OpenLoops 2
Abstract
We present the new version of OpenLoops, an automated generator of tree and one-loop scattering amplitudes based on the open-loop recursion. One main novelty of OpenLoops 2 is the extension of the original algorithm from NLO QCD to the full Standard Model, including electroweak (EW) corrections from gauge, Higgs and Yukawa interactions. In this context, among several new features, we discuss the systematic bookkeeping of QCD–EW interferences, a flexible implementation of the complex-mass scheme for processes with on-shell and off-shell unstable particles, a special treatment of on-shell and off-shell external photons, and efficient scale variations. The other main novelty is the implementation of the recently proposed on-the-fly reduction algorithm, which supersedes the usage of external reduction libraries for the calculation of tree–loop interferences. This new algorithm is equipped with an automated system that avoids Gram-determinant instabilities through analytic methods in combination with a new hybrid-precision approach based on a highly targeted usage of quadruple precision with minimal CPU overhead. The resulting significant speed and stability improvements are especially relevant for challenging NLO multi-leg calculations and for NNLO applications.
1 Introduction
Scattering amplitudes at one loop are a mandatory ingredient for any precision calculation at high-energy colliders. At next-to-leading order (NLO), the calculation of hard cross sections requires one-loop matrix elements with hard kinematics, while next-to-next-to leading order (NNLO) predictions require one-loop amplitudes with one additional unresolved particle. Nowadays, thanks to a variety of modern techniques [1, 2, 3, 4, 5, 6, 7, 8, 9], one-loop calculations can be carried out with a number of automated and widely applicable programs [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] that have strongly boosted the field of precision phenomenology. Most notably, such tools have extended the reach of NLO calculations to highly non-trivial multi-particle processes [21, 22, 23, 24, 25] and have opened the door to the automation of multi-purpose Monte Carlo generators at NLO [26, 27, 28, 29, 30, 31, 32].
In this paper we present the new version of OpenLoops,^{1} an automated tool for the calculation of tree and one-loop scattering amplitudes within the Standard Model (SM). The OpenLoops algorithm is based on a numerical recursion^{2} that generates loop amplitudes in terms of cut-open loop diagrams [9, 33]. Such objects, called open loops, are characterised by a tree topology but depend on the loop momentum.
In the original version of the algorithm [9], implemented in OpenLoops 1 [16], loop amplitudes are built in two phases. In the first phase, Feynman diagrams are constructed in terms of tensor integrals using the open-loop recursion, while in the second phase, loop amplitudes are reduced to scalar integrals using external libraries such as Collier [19] or CutTools [10]. The main strengths of this approach are the high speed of the open-loop recursion and the possibility of curing numerical instabilities through the tensor-reduction techniques [4, 34] implemented in Collier [19].
In the original open-loop algorithm [9], the rank of open loops increases at each step of the recursion. As a consequence, the CPU time required for their processing, the memory footprint, and also numerical instabilities, tend to grow rather fast with the number of scattering particles. For these reasons, in OpenLoops 2 the construction of loop amplitudes and their reduction have been unified in a single recursive algorithm [33] that makes it possible to avoid high-rank objects at all stages of the amplitude calculations. This is achieved by interleaving single steps of the construction of open loops with reduction operations at the integrand level [2]. The implementation of this method, called on-the-fly reduction, is one of the main novelties of OpenLoops 2. So far it is restricted to tree–loop interferences at NLO, while squared loop amplitudes are still processed in the same way as in OpenLoops 1.
The on-the-fly reduction algorithm in OpenLoops 2 is equipped with an automated system that avoids numerical instabilities in a highly efficient way. This stability system makes use of analytic techniques that have been introduced in [33] and have meanwhile been extended in various directions, and supplemented by a novel hybrid-precision system. The latter monitors the level of stability by exploiting information on the analytic structure of the reduction identities, and residual instabilities are stabilised on-the-fly through quadruple precision (qp). This system is implemented at the level of individual operations. In this way, the usage of qp is restricted to a minimal part of the calculations, which results in a huge speed-up as compared to complete qp re-evaluations. Thanks to these features, the on-the-fly reduction method makes it possible to achieve an unprecedented level of numerical stability, both for multi-leg NLO calculations with hard kinematics and for NNLO applications with unresolved partons.
The structure of the open-loop recursion [9, 33] is model independent, and the explicit form of its kernels depends only on the Lagrangian of the model at hand. The original implementation [16] was applicable to any SM process at NLO QCD, and the other major novelty of OpenLoops 2 is the extension of NLO automation to the full SM [35, 36], including any correction effect of \(\mathcal {O}(\alpha _{\mathrm {s}})\) and \(\mathcal {O}(\alpha )\).^{3} In this respect, in this paper we present a detailed discussion of the interplay of QCD and EW effects in scattering amplitudes with more than one quark chain, which are relevant for LHC processes with two or more light jets. In that case, Born amplitudes consist of towers of terms of order \(\alpha _{\mathrm {s}}^p\alpha ^q\) with fixed total power \(p+q\) but variable powers in the QCD and EW couplings. In such cases, as is well known, QCD and EW interactions mix through interference effects and, in general, NLO terms of fixed order \(\alpha _{\mathrm {s}}^P\alpha ^Q\) involve correction effects of QCD and EW kind. However, as we will point out, each NLO term of order \(\alpha _{\mathrm {s}}^P\alpha ^Q\) is always dominated either by QCD corrections to Born terms of order \(\alpha _{\mathrm {s}}^{P-1}\alpha ^Q\)or by EW corrections to Born terms of order \(\alpha _{\mathrm {s}}^{P}\alpha ^{Q-1}\).
In this paper the renormalisation of the SM and its implementation in OpenLoops are discussed in detail. In the QCD sector, quark masses and Yukawa couplings can be renormalised in the on-shell and \(\overline{\text {MS}}\) schemes, and the \(\alpha _{\mathrm {s}}\) counterterm can be flexibly adapted to any flavour-number scheme. The renormalisation of masses and couplings at \(\mathcal {O}(\alpha )\) is based on the on-shell scheme [37] and its extension to complex masses [38] for off-shell unstable particles. More precisely, in OpenLoops 2 these two approaches are unified in a generic scheme that can address processes with combinations of on-shell and off-shell unstable particles, such as for \(pp\rightarrow t {\bar{t}} \ell ^+ \ell ^-\), where Z-bosons occur as internal resonances, while top quarks are on-shell external states. Besides UV counterterms, OpenLoops 2 implements also Catani–Seymour’s \({{\mathbf {I}}}\)-operator for the subtraction of infrared (IR) singularities at \(\mathcal {O}(\alpha _{\mathrm {s}})\) [39, 40] and \(\mathcal {O}(\alpha )\) [36, 41, 42, 43, 44].
For the definition of EW couplings, three different schemes based on the the input parameters \(\alpha (0)\), \(\alpha (M_Z^2)\) and \(G_\mu \) are supported. Moreover, OpenLoops 2 implements an automated system for the optimal choice of the coupling of on-shell and off-shell external photons. Concerning the choice of \(\alpha _{\mathrm {s}}\) and the renormalisation scale \(\mu _\mathrm{R}\), a new automated scale-variation mechanism makes it possible to re-evaluate scattering amplitudes for multiple values of \(\alpha _{\mathrm {s}}\) and \(\mu _\mathrm{R}\) with minimal CPU cost.
The OpenLoops 2 program can be combined with any other code by means of its native Fortran and C/Open image in new window interfaces, which allow one to exploit its functionalities in a flexible way. Besides the choice of processes and parameters, the interfaces support the calculation of LO, NLO, and loop-induced matrix elements and building blocks thereof, as well as various colour and spin correlators relevant for the subtraction of IR singularities at NLO and NNLO. Additional interface functions give access to the SU(3) colour basis and the colour flow of tree amplitudes. Besides its native interfaces, OpenLoops offers also a standard interface in the BLHA format [45, 46].
The OpenLoops program can be used as a plug-in by the Monte Carlo programs Sherpa [26, 47], Powheg-Box [27], Open image in new window [32], Geneva [48], and Whizard [49], which possess built-in interfaces that control all relevant OpenLoops functionalities in a largely automated way, requiring only little user intervention. Moreover, OpenLoops is used as a building block of Matrix [50] for the calculation of NNLO QCD observables. In this context, the automation of EW corrections in OpenLoops 2 opens the door to ubiquitous NLO QCD+NLO EW simulations in Sherpa [51, 52] and NNLO QCD+NLO EW calculations in Matrix [53].
The OpenLoops 2 code is publicly available on the Hepforge webpage
https://openloops.hepforge.org
and via the Git repository
https://gitlab.com/openloops/OpenLoops.
It consists of a process-independent base code and a process library that covers several hundred partonic processes, including essentially all relevant processes at the LHC. The desired processes can be easily accessed through an automated download mechanism. The set of available processes is continuously extended, and possible missing processes can be promptly generated by the authors upon request.
The paper is organised as follows. Section 2 presents the structure of the original open-loop recursion and the new on-the-fly reduction algorithm. Numerical instabilities and the new hybrid-precision system are discussed in detail. Section 3 deals with general aspects of NLO calculations and their automation in OpenLoops. This includes the bookkeeping of towers of terms of variable order \(\alpha _{\mathrm {s}}^p\alpha ^q\), the treatment of input parameters, optimal couplings for external photons, the renormalisation of the SM at \(\mathcal {O}(\alpha _{\mathrm {s}})\) and \(\mathcal {O}(\alpha )\), the on-shell and complex-mass schemes, and the \({{\mathbf {I}}}\)-operator. Section 4 provides instructions on how to use the program, starting from installation and process selection, and including the various interfaces for the calculation of matrix elements, colour/spin correlators, and tree amplitudes in colour space. Technical benchmarks concerning the speed and numerical stability of OpenLoops 2 are presented in Sect. 5. A detailed description of the syntax and usage of the OpenLoops interfaces can be found in the appendices.
While the paper as a whole serves as a detailed documentation of the algorithms implemented in OpenLoops 2, Sect. 4 together with Appendix A can be used alone as a manual.
2 The OpenLoops algorithm
The calculation of loop amplitudes in OpenLoops proceeds through the recursive construction of open loops and their reduction to master integrals. In this section we outline two variants of this procedure: the original open-loop method [9], which was used throughout in OpenLoops 1 and is still used for loop-induced processes in OpenLoops 2, and the new on-the-fly reduction method [33] used for tree–loop interferences in OpenLoops 2.
2.1 Scattering amplitudes and probability densities
For standard processes with \(\mathcal {M}_{0}\ne 0\), leading-order (LO) cross sections involve only squared tree contributions \(\mathcal {W}_{00}\), while at next-to-leading order (NLO) virtual one-loop contributions \(\mathcal {W}_{01}\) and real-emission contributions of type \(\mathcal {W}_{00}\) with one additional parton are needed. The squared one-loop probability density \(\mathcal {W}_{11}\) is the main LO building block for loop-induced processes, i.e. processes with \(\mathcal {M}_{0}=0\). For the calculation of such processes at NLO also \(\mathcal {W}_{11}\)-type densities with one additional parton are needed. Otherwise \(\mathcal {W}_{11}\) is relevant as ingredient of next-to-next-to-leading order (NNLO) calculations.
2.2 Tree amplitudes
2.3 One-loop amplitudes
2.4 Reduction to master integrals
In OpenLoops the reduction of loop amplitudes to master integrals is carried out with two different methods. Squared loop amplitudes and tree-loop interferences in the Higgs Effective Field Theory (HEFT)^{8} are handled along the lines of the original open-loop approach [9], where the reduction is performed a posteriori of the dressing recursion. Since every dressing step can increase the tensor rank by one (see Fig. 1 a), this generates intermediate objects of high tensor rank, i.e. high complexity, with a negative impact on CPU speed. In contrast, all other tree–loop interferences are computed with the on-the-fly reduction approach [33], where dressing steps are interleaved with integrand reduction steps in such a way that the tensor rank, and thus the complexity, remain low at all stages of the calculation (see Fig. 1b).
2.4.1 A posteriori reduction
2.4.2 On-the-fly reduction
In the on-the-fly approach, the dressing of open loops is interleaved with reduction steps. The latter are applied in such a way that the tensor rank never exceeds two.
Rank-two open loops with only three loop denominators can be reduced on-the-fly in a similar way as open loops with more than three propagators [33]. The remaining reducible integrals have the following number of propagators N and tensor rank R: \(N\ge 5\) and \(R=1,0\); \(N=4,3\) and \(R=1\); \(N=2\) and \(R=2,1\). For their reduction to master integrals we use a combination of integral reduction and OPP reduction identities [33]. Master integrals are evaluated with Collier [19], which is the default in double precision, or OneLOop [65], which is the default in quadruple precision.
2.5 Tree–loop interference
2.5.1 Parent-child algorithm
- (i)
The numerator of a colour-stripped N-point loop diagram (2.12) is constructed as outlined in Sect. 2.3, i.e. starting from Open image in new window and applying N dressing steps of type (2.15).
- (ii)
In general, open loops with higher number N of loop propagators do not need to be built from scratch, but can be constructed starting form pre-computed open loops with lower N exploiting parent–child relations [9] as illustrated in Fig. 2. The efficiency of the parent–child approach is maximised by means of cutting rules that set the position of the cut propagator and the dressing direction in a way that favours parent–child matching (for details see [9, 33]).
- (iii)After the last dressing step, the loop numerator is closed by taking the trace and, for every helicity state \(h\), the colour-summed Born interference (2.21) is built as$$\begin{aligned}&\mathcal {U}(\mathcal {I}_N,q,h) = \mathcal {U}_0(\mathcal {I}_{N},h) {\mathrm {Tr}}\Big [\mathcal {N}(\mathcal {I}_{N},q,h)\Big ]. \end{aligned}$$(2.23)
- (iv)Helicity sums are performed, and the set of loop diagrams with the same one-loop topology \(t=\{D_{0}\), \(\ldots \), \(D_{N-1}\}\), denoted \(\varOmega _N(t)\), is combined to form a single numerator,$$\begin{aligned}&{\mathcal {V}}(t,q)=\sum \limits _{h}\sum \limits _{\mathcal {I}_N\in \varOmega _N(t)} \mathcal {U}(\mathcal {I}_N,q,h). \end{aligned}$$(2.24)
- (v)The corresponding loop integral,is reduced to master integrals as described in Sect. 2.4.1, and all topologies are summed.$$\begin{aligned}&\mathcal {W}_{01}(t) = \int \!\mathrm {d}^D\!\bar{q}\, \frac{{\mathcal {V}}(t,q)}{\bar{D}_{0} \bar{D}_{1}\cdots \bar{D}_{N-1}}, \end{aligned}$$(2.25)
2.5.2 On-the-fly algorithm
- (i)The generalised open loops (2.26) are constructed through subsequent dressing stepsstarting from \(\mathcal {U}_0(\mathcal {I}_N,q,\check{h}_0)= \mathcal {U}_0(\mathcal {I}_N,h)\). The summation over the helicities \(h_k\) is performed on-the-fly after the dressing of the related segment. This results in a reduction of helicity degrees of freedom, and thus of the number of required operations, at each dressing step.$$\begin{aligned} \mathcal {U}_k(\mathcal {I}_N,q,\check{h}_k)&= \sum _{h_k}\;\mathcal {U}_{k-1}(\mathcal {I}_{N},q,\check{h}_{k-1}) S_k(q,h_{k}), \end{aligned}$$(2.27)
- (ii)Before each new dressing step, the set \(\varOmega _N=\{\mathcal {I}_N^{(n)}\}\) of open loops with the same loop topology and the same undressed segments is combined into a single object,In this way, the remaining dressing operations for the objects in \(\varOmega _N\) need to be performed only once. This procedure, called on-the-fly merging, is illustrated in Fig. 3. It plays an analogous role as the parent-child approach in Sect. 2.5.1, and its efficiency is maximised by means of cutting rules tailored to the needs of merging.$$\begin{aligned} {\mathcal {V}}_k(\varOmega _N,q,\check{h}_k)&= \sum \limits _{n}\, \mathcal {U}_k(\mathcal {I}_N^{(n)},q,\check{h}_k). \end{aligned}$$(2.28)
- (iii)Open-loop objects of type (2.28) with more than three loop propagators are reduced on-the-fly using the integrand-reduction identity (2.20). This generates new open loops of the form(2.29)where \(\tilde{q}^2\) terms that arise from pinched propagators (see Sect. 2.4.2) are retained in all UV divergent integrals and lead to \(R_1\) rational terms.$$\begin{aligned} {\mathcal {V}}_k(\varOmega ,\bar{q})&{=} \sum _{s,r} {\mathcal {V}}^s_{k;\mu _1\dots \mu _r}(\varOmega )\, q^{\mu _1}{\cdots } q^{\mu _r}\, ({\tilde{q}}^2)^s, \end{aligned}$$(2.30)
- (iv)At this stage, the loops are closed by taking the trace, and the resulting loop integrals,are reduced to master integrals upon extraction of \(R_1\) terms, as described at the end of Sect. 2.4.2. Finally, all topologies are summed.$$\begin{aligned} \mathcal {W}_{01}(\varOmega )&= \int \!\mathrm {d}^D\!\bar{q}\, \frac{{\mathrm {Tr}}\left[ {\mathcal {V}}(\varOmega , {\bar{q}})\right] }{\bar{D}_{0} \bar{D}_{1}\cdots \bar{D}_{N-1}}, \end{aligned}$$(2.31)
2.6 Squared loop amplitudes
- (i)
The numerators of colour-stripped loop diagrams are constructed with the dressing recursion (2.15) exploiting parent–child relations.
- (ii)After the last dressing step, loop numerators are closed by taking the trace, and colour-stripped diagrams expressed in terms of integrals \(T_N^{\mu _1\cdots \mu _r}\) (2.18),which are then computed with Collier. While the \(\mathcal {N}_{\mu _1\dots \mu _r}(\mathcal {I}_N, h)\) coefficients need to be evaluated for every helicity state h, the reduction is done only once – and thus very efficiently – at the level of the h-independent tensor integrals.$$\begin{aligned} {{\mathcal {A}}}_{1}(\mathcal {I}_N,h)&= \int \!\mathrm {d}^D\!\bar{q}\, \frac{{\mathrm {Tr}}\Big [{\mathcal {N}}({\mathcal {I}}_{N},q,h)\Big ]}{\bar{D}_{0} \bar{D}_{1}\cdots \bar{D}_{N-1}} \nonumber \\&= \sum \limits _{r}\, {\mathrm {Tr}}\Big [\mathcal {N}_{\mu _1\dots \mu _r}(\mathcal {I}_N, h)\Big ]\, T_N^{\mu _1\cdots \mu _r}, \end{aligned}$$(2.32)
- (iii)Individual colour-stripped diagram amplitudes are combined with the corresponding colour structure and converted into colour vectors in the colour basis \(\{{\mathcal {C}}_i\}\),Then, summing all diagrams yields the full one-loop colour vector$$\begin{aligned} \mathcal {M}_1(\mathcal {I}_N,h)&= {\mathcal {C}}(\mathcal {I}_N) \mathcal {A}_1(\mathcal {I}_N,h) \nonumber \\&= \sum _i {\mathcal {C}}_i\,\mathcal {A}^{(i)}_1(\mathcal {I}_N,h). \end{aligned}$$(2.33)$$\begin{aligned} \mathcal {A}_1^{(i)}(h)&= \sum _{\mathcal {I}} \mathcal {A}^{(i)}_1(\mathcal {I},h). \end{aligned}$$(2.34)
- (iv)Finally, the helicity/colour summed squared loop amplitude is built though the colour-interference matrix (2.8) as$$\begin{aligned} \mathcal {W}_{11}&= \frac{1}{N_{\mathrm {hcs}}} \sum _{h}\sum _{\mathrm {col}} \mathcal {M}_1^*(h)\mathcal {M}_1(h) \nonumber \\&= \frac{1}{N_{\mathrm {hcs}}} \sum \limits _{h}\sum \limits _{i,j} K_{ij}\, \big [\mathcal {A}_1^{(i)}(h)\big ]^* \mathcal {A}_1^{(j)}(h). \end{aligned}$$(2.35)
2.7 Numerical stability
In principle, numerical accuracy can be augmented through quadruple precision (qp) arithmetic. But the resulting CPU overhead, of about two orders of magnitude, is often prohibitive. In OpenLoops, numerical instabilities are thus addressed as much as possible in double precision (dp) using analytic methods. In OpenLoops 1, as detailed below, numerical instabilities are avoided by means of the Collier library [19] in combination with a stability rescue system that makes use of CutTools [10] in qp. In OpenLoops 2, loop-induced processes are handled along the same lines, while standard NLO calculations are carried out with the new on-the-fly reduction algorithm, which is equipped with its own stability system (see Sect. 2.7.2). The latter combines analytic techniques together with a new hybrid-precision system that uses qp in a highly targeted way, requiring only a tiny CPU overhead as compared to a complete qp re-evaluation.
An additional source of numerical instabilities originates from the violation of on-shell relations or total momentum conservation of external particles, i.e. due to the quality of the provided phase-space point. To this end before amplitude evaluation on-shell conditions and momentum conservation are checked. A warning is printed when these conditions are violated beyond a certain relative threshold, which can be altered via the parameter psp_tolerance (\(\hbox {default}=10^{-9}\)). Additionally, we apply a “cleaning procedure” which ensures kinematic constraints of the phase-space up to double precision, rsp. qp where applicable.
2.7.1 Stability rescue system
In the original open-loop algorithm – which was used throughout in OpenLoops 1 and is still used in OpenLoops 2 for squared loop amplitudes and tree–loop interferences in the HEFT – the reduction to scalar integrals is entirely based on external libraries, and the best option is to carry out the reduction of tensor integrals using the Collier library [19]. In the vicinity of spurious poles, Collier cures numerical instabilities by means of expansions in the Gram determinants and alternative reduction methods [4, 34]. Such analytic techniques are applied in a fully automated way, and the resulting level of numerical stability is generally very good. Alternatively, the reduction can be performed at the integrand level using CutTools [10], but this option is mainly used as rescue system in qp, since CutTools does not dispose of any mechanism to avoid instabilities in dp.
- (i)
The stability of tensor integrals is assessed by comparing the two independent Collier implementations of the tensor reduction, Coli-Collier (default) and DD-Collier. This test can be applied to all phase-space points or restricted to a certain fraction of points with the highest virtual K-factor^{10} Given the desired fraction, the points to be tested are automatically selected by sampling the distribution in the K-factor at runtime.
- (ii)
Points that are classified as unstable are re-evaluated in qp using CutTools and OneLOop.
- (iii)
In CutTools, numerical instabilities can remain significant even in qp. Their magnitude is estimated through a so-called rescaling test, where one-loop amplitudes are computed with rescaled masses, dimensionful couplings and momenta and scaled back according to the mass dimensionality of the amplitude.
In the case of squared loop amplitudes, the qp rescue with CutTools is disabled, because of the inefficiency of OPP reduction for loop-squared amplitudes. This is due to the fact that all helicity and colour configurations must be reduced independently. Thus the above stability system is restricted to stage (i). Moreover, due to the fact that a K-factor is not available for loop-squared amplitudes, the comparison of Coli -Collier versus DD -Collier to assess numerical stability is extended to all phase-space points. Details on the usage of the stability rescue system can be found in Sect. 4.6.
2.7.2 On-the-fly stability system
In this way, rank-two Gram instabilities are delayed to later stages of the reduction, where three-point objects with a single Gram determinant \(\varDelta _{12}\) are encountered. In this case, instabilities at small \(\varDelta _{12}\) are cured by means of an analytic \(\varDelta _{12}\)-expansion, which have been introduced in [33] for the first few orders in \(\varDelta _{12}\) and are meanwhile available to any order [66].
Rank-three Gram determinants
OPP reduction The OPP method, used for five- and higher-point objects of rank smaller than two, is based on the same auxiliary momenta \(\ell _i\) mentioned above. Related rank-two Gram instabilities are avoided by permuting the propagators of the resulting scalar boxes according to (2.38).
global and numerically stable implementation of all kinematic quantities, including the basis momenta \(\ell ^\mu _i\) used for the reduction, in special regions;
analytic expressions for renormalised self-energies to avoid numerical cancellations between bare self-energies and counterterms in the limit of small \(p^2\). This is relevant for self-energy insertions into propagators that are connected to two external partons via soft or collinear branchings.
Quad precision is triggered and used at the level of individual reduction steps, based on the kinematics of the actual phase-space point and the loop topology of the individual open-loop object that is being processes at a given stage of the recursion.
Reduction steps that are identified as unstable and all consecutive connected operations are carried out in quad precision until spurious singularities are cancelled. Quad precision is thus used for all subsequent operations (dressing, merging, reduction, master integrals) that are connected to an instability.
For each type of reduction step, the magnitude of potential instabilities is estimated based on the actual kinematics and the analytical form of the reduction identity. This information leads to an error estimate that is attributed to each processed object and is propagated and updated through all steps of the algorithm.
Quad precision is triggered when the cumulative error esimate for a certain object exceeds a global accuracy threshold, which can be adjusted by the user (see Sect. 4.6) depending on the required numerical accuracy.
The efficiency of the hp system strongly benefits from the above mentioned analytical treatments of Gram determinants and soft regions, which avoid most of the instabilities and delay the remaining ones to later stages of the recursion, minimising the number of subsequent qp steps. As a result, for one-loop calculations with hard kinematics qp is typically needed only for a tiny fraction of the phase-space points, and for a very small part of the calculation of an amplitude. The usage of qp can become significantly more important in NNLO calculations, especially when local subtraction methods are used. In this case, one-loop amplitudes need to be evaluated in deep IR regions, where new types of instabilities occur for which no analytic solution is available at the moment. Such instabilities are automatically detected and cured by the hp system. This may lead, depending on the process and kinematic region, to a significant CPU overhead. In such cases, the accuracy threshold parameter should be tuned such as to achieve an optimal trade-off between performance and numerical stability.
Technical details and usage of the on-the-fly stability system are described in Sect. 4.6.
External libraries Finally, OpenLoops 2 benefits from improvements in Collier 1.2.3 [19], which is used for dp evaluations of scalar integrals and for tensor reduction in loop-induced processes, as well as in OneLOop [65], which is used to evaluate scalar integrals in qp.
3 Automation of tree- and one-loop amplitudes in the full SM
3.1 Power counting
In summary, apart from the leading QCD and EW terms, NLO SM contributions at a given order \(\alpha _{\mathrm {s}}^{n+1-r}\alpha ^{m+r}\) cannot be regarded as pure QCD or pure EW corrections. Nevertheless, the orders \(r=2R\) and \(2R+1\) are typically dominated, respectively, by QCD and EW corrections to the squared Born amplitude \(\mathcal {W}_{00}^{2R}\sim \big \langle \mathcal {M}_0^R|\mathcal {M}_0^R\big \rangle \). Thus, keeping in mind that all relevant EW–QCD mixing and interference effects must always be included, each NLO order can be labelled in a natural and unambiguous way either as QCD or EW correction as illustrated in Fig. 4.
3.2 Input schemes and parameters
In this section we discuss the different input schemes and the SM input parameters that are used for the calculation of scattering amplitudes in OpenLoops. All parameters are initialised with physical default values, and can be adapted by the user by calling the Fortran routine set_parameter or the related C/Open image in new window functions as detailed in Appendix A.2. Table 10 in Appendix C summarises input parameters and switchers that can be controlled through set_parameter. Parameters with mass dimension should be entered in GeV units. The values of specific parameters in OpenLoops can be obtained by calling the routine get_parameter, and the full list of parameter values can be printed to a file by calling the function printparameter (see Appendix A.2).
For performance reasons, the public OpenLoops libraries are typically generated with \(m_e=m_\mu =m_\tau =0\) and \(m_u=m_d=m_s=m_c=0\), while generic mass parameters \(m_q\) are used for the heavy quarks \(q=b,t\). By default, heavy-quark masses are set to \(m_b=0\) and \(m_t=172\) GeV, but their values can be changed by the user as desired. Dedicated process libraries with additional fermion-mass effects (any masses at NLO QCD and finite \(m_\tau \) at NLO EW) can be easily generated upon request. For efficiency reasons, when \(m_Q\) is set to zero for a certain heavy quark, whenever possible amplitudes that involve Q as external particle are internally mapped to corresponding (faster) massless amplitudes. To this end the desired fermion masses have to be specified before any process is registered, see Sect. 4.2.
Available EW input schemes and corresponding values of the \(\texttt {ew\_scheme}\) selector. For each scheme the default values of the corresponding input parameter is indicated. Note that instead of \(\alpha (M_Z^2)=1/127.94\) [69] we use 1 / 128. Assuming the default weak-boson mass values \(M_W=80.399\) GeV, \(M_Z=91.1876\) GeV and \(\varGamma _W=\varGamma _Z=0\). For the weak mixing angle, \(\sin ^2\theta _{\mathrm {w}}=0.22262651564387248\) in all three schemes, while the derived value of \(\alpha \vert _{G_{\mu }}\) is reported in the table
ew_scheme | Scheme | Input parameter | Default input | Value of \(\alpha \) |
---|---|---|---|---|
0 | \(\alpha (0)\) | alpha_qed_0 | 1 / 137.035999074 | Idem |
1 (default) | \(G_{\mu }\) | Gmu | \(1.16637\cdot 10^{-5}\) \(\hbox {GeV}^{-2}\) | 1/132.34890452162441 |
2 | \(\alpha (M_Z^2)\) | alpha_qed_mz | 1 / 128 | Idem |
Number of colours By default, in OpenLoops colour effects and related interferences are included throughout, i.e. scattering amplitudes are evaluated by retaining the exact dependence on the number of colours \(N_c\). In addition, dedicated process libraries with large-\(N_c\) expansions can be generated by the authors upon request. When available, leading-colour amplitudes can be selected at the level of process registration (see Sect. 4.2) via the parameter \(\texttt {leading\_colour}=1\) (default=0).
- (i)
\(\varvec{\alpha (0)}\)-scheme: as input for \(\alpha \) the parameter alpha_qed_0 is used, which corresponds to the QED coupling in the \(Q^2\rightarrow 0\) limit. This scheme is appropriate for pure QED interactions at scales \(Q^2\ll M_{W}^2\), and for the production of on-shell photons (see below).
- (ii)\({\varvec{G}}_{{\varvec{\mu }}}\)-scheme: the input value of \(\alpha \) is derived from the matching conditionwhich relates squared matrix elements for the muon decay in the Fermi theory to corresponding W-exchange matrix elements in the low-energy limit. This results into^{17}$$\begin{aligned} \Big \vert \frac{8}{\sqrt{2}} G_{\mu }\Big \vert ^2 = \Big \vert \frac{g_2^2}{\mu _W^2} \Big \vert ^2, \end{aligned}$$(3.29)As input for \(\alpha \vert _{G_{\mu }}\) the parameter Gmu is used, which corresponds to the Fermi constant \(G_{\mu }\). The \(G_{\mu }\)-scheme resums large logarithms associated with \(\alpha (M_Z^2)\) as well as universal \(M_t^2/M_W^2\) enhanced corrections associated with the \(\rho \) parameter. This guarantees an optimal description of the strength of the SU(2) coupling, i.e. W-interactions, at the EW scale.$$\begin{aligned} \alpha \vert _{G_{\mu }}= \frac{\sqrt{2}}{\pi } \, G_{\mu }\Big \vert \mu _W^2 \sin ^2\theta _{\mathrm {w}}\Big \vert . \end{aligned}$$(3.30)
- (iii)
\({\varvec{\alpha }}{\varvec{(}}{{\varvec{M}}}_{{\varvec{Z}}}^{\mathbf{2}}{\varvec{)}}\)-scheme: as input for \(\alpha \) the parameter alpha_qed_mz is used, which corresponds to the QED coupling at \(Q^2=M_Z^2\). This scheme is appropriate for hard EW interactions around the EW scale, where it guarantees an optimal description of the strength of QED interactions and a decent description of the strength of weak interactions.
External photons The high-energy couplings \(\alpha \vert _{G_{\mu }}\) and \(\alpha (M_Z^2)\) are appropriate for the interactions of EW gauge bosons with virtualities of the order of the EW scale. In contrast, the appropriate coupling for external high-energy photons is \(\alpha (0)\) [70]. More precisely, for photons of virtuality \(Q_\gamma ^2\) the coupling \(\alpha (Q_\gamma ^2)\) should be used. For initial- or final-state on-shell photons this corresponds to \(\alpha (0)\). However, in photon-induced hadronic collisions, initial-state photons inside the hadrons effectively couple as off-shell partons with virtuality \(Q^2_\gamma = \mu _F^2\), where \(\mu _\mathrm{F}\) is the factorisation scale of the parton distribution functions (see Appendix A.3 of [36]), Thus, at high \(\mu _F^2\) the high-energy couplings \(\alpha \vert _{G_{\mu }}\) or \(\alpha (M_Z^2)\) should be used.
Unresolved photons (iPDG = 22): extra photons (absent at LO) in NLO EW bremsstrahlung.
Hard photons of on-shell type (iPDG = 2002): standard hard final-state photons that do not undergo \(\gamma \rightarrow f{\bar{f}}\) splittings at NLO EW, or initial-state photons at photon colliders;
Hard photons of off-shell type (iPDG\(= -2002\)): hard final-state photons that undergo \(\gamma \rightarrow f{\bar{f}}\) splittings at NLO EW, or initial-state photons from QED PDFs in high-energy hadronic collisions.
By default, the \(R^{({\mathrm {on}})}_\gamma \) and \(R^{({\mathrm {off}})}_\gamma \) rescaling factors in (3.32)–(3.33) are applied to all on-shell and off-shell photons. They can be deactivated independently of each other by setting, respectively, onshell_photons_lsz=0 (default=1) and offshell_photons_lsz=0 (default=1).
By default, according to the SM relation between Yukawa couplings and masses, the Yukawa masses \({\mu }_{f,{\mathrm {Y}}}\) are set equal to the complex masses \(\mu _f\) in (3.26). More precisely, each time that mass(PID) and width(PID) are updated, the corresponding Yukawa mass parameters yuk(PID) and yukw(PID) are set to the same values. Thus, modified Yukawa masses should always be set after physical masses. This interplay, can be deactivated by setting \(\texttt {freeyuk\_on}=1\) (\(\hbox {default}=0\)). In this case, yuk(PID) and yukw(PID) are still initialised with the same default values as mass parameters, but are otherwise independent. This switcher acts in a similar way on the Yukawa renormalisation scale \(\mu _{f,{\mathrm {Y}}}\) in (3.52). At NLO EW, modified Yukawa masses are not allowed.^{19}
Wherever present, the imaginary parts of \(\mu _f\), \(\mu _H\), \(\mu _W\) and \(\sin \theta _{\mathrm {w}}\) are consistently included throughout in (3.36)–(3.38).
CKM matrix The OpenLoops program can generate scattering amplitudes with a generic CKM matrix \(V_{ij}\). However, for efficiency reasons, most process libraries are generated with a trivial CKM matrix, \(V_{ij}=\delta _{ij}\). Process libraries with a generic CKM matrix are publicly available for selected processes, such as charged-current Drell-Yan production in association with jets, and further libraries of this kind can be generated upon request. When available, such libraries can be used by setting \(\texttt {ckmorder=1}\) before the registration of the process at hand (see Sect. 4.2). In this case the default values of \(V_{ij}\) remain equal to \(\delta _{ij}\), but the real and imaginary parts of the CKM matrix can be set to any desired value by means of the input parameters VCKMdu, VCKMsu, VCKMbu, VCKMdc, VCKMsc, VCKMbc, VCKMdt, VCKMst, VCKMbt for \({\mathrm {Re}}(V_{ij})\) and VCKMIdu, VCKMIsu, etc. for \({\mathrm {Im}}(V_{ij})\).
3.3 Renormalisation
For unstable particles, as discussed in Sect. 3.3.2, OpenLoops implements a flexible combination of the on-shell scheme [37] and the complex-mass scheme [38]. In this approach, the width parameters \(\varGamma _i\) of the various unstable particles can be set to non-zero or zero values independently of each other. Depending on this choice, the corresponding particles are consistently renormalised as resonances with complex masses or as on-shell external states with real masses.
In the following, we discuss the various counterterms needed at NLO QCD and NLO EW. In general, as discussed in Sect. 3.1, one-loop contributions of \(\mathcal {O}(\alpha _{\mathrm {s}}^P\alpha ^Q)\) can require \(\mathcal {O}(\alpha _{\mathrm {s}})\) counterterm insertions in Born terms of \(\mathcal {O}(\alpha _{\mathrm {s}}^{P-1}\alpha ^Q)\) as well as \(\mathcal {O}(\alpha )\) counterterm insertions in Born terms of \(\mathcal {O}(\alpha _{\mathrm {s}}^{P}\alpha ^{Q-1})\).
3.3.1 QCD renormalisation
The SM parameters that involve one-loop counterterms of \(\mathcal {O}(\alpha _{\mathrm {s}})\) are the strong coupling, the quark masses, and the related Yukawa couplings.
Strong coupling The renormalisation of the strong coupling constant is carried out in the \(\overline{\text {MS}}\) scheme, and can be matched in a flexible way to the different flavour-number schemes that are commonly used in NLO QCD calculations. To this end, the full set of light and heavy quarks that contribute to one-loop amplitudes and counterterms is split into a subset of active quarks (\(q\in {\mathcal {Q}}_{\mathrm {active}}\)) and a remaining subset of decoupled quarks (\(q\notin {\mathcal {Q}}_{\mathrm {active}}\)). Active quarks with mass \(m_q\ge 0\) are assumed to contribute to the evolution of \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\) above threshold. Thus they are renormalised via \(\overline{\text {MS}}\) subtraction at the scale \(\mu =\max (\mu _\mathrm{R},m_q)\). The remaining heavy quarks (\(q\notin {\mathcal {Q}}_{\mathrm {active}}\)) are assumed to contribute only to loop amplitudes and counterterms, but not to the running of \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\). Thus, they are renormalised in the so-called decoupling scheme, which corresponds to a subtraction at zero momentum transfer.
The number of active and decoupled quarks included in (3.46) is determined as explained in the following.
Choice of flavour-number scheme In NLO QCD calculations, the logarithms of \(\mu _\mathrm{R}\) in the counterterm (3.46)–(3.47) should cancel the leading-order \(\mu _\mathrm{R}\) dependence associated with \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\). To this end, the number \(N_{q,{\mathrm {active}}}\) of active quark flavours in (3.46) should be set equal to the number \(N_{\mathrm {F}}\) corresponding to the flavour-number scheme of the calculation at hand. More precisely, when using a running \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\) with \(N_{\mathrm {F}}\) quark flavours, the user^{20} should set \(N_{q,{\mathrm {active}}}=N_{\mathrm {F}}\). In variable-flavour number schemes, \(N_{\mathrm {F}}\) corresponds to the maximum number of quark flavours in the evolution, and typically \(N_{\mathrm {F}}=4,5\) or 6.^{21}
Total number of quark flavours By default, most public OpenLoops libraries involve quark-loop contributions with \(N_{q,{\mathrm {loop}}}=6\) quark flavours. Such libraries can be used for NLO calculations in any flavour-number scheme with \(N_{\mathrm {F}}=N_{q,{\mathrm {loop}}}\) or \(N_{\mathrm {F}}<N_{q,{\mathrm {loop}}}\). In the latter case, heavy-quark loop contributions that do not contribute to the evolution of \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\) are consistently accounted for by the \(N_{q,{\mathrm {loop}}}-N_{\mathrm {F}}\) decoupled quarks in the one-loop matrix elements.
Extra libraries without top-quark loops (\(N_{q,{\mathrm {loop}}}=5\)) can be easily generated upon request and are publicly available for selected processes. When available, libraries with \(N_{q,{\mathrm {loop}}}<6\) can be used by setting the parameter \(\texttt {nf}\) (default=6) to the desired value of \(N_{q,{\mathrm {loop}}}\) at the moment of the process registration.
Renormalisation and regularisation scales At the level of the user interface, the UV and IR regularisation scales are treated as a common scale \(\mu _{D}=\mu _{\mathrm {UV}}=\mu _{\mathrm {IR}}\), and the logarithms of \(\mu _{\mathrm {UV}}^2/\mu _{\mathrm {IR}}^2\) in (3.55) are set to zero. In the literature, also the logarithms of \(\mu _{\mathrm {UV}}^2/\mu _\mathrm{R}^2\) in (3.46)–(3.47) are often omitted by assuming \(\mu _{\mathrm {UV}}=\mu _\mathrm{R}\) in the \(\overline{\text {MS}}\) scheme. On the contrary, in OpenLoops the values of \(\mu _{D}\) and \(\mu _\mathrm{R}\) are controlled by two independent parameters, mureg and muren, respectively. Their default values are \(\mu _{D}=\mu _\mathrm{R}=100\) GeV. For convenience it is possible to simultaneously set \(\mu _\mathrm{R}=\mu _{D}=\mu \) by means of the auxiliary OpenLoops parameter mu. As described in Sect. 4.3, variations of \(\mu _\mathrm{R}\) and \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\) can be carried out in a very efficient way in OpenLoops 2.
3.3.2 EW renormalisation
The renormalisation of UV divergences in the EW sector is based on the scheme of [37] for on-shell particles, and on the complex-mass scheme [38] for the treatment of off-shell unstable particles. In OpenLoops 2 these two schemes are combined into a flexible renormalisation scheme that makes it possible to deal with processes such as \(pp\rightarrow t {\bar{t}} \ell ^+ \ell ^-\), where some unstable particles (\(t,{\bar{t}}\)) are treated as on-shell external states, while other ones (Z) play the role of intermediate resonances. This is achieved through a refined definition of field-renormalisation constants, and by adapting the mass-renormalisation prescriptions for unstable particles on a particle-by-particle basis, depending on whether the individual width parameters \(\varGamma _i\) are set to non-zero values or not by the user. As explained in the following, the \(\mathcal {O}(\alpha )\) renormalisation in OpenLoops involves also a non-standard treatment of \(\varDelta \alpha (M_Z^2)\) and special features related to external photons.
Wave functions The wave-function renormalisation constants (WFRCs) \(\delta Z_{ij}\) are defined in a way that one-loop propagators do not mix, and their residues are normalised to one. Thus renormalised amplitudes correspond directly to S-matrix elements and do not require additional LSZ factors. On the one hand, due to the presence of absorptive contributions and complex parameters, in the complex-mass scheme the \(\delta Z_{ij}\) constants can acquire complex values. On the other hand, the WFRCS for on-shell particles are usually defined as real parameters [37]. As explained in detail below, in OpenLoops these two approaches are reconciled by implementing WFRCs in a way that is consistent with [37] when the width parameters \(\varGamma _i\) are set to zero for all particles, while imaginary \(\delta Z_{ij}\) contributions are taken into account wherever they are strictly needed for the consistency of the complex-mass scheme at \(\mathcal {O}(\alpha )\).
- (i)
complex_mass_scheme=1 (default) corresponds to the implementation described above: the complex-mass counterterms (3.62)–(3.65) are used when \(\varGamma _i>0\), and the on-shell mass counterterms (3.71)–(3.73) are used when \(\varGamma _i=0\), while for WFRCs the generic formulas (3.76)–(3.80) are applied. As discussed above, this flexible approach guarantees a consistent one-loop description of processes like \(pp\rightarrow t {\bar{t}} \ell ^+ \ell ^-\), where unstable particles occur both as internal resonances and as on-shell external states.
- (ii)
complex_mass_scheme=0 keeps the complex masses (3.26) unchanged but deactivates the complex-mass scheme at the level of all \(\mathcal {O}(\alpha )\) counterterms: for mass counterterms the on-shell formulas (3.71)–(3.73) are used throughout; moreover, the \({\widetilde{{\mathrm {Re}}}}\) operations in (3.71)–(3.73) and (3.76)–(3.80) are replaced by a complete truncation of the imaginary parts at the level of the full counterterms. This option is implemented for validation purposes. Depending on the process, it can result in incomplete pole cancellations or other inconsistencies, in particular when internal or external particles with \(\varGamma _i>0\) are present.
- (iii)
complex_mass_scheme=2 corresponds to the implementation of the complex-mass scheme in Recola [20]. In this case all mass counterterms are evaluated with the complex-mass scheme formulas (3.62)–(3.65), while all \({\mathrm {Re}}\) and \({\widetilde{{\mathrm {Re}}}}\) operators are removed from (3.76)–(3.80), i.e. all imaginary parts of WFRCs are kept exact.
- (i)\({\varvec{\alpha (0)}}\)-scheme: the parameter \(\alpha \) is identified with the strength of the photon coupling at \(Q^2\rightarrow 0\). The resulting counterterm reads$$\begin{aligned} \delta Z_e \vert _{\alpha (0)}&= -\frac{1}{2}\,{\mathrm {Re}}\,\left( \delta Z_{AA} + \frac{s_W}{c_W} \delta Z_{ZA}\right) \nonumber \\&= \frac{1}{2}\,{\mathrm {Re}}\, \bigg [\varPi ^{\gamma \gamma }_{\mathrm{heavy}}(0) + \varPi ^{\gamma \gamma }_{{\mathrm{light}}}\left( M_Z^2\right) \nonumber \\&\quad + \varDelta \alpha (M_Z^2)- \frac{2 s_W}{c_W} \frac{\varSigma _\mathrm {T}^{AZ}(0)}{\mu _Z^2}\bigg ]. \end{aligned}$$(3.87)
- (ii)\({\varvec{G}}_{{\varvec{\mu }}}\)-scheme: the QED coupling is related to the Fermi constant through (3.30). This relation can be connected to the \(\alpha (0)\)-scheme viawhere \(\varDelta r\) represents the radiative corrections to the muon decay, i.e. to the Fermi constant, in the \(\alpha (0)\)-scheme [37]. This leads to the \(G_\mu \)-scheme counterterm$$\begin{aligned}&\frac{\alpha \vert _{G_{\mu }}}{\big \vert s_W^2 \mu _W^2\big \vert } = \frac{\sqrt{2}G_\mu }{\pi } = \alpha (0)\left| \frac{1+\varDelta r}{s_W^2 \mu _W^2}\right| , \end{aligned}$$(3.88)Note that, since \(\alpha \vert _{G_{\mu }}\) is effectively defined at the EW scale, its counterterm (3.89) does not depend on \(\varPi ^{\gamma \gamma }(0)\).$$\begin{aligned} \delta Z_e \vert _{G_{\mu }}&= \delta Z_e \vert _{\alpha (0)} - \frac{1}{2}{\mathrm {Re}}\left( \varDelta r\right) \nonumber \\&=\frac{1}{2}\,{\mathrm {Re}}\, \bigg \{ \frac{\delta s_W^2}{s_W^2} +\frac{\delta \mu _W^2-\varSigma ^{W}_\mathrm {T}(0)}{\mu _W^2} \nonumber \\&\quad {}-\frac{\alpha }{\pi s_W^2}\left[ \frac{C_\epsilon }{{\varepsilon }_{\mathrm {UV}}}+ \ln \left( \frac{\mu ^2_{\mathrm {UV}}}{\mu _Z^2}\right) + \frac{3}{2}\right. \nonumber \\&\quad \left. + \frac{7-12 s_W^2}{8 s_W^2} \ln \left( \frac{\mu _W^2}{\mu _Z^2}\right) \right] \bigg \}. \end{aligned}$$(3.89)
- (iii)\({\varvec{\alpha }}({\varvec{M}}_{\varvec{Z}}^\mathbf{2})\)-scheme: the photon coupling is defined as the strength of the pure QED interaction at \(Q^2=M_Z^2\). This corresponds to the countertermAlso in this case \(\varPi ^{\gamma \gamma }(0)\) drops out.$$\begin{aligned}&\delta Z_e \vert _{\alpha (M_Z^2)} \nonumber \\&\quad =\,\delta Z_e \vert _{\alpha (0)} -\frac{\varDelta \alpha (M_Z^2)}{2} \,=\, \frac{1}{2}\,{\mathrm {Re}}\, \bigg [\varPi ^{\gamma \gamma }_\mathrm{heavy}(0) \nonumber \\&\quad \qquad {}+\varPi ^{\gamma \gamma }_{{\mathrm{light}}}(M_Z^2) - \frac{2 s_W}{c_W} \frac{\varSigma _\mathrm {T}^{AZ}(0)}{\mu _Z^2} \bigg ]. \end{aligned}$$(3.90)
PDG identifiers for photons and switchers that control the coupling factors and renormalisation constants for the different types of external photons introduced in Sect. 3.2. The high-energy coupling \(\alpha _{\mathrm {off}}\) is defined in (3.34). If the switchers are set to zero (\(\hbox {default}=1\)) the standard user-defined coupling \(\alpha \) is used, and the related \(\delta Z^{(\mathrm {on/off})}\) factors are deactivated. As indicated in the last column, contributions from collinear \(\gamma \rightarrow f {\bar{f}}\) splittings are included in Catani–Seymour’s \({{\mathbf {I}}}\)-operator (see Sect. 3.4) only for off-shell photons
Photon type | iPDG | Switcher (\(1=\hbox {on}, 0=\hbox {off}\)) | Coupling | \(\varDelta \alpha \) | \(\gamma \rightarrow f {\bar{f}}\) |
---|---|---|---|---|---|
Unresolved | 22 | \(\alpha \) | \(\varDelta \alpha (M_Z^2)\) | Off | |
On-shell | 2002 | onshell_photons_lsz | \(\alpha (0)\) | \(\varDelta \alpha (M_Z^2)\) | off |
Off-shell | \(-\) 2002 | offshell_photons_lsz | \(\alpha _{\mathrm {off}}\) | \(\varDelta \alpha ^{\mathrm {reg}}(M_Z^2)\) | on |
- (i)For on-shellphotons the coupling \(\alpha (0)\) is used. Thus,and \(\delta K^{({\mathrm {on}})}_\gamma =2\,\delta Z_e\vert _{\alpha (0)}+\delta Z_{AA}\) yields the correct coupling counterterm \(\delta Z_e\vert _{\alpha (0)}\). Note that, as a result of the choice of a low-energy coupling, the \(\varDelta \alpha (M_Z^2)\) contributions to \(\delta Z_{AA}\) and \(\delta Z_e\vert _{\alpha (0)}\) cancel out in \(\delta K^{({\mathrm {on}})}_\gamma \).$$\begin{aligned} \delta Z^{({\mathrm {on}})}_\gamma&= 2\left[ \delta Z_e \vert _{\alpha (0)} - \delta Z_e\right] , \end{aligned}$$(3.93)
- (ii)For off-shellphotons the high-energy coupling \(\alpha _{\mathrm {off}}\) defined in (3.34) is used. As a result, the \(\varDelta \alpha (M_Z^2)\) contribution to \(\delta Z_{AA}\) remains uncancelled, and the renormalised scattering amplitude depends on large logarithms of the light-fermion masses. In photon-induced hadronic collisions, such logarithmic mass singularities are cancelled by collinear singularities associated with virtual \(\gamma \rightarrow f {\bar{f}}\) splitting contributions to the photon-PDF counterterm [36] (see Sect. 3.4). The latter are typically handled in dimensional regularisation with massless light fermions, which results in collinear singularities of the form \(1/{\varepsilon }_{\mathrm {IR}}\). For consistency, the same regularisation must be used also for the related light-fermion contributions from \(\varDelta \alpha (M_Z^2)\). To this end, the finite renormalisation factor for off-shell photons is defined aswhere the counterterm \(\delta Z_e \vert _{\alpha _{\mathrm {off}}}\) corresponds to the renormalisation scheme associated with \(\alpha _{\mathrm {off}}\) according to (3.34)–(3.35), while \({\mathcal {D}}\alpha ^{(\mathrm {reg})}(M_Z^2)\), defined in (3.85), converts \(\varDelta \alpha (M_Z^2)\) into its dimensionally regularised variant (3.84). The resulting overall renormalisation factor for off-shell photons reads$$\begin{aligned} \delta Z^{({\mathrm {off}})}_\gamma&= 2\,\left[ \delta Z_e \vert _{\alpha _{\mathrm {off}}} - \delta Z_e\right] -{\mathcal {D}}\alpha ^{(\mathrm {reg})}(M_Z^2), \end{aligned}$$(3.94)with$$\begin{aligned} \delta K^{({\mathrm {off}})}_\gamma&= 2\delta Z_e \vert _{\alpha _{\mathrm {off}}} +\delta Z^{(\mathrm {reg})}_{AA}, \end{aligned}$$(3.95)$$\begin{aligned} \delta Z^{(\mathrm {reg})}_{AA}&= \delta Z_{AA}- {\mathcal {D}}\alpha ^{(\mathrm {reg})}(M_Z^2)\nonumber \\&=\, {-}\left[ \varPi ^{\gamma \gamma }_{\mathrm{heavy}}(0) {+} \varPi ^{\gamma \gamma }_{{\mathrm{light}}}\left( M_Z^2\right) {+} \varDelta ^{(\mathrm {reg})} \alpha (M_Z^2)\right] . \end{aligned}$$(3.96)
For the various \(\varDelta \alpha (M_Z^2)\) terms that enter the factors \(\delta Z_e\), \(\delta Z_{AA}\) and \(\delta Z^{({\mathrm {on/off}})}_\gamma \) associated with external photons, depending on the type of photon, either the numerical expression (3.83) or the dimensionally regularised form (3.84) are used as explained above. Alternatively, it is possible to enforce the usage of \(\alpha ^{(\mathrm {reg})}(M_Z^2)\) in all terms associated with external photons by setting all_photons_dimreg=1 (default=0).
3.4 Infrared subtraction
One-loop matrix elements with on-shell external legs involve divergences of IR (soft and collinear) origin, which take the form of double and single \(1/{\varepsilon }_{\mathrm {IR}}\) poles in \(D=4-2{\varepsilon }_{\mathrm {IR}}\) dimensions. In OpenLoops such divergences can be subtracted through an automated implementation of Catani–Seymour’s \({{\mathbf {I}}}\)-operator that accounts for QCD singularities [39, 40] as well as for singularities of QED origin [36, 41, 42, 43, 44]. The singular part of the \({{\mathbf {I}}}\)–operator is universal and can be used to check the cancellation of IR poles in any one-loop calculation. Moreover, the full \({{\mathbf {I}}}\)–operator provides a useful building block for NLO calculations based on Catani–Seymour’s dipole subtraction.
In addition to the \({{\mathbf {I}}}\)-operator, as documented in Sect. 4.3 and Appendix A.5, OpenLoops provides also routines for more general building blocks of IR divergences, namely colour- and gluon-helicity correlated Born matrix elements for QCD singularities, and corresponding charge- and photon-helicity correlations for QED singularities.
Here \(Q_i\) denotes the electromagnetic charge of parton i, while \(n_{\mathrm{I},j}\) is the number of initial-state partons in \({\mathcal {S}}_{\mathrm{in}}\backslash \{j\}\). By definition, on-shell photons do not undergo collinear splittings at NLO. Thus, \({\mathcal {T}}^{\mathrm {QED}}_{jk}\) vanishes when the emitter j is an on-shell photon. Vice versa, off-shell photons are subject to final-state \(\gamma \rightarrow f\bar{f}\) and initial-state \(f\rightarrow f\gamma \) splittings at NLO. The related \(-1/n_{\mathrm{I},j}\) term in (3.100) is such that the recoil of the collinear radiation is shared by all initial-state partons that belong to \({\mathcal {S}}_{\mathrm{in}}\backslash \{j\}\) [36].
Here \(N_{f,u},N_{f,d},N_{f,l}\) are the numbers of massless up-type quarks, down-type quarks and leptons, respectively, while \(N_f=N_{f,u}+N_{f,d}\). Since massive external legs induce only soft singularities, external \(W^\pm \)-bosons are treated in the same way as massive fermions with mass \(M_W\) and charge \(\pm 1\)
Interaction | j | \({\mathcal {Q}}_j^{2,{\mathrm {QCD}}/\mathrm {QED}}\) | \(\gamma _j^{{\mathrm {QCD}}/\mathrm {QED}}\) | \(K_j^{{\mathrm {QCD}}/\mathrm {QED}}\) |
---|---|---|---|---|
QCD | Quark | \(C_F\) | \(\frac{3}{2} C_F\) | \((\frac{7}{2}-\frac{\pi ^2}{6})C_F\) |
QCD | Gluon | \(C_A\) | \(\frac{11}{6}C_A - \frac{2}{3}T_R N_f\) | \((\frac{67}{18} - \frac{\pi ^2}{6})C_A -\frac{10}{9}T_RN_f\) |
QED | Fermion or \(W^\pm \) | \(Q_j^2\) | \(\frac{3}{2}Q_j^2\) | \((\frac{7}{2}-\frac{\pi ^2}{6}) Q_j^2\) |
QED | \(\gamma \) | 0 | \(- \frac{2}{3}\Big [ N_C \big (N_{f,u}\, Q_u^2 +N_{f,d}\, Q_d^2\big )+ N_{f,l}\,Q_l^2\Big ]\) | \(\frac{5}{3}\gamma _{\gamma }^{\mathrm{QED}}\) |
4 Overview of the program
This section describes various aspects that are relevant for the usage of OpenLoops in the context of external programs. Once installed and linked to an external program, OpenLoops can be controlled through its native interfaces for Fortran and C/Open image in new window codes, or using the standard BLHA interface [45, 46]. In the following, we introduce various functionalities of the OpenLoops interfaces, such as the registration of processes, the setting of parameters, and the evaluation of different types of matrix elements. In doing so we will always refer to the names of the relevant Fortran interface functions. The corresponding C functions are named in the same way with an extra ol_ prefix.
Further technical aspects, such as the signatures of the interfaces, can be found in Appendix A and Appendix B. As discussed there, the multi-purpose Monte Carlo programs Munich/Matrix [50], Sherpa [26, 47], Open image in new window [32], Powheg-Box [27], Whizard [49] and Geneva [48] dispose of built-in interfaces that control all relevant OpenLoops functionalities in a largely automated way requiring only little user intervention. Besides the Fortran and C/Open image in new window interfaces the OpenLoops package also contains a Python wrapper and a command line tool. Further details and examples of the Python interface are given in Appendix B.4.
The OpenLoops program itself is written in Fortran and consists of process-independent main code and process-dependent code provided in the form of process libraries, which can be downloaded and automatically installed within the OpenLoops program for a wide range of processes in the Standard Model (SM) and Higgs effective theory (HEFT), as detailed in the following. The process libraries are automatically generated based on a (private) process generator implemented in Mathematica.
4.1 Download and installation
4.1.1 Installation of the main program
This section describes the installation of the process-independent part of the OpenLoops program, which is denoted as base code. The calculation of specific scattering amplitudes requires additional process-specific libraries, denoted as process code. Their installation is discussed in Sect. 4.1.2.
Prerequisites To install OpenLoops a Fortran compiler (gfortran 4.6 or later, or ifort) and Python 2.7 or 3.5 or later are needed.
where the user can also find a detailed list of the available process libraries and extra documentation, as well as an up-to-date version of this paper.
The default compiler is gfortran, alternatively ifort can be used. To change the compiler and set various other options, rename the sample configuration file openloops.cfg.tmpl in the OpenLoops directory to openloops.cfg and set the options in there. The sample configuration file lists various available options and describes their usage.
4.1.2 Installation of process libraries
The calculation of scattering amplitudes for specific processes requires the installation of corresponding process libraries. The available collection of OpenLoops process libraries supports the calculation of QCD and EW corrections for a few hundred different partonic reactions, which cover essentially all interesting processes at the LHC, as well as several lepton-collider processes. This includes \(pp\rightarrow \) jets, \(t{\bar{t}}\hbox {+jets}\), \(V\hbox {+jets}\), \(VV\hbox {+jets}\), \(HV\hbox {+jets}\), \(H\hbox {+jets}\) and various other classes of processes with a variable number of extra jets. Process libraries for a large variety of loop-induced processes such as \(gg\rightarrow \ell \ell \ell \ell \hbox {+jets}\), \(gg\rightarrow HV\hbox {+jets}\), \(gg\rightarrow HH(H)\hbox {+jets}\), etc. are also available.
New processes libraries, especially with EW corrections, are continuously added to the collection by the authors. Moreover, extra processes libraries can be easily made available upon request, either through an online form on the OpenLoops webpage or by contacting the authors. In particular this allows for the generation of dedicated process libraries tailored to specific user requirements. For example, it is possible to generate dedicated process libraries with special filters for the selection of certain classes of diagrams/topologies or various approximations related to the treatment of heavy-quark flavours, the expansion in the number of colours, the selection of resonances, non-diagonal CKM matrix elements, and so on.
Download and installation The web page
https://openloops.hepforge.org/process_library.php
where \({\texttt {<processes>}}\) is either a predefined process collection (see below) or a list of white-space or comma separated names of process libraries. A single process library typically contains the full set of parton-level scattering amplitudes that is needed for the calculation of a certain family of hadron-collider processes, either at NLO QCD or including EW corrections. For instance, the libraries named ppllll and ppllll_ew include, respectively, the NLO QCD and NLO EW matrix elements for the production of four leptons, i.e. the processes \(pp\rightarrow \ell _i^+\ell _i^-\ell _k^+\ell _k^-\), \(\ell _i^+\ell _i^-\ell _k^+\nu _k\), \(\ell _i^+\ell _i^-{\bar{\nu }}_k\ell ^-_k\), \(\ell _i^+\ell _i^-{\bar{\nu }}_k\nu _k\), \(\ell _i^+\nu _i\ell ^+_k\nu _k\), \(\ell _i^+\nu _i{\bar{\nu }}_k\ell ^-_k\), \({\bar{\nu }}_i\ell _i^-{\bar{\nu }}_k\ell ^-_k\), \(\ell _i^+\nu _i{\bar{\nu }}_k\nu _k\), and \({\bar{\nu }}_i\ell _i^-{\bar{\nu }}_k\nu _k\), with lepton flavours \(i\ne k\) or \(i=k\).
Each process library includes all relevant LO and NLO ingredients for the partonic processes at hand, i.e. all Born, one-loop and real-emission amplitudes at the specified order. More precisely, NLO QCD libraries contain LO contributions of a given order \(\alpha _{\mathrm {s}}^p\alpha ^q\) and corrections of order \(\alpha _{\mathrm {s}}^{p+1}\alpha ^q\), while NLO EW libraries contain the full tower of LO and NLO contributions apart from the NLO terms with the highest possible order in \(\alpha _{\mathrm {s}}\). Real-emission matrix elements are available throughout, but are not installed by default. This can be changed by using the option compile_extra=1 (\(\hbox {default}=0\)) when installing the process. This option can also be set in the openloops.cfg file in order to enable real corrections for every process installation.
With the libinstall command it is also possible to install pre-defined or user-defined process collections. The pre-defined collection lhc.coll covers the most relevant LHC processes.^{28} In particular, it includes matrix elements for \(V+\hbox {jets}\), \(VV+\hbox {jets}\), \(t{\bar{t}}+\hbox {jets}\), \(HV+\hbox {jets}\) and \(H+\hbox {jets}\) (for finite and infinite \(m_t\)), where V stands for photons as well as for the various leptonic decay products of off-shell Z and \(W^\pm \) bosons. Additional user-defined collections can be created as plain text files with the file extension .coll, listing the desired process-library names, one per line.
4.2 Selection of processes and perturbative orders
Values of amptype to register different types of perturbative contributions and corresponding probability densities that can be computed by OpenLoops. Objects of LO and NLO kind are evaluated at order \(\alpha _{\mathrm {s}}^p\alpha ^q\) and \(\alpha _{\mathrm {s}}^P\alpha ^Q\), respectively, according to the values p, q, P, Q of the LO and NLO power selectors in Table 5. The symbols \({\mathcal {B}}\) and \({\mathcal {C}}\) stand for the various spin and colour/charge correlators defined in Sect. 4.4
amptype | Amplitude type | LO output | NLO output |
---|---|---|---|
1 | Tree–tree | \(\mathcal {W}_{00}^{(p,q)}\), \({\mathcal {C}}^{(p,q)}_{00,{\scriptscriptstyle \mathrm {LO}}}\), \({\mathcal {B}}^{(p,q)}_{00,{\scriptscriptstyle \mathrm {LO}}}\), | |
11 | Tree–loop | \(\mathcal {W}_{00}^{(p,q)}\) | \(\mathcal {W}_{01}^{(P,Q)}\), \(\mathcal {W}_{00, \hbox {I-op}}^{(P,Q)}\), \({\mathcal {C}}^{(P,Q)}_{01,{\scriptscriptstyle \mathrm {NLO}}}\), \({\mathcal {B}}^{(P,Q)}_{01,{\scriptscriptstyle \mathrm {NLO}}}\) |
12 | Loop–loop | \(\mathcal {W}_{11}^{(p,q)}\), \({\mathcal {C}}^{(p,q)}_{11,{\scriptscriptstyle \mathrm {LO}}}\), \({\mathcal {B}}^{(p,q)}_{11,{\scriptscriptstyle \mathrm {LO}}}\), | \(\mathcal {W}_{11,\hbox {I-op}}^{(P,Q)}\) |
- (a)
Setting \(\texttt {order\_ew}=q\) selects contributions of fixed EW order, i.e. LO terms of \(\mathcal {O}(\alpha _{\mathrm {s}}^p\alpha ^q)\) and NLO QCD corrections of \(\mathcal {O}(\alpha _{\mathrm {s}}^{p+1}\alpha ^q)\). In this case, the QCD order p is automatically fixed according to \(p+q=N_{\mathrm {p}}-2\).
- (b)
Similarly, \(\texttt {order\_qcd}=p\) selects a fixed QCD order, i.e. LO terms of \(\mathcal {O}(\alpha _{\mathrm {s}}^p\alpha ^q)\) and NLO EW corrections of \(\mathcal {O}(\alpha _{\mathrm {s}}^{p}\alpha ^{q+1})\). In this case, q is automatically derived from \(p+q=N_{\mathrm {p}}-2\).
- (c)
Alternatively, NLO terms of \(\mathcal {O}(\alpha _{\mathrm {s}}^{P}\alpha ^Q)\) can be selected by setting \(\texttt {loop\_order\_qcd}=P\)or\(\texttt {loop}{} \texttt {\_order\_ew}=Q\). This option is supported only for the evaluation of tree-loop interferences (amptype=11). In that case, the output includes also the dominant underlying Born contribution of \(\mathcal {O}(\alpha _{\mathrm {s}}^{p}\alpha ^q)\), which is chosen between \(\mathcal {O}(\alpha _{\mathrm {s}}^{P}\alpha ^{Q-1})\) and \(\mathcal {O}(\alpha _{\mathrm {s}}^{P}\alpha ^{Q-1})\) as indicated in Fig. 4. When the loop order P or Q is specified, the complementary order Q or P is fixed internally according to \(P+Q=N_{\mathrm {p}}-1\).
Selection of the orders \(\alpha _{\mathrm {s}}^p\alpha ^q\) and \(\alpha _{\mathrm {s}}^P\alpha ^Q\) for the LO and NLO objects defined in Table 4. Each selector takes one of the powers p, q, P, Q as input and derives all other powers as indicated in columns 2–5. The QCD and EW coupling powers at LO and NLO are related through \(p+q=N_{\mathrm {p}}-2\) and \(P+Q=N_{\mathrm {p}}-1\), where \(N_{\mathrm {p}}\) is the number of external particles. The \(\texttt {loop\_order}\) selectors are supported only for amptype=11. They return the desired loop–tree interference of \(\mathcal {O}(\alpha _{\mathrm {s}}^P\alpha ^Q)\) together with the dominant underlying squared Born term of \(\mathcal {O}(\alpha _{\mathrm {s}}^{p}\alpha ^q)\) whose powers, \((p,q) = (p_{\tiny \hbox {Born}},q_{\tiny \hbox {Born}}) = (P-1,Q)\) or \((P,Q-1)\), are selected in a unique way as indicated in Fig. 4
Power selection\(\backslash \)derived powers | LO power \(\alpha _{\mathrm {s}}^p\alpha ^q\) | NLO power \(\alpha _{\mathrm {s}}^P\alpha ^Q\) | ||
---|---|---|---|---|
\(\texttt {order\_qcd}=p\) | \(p\) | \(N_{\mathrm {p}}-p-2\) | \(p\) | \(q+1\) |
\(\texttt {order\_ew}=q\) | \(N_{\mathrm {p}}-q-2\) | \(q\) | \(p+1\) | \(q\) |
\(\texttt {loop\_order\_qcd}=P\) | \(p_{\tiny \hbox {Born}}\) | \(q_{\tiny \hbox {Born}}\) | \(P\) | \(N_{\mathrm {p}}-P-1\) |
\(\texttt {loop\_order\_ew}=Q\) | \(p_{\tiny \hbox {Born}}\) | \(q_{\tiny \hbox {Born}}\) | \(N_{\mathrm {p}}-P-1\) | \(Q\) |
Particle identifiers (PID) for process specification in OpenLoops. The numerical and string PID representations can be mixed. As explained in Sect. 3.2, for an optimal treatment of the coupling of on-shell and off-shell hard external photons the special PIDs \(\pm 2002\) should be used
Particle | \(q_d/{{\tilde{q}}}_d\) | \(q_u/{{\tilde{q}}}_u\) | \(q_s/{{\tilde{q}}}_s\) | \(q_c/{{\tilde{q}}}_c\) | \(q_b/{{\tilde{q}}}_b\) | \(q_t/{{\tilde{q}}}_t\) |
---|---|---|---|---|---|---|
PID | 1/-1 | 2/-2 | 3/-3 | 4/-4 | 5/-5 | 6/-6 |
String-PID | \(\texttt {d/d}\sim \) | \(\texttt {u/u}\sim \) | \(\texttt {s/s}\sim \) | \(\texttt {c/c}\sim \) | \(\texttt {b/b}\sim \) | \(\texttt {t/t}\sim \) |
Particle | \(l_e/{{\tilde{l}}}_e\) | \(\nu _{e}/{{\tilde{\nu }}}_{e}\) | \(l_{\mu }/{{\tilde{l}}}_{\mu }\) | \(\nu _{\mu }/{{\tilde{\nu }}}_{\mu }\) | \(l_{\tau }/{{\tilde{l}}}_{\tau }\) | \(\nu _{\tau }/{{\tilde{\nu }}}_{\tau }\) |
---|---|---|---|---|---|---|
PID | 11/-11 | 12/-12 | 13/-13 | 14/-14 | 15/-15 | 16/-16 |
String-PID | e-/e+ | \(\texttt {ve/ve}\sim \) | mu-/mu+ | \(\texttt {vm/vm}\sim \) | ta-/ta+ | \(\texttt {vt/vt}\sim \) |
Particle | g | \(\gamma \) | On-/off-\(\gamma \) | Z | \(W^{\pm }\) | Higgs |
---|---|---|---|---|---|---|
PID | 21 | 22 | 2002/-2002 | 23 | 24/-24 | 25 |
String-PID | g | a | aon/aoff | \(\texttt {z}\) | \(\texttt {w+/w-}\) | h |
Together with the external particles, also a specific type of perturbative output (amptype) should be selected. As summarised in Table 4, the available options correspond to the various scattering probability densities defined in (2.1)–(2.3), i.e. squared tree amplitude (\(\mathcal {W}_{00}\)) tree–loop interference (\(\mathcal {W}_{01}\)), and squared one-loop amplitude (\(\mathcal {W}_{11}\)), but each amptype supports also the calculation of various related objects.
4.3 Evaluation of scattering amplitudes
In this section we introduce various OpenLoops interface functions for the evaluation of the scattering probability densities (2.1)–(2.3), the \({{\mathbf {I}}}\)-operators (3.97), and some of their building blocks.
The input required by the various interface functions consists of a phase-space point together with the integer identifier for the desired (sub)process. The output is always returned according to the normalisation conventions of Eqs. (2.1)–(2.3), i.e. symmetry factors, external colour and helicity sums, and average factors are included throughout. This holds also for the interface functions discussed in Sects. 4.4–4.5. The syntax of the various interfaces is detailed in Appendix A.
In general, the output depends on the values of all relevant physical and technical input parameters (see Sects. 3.2–3.3) at the moment of calling the actual OpenLoops interface routine. All parameters and settings are initialised with physically meaningful default values, which can be updated at any moment by means of set_parameter. In principle, all parameters can be changed before any new amplitude evaluation. As explained below, thanks to a new automated scale-variation system, scattering amplitudes can be re-evaluated multiple times with different values of \(\mu _\mathrm{R}\) and \(\alpha _{\mathrm {s}}\) in a very efficient way.
The calculation of the probability densities (2.1)–(2.3) is supported by the following interfaces.
Squared born amplitudes\(\mathcal {W}_{00}=\langle \mathcal {M}_0|\mathcal {M}_0\big \rangle \) are evaluated by the function evaluate_tree.
- (i)
The bare loop amplitudes\(\mathcal {W}_{01,4\mathrm {D}}\), with four- dimensional numerator, are evaluated by evaluate_loopbare, which returns a Laurent series similar to (4.2). As for evaluate_loop, pole residues are derived from the related UV and IR counterterms (default) or explicitly reconstructed, depending on the value of truepoles_on.
- (ii)
The UVcounterterms \(\mathcal {W}_{01,\mathrm {CT}}\) are evaluated by evaluate_loopct, which returns a Laurent series similar to (4.2). In this case, UV pole coefficients are always obtained via two-fold evaluation. The more efficient function evaluate_ct restricts the calculation of the counterterm to its finite part \(\mathcal {W}^{(0)}_{01,\mathrm {CT}}\).
- (iii)
The \(R_2\)rational part\(\mathcal {W}_{01,R_2}\) is free from UV and IR divergences. It is evaluated by evaluate_r2, which returns a single-valued output.
- (iv)
Tree–tree\({{\mathbf {I}}}\)-operator insertions, \(\mathcal {W}_{00, \hbox {I-op}}=\)\(=\langle M_0| {{\mathbf {I}}}(\{p\};{\varepsilon }_{\mathrm {IR}}) | M_0\big \rangle \), are evaluated by the function evaluate_iop. The output is a Laurent series similar to (4.2).
- (v)
The poles of all divergent building blocks of (4.4) can be accessed with a single call of evaluate_poles, which returns the residues of the \(1/{\varepsilon }_{\mathrm {UV}}\), \(1/{\varepsilon }_{\mathrm {IR}}\) and \(1/{\varepsilon }_{\mathrm {IR}}^2\) poles for each building block. In this case, irrespectively of the value of truepoles_on, all residues are always computed explicitly.
Squared loop amplitudes\(\mathcal {W}_{11}=\langle \mathcal {M}_1|\mathcal {M}_1\big \rangle \) are evaluated by the function evaluate_loop2. Since we assume that it is used for loop-squared processes, which are free from UV and IR divergences at LO, evaluate_loop2 returns a single-valued finite output. The calculation of \({{\mathbf {I}}}\)-operator insertions in loop-squared amplitudes, \(\mathcal {W}_{11,\hbox {I-op}}=\langle M_1| {{\mathbf {I}}}(\{p\};{\varepsilon }_{\mathrm {IR}}) | M_1\big \rangle \), is supported by evaluate_loop2iop. Since we assume loop-induced processes, the output is a Laurent series of type (4.2) with poles up to order \(1/{\varepsilon }^2\). In general, \(\mathcal {W}_{11}\) and \(\mathcal {W}_{11,\hbox {I-op}}\) are evaluated using only the finite part of \(\mathcal {M}_1\), and possible UV and IR poles are simply amputated at the level of \(\mathcal {M}_1\).
Efficient QCD scale variationsOpenLoops 2 implements a new automated system for the efficient assessment of QCD scale uncertainties. This system is designed for the case where scattering amplitudes are re-evaluated multiple times with different values of \(\mu _\mathrm{R}\) and \(\alpha _{\mathrm {s}}\), while all other input and kinematic parameters are kept fixed. This type of variations are automatically detected by keeping track, on a process-by process basis, of the pre-evaluated phase-space points, and possible variations of parameters. For each new phase-space point, matrix elements are computed from scratch and stored in a cache, which is used for \((\mu _\mathrm{R},\alpha _{\mathrm {s}})\) variations. In that case, the previously computed bare amplitude is reused upon appropriate rescaling of \(\alpha _{\mathrm {s}}\), and only the \(\mu _\mathrm{R}\)-dependent QCD counterterms are explicitly recomputed. This mechanism is implemented for both types of loop contributions (2.2)–(2.3).
4.4 Colour- and spin-correlators
This section presents interface functions for the evaluation of colour- and helicity-correlated quantities that are needed in the context of NLO and NNLO subtraction methods, both for tree- and loop-induced processes. For efficiency reasons, colour/spin correlations are always computed in combination with the related squared tree or loop matrix elements, in such a way that the former are obtained with a minimal CPU overhead.
4.5 Tree-level amplitudes in colour space
Particle and colour numbering scheme. The external particles are labelled through consecutive integers \(1,2,\dots ,N_{\mathrm {p}}\) according to the ordering (4.1) specified through the process registration. The symbols \(\sigma _k\) are used in (4.26)–(4.29) to represent the integer labels of external gluons, while \(a_{\sigma _k}\) are the corresponding colour indices. Similarly, \(\alpha _k\,(\beta _l)\) represent the integer labels of incoming quarks (antiquarks) or outgoing antiquarks (quarks), and their colour indices are \(i_{\alpha _k}\,(\bar{j}_{\beta _l})\). For the process considered in the table, \(q{\bar{q}} \rightarrow \gamma q \bar{q} {Z} g g g\), we have \((\alpha _1,\alpha _2)\,=\,(1,5)\), \((\beta _1,\beta _2)\,=\,(2,4)\), \((\sigma _1,\sigma _2,\sigma _3)\,=\,(\alpha _3,\alpha _4,\alpha _5)\,=\,(7,8,9)\). The last row illustrates the notation of the colour-flow basis. In this case, as explained in the text, the antiquark indices \(\beta _k\) are replaced by a permutation \({{\tilde{\alpha }}}_k=\pi (\alpha _k)\) of the quark indices according to the actual colour flow. Moreover, gluons are represented by a pair of indices (\(\alpha _k,{{\tilde{\alpha }}}_k\)) corresponding to a quark–antiquark pair
External particles | q | \({\bar{q}}\) | \(\scriptstyle \rightarrow \) | \(\gamma \) | q | \({{\bar{q}}}\) | Z | g | g | g |
---|---|---|---|---|---|---|---|---|---|---|
Integer labels | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
\(\alpha _1\) | \(\beta _1\) | \(\beta _2\) | \(\alpha _2\) | \(\sigma _{1}\) | \(\sigma _2\) | \(\sigma _3\) | ||||
\(\alpha _{3}\) | \(\alpha _4\) | \(\alpha _5\) | ||||||||
SU(3) indices | \(i_1\) | \(\bar{j}_{2}\) | \(\bar{j}_{4}\) | \(i_5\) | \(a_{7}\) | \(a_8\) | \(a_9\) | |||
Colour flow | \({\scriptstyle (\alpha _{1},0)}\) | \({\scriptstyle (0,{{\tilde{\alpha }}}_{1})}\) | \({\scriptstyle (0,0)}\) | \({\scriptstyle (0,{{\tilde{\alpha }}}_{2})}\) | \({\scriptstyle (\alpha _{2},0)}\) | \({\scriptstyle (0,0)}\) | \({\scriptstyle (\alpha _{3},{{\tilde{\alpha }}}_{3})}\) | \({\scriptstyle (\alpha _{4},{{\tilde{\alpha }}}_{4})}\) | \({\scriptstyle (\alpha _{5},{{\tilde{\alpha }}}_{5})}\) |
The explicit form of the colour-flow basis for a given process can be accessed through the interface function tree_colourflow, which returns an array of basis elements \(\{{\mathcal {C}}_i^{\mathrm {flow}}\}\) in a format corresponding to (4.35).
The interface functions described in this section are supported under amptype=1,11. So far they are implemented in a way that guarantees consistent results only for leading-QCD Born quantities, i.e. terms of order \(\alpha _{\mathrm {s}}^p\alpha ^q\) with maximal power p, which involve a single Born term of order \(g_\mathrm {s}^p e^q\).
4.6 Reduction methods and stability system
As discussed in Sect. 2.7, tree–loop interferences and squared loop amplitudes are computed using different methods for the reduction to scalar integrals and the treatment of related instabilities.
For all types of amplitudes, OpenLoops chooses default settings for the stability system that require adjustments only in very rare cases.
On-the-fly stability system For tree–loop interferences, with the only exception of the Higgs Effective Field Theory, the reduction to scalar integrals is based on the on-the-fly method and the stability system described in Sect. 2.7.2. Each processed object carries a cumulative instability estimator^{37} that is propagated through the algorithm and updated when necessary. If the estimated instability exceeds a threshold value, the object at hand and all subsequent operations connected to it are processed through the qp channel. The stability threshold is controlled by the interface parameter hp_loopacc, which plays the role of target numerical accuracy for the whole Born–loop interference \(\mathcal {W}_{01}\). Its default value is 8 and corresponds to \(\delta \mathcal {W}_{01}/\mathcal {W}_{01}\sim 10^{-8}\).
In order to find an optimal balance between CPU performance and numerical accuracy, certain aspects of the stability system can be activated or deactivated using the parameter hp_mode. Setting hp_mode=1 (default) enables all stability improvements described in Sect. 2.7.2 and is recommended for NLO calculations with hard kinematics. Setting hp_mode=2 activates qp also for additional types of rank-two Gram-determinant instabilities that occur exclusively in IR regions. This mode is supported only for QCD corrections and is recommended for real–virtual NNLO calculations. Finally, hp_mode=0 deactivates the usage of qp through the hybrid-precision system, while keeping all stability improvements of analytic type in dp.
Stability rescue system For tree–loop interferences in the Higgs Effective Field Theory, the reduction to scalar integrals is based on external libraries. The primary reduction library redlib1 (default: Coli -Collier) is used to evaluate all points in dp. The fraction stability_triggerratio (default: 0.2, meaning 20 %) of the points with the largest K-factor is re-evaluated with the secondary reduction library redlib2 (default: DD -Collier). If the relative deviation of the two results exceeds stability_unstable (default: 0.01, meaning 1 %), the point is re-evaluated in qp with CutTools including a qp scaling test to estimate the resulting accuracy. If the estimated relative accuracy \(\delta \mathcal {W}_{01}/\mathcal {W}_{01}\) in qp is less than stability_kill (default: 1, meaning 100 %), the result is set to zero, otherwise the smaller of the scaled and unscaled qp results is returned. The accuracy argument of the matrix element routines (e.g. evaluate_loop) returns the relative deviation of the Coli -Collier and DD -Collier results or, if qp was triggered, of the scaled and unscaled qp result. In case of a single dp evaluation, the accuracy argument is set to \(-1\).
Also squared loop amplitudes are reduced to scalar integrals using external libraries. To asses related instabilities, for all phase-space points the reduction is carried out twice, using redlib1 and redlib2. The option stability_kill2 (default: 10) sets the relative deviation of the two results beyond which the result is set to zero. Due to the double evaluation of all points, an accuracy estimate is always returned by the matrix element routine evaluate_loop2.
Setting redlib1 and redlib2, as well as various other options to control the stability system, is only possible in the so-called “expert mode”. Further details can be obtained from the authors upon request.
5 Technical benchmarks
In this section we present speed and stability benchmarks obtained with OpenLoops 2 and compare them with the performance of OpenLoops 1.
5.1 CPU performance
Runtimes for the calculation of the NLO QCD and NLO EW virtual corrections (with respect to the leading QCD Born order) for various partonic processes at the LHC. Timings are given per phase-space point, including colour and helicity sums, and averaged over a sample of random points generated with Rambo [77] at \(\sqrt{s}=1\) TeV without cuts. The measurements have been carried out on a single Intel i7-4790K @ 4.00GHz core using gfortran 7.4.0. The reference OpenLoops 2 timings (\(t^{\mathrm {def}}_{\mathrm {OL2}}\)) correspond to the on-the-fly approach with default stability settings, while \(t^{\mathrm {11\,digits}}_{\mathrm {OL2}}\) illustrates the CPU overhead caused by augmenting the hybrid-precision target accuracy from 8 to 11 digits. Default OpenLoops 1 timings (\(t^{\mathrm {preset2}}_{\mathrm {OL1}}\)) correspond to the recommended stability setting (preset=2), where tensor reduction is done with Coli-Collier and compared against DD-Collier for 20% of the points with the largest K-factor; differences beyond one percent between Coli-Collier and DD-Collier trigger qp re-evaluations with CutTools +OneLOop and a further stability test via qp-rescaling. For comparison, also OpenLoops 1 timings with disabled stability system (\(t^{\mathrm {no\, stab}}_{\mathrm {OL1}}\)) are shown within parentheses
Process | \(t^{\mathrm {def}}_{\mathrm {OL2}}\) [ms] | \(t^{\mathrm {11digits}}_{\mathrm {OL2}} /t^{\mathrm {def}}_{\mathrm {OL2}}\) | \(t^{\mathrm {preset2}}_{\mathrm {OL1}} \,(t^{\mathrm {no\,stab}}_{\mathrm {OL1}})/t^{\mathrm {def}}_{\mathrm {OL2}}\) | ||||
---|---|---|---|---|---|---|---|
QCD | EW | \(\frac{\mathrm {EW}}{{\mathrm {QCD}}}\) | QCD | EW | QCD | EW | |
\(gg\rightarrow t{\bar{t}}\) | 0.80 | 1.17 | 1.46 | 1.01 | 1.01 | 1.82 (1.67) | 2.22 (2.02) |
\(gg\rightarrow t{\bar{t}}g\) | 21.4 | 24.0 | 1.12 | 1.04 | 1.07 | 1.68 (1.56) | 2.16 (2.10) |
\(gg\rightarrow t{\bar{t}}gg\) | 600 | 582 | 0.97 | 1.15 | 1.22 | 2.18 (2.17) | 2.64 (2.59) |
\(gg\rightarrow t{\bar{t}}ggg\) | 21,145 | 16,928 | 0.80 | 1.09 | 1.14 | 2.59 (2.55) | 3.06 (3.06) |
\(u{\bar{u}}\rightarrow t{\bar{t}}\) | 0.23 | 0.43 | 1.87 | 1.0 | 1.02 | 1.22 (0.93) | 1.65 (1.37) |
\(u{\bar{u}}\rightarrow t{\bar{t}} g\) | 3.1 | 8.0 | 2.58 | 1.06 | 1.08 | 1.28 (1.19) | 1.36 (1.28) |
\(u{\bar{u}}\rightarrow t{\bar{t}} gg\) | 73 | 176 | 2.41 | 1.16 | 1.19 | 1.45 (1.45) | 1.64 (1.63) |
\(u{\bar{u}}\rightarrow t {\bar{t}}ggg\) | 2085 | 4862 | 2.33 | 1.26 | 1.28 | 1.88 (1.88) | 2.05 (2.04) |
\(b{\bar{b}} \rightarrow t{\bar{t}}\) | 0.22 | 0.92 | 4.18 | 1.01 | 1.01 | 1.78 (1.53) | 2.01 (1.73) |
\(b{\bar{b}} \rightarrow t{\bar{t}} g\) | 3.53 | 18.1 | 5.13 | 1.04 | 1.07 | 2.04 (1.90) | 1.92 (1.84) |
\(b{\bar{b}} \rightarrow t{\bar{t}} gg\) | 95 | 415 | 4.37 | 1.18 | 1.23 | 2.15 (2.05) | 2.49 (2.40) |
\(d{\bar{u}}\rightarrow W^- g\) | 0.33 | 0.71 | 2.15 | 1.03 | 1.03 | 0.96 (0.79) | 1.45 (1.17) |
\(d{\bar{u}}\rightarrow W^- gg\) | 5.6 | 12.9 | 2.30 | 1.05 | 1.10 | 0.99 (0.92) | 1.14 (1.05) |
\(d{\bar{u}}\rightarrow W^- ggg\) | 134 | 269 | 2.01 | 1.16 | 1.22 | 1.33 (1.28) | 1.44 (1.44) |
\(d{\bar{u}}\rightarrow W^- gggg\) | 3760 | 7442 | 1.98 | 1.14 | 1.18 | 1.41 (1.41) | 1.69 (1.68) |
\(d{\bar{u}}\rightarrow e^-{\bar{\nu }}_e\) | 0.024 | 0.23 | 9.58 | 1.02 | 1.02 | 1.60 (0.92) | 1.98 (1.37) |
\(d{\bar{u}}\rightarrow e^-{\bar{\nu }}_eg\) | 0.29 | 1.40 | 4.83 | 1.04 | 1.11 | 1.00 (0.81) | 1.31 (1.09) |
\(d {\bar{u}} \rightarrow e^- {\bar{\nu }}_e g g\) | 4.0 | 13.3 | 3.33 | 1.13 | 1.27 | 0.80 (0.75) | 1.11 (1.11) |
\(u {\bar{u}} \rightarrow W^+ W^-\) | 0.19 | 3.34 | 17.6 | 1.00 | 1.00 | 1.47 (1.19) | 1.42 (1.36) |
\(u {\bar{u}} \rightarrow W^+ W^- g\) | 6.7 | 25.7 | 3.84 | 1.16 | 1.06 | 1.31 (1.24) | 1.46 (1.40) |
\(u {\bar{u}} \rightarrow W^+ W^- gg\) | 154 | 379 | 2.46 | 1.19 | 1.15 | 1.63 (1.60) | 2.03 (2.01) |
\(u {\bar{u}} \rightarrow W^+ W^- g g g\) | 3660 | 8606 | 2.35 | 1.17 | 1.15 | 2.18 (2.18) | 2.44 (2.44) |
\(d {\bar{d}} \rightarrow e^- {\bar{\nu }}_e \mu ^+ \nu _\mu \) | 0.19 | 9.02 | 47.5 | 1.02 | 1.68 | 0.80 (0.58) | 1.67 (1.34) |
\(d {\bar{d}} \rightarrow e^- {\bar{\nu }}_e \mu ^+ \nu _\mu g\) | 5.6 | 42.2 | 7.54 | 1.23 | 1.85 | 0.57 (0.51) | 1.36 (1.15) |
The observed timings are roughly proportional to the number of one-loop Feynman diagrams, which ranges from \(\mathcal {O}(10)\) for the simplest \(2\rightarrow 2\) processes to \(\mathcal {O}(10^5)\) for the most complex \(2\rightarrow 5\) processes. Absolute timings correspond to OpenLoops 2 with default settings, i.e. with all stability improvements in dp plus the hybrid-precision system with a target accuracy of 8 digits. Augmenting the target accuracy to 11 digits causes a CPU overhead of 1% to 50%, depending on the process, while we have checked that switching off hybrid precision (hp_mode=0) yields only a speed-up of order one percent.
Comparing QCD to EW corrections, for processes without leptonic weak-boson decays we observe timings of the same order. More precisely, the QCD (EW) corrections tend to be comparatively more expensive in the presence of more external gluons (weak bosons). In contrast, in processes with off-shell weak bosons decaying into leptons EW corrections are drastically more expensive than QCD corrections. This is due to the fact that, for each off-shell W / Z decay to leptons, at NLO EW the maximum number of loop propagators increases by one, while at NLO QCD it remains unchanged. Due to Yukawa interactions, also the presence of massive quarks tends to increase the CPU cost of EW corrections.
Timings of OpenLoops 2 are compared against OpenLoops 1 with recommended stability settings (\(\texttt {preset}=2\), preset is deprecated in OpenLoops 2) and, alternatively, with the stability rescue system switched off (“no stab”) in OpenLoops 1. The difference reflects the cost of stability checks in OpenLoops 1, which is significantly higher than in OpenLoops 2. Note that this cost depends very strongly on the kinematics of the considered phase-space sample, and the values reported in Table 8 should be understood as a lower bound.
Apart from few exceptions, OpenLoops 2 is similarly fast or significantly faster than OpenLoops 1. In particular, for the most complex and time consuming processes the new on-the-fly approach yields speed-up factors between two and three.
5.2 Numerical stability
In Fig. 6 the average level of instability and its spread are plotted versus \(\xi _{{\mathrm {coll}}}\) in \(gg\rightarrow t{\bar{t}}\) and \(\xi _{{\mathrm {soft}}}\) in \(u{\bar{u}} \rightarrow W^+W^-g\). The stability of qp benchmarks is again very high in the whole phase space. In the deep IR regions numerical instabilities grow at a speed that depends on the process, the type of region (soft/collinear), and the employed method. For initial-state collinear radiation in \(gg\rightarrow t{\bar{t}} g\), CutTools loses three digits per order of magnitude in \(\xi _{{\mathrm {coll}}}\), resulting in huge average instabilities of \(\mathcal {O}(10^{10})\) in the deep unresolved regime. Using the Collier library in dp we observe a more favourable scaling, with losses of only one digit per order of magnitude in \(\xi _{{\mathrm {coll}}}\), and an average of three stable digits in the tail. Thanks to the hybrid-precision system, the level of stability of OpenLoops 2 is even much higher. It stays always above 10 digits and is roughly independent of \(\xi _{{\mathrm {coll}}}\). For soft radiation in \(u{\bar{u}} \rightarrow W^+W^-g\), apart from the fact that numerical instabilities are generally milder, the various tools behave in a qualitatively similar way.
Similar tests of the OpenLoops 2 stability system as the ones presented here have been carried out for various \(2\rightarrow 3,4, 5\) hard processes and \(2\rightarrow 3\) processes with an unresolved parton, finding similar stability curves as shown here, and not a single fully unstable result, i.e. one with zero correct digits. A more comprehensive study on numerical instabilities will be presented in a follow-up paper [66].
6 Summary and conclusions
We have presented OpenLoops 2, the latest version of the OpenLoops tree and one-loop amplitude provider based on the open-loop recursion. This new version introduces two significant novelties highly relevant for state-of-the art precision simulations at high-energy colliders. First, the original algorithm has been extended to provide one-loop amplitudes in the full SM, i.e. including, besides QCD corrections, also EW corrections from gauge, Higgs and Yukawa interactions. The inclusion of EW corrections becomes mandatory for the control of cross sections at the percent level, and even more importantly in the tails of distributions at energies well above the EW scale. Second, the original algorithm has been extended to include the recently proposed on-the-fly reduction method, which supersedes the usage of external reduction libraries for the calculation of tree–loop interferences. In this approach, loop amplitudes are constructed in a way that avoids high tensorial rank at all stages of the calculation, thereby preserving and often ameliorating (by up to a factor of three) the excellent CPU performance of OpenLoops 1. The on-the-fly reduction algorithm has opened the door to a series of new techniques that have reduced the level of numerical instabilities in exceptional phase-space regions by up to four orders of magnitude. These speed and stability improvements are especially significant for challenging multi-leg NLO calculations and for real-virtual contributions in NNLO computations.
In this paper we have presented the algorithms implemented in OpenLoops 2 for the calculation of squared tree, tree–loop interference and squared loop amplitudes. This entails a summary of the on-the-fly reduction method [33] and its stability system, which automatically identifies and cures numerical instabilities in exceptional phase-space regions. This is achieved by means of Gram-determinant expansions and other analytic methods in combination with a hybrid double-quadruple precision system. The latter ensures an unprecedented level of numerical stability, while making use of quadruple precision only for very small parts of the amplitude construction. Details of these stability improvements and hybrid precision system will be presented in an upcoming publication [66].
In the context of the extension to calculations in the full SM, we presented a systematic discussion of the bookkeeping of QCD–EW interferences and sub-leading one-loop contributions, which are relevant for processes with multiple final-state jets. We also detailed the input parameter schemes and one-loop \(\mathcal {O}(\alpha _{\mathrm {s}})\) and \(\mathcal {O}(\alpha )\) renormalisation as implemented in OpenLoops 2. Here we emphasised crucial details in the implementation of the complex-mass scheme for the description of off-shell unstable particles. The flexible implementation of the complex-mass scheme in OpenLoops 2 is applicable to processes with both on-shell and off-shell unstable particles at NLO. We also introduced a special treatment of processes with external photons, handling photons of on-shell and off-shell type in different ways, which is inherently required by the cancellation of fermion-mass singularities associated with the photon propagator and with collinear splitting processes.
While this manuscript as a whole provides detailed documentation of the algorithms implemented in OpenLoops 2, Sect. 4 together with Appendix A can be used as a manual, both in order to use OpenLoops 2 as a standalone program or to interface it to any Monte Carlo framework. Calculations at NLO and beyond require, besides squared amplitude information, also spin and colour correlators for the construction of infrared subtraction terms. To this end we documented the available correlators and conventions available in OpenLoops 2, which comprise tree-tree and loop-loop correlators as well as tree-loop correlators. The former are necessary for the construction of NLO subtraction terms for standard and loop-induced processes. The latter are necessary in NNLO subtraction schemes. Furthermore, conventions and interfaces for the extraction of full tree amplitude vectors in colour space are given. These are necessary ingredients for parton shower matching at NLO.
The new functionalities of OpenLoops 2 and their future improvement will open the door to a wide range of new precision calculations in the High-Luminosity era of the LHC.
Footnotes
- 1.
- 2.
This type of recursion was first proposed in the context of off-shell recurrence relations for colour-ordered gluon-scattering amplitudes [8].
- 3.
In the following, by \(\mathcal {O}(\alpha )\) or EW corrections we mean the full set of NLO corrections in the EW, Higgs and Yukawa couplings.
- 4.
- 5.
Quartic gluon couplings involving three different colour structures are split into colour-factorised contributions which are treated as separate diagrams.
- 6.
The Feynman diagrams in this paper are drawn with Axodraw [55].
- 7.
See [33] for more details.
- 8.
By HEFT we mean effective Higgs–gluon and Higgs–quark interactions in the heavy-top limit.
- 9.
Note that it is also possible to apply only (i)–(ii). This leads to the same objects \({\mathcal {V}}(t,q)\) as in (2.24), which can then be reduced a posteriori.
- 10.
This approach allows one to trigger the most extreme instabilities, where the K-factor is altered by \(\mathcal {O}(1)\) or more.
- 11.
The implementation of such NLO EW expansions is in progress and will be completed in a future update of the code. In the meanwhile, Gram-determinant instabilities for which no expansion is implemented are cured by means of the hybrid-precision system (see below).
- 12.
For simplicity, here we regard Yukawa and Higgs couplings as parameters of order e, keeping in mind that a separate power counting in \(\lambda _{\mathrm {Y}}\) and \(\lambda _{\mathrm {H}}\) is possible.
- 13.
In the following, for convenience, we refer to the the full amplitude \(\mathcal {M}_0^{(2R)}\) as squared Born term.
- 14.
In the absence of extenal quarks and gluons, tree and one-loop amplitudes have a trivial purely EW coupling structure, \(\mathcal {M}_0=e^m\mathcal {M}_0^{(0)}\) and \(\mathcal {M}_1=e^{m+2}\mathcal {M}_1^{(1)}\).
- 15.
In [67] the contributions \(\mathcal {W}_{\mathrm{tree}}^{(r)}\) and \(\mathcal {W}_{01}^{(r)}\) are rsp. denoted as \(\hbox {LO}_{r+1}\) and \(\hbox {NLO}_{r+1}\).
- 16.
For historical reasons their default values are \(\mu _\mathrm{R}=100\) GeV and \(\alpha _{\mathrm {s}}=0.1258086856923967\).
- 17.
In the literature, the coupling \(\alpha \) in the \(G_\mu \)-scheme is often defined as \(\alpha \vert _{G_{\mu }}=\sqrt{2}/\pi \, G_{\mu }{\mathrm {Re}}\left( \mu _W^2 \sin \theta _{\mathrm {w}}^2\right) \), where the truncation of the imaginary part is an ad-hoc prescription aimed at keeping \(\alpha \in {\mathbb {R}}\) in the complex-mass scheme. However, from the matching condition (3.29) it should be clear that (3.30) is the natural way of defining \(\alpha \vert _{G_{\mu }}\) as real-valued parameter.
- 18.
In the case of NLO EW contributions \(\mathcal {W}_{01}\), the rescaling factors are renormalised according to (3.91).
- 19.
More precisely, Yukawa masses are always renormalised like physical masses at \(\mathcal {O}(\alpha )\). Moreover, when \({\mu }_{f,{\mathrm {Y}}}\ne \mu _f\) for any particle during process registration NLO EW process libraries cannot be loaded and if \({\mu }_{f,{\mathrm {Y}}}\ne \mu _f\) is set at a later stage a warning is printed.
- 20.
Note that \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\) and \(\mu _\mathrm{R}\) are separate input parameters controlled by the user, i.e. OpenLoops does not control the evolution of \(\alpha _{\mathrm {s}}(\mu _\mathrm{R}^2)\) but only the related counterterm. Thus it is the role of the user to set \(N_{q,{\mathrm {active}}}\) to the correct value \(N_{\mathrm {F}}\).
- 21.
In case the running of \(\alpha _{\mathrm {s}}\) is obtained from LHAPDF the information about the number of quark flavours contributing to the evolution of \(\alpha _{\mathrm {s}}\) is available in the PDF info file as the tag \(\texttt {NumFlavors}\) for LHAPDF versions \(\ge 6.0\).
- 22.In practice, the truncation of absorptive contributions is implemented at the level of the scalar two-point integrals throughand in the same way for \(B'_0\). For the derivative of self-energies also the following formulas for \(B_1\) and \(B'_1\) functions are used
- 23.
Note that \({\widetilde{{\mathrm {Re}}}}\, \varSigma _\mathrm {T}^{AZ}(0)= \varSigma _\mathrm {T}^{AZ}(0)\) since \(\varSigma _\mathrm {T}^{AZ}(0)\) is free from absorptive parts.
- 24.
Here all SU(3) generators as well as electromagnetic charges should be understood in terms of incoming charge flow.
- 25.
Due to our recoil conventions for (off-shell) photon emitters in (3.100), \({\hat{\gamma }}^{\mathrm {QED}}_{\gamma }\) contributions are only relevant for massive initial-state spectators.
- 26.
A version of SCons (“scons-local”) is shipped with OpenLoops, but a system-wide installation may be used as well.
- 27.
An installation routine to move the library to a different location is currently not available.
- 28.
The collection all.coll makes it possible to download the full set of available processes libraries at once. However, due to the large overall number of processes and the presence of several complex processes, this is requires a very large amount of disk space and very long CPU time for compilation. Thus all.coll should not be used for standard applications.
- 29.
In general, base code and process code can be combined in a rather flexible way, but care must be taken that they remain mutually consistent. The API compatibility between base code and process code is typically guaranteed across many sub-versions, both in the forward and backward directions. To this end, all mutually consistent versions are labelled with the same (internal) API version number, and OpenLoops accepts to use only combinations of process code and base code that belong to the same API version.
- 30.
The registration procedure through the BLHA is explained in Appendix B.1.
- 31.
This corresponds to the normalisation convention used by the Collier [19] library.
- 32.
The values of \(\varDelta _2\), \(\varDelta _{\mathrm {IR},1}\) and \(\varDelta _{\mathrm {UV},1}\) are controlled internally by OpenLoops. For validation purposes they can be changed using the parameters pole_IR2, pole_IR1 and pole_UV1, respectively. However such modifications may jeopardise the calculation of UV and IR divergent quantities.
- 33.
As usual, the corresponding SU(3)\(\times \)U(1) quantum numbers should be understood in terms of incoming charge flow, in such a way that \(\sum _k T^a_k | \mathcal {M}\big \rangle =\sum _k Q_k | \mathcal {M}\big \rangle =0\).
- 34.
Explicit expression for \(k_\perp ^{\mu }\) in the dipole subtraction formalism are for example listed in Tab. 1 of [73] for all relevant splittings.
- 35.
OpenLoops automatically amputates possible non-orthogonal parts of \(k_\perp \) by projecting \(k_\perp ^\mu \) onto \({\varepsilon }^\mu _{\pm }(p_j)\).
- 36.
Note that (4.40) is computed in the trace basis excluding off-diagonal \(\mathcal {K}_{ij}\) terms but including any other sub-leading-colour contributions.
- 37.
This estimate is based on the analytic form of all presently known spurious singularities. So far it was found to be quite reliable. However, it may have to be improved if new types of instabilities are encountered.
- 38.
In the tail of the CutTools curve (not shown) numerical instabilities can reach and largely exceed \(\mathcal {O}(10^{10})\).
- 39.
For performance reasons, by default the (negative) IR poles of the \({{\mathbf {I}}}\)-operator, Eq. (3.98), are returned as IR poles in m2l1. The true poles of the virtual amplitudes can be obtained by setting the parameter \(\texttt {truepoles=1}\). Alternatively setting \(\texttt {truepoles=2}\) sums the virtual amplitude including its true poles and the \({\mathbf {I}}\)-operator including its finite part and poles, which allows for easy pole cancellation checks. See more details in Sect. 4.3.
- 40.
For performance reasons, by default the (negative) IR poles of the \({\mathbf {I}}\)-operator and UV counterterm are returned as poles in m2l1bare. The true poles of the bare virtual amplitudes can be obtained by setting the parameter \(\texttt {truepoles=1}\).
Notes
Acknowledgements
We are thankful to Andreas van Hameren for supporting OneLOop, and to Ansgar Denner and Stefan Dittmaier for supporting Collier . We are indebted to Stefan Kallweit for numerous bug reports and pre-release tests. Also we would like to thank the Sherpa and Matrix collaborations for continuous collaboration and discussions. We thank the ATLAS and CMS Monte Carlo groups for valuable feedback. J.M.L. would like to thank the Theoretical Particle Physics group at Sussex University for the hospitality during the completion of this work. F.B., J.-N.L., S.P., H.Z., M.Z. acknowledge support from the Swiss National Science Foundation (SNF) under contract BSCGI0-157722. M.Z. acknowledges support by the Swiss National Science Foundation (Ambizione grant PZ00P2-179877).
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