. Technical checks are thusspared. Optimal solution for the iteration was chosen andverified with alternative calculations [55, 56] based on thesecond order matrix element. It was later extended to themultiphoton case for Z decays in Ref. [27]. Numerical testsof that paper, for distributions of generic kinematical observ-ables pointed to the theoretical precision for the simulationof photon bremsstrahlung of the 0.1 % level.

When presenting numerical results from PHOTOS we al-ways refer to its C++ version [29] with matrix elements forW and Z decays switched on. If the LO level is explicitlymentioned, matrix elements are replaced by universal ker-nels and algorithm as of FORTRAN version 2.14 [25], orhigher is used. At present, version [29] represents the up-to-date version of PHOTOS; it is available, with few technicalupdates, from LCG library [57] as well.

3 Definitions and results of the first order calculations

As a first step of our tests we have compared numerical re-sults obtained from PHOTOS and from SANC programs incase of the single photon emission. These tests cross-checkconventions used and numerical stability of the two calcu-lations. They also verify the proper choice of parameters inPYTHIA8 generator, which is used to produce electroweakBorn level events, on which PHOTOS is activated. We havemonitored distributions for the following observables: pseu-dorapidity η of −, transverse mass MT of −ν̄ pair, andtransverse momentum pT of − in the case of charged cur-rent; and pseudorapidity η of −, invariant mass M of +−pair, and transverse momentum pT of − in the case of neu-tral current.

The following set of input parameters was used:

Gμ = 1.6637 × 10−5 GeV−2,MW = 80.403 GeV, ΓW = 2.091 GeV,MZ = 91.1876 GeV, ΓZ = 2.4952 GeV, (2)Vud = 0.9738, Vus = 0.2272,

the event record which was available from PHOTOS interface in FOR-TRAN.


Vcd = 0.2271, Vcs = 0.9730,me = 0.511 MeV, mμ = 0.10566 GeV,and the following experiment motivated cuts were appliedon momenta of the final-state leptons:

NC: ∣∣η(+)∣∣ < 10, ∣∣η(−)∣∣ < 10,p⊥

(

+

)> 15 GeV, p⊥

(

−

)> 15 GeV,

70 < M(

+−

)< 110 GeV; (3)

CC: ∣∣η(−)∣∣ < 10, p⊥(

−

)> 0.1 GeV,

p⊥(ν̄) > 0.1 GeV.

We work in the running width scheme for W and Z bo-son propagators and fix value of the weak mixing angle:

Fig. 1 Ratios for Born-level distributions in W → eν decay

cos θW = MW/MZ , sin2 θW = 1 − cos2 θW . The value of theelectromagnetic coupling α is evaluated in the Gμ-schemeusing the Fermi constant Gμ: the effective coupling is de-fined by αGμ =

√2GμM2W sin

2 θW/π .To compute the hadronic cross section we have used

CTEQ6L1 set of parton distribution functions with runningfactorization scale μ2r = ŝ, where ŝ is squared total energyof the colliding partons in their center-of-mass system.

Comparison is performed for muon and electron finalstates. For the electron final states and for each observablethe bare and calo results are provided. In the calo case thefour-momenta of the final electron and photon are combinedinto effective four-momentum of the electron when the sep-

Fig. 2 Ratios for Born-level distributions in Z → ee decay

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Fig. 3 O(α) corrections for basic kinematical distributions fromPYTHIA+PHOTOS and SANC in W → eν decay

aration

�R =√

(�η(e, γ ))2 + (�φ(e, γ ))2 < 0.1.To check that the normalization of Born cross sections

is properly adjusted between simulations using SANC andPHOTOS, we have completed tests at a sub-permille preci-sion level. The corresponding results for electrons and ratiosof differential cross sections are shown in Fig. 1 for CC andin Fig. 2 for NC. Errorbars in this plot and in all that fol-low represent the statistical fluctuations of the correspond-ing Monte Carlo integration.

For SANC and PYTHIA+PHOTOS cases, the resultsfor O(α) corrections which are defined by δ = (σO(α) −σ Born)/σ Born are presented on Figs. 3–8. As we can see fromthese figures agreement at the one-loop level was found to

Fig. 4 O(α) corrections for basic kinematical distributions fromPYTHIA+PHOTOS and SANC in W → μν decay

be excellent, at the level of 0.01 % for both Z and W decays,once biases due to the technical parameter separating hardand soft photon emission were properly tuned between thetwo calculations.

3.1 Dependence on technical parameters

While performing tests we had to address well known tech-nical problem of the so called “k0 bias”. In case of fixedorder correction implemented into Monte Carlo algorithm, athreshold on energy for emitted photon, typically in the rest-frame of the decaying particle, has to be introduced. It regu-larizes infrared singularity. Below this threshold photons aresimply integrated out and resulting contribution is summedup with virtual corrections to cancel the infrared singular-ity. Unfortunately k0 → 0 limit can not be reached, unless


Fig. 5 O(α) corrections for basic kinematical distributions fromPYTHIA+PHOTOS and SANC in W → eν decay (calo electrons)

one accept working with negative event weights. The de-pendence on the technical parameter k0 is however small forinclusive quantities such as the total cross section. The effectbecomes enhanced on differential distributions near the res-onance peaks. Such an effect can be observed in Fig. 9 forinvariant mass mμμ in Z → μμ(γ ) decay. Generally thisdependence is nowadays of no interest, since in practicalapplication options of the programs with multiple photonemission should be used. This example however may be in-structive for studies of ambiguities in implementation like in[58] where PHOTOS is used for soft photon emission only,while the hard photon phase space is populated with the helpof genuine POWHEG simulation for the final and initial stateemissions simultaneously.

Fig. 6 O(α) corrections for basic kinematical distributions fromPYTHIA+PHOTOS and SANC in Z → ee decay

4 Multiple photon emissions

Let us now turn to the same observables, but calculated withthe multiple photon emission option of SANC and PHOTOS,suitable for the actual comparisons with the data. One cansee from Figs. 10, 11, 12 and 13, that agreement is a bitworse than for the single photon case, but well within theexpected theoretical precision. The relative contribution ofhigher order corrections is defined as δ = (σO(α2+higher) −σO(α))/σ Born, so that it can be directly summed with thefirst order effect considered in the previous section. Oneshould stress that these results represent quantitative com-parisons of different approximations used in the SANC andPHOTOS as well. In fact, the approximations used, con-trary to the single photon case, are not identical. Results of

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Fig. 7 O(α) corrections for basic kinematical distributions fromPYTHIA+PHOTOS and SANC in Z → μμ decay

PHOTOS represent the NLO calculations with exponentia-tion and resummation of the collinear terms of the first or-der photon emission matrix element. Results of SANC usethe collinear leading logarithmic approximation which in-troduces by construction numerical dependence on the cor-responding QED factorization scale μ2. Differences due tonon-optimal choice of the scale μ2 in SANC are below sev-eral permille for differential distributions and below 0.1 %otherwise, see Figs. 10, 11, 12 and 13. Additional effort andcare in estimation of the size of seemingly minor effects maybe required, if further improvements on theoretical preci-sion, beyond 0.1 % are needed.

One should keep in mind that in precision measurementsof LHC experiments unfolding procedure is applied (see e.g.Table 1 and discussion in Sect. 7 of Ref. [65]) to obtain theso called dressed leptons (charged leptons and accompany-

Fig. 8 O(α) corrections for basic kinematical distributions fromPYTHIA+PHOTOS and SANC in Z → ee decay (calo electrons)

ing them collinear photons are recombined into a single ef-fective lepton). In this way the conditions of the Kinoshita–Lee–Nauenberg theorem [59, 60] are fulfilled and the largecorrections proportional to logarithms of the lepton masscancel out. Numerically this effect can be seen from a com-parison of Figs. 6 and 8 for the first order RC. Such a reduc-tion happens with the higher order photonic corrections aswell.

4.1 Comparisons of PHOTOS with KKMC

In Ref. [27] we have demonstrated physics reasons behindvery good, 0.1 % level, agreement between PHOTOS andKKMC results for final-state photonic bremsstrahlung. Stilluntil now, only in case of Z/γ ∗ intermediate state rigoroustests with second order QED matrix element Monte Carlo


Fig. 9 Ratio of invariant massdistributions from SANC andPHOTOS in Z decay as functionof k0 = � = 2Eγ,min/√s

are available, and only in restricted condition of no hadronicactivities in the initial state. For the reference simulationsof qq̄ → Z/γ ∗ → l+l−(nγ ) processes KKMC Monte Carlo[14] was used.5 Also the Born level qq̄ → Z/γ ∗ → l+l−events from KKMC were used for simulations with PHOTOS.For both cases the monochromatic series of events with fixedvirtuality of intermediate Z/γ ∗ state have been generated.This provides source of particularly valuable benchmarks asKKMC is the only program which features exclusive expo-nentiation combined with spin amplitudes for double pho-ton emissions. As the numerical results of such quite exten-sive tests, more than 1000 figures are collected on our webpage [62]: plots for Z → μ+μ−(nγ ) and Z → e+e−(nγ )are presented there.

For all plots of collection [62], before selection cuts areapplied, events are boosted to the laboratory frame assum-ing fixed 4-vector of Z defined by its pZT and η

Z ; the twodimensional grid in (pZT , η

Z) is constructed, with 7 bins inpZT spanning region 0–50 GeV and 4 bins in η

Z spanningfrom 0–2, for each point in the grid 40M events are sim-ulated. Kinematical selection is applied on lepton and pho-ton 4-momenta and kinematical distributions are constructedfrom accepted events. It is required that each lepton hasplT > 20 GeV and |η±| < 2.47, with the gap 1.37 < |η±| <1.52 excluded. The gap in η± corresponds to transition re-gion between central and end-cap calorimeter in ATLASdetector, and is used here to somewhat arbitrary enhancepossible effect of exclusive selection. Then, straightforwardcomparison between electron and muon cases is available;the only difference originating from leptons’ masses. Forphoton |ηγ | < 2.37 and region 1.37 < |ηγ | < 1.52 is ex-cluded again, pγT > 15 GeV is required. In example shownin Fig. 14, angle between photon and a closer lepton is

5This program, at present can not be used with simulation of the wholeprocesses at LHC. An effort in this direction should be mentioned, seeRef. [61] but the corresponding results are not available for us at thismoment.

shown for pZT = 9 GeV and ηZ = 2. Results from KKMCand PHOTOS NLO are compared. In both cases multipho-ton emission is generated in Z/γ ∗ rest frame for uū →Z/γ ∗ → μ+μ−(nγ ) production process, with no initialstate activity of any sort and virtuality of intermediate stateMZ + 6 GeV = 97.187 GeV, where MZ is the Z-bosonmass.

For each choice of pZT and ηZ used to define Z/γ ∗ state

4-momentum a set of three observables: angle between pho-ton and closer lepton, directions of leptons and an overallacceptance rate is monitored in [62]. A general agreementof 0.1 % can be concluded for all distributions. The resultsfor LO restricted PHOTOS are collected there as well.

For all cases one has to bear in mind that overall normal-ization correction factors for cross section, like (1 + 34 απ ) inthe case of Z decay, have to be included when using PHO-TOS package.

4.2 The φ∗η observable

Motivated by [63, 64] and by recent ATLAS publication [65]on precise observable φ∗η , representing important improve-ment for the measurement of Z boson transverse momentum(pZT ) at LHC, we have decided to devote section of this pa-per to discussion on the respective QED FSR corrections.

The measurement of the Z boson transverse momentum(pZT or φ

∗η ) offers a very sensitive way for studying dynam-

ical effects of the strong interaction, complementary to themeasurements of the associated production of bosons withjets. The knowledge of the pZT distribution is crucial also toimprove the modeling of the W boson production needed fora precise measurement of the W mass [6], in particular in thelow pZT region which dominates the cross section. The studyof the low pZT spectrum (p

ZT < MZ), has also an important

implication for the understanding of the Higgs signatures [1]as well as for the New Physics searches at the LHC [66].

The precision of the direct measurement of the spectrumat low pZT at the LHC and at the Tevatron using the Z lep-

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Fig. 10 Higher order corrections for basic kinematical distributionsfrom PYTHIA+PHOTOS and SANC in W → eν decay

tonic decay is limited by systematic uncertainties related tothe knowledge and unfolding of the experiments resolution,in particular lepton energy scale [67, 68].

In recent years, additional observables with better experi-mental resolution and less sensitive to experimental system-atic uncertainties have been investigated [69–72]. The opti-mal experimental observable to probe the low pZT domainof Z/γ ∗ production at hadron colliders was found to be φ∗ηwhich is defined as

φ∗η = tan(φacop/2) · sin(Θ∗η

), (4)

where φacop is an azimuthal opening angle between the twoleptons and the angle Θ∗η is the scattering angle of the lep-tons with respect to the proton beam direction in the rest

Fig. 11 Higher order corrections for basic kinematical distributionsfrom PYTHIA+PHOTOS and SANC in W → μν decay

frame of the dilepton system. The Θ∗η angle is defined as

cos(Θ∗η

) = tanh(

η− − η+2

), (5)

where η− and η+ are the pseudorapiditities of the negativelyand positively charged lepton, respectively. The variable φ∗ηis correlated to the quantity pZT /mll , where mll is the in-variant mass of the lepton pair. It therefore probes the samephysics as the transverse momentum pZT and can be approxi-mately related to it by pZT � MZ ·φ∗η . From the experimentalpoint of view the variable φ∗η relies entirely on the angle re-construction of the leptons in pair production, therefore onthe tracking devices of high precision.

We have studied the theoretical error on φ∗η distributionsdue to photonic bremsstrahlung effects. As in Sect. 4.1 com-


Fig. 12 Higher order corrections for basic kinematical distributionsfrom PYTHIA+PHOTOS and SANC in Z → ee decay

parison of results from PHOTOS and KKMC generators wasperformed and stored on web page [62]. Let us list the ap-propriate cuts and give sample results.

We have requested that for both leptons plT > 20 GeV

and |η±| < 2.4. Distributions of dN(Z→l+l−)dφ∗η

from KKMC

and PHOTOS were monitored. As before, the monochro-matic samples of qq̄ → Z/γ ∗ → l+l−(nγ ) were generatedfor two virtualities: MZ + 6 GeV (97.187 GeV) and MZ − 4GeV (87.187 GeV), for incoming up and down quarks. Gen-erated events were boosted to the laboratory frame assumingfixed pZT and η

Z of intermediate Z/γ ∗ state. Again the gridof 7 bins in pZT spanning region 0–50 GeV and 4 bins in η

Z

spanning from 0–2 was used.In Fig. 15 an example for one point of (pZT , η

Z) grid ischosen and the φ∗η distribution is shown. An agreement sig-

Fig. 13 Higher order corrections for basic kinematical distribu-tions from PYTHIA+PHOTOS and SANC in Z → μμ decay

nificantly better than 0.1 % between KKMC and PHOTOS isobserved. For other choices of flavor of incoming quarks,Z/γ ∗ momentum and virtuality agreement is equally good[62].

4.3 Case of universal kernel

So far, in all numerical results PHOTOS with first order ma-trix elements as available in C++ version were used bothin Z and W decays. If only the universal kernel was used,as available in public FORTRAN PHOTOS version 2.14 orhigher, the loss of precision would be noticeable, but theuncertainty calculated with respect to the total rate wouldremain at 0.2 % level, for photonic bremsstrahlung correc-tions to the shapes of distributions [62]. As in the NLO case,overall normalization factor has to be corrected separately.

of 18 Eur. Phys. J. C (2013) 73:2625

Fig. 14 The distribution of the angle in the laboratory frame betweenphoton and the closer muon, as generated from KKMC (thick line) andPHOTOS (thin line); selection cuts are applied, see the text. The inter-mediate state of virtuality MZ + 6 GeV = 97.187 GeV with the trans-verse momentum pT = 9 GeV and pseudorapidity η = 2 was createdin the uū annihilation. The samples of 40M events were simulated.Rates of events with photon passing selection cuts defined in the text,differ for the two programs by a factor 0.9991. Relatively large (0.1 %)difference to unity is typical for the larger values of pseudorapidity.For LO results (see web page [62] for extended results) the ratio forthe surfaces, is of the same order, for electrons it is closer to 1. Thedistribution would feature plateau if the angle and cuts were calculatedin the rest frame of Z/γ ∗ state

Fig. 15 The distribution of the φ∗η in Z/γ ∗ → μ+μ−(nγ ) as gener-ated from KKMC and PHOTOS; selection cuts are applied, see the text.The intermediate state of virtuality 97.187 GeV with the transverse mo-mentum pT = 9 GeV and pseudorapidity η = 2 decaying into muonpair was created in the uū annihilation. The samples of 40M eventswere used, ratio of the surfaces under distributions is 0.9991. Rela-tively large (0.1 %) difference to unity is typical for the larger value ofpseudorapidity. For LO results (see web page [62] for extended results)the ratio for the surfaces, is of the same order. For electrons, both in LOand NLO cases, this ratio is closer to 1

5 Non-photonic final-state bremsstrahlung

In this section we concentrate on those effects which go be-yond multiple photon emissions. They can be divided intothree groups. Emission of additional pairs, the effect whichcertainly belongs to final-state emissions, effect of interfer-

ence of initial-final-state QED effects and finally all effectswhich are not directly related to final-state radiation, butnonetheless may affect their matrix element calculations.

5.1 Emission of pairs

Emission of light fermion pairs should be included startingfrom the second order of QED, i.e. from the O(α2) cor-rections. There are two classes of diagrams which need tobe taken into account. Emission of real pairs (Fig. 16) andthe corresponding correction to the vertex (Fig. 17). Thesetwo effects cancel each other to a large degree due to theKinoshita–Lee–Nauenberg theorem. The generic size of theeffect can be expected to be of the order of higher order pho-tonic bremsstrahlung corrections discussed so far. Moreover,it is well known from direct calculations in particular casesthat the pair corrections are typically several times smallerthan the photonic bremsstrahlung ones in the same order inα. Let us recall that careful studies of pair radiation effectswere performed at LEP times [39, 73].

In the context of this paper we will estimate theoreticaluncertainty due to neglecting of pair effects in the Drell–Yanobservables.

The SANC integrators allow to perform a quick calcu-lation of the pair corrections within the leading logarith-mic approximation. We apply here the formalism of elec-tron structure (fragmentation) functions [38, 44, 45] whichdescribe radiation in the approximation of collinear kine-matics.

The leading logarithmic approximation (LLA) was ap-plied to take into account the corrections of the ordersO(αnLn), n = 2,3. Let us remind that in the first orderO(α) SANC has the complete calculation. The large log-arithm L = ln(μ2/m2l ) depends on the lepton mass ml andon the factorization scale μ. The latter is taken to be equalto the c.m.s. incoming parton energy (other choices are alsopossible).

The pure photonic contribution to the non-singlet elec-tron fragmentation function in the collinear leading logarith-mic approximation reads:

Dγee(y,L) = δ(1 − y) + α2π (L − 1)P(1)(y)

+ 12

(α

2π(L − 1)

)2P (2)(y)

+ 16

(α

2π(L − 1)

)3P (3)(y) +O(α4L4). (6)

Analytic expressions for the relevant higher order splittingfunctions can be found in Refs. [38, 45].

For numerical evaluations of the splitting functions regu-larized by the plus prescription, we applied the phase space


Fig. 16 A typical example ofreal pair correction

Fig. 17 A typical example of virtual pair correction

slicing as follows:

P (1)(y) =[

1 + y21 − y

]

+

= lim�→0

{δ(1 − y)

(2 ln� + 3

2

)

+ Θ(1 − y − �)1 + y2

1 − y}, (7)

where an auxiliary small parameter � is introduced. In ac-tual computations we used � = 10−4 and verified the inde-pendence of the numerical results from the variations of thisparameter, definition of P i(y), i > 1 is given in Refs. [38,45].

The effect due to emission of real and virtual electron-positron pairs can be estimated using the non-singlet andsinglet pair contributions to the LLA electron fragmentationfunctions:

Dpairee (y,L) =(

α

2π(L − 1)

)2[13P (1)(y) + 1

2Rs(y)

]

+(

α

2π(L − 1)

)3

×[

1

3P (2)(y) + 4

27P (1)(y)

+ 13Rs ⊗ P (1)(y) − 1

9Rs(y)

]+O(α4L4).

(8)

Expressions for the relevant singlet splitting functions Rs

and Rs ⊗ P (1) can be found in Ref. [45].The differential cross section of the neutral current Drell–

Yan process with FSR leading logarithmic corrections takes

the factorized form

d5σ FSRLLA(p + p → X + l+(y1p+

) + l−(y2p−))

= d3σ Born(p + p → X + l+(p+) + l−(p−))

× dy1(Dγ (y1,L) +Dpair(y1,L)

)

× dy2(Dγ (y2,L) +Dpair(y2,L)

). (9)

For consistency we expand the product of the fragmenta-tion functions in α and take in only terms of the orderO(α2L2) and O(α3L3). Terms coming from the product ofthe photonic and pair parts of the fragmentation functionsare treated as a part of pair corrections.

Numerical results for the contribution of pair correctionsare presented in Figs. 18, 19, for W → lν decays.

If further improvement in precision would be required,pair emission can be implemented e.g. into C++ PHOTOSgenerator [29].

We assume that in the experimental analysis, there is nospecific implicit cut rejecting Z → l+l−f f̄ events. Our cuton plT may reject some Z → l+l−f f̄ events, since with thereal f f̄ pair emission, leptons l will have a somewhat softerenergy spectrum. But this require relatively high energy tobe carried out by the f f̄ pair, the dominant triple logarith-mic term will thus cancel between contributions from dia-grams of Figs. 16 and 17. Let us stress that special care isneeded in case of selection cuts sensitive to the soft, smallvirtuality and collinear to primary lepton pairs. Only this re-gion of phase space may contribute significantly to pair cor-rections. Cuts affecting additional leptons of larger energiesare thus of no concern.

The direction of the leptons originating from Z decays(and therefore our φ∗η observable) may be affected by partialreconstruction of two leptons (or two hadronized quarks) ofthe extra pair. Resulting phenomena may be important onlyif the pair f f̄ is not collinear to any of the primary leptonsl. Again this represents a non dominant effect, thus substan-tially below required precision goal of a permille level.

Summarizing, details of pair corrections are still not im-portant for precision tag at the level of 0.1–0.2 %.

5.2 Initial-final-state bremsstrahlung interference

The effect of QED-type interference between spin ampli-tudes for emission from the initial and final states repre-

of 18 Eur. Phys. J. C (2013) 73:2625

Fig. 18 Higher order photonic and pair corrections (δ in %) for basicdistributions from PYTHIA+PHOTOS and SANC in W− → μ−ν̄ decay

sents important, even if numerically small, class of cor-rections. Even if separation of the initial and final-statebremsstrahlung at the spin amplitude level is clear, large in-terference effects may make this separation of limited prac-tical convenience. However in certain approaches interfer-ences can be combined with QED final-state effects in a con-venient way.

Before we will present numerical results, let us recall firstsome details of the discussion from LEP times, see Ref.[73]. The discussion was devoted to energies higher thanresonance peak, that is why interference cancellations as ex-plained e.g. in [74, 75] did not apply. In the case of LHCobservables discussed in this paper, oriented on productionof W and Z resonances, such suppression is nonetheless ex-pected. The reason is of a physical origin; time separationbetween boson production and decay. However, as a conse-

Fig. 19 Higher order photonic and pair corrections (δ in %) for basicdistributions from PYTHIA+PHOTOS and SANC in W− → e−ν̄ decay

quence of the uncertainty principle this suppression can bebroken with strong event selection cuts if these cuts wouldconstrain the final-state energies.

In the studies of Drell–Yan processes at LHC one can re-strict discussion of the interference to the first order only.On the technical level control of the QED O(α) interferencecontribution is realized in the SANC Monte Carlo integra-tor rather simply. The corresponding effect is computed byswitching a respective flag in the code.

For the W decays similar arguments related to intermedi-ate state life-time apply. In this case however, size of cor-rections is calculation scheme dependent, i.e. depends onthe way how diagrams of photon emission off W line aretreated. Studies with SANC demonstrated that the interfer-ence is below 0.1 % for LHC applications. They were com-pleted not only for W but for Z as well, see e.g. [76].


Fig. 20 IFI/FSR ratio in Z decay for φ∗ distribution. For φ∗η > 0.2interference effects become sizable

Let us show in Fig. 20, as an example, corrections frominterference to the φ∗η observable discussed previously. Theeffect is small, below 0.1 % for φ∗η < 0.15 and rises to 0.5 %for φ∗η � 0.3. After combining the interference and FSR cor-rections with QCD parton showers (for this purpose we usedSANC Monte Carlo generator interfaced to Pythia8 pro-gram) the effect becomes significantly smaller (see Fig. 21).This is due to the fact that parton showers strongly affect thedistribution of φ∗η variable, increasing the population of binswith φ∗η > 0.004 by a few orders of magnitude.

Recent measurement from ATLAS collaboration [65] ofφ∗η observable extends to 1.3 but with large so far, statisticalerrors in range φ∗η > 0.2. If precision requirements wouldbecome more demanding, the effect should be included to-gether with the FSR corrections. At present, size of the in-terference effect can be used to estimate the size of the cor-responding theoretical uncertainty due to its omission.

We can conclude that the initial-final-state interferencedoes not represent a problem for separating final-state pho-tonic bremsstrahlung from the remaining electroweak cor-rections in processes of W and Z production at LHC. Thisconclusion is justified for the precision of 0.1 %, but it willhave to be studied in more detail for more exclusive con-figurations, like e.g. larger values of φ∗η distribution or forobservables defined for off resonance peak regions of leptonpair invariant masses.

Fig. 21 IFI/FSR ratio in Z decay for φ∗ distribution combined withparton showers

5.3 Relations with other electroweak and hadronicinteractions

In calculation of final-state radiation matrix element, depen-dence on the direction of incoming quarks is present. How-ever one can see from [27] that such effects due to e.g. ini-tial state interactions, affect numerical results for final-statebremsstrahlung in a minor way, through the term which is initself at the permille level, thus well below present precisionrequirement of 0.1 %.

For decays of narrow width states or when gauge sym-metry can be used to separate phenomena from other partsof the interactions, there are no major difficulties to iden-tify QED effects at the spin amplitude level. In the generalcase, QED FSR can be defined and its systematic error canbe discussed as well. However, if e.g. contribution of dia-grams featuring t-channel exchange of bosons complicatethe separation, discussion of systematic errors of other partsof calculations may become scheme dependent. Our discus-sion on QED FSR corrections only will nonetheless be stilluseful for experimental applications.

6 Summary

We have addressed question of theoretical error for predic-tions of QED final-state bremsstrahlung in decays of W andZ bosons, used in precision measurements at LHC and Teva-tron experiments.

Tests and comparisons of PHOTOS versus SANC pro-grams for final-state photonic bremsstrahlung were per-formed in realistic conditions. Hard processes and initialstate hadronic interactions were simulated with the help ofPYTHIA8 [77] Monte Carlo program, for results with PHO-TOS. For SANC its own set-up was used. Related differencesin electroweak Born-level processes required careful tuninguntil agreement was established.

We have started our discussion with technical tests andresults obtained at the first order. Separation into QED FSR

of 18 Eur. Phys. J. C (2013) 73:2625

and remaining electroweak corrections have been studiedand verified at the spin amplitude level. We have checkedthat in PHOTOS and SANC numerically compatible, downto 0.01 % precision level, schemes of such separation weredefined. This agreement confirms proper installation of ma-trix elements and numerical stability of SANC and PHOTOSas well. Then the comparison was repeated after allowingmulti-photon emission in both programs. An agreement nec-essary to estimate systematic error in implementation of theQED final-state photonic bremsstrahlung for the precisionlevel of 0.1 % was found. This conclusion holds for the de-cays of intermediate states, produced from annihilation oflight quarks, predominantly close to the W and Z resonancepeaks but with tails of the distributions taken into account.The conclusion holds if NLO kernel is active in PHOTOSand for SANC multiphoton option. For PHOTOS with the LOkernel theoretical precision is estimated to be 0.2 %. Thisconclusion is limited to effects resulting from shapes of dis-tributions and for the selection cuts discussed in the paper.In principle, whenever new type of cuts is applied such com-parison needs to be repeated. Effects on normalization haveto be taken into account independently, either as part of gen-uine electroweak corrections (thanks to the proper choiceof μPW in W decays), or as an simple overall factor, like(1 + 34 απ ) in case of the Z decay.

We have estimated the size of the higher orders QED pho-tonic bremsstrahlung corrections using other programs. TheKKMC [14, 15] Monte Carlo program of LEP era, featuringexclusive exponentiation and second order matrix elementfor final-state photonic bremsstrahlung was used for refer-ence results. With the help of this program monochromaticintermediate Z/γ ∗ states of fixed virtuality were producedfrom annihilation of light quarks. This provided interestingtest while grid of predefined values of (pT , η) was popu-lated, in particular for φ∗η observable. The differences versusNLO PHOTOS was found below 0.1 % (0.2 % for PHOTOSkernel restricted to LO only).

Contributions due to interference of the initial and fi-nal state QED radiation were found to be below 0.1 %for selected W and Z observables, as expected from thephysics arguments. Separation of the final-state radiationfrom the remaining electroweak effects is of a practical im-portance as it facilitate phenomenological work. Our calcu-lation schemes are convenient from that point of view. In-terference effect was found to be below required precisionlevel.

We estimate precision level of photonic final-state correc-tions at 0.1 %. With such precision tag separation of QEDFSR from the rest of the process can be used for the sakeof detector studies on final-state leptons. Such detector stud-ies represent also a well defined segment in comparison oftheoretical predictions with the measured data. One excep-tion is φ∗η distribution in region of large φ∗η > 0.15 if ini-tial state parton shower is not taken into account. Already at

φ∗η � 0.3 interference reach 0.5 %. In this region of phasespace spin amplitudes for bremsstrahlung in the initial andfinal states become gradually of comparable size. Emissionof additional pairs was discussed as well and a size of effectwas estimated at 0.1 % level.

We estimate an overall systematic error for FSR imple-mentation in PHOTOS and SANC at 0.2 % (0.3 % for PHO-TOS with LO kernel). Further improvement of precision ispossible, but requires more detailed discussion. Details ofexperimental acceptance have to be taken into account.

At a margin of the discussion we entered investigationof dependence on scheme specific parameters such as elec-tromagnetic factorization scale μ2 or photon energy thresh-old k0 = � used in fixed order simulations. This may be ofsome interest for further studies of uncertainties resultingfrom some choices of matching of FSR with hard processand/or initial state interactions and/or hard emission matrixelements.

Acknowledgements This project is financed in part from funds ofPolish National Science Centre under decisions DEC-2011/03/B/ST2/00220, DEC-2012/04/M/ ST2/00240 and by Russian Foundation forBasic Research, grant No 10-02-01030-a. A.A. is grateful for a finan-cial support to the Dynasty Foundation.

Part of this work devoted to observable of φ∗η angle has been in-spired by the discussion with Lucia di Ciaccio Elzbieta Richter-Wasand other members of ATLAS LAPP-Annecy group, continuous en-couragements and comments on intermediate steps of the work are ac-knowledged. Work was performed in frame of IN2P3 collaboration be-tween Krakow and Annecy. Useful discussion with Ashutosh Kotwalon distributions of importance for benchmarking algorithm in contextof its reliability for the simulations of importance for CDF W massmeasurements is to be mentioned as well.

Open Access This article is distributed under the terms of the Cre-ative Commons Attribution License which permits any use, distribu-tion, and reproduction in any medium, provided the original author(s)and the source are credited.

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QED bremsstrahlung in decays of electroweak bosons IntroductionDescription of the programsSANCPHOTOS

Definitions and results of the first order calculationsDependence on technical parameters

Multiple photon emissionsComparisons of PHOTOS with KKMCThe phi*eta observableCase of universal kernel

Non-photonic final-state bremsstrahlungEmission of pairsInitial-final-state bremsstrahlung interferenceRelations with other electroweak and hadronic interactions

SummaryAcknowledgementsReferences

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