Soft gluon evolution and non-global logarithms

Journal of High Energy Physics, May 2018

Abstract We consider soft-gluon evolution at the amplitude level. Our evolution algorithm applies to generic hard-scattering processes involving any number of coloured partons and we present a reformulation of the algorithm in such a way as to make the cancellation of infrared divergences explicit. We also emphasise the special role played by a Lorentz-invariant evolution variable, which coincides with the transverse momentum of the latest emission in a suitably defined dipole zero-momentum frame. Handling large colour matrices presents the most significant challenge to numerical implementations and we present a means to expand systematically about the leading colour approximation. Specifically, we present a systematic procedure to calculate the resulting colour traces, which is based on the colour flow basis. Identifying the leading contribution leads us to re-derive the Banfi-Marchesini-Smye equation. However, our formalism is more general and can systematically perform resummation of contributions enhanced by the t’Hooft coupling α s N ∼ 1, along with successive perturbations that are parametrically suppressed by powers of 1/N . We also discuss how our approach relates to earlier work.

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Soft gluon evolution and non-global logarithms

HJE Soft gluon evolution and non-global logarithms Michael H. Seymour 0 1 2 3 4 7 8 Rene Angeles Mart nez 0 1 3 6 7 8 Matthew De Angelis 0 1 3 4 7 8 Je rey R. Forshaw 0 1 3 4 7 8 Simon Platzer 0 1 3 5 7 8 0 Manchester , M13 9PL , U.K 1 PL-31342 , Krakow , Poland 2 Theoretical Physics Department , CERN 3 Polish Academy of Sciences 4 Consortium for Fundamental Physics, School of Physics & Astronomy, University of Manchester 5 Particle Physics, Faculty of Physics, University of Vienna 6 The Henryk Niewodniczanski Institute of Nuclear Physics in Cracow 7 CH-1211 , Geneva 23 , Switzerland 8 1090 , Wien , Austria We consider soft-gluon evolution at the amplitude level. Our evolution algorithm applies to generic hard-scattering processes involving any number of coloured partons and we present a reformulation of the algorithm in such a way as to make the cancellation of infrared divergences explicit. We also emphasise the special role played by a Lorentzinvariant evolution variable, which coincides with the transverse momentum of the latest emission in a suitably de ned dipole zero-momentum frame. Handling large colour matrices presents the most signi cant challenge to numerical implementations and we present a means to expand systematically about the leading colour approximation. Speci cally, we present a systematic procedure to calculate the resulting colour traces, which is based on the colour ow basis. Identifying the leading contribution leads us to re-derive the Ban - Marchesini-Smye equation. However, our formalism is more general and can systematically perform resummation of contributions enhanced by the t'Hooft coupling sN with successive perturbations that are parametrically suppressed by powers of 1=N . We also discuss how our approach relates to earlier work. Perturbative QCD; Resummation; 1/N Expansion; Scattering Amplitudes 1 Introduction 2 The general algorithm 3 Large-N structures 2.1 2.2 3.1 3.2 3.3 3.4 Working in a non-orthogonal colour basis generators such as Herwig [1, 2], Pythia [3] and Sherpa [4]. Resummation can be based on the direct analysis of contributing Feynman graphs in QCD, and this is the approach we have taken in the past, though e ective eld theories have also been recognized as powerful tools to organize resummed calculations through a renormalization group evolution [5]. For a large class of observables, which are fully inclusive below some resolution scale in all phase-space regions, no contributions originate from unresolved parton emission due to a perfect cancellation of real and virtual corrections. This eases the resummation procedure, { 1 { as typically only one or very few emissions need to be taken into account. Observables of this kind are referred to as global observables and include, for example, many of the event shape variables measured at LEP. On the other hand, observables that are insensitive to emissions into a certain patch of phase space are called non-global and they are subject to contributions from an arbitrary number of emissions, e ectively probing QCD dynamics in the blinded phase-space region [6]. This e ect makes the all-orders resummation (even of the leading contributions) a much more complicated endeavour, mainly because non-trivial colour correlations become unavoidable. Fortunately, these colour correlations simplify dramatically in the leading colour approximation and they can therefore be approximately accounted for in the general purpose event generators. In recent years there has been a good deal of progress in developing the technology to tackle non-global observables and, in many cases, go beyond the leading colour approximation [7{17]. Subleading colour contributions have also been addressed in the context of parton shower algorithms [18{21]. Often, attention has focussed on processes with no coloured particles in the initial state, not least because the simpler colour structure eliminates the need to consider Coulomb (a.k.a. Glauber) gluons. Coulomb gluon interactions are particularly interesting since they have been shown to induce a breakdown in the factorization of wide-angle soft gluon emission from hard-collinear emission [22{26]. In spite of the existing progress, it remains to develop an automated approach to resummation beyond the leading colour approximation for general hard processes. Progress in this direction has been made by Nagy and Soper [19{21]. In this paper, our aim is to present a general framework that can be used as a basis for future automated resummations. To be more precise, we consider algorithmic, recursive de nitions of QCD amplitudes for the radiation of many soft gluons and including leading virtual corrections to all orders. Such an approach is at the heart of direct QCD analyses of observables involving many coloured legs, and it was used to identify the aforementioned violations of strict collinear factorisation that occur at hadron colliders. The present work consists of two main parts, and a number of appendices devoted to more technical details. In section 2 of the paper we lay down the general evolution algorithm, in a form that is suited to the calculation of multiple soft-gluon contributions to any observable in a fully-di erential way. We show how this approach connects to earlier work. We then proceed to reformulate the algorithm in such a way as to make the cancellation of infrared divergences explicit. We also highlight the role of the speci c ordering variable rst identi ed in [27]. In section 3, we focus on the colour structures encountered when solving the evolution equations. We present a systematic procedure to calculate the resulting colour traces, which is based on the colour ow basis and the work presented in [28]. Identifying the leading contribution leads us to re-derive the Ban -Marchesini-Smye equation [29]. However our formalism is more general and can systematically perform resummation of contributions enhanced by the t'Hooft coupling sN parametrically suppressed by powers of 1=N . 1, along with successive perturbations that are { 2 { In appendix A we show how our approach connects to the work presented by Becher et al. [14], as well as Weigert and Caron-Huot [7, 11, 30]. In appendix B we make the cancellation of infrared divergences explicit for observables that are inclusive below a resolution scale, and in appendix C we explicitly calculate contributions in a xed-order expansion. Finally, appendix D sets up the machinery to deal with the fact that most colour bases are not orthogonal (see [31] for a notable exception). 2 The general algorithm Our starting point is the cross section for emitting n soft gluons. At this stage we will If partons i and j are both in the initial state or they are both in the nal state then eij = 1 otherwise eij = 0. The real emission operator and phase-space factor are given by D i = X Tj Ei j p j pj q i and d i = i s 2 dEi d i3 2 E1+2 4 (2 ) 2 : These expressions for the Sudakov and emission operators are guaranteed to capture the leading soft logarithms, though the framework can be extended beyond this level of approximation to include collinear logarithms and next-to-leading soft logarithms. Note that the sum over partons j in the de nition D i is context-speci c, i.e. it runs over any prior soft gluon emissions in addition to the partons in the hard scattering. Likewise, the colour 1Note: d = 2 d=2= (d=2). { 3 { (2.1) (2.2) (2.3) (2.4) where the un are the observable dependent measurement functions and the ki are soft gluon momenta. We suppress dependence on the hard partons and integration over their phase space. In the above, we should take the limit ! 0, though we will consider non-zero values in what follows. The carat on k^ reminds us that !ij(k^) is dependent only upon the direction of the vector k in the ij rest frame. The path-ordering, P, in the de nition of Va;b is not actually needed here, because the expression in curly brackets in eq. (2.2) is independent of the ordering variable, Ek. The cross sections in eq. (2.1) are the general building blocks for any observable and they can be used as the basis for a Monte Carlo computer code to generate partonic events. We can also write an evolution equation:2 E An(E) y + Dn An 1(E) Dyn E (E En) ; where the anomalous dimension operator is = s X( Ti Tj) i<j Z d k !ij(k^) i eij : When the measurement function factorizes, i.e. un(k1; k2; kn) = u(E1; k^1) u(En; k^n), charge operator, Tj, and the Sudakov operator, Va;b, are in a context-speci c representation of SU(3)c. The operators An satisfy the recurrence equation An(E) = VE;EnDn An 1(En) Dyn VEy;En (E En) ; where (E En) is the Heaviside function. A general observable, , can be computed using ( ) = Z X d n un(k1; k2; ; kn) ; = X n n Yn Z i=1 ! d i Tr An( ) un(k1; k2; kn) ; Gn(E) = Yn Z dEi An(E) i=1 4 Ei { 4 { (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) HJEP05(218)4 we can de ne and eq. (2.7) becomes E This can be re-written as In appendix A, we show that eq. (2.11) is the same as the leading-logarithmic accuracy RG equations considered in [7, 11, 14, 30]. 2For simplicity, we work in d = 4 spacetime dimensions unless otherwise stated. = Gn(E) Gn(E) y + Dn Gn 1(E) Dyn u(E; k^n): !ij(k^) Ti Tj Gn + Gn Ti Tj !ij(k^n)(Ti Gn 1Tjy + Tj Gn 1Tiy) u(E; k^n) + Coulomb terms. Note that, if we are interested in a speci c observable that we know is fully inclusive of real emissions with E < Q0 then we can use the Bloch-Nordsieck cancellation in order to x = Q0 and integrate the real-emission phase space over E > Q0. This is proved in appendix B. n after integration over the real emissions. In this section we present a reformulation of the algorithm in which the cancellation of infra-red divergences (both soft and soft-collinear) arising in the eikonal approximation is manifest. General observables may be de ned by dividing the angular phase-space into two complementary sub-regions, which we refer to as the \in" and \out" regions, such that the observable is fully inclusive over emissions in the \out" region. If the \out" region is of zero extent then the observable is referred to as a global observable, otherwise it is known as a non-global observable. Phrased this way, we see that all observables are non-global to some extent (since 4 -detectors do not exist). We note that there is potential for confusion here, because our \in" region corresponds to what some other authors (e.g. [14]) refer to as the \out" region. In order to expose the infra-red cancellation, it is useful to break apart the virtual loop-integral in (see eq. (2.8)) so that we expose the part which is destined to cancel against a corresponding real emission contribution. To this end, it is useful to consider the measurement function in the soft gluon limit: um(q1; ; qm) = u(qj ; fq1; ; qj 1; qj+1; ; qmg) um 1(q1; ; qj 1; qj+1; ; qm) : This is quite general, the important thing is that, u(qj ; fq1; in the limit that gluon j has zero energy. Generally, we can write ; qj 1; qj+1; u(k; fqg) = out(k) + in(k)uin(k; fqg) : The set fqg corresponds to all other real emissions and in/out(k) is de ned to be unity if k is in the \in"/\out" region and zero otherwise. For global observables, the \out" region is of zero extent, in which case u(k; fqg) = uin(k; fqg). Armed with this we de ne = u + u; where the infra-red nite part of the anomalous dimension operator is u = s Z d k (1 4 u(k; fqg)) D2k + i 2 X i<j eij Ti Tj ; and the part containing the infra-red divergence is u = 4 s Z d k u(k; fqg) D2k : 2 { 5 { (2.12) ; qmg) ! 1 (2.13) (2.14) (2.15) (2.16) HJEP05(218)4 Note that we have written these in terms of the squared real-emission operator, 1 2 D2a = X( Ti Tj ) !ij (k^a) ; i<j so as to make the Bloch-Nordsieck cancellation more readily apparent. The virtual evolution operator, which is the exponential of the anomalous dimension, can now be expanded in a manner that makes the cancellation of divergences explicit, i.e. where the operators Bn satisfy the recurrence relation (i.e. the analogue of eq. (2.5)): R Bn(E) = VE;En DnBn 1(En)Dyn n Va;b = Va;b Z Z Notice that the virtual gluons are summed to all orders only if they are in the \in" region, i.e. V involves virtual gluons integrated over the \in" region. Since 1 u(k; fqg) ! 0 when Ek ! 0 it follows that there are no soft singularities in V except those arising from Coulomb gluon exchange. The poles from Coulomb gluon exchange cancel though because they always appear in terms Tr(V0;a Vy0;a). The cyclicity of the trace ensures that the i terms can be combined into the unit matrix, since the real part of V0;a vanishes in the limit E ! 0. We can now re-write the observable as ( ) = X n Yn Z i=1 d i Tr Bn( ) n(q1; q2; ; qn) ; Bn 1(En); nV u(qn; fqg) VyE;En (E En) ; starting from B0(E1) = VE1;Q H VyE1;Q. In the above, it should be understood that f iR = 1; iV = 0g if i is a real emission and f i R = 0; iV = 1g if it is virtual. As before, should be set equal to zero or a global scale below which the observable is known to be fully inclusive. In eq. (2.20), n encodes the measurement functions for any number of real emissions: ! D2n 2 n(q1; q2; : : : ; qn) 0 1; 2n a=1 X um(fqkg) m=jPnaj 0 Y j2Pna i2=Pna 1 V i A : { 6 { (2.17) (2.18) The Pna set is indexed by a indicating which of the n gluons are real, and m is the cardinality of the set. For example, if n = 2 then P21 = f1; 2g indicates that gluons q1 and q2 are both real and m = 2, P22 = f1g indicates that q1 is real and q2 is virtual (m = 1), P23 = f2g indicates that q2 is real and q1 is virtual (m = 1), and P24 = fg indicates that both gluons are virtual (m = 0). In this case, 2(q1; q2) = 1V 2V + u1(q1) 1R 2V + u1(q2) 2R 1V + u2(q1; q2) 1R 2R and, recalling that u(q; fg) = u1(q) and u(q; fq1g)u1(q1) = u2(q1; q), 0 = Tr(V0;Q H Vy0;Q) ; 1 = d 1 u1(q1) Tr V0;E1 D1 VE1;QHVyE1;QDy1 Vy0;E1 V0;QHVyE1;Q 2 2 the pairs of contributions in which the softer gluon is real or virtual (i.e. 2RV ) are proportional to the same expression, ensuring that their soft singularities cancel. For an infra-red safe observable the n are all nite (in the eikonal approximation). In appendix C we show how the cancellation of infra-red poles works out by explicit calculation of the non-global contribution to the hemisphere mass in e+e collisions to order s 3 . 2.1.1 Non-global observables: a simple example In the leading logarithm approximation, many observables can be computed with a factorizable measurement function, i.e. { 7 { n i=1 un(q1; : : : ; qn) = Y u1(qi) : (2.26) Also, in many cases (such as the hemisphere jet mass3 and gaps-between-jets), the measurement function simply vetoes real emissions into some region of phase-space, e.g. d k D2k + i 2 X i<j eij Ti Tj A 19 = ; and replace u(qn; fqg) ! out(qn) in eq. (2.22). This is because of the inclusivity of the observable for E < , which leads to a complete cancellation of the real and virtual contributions (see appendix B). In other words, n is the contribution from n gluons in the \out" region (the gluons can be real or virtual). Speci cally, (2.27) (2.28) (2.29) 0 = Tr(V ;QHVy;Q) 1 = 2 = V ;E2 2 +V ;E2 2 etc: +V ;QHVyE1;Q 2 D21 VyE2;E1 2 D22 Vy;E2 + V ;E1 2 D21 VE1;QHVyE2;Q 2 D22 VE2;QHVyE1;Q 2 D21 Vy;E1 + V ;E2 2 D22 VE2;E1 2 D21 VE1;QHVy;Q : This is the \out of gap" expansion used in [22, 23] to derive the super-leading logarithmic contribution to gaps-between-jets. 2.2 The ordering variable So far we have presumed energy ordering in the virtual gluon operators Va;b. However, this is known not to generate the correct super-leading logarithms and instead transverse 3For the jet mass, we should really take u1(q) = out(q) + in(q) ( > E(1 cos )), which correctly accounts for the double logarithms associated with emissions collinear with the jet axis and ensures that 0 is nite. For n with n 1, this modi cation to u1(q) is unimportant since the non-global corrections are single logarithmic. { 8 { qn+1 i j i qn j qn+1 Z qn(ij) qn(i+j)1 dkT kT Z qn(ij) dkT qn([+n]1j) kT Illustrating how the `dipole transverse momentum' serves to limit the virtual loop integration. The index [n] refers to the gluon with momentum qn. momentum ordering should be used [32]. Interestingly, and working in the eikonal approximation, but only for gluons that couple to the original hard partons, explicit calculation of all relevant Feynman diagrams (at one loop) reveals that the corrections associated with the exact triple and four-gluon vertices can be largely subsumed into a Lorentz invariant ordering variable, in a potentially simple extension of the algorithm that we described in [27, 32]. This intriguing physics may not be so clearly visible in an e ective eld theory treatment, where the evolution is in an arbitrary renormalization scale. gluon loop integral has its transverse momentum limited by the transverse momenta of the two nearest real emissions. The relevant transverse momentum of a gluon, q(ij), is de ned by its Sudakov decomposition over the momenta pi and pj involved in dipole (ij), and is given by (q(ij))2 = 2q pi q pj = pi pj 2E2 !ij (q^) : (2.30) When one of the gluons to which the virtual gluon couples also happens to be one of the nearest emissions, the relevant dipole transverse momentum vanishes. In this case however, the explicit calculation reveals that the relevant dipole momentum is that of the parent of the parton which couples to the virtual gluon, i.e. parton i in the lower graph of gure 1. { 9 { HJEP05(218)4 According to the results in [27, 32], which were performed to one-loop accuracy, the corresponding di erential cross section has a similar structure to eq. (2.1) but with A0(q~1) H(Q) + X i<j Iij (qe1; Q) H(Q) + H(Q) Iiyj (qe1; Q) ; An(qen+1) Tk An 1(qen) Tl !kl(q^n) + Iij (qen+1; qen) Tk An 1(qen) Tl !kl(q^n) h X i<j6=[n] X k6=l + i6=[n] X Ii[n] qen+1; qn(ik) Tk An 1(qen) Tl (!kl(q^n) !il(q^n)) + h.c.i ; (2.31) Lorentz invariant operator Iij (a; b) is given by where it should be understood that qe = q(ij) when placed in the argument of Iij . The Iij (a; b) = s Ti Tj Z b dkT a kT Z d 4 !ij (k^) ij (k) i eij ; where ij (k) = (pj (pi k) > 0) (pi(pj k) > 0) restricts the region of the angular integration to be the same as in the phase space integral for a real gluon with the same transverse momentum. This can be written ln tan < ln 2 p2pi pj kT ; or j sin j > 2kT p2pi pj : Up to the limits on the cos integral, Iij is equal to the exponent in the V operator de ned in eq. (2.2). In [27] we presented the form of this operator in 4 2 dimensions after integration over the solid angle, i.e. Iij (a; b) = 2 s c E , and E is the Euler-Mascheroni constant. This expression is accurate up to non-logarithmic terms of order 0 in the real part and order 1 in the imaginary part. When both scales a and b are non-zero, Iij is nite and given by Iij (a; b) = 2 s Ti Tj 1 2 ln2 2pi pj + b2 1 2 ln2 2pi pj a2 i ~ij ln a2 b 2 : (2.35) Eq. (2.34) would be identical to the result of Catani and Grazzini [33] (see also [34, 35]) ~ if we replaced the factor 1 + i ij in eq. (2.34) by cos( ) + i sin( ~ij ). It should be re-iterated that the calculations in [27, 32] were performed only at one loop and it remains to be seen how the improved resummation proceeds to all orders. (2.32) (2.33) (2.34) In practical calculations, the colour algebra rapidly becomes intractable after only a few real gluon emissions. To simplify matters, we shall now identify the leading contributions in an expansion in the number of colours. We will work in the colour ow basis [36], which is closely related to the way colour is treated in parton shower algorithms [37{40]. In the following subsection, we will show that, to leading order in the number of colours, our algorithm gives rise to a dipole-type parton shower and that it reproduces the Ban Marchesini-Smye equation [41]. We then turn our attention to setting up a framework to calculate the rst subleading-colour corrections. To start with, we collect together some of the key results concerning the colour ow basis. We label the set of basis tensors as fj ig, and we assign a colour or anti-colour index, ci or ci, to each external leg i of any scattering amplitude. Gluons carry both colour and anti-colour and incoming quarks carry anti-colour. We start to count colour index labels from 1, and choose ci = 0 (ci = 0) if i only carries anti-colour (only carries colour). The basis tensors are labelled by permutations of the colour indices and are given by products 1:::n are fundamental and anti-fundamental indices assigned to the colour (anti-colour) legs, taking values in the actual number of colours 1; : : : ; N . There are n = nq + ng = nq + ng possible colour lines and n! colour ows (i.e. there are n! basis tensors). Inner products of colour ow basis tensors are given by h j i = 1 ( 1 ) n (n) 1 ( 1 ) n (n) = N n #transpositions( ; ) ; where #transpositions( ; ) is the number of transpositions by which the permutations and di er. This is equal to n minus the number of loops obtained after contracting the Kronecker symbols, see the right-hand part of gure 2. In gure 2, we show three of the six colour ows that represent the four-parton state on the left, and in table 1 we specify the We also include the binary variables i and i, where i = p corresponding colour and anti-colour indices for each of the four partons (labelled by i). TR; i = 0 for a quark, i = 0; i = p i = p TR for an antiquark and i = TR for a gluon (TR = 1=2 in QCD). Note that in the gure we use the more compact notation: We express amplitudes as jAi = P A j i, where labels the individual basis tensors, and the evolution and traces in colour space can be performed in terms of ordinary complex matrices with elements A , which relate to the basis independent objects via (3.1) (3.2) (3.3) (3.4) left half of the gure shows three out of the six basis tensors required for a qggq leg content, where the grey arrows indicate how leg labels i = 1; : : : ; 4 are mapped onto colour and anti-colour indices (see also table 1). The right hand part of the gure illustrates how inner products, i.e. elements of the scalar product matrix, relate to powers of N depending on how many loops are formed after contraction (see eq. (3.2)). i 1 2 3 4 c i 1 2 3 0 c i 0 1 2 3 p p p i TR TR TR 0 i 0 p p p TR TR TR [ jAj ] A : Tr[A] = Tr[AS] = X[ jAj ] h j i : ; Ti = i tci i tci ( i i) s ; 1 N 1 ( 1 ) n (n) + = t j i = t 1( )j i ; n + 1 are not matrix elements of the operator A since the colour ow basis is not orthonormal. Consequently, we will introduce a dual basis in which We refer to appendix D for more details on the properties of the dual basis vectors (see also [28]). The scalar product matrix S the traces of operators in colour space: = h j i has to be considered when evaluating The colour charge (or emission) operator associated to each leg i can be decomposed as where the colour-line operators t; t and s are de ned through their action on the basis states, i.e. t j i = t 1 ( 1 ) ( ) n n + 1 (n) ( ) + ; (3.5) (3.6) (3.7) (3.8) (3.9) for the inverse permutation 1 for which 1( ( )), and sj i = s 1 ( 1 ) It is useful to note that n (n) + = = 1 ( 1 ) t j i = s ;n+1 s j i ; n (n) n + 1 + where s ; exchanges ( ) and ( ). It is hence obvious that through the action of any of the emission operators we cannot map two distinct basis tensors j i and j i into the same tensor j i. Furthermore, if and di er by n transpositions, then t j i and t j i will di er by n + 2 transpositions if ( ) 6= ( ), and by the original n transpositions if ( ) = ( ) (implying in this case that n 1), t j i and sj i will di er by n + 1 transpositions, sj i and sj i will di er by the original n transpositions. N + X (ab) Colour-line operators and their products, such as t t = t t , are referred to as colour reconnectors in [28]. We note that s t = t s = 1 and s s = N 1. Matrix elements involving colour reconnectors are straightforward to compute because of the important property that R j i = [ jRj i j i ; (3.12) where R is a general reconnector (see appendix D). Note that there is no sum over since reconnectors constitute a unique map from one colour ow to another. The matrix elements of the colour correlators are [ jTi Tj j i = i j ci; 1(cj) + j i cj; 1(ci) + i)( j j ) 1 the elements a and b in the permutation . is zero if the permutations and are not equal and unity otherwise. The sum over (ab) is rather cumbersome, since each of the four terms can be written without any summation after implementing the colour- ow Kronecker delta. However, this way of writing things ensures that the second and third line in eq. (3.13) do not contribute if i and j are colour connected in . By `colour connected' we mean ci = 1(cj ) or cj = 1(ci).4 Note also that the o -diagonal elements in the matrix representation of Ti Tj are non-vanishing only if the permutations labeling the two basis tensors in question di er by at most one transposition. A similar expression can 4If = j321i then (3) = 1 and 1(3) = 1 etc. be obtained for colour charges multiplied to the left and right of a colour matrix, A, which corresponds to real emission: [ nnjAj nn] : i)( j The colour lines associated with the emitted particle, n, are labelled by cn and cn, and nn denotes the permutation with the entries associated with cn and cn merged and removed, i.e. 1 ( 1 ) cn (cn) 1(cn) cn m (m) ! nn = 1 ( 1 ) 1(cn) (cn) m (m) ! : (3.15) Our aim is to organize contributions to the cross section in terms of a series of leading powers in N , to extract both the large-N limit as well as corrections to it. To this end we introduce the operation Leading(l) [A] = 1=Nk #transpositions( ; );l k ; where the notation l X k=0 A A 1=Nk indicates to pick those terms in A spect to the leading power present in A which are suppressed by a factor of 1=N k with re . Contributions to the trace of A then all yield an enhancement or a supression by the same power of N by virtue of either an explicit suppression in A other words, if A or by picking up a subleading element in the scalar product matrix. In A0 + A1=N + : : : is an operator in the space of n colour lines, then Tr hLeading(l) [A]i / N n l : We are speci cally interested in traces originating from soft-gluon evolution: and where P !i ;(n)(!i ;(n)) = 0. Tr[VnAnVny] Tr[ nAn + An yn] where where X TrhDnAn 1 Dn y i refers to all other quantum numbers of the emission, and it is understood that n = X i;j where 0 X i;j i(jn)Ti Tj ; 1 i(jn)Ti Tj A ; Dn = X !i ;(n)Ti ; i where with where Also, we have de ned For single gluon exchange, And the emission contribution is Leading(0) hDnAn 1Dyni = For the leading contributions of the virtual evolution operators we nd Leading(0) hVnAnVnyi = V (n) 2 Leading(0) [An] ; V (n) = exp N X and c.c. here means colour connected, i.e. ri(jn) = 2Re (n) : Leading(0) h nAn + An yn = N i X For a pair of evolution operators, VnAnVny, multiply by exp 2N Re i(jn) + (n) ji i j for each colour connected dipole i; j in . (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) X i;j c.c. in nn i j Ri(jn) Leading(0) nn; nn [An 1] Ri(jn) = 2Re !(n) !(n) i j : We are now able to compute traces in the leading-N limit. We must sum over diagonal colour ow contributions, i.e. A . For each colour ow, , we multiply by N raised to the number of colour lines present in the colour ow . Notice that the number of possible colour ows at this level of approximation is equal to the number present at the level of the hard process plus the number of real emissions. Each of the contributions A can then be computed by a set of recursive rules that correspond to working inwards from the outer matrices (multiplied from the left and right) towards the hard matrix in between. The rules are as follows: [σ| ci c¯j cn ci c¯j c¯n α α¯ [σ\n| ci c¯j α α¯ c¯n ci c¯j | σ\n] | σ] (3.31) (3.32) HJEP05(218)4 An example to illustrate colour- ow evolution. In this case, we consider the leadingcolour contribution, with one real emission and one virtual correction, to [ jAj ]. The vertical dashed lines indicate that it is possible to read o the colour- ow map at any time during the evolution. See the text for the corresponding rules. For a virtual gluon insertion, nAn + An yn, multiply by 2N i j Re and sum over the dipoles (i; j) that are colour connected in . For a pair of emission operators, DnAn 1Dyn, combine the dipoles (i; n) and (n; j) in the colour ow , leaving behind a dipole (i; j) in the colour ow nn, and include a factor i j 2 Re !(n) !(n) i j : This procedure is illustrated in gure 3, where we take the opportunity to show a speci c contribution at leading-colour. This graph would contribute to the soft gluon evolution of qq ! qq scattering with one emission and one virtual correction. If the hard process has been reached, multiply by the square of the corresponding amplitude, jM j2. 3.3 Dipole evolution and the BMS equation We will now show how the rules of the preceding section give rise to the BMS equation [41]. Our rules apply to a general process with any number of outgoing partons. The algorithmic incarnation of the generalized BMS equation that we present here corresponds to a dipole shower algorithm. The evolution of dipoles is universal, i.e. at this level of approximation the process dependence solely enters through selecting an initial colour ow weighted by the modulus squared of the corresponding amplitude jM j2. To illustrate how things work out, we will consider the same example as in section 2.1.1. In this case i(jn) = (jni) = (n) = ji ij (n) = Z in d k (En+1 < E < En) 2 !ij (k^) ; 1 out (En < En 1) 2 !ij (q^n) ; 1 and i !(n) !(n) j = out (En < En 1) !ij (q^n) : The evolution with the in-region anomalous dimension contributes a factor ViEjn+1;En = exp N s Z En dE Z d k En+1 E in 4 per colour ow. The above expressions have a very simple diagrammatic interpretation, illustrated in gure 4. To simplify the discussion, we consider the case of e+e scattering, i.e. we take H = N1 1. Each double line in the gure corresponds to a Sudakov factor, ViEj1;E2 , where i and j label the directions associated with the corresponding colour and anti-colour lines. The shaded circles correspond to a factor !ij (k^), and the vertical dashed line indicates the associated energy. The arguments of the Sudakov are also determined by these vertical dashed lines. We can immediately see how the algorithm maps onto a classical dipole shower at leading N . These diagrammatic rules can be used to compute the leading colour contribution to the non-global logarithms: 0 = Vab;Q 1 = RR = 2 where the hard partons have momenta pa and pb, ti = (N s= ) ln(Ei= ) and we used the notation !aib = !ab(q^i). The n can also be obtained by iteratively solving the BMS equation, as we will now illustrate. The BMS equation can be written as follows, Z d k in 4 !ab(k)Gab(t) + h !ab(k) Gak(t)Gkb(t) Gab(t)i (3.38) Z d k out 4 b ⌃ 1R = ⌃ 2RR = ⌃ 2V V ⌃ 2RV = = Q Q Q a b a b a b a b ⇢ 1 a b 2 ⇢ a E2 1 b E1 1 a a b E1 E2 E1 1 a E2 a b a 1 b ⇢ ⇢ and our observable corresponds to will rst rewrite it by replacing Gij (t) = Vij;E gij (t), which gives with Gab(0) = 1 and tQ = (N s= ) ln(Q= ). To solve the BMS equation iteratively, we Q Z d k out 4 !ab(k) gak(t)gkb(t) gab(t) : (3.40) ⌃ 2V R = a b Q E1 E2 b 1 a ⇢ Q a b 1 E1 b ⌃ 1V = a b Q a b Q 1 E1 b + + ( ) = Gab(tQ) E2 b E2 b a b a b a 1 # ⇢ ⇢ (3.39) HJEP05(218)4 Putting ga(0b)(t) = 1 on the r.h.s. of the BMS equation immediately gives 1, i.e. = Z d k out 4 !ab(k) which gives the desired result after integrating over 0 < t < tQ. The next iteration gives 2, i.e. we substitute gij (t) on the r.h.s. of the BMS equation by ga(1b)(t): Z d k out 4 !ab(k) # where we left ga(0k) = 1 explicit for clarity. It is easy to show that VaQb ; ga(2b)(tQ) = 2V V . So we see that, at leading N , our algorithm generates the iterative solution to the BMS equation. Subleading contributions Subleading colour contributions are substantially more di cult to compute. In this section we present some initial steps towards a systematic approach to including 1=N k corrections to the leading result. Figure 5 illustrates the general structure of the calculation (of which gure 3 is a speci c example). Figure 6 provides an overview of the power counting we use to de ne successive orders | we hope its interpretation will become clear after the following paragraphs. There are subleading colour contributions arising from the hard scattering matrix, from the 1=N and 1=N 2 suppressed terms in the real emission operator (see eq. (3.14)) and the virtual correction operator (see eq. (3.13)), and from o -diagonal contributions to the scalar product matrix. In the following, we will use the general form of the anomalous dimension resulting from eq. (3.13), i.e. [ j j i = N + + 1 N : (3.43) Each of , and are of order s . To compute a correction of order 1=N k we need to consider states and in gure 5 that di er by k l permutations, where 0 l k. Then we must determine the 1=N l corrections arising from the soft gluon evolution and from the hard scattering matrix. The leading colour contributions from the virtual evolution operator come from and so are all enhanced by powers of sN which, owing to the fact that the leading contribution is diagonal, can easily be accounted for to all orders in a simple exponential. This evolution does not result in any di erence between the colour structure in the amplitude and that in its conjugate, and it corresponds to the blue boxes in gure 6. If this evolution is then supplemented by those pieces of the real emission operator that also preserve the identity of the colour structure in the amplitude and its conjugate (such as the example in gure 3) then we recover the leading-N picture of the last two sections. Subleading colour contributions may result in di erences between the colour in the amplitude and that in the conjugate amplitude. To keep track of this, we will count the N3 N2 N1 N0 N−1 N−2 N−3 [τ | Γ Σ ρ1 1 α 0 s α 1 s Γ2 ΣΓ ρΓ Σ2 ρΣ ρ21 α 2 s Γ3 ΣΓ2 ρΓ2 Σ2Γ ρΣΓ Σ3 ρ2Γ ρΣ2 ρ2Σ ρ31 α 3 s σ virtuals (0 flips) × 1 × (αsN)n (1 flip) × αs × (αsN)n (0 flips) × αsN−1 × (αsN)n (t[...]t| 0 flips)r (0 flips) × (2 flips) × αs × (αsN)n 2 αs × (αsN)n 2 (t[...]t| 0 flips)r (t[...]t| 0 flips)r−1 t[...]t| 2 flips | σ] reals (t[...]t| 0 flips)r−1 t[...]t| 2 flips × 1 (t[...]t| 0 flips)r−1 t[...]s| 1 flip × (t[...]t| 0 flips)r−1 s[...]s| 0 flips × NN−−12 (t[...]t| 0 flips)r (t[...]t| 0 flips)r−1 t[...]s| 1 flip × N−1 in all powers of the t'Hooft coupling sN 1. In the set of boxes shown we count, for the virtual evolution operator, increasing powers of s from left to right, and decreasing powers of N from top to bottom, with N 0 in the middle row. The e ect of r real emissions is indicated in the rightmost column of the gure and any 1=N suppression due to the scalar product matrix is indicated by the number of ips. See the text for more details. number of colour reconnections (or transpositions or ips or swings) by which the two colour structures di er. It turns out that pure 1=N corrections can only originate from interference contributions in the hard process matrix. We will ignore subleading colour contributions from this source in what follows, though they could easily be included. The most important subleading colour contributions due to real emission are suppressed by a power of 1=N 2 relative to the leading contribution and they originate as a result of the following three possibilities: (i) two colour ips accompanied by no explicit factor of 1=N (coming from contributions of the type t[ ]t); (ii) one ip and a factor of 1=N (coming from contributions of the type t[ (coming from contributions of the type s[ ]s). See eq. (3.14) to appreciate the factors of 1=N . We note that real emissions never reduce the number of ips by which the amplitude and its conjugate di er. We will present the explicit rules corresponding to these real emission contributions below but rst we consider subleading virtual corrections. A single insertion of a perturbation 0 comes with a factor of ( sN )=N relative to the leading contribution and it results in a single ip, and hence an additional 1=N suppression via the scalar product matrix. This ip can undo that induced by a previous real emission of the type s[ ]s. However, since this will require the action of a single s operator, it re-introduces the additional factor of 1=N . Thus, both of these xed-order contributions, when combined with the all-order summation of contributions from , are suppressed by ( sN )=N 2 1=N 2 relative to the leading contributions. These contributions are illustrated by the dark orange boxes in gure 6. A similar reasoning applies to the contribution of a single perturbation (light orange boxes), which contributes at the same order s=N 1=N 2 since it generates zero ips. We nally need to consider two insertions of 0 combined in such a way that the net number of ips is zero or two. The zero- ip case is clearly proportional to contributes a ( sN )2=N 2 correction (green boxes). The two- ip case can also contribute at this order provided it compensates a t[ ]t two- ip real emission, i.e. so the net result s2 and hence is that the amplitude and its conjugate di er by zero ips and there is no suppression from the scalar product matrix. These two contributions should therefore be included along with the contributions discussed above. However, contributions from the diagonal below the green boxes lead to a factor of ( sN )2=N 4, which means they are beyond the next-to-leading colour approximation. These corrections to soft-gluon evolution can be considered as xed-order corrections to the leading-N rules, though these need to be extended to include the possibility that the permutations and need no longer be equal. This means we should update the rules at the end of section 3.2 as follows. For a pair of evolution operators VnAnVny multiply by V (n) V (n) , i.e. include a factor of the virtual amplitude exponentiated for each colour connected pair of legs in , and each colour connected pair in . For a virtual gluon insertion nAn + An yn, include a factor w(n) + w(n) , where w(n) = X i;j c.c. in i j (n) : (3.44) Real emission operators, DnAn 1Dyn, only contribute if the emitted gluon is connected to the same, identically connected, colour line in and , as otherwise the operation of merging the emitting dipoles would alter the number of ips by which and di er. We will now present the rules to compute the rst 1=N 2 corrections to the leading colour trace. For the virtual contributions we need to consider the next-to-leading colour approximation for the evolution operator, which (using the notation in [28]) is 1 N X VnLC+NLC j i = V (n) j i #transpositions( ; );1 (n) j i ; where the colour ow transition matrix elements can be expressed as and (n) = N e W (n) W (n) X i;k c.c. in j;l c.c. in e W (n) W (n) i j Wi(jn) + k lWk(ln) i lWi(ln) k j Wk(jn) i;l c.c. in k;j c.c. in W (n) = ln V (n) = N X i;j c.c. in i j Wi(jn) : (3.45) (3.46) (3.47) This source of subleading correction contributes when and single ip. In the latter case (one ip), we include a factor of 2 Re are identical or di er by a h (n)i and the other factor of 1=N comes from the scalar product matrix. In the former case there are three possibilities. Speci cally, we may include either one or two factors of (in either the amplitude or the conjugate amplitude) in such a way as to undo the e ect of a one or two- ip real emission (see below for the rules for including real emissions), these contribute to the dark orange and green boxes in the gure. Or else we may include a factor of j in the case that the two ips (one from each ) cancel each other out (green boxes). (n) 2 j At this order we also need to include corrections which are suppressed by 1=N 2 and are proportional to s s (light orange boxes) in the virtual evolution operator. This term is diagonal in colour and VnNLC; j i = (n) N 2 j i ; where (n) = N X( i i<j i)( j j )Wi(jn) : (3.48) As discussed above, we must also consider subleading corrections to real gluon emission. Recall that the 1=N 2 corrections arise when and di er by at most two ips. In this case, we are to include contributions where the gluon is emitted o either colour line ci or ck in the amplitude (ci and ck are colour connected in cj in the conjugate amplitude (cl and cj are colour connected in nn), and either colour line cl or nn). Evolving towards the hard process, we are to combine the dipoles (i; n) and (n; k) in , and (l; n) and (n; j) in , leaving behind dipoles (i; k) and (l; j). The corresponding factor is i l !i(n) !(n) l + k j !k(n) !(n) j i j !i(n) !(n) j k l !k(n) !(n) l ; (3.49) which comes from the rst two lines in eq. (3.14). Notice that a potential 1=N contribution arising when and di er by only one ip vanishes because of the rst bullet point in the list above eq. (3.13), i.e. contributions of the type t[ ]t require and ips.5 If and di er by one ip and the gluon connects to itself in but not in , then we should combine the dipoles (i; n) and (n; k) in and include a factor of i !i(n) k !k(n) X( j j j ) !(n) j ; or the corresponding conjugate. This corresponds to the third line in eq. (3.14). Again a possible 1=N correction arising when and are equal vanishes because contributions of the type s[ ]t require and to di er by one ip (see the second bullet point in the list above eq. (3.13)). Finally, if the gluon is connected to itself in both and , we include a i)( j ; (3.50) (3.51) corresponding to the fourth line in eq. (3.14). Armed with these rules it is possible to go ahead and compute the rst subleading colour contributions to the BMS equation. We leave such a phenomenological study to future work. 4 Accounting systematically for partonic radiation in short-distance scattering processes is of practical importance and theoretical interest. Progress in accurately accounting for this physics has been dominated by coherence-improved parton/dipole shower Monte Carlo programs [2{4] though to date these are all limited to leading N , with some subleading improvements [18]. Probably the main challenge in going beyond leading N arises because of the need to include quantum interference e ects, which would seem to necessitate an amplitude-level approach. This paper represents our rst steps towards the implementation of a general algorithmic approach to amplitude-level parton evolution, which has also been advocated in [19]. We anticipate that numerical results using the technology outlined in this work will be available soon, and we postpone a detailed discussion of the computational methods to a follow-up work. Acknowledgments This work has received funding from the UK Science and Technology Facilities Council (grant no. ST/P000800/1), the European Union's Horizon 2020 research and innovation programme as part of the Marie Sklodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no. 722104). JRF thanks the Institute for Particle Physics Phenomenology in Durham for the award of an Associateship. MDA thanks the UK Science and Technology Facilities Council for the award of a studentship. SP acknowledges partial support by the COST action CA16201 PARTICLEFACE, and is grateful for the kind hospitality of ESI at Vienna, AEC at Bern and MIAPP at Munich, where part of this work has been addressed. We are indebted to Thomas Becher for valuable discussions. 5Recall we are ignoring any o -diagonality due to the hard scattering matrix in the way we count ips. The connection with other approaches In this appendix we show how the colour evolution algorithm de ned by eq. (2.1) relates to the previous work of Becher et al. [14] and Caron-Huot [11, 30]. A.1 Becher et al. In [14], the hard process e+e ! 2 jets is considered with the requirement that the to2Eout < tal energy emitted outside of cones centred on the two (back-to-back) jets should satisfy Q where Q = ps. This observable is of the type described by our eq. (2.11) and, because there are no coloured particles in the initial state, the Coulomb terms can be neglected. Accordingly, in the leading logarithmic approximation they nd (see section 5.2 of [14]). LL( ; ) = 0 Tr (S2(fn1; n2g; Q ; ; h)) 1 X m=2 = 0 Tr (U 2Sm(fn1; n2g; ; s; h) ^ 1) ; where = tan( =2) ( is the opening angle of the jets), h = Q and s = Q . Formally, the evolution operator is given by U lSm(fng; ; s; h) = P exp Z h d s H (fng; ; ) lm ; where fng is the set of l light-like vectors that x the directions of the nal-state par(A.1) (A.2) (A.3) (A.4) H 23 H 33 H 43 . . . m i<j m i<j H 24 H 34 H 44 . .. . . A . 1 C CC = C 4 s BB 0 B BB 0 B 0 V 3 R3 0 0 1 C C C C CC + O( s2) 0 V 4 R4 0 . . . 0 V 5 . . . . .. . . A . Z d k 4 !ij (k^) ; V m = 2 X(Ti;L Tj;L + Ti;R Tj;R) Rm = d q^m+1 !ij (q^m+1) in(q^m+1) : 4 tons, and with H = BB 0 B B 42 We are using the notation of [14], so the in(q^m+1) restricts the emitted gluon with momentum qm+1 to lie inside either the quark or anti-quark jet (de ned by the cones around their directions). In other words, the \in" (jet) region corresponds to what we previously called the \out" (of gap) region. The subscripts \L" and \R" on the colour charge operators denote that they sit to the left or right of the object upon which they operate. Expanding the exponential in eq. (A.2) gives rise to exactly the same series as using eq. (2.5) (see eq. (5.17) of [14]). We can also translate eq. (2.11) into the notation and language of [11]. The starting point is to introduce a rotation matrix, Ui, for each parton in the hard subprocess and each soft gluon. Operators Li and Ri are also de ned such that LiaUi TiaUi; LiaUj = Uj Lia RiaUi UiTiay RiaUj = Uj Ria; and for i 6= j: Their commutation relations are inherited from the colour algebra: [Rj ; Lk] = 0; hLja; Lbki = jkif abcLc; j hRja; Rbki = jkif abcRjc: Now one de nes the one-loop kernel Ki(j1) = s 2 d q3 2 (!ij (q^)) 2 !ij (q^) ij (q) 4 2 2 RiaUqabLjb RjaUqabLib + RiaRja + LiaLja n = En for energy ordering and n = qn(ij) for dipole ordering (see below). Up to the Coulomb gluon term, this is equal to the lowest order resummation contained in equations (2.7) and (2.14) of [11]. This is also very closely related to the work of Weigert [7]. Another way to write eq. (A.9) is via a generating functional: 0 X i6=j 1 d 1+2 Ki(j1)A Z0 ; d1+n2 Ki(j1) = 1 2 i s 2 (2 ) 2 eij 2 2 2 d q3n 2 q3n 2 d n 1+2 n where ! 0 and ij ! 1 gives rise to ordering in energy. Ordering in dipole transverse momentum is obtained with ! and ij ! The corresponding equation for a general observable (i.e. eq. (2.6) for energy ordering and eq. (A.14) for dipole transverse momentum ordering) is X i6=j " d P exp d 1+2 Ki(j1) D M U1 UN M E # U1 UN+m!IN+m ; (A.9) which is fully di erential. Here N is the number of partons in the hard subprocess, the path ordering acts over and the colour matrices Ui should be independent of this parameter. In proving the equivalence of eq. (A.9) with eq. (2.6) and eq. (A.14) it is useful to note that d qn RiaUqanb Ljb + RjaUqanb Lib RiaRja LiaLja !ij (q^n) ij (qn) RiaRja LiaLja ( n 1 < n < n+1) ;(A.10) (A.5) (A.6) (A.7) (A.8) (A.11) HJEP05(218)4 The case n = qn(ij) is reminiscent of (but not the same as) the dipole transverse momentum ordering we discussed in section 2.2 (see eq. (2.30)). Indeed, eq. (A.9) can be re-written as a recurrence relation: Afn(i1;j1);:::;(in;jn)g( ) h V qn(injn) injn (qn)Tin Afn(i11;j1);:::;(in 1;jn 1)g q(injn) Tjn !injn (q^n) Vy qn(injn) ; n where A0( ) = V ;Q H Vy ;Q and V is de ned analogously to eq. (2.2). Formally, each of the gluons should have an energy E < Q and this is imposed via injn (qn). In [27], direct calculation led to ij (q) = q) > 0) and we introduce it here to cut-o arbitrarily high momentum modes. We cannot avoid the long chain of indices because the observable is obtained by integrating over the multi-gluon phase space subject to q(i1;j1) > q2(i2;j2) > 1 , i.e. X Yn Z ! N N+1 X X i1;j1 i2;j2 # N+n 1 X in;jn TrAfn(i1;j1);:::;(in;jn)g( ) ; q(i1j1) > q2(i2j2) > 1 un(k1; k2; ; kn) where Z0 = and Z m = um Z Un+m fUig=0 : where im; jm N + m 1 and N is the number of hard partons. As pointed out in [11, 30], choosing the dipole transverse momentum to order the emissions is, ultimately, a renormalization scheme choice in the e ective theory, albeit one that has the virtue of making Lorentz invariance manifest. The dipole transverse momenta of successive real emissions are ordered and this set of ordered momenta acts to limit the virtual gluon loop integrals in eq. (A.14). We note that dipole ordering avoids all collinear poles except for those associated with the very last emission. This is a very attractive feature. The proof proceeds along the following lines: for a given dipole chain, poles come from (pi q) (pj q) = 0. But this quantity is proportional to the ordering variable, so the only possibility of it equalling zero is the case of the nal emission (when = 0). B On the cancellation of infrared divergences below the inclusivity scale The aim here is to show that, for observables fully inclusive for E < , we can simply impose E > in the algorithm. This fact follows because = Tr H + X1 Z n=0 n+1 Y d m m=1 ! Tr An+1(En+1) (un+1(q1; : : : ; qn+1) un(q1; : : : ; qn)) ; (A.12) (A.13) as the lower bound on the energy integrals for both real emissions and virtual exchanges. To prove eq. (B.1) we make use of the identity Vay;bVa;b 1 = a s Z b dE d q VEy;bD2(q^)VE;b ; where we used the shorthand D2(q^) D (q) D (q) and the Sudakov operator is given in eq. (2.2). Using eq. (B.2), we can rewrite the contribution to the observable from n real emissions as n = = Z Z n m=1 Y d m Y d m m=1 ! ! Tr VEn;0An(En)VEyn;0 Z Tr An(En) d n+1 An+1(En+1) ; (B.2) (B.3) where it should be understood that E0 = Q, i.e. A0(E0) = H. Eq. (B.1) trivially follows from this expression by grouping terms that depend on the same trace. C C.1 Non-global logarithms at s Calculation of 1 at order 2 s In this appendix, we compute the xed-order expansion of the non-global logarithmic contributions to the hemisphere mass. Apart from checking the correctness of the algorithm, this allows us to con rm that the expansion proposed in section 2.1 is indeed free from infra-red divergences at each order, i.e. the n are separately nite. We will be very explicit in the hope that it will be useful to see how a calculation proceeds in detail. First we compute the hemisphere jet mass to xed order, as in Dasgupta-Salam [6]. As they do, we start by computing the lowest order non-global correction to the cumulative event shape where the jet mass is required to be less than . We can do this by using the algorithm with and taking the \out" region to be the region of phase space that does not contribute to the hemisphere jet mass, i.e. it is the wrong-side hemisphere. The \in" region is the complement of this. Note it is only to leading ln(Q= ) accuracy that the observable is fully inclusive over gluon emissions with E < . In which case we may write (see eq. (2.29) with V ! Vin and ! ): 0 d d = out d 1 hTr(Vin;E1 D1 VEin1;QVEin1y;QDy1 Vin;Ey1 )+ Tr(Vin;E1 1VEin1;QVin;Qy) + Tr(Vin;QVEin1y;Q y1Vin;Ey1 ) : We have set the Born matrix element equal to the identity (since we are considering a event shape) and the factor of 1=N removes the colour factor for the lowest two-jet e+e order cross section. To lowest order, i !ij : (C.1) (C.2) Vain;b 1 a E i<j s Z b dE X( Ti Tj ) Z d in 4 Expanding out gives (note the lower case notation, ta, which (only in this appendix) indicates that these operators act on 3-parton objects): Z Q dE1 Z N in 4 E1 out 4 Tr ((ta tq1 !aq1 (k) + tb tq1 !bq1 (k) + ta tb !ab(k)) Ta Tb) 4!ab(q1) 4!ab(q1) Z E1 dE Z Q dE E1 E Z E1 dE E E Z Q dE E1 E Z Q dE E !ab(k) Tr(Ta Tb Ta Tb) !ab(k) Tr(Ta Tb Ta Tb) !ab(k) Tr(Ta Tb Ta Tb) !ab(k) Tr(Ta Tb Ta Tb) which reduces nicely to Z E1 dEk Z d k E1 out 4 Ek in 4 To compare to [6] we write !ab(q1) Tr [(ta tq1 !aq1 (k) + tb tq1 !bq1 (k) + (ta t b Ta Tb) !ab(k))Ta Tb] : Q 2 Q 2 Q 2 Q 2 pa = pb = (1; 0; 0; 1) (1; 0; 0; 1) q1 = x1 (1; 0; sin 1; cos 1) k = x2 (1; sin 2 sin ; sin 2 cos ; cos 2) and !aq1 (k) = !bq1 (k) = !ab(q1) = !ab(k) = (1 cos 2) (1 (1 + cos 2) (1 2 2 sin2 1 : We can do the azimuthal integral using (1 cos 1) sin 1 sin 2 cos (1 + cos 1) sin 1 sin 2 cos cos 1 cos 2) cos 1 cos 2) Z 2 d 0 sin 1 sin 2 cos cos 1 cos 2) = 1 j cos 1 cos 2j 1 (C.3) (C.4) (C.5) (C.6) (C.7) Tr(ta tq1 Ta Tb) +Tr(tb tq1 Ta Tb) (1 cos 1)(1 + cos 2) cos 2 cos 1 (1 cos 2)(1 + cos 1) cos 2 cos 1 # +2 Tr((ta tb Ta Tb) Ta Tb) : Now do the colour traces, i.e. Tr(ta tq Ta Tb) = Tr(tb tq Ta Tb) = N CF2 + CF ; 2 Tr(ta tb Ta Tb) = Tr(Ta Tb Ta Tb) = N CF2 : CF ; 2 s 2 Z 1 dx1 Z 0 d(cos 1) ! =Q x1 1 sin2 1 Z x1 dx2 Z 1 d(cos 2) =Q x2 0 1 cos 2 cos 1 h s 2 ln2(Q= ) (1 cos 2)(1 + cos 1) : (1 cos 1)(1+cos 2) + (1 cos 2)(1+cos 1) 2(cos 2 cos 1) Z 0 d(cos 1) 1 sin2 1 Z 1 d(cos 2) i (C.11) (C.12) So that x)(1 + y) = (2) 2 1( ) N CF (2) 2 s 2 ln2(Q= ) ; which is equal to the result in [6]. C.2 Calculation of 1 and 2 at order s3 The same methodology as in the previous subsection can be used to compute s3, and 1 at the same order. The sum 1 + 2 then gives the non-global contribution at N out E1 4 2CA3CF !ab(q1)(Aqa21k + Aq2k q1b s 3 ln(Q= )3 3! CA2CF (3) : + !ab(q1)!aq1 (q2)!aq2 (k) + !ab(q1)!aq1 (q2)!q1q2 (k) !ab(q1)!bq1 (q2)!bq1 (k) +!ab(q1)!bq1 (q2)!bq2 (k) + !ab(q1)!bq1 (q2)!q1q2 (k)) HJEP05(218)4 s Z Q dE1 d 1d 1 Z s Z E1 dE2 d 2d 2 s Z d kd k Z E2 dEk out Aqa2bk) + E2 4 in 4 2 4CA2CF2 !ab(q1)!ab(q2)(Aqa2bk Aqa1bk) 2( ) = N out s Z Q dE1 d 1 Z s Z E1 dE2 d 2 s Z d k Z E2 dEk h out in 4 Ek 4CA2CF2 (!ab(q1)!ab(q2)!aq1 (k) + !ab(q1)!ab(q2)!bq1 (k)) 2CACF (!ab(q1)!ab(q2)!aq2 (k) + !ab(q1)!ab(q2)!bq2 (k)) + 2CA3CF (!ab(q1)!ab(q2)!ab(k) !ab(q1)!aq1 (q2)!aq1 (k) s Z Q dE1 d 1d 1 Z s Z E1 dE2 d 2d 2 s Z d kd k Z E2 dEk h In order to facilitate comparison with the work of Delenda & Khelifa-Kerfa [8], we have Aiajb = !ab(qi)(!aqi (qj ) + !qib(qj ) !ab(qj )) ; used the notation 3 1 s ( ) = N out see eq. (2.2b) of [8]. In addition, 1 at order s3 is s Z Q dE1 d 1 Z s Z E1 dE2 d 2 s Z d k Z E2 dEk h E1 4 in E2 4 in 4 CACF (!ab (q1) !ab (q2) (!aq1 (k) + !bq1 (k) !ab (k)) + !ab (q1) !ab (k) (!aq1 (q2) + !bq1 (q2) !ab (q2))) + CA3CF !ab (q1) (!ab (q2) !ab (k) !a1 (q2) !aq1 (k) !bq1 (q2) !bq1 (k) !bq1 (q2) !aq1 (k) !aq1 (q2) !bq1 (k)) in Ek (C.13) (C.14) Ek (C.15) (C.16) = 2 N out CA3CF Aqa1bq2 Aqa1bk s 3 ln(Q= )3 3 1 s = 2 = 3! Z Z E1 4 in E2 4 2CA2CF2 (!a3bAqa1bq2 + !a2bAqa1bk)i CA2CF (3) ; x1>x2>x3 x1>x2>x3 out in in ; where Aqaibqj = Aqaibqj =!ab(qi) (see eq. (3.8) and eq. (3.11) in [8]). This is in agreement with the result in [8, 9], which is written as where iin = ( i); iout = ( i) and explicit expressions for W123 RV R and W123 RRR are pre Generally we wish to compute hHjVyVjHi = Tr(O) where O = VjHihHjVy, and jHi represents the hard scattering process while V accounts for the subsequent evolution (real and virtual). Since the basis is non-orthonormal it is useful to introduce dual basis vectors, j ] de ned so that We can now write X X j i[ j = j ]h j = 1 and h j ] = [ j i = : Tr(O) = X Tr( [ jOj ] h j i ) and ; ; O = X[ jOj ] j ih j : Our interest is to compute the matrix elements [ jOj ] for a speci ed pair of external states, and . This we do by evolving inwards from the external states, stripping o soft-gluon operators as we head towards the hard scattering (which lies at the heart of O). The key result in allowing us to accomplish this is the fact that we can write O = LO0R where L and R are colour reconnectors, which means Rj i = CR j i where CR = [ jRj i : Note that there is no sum over on the right-hand side, i.e. reconnectors constitute a unique map from one basis vector into another. A similar relation holds for L. To make the equations slightly simpler, we will put L = 1 in what follows. 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René Ángeles Martínez, Matthew De Angelis, Jeffrey R. Forshaw, Simon Plätzer, Michael H. Seymour. Soft gluon evolution and non-global logarithms, Journal of High Energy Physics, 2018, 44, DOI: 10.1007/JHEP05(2018)044