# Summing large-\(N\) towers in colour flow evolution

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## Abstract

We consider soft-gluon evolution in the colour flow basis. We give explicit expressions for the colour structure of the (one-loop) soft anomalous dimension matrix for an arbitrary number of partons, and we show how the successive exponentiation of classes of large-\(N\) contributions can be achieved to provide a systematic expansion of the evolution in terms of colour-suppressed contributions.

## Keywords

Anomalous Dimension Parton Shower Soft Gluon Anomalous Dimension Matrix Matrix Element Generator## 1 Introduction

In order to reliably interpret current and upcoming measurements at the LHC, precise QCD predictions for multi-jet final states are indispensable. These include both fixed-order calculations, as well as their combination with analytic resummation and/or parton shower event generators, e.g. [1, 2, 3], to sum leading contributions of QCD corrections to all orders, such as to arrive at a realistic final state modelling. Fixed-order calculations at leading and next-to-leading order in the strong coupling are by now highly automated, and frameworks to automatically resum a large class of observables have been pioneered as well [4]. The combination of NLO QCD corrections with event generators [5, 6, 7, 8, 9, 10] is an established research area, and first steps towards combining analytic resummation and event generators have been taken [11].

The efficient treatment of QCD colour structures is central to both fixed-order and resummed perturbation theory. Particularly the use of the colour flow basis has led to tremendously efficient implementations of tree-level amplitudes [12, 13, 14], which can be used both for leading order calculations, as well as one-loop corrections within the context of recent methods requiring only loop integrand evaluation (see [15] for the exact treatment of the colour flow basis in the one-loop case). This colour basis is closely linked to determining initial conditions for parton showering. After evolving a partonic system through successive parton shower emissions, while keeping track of the colour structures (in the large-\(N\) limit), colour flows also constitute the initial condition to hadronisation models; this includes the dynamics of how multiple partonic scatterings are linked to hadronisation. Colour reconnection models, such as those described in [16, 17], are exchanging colour between primordial hadronic configurations like strings or clusters, and have proven to be of utmost phenomenological relevance in the description of minimum bias and underlying event data.

Despite its relevance to event generators, the colour flow basis has typically not been considered in analytic resummation, most probably for the reason of being not the most simple or minimal basis. While recent work has focussed on obtaining minimal (and even orthogonal) colour bases [18], an intuitive connection to the physical picture is hard to maintain in such approaches. It is until now an open question whether amplitudes can be evaluated in a similarly efficient way in such bases. Also, in analytic resummation, matching to a fixed-order calculation is usually mandatory and the use of colour flow bases could allow one to use the full power of automated matrix element generators within this context. Understanding soft-gluon evolution in the colour flow basis thus seems to be a highly relevant problem to address, which can also shed light on colour reconnection models, being so far based on rather simple phenomenological reasoning.

The purpose of the present work is to study soft-gluon evolution in the colour flow basis. While for a fixed, small number of partons the exponentiation of the soft gluon anomalous dimension matrix can be performed either analytically or numerically, the case for a large number of partons is rapidly becoming intractable. This limitation thus prevents insight into the soft gluon dynamics of high-multiplicity systems relevant to both improved parton shower algorithms [19, 20] as well as colour reconnection models. We will derive the general structure of the soft anomalous dimension matrix in the colour flow basis for an arbitrary number of partons, and we tackle its exponentiation by successive summation of large-\(N\) powers in a regime where the kinematic coefficients \(\gamma \) are of comparable size to the inverse of the number of colours, \(\gamma N \sim 1\), leading to a computationally much more simple problem than the full exponentiation. This strategy can well be applied to a large number of partons in an efficient way.

This paper is organised as follows: In Sect. 2 we set our notation and present the general form of the soft gluon anomalous dimension. In Sect. 3 we derive its exponentiation and show how subsequent towers of large-\(N\) contributions can be summed in a systematic way. Section 4 is devoted to a few numerical studies of testing the accuracy of these approximations in a simple setting of QCD \(2\rightarrow 2\) scattering, while Sect. 5 presents an outlook on possible future applications before arriving at conclusions in Sect. 6. A number of appendices is devoted to calculational details and for reference formulae to achieve what we will later call a next-to-next-to-next-to-leading colour (N\(^3\)LC) resummation.

## 2 Notation and soft anomalous dimensions

*colour flow basis*, by translating all colour indices into indices transforming either in the fundamental (\({\mathbf N}\)) or the anti-fundamental (\(\bar{{\mathbf N}}\)) representation. For a thorough derivation of this paradigm, including a list of Feynman rules and their application to fixed-order calculation, see for example [12]. Translating the labelling of physical legs, \(\alpha \), to a labelling of corresponding colour and anti-colour ‘legs’,

^{1}A pictorial representation of these basis tensors is given in Fig. 1. The colour charges (note that \({\mathbf T}_\alpha \cdot {\mathbf T}_\beta = {\mathbf T}_\beta \cdot {\mathbf T}_\alpha \)) translate as (obvious cases relating colour and anti-colour are not shown):

^{2}

^{3}Hence the anomalous dimension reads

^{4}

^{5}

## 3 Summation of large-\(N\) towers

*phase space*region for which \(\Delta ^2\gg 4\varSigma _{1212}\varSigma _{1221}\), \(\kappa \sim |\Delta |\), we recover the NLC’ approximation, i.e., there is a phase space region where

*purely kinematic reasons*give rise to a NLC’ expansion without having actually considered the very size of \(N\) itself. Note that the different treatment of \(\rho \), either absorbing it into a redefinition of the \(\varGamma _\sigma \), or treating it as subleading itself, amounts—for the case of \(q\bar{q}\) singlet—to either keeping \(C_F=(N^2-1)/(2N)\) exactly or doing a strict large-\(N\) limit with \(C_F\sim C_A/2\). An observation that these different prescriptions account for the bulk of subleading-\(N\) effects in a colour-improved parton shower evolution [20] has already been made, though we are far from drawing an ultimate conclusion here.

## 4 Numerical results

For the other configurations contributing to QCD \(2\rightarrow 2\) scattering we find a similar pattern of convergence through successive orders. We note, however, that some of the matrix elements for processes with more and more colour flows are non-zero starting only from a high enough order. Especially for a large number of legs, this will require a minimum order to obtain at least a first, non-zero, contribution for the respective matrix elements. Investigating the impact of these contributions at the level of squared amplitudes will be subject to future work.

## 5 Outlook on possible applications

The work presented here is relevant to cases where soft gluon evolution is a required ingredient for precise predictions, but not feasible in exact form owing to a large number of external legs present. This, in particular, applies to improved parton shower algorithms but also to analytic resummation for observables of multi-jet final states. Looking at the convergence of the N\(^d\)LC expansions, which can easily be implemented in an algorithmic way, one can gain confidence of providing a reliable resummed prediction at some truncation of the exponentiation. As for the case of parton showers, the colour flow basis, being itself ingredient to many highly efficient matrix element generators, offers unique possibilities to perform Monte Carlo sums over explicit colour structures or charges, such that efficient algorithms in this case seem to be within reach. The requirement to study soft gluon dynamics for a large number of legs is as well at the heart of the dynamics behind non-global logarithms [25], when considered to more than the first order in which they appear, and beyond leading colour. Another application (which, in part, triggered the present work) is to gain insight into the dynamics of colour reconnection models. A QCD motivated and feasible colour reconnection model based on summing large-\(N\) towers is subject to ongoing work and will be presented elsewhere.

Let us finally remark that N\(^d\)LC calculations in general do not require matrix exponentiation and at most \(d\) plain matrix multiplications. Owing to the respective matrices being very sparse,^{6} this can be performed very efficient. Indeed, one can imagine to perform a Monte Carlo summation over colour structures by generating subsequent sequences of colour flows to be considered. The number of possible sequences is very limited given the fact that the \(\varSigma \) matrices only contain non-vanishing matrix elements for two colour flows which differ at most by a transposition in the permutations labelling them.

## 6 Conclusions

In this paper we have investigated soft-gluon evolution in the colour flow basis, presenting the structure of the soft anomalous dimension for any number of legs. We have then focussed on systematic summation of large-\(N\) enhanced terms with the aim of providing successive approximations to the exact exponentiation of the anomalous dimension. We generally find a good convergence of these approximations for a simple anomalous dimension in QCD \(2\rightarrow 2\) scattering. The present work can be used to perform soft gluon resummation for a large number of external legs, where the full exponentiation is not feasible anymore. It also forms the basis for improved parton shower evolution and may shed light on the dynamics to be considered for colour reconnection models.

Particularly in conjunction with matrix element generators, making use of the colour flow basis, very efficient and highly automated calculations can be performed owing to the algorithmic structure of N\(^d\)LC approximations, including Monte Carlo sums over individual colour structures. The C++ library CVolver [27], which has been developed within this context provides all required tools to do so.

## Footnotes

- 1.
Notice that we do not impose a limitation to colour structures as appearing for tree-level calculations. Indeed, the gluon exchange will generate all possible structures starting from only tree-level ones.

- 2.
\((t^a)^i{}_j (t^a)^k{}_l = \frac{1}{2}(\delta ^i {}_l \delta ^k {}_j - (1/N) \delta ^i {}_j \delta ^k {}_l)\).

- 3.
Note that appropriate crossing signs have to be included when considering incoming quarks, i.e., a factor of \(-1\) for each correlator involving an incoming quark or anti-quark as long as the anomalous dimension coefficients and amplitudes are evaluated in the physical regime.

- 4.
Note that we did not assume \(\varGamma ^{\alpha \beta } = \varGamma ^{\beta \alpha }\) in the first place, as may be the case due to inclusion of recoil effects or further contributions along the lines of dipole subtraction terms [22].

- 5.
Our notation \([ |...| ]\) indicates that we refer to the matrix element with respect to the given representation of the amplitude as a complex vector, and not the quantity \(\langle \tau | {\mathbf \Gamma }|\sigma \rangle \), which will only coincide with the former in an orthonormal basis, that being not the case for the colour flow basis considered here, nor for most other colour bases.

- 6.
Note that this does not only apply to the colour flow basis, but similar observations have been made for other choices, e.g. [26].

## Notes

### Acknowledgments

I am grateful to Malin Sjödahl and Mike Seymour for many valuable discussions and comments on the work presented here. This work has been supported in part by the Helmholtz Alliance ‘Physics at the Terascale’.

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