1 INTRODUCTION All Order Merging of High Energy and Soft Collinear Resummation Jeppe R. Andersen1 Hitham Hassan1and Sebastian Jaskiewicz1

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1 INTRODUCTION

All Order Merging of High Energy and Soft Collinear Resummation

Jeppe R. Andersen1, Hitham Hassan1?and Sebastian Jaskiewicz1

1Institute for Particle Physics Phenomenology, Durham University, Durham, United Kingdom

* hitham.t.hassan@durham.ac.uk

January 23, 2023

51st International Symposium on Multiparticle Dynamics (ISMD2022)

Pitlochry, Scottish Highlands, 1-5 August 2022

doi:10.21468/SciPostPhysProc.?

Abstract

We present a method of merging the exclusive LO-matched high energy resummation of High

Energy Jets (HEJ) with the parton shower of PYTHIA which preserves the accuracy of the LO

cross sections and the logarithmic accuracy of both resummation schemes across all of phase

space. Predictions produced with this merging prescription are presented with comparisons

to data from experimental studies and suggestions are made for further observables and

experimental cuts which highlight the importance of both high energy and soft-collinear

effects.

1 Introduction

The emergence of complex divergent and slowly-convergent structures in perturbation theory has

signiﬁcant implications on theorists’ ability to produce stable and robust predictions for standard

model processes. The direct implication of these is that expansions in αsconverge slowly with

higher orders of expansion in certain regions of phase space as the size of the logarithms compen-

sates for the smallness of the coupling and an all-order treatment is required. Most pertinent to

these proceedings is the presence of large logarithms to all orders of QCD in different kinematic

limits.

In the high energy (HE) or multi-Regge kinematic (MRK) limit for a 2 →nscattering of partons

with momenta pa,pb→p1,p2. . . , pn−1,pn:

y1y2 ···  yn−1yn,pi,⊥≈k⊥∀i∈ {1, 2, ··· ,n−1, n}, (1)

with k⊥some hard transverse momentum scale, large logarithms log(ˆ

s/k2

⊥)→ |∆y1,n|arise to

all orders in perturbation theory [1]. Without an accounting for these effects, the phase space

corresponding to the MRK limit will be poorly modelled by ﬁxed-order predictions and descriptions

of observables which receive this logarithmic enhancement from semi-hard wide-angle radiation

will not be perturbatively stable.

These logarithms are most often resummed analytically via the Balitsky-Fadin-Kuraev-Lipatov

(BFKL) formalism [1–4]. For this study we work instead with the High Energy Jets [5]Monte

Carlo implementation of high energy resummation which matches the inclusive cross section to

arXiv:2210.06898v2 [hep-ph] 20 Jan 2023

2 MERGING PROCEDURE

the leading order (LO) prediction and applies the leading logarithmic- (LL-)accurate high energy

corrections to processes which at tree level contribute at LL or NLL accuracy [6].

The HEJ-resummable processes at LL (for pure dijet production) are scatterings of the form

f1,f2→f1,g, . . . , g,f2where the ﬁnal state is understood to be ordered in rapidity. We refer to

such conﬁgurations as “Fadin-Kuraev-Lipatov” or “FKL” conﬁgurations. The subleading conﬁgu-

rations which receive LL resummation in HEJ include those where the rapidity ordering between

any two neighbours in the ﬁnal state is relaxed1. Additional subleading processes include those

with t-channel quark propagators (allowing for central qq emissions). The event output of HEJ

is exclusive to its logarithmic accuracy compared to the inclusive event input generated at leading

order.

In the limit of soft-collinear parton splittings a different class of logarithms emerge which

typically manifest as ratios of transverse energy scales:

log2tj

tk, (2)

where thas mass dimension +2. Such double logarithms may be recast into the product of a soft

logarithm and a collinear logarithm, each of which diverges respectively when partons in an event

have low transverse momenta or have small angular separation. Just as with the high energy log-

arithms log(ˆ

s/k2

⊥), the presence of these spoils the rapid convergence of the perturbative series in

the regions of phase space where such logarithms are large. This includes, most plainly expressed

in Eq. (2) for a transverse momentum-based evolution scale, conﬁgurations with hierarchies in p⊥

and collinear splittings inside jet cones.

These effects may be accounted for with Monte Carlo parton showers which resum the leading

logarithmic soft-collinear effects to all orders in perturbation theory. Most often inclusive ﬁxed-

order events are merged with parton shower resummation. Merging events generated at ﬁxed-

order with parton showers is a well-established ﬁeld of research with many well-proven procedures

including CKKW-L [7,8]merging for leading order input events. Methods for matching to higher

orders in perturbation theory, including MC@NLO [9]and the POWHEG [10]methods for NLO events

are also widely used. Further generalisations of these methods for NLO matching which increase

the accuracy below the merging scale to full NLO have additionally been developed, including

MENLOPS [11,12]and UNLOPS [13,14]. The result of these are exclusive showered events with

an all-order description of the soft and collinear splittings.

We discuss here the implementation of a new procedure for merging the exclusive high-energy-

resummed event output of HEJ with the exclusive parton shower resummation of PYTHIA8[15]

which accounts for both missing higher order perturbative effects and systematically removes the

double counted contributions. The software implementation of this procedure is HEJ+PYTHIA.

2 Merging Procedure

To produce high energy- and soft-collinear-resummed predictions, we express the resummation

of HEJ in the language of the parton shower by deﬁning a splitting kernel corresponding to the

1Sec 1.2.3 of ref [6]provides an overview of contributions from subleading conﬁgurations and how these are re-

summed in HEJ.

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1INTRODUCTIONAllOrderMergingofHighEnergyandSoftCollinearResummationJeppeR.Andersen1,HithamHassan1?andSebastianJaskiewicz11InstituteforParticlePhysicsPhenomenology,DurhamUniversity,Durham,UnitedKingdom*hitham.t.hassan@durham.ac.ukJanuary23,202351stInternationalSymposiumonMultiparticleDynamics(ISMD202...

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1 INTRODUCTION All Order Merging of High Energy and Soft Collinear Resummation Jeppe R. Andersen1 Hitham Hassan1and Sebastian Jaskiewicz1

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