A Scattering Phase Space Primer

Introduction

There is something that a 100 TeV future hadron collider [1] and a 10 GeV lepton collider [2] have in common. It is the same thing that brings together the Higgs discovery [3] with a nuclear recoil in a Dark Matter search [4]. Under this respect the collision of muon beams [5, 6] is not that different with respect to that of protons dumped on a target [7]. The common denominator is the kinematic phase space of a scattering [8]. Having a full understanding of the kinematic phase space is at least as important as controlling the details of the theory we want to study, whatever it is. In some sense, we can think of a collider experiment (or of a big tank of Xenon exposed to the Dark Matter wind, etc.) as a sophisticated device for the creation of kinematic phase spaces. In this epoch of great ferment for the creation of future colliders [1, 9, 10, 11, 12, 13] it is mandatory that we know every detail of the thousands of phases spaces these machines will produce every second of their activity. Indeed, the BSM theories a collider can probe, the Dark Matter models a detection experiment can exclude or the nuclear interaction hypotheses an experiment can test are almost countless, but the underlying kinematic phase space is one. Improving our knowledge of it benefits all of the above analyses.

The knowledge accumulated on $2 \to n$ phase spaces is vast, but is often given for granted, scattered across the literature, hidden in specialized or decades-old references, and increasingly obscured by the increasing ease with which modern computational tools can generate and integrate them [14, 15, 16]. However knowing basic properties of phase spaces can guide us to the right answer before many complicated and (computationally and economically) expensive simulations. This is particularly true for Vector Boson Fusion scatterings. Their total cross section is bounded only because the masses in the scattering are non zero. The cross section drastically increases when these become particularly small with respect to the center of mass energy, making Monte Carlo event generators unreliable. Using an Equivalent Vector Approximation is an option, but its applicability needs to be verified case by case and is not guaranteed for any value of the masses in the scattering [17].

In particular, this work stems from the necessity of analytically computing the cross section of a non trivial, massive $2 \to3$ scattering, e.g. $e^{+}e^{-} \to e^{+}e^{-}a$, where $a$ is an Axion-Like Particle (ALP). For this reason throughout this work we will use this scattering as an example. The scattering comes in an ALP-Strahlung (Figure 1, left panel) and a Vector Boson Fusion topology (Figure 1, right panel). The latter would be divergent if the electron mass was identically 0. A phenomenological study of this process at Belle II can be found in [18] but the results of this work can be applied to any $2 \to3$ scattering regardless of the Center of Mass energy, the mass and nature of the particles and their interaction. Indeed a fully scalar theory can be a good approximation to capture the almost divergent nature of VBF. We refer the reader to [19] for a quantification of the errors we make in different forms of scatterings approximations (neglecting spins, neglecting $s$-channels, neglecting phase space regions where the matrix element is subdominant, a.k.a. Equivalent Vector Approximation).

Vector (PDF)

Resorting to a Lorentz invariant treatment stems from the failure to compute the analytic cross section for $e^{+}e^{-} \to e^{+}e^{-}a$ in polar coordinates:

We used the four-momenta naming of Figure 1 and collected all constants in $\sigma_0$. We can integrate the phase space of $p_4$ against the four-momentum conservation Dirac $\delta$ distribution

where we explicitly used polar coordinates, we called $\cos\theta_{3,5}=c_{3,5}$, we rotated the azimuthal angle of $p_5$ away and redefined $\sigma_0$. Now

The easiest way of imposing four-momentum conservation is solving it for $\cos\phi_3$ because the argument of the $\delta$ distribution is linear in it. With the definitions

the Dirac $\delta$ distribution is solved by

The cross section becomes

The two $\theta$ functions come from imposing $0 \le\phi_3 \le2\pi$. They define this scattering phase space but are highly non trivial to solve. The jacobian adds to the complexity of this cross section.

Gram Determinants eliminate this complexity. The aim of this work is walking the reader through all the steps to solve eq. (8). Fundamental to this goal was gathering all the necessary information on the Lorentz invariant phase space scattered across old and cryptic literature. We deliberately organize the presentation around definitions, theorems, corollaries, and proofs so to provide explicit reference points, besides section A, to which the reader can return when revisiting a particular result, while allowing different parts of the discussion to be read at different levels of detail. Readers interested primarily in the practical results may skip proofs and derivations, and in general sections 2.3 to 2.5, 3.1 and 3.3.3 can be omitted on a first reading. In § 2 we introduce the language kinematic phase space are written in, Gram Determinants (GDs). We show how to interpret them geometrically and provide a dictionary with our most natural language for scatterings, energies and angles; we identify the scattering boundary as the place where GDs are 0. In § 4 we use these properties to characterize the $2 \to3$ phase space. Once the phase space is known, we can compute cross section distributions on it. This is what we do in section 4. We conclude in § 7.

Gram Determinants

Overview of Gram Determinants

We can trace back the history of the use of the phase space properties for discovery back to Dalitz’s work [20]. He wanted to characterise the $J^P$ of the $K^+$ from the $K^+ \to\pi^+\pi^-\pi^+$ decay. Similar processes have only two degrees of freedom, that can be expressed in a Lorentz invariant form, for example, as invariant masses of two couples of final state particles. Basing on the population of a two dimensional plot of such variables, $J^P$ could be predicted. Dalitz was working in a non relativistic frame, but soon Fabri [21] improved his work by implementing relativistic corrections.

The need for a mathematical grasp on the $1 \to n$ decay phase space was soon fulfilled by Byers and Yang: in [22] they provide the phase space region and density via Gram Determinants. This is the foundational point of the theory of a generic phase space in a Lorentz invariant language, for this reason we summarize their work in section B. However a $1 \to n$ decay has $3n - 7$ degrees of freedom and dealing with a $(3n - 7)$–dimensional phase space density may be cumbersome. Therefore it is worth to specialize to some $n$ and study some phenomenologically relevant phase space distributions. This is the case of [23], that considers $n = 4$. Phase space distributions are provided for trivial matrix elements, emphasizing how symmetry can help us transform some distributions into some others. [23]

also extend the validity of their results to $2 \to4$ scatterings with special resonances. Another specialization of [22] is due to Morrow [24], studying the $2 \to3$ scattering as a generalization of Kibble’s treatment [25]. He implicitly provides 2 dimensional distributions of the phase space (neglecting the matrix element). Kajantie and Lindblom [26] start from Morrow’s work and add to it the explicit analytic form of 1 and 2 dimensional phase space distributions.

Finally, what Byers and Yang express formally for the $1 \to n$ generic decay is made explicit by Poon: in [27] the author derives all the determinantal relations which are necessary for expressing the $1 \to n$ phase space boundary.

The picture that emerges is that scattering phase spaces and their distributions are dense of information. If a scattering includes a BSM particle, model independent kinematic features can be used to recover the new particle mass. The first quantities that have been exploited in this sense are one dimensional kinematic features, namely edges and endpoints in invariant masses distributions [28, 29, 30, 31, 32, 33]. However [34] show that multi-dimensional features can be way more effective: the authors perform a likelihood analysis exploiting the integrable divergence on the phase space boundary of many-body processes and compare it to a quality of fit analysis of classical edges and endpoints of squared masses distributions. They obtain a great improvement on the latter in two directions: i) via multi-dimensional analysis one can determine not only NP mass differences but also the overall scale of masses; ii) the GD-based analysis is viable in case of low signal statistics too, which can likely be the case of BSM Physics.

Another direction for exploiting the dimension 4 Symmetric Gram Determinant¹ $\Delta_{4}$ is found in [35]: they realize that the phase space boundary is both an accumulation surface for events thanks to the integrable divergence and a sharp limit beyond which signal events can not be found any more. To use this information it is necessary to study the fully derivative phase space, whose technical complication is overcome by the means of Voronoi Tessellations [36]. The $\Delta_{4}$ variable can itself be a Lorentz invariant function to leverage as a discovery variable at colliders, as proposed by [37].

Simplifying working assumptions in phase space literature until now has been neglecting masses and considering a constant matrix element. Such very powerful simplifications can capture fundamental aspects of the scattering but can not provide the full picture. If our goal is constraining feeble BSM signals, we must refine this pictures and take masses and matrix elements into account. For the sake of clarity we focus here on the simplest matrix element (pure propagators), more complicated cases being possible to tackle in this formalism but beyond the scope of this project. At the same time we could verify that there is a large ensemble of configurations for which pure propagators matrix elements yield a cross section with a negligible error with respect to the full matrix element, showing that our improvement over the state of the art is also sufficient.

We remark that the results of this work are not limited to specific topologies. In particular, in the past decades $2 \to$ many scatterings were studied, also through Gram Determinants [29, 33, 37], because decay chains initiated by supersymmetric particles have been of particular interest. In general, when dealing with decay chains, a solid strategy is considering subsystem of final state particles and use recursion relations [38] on the phase space of these subsystems. [39] offer a graphical method to apply recursion with $n$ particles in $d$ dimensions. But New Physics is not necessarily hidden in decay chains: especially in this epoch of future high energy machines, $s$-channel like topologies are penalized with respect to VBF [40] so it is imperative to find strategies to analytically describe the VBF too. It is true that in the extreme case of constant matrix element it can be shown that a decay chain framework is superior to a purely Lorentz invariant method if the aim is the computation of the fully integrated phase space: [41] explicitly show that decay chains naturally indicate invariant masses and angles between subsystems as the best suited set of variables to get to fully integrated cross sections because this set has the easiest phase space boundary. However not always can we completely neglect the matrix element and not always the fully integrated phase space is the aim of our computation, simply because that’s not the most powerful discriminatory variable between signal and background: different propagators make different phase space regions orders of magnitudes more important than other and can guide us to the discovery of new particles. In this sense peaking cross section distributions can be way more important than the total phase space therefore it is mandatory to use the set of coordinates that enhances resonances the most. This work introduces two new topologies to the discussion, specifically realized here in Figure 1. This opens the possibility for many BSM models to be discovered thanks to Gram Determinants.

Definitions

A recent reference for most of these properties is given in appendix C from [42]. A more complete but older reference is appendix A from [43].