How Can Polarised Cross Sections for Loop-Induced Processes Be Computed with SHERPA?

The extension brings polarisation to loop-induced processes

Polarised templates from parton-shower Monte Carlo simulations are needed for vector boson polarisation measurements, but their accuracy does not yet meet the precision requirements of upcoming measurements. Polarised loop-induced processes had so far not been included in most parton-shower event generators, even though they can contribute significantly to processes of interest because of the large gluon content of the proton.

This work extends the polarisation framework in the Monte Carlo event generator SHERPA to polarised loop-induced processes at fixed LO and at LO matched to a parton shower. The methodology is based on polarised amplitudes from the RECOLA matrix element generator. The paper presents polarisation effects in relevant inclusive, loop-induced processes and their jet-associated counterparts, several of them for the first time, and investigates both the influence of different polarisation definitions and of QCD and QED resummation effects.

Polarised cross sections split a resonant propagator into states

The formalism replaces the numerator of a massive vector boson propagator with a sum over four polarisation states: two transverse, one longitudinal, and a fourth unphysical state that vanishes as the propagator momentum approaches its on-shell value. Using the narrow-width approximation for a resonant s-channel massive vector boson, a matrix element factorises into production and decay amplitudes linked through one polarisation state.

Squaring this factorised amplitude gives a sum of two kinds of terms. The first is the classical approximation in which the intermediate vector boson is in a well-defined polarisation state; the second is the quantum interference between different polarisation states. For multiple resonant intermediate vector bosons, the same construction generalises to a production tensor combined with decay matrices, forming the basis of SHERPA's spin-correlation algorithm. The paper notes that the quantum interference of different polarisation states can often, but not always, be neglected relative to the intrinsic O(Gamma/m) uncertainty of the narrow-width approximation.

RECOLA amplitudes must be rotated into SHERPA's basis

Decay matrices are generated at LO accuracy within SHERPA using COMIX helicity amplitudes in COMIX's polarisation basis. The loop-induced polarised production-amplitude tensor, by contrast, is calculated with RECOLA. Because RECOLA employs its own polarisation basis, a basis transformation is required before its amplitudes can be fed into the polarisation-correlation algorithm.

The calculation starts from helicity amplitudes for the scattering process in RECOLA's native colour-flow decomposition and polarisation basis. The colour-summed and colour-averaged spin-amplitude tensor is then formed, still differential in the helicity indices of all external particles. To keep only the polarisation states of the final-state vector bosons of interest, the tensor is summed and averaged over the helicities of all other external states. The resulting tensor is transformed from RECOLA's native basis into SHERPA's native basis, after which spin amplitudes can be computed for all intermediate vector bosons in all desired reference frames.

Two further implementation details matter for interpretation. Dominant kinematic off-shell effects near the resonance are recovered by adjusting the momenta of the resonant vector boson according to the Breit-Wigner distribution of its off-shell propagator. Phase-space integration channels for the underlying unpolarised production process are generated with AMEGIC based on the corresponding tree-level process. The same methodology can also be applied to loop-induced production-amplitude tensors built from OpenLoops helicity amplitudes, but the required interfaces are still under development and are left to a future publication.

Validation against W+W- agrees within the expected approximation error

The basis rotation of RECOLA's polarisation definition into SHERPA's internal definition is validated using tree-level matrix elements, comparing against equivalent calculations with SHERPA's internal matrix element generator COMIX, both at the level of individual phase-space points and at the level of distributions.

The complete polarised loop-induced cross sections are validated against existing literature results for W+W- production using the setup defined therein. For loop-induced W+W- production at LO with polarisation states defined in the laboratory frame, the fiducial cross section is 13.892(8) fb in SHERPA's narrow-width approximation, compared with 13.8(2) fb for STRIPPER in the double-pole approximation. The unpolarised cross section is 13.543(5) fb in SHERPA and 13.28(3) fb in STRIPPER. The polarisation fractions are 5.3% longitudinal-longitudinal, 13.5% longitudinal-transverse, 13.5% transverse-longitudinal, 72.1% transverse-transverse, and -4.4% interference in SHERPA, against 5.2%, 13.5%, 13.4%, 72.1% and not available in STRIPPER.

The paper reports excellent agreement for both the off-shell computation and the individual polarisation fractions. The unpolarised and polarised cross sections computed in the improved narrow-width approximation differ from the double-pole approximation of the reference calculation, but never by more than the expected intrinsic uncertainty of O(Gamma/m). First results computed with an early incarnation of this implementation are part of a separate study comparing predictions of polarised cross sections for quark- and loop-induced inclusive ZZ production between all available tools, where the results are in very good agreement with the other tools within the accuracy expected from the different approximations applied.

Frame choice reshapes transverse and mixed contributions

For loop-induced inclusive ZZ production, the pure transverse polarisation contribution dominates over the entire phase space for the muon decay angle, the invariant mass of the electron-positron pair and the muon, and the other observables studied. The lepton decay angle is one of the most polarisation-sensitive observables, with very different shapes for the pure transverse contribution, which has a maximum near cos(theta*) = 1, and for the longitudinal contributions, which peak at cos(theta*) = 0.

The polarisation definition has a pronounced effect on the invariant mass distribution. While pure longitudinal polarisation contributions differ only slightly in the peak region between the laboratory and centre-of-mass definitions, pure transverse and mixed polarisation states show substantial differences there. In the laboratory frame definition, the pure transverse polarisation contribution is significantly reduced at the peak, reaching only around 60%, compensated by a high mixed polarisation fraction of around 30%. For high invariant masses, the two definitions of polarisation approach each other. The pure longitudinal contribution increases in this region, while the mixed contribution is reduced in the laboratory frame definition and the pure transverse contribution is reduced in the centre-of-mass definition. Only in the laboratory frame definition does the interference exhibit a non-trivial shape.

The same pattern appears in jet-associated ZZ production. In the bulk of the distribution, the pure transverse contribution is reduced in the laboratory compared with the centre-of-mass definition, featuring a significantly increased mixed-polarisation fraction in this region. The pure longitudinal state and the interference coincide for both polarisation definitions, and the transverse-transverse and mixed contributions defined in the different frames approach each other in the high transverse momentum tails.

Resummation distorts differential fractions more than integrated ones

For loop-induced inclusive ZZ production, the largest shape differences compared with the fixed-order result are caused by QCD radiation and reach up to 60% at large angular separation between the electron and positron and 15% at small muon transverse momentum. While the pure transverse polarisation fraction remains almost unchanged, the pure longitudinal contribution is significantly enhanced around an angular separation of 2.1 and a muon transverse momentum of 70 GeV, and the mixed contribution around an angular separation of 1.5 and a transverse momentum of 120 GeV, as well as for large angular separations. The interference contribution shows large changes at small muon transverse momentum and around an angular separation of 2.9. Additional QED radiation mainly produces an overall cross section reduction that is similar for all polarisation components, in agreement with the integrated result.

The paper's discussion states that resummation effects can distort the polarisation decomposition significantly compared with the fixed-order result, making their inclusion in realistic polarisation templates imperative. This is especially evident for jet-associated processes, where resummation effects are even visible at the level of fiducial polarisation fractions. For inclusive processes, changes in the fiducial fractions remain below 1%, but differential distributions, especially of the longitudinal components, are also distorted substantially.

The size of resummation effects is not uniform across processes. In the jet-associated case studied with strict jet selection criteria, resummation effects on the polarisation fractions remain rather small because those criteria preferentially select fixed-order-like kinematic configurations; among the pure-polarisation components, visible shape changes occur only for the longitudinal contribution, and these modifications are compensated almost entirely by corresponding changes in the interference distribution.

Interference can rival subleading pure polarisation states

The paper reports that interference contributions become significant in certain regions of phase space, reaching magnitudes comparable to, or in some regions larger than, those of the subleading pure-polarisation components. In the azimuthal angle difference between the positron and muon for loop-induced inclusive W+W- production, the interference contribution reaches nearly 10% over broad regions of phase space, becoming the second-dominant contribution.

The choice of reference frame affects not only the longitudinally polarised components but also the interference and transverse-transverse contributions. This is especially apparent in the muon rapidity distribution, where both the transverse-transverse and interference fraction contributions are nearly flat in the centre-of-mass definition but acquire a pronounced shape dependence in the laboratory frame. In the same process, the longitudinal contribution is subdominant but exceeds the mixed contributions over broad regions in the centre-of-mass frame definition.

The level of agreement between on-shell and off-shell calculations also depends strongly on the observable. For loop-induced inclusive Zgamma production, the discrepancy is negligible in the rapidity of the positron-photon system, whereas the dilepton angular separation shows large differences in suppressed phase-space regions: the off-shell distribution falls off substantially faster at both large and small separations compared with the on-shell result.

Frame and resummation choices decide the template

The paper's own priority is that resummation effects can distort the polarisation decomposition significantly compared with the fixed-order result, so their inclusion in realistic polarisation templates is imperative. The evidence for this priority is strongest where the distortion is largest: jet-associated processes, where resummation effects are visible even at the level of fiducial polarisation fractions, and the differential distributions of the longitudinal components in inclusive processes, where fiducial fractions change by less than 1% but shapes are still substantially distorted.

The second decision-relevant split is the reference frame. For loop-induced inclusive ZZ production, the laboratory frame definition reduces the pure transverse contribution at the peak of the invariant mass distribution to around 60% and raises the mixed fraction to around 30%, while the two definitions converge at high invariant masses. For loop-induced inclusive W+W- production, the transverse-transverse and interference fractions are nearly flat in the centre-of-mass definition but acquire a pronounced shape dependence in the laboratory frame.

A reader comparing polarised predictions should therefore check which frame defines the polarisation, whether the calculation is fixed order or matched to a parton shower, and whether the observable is an integrated fiducial fraction or a differential distribution. The paper's validation supports the new implementation for W+W- within the expected O(Gamma/m) approximation error, and its results for jet-associated loop-induced processes allow, for the first time, judgement of higher-order effects on the polarisation decomposition in these processes. The authors also note that the availability of jet-associated loop-induced processes with parton showering will enable multijet-merged samples for higher-order predictions of the inclusive processes, and that public release of the framework and its integration with experimental workflows is intended to facilitate application to real LHC data.

Sources used in this answer