Electroweak superpartner production at 13.6 Tev with Resummino

Abstract

Due to the greater experimental precision expected from the currently ongoing LHC Run 3, equally accurate theoretical predictions are essential. We update the documentation of the Resummino package, a program dedicated to precision cross section calculations for the production of a pair of sleptons, electroweakinos, and leptons in the presence of extra gauge bosons, and for the production of an associated electroweakino-squark or electroweakino-gluino pair. We detail different additions that have been released since the initial version of the program a decade ago, and then use the code to investigate the impact of threshold resummation corrections at the next-to-next-to-leading-logarithmic accuracy. As an illustration of the code we consider the production of pairs of electroweakinos and sleptons at the LHC for centre-of-mass energies ranging up to 13.6 TeV and in simplified model scenarios. We find slightly increased total cross section values, accompanied by a significant decrease of the associated theoretical uncertainties. Furthermore, we explore the dependence of the results on the squark masses.

1 Introduction

The Run 3 data taking period of the LHC has started in 2022 at an unprecedented centre-of-mass energy of $\sqrt{S} ={13.6}\,\textrm{TeV}$. The two main general-purpose LHC experiments ATLAS and CMS are expected to collect an integrated luminosity of about 300 $\hbox {fb}^{-1}$, which will complement the 140 $\hbox {fb}^{-1}$ already collected at 13 TeV during Run 2. This increase in luminosity and centre-of-mass energy will thus make it possible to further explore extensions of the Standard Model (SM) of particle physics, such as supersymmetry (SUSY) and its minimal incarnation dubbed the Minimal Supersymmetric Standard Model (MSSM) [1, 2]. The latter is one of the most appealing options for new physics, as it answers various open questions of the SM and provides explanations to several of its conceptual limitations. These include, among others, the presence of dark matter in the universe, the unification of the strong and electroweak forces at large energy scales, and the stabilisation of the mass of the Higgs boson with respect to radiative corrections.

Experimental searches at the LHC have mainly focused on signatures arising from the production and decay of squarks and gluinos, as these particles can be strongly (and thus generally copiously) produced. For a long time, the signatures of these QCD-sensitive superparticles were consequently expected to be the first visible sign of supersymmetry in LHC data. However, with the associated mass limits being now deeply in the TeV regime, searches for electroweakinos and sleptons received more attention and became equally important. Subsequently, accurate theoretical calculations of signal cross sections and key kinematic distributions for all supersymmetric processes became imperative, and in particular for processes in which at least one non-strongly interacting superpartner is present in the final state.

The Resummino program [3] has been developed in this context. It consists of a public tool computing precision predictions including soft-gluon radiation resummation effects for the production of a pair of sleptons, electroweakinos, and for the associated production of one electroweakino and either one squark or one gluino. Moreover, Resummino can also be used to calculate cross section predictions for the neutral-current or charged-current production of a pair of leptons in the presence of extra gauge bosons. So far, its predictions have been used by both the ATLAS and CMS collaborations in order to extract bounds on sleptons and electroweakinos. In particular, the most stringent constraints on simplified models inspired by the MSSM enforce viable slepton and electroweakino masses to be larger than about 700 GeV and 800–1200 GeV respectively, for a not too heavy lightest SUSY particle (see e.g. [4,5,6]). The exact values of these mass limits depend on the details of the search channels, and bounds can always be evaded by either compressing the particle spectrum (thus increasing the mass of the lightest SUSY state) or reducing the branching ratio in the final state of interest (by allowing numerous potential decay modes for a given SUSY particle).

The production cross section of a pair of electroweakinos at hadron colliders has been studied in numerous renowned works, in which fixed-order predictions at leading order (LO) [7, 8] and next-to-leading order (NLO) [9] have been considered. Further precision was obtained by matching these fixed-order results with either parton showers [10, 11] (NLO + PS) or the threshold resummation of the next-to-leading logarithms [12,13,14] (NLO + NLL). Furthermore, approximate next-to-next-to-leading-order predictions have been recently matched with threshold resummation at the next-to-next-to-leading logarithmic accuracy [15] (aNNLO + NNLL). Similarly, slepton pair production total cross sections are known at LO [8, 16], NLO [9], NLO + PS [11, 17], NLO + NLL [18,19,20] and aNNLO + NNLL [21].

Whereas bounds on viable mass regimes for squarks and gluinos imply that their pair production is now phase-space suppressed, their single production with a typically lighter electroweakino is still relevant and could even provide the best insights on both supersymmetric masses and interactions. Associated signals include in particular the associated production of hard jets with missing transverse energy, a signature well studied at the LHC in the context of searches for dark matter [22, 23]. Associated production rates are known at LO [8] and NLO [24, 25], and the matching of these fixed-order results with threshold resummation has been recently achieved [26, 27]. Finally, fixed-order predictions for charged-current and neutral-current lepton pairs in the presence of additional gauge bosons are known at next-to-next-to-leading-order (NNLO) [28, 29], and matched results at NLO + NLL accuracy are available as well [30, 31].

Resummino takes advantage of these developments of the last few decades. It combines the LO calculations for slepton-pair, electroweakino-pair, associated gluino-electroweakino and squark-electroweakino production available from [32,33,34] with the associated NLO SUSY-QCD corrections obtained in [12, 18, 26, 27], and with the aNNLO QCD corrections of [15, 21]. These fixed-order predictions are next matched with the threshold resummation of soft gluon radiation to all orders and at varied accuracies [13, 15, 19, 21, 26, 27], according to the standard formalism introduced in [35,36,37,38,39,40] or the collinear-improved one of [41,42,43,44]. NLO + NLL implementations of rates associated with lepton pair production in the presence of extra gauge bosons are available as well [30, 31]. In addition, the code can be employed to achieve NLO + NLL cross section computations in which fixed-order predictions are matched with soft gluon resummation in the small transverse-momentum ($p_T$) regime [45, 46] following the formalism of [47,48,49], or jointly at small $p_T$ and close to threshold [50, 51] following the formalism of [52,53,54].

In the remainder of this manuscript, we begin by briefly outlining in Sect. 2 the threshold resummation formalism that is implemented in Resummino, and that is relevant for electroweakino-pair, slepton-pair and squark-electroweakino and gluino-electroweakino total production cross section calculations at the LHC. For interested readers, this short description is further complemented by additional details including self-contained analytical formulas in Appendix A. The installation and running of the Resummino code is described in Sect. 3 and in Appendix B-Appendix D. In Sect. 4 and Appendix E, we make use of Resummino to compute and document for the first time total production rates obtained for the LHC Run 3, operating at a centre-of-mass energy of 13.6 TeV. We consider simplified model scenarios in which all superparticles are decoupled, excepted for those produced in the final state. Our results highlight how theoretical uncertainties are reduced relative to the perturbative order of the fixed-order and resummed component of the matched predictions and how they compare with predictions at a centre-of-mass energy of 13 TeV. Moreover, we explore next-to-minimal scenarios, and discuss the impact of internal squark masses on the predictions for configurations in which squarks are not decoupled but only slightly heavier than the lighter electroweakinos. We summarise our work in Sect. 5.

2 Threshold resummation at aNNLO + NNLL

Within its first public release, the Resummino package was suitable for NLO + NLL resummation calculations in the threshold regime, as well as at small transverse momentum or in both regimes simultaneously. Since then, the implementation of threshold-resummed cross sections has been updated so that corrections up to aNNLO + NNLL could be calculated. For that reason, we provide below a description of the formalism used for threshold resummation up to aNNLO + NNLL, and we refer instead to [3] for details on $p_T$ and joint resummation.

We consider the production of a pair of heavy particles i and j in hadronic collisions through the process $AB\rightarrow ij$ (in which A and B stand for the initial hadrons). Whereas soft and collinear divergences originating from real and virtual corrections cancel in the perturbative expansion of the associated production cross section, logarithmic terms remain due to the different phase spaces inherent in the different ingredients of the calculation [55, 56]. These logarithmic contributions encode the effects of soft and collinear emission from either initial-state or final-state coloured particles. As the partonic energy approaches the production energy threshold, they become large and can hence spoil the convergence of fixed-order calculations.

This can be cured through QCD resummation techniques that rely on dynamical and kinematic factorisation of the cross section to account for soft and collinear radiation to all orders [57]. In practice, these factorisation properties are exploited in Mellin space. The hadronic differential cross section $\textrm{d}\sigma _{AB}/\textrm{d}M^2$ is expressed in terms of the Mellin variable N, that is conjugate to the quantity $\tau =M^2/S$ (with M being the invariant mass of the produced particles in the Born process and S the hadronic centre-of-mass energy). This hadronic cross section can be written as a product of the densities $f_{p/H}$ of parton p in hadron H (in Mellin space, where the corresponding Mellin moments are obtained relative to the momentum fraction x), and of the corresponding partonic cross section $\sigma _{ab\rightarrow ij}$ (whose Mellin moments are derived relative to the variable $z=M^2/s$ with s being the partonic centre-of-mass energy),

$$\begin{aligned}{} & {} M^2 \frac{\textrm{d}\sigma _{AB\rightarrow ij}}{\textrm{d}M^2} (N-1) = \sum _{a,b}\ f_{a/A}(N,\mu _F^2) \nonumber \\{} & {} \quad \times f_{b/B}(N,\mu _F^2)\ \sigma _{ab\rightarrow ij}(N,M^2,\mu _F^2,\mu _R^2). \end{aligned}$$

(1)

Here the logarithms depend on the Mellin variable N and they become large in the large-N limit, and $\mu _F$ and $\mu _R$ stand for the usual factorisation and renormalisation scales.

After accounting for soft-gluon emission to all orders, the partonic cross section can be re-expressed in a closed exponential form scaled by a hard function $\mathcal H$ [35,36,37,38,39,40],

$$\begin{aligned}{} & {} \sigma ^{\text {res.}}_{ab\rightarrow ij}(N, M^2, \mu _F^2, \mu _R^2) = {\mathcal {H}}_{ab \rightarrow ij}(M^2,\mu _F^2, \mu _R^2)\nonumber \\{} & {} \quad \times \exp \Big [ G_{ab\rightarrow ij}(N,M^2,\mu _F^2, \mu _R^2) \Big ]. \end{aligned}$$

(2)

The N-independent hard function ${\mathcal {H}}$ can be written in terms of the LO Mellin-transformed cross section $\sigma ^{(0)}_{ab\rightarrow ij}$ and the hard matching coefficient $C_{ab\rightarrow ij}$,

$$\begin{aligned} \begin{aligned} {\mathcal {H}}_{ab \rightarrow ij}(M^2, \mu _F^2, \mu _R^2) = \sigma ^{(0)}_{ab\rightarrow ij} C_{ab\rightarrow ij} (M^2,\mu _F^2, \mu _R^2), \end{aligned} \end{aligned}$$

(3)

where the coefficient $C_{ab\rightarrow ij}$ can be computed perturbatively,

$$\begin{aligned} {C}_{ab\rightarrow ij} (M^2\!, \mu _F^2, \mu _R^2) \!=\! \sum _{n=0} \left( \frac{\alpha _s}{2\pi }\right) ^{\!\!n}\! {C}_{ab}^{(n)} (M^2\!, \mu _F^2, \mu _R^2). \end{aligned}$$

(4)

The hard matching coefficients ${C}_{ab}^{(n)}$ are then derived from fixed-order predictions in Mellin space at a given order in the strong coupling $\alpha _s$. They correspond to the ratio of the finite N-independent pieces of the $\hbox {N}^n$LO correction terms over the LO cross section,

$$\begin{aligned} {C}_{ab}^{(n)} (M^2, \mu _F^2, \mu _R^2) = \left( \frac{2\pi }{\alpha _s}\right) ^n \left[ \frac{\sigma ^{(n)}_{ab\rightarrow ij}}{\sigma ^{(0)}_{ab\rightarrow ij}}\right] _{\text {N-ind.}} \end{aligned}$$

(5)

The soft and collinear gluon radiation contributions appearing in (2) are included in the so-called Sudakov exponent $G_{ab\rightarrow ij}$. They depend on the quark/gluon nature of the initial state and can be written at NNLL accuracy as a sum of leading logarithmic (LL) terms $G^{(1)}_{ab}$, NLL terms $G^{(2)}_{ab\rightarrow ij}$ and NNLL terms $G^{(3)}_{ab\rightarrow ij}$. This yields

$$\begin{aligned}{} & {} G_{ab\rightarrow ij}(N,M^2, \mu _F^2, \mu _R^2) \approx \underbrace{L\ G^{(1)}_{ab}(N)}_{\text {LL}} \nonumber \\{} & {} \quad + \underbrace{G^{(2)}_{ab\rightarrow ij}(N,M^2,\mu _F^2, \mu _R^2)}_{\text {NLL}} \nonumber \\{} & {} \quad + \underbrace{\alpha _s G^{(3)}_{ab\rightarrow ij}(N,M^2,\mu _F^2, \mu _R^2)}_{\text {NNLL}}, \end{aligned}$$

(6)

with $L=\ln (N e^{\gamma _E})$.

For all processes implemented in Resummino, we provide in Appendix A explicit expressions for the resummation coefficients $G_{ab\rightarrow ij}^{(n)}$ and the hard matching coefficients ${C}_{ab\rightarrow ij}^{(n)}$. In principle, all above expressions should be generalised and include an index referring to a given irreducible colour representation, and the total rate given in (2) should embed a sum over all possible structures emerging from the colour decomposition of the LO partonic cross section in Mellin space [58]. However, all the processes considered, namely the production of a pair of colourless particles (electroweakino-pair and slepton-pair production) and that of the associated production of one colourless and one coloured particle (electroweakino-squark and electroweakino-gluino production), only feature a single colour representation. All analytical expressions therefore simplify, and the colour representation indices can be omitted.

In order to obtain meaningful predictions over the entire phase space, the resummed cross section has to be matched to its fixed-order counterpart. This is achieved by subtracting from the sum of the resummed total rate $\sigma _{ab}^{\text {res.}}$ and the fixed-order one $\sigma _{ab}^{\text {f.o.}}$ their overlap, that is given by the expansion $\sigma _{ab}^\text {exp.}$ of the resummed result at the same order in $\alpha _s$ as that of the fixed-order calculation,

$$\begin{aligned} \sigma _{ab} = \sigma _{ab}^\text {f.o.} + \sigma _{ab}^{\text {res.}} - \sigma _{ab}^\text {exp.}\,. \end{aligned}$$

(7)

In this expression, fixed-order predictions are computed in physical space, whereas the resummed component and its expansion to given order in $\alpha _s$ are calculated in Mellin space. This therefore requires to convolve the associated partonic cross sections with Mellin-transformed parton distribution functions (PDFs). In Resummino, this is achieved by employing the parametrisation of the MSTW collaboration [59] to fit the used PDFs in x-space, and thereby obtain an analytic formula for its expression in Mellin space.¹ Finally, an inverse Mellin transformation has to be performed in order to go back to physical space. This is achieved by choosing a distorted integration contour in the complex plane inspired by the principal value and minimal prescription procedure [64, 65]. Along this contour, the Mellin variable N is parameterised by

$$\begin{aligned} N = C y \ \exp \big [ \pm i \varphi \big ] \quad \text {with}\ \varphi \in [\pi /2, \pi ] \ \text {and}\ y \ge 0, \end{aligned}$$

(8)

so that all singularities that may appear in the inverse transform process are correctly taken care of. The parameter C is indeed chosen so that the poles in the PDF Mellin moments originating from the Regge singularity are on the left of the contour, and that stemming from the Landau pole of the running of $\alpha _s$ is on its right.

3 Running Resummino

3.1 Installation

The Resummino package is a high-energy physics program whose source code is written in C++, that is publicly available for download at https://resummino.hepforge.org/, and that is licensed under the European Union Public Licence v1.1. It can be compiled with the GNU compiler collection (GCC) and a working installation of CMake (version 3.0 or more recent) [66]. It requires as external dependencies the GNU Scientific Library GSL (version 2.0 or more recent) [67] and the Boost package (version 1.70.0 or more recent) [68]. The former allows Resummino to make use of the VEGAS Monte Carlo routines for numerical integration [69], and of the Levenberg–Marquardt algorithm [70, 71] to fit non-linear functions like parton distribution functions [72]. On the other hand, the Boost package is a dependency necessary to read supersymmetric spectra encoded in the SLHA format [73, 74] through the SLHAea library [72]. In addition, Resummino must be linked to LHAPDF (version 6.0 or more recent) [75] for PDF handling, and LoopTools (version 2.15 or more recent) [76] for the evaluation of one-loop integrals.

The code can easily be installed by typing the following commands in a shell, once the Resummino zipped sources have been downloaded:

The last step is optional, and it is only relevant to install the code system-wide. The cmake command can be cast with options dictating the behaviour of Resummino relative to its two external dependencies LHAPDF and LoopTools. The program is equipped with LHAPDF version 6.2.3 and LoopTools version 2.15, and the usage of those built-in libraries can be enforced through the cmake options -DBUILD_LHAPDF=ON (default: OFF) and -DBUILD_LOOPTOOLS=ON (default: ON) respectively. Different versions of these packages can be used by providing information about the path to existing installations through the options -DLHAPDF=PATH and -DLOOPTOOLS=PATH if libraries and headers are installed in the same folder, or through LHAPDF_INCLUDE_DIR/LHAPDF_LIB_DIR and LOOPTOOLS_INCLUDE_DIR/LOOPTOOLS_LIB_DIR if not.

The Resummino binary, called resummino, is located in the folder build/bin. After completion of compilation, users can test their local installation by running (from the installation folder)

The source files of the code are collected in the folder src whose architecture and structure is described in Appendix B. The folder input includes exemplary input files, whereas the folder external contains necessary external packages such as LoopTools and LHAPDF. Finally, the folder scripts provides a set of docker and apptainer scripts related to the usage of portable installations of Resummino.

3.2 Running Resummino and input parameters

Resummino can be run as exemplified at the end of the previous section, by typing in a shell, from the installation folder, the command

The keyword <some-input-file> provides the path to a configuration file indicating what to calculate and how. Details on all the options available to write such an input configuration file are briefly provided in the rest of this subsection, and they are additionally extensively documented in Appendix C. Moreover, an example input file input/resummino.in is shipped with the code, and the output obtained from running the code with this input file is described in Appendix D.

3.2.1 Calculation, process and collider settings

A configuration for Resummino includes a list of equalities fixing the values of certain variables acting on how the code functions. This file begins with a definition of the process to be considered, including details on the collider environment. Users must specify the type of colliding particles (protons or antiprotons), the centre-of-mass energy, and the nature of the final-state particles. This is achieved through the self-explanatory variables collider_type (to be set to proton–proton or proton–antiproton), center_of_mass_energy (to be given in GeV), particle1 and particle2 (provided through their Particle Data Group (PDG) identifiers [77]). For example, the following settings,

would correspond to using Resummino for calculations relevant to LHC proton–proton collisions at a centre-of-mass energy of 13.6 TeV, and leading to the production of a pair of charginos ($pp\rightarrow {\tilde{\chi }}_1^+{\tilde{\chi }}_1^-$ at $\sqrt{S}={13.6}\,\textrm{TeV}$).

Table 1 List of processes included in Resummino, given together with the orders of the different calculations supported by the code (defined as the value of the result variable discussed in the text) and the associated references

Information on the observable to calculate is passed through the variable result, that can be set to total (total cross section according to the threshold resummation formalism), pt or ptj (differential cross section at fixed transverse momentum $p_T$ according the $p_T$ or joint resummation formalism respectively), or m (differential cross section at fixed invariant mass M according to the threshold resummation formalism). Differential calculations make use of the variables pt and M to get the numerical values to employ for $p_T$ and M, respectively, that are provided in GeV. For example,

defines a total cross section calculation, whereas

estimates $\textrm{d}\sigma /\textrm{d}p_T$ for a final-state transverse momentum of 50 GeV, using the joint resummation formalism (including therefore integration upon the invariant mass M). In addition, for the Drell–Yan production of a lepton pair, an invariant-mass cut is required to regularise phase-space integration when total rates are evaluated. Information on such a cut is passed through the parameters Minv_min and Minv_max. The list of calculations supported in Resummino is summarised in Table 1, together with the key references documenting their implementation.

The numerical precision of the performed calculation can be controlled through the input variables precision and max_iters acting on the VEGAS algorithm. The former is related to the relative precision of the numerical integration process, whereas the latter refers to the maximum number of integration iterations (excluding the warm-up phase) allowed before stopping the calculation. The settings

enforces a numerical precision of 1% and imposes that at most five VEGAS iterations are used (even if the desired precision is not reached).

Next, details about the PDF set to be used in LO and higher-order calculations are provided through a few variables. These variables define the name of the LHAPDF set of parton densities to employ (through the self-explanatory variables pdf_lo and pdf_nlo²), and the exact identifier of the set member considered (specified as an integer through the variables pdfset_lo and pdfset_nlo). The cross section values spanned after considering all set members included in a given LHAPDF set (obtained by means of multiple runs of Resummino) then allow for PDF error assessments following standard Hessian or Monte Carlo prescriptions [78]. For instance, including in the input file

enforces the usage of CT14 parton densities [60] for all calculations. The central LO set CT14lo will be used for LO predictions, whereas both higher-order fixed-order predictions and resummed ones will rely on the central NLO set CT14nlo.

As briefly discussed in Sect. 2, Mellin-transformed PDF are obtained from a fit of the PDFs considered to the MSTW parameterisation of [59]. This relies on a logarithmic sampling of Bjorken-x values from a minimum value $x_\textrm{min}$ to 1, and on a fitting method using weighted least squares as achieved by the Levenberg–Marquardt algorithm (handled through the GSL library). The minimum value $x_\textrm{min}$ can be provided through the input variable xmin, and the weights used in the fitting procedure are given through the variables weight_valence, weight_sea and weight_gluon for valence quark, sea quark and gluon PDFs respectively. The default option corresponds to

Finally, the value of the renormalisation and factorisation scales is taken to be $\mu _F = \mu _R = (m_1+m_2)/2$ for the computation of total cross sections and $p_T$ distributions, with $m_1$ and $m_2$ being the masses of the two final-state particles. For invariant-mass distributions, the choice $\mu _F = \mu _R = M$ is adopted instead, unless the FIXED_SCALE flag is turned on in the file src/resummino.cc. As required for the evaluation of scale uncertainties, users have the possibility to multiply these values by specific factors via the input variables mu_f and mu_r. For instance, a calculation corresponding to $\mu _F = (m_1+m_2)/4$ and $\mu _R = (m_1+m_2)$ would be achieved through

3.2.2 Model free parameters

Information on the SUSY spectrum and interactions is passed to Resummino through a standard SLHA [73, 74] file whose path is specified in the input file through the variable slha. In the case of the charged-current or neutral-current Drell–Yan process in the presence of additional gauge bosons, information on the SM electroweak parameters and on the $W'$ and $Z'$ properties is provided through a file encoded following an SLHA-like structure, and whose path is passed through the variable zpwp as described in [31]. An example of such a file and its structure is provided in input/ssm.in.

By default, the electroweak input parameters are determined from the information included in the provided SLHA file.³ For SUSY calculations Resummino makes use of tree-level formulas,

$$\begin{aligned}{} & {} G_F = {1.166379e-5}{\textrm{GeV}^{-2}}, \nonumber \\{} & {} \sin ^2 \theta _W = 1-\frac{M_W^2}{M_Z^2}, \nonumber \\{} & {} g_2 = 2 M_W \sqrt{\sqrt{2} G_F}. \end{aligned}$$

(9)

following the prescription introduced in Prospino [79]. In these expressions, $M_W$ and $M_Z$ stand for the W-boson and Z-boson masses, $G_F$ is the Fermi constant, $g_2$ is the weak coupling constant, and $\theta _w$ is the electroweak mixing angle. All Yukawa couplings of light quarks (for a number of active quark flavours $n_f=5$) are set to zero, and the strong coupling constant $\alpha _s$ is computed using the value returned by LHAPDF. Consequently, Resummino does not use the values of $\alpha _s$ and $\alpha$ included in the SMINPUTS block of the SLHA input file. For computations in the presence of extra gauge bosons, the code uses instead

$$\begin{aligned}{} & {} \sin ^2_{\theta _W} = 1 - \frac{M_W^2}{M_Z^2}, \nonumber \\{} & {} g_1 = \sqrt{4\pi \alpha _\textrm{em}} / \cos \theta _W, \nonumber \\{} & {} g_2 = \sqrt{4\pi \alpha _\textrm{em}} / \sin \theta _W, \end{aligned}$$

(10)

where $g_1$ stands for the hypercharge coupling constant. Users have the possibility to change the electroweak parameter scheme through the different macros implemented in the file src/params.cc. For instance, the flag GAUGE_GAUGECOUPLING can be switched on to use as inputs the values of the SM gauge couplings encoded in the SLHA block GAUGE, or the flag MADGRAPH_COUPLING can be turned on to select the electroweak parameters as in [11].

3.3 The command line interface of Resummino

Several of the configuration settings of the code can be fixed through optional arguments when casting the Resummino executable command in a shell, as in

Available options facilitate, for example, iterations over different PDF sets, calculations at different perturbative orders, or the implementation of similar calculations for various processes. Users do not therefore have to manually modify the input file (generically denoted by <input-file> in the above command), as any parameter value passed through the command line supersedes that included in the input file. The full list of options being detailed in Appendix D, we only briefly introduce them below.

The perturbative order of the calculation can be fixed through the options --lo (or -l, for LO-accurate calculations), --nlo (or -n, for NLO-accurate calculations) and --nll (or -s, for NLO + NLL-accurate calculations), as well as through --nnll (or -z) for aNNLO + NNLL-accurate calculations when available (namely for implemented Drell–Yan-like processes). Casting the Resummino command with the --center_of_mass_energy flag (or -e) followed by a double-precision number allows for a modification of the collider hadronic centre-of-mass energy value (in GeV). Similarly, the $p_T$ and M values (in GeV) relevant for calculations at fixed transverse momentum or fixed invariant mass can be fixed through the two options --transverse-momentum (or -t) and --invariant-mass (or -m), both followed by a double-precision number. Additionally, the PDG identifiers of the final state particles can be modified through the options --particle1 (or -c) and --particle2 (or -d), both followed by an integer.

The evaluation of the theory uncertainties associated with any given calculation can be easily performed through the modification of the PDF set considered and the chosen unphysical scales. This is achieved through the options --pdfset_lo (or -a) and --pdfset_nlo (or -b), both followed by an integer, and --mu_f (or -f) and --mu_r (or -r), both followed by a double-precision number. The former two flags allow users to change the identifier of the PDF member set (within a specific LHAPDF collection of parton densities) used for calculations at LO and beyond, whereas the latter two flags provide multiplicative factors to include when fixing the factorisation and renormalisation scales.

In addition, the code can also be started with the options --version (or -v), --help (or -h), --parameter-log (or -p) followed by a string, and --output (or -o) followed by a string. The first of these displays the Resummino release number to the screen, whereas the second of them prints a help message indicating how to run the code. The third possibility makes the code writing all the parameters inherent to the calculation considered in a file (whose path is provided through the included string), and the last one defines the directory (defined through the included string) in which all files created by the code on run time are stored.

3.4 Docker containers

We provide convenient ‘docker’ containers that allow users to employ Resummino without the need to compile it from its source code. This requires to have either Docker or Singularity/Apptainer available on the system, the latter choice being recommended for secured high-performance computing. In this case, Resummino can be run from the scripts localised in the scripts folder,

These scripts automatically detect a local LHAPDF installation, and then enable the usage of any PDF set available within the system. If no installed version of LHAPDF is found, then Resummino is restricted to make use of the default PDF sets included within the docker images. New PDFs can always be added via the command ‘lhapdf install’, to be cast within a shell. In addition, when it is run from a script, Resummino only accepts input files located in the current directory. Such a behaviour can however be circumvented through modifications of the docker run command’s bound directories, through its usual -v option.

4 Precision predictions for electroweak SUSY processes in simplified models

In this section we provide state-of-the-art cross section predictions for slepton and electroweakino pair production at the LHC Run 3 operating at an increased centre-of-mass energy $\sqrt{S}$ = 13.6 TeV, and we compare our findings with predictions relevant for the LHC Run-2 at $\sqrt{S}$ = 13 TeV. We employ the PDF4LHC21_40 set of parton distribution functions [80], and we provide results together with the associated PDF and scale uncertainties (the latter being obtained with the seven-point method) added in quadrature. Complementary to the figures shown in this section, the complete collection of numerical predictions are shown in the tables of Appendix E.

We consider simplified SUSY models inspired by the MSSM, and we explore several typical scenarios. In the context of slepton pair production, we focus on a new physics configuration in which all SUSY particles are decoupled by setting their masses at 100 TeV, with the exception of a single slepton species that is taken either left-handed (${\tilde{e}}_L$), right-handed (${\tilde{e}}_R$) or maximally mixed (${\tilde{\tau }}_1 = 1/\sqrt{2} \big [{\tilde{\tau }}_L + {\tilde{\tau }}_R\big ]$). This last scenario is representative of models featuring light tau sleptons, as originating from many SUSY scenarios [19].

Electroweakino pair production rates are estimated in similar scenarios, in which all SUSY particles are decoupled with the exception of the lightest electroweakinos [13]. We begin our study with scenarios in which the three lightest electroweakinos are all higgsinos. In a first setup, all higgsinos are taken mass-degenerate and the lightest states are defined by

$$\begin{aligned} {\tilde{\chi }}_{1,2}^0 \sim \frac{1}{\sqrt{2}} \left( {\tilde{H}}^0_u \pm {\tilde{H}}^0_d \right) ,\quad {\tilde{\chi }}_{1}^\pm \sim {\tilde{H}}_{u,d}^\pm . \end{aligned}$$

(11)

Whereas this choice of a real neutralino mixing matrix implies a negative $m(\chi ^0_1)$ eigenvalue, it can always be transformed back to positive a mass eigenvalue through a chiral rotation [73]. In a second setup, we introduce some mass splitting between the three electroweakinos, such a splitting being typical of next-to-minimal electroweakino simplified models studied at the LHC, and that turn out to be more realistic in the light of concrete MSSM scenarios [81]. In this case, we define the three lightest electroweakinos by

$$\begin{aligned} {\tilde{\chi }}_{1,2}^0 \sim -\frac{1}{\sqrt{2}} \left( \mp {\tilde{H}}^0_u + {\tilde{H}}^0_d \right) ,\quad {\tilde{\chi }}_{1}^\pm \sim {\tilde{H}}_{u,d}^\pm . \end{aligned}$$

(12)

Finally, we consider a scenarios in which all lightest electroweakinos are mass-degenerate gauginos,

$$\begin{aligned} {\tilde{\chi }}_{1}^0 \sim i {\tilde{B}},\quad {\tilde{\chi }}_{2}^0 \sim i {\tilde{W}}^3,\quad {\tilde{\chi }}_{1}^\pm \sim {\tilde{W}}^\pm . \end{aligned}$$

(13)

In this last SUSY configuration, we additionally investigate squark mass effects on gaugino pair production. Here, we consider an eight-fold degeneracy of all first-generation and second-generation squarks, the spectrum featuring thus a large number of states reachable at the LHC.

Fig. 1

Total cross sections for slepton pair production at the LHC, operating at centre-of-mass energies $\sqrt{S}$ = 13.6 and 13 TeV (upper insets), shown together with their ratios to the 13 TeV total rates (lower insets) in which combined scale and PDF uncertainties are included. We consider the production of a pair of left-handed sleptons (top), right-handed sleptons (centre) and maximally-mixed sleptons (bottom), and predictions are presented as a function of the slepton mass $m_{{\tilde{\ell }}}$

4.1 Slepton pair production

In Fig. 1 we display aNNLO + NNLL cross section predictions for slepton pair production at the LHC as a function of the slepton mass $m_{{\tilde{\ell }}}$. We consider two centre-of-mass energies fixed to $\sqrt{S}$ = 13.6 TeV (orange) and 13 TeV (blue), and we focus in the upper, central and lower panel of the figure on the respective processes

$$\begin{aligned} p p \rightarrow {\tilde{e}}^+_L {\tilde{e}}^-_L, \quad {\tilde{e}}^+_R {\tilde{e}}^-_R,\quad {\tilde{\tau }}^+_1 {\tilde{\tau }}^-_1. \end{aligned}$$

(14)

In the figures, we restrict the mass range shown to $m_{{\tilde{\ell }}} \lesssim {1}\,\textrm{TeV}$. This corresponds to cross section values larger than 0.01 fb, to which the LHC Run 3 is in principle sensitive as dozens signal events could populate the signal regions of the relevant ATLAS and CMS analyses.⁴

Whereas cross sections for $\sqrt{S}$ = 13 and 13.6 TeV are both shown in the upper insets of the three subfigures, the gain in rate at Run 3 is more visible from the ratio plots presented in their lower insets. This indeed illustrates better how cross section increases ranging up to 20% can be obtained in the three classes of scenarios considered, especially for large slepton masses. As expected from the structure of the slepton couplings to the Z-boson (see e.g. in [32]), left-handed sleptons are more easily produced in high-energy hadronic collisions than their right-handed counterparts that only couple through their hypercharge. Consequently, cross sections corresponding to mixed scenarios lie between the two extreme non-mixing cases for a given slepton mass $m_{{\tilde{\ell }}}$.

The different ratio plots of the lower insets of the subfigures also show the dependence of the theoretical systematic uncertainty bands on the slepton mass. The collider energy upgrade achieved at Run 3 naturally leads to a reduction of the PDF uncertainties as the gain in centre-of-mass energy yields a smaller relevant Bjorken-x regime in which parton distribution functions are better fitted. As shown in table 3 (see Appendix E), scale uncertainties contribute to at most 2% of the combined theoretical uncertainty in the entire mass range probed, regardless of the centre-of-mass energies considered. In contrast, PDF errors vary from 3 to 18% at 13.6 TeV, which must be compared to a variation ranging from 3 to 20% at 13 TeV. For a phenomenological study on the reduction of PDF uncertainties in slepton pair production see [20].

Fig. 2

Total cross sections for mass-degenerate higgsino pair production at the LHC, operating at centre-of-mass energies $\sqrt{S}$ = 13.6 and 13 TeV (top panels), shown together with their ratios to the 13 TeV total rates (bottom panels) in which combined scale and PDF uncertainties are included. We consider the production of associated ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^+$ (upper left) and ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^-$ (upper right) pairs, as well as that of a pair of charginos ${\tilde{\chi }}_1^+{\tilde{\chi }}_1^-$ (lower left) and neutralinos ${\tilde{\chi }}_2^0{\tilde{\chi }}_1^0$ (lower right). Predictions are presented as a function of the electroweakino mass $m_{{\tilde{\chi }}}$

4.2 Higgsino pair production

We now turn to scenarios in which all SUSY particles are decoupled by setting their masses at 100 TeV, with the exception of all higgsino states. We begin with a calculation of aNNLO + NNLL predictions relevant for higgsino pair production in a scenario in which all higgsinos, defined as in Eq. (11), are mass-degenerate, i.e. in which

$$\begin{aligned} m({\tilde{\chi }}^0_{1})=m({\tilde{\chi }}^0_{2})=m({\tilde{\chi }}^\pm _1) \equiv m({\tilde{\chi }}). \end{aligned}$$

(15)

Our results are shown in Fig. 2 for the four processes

$$\begin{aligned} pp \rightarrow {\tilde{\chi }}_1^0{\tilde{\chi }}_1^+, \quad {\tilde{\chi }}_1^0{\tilde{\chi }}_1^-, \quad {\tilde{\chi }}_1^+{\tilde{\chi }}_1^-, \quad {\tilde{\chi }}_2^0{\tilde{\chi }}_1^0, \end{aligned}$$

(16)

since in the case of a degenerate spectrum $\sigma ({\tilde{\chi }}_1^0 {\tilde{\chi }}_1^\pm ) = \sigma ({\tilde{\chi }}_2^0 {\tilde{\chi }}_1^\pm )$. In the results displayed, we restrict the mass range considered to $m_{{\tilde{\chi }}}\lesssim {1.5}\,\textrm{TeV}$, which corresponds to production rates at the LHC larger than 0.01 fb and therefore potentially reachable at Run 3. The largest cross sections are obtained for the charged-current process $pp\rightarrow {\tilde{\chi }}_1^0 {\tilde{\chi }}_1^+$, such an effect originates from a PDF enhancement related to the ratio of valence and sea quarks in the proton and from the structure of the higgsino gauge couplings. This additionally leads to similar neutral-current higgsino production rates ($pp\rightarrow {\tilde{\chi }}_1^+ {\tilde{\chi }}_1^-$ and $pp\rightarrow \tilde{\chi }_2^0 {\tilde{\chi }}_1^0$), and the cross section of the negative charged-current process $pp\rightarrow {\tilde{\chi }}_1^0 {\tilde{\chi }}_1^-$ is then smaller.

Fig. 3

Total cross sections for ${\tilde{\chi }}_2^0{\tilde{\chi }}_1^+$ production at the LHC, operating at centre-of-mass energies $\sqrt{S}$ = 13 TeV (left) and 13.6 TeV (right), in a scenario where all SUSY particles are decoupled with the exception of the non-degenerate produced states. Similar figures can be expected for other higgsino production modes (see Table 5)

As in the slepton case explored in Sect. 4.1, we find an enhancement of total production cross sections at 13.6 TeV relative to those at 13 TeV thanks to the modest gain in phase space. Rates are indeed found to be 10 to 30% larger at $\sqrt{S}$ = 13.6 TeV than at $\sqrt{S}$ = 13 TeV, for low and high electroweakino masses respectively.

Still similarly to the slepton case, theoretical uncertainties get reduced with the increase in centre-of-mass energy. As shown in table 4 (see Appendix E), scale uncertainties negligibly contribute to the total theory errors for both centre-of-mass energies, scale variations indeed leading to errors of about 1–2% for low higgsino masses and lying in the permille range for $m_{{\tilde{\chi }}} > rsim {300}\,\textrm{GeV}$. In contrast, total rates at 13 TeV are plagued with PDF uncertainties varying from a few percent at low masses to more than 20–40% for higgsino masses larger than about 1.2 TeV. The reduction of the average Bjorken-x value inherent to the larger centre-of-mass energy of 13.6 TeV subsequently leads to smaller PDF errors that are found reduced by about a few permille at low masses, to up to 5–7% at large masses. For a phenomenological study on the reduction of PDF uncertainties in higgsino pair production see [14].

We now move on with a second higgsino scenario in which the three lightest higgsino states are defined as in Eq. (12). Moreover, their spectrum is enforced to feature a significant level of compression, so that all three higgsino states exhibit a mass splitting of a few percent⁵. In the following, we impose that

$$\begin{aligned} m(\chi ^\pm _1) = \frac{m(\chi ^0_2) - m(\chi ^0_1)}{2}, \end{aligned}$$

(17)

with all masses being taken positive.

Given these mass relations, we present in Fig. 3 aNNLO + NNLL total cross sections for the process $pp\rightarrow {\tilde{\chi }}_1^0{\tilde{\chi }}_1^+$ at the LHC, for centre-of-mass energies of $\sqrt{S}$ = 13 TeV (left) and 13.6 TeV (right). We consider the mass range in which all higgsinos are lighter than 300 GeV, as this consists of the relevant mass configurations in terms of LHC sensitivity to compressed SUSY higgsino scenarios [82,83,84].

In the figures, associated rates are shown logarithmically through a colour code. In this scheme, other higgsino production modes yield almost identical figures, that we therefore omit for brevity. Numerical results are nevertheless provided for all processes in table 5 (see Appendix E), together with separate scale and PDF uncertainties. For the considered mass range, theoretical systematics are found in very good control, the combined uncertainties being of about 5% for all mass configurations explored. Whereas scale uncertainties decrease from 2–3% in the lightest configurations considered to a few permille for higgsinos of about 200–300 GeV, PDF errors increase from 2–3% in the lightest scenarios to 4–5% in the heavier cases. The combined theory errors are thus similar in size for all scenarios studied.

As for the previous calculations achieved, a cross section increase at 13.6 TeV results from the phase-space enhancement inherent to the increased centre-of-mass energy relative to the 13 TeV case, these findings being numerically testified by the results displayed in table 5. Moreover, a rate hierarchy similar to that observed in the mass-degenerate case is obtained, phase-space effects being minimal for compressed scenarios with a non-degenerate spectrum compared to mass-degenerate scenarios in which all higgsinos have exactly the same mass. The process $pp\rightarrow {\tilde{\chi }}_1^0{\tilde{\chi }}_1^+$ hence dominates, followed by the neutral current modes ($pp\rightarrow {\tilde{\chi }}_1^0{\tilde{\chi }}_2^0$ and $pp\rightarrow {\tilde{\chi }}_1^+{\tilde{\chi }}_1^-$) and finally the charged-current channel $pp\rightarrow {\tilde{\chi }}_1^0{\tilde{\chi }}_1^-$. We remind that such a hierarchy is dictated by the structure of the higgsino gauge couplings to the W and Z bosons, and by the PDF ratio of valence and sea quarks in the proton.

Fig. 4

Total cross sections for wino pair production at the LHC, operating at centre-of-mass energies $\sqrt{S}$ = 13.6 and 13 TeV (upper insets), shown together with their ratios to the 13 TeV total rates (lower insets) in which combined scale and PDF uncertainties are included. We consider the charged-current production of a ${\tilde{\chi }}_2^0{\tilde{\chi }}_1^+$ (top) and ${\tilde{\chi }}_1^0{\tilde{\chi }}_1^-$ (centre) wino pair, as well as the neutral-current production of a ${\tilde{\chi }}_1^+{\tilde{\chi }}_1^-$ pair (bottom). Predictions are presented as a function of the wino mass $m_{{\tilde{\chi }}}$

4.3 Gaugino pair production

We now turn to the analysis of the gaugino scenarios introduced in Eq. (13), in which the lightest electroweakinos consist of bino and wino eigenstates. In our analysis, we consider that the two wino eigenstates are mass-degenerate, with respective masses satisfying

$$\begin{aligned} m({\tilde{\chi }}_2^0) = m({\tilde{\chi }}^\pm _1) \equiv m({\tilde{\chi }}). \end{aligned}$$

(18)

In the following, we then focus on the processes

$$\begin{aligned} pp \rightarrow {\tilde{\chi }}^0_2 {\tilde{\chi }}^+_1, \quad {\tilde{\chi }}^0_2 {\tilde{\chi }}^-_1, \quad {\tilde{\chi }}^+_1 {\tilde{\chi }}^-_1, \end{aligned}$$

(19)

and we present aNNLO + NNLL predictions for the associated total rates in Fig. 4. Other processes are irrelevant as the corresponding cross sections vanish due to the structure of the bino and wino gauge couplings. As in the previous subsections, we consider results at centre-of-mass energies of 13 TeV (blue) and 13.6 TeV (orange) in the upper insets of the figures. This time, however, we display predictions for wino masses ranging up to 2 TeV, the cross sections being much larger than in the higgsino case by virtue of the weak triplet nature of the winos. Consequently, we can expect a better LHC sensitivity to signatures of wino production and decays, due to the machine being capable to naturally probe a larger mass regime.

In accord with parton density effects, the charged-current process $pp\rightarrow {\tilde{\chi }}^0_2 {\tilde{\chi }}^+_1$ dominates for a given wino mass, its rate being a factor of 1.5–3 larger than that of the other charged-current process $pp\rightarrow {\tilde{\chi }}^0_2 {\tilde{\chi }}^-_1$ in the case of lighter and heavier mass setups respectively. Furthermore, total cross sections for the neutral current process $pp\rightarrow {\tilde{\chi }}^+_1 {\tilde{\chi }}^-_1$, mediated by virtual photon and Z-boson exchanges, are usually 1.25–1.5 smaller than rates corresponding to the charged-current mode $pp\rightarrow {\tilde{\chi }}^0_2 {\tilde{\chi }}^+_1$, in which the final state is produced from virtual W-boson exchanges. On the other hand, the increase in cross section observed when the hadronic centre-of-mass energy is modified from 13 to 13.6 TeV can be quite substantial in such mass-degenerate wino scenarios. While for light produced particles the increase is only modest and lies in the 5–10% range for all three processes, it increases with the wino mass $m_{{\tilde{\chi }}}$ and reaches 35–40% for wino masses of about 2 TeV.

Fig. 5

Total cross sections for wino pair production at the LHC, operating at centre-of-mass energies $\sqrt{S}$ = 13 TeV (upper) and 13.6 TeV (lower). We consider the charged-current production of a ${\tilde{\chi }}_2^0{\tilde{\chi }}_1^+$ wino pair, and present predictions as a function of the wino mass $m_{{\tilde{\chi }}}$ and the common first-generation and second-generation squark mass $m_{\tilde{q}}$

In the lower insets of the three subfigures, we display the ratio of the total production rates at 13 and 13.6 TeV to that at 13 TeV, including the combined theory systematic error. We remind that numerical values for the cross sections are presented together with separate scale and PDF uncertainties in table 6 (see Appendix E). For the entire mass range considered, scale uncertainties are under good control. They are about 1–2% at low masses, and then decrease to a few permille for wino masses larger than 300 GeV. In contrast, PDF errors are smaller than 10% for wino masses smaller than about 1 TeV, but quickly increase for heavier mass configurations. In this case, typical Bjorken-x values are large and correspond to phase space regimes in which parton densities are poorly constrained, as already pointed out in [11]. This issue will nevertheless be automatically cured with time. Time will indeed allow the LHC collaborations to collect better-quality SM data at large scales, which will consequently help in reducing the PDF errors.

In order to explore the phenomenological consequences of next-to-minimality, we now consider scenarios in which first-generation and second-generation squarks are mass-degenerate, but not decoupled. We introduce the squark mass parameter $m_{\tilde{q}}$ defined by

$$\begin{aligned} m({\tilde{u}}_{L,R}) = m({\tilde{d}}_{L,R})=m({\tilde{s}}_{L,R})=m({\tilde{c}}_{L,R}) \equiv m_{{\tilde{q}}}, \end{aligned}$$

(20)

that we then vary between 800 GeV and 4 TeV. For illustrative purposes, we focus on the charged-current process

$$\begin{aligned} p p \rightarrow {\tilde{\chi }}^0_2 {\tilde{\chi }}^+_1, \end{aligned}$$

(21)

that gives rise to the largest wino production cross sections for a specific mass spectrum. The discussion and the results below are however applicable to other wino production modes as well. This is further numerically depicted in Table 7 (see Appendix E). The latter indeed includes a complete set of numerical predictions for all wino pair production processes in the presence of not too heavy squarks, once again together with separate information on the total rate values, and the associated scale and PDF uncertainties presented separately.

In Fig. 5 we present aNNLO + NNLO total cross sections for the process of Eq. (21) as a function of the wino and squark masses $m_{{\tilde{\chi }}}$ and $m_{\tilde{q}}$.⁶ Results are presented for collider energies of $\sqrt{S}$ = 13 TeV and 13.6 TeV in the top and bottom row of the figure respectively. In the left subfigures, the rates are represented through a logarithmic colour code, which shows that they exhibit a non-trivial dependence on the squark mass. Due to the destructive interference of s-channel gauge boson exchange diagrams with t/u-channel squark exchange diagrams, the cross section starts by decreasing when the SUSY spectrum is varied from a configuration in which $m_{{\tilde{\chi }}}=m_{\tilde{q}}$ to one with a larger squark mass value still in the vicinity of the same wino mass $m_{{\tilde{\chi }}}$. The rate then gets larger and larger with increasing squark masses (the wino mass being constant), and it finally saturates when squarks decouple.

This feature is further illustrated in the two upper insets of the right subfigures, that display aNNLO + NNLL production cross sections for ${\tilde{\chi }}^0_2 {\tilde{\chi }}^+_1$ production at the LHC as a function of the squark mass for several choices of wino masses. For a given wino mass $m_{{\tilde{\chi }}}$, we observe that the cross section always begins by decreasing before quickly reaching a minimum, and then increases for larger and larger squark masses. In the limit of very heavy squarks, the latter decouple and the rates become independent of the squark properties. They hence solely depend on the wino mass.

In the lower inset of these two right subfigures, we present K-factors defined as the ratio of the most precise aNNLO + NNLL rates to the LO ones for a given mass configuration,

$$\begin{aligned} K = \frac{\sigma _\text {aNNLO + NNLL}}{\sigma _\text {LO}}. \end{aligned}$$

(22)

This further illustrates the squark decoupling at large $m_{\tilde{q}}$ values, the K-factor becoming constant. Moreover, the dependence of the K-factors on the squark mass additionally shows how the minimum of the cross section, for a given wino mass, is shifted by tens of GeV by virtue of the higher-order corrections.

In all the insets included in the right subfigures, we additionally include combined theory uncertainties. In general, those uncertainties, that are numerically given in Table 7, are under good control, with the exception of the heaviest scenarios in which the PDF errors get very large, as already above-mentioned. On the contrary, scale uncertainties always lie in the percent or permille level for all scenarios considered, the error bars being in fact drastically impacted (and reduced) by QCD resummation.

5 Conclusion

We have updated the documentation of the Resummino package dedicated to precision calculations of total rates and invariant-mass and transverse-momentum spectra in the context of slepton and electroweakino pair production, the production of a pair of leptons in the presence of additional gauge bosons, and that of an associated electroweakino-squark or electroweakino-gluino pair. This update of the documentation is motivated by the significant extensions that have been implemented since the initial release of the code a decade ago, and that have never been collected in a single document. Whereas these new features of Resummino have already been used for various phenomenological studies and experimental searches for particles beyond the Standard Model at the LHC, we have taken the opportunity of this paper to provide a new useful illustration of the capabilities of the program. We have computed and tabulated for the first time precision predictions of total rates relevant to searches for sleptons and electroweakinos at the ongoing Run 3 of the LHC, operating at a centre-of-mass energy of $\sqrt{S}={13.6}\,\textrm{TeV}$.

Table 2 Behaviour of aNNLO + NNLL total cross sections for slepton and electroweakino pair production at the LHC, for a centre-of-mass energy of $\sqrt{S}={13.6}\,\textrm{TeV}$. The results are given as a function of the average mass of the final-state particles, and we provide the typical increase in cross section related to predictions at $\sqrt{S}={13}\,\textrm{TeV}$, and information about the size of the theoretical PDF and scale uncertainties inherent to the calculations achieved

We have considered several simplified models inspired by the MSSM in which all superpartners are decoupled, with the exception of a few states whose production at the LHC has been explored. We have presented predictions matching fixed-order calculations at approximate NNLO in QCD (in which SUSY two-loop contributions have been neglected) with predictions including the resummation of soft-gluon radiation at NNLL. In our results, we have put emphasis on the gain obtained from updating the centre-of-mass energy from 13 to 13.6 TeV, and on the theory uncertainties inherent to all calculations achieved. A brief summary of our findings is given in Table 2.

As a rule of thumb valid for all investigated processes, QCD threshold resummation reduces the typical scale dependence to less than 1% at aNNLO + NNLL. These contributions to the combined theory error hence become generally negligible, or at least subleading. For the production of low-mass SUSY states, PDF and scale uncertainties are of the same order of magnitude so that theory systematics are under good control. In our predictions, we have employed the PDF4LHC21 set of parton densities that originates from a combination of variants of the CT18 [61], MSHT20 [62] and NNPDF3.1 [85] global PDF fits. Such a set generally leads to larger PDF uncertainties compared to the individual global sets, particularly at large Bjorken $x > rsim 0.4$ [80]. In this region, yet poorly constrained by experimental data, discrepancies between predictions relying on the different global sets can be significant as the parton density behaviour is mainly driven by the theoretical assumptions made. For instance, recent studies have shown how the recent NNPDF4.0 [63] set predicts (anti)quark distributions falling much faster (slower) with respect to the other sets [86]. Moreover, the sea over valence quark ratio is a determining factor for Drell–Yan-like processes [87], and in turn it also has a strong impact on the determination of the gluon density at high x values and large scale [88]. Predictions for (differential) cross section can therefore vary substantially with the choice of the PDF set, especially in scenarios featuring heavy SUSY particles. This issue will however be naturally fixed with time, as more data gets collected at high scale by the LHC collaborations.

We have observed that the modest increase in centre-of-mass energy at the LHC Run 3 brings a typical increase in the rates associated with colourless SUSY processes of 20% for light spectra to 40% in the heavier cases considered. In addition, we have explored the impact of next-to-minimality following two aspects. First, we have considered the production of a pair of non-degenerate compressed higgsinos, and next we have studied the impact of squark decoupling on wino pair production. Both setups have led to visible effects, so that their impact will have to be considered for non-minimal excursions in the interpretation of future search results.

These findings additionally pave the way to future work. In particular, it will be important to assess the phenomenology of non-minimal model spectra exhibiting several light strongly-interacting and non-strongly-interacting SUSY states. In this case, rarely-studied semi-strong processes (such as squark-electroweakino associated production) could be relevant, and worth to be explored.

The package Resummino and its current release 3.1.2 is ready to achieve related calculations. The source code is publicly available from HEPForge and can be used for aNNLO + NNLL calculations for slepton and electroweakino pair production (as studied in this article), as well as for NLO + NLL calculations for the associated production of a squark-electroweakino and gluino-electroweakino pair. Moreover, non-supersymmetric models featuring additional $W'$ and $Z'$ boson could also be investigated at aNNLO + NNLL accuracy, and in addition to total rates Resummino can be used to calculate invariant-mass and transverse-momentum distributions for all implemented processes (with an NLO + NLL precision).

Appendices

Appendix A: Details on the threshold resummation formulas implemented in Resummino

In this appendix we collect analytic expressions for the exponentiated logarithmic contributions and the hard matching coefficients relevant for all processes supported in Resummino.

The logarithmic coefficients of (6) are given, for a Born process initiated by the initial-state partons a and b by [35,36,37, 40, 41]

$$\begin{aligned} G^{(1)}_{ab}(\lambda )= & {} \sum _{c=a,b} g_c^{(1)}(\lambda ),\nonumber \\ G^{(2)}_{ab \rightarrow ij}(\lambda ,M^2,\mu _F^2,\mu _R^2)= & {} \sum _{c=a,b} g_c^{(2)}(\lambda ,M^2,\mu _F^2,\mu _R^2)\nonumber \\ {}{} & {} + h^{(2)}_{ab\rightarrow ij}(\lambda ),\nonumber \\ G^{(3)}_{ab \rightarrow ij}(\lambda ,M^2,\mu _F^2,\mu _R^2)= & {} \sum _{c=a,b} g_c^{(3)}(\lambda ,M^2,\mu _F^2,\mu _R^2) \nonumber \\ {}{} & {} + h^{(3)}_{ab\rightarrow ij}(\lambda ), \end{aligned}$$

(A.1)

where $\lambda = \alpha _s b_0 L$ with $L = \ln ({\bar{N}})$ and ${\bar{N}} = Ne^{\gamma _E}$, and where $b_0$ is defined from the first coefficient of the QCD beta function. We provide its expression together with that of the next two coefficients of the QCD beta function, that are relevant for the formulas given below. They are normalised according to $b_n=\beta _n/(2\pi )^{n+1}$ [89, 90], which gives

$$\begin{aligned} b_0= & {} \frac{1}{12 \pi } (11 C_A - 2 n_f), \nonumber \\ b_1= & {} \frac{1}{24 \pi ^2} (17 C_A^2 - 5 C_A n_f - 3 C_F n_f), \nonumber \\ b_2= & {} \frac{1}{64 \pi ^3} \bigg (\frac{2857}{54} C_A^3 - \frac{1415}{54} C_A^2 n_f + C_F^2 n_f\nonumber \\{} & {} - \frac{205}{18} C_A C_F n_f + \frac{79}{54} C_A n_f^2 + \frac{11}{9} C_F n_f^2 \bigg ), \end{aligned}$$

(A.2)

with $C_A=N_C=3$, $C_F=(N^2-1)/(2N_C)=4/3$ and the number of active quark flavours $n_f=5$.

The individual process-independent resummation coefficients $g_c^{(n)}$ for quarks ($c=q$) and gluons ($c=g$) only depend on the nature of the initial state. They are given as compact functions of the universal process-independent resummation coefficients $A_c^{(n)}$, the first two of these being written as

$$\begin{aligned} A_c^{(1)}= & {} \ 2 C_c,\nonumber \\ A_c^{(2)}= & {} \ 2 C_c \left( C_A\frac{67}{18} - \frac{\pi ^2}{6} C_A - \frac{5}{9} n_f \right) , \end{aligned}$$

(A.3)

with $C_c=C_F$ for quarks and $C_c=C_A$ for gluons. As in Resummino NNLL precision is only achieved for slepton and electroweakino pair production, the implemented formulas only require the knowledge of the third universal quark coefficient $A_q^{(3)}$,

$$\begin{aligned} A_q^{(3)}= & {} \frac{1}{2} C_F \Bigg [C_A^2 \left( \frac{245}{24} - \frac{67}{9}\zeta _2 + \frac{11}{6}\zeta _3 + \frac{11}{5}\zeta _2^2 \right) \nonumber \\{} & {} \ + C_A n_f \left( \frac{10}{9}\zeta _2 - \frac{7}{3} \zeta _3 - \frac{209}{108} \right) - \frac{n_f^2}{27}\nonumber \\ {}{} & {} \ +C_F n_f \left( 2\zeta _3 - \frac{55}{24} \right) \Bigg ]. \end{aligned}$$

(A.4)

The first two resummation coefficients $g_c^{(n)}$ read

$$\begin{aligned} g_c^{(1)}= & {} \frac{A_c^{(1)}}{4\pi b_0 \lambda } \left[ 2 \lambda + (1- 2 \lambda ) \ln (1- 2\lambda )\right] ,\nonumber \\ g_c^{(2)}= & {} \frac{A_c^{(1)} b_1}{4\pi b_0^3 } \left[ 2 \lambda + \ln (1- 2 \lambda ) + \frac{1}{2} \ln ^2 (1- 2\lambda )\right] \nonumber \\{} & {} - \frac{A_c^{(2)}}{2(2\pi )^2 b_0^2} \left[ 2 \lambda + \ln (1- 2 \lambda )\right] \nonumber \\{} & {} + \frac{A_c^{(1)}}{4\pi b_0} \left[ \ln (1 \!-\! 2 \lambda ) \ln \left( \frac{M^2}{\mu _R^2}\right) \!+\! 2 \lambda \ln \left( \frac{\mu _F^2}{\mu _R^2}\right) \right] ,\nonumber \\ \end{aligned}$$

(A.5)

whereas the third quark coefficient (the only coefficient needed for the processes implemented at aNNLL + NNLL in Resummino) is given by

$$\begin{aligned} g_q^{(3)}= & {} \frac{A_q^{(3)}}{\pi ^3 b_0^2} \frac{\lambda ^2}{1 - 2 \lambda } + \frac{2A_q^{(1)}}{\pi } \zeta _2 \frac{\lambda }{1 - 2 \lambda } \nonumber \\{} & {} - \frac{A_q^{(2)} b_1}{(2 \pi )^2 b_0^3}\ \frac{2 \lambda ^2 + 2 \lambda + \ln (1-2\lambda )}{1 - 2 \lambda }\nonumber \\{} & {} - \frac{A_q^{(2)}}{2 \pi ^2 b_0} \left[ \frac{\lambda }{1 - 2 \lambda }\ln \left( \frac{M^2}{\mu _R^2}\right) - \lambda \ln \left( \frac{\mu _F^2}{\mu _R^2}\right) \right] \nonumber \\{} & {} + \frac{A_q^{(1)} b_1^2}{2 \pi b_0^4}\ \frac{2 \lambda ^2 + 2 \lambda \ln (1-2\lambda ) + \frac{1}{2}\ln ^2(1-2\lambda )}{1 - 2 \lambda } \nonumber \\{} & {} + \frac{A_q^{(1)} b_2}{2 \pi b_0^3} \left[ 2 \lambda + \ln (1-2\lambda ) + \frac{2\lambda ^2}{1 - 2 \lambda }\right] \nonumber \\{} & {} + \frac{A_q^{(1)} b_1}{2 \pi b_0^2} \frac{2 \lambda + \ln (1-2\lambda )}{1 - 2 \lambda }\ \ln \left( \frac{M^2}{\mu _R^2}\right) \nonumber \\{} & {} +\frac{A_q^{(1)}}{2 \pi } \left[ \frac{\lambda }{1 - 2 \lambda }\ln ^2\left( \frac{M^2}{\mu _R^2}\right) - \lambda \ln ^2\left( \frac{\mu _F^2}{\mu _R^2}\right) \right] . \end{aligned}$$

(A.6)

The process dependent term $h^{(2)}_{ab\rightarrow ij}$ appearing in (A.1) can be expressed in terms of the soft anomalous dimension ${\varGamma }_{ab\rightarrow ij}$ associated with the partonic process $ab\rightarrow ij$, from which the Drell–Yan soft anomalous dimension ${\varGamma }^\textrm{DY}_{ab}$ has been subtracted. The coefficient $h^{(2)}_{ab\rightarrow ij}$ hence reads⁷

$$\begin{aligned} h^{(2)}_{ab\rightarrow ij}(\lambda )= & {} \ \frac{\ln (1-2\lambda )}{b_0} \ \frac{\Re ({\varGamma }_{ab\rightarrow ij} - {\varGamma }^\textrm{DY}_{ab})}{\alpha _s}\nonumber \\\equiv & {} \ \frac{\ln (1-2\lambda )}{b_0} \ \frac{\Re (\bar{\varGamma }_{ab\rightarrow ij})}{\alpha _s}, \end{aligned}$$

(A.7)

with

$$\begin{aligned} {\varGamma }^\textrm{DY}_{ab}= & {} \ \frac{\alpha _s}{2\pi } \sum _{k=\{a,b\}} C_k \bigg [1 - \ln (2) - i\pi \nonumber \\{} & {} - \ln \left( \frac{(v_k \cdot n)^2}{|n|^2}\right) \bigg ]. \end{aligned}$$

(A.8)

For Drell–Yan-like processes computed at the aNNLO + NNLL accuracy (namely electroweakino and slepton pair production), the coefficient $h^{(3)}_{ab\rightarrow ij}$ must be included too. It is in this case universal, and it is given by

$$\begin{aligned} h^{(3)}_{ab\rightarrow ij}(\lambda )= & {} - \frac{2C_F}{2 \pi ^2 b_0}\ \frac{\lambda }{1 - 2 \lambda } \bigg [n_f\left( \frac{14}{27}-\frac{2}{3}\zeta _2\right) \nonumber \\{} & {} \ + C_A\left( -\frac{101}{27}+\frac{11}{3}\zeta _2+\frac{7}{2}\zeta _3\right) \bigg ]. \end{aligned}$$

(A.9)

In the next subsections, we collect the process-dependent hard matching coefficients appearing in (3) and (4), expressions for the modified soft anomalous dimension $\bar{\varGamma }_{ab\rightarrow ij}$, as well as the ingredients allowing for the computation of the resummed cross section at a given order in $\alpha _s$, the latter being necessary for the matching of the resummed and fixed-order predictions shown in (7).

1.1 Supersymmetric Drell–Yan-like processes

This subsection collects the formulas relevant for the purely electroweak production of pairs of sleptons and electroweakinos.

These processes do not feature any coloured final-state particle, and they only involve a quark-antiquark initial state at Born level. The associated hard matching coefficients $C^{(n)}_{q\bar{q}}$ are hence identical for all Dell-Yan-like processes, the process dependence being factorised in the Born cross section in (3). The first two $C^{(n)}_{q\bar{q}}$ coefficients can be written in a compact form in Mellin space,

$$\begin{aligned} C^{(0)}_{q\bar{q}}= & {} 1,\nonumber \\ C^{(1)}_{q\bar{q}}= & {} C_F \left[ \frac{4}{3}(\pi ^2 - 6) - 3 \ln \left( \frac{\mu _F^2}{M^2}\right) \right] , \end{aligned}$$

(A.10)

whereas the third coefficient is

$$\begin{aligned} C^{(2)}_{q\bar{q}}= & {} \frac{C_F}{720} \bigg \{ 5(762~n_f - 4605~C_A + 4599~C_F) \nonumber \\{} & {} + 80 (151~C_A - 135~C_F + 2~n_f) \zeta _3\nonumber \\{} & {} + 20 \pi ^2 (188~C_A - 297~C_F - 32~n_f) \nonumber \\{} & {} - 92 \pi ^4 (C_A - 6~C_F) \nonumber \\{} & {} + 180 (11~C_A + 18~C_F - 2~n_f) \ln ^2\left( \frac{\mu _F^2}{M^2}\right) \nonumber \\{} & {} - 160 (11~C_A - 2~n_f)(6 - \pi ^2)\ln \left( \frac{\mu _R^2}{M^2}\right) \nonumber \\{} & {} + 20 \ln \left( \frac{\mu _F^2}{M^2}\right) \bigg [-51~C_A + 837~C_F + 6~n_f\nonumber \\{} & {} - 4 \pi ^2 (11~C_A + 27~C_F - 2~n_f) \nonumber \\{} & {} + (-198~C_A + 36~n_f)\ln \left( \frac{\mu _R^2}{M^2}\right) \nonumber \\{} & {} + 216 (C_A - 2~C_F)\zeta _3\bigg ] \bigg \}. \end{aligned}$$

(A.11)

For SUSY Drell–Yan-like processes, fixed-order predictions are available at aNNLO in QCD. Calculations hence include SM NNLO contributions, but neglect NNLO SUSY-QCD corrections. The latter are however expected to be small, given current bounds on SUSY particles. The matching of the resummed cross sections at NNLL with the aNNLO fixed-order ones is performed by expanding the resummed cross section up to ${{\mathcal {O}}}(\alpha _s^2)$. Starting from the expressions given in (2), (3) and (4), we get

$$\begin{aligned}{} & {} \sigma ^\mathrm{(exp.)}_{ab}(N,M^2,\mu ^2_F, \mu _R^2) =\sigma ^{(0)}_{ab\rightarrow ij}\ C_{ab\rightarrow ij}(M^2,\mu ^2_F, \mu _R^2)\nonumber \\{} & {} \quad \quad \times \exp [G_{ab\rightarrow ij}(N,M^2,\mu ^2_F, \mu _R^2)] \nonumber \\{} & {} \quad = \sigma ^{(0)}_{ab\rightarrow ij} \bigg [1 + \left( \frac{\alpha _s}{2\pi }\right) C_{ab}^{(1)} + \left( \frac{\alpha _s}{2\pi }\right) ^2 C_{ab}^{(2)} +\dots \bigg ]\nonumber \\{} & {} \quad \quad \times \left[ 1 + \left( \frac{\alpha _s}{2\pi }\right) K^{(1)} + \left( \frac{\alpha _s}{2\pi }\right) ^2 K^{(2)} +\dots \right] \nonumber \\{} & {} \quad = \sigma ^{(0)}_{ab}\bigg [1 + \left( \frac{\alpha _s}{2\pi }\right) \left( C_{ab}^{(1)} + K^{(1)}\right) \nonumber \\{} & {} \quad \quad + \left( \frac{\alpha _s}{2\pi }\right) ^2 \left( C_{ab}^{(2)} + K^{(2)} + C_{ab}^{(1)} K^{(1)}\right) + \dots \bigg ]. \end{aligned}$$

(A.12)

In the above expressions, we have omitted the arguments of the various functions from the second equality onward, in order to simplify the notation. The coefficients $K^{(n)}$ of the expanded exponential factor can be organised in powers of the logarithm $L = \ln (Ne^{\gamma _E})$,

$$\begin{aligned} K^{(1)}= & {} K^{(1,1)} L + K^{(1,2)} L^2, \nonumber \\ K^{(2)}= & {} K^{(2,1)} L \!+\! K^{(2,2)} L^2 \!+\! K^{(2,3)} L^3 \!+\! K^{(2,4)} L^4, \end{aligned}$$

(A.13)

with the various N-independent contributions being given by

$$\begin{aligned} K^{(1,1)}= & {} 4 C_F \ln \left( \frac{\mu _F^2}{s}\right) , \nonumber \\ K^{(1,2)}= & {} 4 C_F, \nonumber \\ K^{(2,1)}= & {} -\frac{C_F}{27} \bigg \{56~n_f - 404~C_A + 378~C_A~\zeta _3 \nonumber \\{} & {} + 3\ln \left( \frac{\mu _F^2}{s}\right) \Bigg [20~n_f + 2~C_A (-67 + 3\pi ^2) \nonumber \\{} & {} + 3 (11~C_A - 2~n_f) \left( \ln \left( \frac{\mu _F^2}{\mu _R^2}\right) - \ln \left( \frac{\mu _R^2}{s}\right) \right) \Bigg ] \bigg \}, \nonumber \\ K^{(2,2)}= & {} \frac{2}{9} C_F \Bigg [-10~n_f + 67~C_A - 3~C_A \pi ^2\nonumber \\{} & {} +\! 36~C_F \ln ^2\left( \frac{\mu _F^2}{s}\right) \!+\! (33~C_A \!-\! 6~n_f) \ln \left( \frac{\mu _R^2}{s}\right) \Bigg ], \nonumber \\ K^{(2,3)}= & {} \frac{4}{9} C_F \left[ 11~C_A - 2~n_f + 36~C_F \ln \left( \frac{\mu _F^2}{s}\right) \right] , \nonumber \\ K^{(2,4)}= & {} 8 C_F^2. \end{aligned}$$

(A.14)

1.2 Gluino-electroweakino associated production

The soft anomalous dimension ${{\varGamma }}_{q{\bar{q}}\rightarrow \tilde{g}\tilde{\chi }}$ associated with the process $q{\bar{q}}\rightarrow \tilde{g}\tilde{\chi }$ can be calculated following standard techniques, from eikonal Feynman rules. The corresponding modified soft anomalous dimension $\bar{{\varGamma }}_{q{\bar{q}}\rightarrow \tilde{g}\tilde{\chi }}$ is given, after that the Drell–Yan contribution has been subtracted, by

$$\begin{aligned} \bar{{\varGamma }}_{q{\bar{q}}\rightarrow \tilde{g}\tilde{\chi }}= & {} \frac{\alpha _s}{2\pi } C_A \Bigg [ \ln \left( \frac{m_{\tilde{g}}^2 \!-\! t}{m_{\tilde{g}} \sqrt{s}}\right) \nonumber \\{} & {} + \ln \left( \frac{m_{\tilde{g}}^2 \!-\! u}{m_{\tilde{g}} \sqrt{s}}\right) -1 + i\pi \Bigg ]. \end{aligned}$$

(A.15)

For such a process and predictions at NLO + NLL, the hard matching coefficients can be derived from the collinear remainder originating from the $\textbf{P}$ and $\textbf{K}$ insertion operators defined in the language of dipole subtraction [91,92,93]. This is achieved by restricting these operators to their N-independent pieces, and by adding to the result the finite part of the virtual corrections,

$$\begin{aligned}{} & {} {\mathcal {C}}^{(0)}_{q{\bar{q}}} = 1,\nonumber \\{} & {} \frac{\alpha _s}{2\pi } {\mathcal {C}}^{(1)}_{q{\bar{q}}} \!=\! 2 \Big \langle \textbf{P} \!+\! \textbf{K}\Big \rangle _{q, N\text {-ind.}} \!+\! \Big (\sigma ^\text {V} \!+\! \int _1 \textrm{d}\sigma ^\text {A} \Big ). \end{aligned}$$

(A.16)

In this expression, the arguments of the various functions are omitted for brevity. Moreover, $\sigma ^V$ stands for the virtual corrections contributing at NLO, and $\sigma ^A$ consists of the auxiliary cross section shifting the divergences so that integration over phase space can be numerically achieved. Explicit expressions for the insertion operators $\textbf{P}$ and $\textbf{K}$ for an initial quark are

$$\begin{aligned} \langle \textbf{P}\rangle _q= & {} \frac{\alpha _s}{2\pi }\ \left( \ln \bar{N} - \frac{3}{4} \right) \nonumber \\{} & {} \times \left[ (2C_F \!-\! C_A)\ln \frac{\mu _F^2}{s} \!+\! C_A \ln \frac{\mu _F^2}{m_{\tilde{g}}^2 \!-\! t}\right] ,\nonumber \\ \langle \textbf{K}\rangle _q= & {} \frac{\alpha _s}{2\pi } \Bigg \{ 2 C_F \ln ^2\bar{N} + C_F \left( \frac{2}{3} \pi ^2 - 5\right) \nonumber \\{} & {} + C_A \ln \bar{N}\left( 1+\ln \frac{m_{\tilde{g}}^2}{m_{\tilde{g}}^2-t}\right) \nonumber \\{} & {} + \frac{C_A}{4}\Bigg [1 + 4\textrm{Li}_2\left( \frac{2m_{\tilde{g}}^2 - t}{m_{\tilde{g}}^2}\right) \nonumber \\{} & {} + \left( 1 \!+\! 2\frac{m_{\tilde{g}}^2}{m_{\tilde{g}}^2-t} \!+\! 4\ln \frac{m_{\tilde{g}}^2}{m_{\tilde{g}}^2-t} \right) \ln \frac{m_{\tilde{g}}^2}{2m_{\tilde{g}}^2-t}\nonumber \\{} & {} + 3\ln \frac{3m_{\tilde{g}}^2-t-2 m_{\tilde{g}} \sqrt{2m_{\tilde{g}}^2-t}}{m_{\tilde{g}}^2-t} \nonumber \\{} & {} + 6\frac{m_{\tilde{g}}}{\sqrt{2m_{\tilde{g}}^2-t}+m_{\tilde{g}}} - 3 \bigg ] \bigg \}, \end{aligned}$$

(A.17)

where the Mandelstam variables are defined according to the ordering of the two-to-two partonic process $q{\bar{q}}\rightarrow \tilde{g}\tilde{\chi }$. Similar equations hold for u-channel contributions, that must be included as well when computing the $\langle \textbf{P} \!+\! \textbf{K}\rangle _q$ term in (A.16),

$$\begin{aligned} \Big \langle \textbf{P} \!+\! \textbf{K}\Big \rangle _q = \langle \textbf{P}\rangle _q + \langle \textbf{K}\rangle _q + \langle \textbf{P}\rangle _q|_{t\leftrightarrow u} + \langle \textbf{K}\rangle _q|_{t\leftrightarrow u}. \end{aligned}$$

(A.18)

1.3 Squark-electroweakino associated production

Similarly to 5.2, the modified soft anomalous dimension $\bar{{\varGamma }}_{qg\rightarrow \tilde{q}\tilde{\chi }}$ associated with the process $qg\rightarrow \tilde{q}\tilde{\chi }$ is obtained from eikonal Feynman rules, and it is given by

$$\begin{aligned} \bar{{\varGamma }}_{qg\rightarrow \tilde{q}\tilde{\chi }}= & {} \frac{\alpha _s}{2\pi } C_F \left[ 2\ln \left( \frac{m_{\tilde{q}}^2 - t}{\sqrt{s} m_{\tilde{q}}}\right) - 1 + i\pi \right] \nonumber \\{} & {} + \frac{\alpha _s}{2\pi } C_A \ln \left( \frac{m_{\tilde{q}}^2 - u}{m_{\tilde{q}}^2 - t}\right) . \end{aligned}$$

(A.19)

The hard-matching coefficients are similarly derived from the N-independent parts of the insertion operators $\textbf{P}$ and $\textbf{K}$,

$$\begin{aligned} {\mathcal {C}}^{(0)}_{qg}= & {} \ 1,\nonumber \\ \frac{\alpha _s}{2\pi } {\mathcal {C}}^{(1)}_{qg}= & {} \Big \langle \textbf{P} \!+\! \textbf{K}\Big \rangle _{q, N\text {-ind.}} \!+\! \Big \langle \textbf{P} \!+\! \textbf{K}\Big \rangle _{g, N\text {-ind.}}\nonumber \\{} & {} + \Big (\sigma ^\text {V} \!+\! \int _1 \textrm{d}\sigma ^\text {A} \Big ), \end{aligned}$$

(A.20)

in which we use the explicit expressions

$$\begin{aligned} \langle \textbf{P}\rangle _q= & {} \frac{\alpha _s}{2\pi }\left( \ln \bar{N} - \frac{3}{4}\right) \nonumber \\{} & {} \times \left( 2 C_F \ln \frac{\mu _F^2}{m_{\tilde{q}}^2 - t} - C_A \ln \frac{s}{{m_{\tilde{q}}^2 - t}}\right) , \nonumber \\ \langle \textbf{P}\rangle _{g}= & {} \frac{\alpha _s}{2\pi }\left( C_A \ln \bar{N} \!-\! \frac{11}{4} \!+\! \frac{n_f}{6} \right) \ln \frac{\mu _F^4}{s(m_{\tilde{q}}^2 \!-\! u)},\nonumber \\ \langle \textbf{K}\rangle _q= & {} \frac{\alpha _s}{2\pi }\Bigg \{ C_F \left( 2\ln ^2\bar{N} + \frac{2}{3}\pi ^2 - 5 \right) \nonumber \\{} & {} +\! \left( C_F \!-\! \frac{C_A}{2}\right) \! \left[ 2\ln \bar{N}\left( \!1 \!+\! \log \frac{m_{{\tilde{q}}}^2}{m_{\tilde{q}}^2\!-\!t}\!\right) \!+\! \mathcal {Q}\right] \! \Bigg \},\nonumber \\ \langle \textbf{K}\rangle _g= & {} \frac{\alpha _s}{2\pi }\frac{C_A}{2}\Bigg [ 4\ln ^2\bar{N} \!+\! 2\ln \bar{N}\left( \!1 \!+\! \ln \frac{m_{\tilde{q}}^2}{m_{{\tilde{q}}}^2\!-\!u}\!\right) \nonumber \\{} & {} -\frac{100}{9} + \frac{4}{3}\pi ^2 + \frac{16}{9}\frac{n_f}{C_A} + \mathcal {G}\Bigg ]. \end{aligned}$$

(A.21)

The two quantities $\mathcal {Q}$ and $\mathcal {G}$ appearing in these formulas are given by

$$\begin{aligned} \mathcal {Q}= & {} \frac{m_{{\tilde{q}}}^2 - t}{2 m_{{\tilde{q}}}^2-t} + \frac{3 m_{{\tilde{q}}}}{m_{{\tilde{q}}} + \sqrt{2 m_{{\tilde{q}}}^2-t}} - 2 \nonumber \\{} & {} + \ln \frac{m_{{\tilde{q}}}^2}{2 m_{{\tilde{q}}}^2\!-\!t} \left( 1 \!+\! 2\ln \frac{m_{{\tilde{q}}}^2}{m_{{\tilde{q}}}^2\!-\!t}\right) + 2\textrm{Li}_2 \frac{2 m_{{\tilde{q}}}^2\!-\!t}{m_{{\tilde{q}}}^2}\nonumber \\{} & {} - \frac{3}{2}\ln \frac{3m_{{\tilde{q}}}^2 -t - 2m_{{\tilde{q}}} \sqrt{2 m_{{\tilde{q}}}^2-t}}{m_{{\tilde{q}}}^2-t},\nonumber \\ \mathcal {G}= & {} \frac{m_{{\tilde{q}}}^2\!-\!u}{2m_{{\tilde{q}}}^2\!-\!u} \!+\! \ln \frac{m_{{\tilde{q}}}^2}{2 m_{{\tilde{q}}}^2 \!-\!u}\left( \!1 \!+\! 2 \ln \frac{m_{{\tilde{q}}}^2}{m_{{\tilde{q}}}^2-u}\!\right) \!-\! 2 \nonumber \\{} & {} + 2\textrm{Li}_2 \frac{2m_{{\tilde{q}}}^2\!-\!u}{m_{{\tilde{q}}}^2} \!+\! \left( \! \frac{11}{6} \!-\! \frac{n_f}{3 C_A}\right) \frac{2m_{{\tilde{q}}}}{\sqrt{2m_{{\tilde{q}}}^2\!-\!u} \!+\! m_{\tilde{q}}}\nonumber \\{} & {} + \left( \frac{11}{6} \!-\! \frac{n_f}{3 C_A}\right) \ln \frac{3m_{{\tilde{q}}}^2\!-\!u \!-\! 2m_{{\tilde{q}}}\sqrt{2m_{\tilde{q}}^2\!-\!u}}{m_{{\tilde{q}}}^2\!-\!u}, \end{aligned}$$

(A.22)

where the Mandelstam variables are defined from the two-to-two partonic process $qg\rightarrow \tilde{q}\tilde{\chi }$.

Appendix B: Architecture and structure of the Resummino sources files

The source code of Resummino is available from the folder src of any local installation of the code. It includes the following set of files, that we briefly describe.

• constants.h: Constants useds throughout the code, such as colour factors and the regulator width used for on-shell subtraction [94].

• dipoles.cc: NLO dipoles in the Catani-Seymour formalism for the production of heavy SUSY particles [91, 92].

• hxs.cc: Phase space implementations and integration routines relevant for resummed cross sections matched to NLO fixed-order predictions evaluated within the dipole formalism. It thus includes separate components for the partonic LO cross section, the associated virtual corrections (to which integrated dipoles are added), the collinear remainders, the real emission corrections (from which dipole contributions are subtracted), and threshold resummation components. The integration relevant for total rate calculations is performed through the function hadronic_xs. Integrations associated with transverse momentum and invariant mass differential cross sections are similarly implemented in the files hxs_dpt2.cc and hxs_dlnm2.cc respectively.

• kinematics.cc: Calculation of the kinematics inherent to the different partonic configurations relevant for a given process, and momentum reshuffling as required by the regularisation of infrared divergences in the dipole subtraction formalism.

• integration_method.cc: General-purpose wrapper for Monte Carlo integration through the GSL routine VEGAS.

• main.cc: Initialisation of the code within its main function.

• maths.h: Useful mathematical functions.

• options.h: Different options for the implementation of on-shell resonance subtraction from the real emission corrections to squark-electroweakino associated production.

• params.cc: Values of all SUSY and SM couplings and masses (see also details in Sect. 3.2.2).

• pdf.cc: Initialisation of the PDF functions, interface with LHAPDF, and fit of the PDFs in Mellin space.

• resummation.cc: Implementation of the resummed exponent defined in (6), and the hard-matching coefficients defined in (5).

• resummino.cc: Parsing of the input file and calls to the functions appropriate for the process and calculation chosen (total rate, $p_T$ distribution, etc.).

• utils.h: Additional functions allowing for internal checks.

Specific functions relevant for each of the supported processes, and including Born, NLO-real and NLO-virtual contributions, are collected in the sub-folders gaugino_gluino (${\tilde{\chi }}{\tilde{g}}$ production), gaugino_squark (${\tilde{\chi }}{\tilde{q}}$ production), gauginos (${\tilde{\chi }}{\tilde{\chi }}$ production), sleptons (${\tilde{l}}{\tilde{l}}$ production) and leptons (dilepton production including $Z^\prime$ and $W^\prime$ exchange contributions).

Appendix C: Configuration of the Resummino run

In this appendix, we describe how to write a configuration file for Resummino. Such a file includes a set of equalities allowing to set keywords to user-defined values. A first ensemble of keywords is dedicated to the definition of the hadronic process and the type of calculation to be performed. They are given by:

• collider_type = <string>: information on the initial-state hadrons. Allowed values are proton–proton or proton–antiproton.

• center_of_mass_energy = <double>: hadronic centre-of-mass energy in GeV.

• particle1 = <int> and particle2 = <int>: nature of the first and second outgoing particles, given through their PDG identifiers [77].

• result = <string>: defines the cross section to compute. Allowed values are total (total cross section using the threshold resummation formalism), pt (differential cross section $\textrm{d}\sigma /\textrm{d}p_T$ using the $p_T$ resummation formalism), ptj (differential cross section $\textrm{d}\sigma /\textrm{d}p_T$ using the joint resummation formalism) and m (differential cross section $\textrm{d}\sigma /\textrm{d}M$ using the threshold resummation formalism).

• pt = <double>: transverse momentum in GeV, used for differential cross section calculations at fixed $p_T$. This variable is only relevant when the result parameter is set to pt or ptj.

• M = <double>: invariant mass in GeV, used for differential cross section calculations at fixed M. The invariant mass of the final-state system is used by enforcing the specific value auto.

• Minv_min = <double> and Minv_max = <double>: invariant-mass window, given in GeV, that is used for total cross section computations relevant for the production of a Drell–Yan pair of leptons. Such parameters can also be set to auto, which corresponds to $M_\textrm{min} = (3/4) M_{Z^\prime }$ and $M_\textrm{max} = (5/4) M_{Z^\prime }$.

Information on the SUSY properties or on the extra gauge boson properties is provided through files encoded in an SLHA(-like) format. Two variables allow users to provide information about the path to these files:

• slha = <string>: path to the SLHA input file containing information on the SUSY scenario considered (masses, widths, parameters, etc.).

• zpwp = <string>: path to the SLHA-like input file containing information on the properties of extra $Z^\prime$ and $W^\prime$ bosons.

A third ensemble of keywords are related to the choice of the parton density sets to be used in given calculations, and how their Mellin transform is handled. They are:

• pdf_format = <string>: format according to which the LHAPDF grids are encoded. Allowed values include LHpdf (default) and LHgrid (deprecated option relevant for LHAPDF versions anterior to 6).

• pdf_lo = <string> and pdf_nlo = <string>: name of the PDF set to be employed for LO and higher-order calculations respectively, according to the naming scheme of the webpage https://lhapdf.hepforge.org/pdfsets.

• pdfset_lo = <int> and pdfset_nlo = <int>: specific PDF set member to be used in LO and higher-order calculations respectively. These integers allow for the selection of the best-fit set or of one of the next-to-best-fit sets provided within a given LHAPDF set, in the context of PDF error estimates.

• weight_gluon = <double>, weight_sea = <double> and weight_valence = <double>: weights used in the respective fits of the gluon, sea quark and valence quark densities according to the Levenberg–Marquard algorithm. The (recommended) default values are − 1.6.

• xmin = <double>: minimum Bjorken-x value to consider in the PDF fitting procedure. When set to auto (default), the code uses $x_{\text {m}in}=M^2/S$ where M is the invariant mass of the pair of produced particles.

Renormalisation and factorisation scales are automatically set to specific values for any calculation achieved by Resummino (see Sect. 3.2.1). Those values can however be multiplied by user-defined factors given through two self-explanatory variables:

• mu_f = <double> and mu_r = <double>: factors multiplying the factorisation and renormalisation scale values specific to the calculation considered. Default values are 1.0.

Finally, the numerical precision of the Monte Carlo integration achieved with the VEGAS algorithm is controlled through two input variables:

• precision = <double>: relative precision of the Monte Carlo integration (recommended value: 0.001 $\equiv 0.1\%$). The integration routine stops when the relative numerical error of the result matches the requested precision. For contributions beyond LO, this error is estimated with respect to the tree-level result, which ensures that no computing power is wasted on small contributions.

• max_iters = <int>: maximum number of integration iterations, excluding the warm-up phase (recommended value: 50). After the maximum number of iterations, Monte Carlo integration is stopped, even if the desired precision is not reached.

Fig. 6

Resummino screen output relevant for the calculation of the LO total rate associated with slepton pair production at the LHC, using the SUSY spectrum encoded in the file input/slha.in and the collider settings provided in the file input/resummino.in, both files being shipped with the program

Fig. 7

Screen output relevant for the fit of the gluon density in the case of slepton pair production at the LHC operating at $\sqrt{S}={13}\,\textrm{TeV}$, the details on the calculation being encoded in the file input/resummino.in and the SUSY spectrum in the SLHA file input/slha.in

Appendix D: Resummino output

In this appendix, we detail the (screen) output generated by Resummino in the context of the calculations of the total cross section associated with slepton pair-production at the LHC, operating at a centre-of-mass energy of 13 TeV. We choose to make use of the LO and NLO sets of CT14 parton densities [60], and to rely on one of the SUSY scenarios whose SLHA input file is provided in the input folder of the Resummino source files.

The used input file including all computational settings is available from the folder input/resummino.in. It reads:

We remind that several entries (like zpwp or Min_min and Minv_max) are irrelevant for the SUSY process considered. The code is started by typing in a shell

When executed, Resummino begins by printing to the screen its banner (that includes information on the articles to cite when using the code), as well as LHAPDF settings (path to the PDF grids, information on the chosen PDF set, value of the strong coupling, etc.) used for the first calculation achieved by the code, namely the LO rate associated with the process considered. This gives in our case

where <lhapth> corresponds to the path to the folder containing the PDF grid CT14lo_0000.dat, that is used for the evaluation of the LO cross section.

Subsequently, Resummino displays to the screen the results of each VEGAS iteration, together with its numerical error, as well as the accumulated result obtained by combining all VEGAS iterations. The input file provided above leads to the screen output shown in Fig. 6, in which the first five iterations are related to the warm-up stage of the numerical integration process, and the last five to the actual calculation. The latter stops when the required precision is reached.

Next, LHAPDF information on the NLO PDF set used is printed to the screen, followed by a similar output as in Fig. 6 for each component of the full fixed-order calculation. These consist of a recalculation of the LO rate with NLO parton densities, the sum of the virtual corrections and the integrated dipoles (in the language of the dipole subtraction formalism [91,92,93]), the collinear remainder stemming from the $\textbf{P}$ and $\textbf{K}$ insertion operators (still in the language of the dipole subtraction formalism), and the difference between the real gluon emission corrections and the associated dipoles, and that between the real quark emission corrections and the associated dipoles. In the case in which on-shell resonant contributions appear in the real emission contributions, their integration is handled separately for improved convergence.

The code then moves with the calculation of the resummed components of the cross section, which starts with a fit of the PDFs so that they could be transformed to Mellin space (see Sect. 2). Information on this fit is printed to the screen, as shown in Fig. 7 for the case of the gluon density. The cross section is then evaluated at NLO + NLL using the collinear-improved resummation formalism of [41,42,43,44], or at aNNLO + NNLL using the standard threshold resummation formalism of [35,36,37,38,39,40] if the code is run with its –nnll option (see Appendix D).

The matched results are finally displayed to the screen,⁸

the last line being replaced by an NLO + NLL prediction if the aNNLO + NNLL one is not required or unavailable.

Appendix E: Options available from the command line interface of Resummino

The Resummino program can be executed with several options, by typing in a shell

The keyword <input-file> provides the path to a configuration file detailing the calculation to be achieved, and that must be encoded following the guidelines sketched in Sect. 3.2 and Appendix C. All allowed options are listed below.

• --center_of_mass_energy (or -e), followed by a double-precision number: this sets the hadronic centre-of-mass energy, given in GeV.

• --help (or -h): this displays a help message to the screen, indicating how to run the code.

• --invariant-mass (or -m), followed by a double-precision number: this sets the value, in GeV, of the final-state invariant mass for calculations at fixed invariant mass.

• --lo (or -l): this enforces a calculation at LO.

• --mu_f (or -f) and --mu_r (or -r), both followed by a double-precision number: this determines the factors multiplying the factorisation and renormalisation scales, relatively to central scale choices.

• --nlo (or -n): this enforces a calculation at NLO.

• --nll (or -s): this enforces a calculation at NLO + NLL.

• --nnll (or -z): this enforces a calculation at aNNLO + NNLL.

• --output (or -o), followed by a string referring to the path to a specific folder: this defines the folder in which all output files created by Resummino will be stored.

• --parameter-log (or -p), followed by a string representing the path to a file: this makes the code writing all the parameters inherent to the calculation considered in the provided file.

• --particle1 (or -c) and --particle2 (or -d), followed by integer numbers: modifications of the nature of the first and second final-state particles through a change in their PDG identifiers.

• --pdfset_lo (or -a) and --pdfset_nlo (or -b), both followed by an integer: this determines the exact PDF set, within a given collection of LHAPDF parton densities, to employ for calculations at LO and beyond respectively.

• --transverse-momentum (or -t), followed by a double-precision number: this sets the value, in GeV, of the final-state transverse momentum, for calculations at fixed $p_T$.

• --version (or -v): this displays the Resummino version number to the screen.

Appendix F: Lists of total cross sections

In this appendix we collect total cross section predictions for the different SUSY processes studied in this article, and present them in a tabulated form in which scale and PDF uncertainties are separate. Their digitised version can be found on the webpage https://github.com/APN-Pucky/HEPi/tree/master/hepi/data/json.

Table 3 aNNLO + NNLL total cross sections for slepton pair production at the LHC, for $\sqrt{S} = {13.0}\,\textrm{TeV}$ (upper) and 13.6 TeV (lower). All other SUSY particles are decoupled

Table 4 aNNLO + NNLL total cross sections for the production of a pair of mass-degenerate higgsinos at the LHC, for $\sqrt{S} = {13.0}\,\textrm{TeV}$ (upper) and 13.6 TeV (lower). Higgsinos are defined as in Eq. (11), and all other SUSY particles are decoupled

Table 5 aNNLO + NNLL total cross sections for the production of a pair of compressed higgsino states at the LHC, for $\sqrt{S} = {13.0}\,\textrm{TeV}$ (upper) and 13.6 TeV (lower). Higgsinos are defined as in Eq. (12), and all other SUSY particles are decoupled

Table 6 aNNLO + NNLL total cross sections for the production of a pair of mass-degenerate winos at the LHC, for $\sqrt{S} = {13.0}\,\textrm{TeV}$ (upper) and 13.6 TeV (lower). Winos are defined as in Eq. (13), and all other SUSY particles are decoupled

Table 7 aNNLO + NNLL total cross sections for the production of a pair of mass-degenerate winos at the LHC, for $\sqrt{S} = {13.0}\,\textrm{TeV}$ (upper) and 13.6 TeV (lower). Winos are defined as in Eq. (13), and the spectrum features mass-degenerate first- and second-generation squarks. All other SUSY particles are decoupled