1 Introduction

Event generators [1,2,3,4,5] for scattering amplitudes are indispensable tools for calculating cross sections and understanding event topologies at collider experiments.

At the core of amplitude calculations is the evaluation of the hard scattering matrix element, typically calculated using Feynman diagram techniques as helicity amplitudes [1, 2, 6], i.e., amplitudes with assigned helicities.

While such calculations may well be performed using the full four-dimensional Dirac spinors, simplifications can be achieved using the spinor-helicity [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and Weyl-van der Waerden [1, 23,24,25,26,27,28,29,30,31] formalisms, in which spinors are decomposed into their left- and right-chiral parts which transform separately as different \(SL(2,{\mathbb {C}})\) copies (see e.g. [32,33,34,35,36] for pedagogical introductions). The Dirac spinors are thus split into

for some \(p_1\) and \(p_2\), one of which reduces to p in the ultrarelativistic/massless case, while the other vanishes.Footnote 1

Here the brackets are Weyl spinors transforming under and the brackets are Weyl spinors transforming under .

Since the only invariant tensor for \(SL(2,{\mathbb {C}})\) is the fully antisymmetric Levi-Civita tensor, \(\epsilon ^{12} = -\epsilon ^{21} = \epsilon _{21} = -\epsilon _{12} = 1\), invariant spinor inner products are formed by contractions with this tensor

where we denote etc., for brevity, and where (up to a phase) .

Since these are the only invariant structures at hand, it can be anticipated that all scattering amplitudes should be expressible in terms of these spinor inner products, and that depicting the contraction with a connecting line, one can obtain a “flow” picture for the Lorentz structure.

This flow picture, the chirality-flow formalism is introduced in the next section, along with an illuminating example of how to write down amplitudes. In the subsequent section, Sect. 3, we describe our chirality-flow implementation based on the MadGraph5_aMC@NLO framework [2]. After that, the obtained speed gain is digested in Sect. 4. Finally, concluding remarks and an outlook are given in Sect. 5.

2 Chirality flow

In the chirality-flow formalism [37,38,39,40]   we take the simplifications of the spinor-helicity formalism one step further. By proving that we can recast Feynman rules to be represented in terms of flows between external spinors, we manage to simplify Feynman rules and diagrams to the extent that helicity amplitudes can often be immediately written down given a Feynman diagram.

Introducing graphical flow representations [39] for the external spinors,

we can — in analogy with the color-flow representation of gluons in QCD — obtain a double line representation for external spin-1 particles. Letting \(\epsilon _{ {L}}(p_i,r)\) denote a left-chiral (negative helicity incoming or positive helicity outgoing) photon of momentum \(p_i\) and with gauge reference vector r, and similarly \(\epsilon _{ {R}}(p_i,r)\) denote a right-chiral (positive helicity incoming or negative helicity outgoing) photon, we have

Note that the unphysical reference momentum r is carried by the right-chiral line for a left-chiral photon, and vice versa.

In [39], we proved that we can always use the Fierz identity

on Dirac matrices decomposed into the Pauli matrices,Footnote 2\(\tau ^\mu =\sigma ^\mu /\sqrt{2}\), combined with charge conjugation (see e.g. [33, 35])

to replace a photon (spin-1) propagator by a solid and a dotted line with arrows opposing

The arrow direction, for internal as well as external photons, has to be chosen such that the arrows in the diagram align with each other (rather than oppose each other).

This also enables us to recast the fermion-photon (spin-1) vertex into a simple flow form

The fermion propagator requires some more consideration, but the parts contracted with \(\sigma \) and \({\bar{\sigma }}\) can be represented graphically by

respectively.

Decomposed into massless momenta \(p_i\), with \(p=\sum _i p_i\), \(p_i^2=0\), we have for the first term

and similar for the second term.

Applying these rules, it is possible to directly write down scattering amplitudes, either in terms of slashed momenta or in terms of Lorentz-invariant spinor inner products.

Calculations with Feynman diagrams can then be simplified in an unprecedented manner, making them trivial. To illustrate this, we consider a Feynman diagram relevant for \(e^+_Re^-_L\rightarrow n\) photons, and overlay the chirality-flow representation

Here, for a left-chiral photon \(1_L=p_1\) and \(1_R=r_1\), and vice versa for a right-chiral photon, and we have used the freedom to assign chirality-flow arrows in any consistent direction.

Writing down the amplitude (as an example for photon 1 right-chiral, and photons 2 and 3 left-chiral), either in terms of \(\sigma /\bar{\sigma }\) matrices,

(1)

or with the spinor structure directly expressed in terms of spinor inner products,

(2)

we see that this diagram vanishes if the reference momentum \(r_1=1_L\) is chosen to be \(p_{e^-}\), since , i.e. for a right-chiral photon, this diagram can be chosen to disappear. By picking the gauge vector to be \(p_{e_-}\) for all right-chiral photons, we can make all diagrams with a right-chiral photon attached directly to the electron disappear. Similarly, by letting the reference momentum be \(p_{e^+}\) for all left-chiral photons, diagrams with a left-chiral photon attached next to \(p_{e^+}\) vanish.

If a given assignment of photon chiralities has \(n_L\) left-chiral photons and \(n_R\) right-chiral photons, then we have \(n_L\) non-vanishing ways of placing a photon next to the electron, \(n_R\) ways to place a photon next to the positron, and \((n-2)!\) possible ways to order the remaining photons. This leaves us with \(n_L n_R (n-2)!\), rather than n! diagrams to consider for this chirality assignment, a simplification which turns out to reduce the computation time significantly.

By consistently using this gauge choice, diagram generation can be constructed to recognize any vertices coupling a left-chiral (right-chiral) photon with the right-chiral (left-chiral) fermion as not contributing to the amplitude, and the diagrams can be removed already before compile time. We refer to this process as gauge based diagram removal.

We note that the same simplification could have been achieved within the spinor-helicity or Weyl-van der Waerden formalisms, but with chirality flow it is completely transparent.

3 MadGraph implementation

To test the viability of a numerical implementation of chirality flow, we create a UFO [41] model with chiral particles and vertices, feed this into the software framework MadGraph5_aMC@NLO(MG5aMC) [2] in standalone mode, and repurpose the amplitude evaluations to work within the chirality-flow formalism.

To make the current helicity amplitude evaluation and our implementation as comparable as possible, we follow the structure of MG5aMC wherever possible. We therefore only (1) modify MG5aMC’s diagram generation in order to produce chirality-flow diagrams, and (2) replace the underlying numerical HELAS-like routines generated by ALOHA [42] for calculating off-shell currents and amplitudes with a similar library performing these calculations based on chirality flow.

Although this implementation does not lend itself immediately to all the possible benefits of the chirality-flow formalism, it does make runtime comparisons as fair as possible,Footnote 3 as we are performing the same type of evaluations of the same type of processes using the same type of program. Any advantage in evaluation time will thus be due to simplified calculations (involving smaller Lorentz structures) or a reduced number of evaluations (as for gauge based diagram removal).

The evaluation process performed in our implementation is identical to the MG5aMC version, although with explicitly chiral particles and vertices. As MG5aMC treats different helicity states of a particle as the same type of particle, this means that our implementation runs what in MG5aMC is a helicity state as its own process. If we wish to perform a helicity summed calculation, as in standalone MG5aMC, we need to run each chirality configuration as its own subprocess. This brings about some overhead.Footnote 4

Aside from generating chirality-flow diagrams, rather than standard Feynman diagrams, we also replace the libraries for the matrix element evaluations with a library performing the corresponding calculations using chirality flow. However, as MG5aMC stores particle momenta locally at each vertex evaluation, the decomposition of the fermion propagator momentum is based on Eq. (1) rather than Eq. (2). Changing this and implementing caching of spinor inner products could likely offer additional speed gain.

To validate our implementation, several classes of processes were evaluated at random points in phase space and compared with the same processes evaluated at the same phase space points using default MG5aMC. The processes validated include \(e^+ e^- \rightarrow n \gamma \), \(2\le n \le 4\); \(e^+ e^- \rightarrow \mu ^+ \mu ^- n\gamma \), \(0\le n \le 2\); \(e^+ e^- \rightarrow e^+ e^- n\gamma \), \(0\le n \le 2\); \(e^-\gamma \rightarrow e^- n\gamma \), \(1\le n \le 3\); \(e^+ e^- \rightarrow 2 \mu ^+ 2\mu ^-\); \(e^+ e^- \rightarrow e^+ e^- \mu ^+ \mu ^-\); and \(e^- \mu ^- \rightarrow 2e^- \mu ^- e^+\), for all possible helicity configurations. All amplitudes were found to be equal within numerical precision.

4 Results

Fig. 1
figure 1

Measured runtimes for evaluation of 100,000 matrix elements for MG5aMC and our implementation, as a function of photon multiplicity n. The dotted orange curve depicts the chirality-flow implementation without gauge based diagram removal, whereas the dashed red curve includes gauge based diagram removal. The same number of helicity/chirality configurations are evaluated in all cases. The chirality-flow implementations evaluate matrix elements faster than MG5aMC for all n. For small n the gain can be explained almost fully by simplified calculations, but for larger n the effects of gauge based diagram removal become readily apparent

Full size image

Figure 1 depicts measured runtimes for evaluation of 100,000 matrix elements for the process \(e^+ e^- \rightarrow n\gamma \) as a function of photon multiplicity n (measured on an AMD Ryzen 5 1600 CPU). Three implementations are depicted: MG5aMC (solid blue line), our implementation without gauge based diagram removal (dotted orange line), and our implementation with gauge based diagram removal (dashed red line). These matrix elements were evaluated for phase space points generated by RAMBO [44], all using the same seed. The comparison of the dotted and solid lines shows the improved evaluation speed obtained by performing calculations using the simplified Lorentz structure, whereas the difference between the dotted line and the dashed line is due to gauge based diagram removal.

For this comparison, MG5aMC has been manually set to consider only contributing helicity configurations, and subprocesses where all external photons have the same chirality have been discarded for the implementation without diagram removal.Footnote 5 With this setup, all three program versions will evaluate the same number of matrix elements for the same number of helicity/chirality configurations.

As can be seen in Fig. 1, the chirality-flow implementations perform the evaluations faster than Mad Graph5 for all photon multiplicities n. Additionally, for small n both chirality-flow versions scale better with n than MG5aMC, and chirality flow with gauge based diagram removal maintains this gentler slope with n into the region of large n.

In the small n region, \(n \lesssim 4\), the difference between MG5aMC and chirality flow is explained almost entirely by the simplified Lorentz structures, but as the large n region is approached, \(n > rsim 5\), the benefit of gauge based diagram removal becomes clear. At \(n=6\), chirality flow with diagram removal is roughly a factor 10 faster than MG5aMC.

5 Conclusion and outlook

In previous articles, we have developed chirality flow and shown its benefits for analytic calculations. Here, we have further demonstrated the viability of the chirality-flow formalism in a numerical MG5aMC implementation.

As Fig. 1 demonstrates, our chirality-flow based implementation evaluates matrix elements faster than standalone MG5aMC. This speed increase is due to two different factors. The first is the simplified calculations obtained by performing evaluations of Lorentz structures in the chirality-flow formalism. The second, however, is an additional benefit of chirality flow: the effect of gauge reference vector choice becomes very transparent. Since the effects of a given reference momentum can be seen directly from the chirality-flow diagrams, a good choice is immediately discernible. Combining these two effects, the process \(e^+ e^- \rightarrow n \gamma \) ends up being evaluated roughly ten times faster for \(n \ge 6\) in our implementation, with speed gain increasing for an increasing number of photons.

In this paper, the MG5aMC structure has been maintained for the purpose of comparison. However, the HELAS-based structure used by MG5aMC is not naturally suited for chirality flow, since it evaluates diagrams by calculating off-shell particle wavefunctions at vertices, before combining them to calculate an amplitude. This general structure allows recycling of currents for each diagram where the given current enters.

For chirality flow, another natural object to recycle is the spinor inner products (cf. Eq. (2)). Since the amplitude corresponding to a given chirality-flow diagram can be expressed by a small number of these scalars, caching them is likely to increase the evaluation speed even further in future implementations.

While this implementation has concerned only massless QED, the chirality-flow formalism has been developed for the full massive Standard Model [37], and sizable speed gains could likely be attained for a large class of phenomenologically relevant Standard Model processes.