The Electroweak Sudakov approximation in SHERPA
TThe Electroweak Sudakov approximation in SHERPA
Jennifer M Thompson ∗ II. Physikalisches InstitutUniversity of GoettingenFriedrich-Hund-Platz 137077 GöttingenE-mail: [email protected]
As experimental particle physics becomes more and more precise, it is becoming increasinglyimportant for Monte Carlo simulations to improve the precision of their predictions. In terms ofthe hard matrix element, this means calculating to a higher order in perturbation theory. To beconsistent this requires both NNLO QCD corrections and NLO EW corrections to be included.There are also interference effects between these processes that are not simple to handle consis-tently. For a broad description of the behaviour of NLO EW corrections at high energies, theSudakov logarithmic approach provides a good approximation, and is much less computationallyexpensive than the full calculation. The implementation of EW Sudakov logarithms within theSHERPA program are outlined here along with some initial results. As well as this, an overviewof the status of full NLO EW computations with SHERPA is presented.
XXIV International Workshop on Deep-Inelastic Scattering and Related Subjects11-15 April, 2016DESY Hamburg, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] J un he EW Sudakov approximation in SHERPA Jennifer M Thompson
1. Introduction
Monte Carlo event generators provide an important bridge between collider experiments andthe underlying physics of the SM. They enable low-multiplicity fixed-order calculations to be com-pared against the busy hadronic environment that characterises modern-day experimental particlephysics. This is achieved by dressing the fixed-order calculation with a parton shower and introduc-ing underlying events and hadronisation effects. The current state-of-the art for these predictionsinvolves matching NLO QCD matrix elements to a parton shower. Moving beyond this perturba-tive accuracy requires either performing an NNLO QCD or an NLO EW calculation, both of whichrequire new technology and often contribute a similar order of magnitude to the correction. Therehas been a lot of work on both calculating fixed order NNLO QCD corrections such as [1, 2, 3] andon the matching of NNLO QCD cross sections with a parton shower [4, 5, 6]. However, the workpresented in the following considers the extension of Monte Carlo event generators, specificallythe SHERPA [7, 8] event generator, to include NLO EW corrections. There is a particular focuson the EW Sudakov approximation, the origin of which is outlined and the implementation withinSHERPA briefly explained. Some initial results are presented for a 14 TeV LHC.
2. NLO QCD
To begin a discussion on NLO EW, it is instructive to consider the comparative case of NLOQCD. Calculating the NLO QCD corrections to a process analytically is given by, σ QCDNLO = (cid:90) ( B + V ) d Φ B + (cid:90) Rd Φ R , (2.1)which cannot be easily transferred to a numerical calculation due to the divergence of thevirtual (V) and real (R) integrals on the RHS. The born contribution, B, obviously does not containany such divergences. Each term in eqn. (2.1) is integrated over its appropriate phase-space togive the full NLO cross section, σ QCDNLO . The solution to this problem which is almost universallyadopted is to use a subtraction scheme [9, 10, 11, 12, 13]. This subtracts a quantity, S which exactlymatches the divergent behaviour of the integral over the real emission. This must be analyticallyintegrable over the 1 parton sub-space and not introduce any new divergences [9]. Once theseconditions are met, the subtraction term can be added to the virtual integral. Once it is appropriatelyintegrated it will then also match exactly the divergent structure of the virtual contribution. Withthis introduction, eqn (2.1) becomes a sum of finite integrals, σ QCDNLO = (cid:90) ( B + V + (cid:90) Sd Φ ) d Φ B + (cid:90) ( R − S ) d Φ R . (2.2)
3. NLO EW
There are several differences between an NLO QCD calculation and an NLO EW calculation.One obvious difference is the appearance of masses of the EW gauge bosons, which regulate thedivergences from soft and collinear radiation. The real emission of W ± and Z bosons decay into2 he EW Sudakov approximation in SHERPA Jennifer M Thompson other particles and are, theoretically, distinguishable processes from the underlying Born term. Thisreduces the problem of including real radiation to that of QED, which can be treated in a similarway to the NLO QCD calculation above. Because the real radiation can be, at least to a large extent,classified as a distinct process, the large corrections in the high energy regime, where the mass ofthe weak bosons becomes negligible, are physically meaningful. This high energy regime is thelimit where the Sudakov logarithmic approximation becomes valid.Another difference introduced in NLO EW calculations is the dependence of the EW bosonson the helicity of the particles involved, particularly clear in W ± boson emission. Unlike in QCD,the couplings of the weak bosons are strongly dependent of the helicity of the particle. Further-more, the exchange of weak particles can change the underlying Born term, for example mixingelectrons with neutrinos, and therefore create interferences between previously distinct processes.These differences affect both the NLO EW calculation and the implementation of the EW Sudakovlogarithmic approximation. A full NLO EW calculation is very computationally intensive, whereas the EW Sudakov ap-proximation does not introduce much overhead and is therefore comparatively cheap. It is also eas-ier to include on top of NLO QCD calculations. This approximation considers only the logarithmiccontribution to the correction, and is dominant in the high-energy regime. The EW Sudakov ap-proximation in SHERPA follows an excellent and clear paper by Denner and Pozzorini [14], whichprovides a break down of how to implement Sudakov logarithms in a process-independent way.These logarithms, L , typically take the form L ∼ log (cid:18) sM V (cid:19) , (3.1)which captures the high energy behaviour of the NLO EW calculations. In eqn (3.1), s isthe centre of mass energy, and M V is the mass of the relevant weak boson. There is always anon-logarithmic piece which is neglected, and this creates a limit to the accuracy EW Sudakovlogarithms can reach, typically ∼ i and j and EW bosons with typical mass M ,3 he EW Sudakov approximation in SHERPA Jennifer M Thompson
Figure 1:
Diagrams which lead to mass singularities and contribute to the logarithmic approximation. L = α π (cid:20) A log (cid:18) ( p i + p j ) M (cid:19) + B log (cid:18) ( p i + p j ) M (cid:19)(cid:21) , (3.2)where p i denotes the momentum of leg i . Within the SHERPA framework, the EW Sudakov approximation is included as a K-factor thatis applied at each phase-space point. As is implied by eqn (3.2), it depends only on the final statelegs, and iterates over all possible exchanges of EW bosons. All bosons are assumed to have a massequal to the W ± boson mass. This introduces only a small logarithmic correction for the Z boson,which can be neglected to the order considered here. The mass difference between the photon andthe W ± boson introduces large logarithms, but these largely cancel against real photon radiation.Therefore, the assumption that the EW bosons all have equal mass does not have a significantimpact on the correction. The EW Sudakov approximation only affects the hard process and canbe easily employed with the parton shower or applied to an NLO QCD computation. However, theimplementation does rely on the COMIX [17] matrix element generator.The results shown here are the first results with the implementation of NLO EW Sudakovlogarithms within SHERPA. The calculations are performed for a 14 TeV LHC collider. The left-hand side of fig. 2 shows the effect of including NLO EW Sudakov corrections at 14 TeV in off-shell W ± +jets production. It is clear that as the p T of the leading jet increases, the relative correctionfrom the EW Sudakov logarithms becomes larger and more negative. This reaches almost 40% at1 TeV. The behaviour is similar on the right-hand side of fig. 2, throughout the p T spectrum, whichshows the same distribution but for on-shell production of the W ± boson. Although the NLO EW Sudakov approximation is comparatively quick and easy to includeon top of NLO QCD computations, for some studies, either to a much higher precision or outsideof the high energy regime, a full NLO EW computation must be employed. This faces the samechallenges and subtleties that the NLO EW Sudakov approach dealt with, alongside ambiguities inprocess definition. It must be decided, for example, what counts as a photon emitted in the NLOEW calculation and what is simply radiation from a jet. Also, the difference between NLO QCDcorrections to EW processes and NLO EW corrections to QCD processes must be defined in orderto avoid double counting.Within the SHERPA event generator, there is currently a working interface to the Open-Loops [16] loop provider for the NLO EW virtual amplitude and QED subtraction handled within4 he EW Sudakov approximation in SHERPA
Jennifer M Thompson
LOEW − − − − Off-Shell l + ν j d σ / d p ⊥ , j [ p b / G e V ] . . . . . . . p ⊥ , j [GeV] E W / L O LOEW − − − − On-Shell W + ( → l + ν ) j d σ / d p ⊥ j [ p b / G e V ] . . . . . . . p ⊥ j [GeV] E W / L O Figure 2: p T of the leading jet in off-shell (left-hand side) and on-shell (right-hand side) production of a W ± boson with a jet. SHERPA. There are already publications on NLO EW corrections, including multijet merging,to V +jets [18, 19], both with an on-shell W ± boson and including off-shell effects, however thecode is not yet public. There is also ongoing effort to implement an NLO EW interface betweenSHERPA and Recola [20], as another one-loop provider for both NLO QCD and NLO EW virtualamplitudes.
4. Conclusions
Improving the perturbative accuracy of the hard interaction in Monte Carlo event simulationnow involves the calculation of NLO EW contributions, which are often of a comparable sizeto NNLO QCD. This includes several new challenges, and the full computation is quite time-intensive. EW Sudakov logarithms provide a simple way for the dominant behaviour of the NLOEW calculation to be taken into account without the computational overhead. It is also trivial toinclude on top of QCD corrections, unlike the full calculation where the interference terms mustbe carefully considered to avoid double counting. This is an important complementary approachto be implemented alongside the full calculation. There is also a lot of promising progress in theautomated evaluation of the full NLO EW correction.
Acknowledgments
This work has been supported by the European Commission through the networks MCnetITN(PITN–GA–2012–315877).
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