5.14. Approximate Electroweak Corrections

As an alternative to the complete set of NLO EW corrections, methods restricted to the leading effects due to EW loops are available in Sherpa. In particular at energy scales \(Q\) large compared to the masses of the EW gauge bosons, contributions from virtual W- and Z-boson exchange and corresponding collinear real emissions dominate. The leading contributions are Sudakov-type logarithms of the form [Sud56][CC99].

\[\frac{\alpha}{4\pi \sin^2\theta_W}\log^2\left(\frac{Q^2}{M^2_W}\right)\quad\text{and}\quad \frac{\alpha}{4\pi \sin^2\theta_W}\log\left(\frac{Q^2}{M^2_W}\right)\,.\]

The one-loop EW Sudakov approximation, dubbed EWsud, has been developed for general processes in [DP01a][DP01b]. A corresponding automated implementation in the Sherpa framework, applicable to all common event generation modes of Sherpa, including multijet-merged calculations, has been presented in [BN20] and [BNSchonherr+21].

Another available approximation, dubbed EWvirt, was devised in [KLMaierhofer+16]. It comprises exact renormalised NLO EW virtual corrections and integrated approximate real-emission subtraction terms, thereby neglecting in particular hard real-emission contributions. However, both methods qualify for a rather straightforward inclusion of the dominant EW corrections in state-of-the-art matrix-element plus parton-shower simulations.

In the following we will discuss how to enable the calculation of thew EWsud and EWvirt corrections, and what options are available to steer their evaluation, beginning with EWvirt.

5.14.1. EWvirt

One option to enable EWvirt corrections is to use KFACTOR: EWvirt. Note that this only works for LO calculations (both with and without the shower, including MEPSatLO). The EW virtual matrix element must be made available (for all process multiplicities) using a suitable Loop_Generator. The EWvirt correction will then be directly applied to the nominal event weight.

The second option, which is only available for MEPSatNLO, applies the EWvirt correction (and optionally subleading LO corrections) to all QCD NLO multiplities. For this to work, one must use the the following syntax:

ASSOCIATED_CONTRIBUTIONS_VARIATIONS:
- [EW]
- [EW, LO1]
- [EW, LO1, LO2]
- [EW, LO1, LO2, LO3]

Each entry of ASSOCIATED_CONTRIBUTIONS_VARIATIONS defines a variation and the different associated contributions that should be taken into account for the corresponding alternative weight. Note that the respective associated contribution must be listed in the process setting Associated_Contributions.

The additional event weights can then be written into the event output. However, this is currently only supported for HepMC_GenEvent and HepMC_Short with versions >=2.06 and HEPMC_USE_NAMED_WEIGHTS: true. The alternative event weight names are either ASS<contrib>, MULTIASS<contrib>, or EXPASS<contrib> for additive, multiplicative, and exponentiated combinations, correspondingly. See On-the-fly event weight variations for more information on variation weights and the variation weight naming scheme.

5.14.2. EWsud

The EWsud module must be enabled during configuration of Sherpa using the --enable-ewsud switch.

Similar to EWvirt, also with the EWsud corrections there is the option to use it via KFACTOR: EWsud, which will apply the corrections directly to the nominal event weight, or as on-the-fly variations adding the following entry to the list of variations (also cf. On-the-fly event weight variations):

VARIATIONS:
- EWsud

Using the latter, corrections are provided as alternative event weights. The most useful entries of the event weight list are accessed using the keys EWsud and EWsud_Exp. The first is the nominal event weight corrected by the NLL EWsud corrections, while the latter first exponentiates the corrections prior to applying it to the nominal event weight, thus giving a resummed NLL result.

The following configuration snippet shows the options steering the EWsud calculation, along with their default values:

EWSUD:
  THRESHOLD: 5.0
  INCLUDE_SUBLEADING: false
  CLUSTERING_THRESHOLD: 10.0
  • THRESHOLD gives the minimal invariant mass (in units of the W mass) for each external pair of particles \(k\) and \(l\), \(r_{kl}\), defining the high energy limit. If any of the invariant masses is below this value for a given event, then no EWsud correction is calculated.

  • INCLUDE_SUBLEADING determines whether a formally subleading term proportional to \(\log^2(r_{kl} / \hat s)\) is included, where \(\hat s\) is the Mandelstam variable for the partonic process, see [BNSchonherr+21].

  • CLUSTERING_THRESHOLD determines the number of vector boson decay widths, for which a given lepton pair with the right quantum numbers is still allowed to be clustered prior to the calculation of the EWsud correction. For reasoning, see again [BNSchonherr+21].

We next list all possible technical parameters under the scope of EWSUD. They are mostly meant for internal or consistency checks and are advisable only to expert users.

  • RS boolean flag to determine whether or not to apply the EWSudakov corrections to RS type events, defaults to true.

  • CHECK boolean flag to enable/disable internal checks on the logarithmic coefficients for various simple processes. Defaults to false and prevents normal running when set to true, in that it terminates the run after having checked the coefficients.

  • CHECK_KFACTOR Same as CHECK but at the level of KFACTOR.

  • CHECK_LOG_FILE Specify a filename in which to store the result of CHECK, defaults to a null string.

  • CHECKINVARIANTRATIOS boolean flag used to enforce a stricter definition of High Energy Limit, defaults to false.

  • COEFF_REMOVED_LIST list of logarithic coefficients that can be ignored, defaults to empty, meaning that all coefficients are included. The available options are: LSC, Z, SSC, C, Yuk, PR and I. See [BN20] for further details.

  • C_COEFF_IGNORES_VECTOR_BOSONS boolean flag to control whether or not Vector Boson contributions should be included in the calculation of the C coefficient. Defaults to false, and can be used to check the PR logarithms, given that for some procs the contributions to C from vector bosons and the PR coefficients cancel.

  • HIGH_ENERGY_SCHEME different implementations of the High Energy limit conditions. At the moment only Default is fully implemented, all other available options imply that no check is enforced on the configurations, and a contribution is calculated independently on whether or we are in the high energy limit.

  • PRINT_GRAPHS sets the name of the directory where to save graphs associated to processes generated by the EWSudakov calculation. Same as Print_Graphs.