5.16. 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 [CC99, Sud56].

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

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 the EWSud and EWVirt corrections, and what options are available to steer their evaluation, beginning with EWVirt.

5.16.1. 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 multiplicities. 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. The alternative event weight names are either ASSOCIATED_CONTRIBUTIONS.<contrib>, ASSOCIATED_CONTRIBUTIONS.MULTI<contrib>, or ASSOCIATED_CONTRIBUTIONS.EXP<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.

5.16.2. EWSud

The EWSud module must be enabled during configuration of Sherpa using the -DSHERPA_ENABLE_EWSUD=ON 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.KFactor and EWSud.KFactorExp. 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.

In order for the EWSud corrections to make sense, Goldstone bosons need to be made available. This is achieved by ensuring that the following is set

MODEL: SMGold

Additionally, a coupling order must be set to correctly initialize the couplings for this model, see Processes for more details

PROCESSES:
  ...
  Order{QCD:xx, EW:yy, SMGold: 0}
  ...

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

EWSUD:
  THRESHOLD: 1.0
  INCLUDE_SUBLEADING: true
  CLUSTERING_THRESHOLD: 10.0

  • THRESHOLD . Strictly speaking the EWSudakov corrections are only valid in the high-energy limit, that is where all possible invariant masses, formed by pairing external particles, are much larger than the W mass. In practice, we need to define how much is much larger. The THRESHOLD option, gives the minimal invariant mass (in units of :math:m_W) that each pairing of external particles can have to respect the high energy limit, and below which no EWSudakov correction is computed. To clarify, a large threshold, say for example 10 (10 times the W mass), would result in little to no corrections at all, except for regions of phase-space truly in the high-energy limit. This result is thus only expected to match exact EW corrections only when all invariants are larger than this threshold. Conversely a lower value, say 1, would apply the correction more uniformly at the price of violating the theoretically sound region where these corrections are derived, but is seen to better reproduce the effect of exact EW corrections across kinematical distributions.

  • 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]. Note that depending on the value of THRESHOLD these may become numerically significant. For lower threshold values, it is recommended to leave this option true, as default.

  • INCLUDE_I_PI determines whether to include imaginary part of high energy logs [PZ22]. This can have a significant numerical effect and it should be set to true, as per default. However, together with INCLUDE_SUBLEADING, they should be turned off when testing the coefficients with older references that did not include them.

  • 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 logarithmic 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.

NOTE that at the moment EW Sudakov corrections do not work for processes that feature a four-vector boson vertex, such as a four-gluon vertex.