DCPT/21/86, DESY-21-174, IPPP/21/43, MCNET-21-14, SAGEX-21-33
HEJ 2.1: High-energy Resummation with Vector Bosons and Next-to-Leading Logarithms

In HEJ 2, fixed-order predictions are supplemented with high-energy resummation in the following way. First, a number of leading-order events are generated. Then, for each event, it is determined whether the corresponding matrix element contributes at a logarithmic accuracy that is included in the current state-of-the art HEJ resummation. While HEJ currently includes partial NLL resummation, we restrict the following discussion to LL resummation for the sake of clarity.

To identify event configurations contributing at LL accuracy, we order incoming and outgoing particles according to rapidity, see figure 1. We then draw a leading-order auxiliary diagram with as many $t$-channel gluons as possible. It should be stressed that this diagram only serves to determine the logarithmic order and is never used to compute the actual matrix element. A configuration contributes at LL accuracy iff the number of $t$-channel gluons is maximal, i.e. if no other rapidity ordering of an equivalent set of final-state particles allows a diagram with a larger number of $t$-channel gluon exchanges. By “equivalent” we mean that we do not distinguish between parton flavours, such that all final states shown in figure 1 are treated on the same footing.

If the matrix element of the currently considered event contributes at a logarithmic order that is covered by the resummation, we replace the original event with a number of newly generated events. These new events add further gluon emissions, while preserving the rapidities of the original jets and non-parton particles. Details of the procedure are given in [15]. In order to obtain an all-order resummed prediction, while at the same time retaining leading-order accuracy, the newly generated events are reweighted by a factor

where ${\cal M}^{\text{{\tt HEJ}}}$ is the all-order resummed matrix element in the high-energy limit and ${\cal M}^{\text{{\tt HEJ}}}_{\text{LO}}$ its leading-order truncation. The bar denotes the sum (average) over outgoing (incoming) helicities and colours.

The HEJ matrix elements are the only process-specific ingredient in the resummation procedure. In the following, we recall their general structure and give results for the new processes included in HEJ 2.1.

We are interested in the LL HEJ matrix elements for processes $f_{a}f_{b}\to Xf_{a^{\prime}}\cdot ng\cdot f_{b^{\prime}}$, where $X$ denotes any number of additional colourless outgoing particles. With the exception of $X$, particles are ordered according to rapidity. $f_{a}$ is therefore the incoming parton in the backward direction, with flavour $a$, and $f_{b}$ the parton in the forward direction. The most backward outgoing parton is $f_{a^{\prime}}$ with rapidity $y_{1}$, followed by $n$ gluons with rapidities $y_{2}<\dots<y_{n+1}$ and the most forward outgoing parton $f_{b^{\prime}}$ with rapidity $y_{n+2}$.

In the absence of $X$, the outgoing flavours match the incoming ones, i.e. $a^{\prime}=a,b^{\prime}=b$. This is still the case if a charged lepton-antilepton pair $X=l\bar{l}$ is produced. However, for this process we require a virtual photon or Z boson, which implies that at least one of $f_{a}$ and $f_{b}$ has to be a quark or antiquark. When producing a pair of charged lepton and a neutrino, $X=l\bar{\nu}_{l}$ or $X=\bar{l}\nu_{l}$, a quark or antiquark couples to a virtual W boson. This means that the respective flavour is changed, so that either $a^{\prime}\neq a$ or $b^{\prime}\neq b$.

There are two sources of NLL contributions. First, the matrix elements for LL configurations, discussed in section 2.2, receive NLL corrections. These types of corrections will be considered in future HEJ versions. Second, configurations that do not allow the maximal number of gluonic $t$-channel exchanges according to the discussion in section 2.1 only contribute at higher logarithmic orders. For instance, this is the case for configurations where a gluon is produced outside the rapidity order mandated for LL configurations. Ordering all particles apart from $X$ according to their rapidity, these processes are denoted by $f_{a}f_{b}\to Xgf_{a^{\prime}}\cdot ng\cdot f_{b^{\prime}}$, where $f_{a^{\prime}}$ is an (anti-)quark and $f_{a}f_{b}\to Xf_{a^{\prime}}\cdot ng\cdot f_{b^{\prime}}g$ with an (anti-)quark $f_{b^{\prime}}$. These “unordered gluon” NLL configurations were considered in [4] and are already accounted for in HEJ 2.0.

Studying the production of a W boson together with jets, it is found that further numerically important NLL configurations arise through the production of additional pairs of a quark $q$ and an antiquark $\bar{q}\mkern 1.0mu^{\prime}$. Note that in general the flavours $q$ and $q^{\prime}$ can be different if there is a W boson coupling. We distinguish between “central” production,

and “extremal” production

In either case, the description is independent of the relative rapidity ordering between $q$ and $\bar{q}$. In fact, the squares of the matrix elements for these configurations have the same structure as at LL accuracy, see equation (2). Differences only arise in the Born function ${\cal B}$ and in the expressions for the $t$-channel momenta $q_{i}$. HEJ 2.1 supports resummation of these configurations for pure multijet production ($X$ is absent) and the production of a leptonically decaying W boson together with multiple jets ($X=l\bar{\nu}$ or $X=\bar{l}\nu$).

In order to run ${\tt HEJ}$, we have to generate leading-order input events. Any generator producing event files in the Les Houches Event File (LHEF) [20] format can be used to this end. Here, we use Sherpa [21] with the following run card. The entries are explained in detail in the Sherpa documentation.

Note that we have chosen a minimum jet transverse momentum of $18$ GeV instead of the $20$ GeV listed in table 2. The reason for this is that the HEJ resummation can change the transverse momenta of the jets slightly compared to the input leading-order events. It is therefore recommended to choose a minimum transverse momentum that is at least $10\%$ smaller in the fixed-order generation. It is also prudent to ensure that the final predictions do not change when choosing an even smaller value.

The contribution from events containing such “soft” jets with transverse momenta below the value required in the final analysis is numerically small. This means that a significant amount of computing time can be saved by generating two separate input event samples: one sample with low statistics in which each event contains at least one soft jet, and one large sample without any soft jets. We then apply resummation separately to each of the samples. Since the two samples cover distinct regions of the fixed-order phase space, the final prediction is simply the set of all generated events. For simplicity, we do not consider this optimisation in the following.

With the above Run.dat run card, an event file events_W2j.lhe can be generated by running

in the same directory. The generated events will contain exactly two jets, but it is straightforward to generate higher-multiplicity samples by increasing the required number of jets in FastjetFinder, adding the corresponding number of 93 entries to the Process final state, and adjusting the name of the output file in the EVENT_OUTPUT. HEJ can then be used to merge samples with different multiplicities. Typically, the contribution of higher jet multiplicities in the analysis will be decreasing, and it is often enough to consider at most four or five jets. One can save computing time at the price of sacrificing leading-order accuracy by using the HEJ fixed-order generator (HEJFOG) for high multiplicities. An example is shown in appendix A.

In addition to the fixed-order input, HEJ also requires a configuration file. A template config.yml is included in the HEJ source code. Adopting the current set of parameters from table 2 and enabling resummation for all supported NLL configurations we get

We can now generate HEJ events and calculate the total resummed cross section. If HEJ is not installed, it can still be run using the Docker virtualisation software [22] and the following command.

If a HEJ installation is available, we can instead run

If HEJ was compiled with support for Rivet [23],²²2The HEJ Docker container includes all optional dependencies. the generated events can be forwarded directly to the MC_WJETS analysis by adding the following lines to config_Wjets.yml before running HEJ:

Another possibility is to create an output event file for manual analysis. After compiling HEJ with support for the HepMC 2 [24] format, we can add the following entry to config_Wjets.yml:

We can then run HEJ as before and feed the generated events into the MC_WJETS analysis with

Results for higher jet multiplicities are obtained in the same way after adjusting the file names in the configuration files and run commands. To guarantee statistical independence it is also recommended to change the seed in config_Wjets.yml. To obtain a prediction that is inclusive in jet multiplicity, we combine the Rivet output with³³3In older Rivet versions yodamerge –add can be used instead of yodastack.

Finally,

produces analysis plots. As examples, we show the inclusive $N$-jet cross sections and the invariant mass distribution of the two hardest jets obtained from fixed-order input events with up to four jets in figure 4.