Non-global logarithms at finite N ^sub c^ beyond leading order

Abstract

Abstract

We analytically compute non-global logarithms at finite N ^sub c^ fully up to 4 loops and partially at 5 loops, for the hemisphere mass distribution in e ^sup +^ e ^sup -^ [arrow right] di-jets to leading logarithmic accuracy. Our method of calculation relies solely on integrating the eikonal squared-amplitudes for the emission of soft energy-ordered real-virtual gluons over the appropriate phase space. We show that the series of non-global logarithms in the said distribution exhibits a pattern of exponentiation thus confirming -- by means of brute force -- previous findings. In the large-N ^sub c^ limit, our results coincide with those recently reported in literature. A comparison of our proposed exponential form with all-orders numerical solutions is performed and the phenomenological impact of the finite-N ^sub c^ corrections is discussed.

Published for SISSA by Springer

Received: January 9, 2015

Accepted: February 23, 2015

Published: March 18, 2015

Kamel Khelifa-Kerfaa and Yazid Delendab

aDpartement de Physique, Facult des Sciences, Universit Hassiba Benbouali de Chlef, Chlef, Algeria

bDpartement des Sciences de la Matire, Facult des Sciences, Universit Hadj Lakhdar, Batna, Algeria

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We analytically compute non-global logarithms at nite Nc fully up to 4 loops and partially at 5 loops, for the hemisphere mass distribution in e+e di-jets

to leading logarithmic accuracy. Our method of calculation relies solely on integrating the eikonal squared-amplitudes for the emission of soft energy-ordered real-virtual gluons over the appropriate phase space. We show that the series of non-global logarithms in the said distribution exhibits a pattern of exponentiation thus conrming by means of brute force previous ndings. In the large-Nc limit, our results coincide with those recently reported in literature. A comparison of our proposed exponential form with all-orders numerical solutions is performed and the phenomenological impact of the nite-Nc corrections is discussed.

Keywords: QCD Phenomenology, Jets

ArXiv ePrint: 1501.00475

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP03(2015)094

Web End =10.1007/JHEP03(2015)094

Non-global logarithms at nite Nc beyond leading order

JHEP03(2015)094

Contents

1 Introduction 1

2 Hemisphere mass distribution at one and two-loops 42.1 Observable denition and kinematics 42.2 One-loop calculation and the Sudakov exponentiation 82.3 Two-loops calculation and non-global logarithms 9

3 Non-global logarithms beyond leading order 113.1 Three-loops calculation 113.2 Four-loops calculation 143.2.1 Reducible parts 153.2.2 Irreducible part 163.3 Five-loops and beyond 18

4 Comparison with large-Nc results 214.1 Comparison with analytical results at large Nc 21

4.2 Comparison with all-orders numerical results 22

5 Conclusions 25

A Angular integrations 26

B A note on NGLs- n relation 28

1 Introduction

As the LHC started colliding protons with unprecedented beam energy, interest has risen in topics that had not received much attention previously, with the aim of uncovering new physics signals. Of these topics, that have recently seen substantial development, is the substructure of fat jets originating from the almost-collinear decay products of heavy resonances that are highly boosted (see for example refs. [113]). Many substructure techniques, such as ltering [1], pruning [4] and trimming [14], have been developed for the purpose of improving the discrimination of signals from QCD background. The latter substructure techniques aid in providing cleaner and more accurate measurements of the properties of these resonances through: rst, identifying the origin of the jet (decayed massive particle signal or plain QCD radiation background). Second, mitigating away the jet constituent particles that have most likely originated from initial-state radiation, underlying-event and pile-up.

1

JHEP03(2015)094

These techniques require in many situations calculations of QCD observables (e.g., jet mass, jet shapes, etc) which need special attention particularly in the vicinity of the threshold limit where they become highly contaminated with perturbative large logarithms as well as non-perturbative corrections. There has recently been some analytical work on a handful of the said substructure techniques with the aim to pave the way to a better understanding of their analytical properties (see [15] and references therein). Nonetheless and for the majority of QCD observables and substructure techniques the only other option available is resorting to numerical simulations which are based on Monte Carlo (MC) integration methods, and which use several approximations, e.g., Herwig [16, 17], Pythia [18, 19] and Sherpa [20]. These MC event generators have been very successful in describing collider data and are commonly used in the extraction of crucial information to boost the search for new physics.

An important issue that needs addressing is the accuracy of the said MC algorithms and the range of validity of the approximations used therein. For instance, amongst the widely adopted approximations in the said MC generators is that of large-Nc limit (with

Nc the dimension of SU(Nc) group). The latter limit, which corresponds to neglecting non-planar Feynman diagrams, greatly simplies the otherwise tremendously complex colour structure, especially at high multiplicities. However, MC generators are generally tuned with data from collider experiments for parameters that account for non-perturbative effects such as hadronisation, underlying event, etc. The process of tuning is itself vulnerable to erroneously ascribing neglected perturbative (observable-dependent) components, which might be originating from nite-Nc corrections, to universal non-perturbative parameters.

This could then potentially be a source of major discrepancy between the data and the predictions by the MC generators. It is thus of great importance to assess the validity of these approximations and make sure that neglected terms would not a ect precision measurements.

Amongst the issues that MC generators are meant to tackle is that of the resummation of large logarithms typically inherent in the distributions of most observables. These large logarithms are a manifestation of the miscancellation of infrared/collinear singularities at the matrix-element level, due to the exclusion of real-emission events in certain regions of phase space. For several observables of su ciently inclusive nature,1 i.e., global observables, the resummation of these logarithms is relatively straightforward and has even been achieved analytically to NNNLL accuracy [21]. In fact semi-numerical programs have been developed with the power of resumming a wide range of global observables up to NLL (CAESAR [22]) and even to NNLL recently (ARES [23]). However the extension of the resummation programme, up NNLL or even to just NLL accuracy in some cases, has seen slow progress for another class of observables, namely non-global observables [24, 25].

Non-global observables are observables that are sensitive to emissions in restricted angular regions of the phase space. The distributions of such observables contain logarithms (named non-global logarithms (NGLs)) of the scales present in the process. For instance, the hemisphere mass distribution contains logarithms of the ratio Q/(Q) , where Q is the

1By su ciently inclusive one means observables that are inclusive over emissions in the entire angular phase space.

2

JHEP03(2015)094

hard scale of the process and is the normalised hemisphere mass squared. In the region where 1, these NGLs can form large contributions to the said distributions, and

should thus be resummed to all-orders. Up to very recently, their resummation was only possible numerically at large Nc by means of: an MC program [24, 25] or solutions to the non-linear integro-di erential Ban-Marchesini-Smye (BMS) evolution equation [26]. The large-Nc approximation signicantly simplies the colour ow in multiple gluon branchings enabling the possibility of the resummation of NGLs at least numerically. Much e ort has recently been advocated to achieving numerical (analytical) resummation of NGLs at nite (large) Nc. The work of Hatta and Ueda [27] exploits the suggestion of Weigert [28]

to use an analogy between the resummation of small-Bjorken-x (BFKL) logarithms and that of NGLs at nite Nc in a numerical fashion. They have noticed that the neglected nite-Nc corrections are indeed negligible in the context of e+e di-jets. They have

however speculated that the situation may be drastically di erent for hadronic collisions. Furthermore, Rubin [29] numerically computed the NGLs series for both ltered Higgs-jet mass as well as interjet energy ow observables up to ve- and six-loops, respectively, at large Nc. In the same limit, Schwartz and Zhu [30] worked on the analytical solution to the BMS equation by means of an iterative series-solution up to ve-loops.

The major hindrance that one inevitably faces when attempting to compute NGLs analytically at nite Nc is twofold. Firstly, the colour topology of a multi-gluon event requires evaluations of non-trivial traces of colour matrices in SU(Nc), which become increasingly cumbersome starting from four-loops. Secondly and not less important, the non-Abelian gluon branchings increase the number of Feynman diagrams factorially at each escalating order to the extent that an automated way of accounting for all possible branchings becomes inescapable.2 Besides, there is also the issue of the various possible real, virtual and real-virtual gluon congurations that are eventually responsible for the miscancellation of soft singularities, thus leading to the appearance of large logarithms. These di culties may have been the main reason for the slow progress in the resummation of NGLs at nite Nc.

In this paper we overcome the above-mentioned di culties and present the rst analytical calculation of NGLs at nite Nc beyond leading order. Working in the eikonal approximation [3136] for soft (strongly) energy-ordered partons, the rst problem, i.e, that of colour structure, is resolved via the use of the Mathematica package ColorMath developed by Sjdahl [37, 38]. The latter program performs the summation of SU(Nc)

colour matrices in an automated way at any loop order. For the second obstacle we developed a Mathematica code that accounts for all possible gluon branchings (and thus for all possible antenna functions) in an automated way.3 Consequently we have been able to analytically calculate all squared amplitudes for the emission of soft energy-ordered gluons (for all possible real, virtual and real-virtual congurations) in the eikonal approximation fully (at nite Nc) up to ve-loops. We leave the presentation of these squared-amplitudes and the method of calculation to a forthcoming paper [39].

2The number of cut diagrams to consider at n loops is formally ((n + 1)!)2 for real gluon emission. This number is slightly reduced by considering on-shell particles and exploiting available symmetries.

3This code will be improved, in the near future [39], into a full Mathematica package capable of analytically computing QCD eikonal amplitudes at (theoretically) any loop order.

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JHEP03(2015)094

With these squared-amplitudes at hand, we provide in this paper a calculation of NGLs at nite Nc to single logarithmic accuracy for single-hemisphere mass distribution in e+e di-jets up to ve-loops. While our calculation is full at four-loops it is incomplete

at ve-loops due to missing terms for which the squared amplitudes are not so simple to simplify and/or integrate. We nd that the aforementioned distribution exhibits a pattern of exponentiation both for global and non-global logarithms. We consequently write the all-orders distribution as a product of two exponentials; the rst being the usual Sudakov form factor and the second represents the resummed form factor for NGLs. For the sake of cross-checking we take the large-Nc limit of our result and compare it with previous calculations obtained by Schwartz and Zhu [30]. We nd complete agreement up to our accuracy, which is four-loops. Furthermore, we compare our analytical resummed factor to the available all-orders numerical results [24] and discuss the phenomenological implications of our ndings, particularly the issue of the accuracy of the large-Nc limit, by assessing the importance of neglected nite-Nc corrections up to four-loops.

This paper is organised as follows. In the next section we outline the usual procedure of calculating NGLs by dening the observable, kinematics and the general relation for the hemisphere mass distribution in terms of the squared amplitudes and a measurement operator.4 We present, in the same section, the calculation of NGLs at leading order (two-loops) to warm up for higher loops. In section 3 we explicitly calculate NGLs beyond leading order at three, four and ve-loops. The di culties associated with calculations at ve-loops will be addressed therein. We compare our ndings, in section 4, to those obtained at large Nc in ref. [30] as well as to the all-orders parametrised form reported by Dasgupta and

Salam, which they obtained by tting to the output of their MC program [24]. We also assess the relative size of the corrections due to nite Nc up to four-loops and discuss our ndings in the same section. Finally, we conclude our work in section 5.

2 Hemisphere mass distribution at one and two-loops

Our aim in this paper is to calculate NGLs at nite Nc to single logarithmic accuracy up to the fth order in the strong coupling s (or equivalently up to ve-loops). Our calculation is performed using QCD squared-amplitudes for the emission of energy-ordered gluons in the eikonal approximation. The latter is su cient to capture all single logarithms nsLn, with L being the large NGL. As stated in the introduction, we do not show herein explicit formulae for the said squared-amplitudes and refer the reader to our coming paper [39]. Moreover, for the purpose of this paper, we do not consider the role of any jet algorithm, and postpone such work to future publications.

2.1 Observable denition and kinematics

For the sake of illustration and to avoid unnecessary complications from a hadronic environment, we choose to work with the same observable that was used in the original paper

4The idea of the measurement operator was introduced by Schwartz and Zhu in their paper [30] which we found very helpful in organizing the real-virtual contributions to NGLs.

4

JHEP03(2015)094

on NGLs by Dasgupta and Salam [24] within the framework of QCD, that is, the hemisphere mass distribution in e+e di-jets. This very observable was also considered in

refs. [4042] in the context of soft-collinear e ective theory (SCET). In both refs. [24, 4042] NGLs were only computed up to two-loops. In the eikonal approximation, su cient for our purpose, we consider energy-ordered soft gluon emissions:5 Q kt1 kt2 ktn , with Q the centre of mass energy and kti the transverse momenta of emitted gluons ki. We note that gluon decay into quarks has a sub-leading contribution to NGLs as was found at two-loops in refs. [40, 43].

The four-momenta of the outgoing quark, anti-quark and gluons are expressed in rapidity parametrisation as:

pq = Q2 (1, 0, 0, 1) , (2.1a)

pq = Q

2 (1, 0, 0, 1) , (2.1b) ki = kti(cosh i, cos i, sin i, sinh i) , (2.1c)

where recoil e ects are negligible to single logarithmic accuracy. Here i and i are the rapidity and azimuthal angle of the ith emission and kti its transverse momentum with respect to the z-axis, which we choose to coincide with the outgoing quark direction. We have kti = !i sin i , with !i the energy of gluon ki and i its polar angle. The rapidity is related to the polar angle i through the relation i = ln tan( i/2).

We dene the following antenna functions relevant to the squared amplitudes that we use in this paper:

wiab = k2ti pa.pb

(pa.ki) (ki.pb) , (2.2a)

Aijab = wiab

wjai + wjib wjab

, (2.2b)

Bijkab = wiab Ajkai

JHEP03(2015)094

+ Ajkib Ajkab

. (2.2c)

The quark and anti-quark directions dene two coaxial back-to-back hemispheres (HL and HR) whose axis coincides with the thrust axis at single logarithmic accuracy (see gure 1). We pick for measurement the hemisphere pointing in the positive z-axis (quark direction). The normalised hemisphere mass (squared) is then dened by:

= pq +

XiHRki

2/Q2 2

XiHRki.pq/Q2 =

Xii ,

(2.3)

where we introduced the transverse momenta fractions xi = kti/Q , and the sum over the index i extends over all emitted real gluons in the measured hemisphere HR.

5Since gluons must satisfy Bose statistics one should normally allow for the permutations of the gluons and divide by a factor n!. This is however equivalent to choosing a specic ordering and removing the 1/n! factor.

i 2 ki pq/Q2 = xi e i ,

5

z axis

Figure 1. Schematic diagram for an outgoing qq pair associated with multiple gluon emission. The measured hemisphere is the one pointing in the quark direction (HR).

We compute the integrated hemisphere mass distribution (cross-section) normalised to the Born cross-section, dened by:

() = Z

q

q

HL

HR

Measured Hemisphere

JHEP03(2015)094

1 0

dd d

= 1 + 1() + 2() + , (2.4)

with

0

m() =

XX

e i

!

m

fWX12m , (2.5)

Zx1>x2>>xm

m

Yi=1d

where

fWX12m = fWX(k1, k2, , km) is the eikonal matrix-element squared for the emission of m energy-ordered soft gluons of conguration X o the primary qq pair at mth order, normalised to the Born cross-section. The sum over X extends over all possible real (R) and/or virtual (V) congurations of all the gluons {kj}. For instance, at 2 loops (m = 2)

the eikonal squared-amplitudes

fWX12 over which the sum is taken are: fWRR12, fWRV12, fWVR12,

and

fWVV12. The quantity fWRV12, for example, is read as: the squared amplitude for the emission of two energy-ordered gluons, k1 and k2, with gluon k1 real and gluon k2 virtual.

Notice that, in the eikonal approximation, the squared amplitude for the softest gluon being virtual is simply minus the squared amplitude for it being real. In other words:

fWxxV12m = fWxxR12m , (2.6)

where x could either be R or V. At one- and two-loops, for example, one has:

fWV1 = fWR1 ,

fWRV12 = fWRR12 , fWVV12 = fWVR12 .

(2.7)

The phase space factor for the emission of m gluons is:

m

Yi=1d

e i =

m

Yi=1d3ki (2)32!i = ms

m

Yi=1dxixi d idi2k2ti2g2s , (2.8)

6

where gs = 4 s and

s = s/. The factor

Qmi=1 k2ti/2g2s multiplies the squared am-

plitude

fWX12m to produce the purely angular squared-amplitude WX12m (i.e., WX12m depends only on and variables and the corresponding colour factor). In other words, one may write:

m

Yi=1d

e i

fWX12m =

m

Yi=1d i WX12m , (2.9)

where

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m

Yi=1d

e i 2g2sk2ti= ms

m

Yi=1d i =

m

Yi=1dxixi d idi 2 ,

WX12m =

(2.10)

The non-linear measurement operatorm acts on the squared amplitudes WX12m and

plays the role of excluding gluon emission events for which the hemisphere mass is greater than . It is not, however, equivalent to a simple heaviside step function (

fWX12m

m

Yi=1k2ti 2g2s .

Pi xie i), since it requires non-numerical input (information about the real-virtual nature of the various gluons). Due to strong ordering, the measurement operatorm factorises into a product

of individual measurement operators;m =

Qmi=1i. The squared amplitudes WX12m are eigenfunctions of the measurement operatorsi with eigenvalues 0 or 1 such that:

if gluon ki is virtual theni WX12m = WX12m ,

if gluon ki is real and outside HR theni WX12m = WX12m ,

if gluon ki is real and inside HR with i < theni WX12m = WX12m ,

if gluon ki is real and inside HR with i > theni WX12m = 0 .

This means that events with real emissions inside the measured hemisphere and which contribute more than to the hemisphere mass are excluded (i.e., not integrated over). That is, () represents the probability that the measured hemisphere mass be less than , as is expressed in eq. (2.4). We therefore write the measurement operator as:

i =

V

i +

R

i

outi + ini ( i) = 1 i ini

R

i , (2.11)

with i = (i ) = (xie i ) , ini = ( i) , and outi = ( i) . The heaviside

step functions ini and outi respectively indicate whether gluon ki is inside or outside the measured hemisphere region. The operator

R

i (

V

i ) equals 1 if gluon ki is real (virtual)

and 0 otherwise. If gluon ki is real then

R

i

WX12m = WX12m and

V

i

WX12m = 0, and vice

versa. In the above we used the relations ini + outi = 1 and

R

i +

V

i = 1.

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2.2 One-loop calculation and the Sudakov exponentiation

Having properly set up the form of the integrated distribution, we can now proceed with the calculation at each loop level. To warm up for higher loops we start with the one-loop case. At one-loop the squared amplitude for the emission of a single real gluon, multiplied by the corresponding phase space, is given by:

d 1 WR1 =

s dx1

x1 d 1

d1

2 CFw1qq

=

s dx1

x1 d 1

d1

2 2 CF . (2.12)

The corresponding virtual contribution is WV1 = WR1. The measurement operator reads:1 =1 = 1 1 in1

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1 , (2.13)

which when acting on the squared eikonal amplitudes yields:

1WR1 +1WV1 = WR1 1 in1

R

R 1

WR1 + WV1 1 in1

R 1

WV 1

= 1 in1WR1 , (2.14)

where we used

R

1

WV1 = 0 ,

R

1

WR1 = WR1 and WR1+WV1 = 0 (which means that the real and

virtual contributions completely cancel out in su ciently inclusive cross-sections). Substituting into the expression of 1 (e.q. (2.5) with m = 1), we are left with the uncancelled integration:

1() =

Z

d 1 1 in1WR1

= 2 CF

s

Z

L

0 d 1 Z

2

d1

1 dx1 x1

Z

2 , (2.15)

where the step function 1 restricts x1 to be greater than e 1, and since x1 < 1 then one also has the restriction on the rapidity such that 1 < L, with L = ln(1/). Performing the integration to single logarithmic accuracy we nd:

1() = CF sL2 P1() . (2.16)

We note that the leading logarithms in the hemisphere mass distribution are double logarithms, which originate from soft and collinear (to the direction of the outgoing quark) singularities of the squared amplitudes for primary gluon emissions o the initiating hard qq pair. It has long been known that the resummed distribution accounting for these primary emissions (or global logarithms) to all-orders is entirely generated from the leading-order result by simple exponentiation (Sudakov form factor). However, Dasgupta and Salam [24] showed that a new class of large single logarithms appears starting at two gluons emission, and which they termed NGLs. We may thus express the resummed hemisphere mass distribution as follows:

() = P() NG() , (2.17)

8

0

where P is the primary Sudakov form factor,

P() = 1 + P1 + 12! P1

2 + 13! P1

3 +

= exp P1

= exp CF

sL2

, (2.18)

and NG is the resummed non-global factor,

NG() = 1 + NG2() + NG3() + . (2.19)

Our aim in this paper is to compute the non-global functions NGm() for m=2, 3, 4 and 5.

2.3 Two-loops calculation and non-global logarithms

The various squared-amplitudes for the emission of two energy-ordered (real or virtual) gluons with x2 x1 1 are expressed as (see for instance [44]):

WRR12 = WR1WR2 + WRR12 , WRV12 = WRR12 , (2.20a)

WVR12 = WR1WR2 , WVV12 = WVR12 , (2.20b)

where the irreducible term WRR12 is:

WRR12 =

JHEP03(2015)094

1

2CFCAA12qq , (2.21)

and the measurement operator in this case is given by:

2 =12 =

1 1 in1

R

1 1 2 in2

R

2

, (2.22)

where 2 = 1 2 since x1 > x2. Acting on the squared amplitudes yields:

12WRR12 +12WRV12 = 1 2 in2 out1WRR12 , (2.23a) 12WVR12 +12WVV12 = 1 2 in2WVR12 . (2.23b)

The sum of these terms then gives:

=1 1 in1

V

R

1 1 2 in2

R

2

1 + out1

R

1

XX2WX12 = 1 2 in2 WVR12 + out1WRR12

= 1 2 in2

in1 WR1WR2 + out1 WRR12

, (2.24)

where we used the squared amplitudes from eqs. (2.20a) and (2.20b). Substituting into the expression of 2() we obtain:

2() =

Zx1>x2 d 1 1 in1 WR1 d 2 2 in2 WR2 Zx1>x2d 12 out1 in2 WRR12 , (2.25)

where we introduced the shorthand notation d 12m =

Qmi=1 d i i. In the rst integral in the right-hand-side of eq. (2.25), the integrand is symmetric under the exchange k1 k2,

9

which means that we can relax the condition x1 > x2 and divide the integral by a factor 2!. Hence this integral factors out into the product of two separate identical contributions from gluons k1 and k2, which both have exactly the same form as the one-loop result (eq. (2.15)). Thus we nd for this term:

P2() =

Zx1>x2 d 1 1 in1 WR1 d 2 2 in2 WR2

= 1

Z

2! d 1 1 in1 WR1 2

= 1

2! CF

2 , (2.26)

which is just the expansion of the Sudakov P() at second order. Hence:

2() = P2() + NG2() , (2.27a)

NG2() = Zx1>x2

JHEP03(2015)094

sL2

2 = 1 2!

P1

d 12 out1 in2 WRR12 . (2.27b)

The latter expression is the pure non-global contribution at this order. It is given by:

NG2() =

1

2CFCA

2s

Zx1>x2dx1

x1

dx2

x2 d 1d 2

d1 2

d2

2 (x2 ) ( 1) ( 2)A12qq , (2.28)

where we have 2 = (x2e 2 ) (x2 ), since no collinear (double logarithms) are

present for the pure non-global contribution. Hence the x integration easily factors from the rapidity integration and we just set the lower limit on x2 to (to single logarithmic accuracy). Performing the trivial integration over xi we obtain the result L2/2!. We note that at nth order we have:

Z

1 dx1 x1

Z

1 dx2 x2

Z

2 dx3 x3

Z

xn1

Lnn! . (2.29)

Using the result of integration over 2 and 2 from eq. (A.6a) of appendix A (with {j, m} {1, 2}), we obtain:

NG2() =

1

2CFCA

dxn xn =

0

2! d 14 ln

L2

Z

11 e2

1

=

1

2CFCA

L2 2!

2

3 =

L2

2! CFCA 2 , (2.30)

where L =

sL and is the Riemann-Zeta function. This is exactly the result obtained by Dasgupta and Salam [24] for NGLs at two-loops. To the best of our knowledge the analytical calculation of NGLs at nite Nc beyond this order has not been performed before, and it is this very task that we do in the next section for the rst time in the literature.

10

3 Non-global logarithms beyond leading order

3.1 Three-loops calculation

Having reproduced the well-known result for NGLs at leading order (two-loops), we proceed to compute NGLs at nite Nc at next-to-leading order, namely triple gluons emission. As usual we begin by the simplication of the measurement operator which will help us identify both angular and real-virtual congurations giving rise to large logarithms:

3 =123 =

1 1 in1

R

1 1 2 in2

R

2 1 3 in3

R 3

= eU3

1 2 3 in3

V

R 3

2 + out2

R 2

, (3.1)

V

1 + out1

R

1

where eU3

is the collection of terms which when acting on the squared amplitudes WX123, with X summed over, yields a zero. The action of the measurement operator on the various squared-amplitudes summed over X gives:

XX3WX123 = 1 2 3 in3 WVVR123 + out2 WVRR123 + out1 WRVR123 + out1 out2 WRRR123

. (3.2)

As stated in the introduction, the explicit expressions for the various squared amplitudes above (together with those at higher loops) will be presented in our forthcoming work [39]. Here we restrict ourselves to showing the simplication of the above squared-amplitudes in terms of the antenna functions dened previously in eq. (2.2). We have:

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XX3WX123 = 1 2 3 in3 in1 in2 WR1WR2WR3 in1 out2 WR1WRR23

out1 in2 WR2WRR13 out1 in2 WR3WRR12+ + out1 WRVR123 + out1 out2 WRRR123

. (3.3)

In eq. (3.3) WRi and WRRij are dened at previous orders (see eqs. (2.12) and (2.21)), and WRVR123 and WRRR123 are the new irreducible terms of the squared amplitudes at this loop order

proportional to the colour factor CFC2A.

Thus the hemisphere mass distribution at O( 3s), 3(), may be written as a sum of

three contributions: 3() = A3() + B3() + C3(). The rst contribution is:

A3() = Zx1>x2>x3

d 1 1 in1 WR1 d 2 2 in2 WR2 d 3 3 in3 WR3

= 1

Z

3! d 1 1 in1 WR1 3

= 1

3! P1

3 = P3() , (3.4)

which is simply the expansion of the Sudakov P() at this order. The factor 1/3! accounts for the fact that the integrand in eq. (3.4) is completely symmetric under the exchange of gluons, which means that the condition x1 > x2 > x3 can be relaxed and the result multiplied by 1/3!. The integral is then factored out into the product of three identical integrals, each of them resembling the one-loop result eq. (2.15).

11

The second contribution to 3() is:

B3() =

Zx1>x2>x3 d 123 in1 WR1 out2 in3 WRR23 + Zx1>x2>x3d 123 in2 WR2 out1 in3 WRR13+

+ Zx1>x2>x3 d 123 in3 WR3 out1 in2 WRR12 . (3.5)

By swapping k1 k2 in the second integral of eq. (3.5), and performing the successive

permutations: k1 k2 then k1 k3 in the third integral of the same equation, B3()

becomes:

B3() =

Zx1>x2>x3 d 123 in1 WR1 out2 in3 WRR23 + Zx2>x1>x3d 123 in1 WR1 out2 in3 WRR23+

+ Zx2>x3>x1 d 123 in1 WR1 out2 in3 WRR23 . (3.6)

The three integrands in eq. (3.6) are identical except for the region of integration over transverse momenta fractions. Thus we unify them into a single integral with the region of integration expressed by:

(x1x2) (x2x3) + (x2 x1) (x1x3) + (x2x3) (x3x1) = (x2 x3) . (3.7)

Hence we write:

B3() =

Z

d 23 out2 in3WRR23 = P1() NG2() . (3.8)

Thus B3() factors out into a product of the one-loop primary cross-section and the two-loop NGLs cross-section (eqs. (2.15) and (2.27b) respectively). This result is expected from the expansion of the Sudakov form factor at one-loop times the leading NGLs. Said di erently, B3() is just an interference term related to previous orders.

The remaining term C3(), which is the pure irreducible NGLs contribution at this order, NG3(), is proportional to CFC2A and given by:

C3() = NG3() = Zx1>x2>x3

d 1 1 in1 WR1 Zx2>x3

d 123 out1 in3

JHEP03(2015)094

WRVR123 + out2 WRRR123 . (3.9)

From the above equation one sees that the new irreducible NGLs contribution at three-loop order is generated by two mechanisms:

(a) the energy-ordered real gluons k1 and k2 outside HR coherently emit the softest gluon

k3 into HR the term WRRR123 ,

(b) the hardest real gluon k1 outside HR emits the softest gluon k3 inside, while k2 is

virtual (inside or outside HR) the term WRVR123.

12

In both cases NGLs result from the miscancellation with the corresponding squared amplitude for k3 virtual, i.e., the miscancellation between WRRV123 and WRRR123 on the one hand, and

between WRVV123 and WRVR123 on the other hand. Both of these contributions are not related

to previous orders. It seems, at rst inspection, that the second mechanism mentioned above (particularly the case where gluon k2 is inside HR, as clearly shown in eq. (3.10)

below) is in contradiction with the common picture about the origin of NGLs. The latter picture dictates that, to all-orders, NGLs are entirely generated from a soft emission into HR that is coherently radiated by arbitrary ensembles of soft, but harder, large-angle

energy-ordered gluons outside HR [24, 25]. Nonetheless, and though NGLs contribution

from the said mechanism comes from both gluons k2 and k3 inside HR, gluon k2 is actu

ally virtual. We shall see later that this mechanism persists at higher loops too. Hence whenever a contribution to NGLs comes from congurations whereby gluons other than the softest are inside HR, then these other gluons must be virtual.6

In fact the contributions of the two terms WRVR123 and WRRR123 in eq. (3.9) are separately divergent but their sum is nite. The integral (3.9) can be expressed as a sum of two nite terms in the following way:

NG3() = Zx1>x2>x3

d 123 out1 in3

in2 WRVR123 + out2 hWRVR123+ WRRR123i . (3.10)

Substituting the explicit expressions of the irreducible terms WRVR123 and WRRR123 in terms of the antenna functions yields:

NG3() =14CFC2A

L3

JHEP03(2015)094

Z

0

3! d 1 8 ln2(1 e2 1)

1 4CFC2A

0

L3

Z

3! d 1 2 A23q1

( 1) + A231q( 1) 2 2

, (3.11)

where we performed the trivial integration over transverse momenta fractions to obtain L3/3! and used the results of rapidity and azimuthal integrations shown in appendix A. The terms A23q1( 1) and A231q( 1) are given in eqs. (A.7b) and (A.7c) with {i, j, m} {1, 2, 3}.

The integration over 1 in the rst line of eq. (3.11) yields the result 8 3 and that in the second line gives 4 3. Thus the pure non-global contribution at this order reads:

NG3() =

L3

3! CFC2A 3 . (3.12)

Hence, up to this order we have:

NG() =1

L2

2! CFCA 2 +

L3

3! CFC2A 3 + O( 4s) . (3.13)

It is intriguing to note that the coe cients of NGLs at nite Nc for two and three-loops(i.e., 2 and 3) are identical to those found at large Nc [26, 30].7 This is due to the fact

6This observation was also made in ref. [30].

7The appearance of 2 and 3 at two- and three-loop orders for NGLs is intriguing too. In appendix B we present some useful observations regarding possible relations between the two (NGLs and Zeta function) seemingly distinct quantities.

13

that the combination of real/virtual squared amplitudes strangely produces identical integrands in the expressions of the NGLs contributions NG2 and NG3 (eqs. (2.27b) and (3.10)

respectively). It would have been tremendously easy to resum NGLs for the hemisphere mass distribution to all-orders if the pattern in (3.13) persisted at higher loops. Although, we shall encounter Zeta functions at higher loop orders, the pattern itself unfortunately breaks down starting at four-loops, as we shall see in the next subsection.

3.2 Four-loops calculation

For four gluons emission, the computation of NGLs for the hemisphere mass distribution proceeds in an analogous manner to that of two and three gluons emission. At this (four-loops) order the measurement operator reads:

4 =1234 =

1 1 in1

JHEP03(2015)094

R

1 1 2 in2

R

2 1 3 in3

R

3 1 4 in4

R 4

= eU4

1 2 3 4 in4

R 4

V

3 + out3

R 3

V

2 + out2

R

2

V

1 + out1

R

1

, (3.14)

where eU4

is the sum of all terms which when operate on the squared amplitudes WX1234 and X is summed over give zero. Acting by the measurement operator on the various squared amplitudes and summing over congurations we obtain:

XX4WX1234 = 1 2 3 4 in4 WVVVR1234 + out1 WRVVR1234 + out2 WVRVR1234+

+ out3 WVVRR1234 + out1 out2 WRRVR1234 + out2 out3 WVRRR1234+ + out1 out3 WRVRR1234 + out1 out2 out3 WRRRR1234

. (3.15)

From eqs. (3.2) and (3.15), it should be clear how the result of the action of the measurement operator on the squared amplitudes at mth loop order would look like:

the softest gluon is always inside HR ,

each real gluon ki is associated with a step function outi ,

virtual gluons are associated with neither in nor out (i.e., they can either be in or

out of HR).

The hemisphere mass distribution at fourth order may then be cast in the form (2.5). Substituting the various matrix-elements squared we can split the hemisphere mass distribution, 4(), into ve parts: 4 = A4 + B4 + C4 + D4 + E4, with:

A4 =

Zx1>x2>x3>x4 d 1234 in1 in2 in3 in4 WR1WR2WR3WR4 ,

B4 = Zx1>x2>x3>x4

d 1234

out1 in2 in3 in4 WRR12WR3WR4 + 2 3 + 2 4+

+ [1 3 and 2 4] + [2 1 then 1 4] + [2 1 then 1 3]

,

14

Zx1>x2>x3>x4 d 1234 n out1 in3 in4 WR4 WRVR123+ out2 WRRR123 + 3 4+

+ [3 2 then 2 4] + [1 2 then 1 3 then 1 4]

o

C4 =

D4 =

WRVVR1234 + out3 WRVRR1234 + out2 WRRVR1234 + out2 out3 WRRRR1234 , (3.16)

where terms in the last line are the four-loop irreducible components of the squared amplitudes for the corresponding gluon congurations. The parts A4, B4, C4, and D4

completely reduce to integrals we calculated at previous orders, while the remaining E4

part is the new NGLs contribution. Let us evaluate each of these integrals separately starting with the reducible parts.

3.2.1 Reducible parts

For the rst part A4 we can, as usual, relax the condition x1 > x2 > x3 > x4 and multiply the result by a factor of 1/4!. This part then factors out into the product of four identical integrals of the form we met at O( s) (eqs. (2.15) and (2.16)), thus we obtain for this term:

A4() = 1

Z

4! d 1 1 in1 WR1 4

2!( P1)2 NG2, (3.18)

where we have used eqs. (2.26) and (2.27b) to arrive at the second line of the above equation. Eq. (3.18) is in fact the interference of the expansion of the Sudakov form factor P with NGLs at two-loops NG2.

Performing the integrations in the third part C4 along the same lines outlined above for B4 (i.e., by making appropriate changes of variables) we obtain:

C4() =

Z

d 123 out1 in3

hWRVR123 + out2WRRR123i = P1() NG3() , (3.19)

d 4 4 in4 WR4 Zx1>x2>x3

15

,

Zx1>x2>x3>x4 d 1234 out1 in2 out3 in4 WRR12WRR34 + 2 3 + 2 3 then 3 4 ,

E4 = Zx1>x2>x3>x4

d 1234 out1 in4

JHEP03(2015)094

4 = P4() , (3.17)

which is just the expansion of the Sudakov at the fourth order.

The second part B4 is carried out in a fashion analogous to that of B3 in eq. (3.5). The ve integrands are transformed into the rst integrand, WRR12WR3WR4, with the appropriate changes in the integration limits. Thus we have six integrals having identical integrands but di erent regions of integrations. Writing these integration regions as step functions and simplifying we obtain (x1 x2) (x3 x4). This means that B4 factors out into the

product of two integrals, one over k1 and k2 and the other over k3 and k4, as follows:

B4() =

Zx3>x4

d 3 3 in3 WR3 d 4 4 in4 WR4 Zx1>x2

d 12 out1 in2 WRR12

= 1

4! P1

= 1

where we used eqs. (2.15) and (3.9). This result is the interference between the expansion of the Sudakov and NG3.

The part D4 can be written as follows:

D4() = 1

2 Zx1>x2 d 12 out1 in2 WRR12 Zx3>x4d 34 out3 in4 WRR34

= 12 NG2

2 , (3.20)

which indicates a possible pattern of exponentiation of NGLs since this term resembles the structure of the expansion of exp{ NG2} at this (fourth) order.

3.2.2 Irreducible part

The irreducible part at four-loops, E4(), is given by:

E4() = Zx1>x2>x3>x4

WRVVR1234 + out3 WRVRR1234 + out2 WRRVR1234 + out2 out3 WRRRR1234 . (3.21)

The rst irreducible squared-amplitudes WRVVR1234 and WRVRR1234 are proportional to CFC3A,

whereas the other two amplitudes, WRRVR1234 and WRRRR1234, contain both CFC3A and C2FC2A

terms. The phase space integration of all terms in eq. (3.21) breaks the simple pattern observed in eq. (3.13), with the last two terms (WRRVR1234 and WRRRR1234) even breaking the colour pattern. Therefore the loss of the pattern of NGLs is in fact a manifestation of the break in the structure of the eikonal amplitudes. This might be related to the failure of the probabilistic scheme discussed by Dokshitzer et al. in ref. [32], where such (irreducible) contributions to the eikonal squared-amplitude were dubbed monster terms. They were traced back to be originating from the colour polarisability of jets [32].8 Moreover, we note here, as can be seen from the form of E4, that contributions to NGLs at this order are generated when the softest gluon k4 is emitted inside the measured hemisphere HR,

whilst the hardest gluon k1 is always outside. The other two gluons, k2 and k3, may be emitted inside HR provided they are virtual (in accordance with the observation made in

the previous subsection).

The contribution E4 is not related to the expansion of the Sudakov nor to NGLs at previous orders, and represents the new non-global contribution at four-loops. Together with D4 they form the total non-global contribution, NG4, to the hemisphere mass distribution at this order. To evaluate the part E4 we rst integrate over transverse momenta fractions, which as usual yields L4/4!, and then perform the azimuthal and rapidity integrations according to the angular congurations indicated in eq. (3.21). The azimuthal integrations are not as straightforward as at three-loops and we nd the method used in ref. [30] of contour integration very useful in reducing the number of integrals to be performed. The reduced integrals are then carried out analytically whenever possible, otherwise numerically. Since all analytical integrations yield results that are explicitly proportional to 4,

8More details are to be found in our forthcoming paper [39].

16

JHEP03(2015)094

d 1234 out1 in4

the resultant values from numerical integrations were interpreted in terms of 4. In addition to this semi-numerical approach we also verify our results by numerically integrating each of the nite terms in the following form of E4():

E4() = Zx1>x2>x3>x4

d 1234 out1 in4

n in2 in3 WRVVR1234+

+ in2 out3

WRVVR1234 + WRVRR1234 + out2 in3

WRVVR1234 + WRRVR1234 +

+ out2 out3

WRVVR1234 + WRVRR1234 + WRRVR1234 + WRRRR1234 o, (3.22)

over the full (7-dimensional) phase space using the multi-dimensional numerical-integration library Cuba [45]. The nal result reads:

E4() =

L4 4!

258 CFC3A + C2FC2A

4 , (3.23)

which may also be rewritten in the following two alternative forms:

E4() =

L4

4! CFC3A 4

29

8 +

JHEP03(2015)094

CF CA

1

2

(3.24a)

258 CFC3A 4 +25 (CFCA 2)2 . (3.24b)

The expression (3.24a) explicitly shows the nite-Nc correction to the large-Nc result, while (3.24b) emphasises the pattern CFCAn n+1 seen at two and three-loops. It also reveals that even though E4 is a new irreducible contribution at four-loops, it still contains factors related to lower-order NGL contributions (the term (CFCA 2)2). Observe that the size of the nite-Nc correction in (3.24a) is about 1.5% that of the large-Nc result. This is

in agreement with the conclusion arrived at in [46] for the impact of nite-Nc corrections at all-orders for e+e processes. The total non-global contribution at this order is then given by:

NG4() = D4 + E4 =

L4 4!

=

L4 4!

258 CFC3A 4 135 C2FC2A 22 , (3.25)

and thus the hemisphere mass distribution up to this order is expressed as:

() = P() NG() , (3.26)

NG() =1

L2

2! CFCA 2 +

L3

3! CFC2A 3

258 CFC3A 4 135 C2FC2A 22 + O( 5s) .

In the next subsection we discuss the ve-loops case and the possibility of resummation of NGLs.

17

L4 4!

3.3 Five-loops and beyond

Following the same steps as before we write the measurement operator at ve-loops as follows:

5 =

1 1 in1

R

1 1 2 in2

R

2 1 3 in3

R

3 1 4 in4

R

4 1 5 in5

R 5

= eU5

1 2 3 4 5 in5

R 5

4 + out4

R 4

V

V

3 + out3

R

3

2 + out2

R

2

V

1 + out1

R

1

, (3.27)

where, as usual, eU5

is the sum of all terms that yield vanishing contributions to the hemisphere mass distribution. Acting by the measurement operator on the various squared amplitudes and summing over congurations we obtain:

XX5WX12345 =

5

V

Yi=1 i in5

WVVVVR12345 + out1WRVVVR12345 + out2WVRVVR12345 + out3WVVRVR12345 + out4WVVVRR12345+

+ out1 out2WRRVVR12345 + out1 out3WRVRVR12345 + out1 out4WRVVRR12345 + out2 out3WVRRVR12345+ + out2 out4WVRVRR12345 + out3 out4WVVRRR12345 + out1 out2 out3WRRRVR12345++ out1 out2 out4WRRVRR12345 + out1 out3 out4WRVRRR12345 + out2 out3 out4WVRRRR12345+ + out1 out2 out3 out4WRRRRR12345

JHEP03(2015)094

. (3.28)

The hemisphere mass distribution at ve-loops is then given in eq. (2.5) with m = 5. Substituting the various matrix-elements squared and following the same procedure outlined at four-loops we again obtain two types of contributions; reducible, r5, and irreducible, irr5. The former contains all the interference terms between the Sudakov factor P and NGLs at previous orders as well as interference terms between two and three-loops NGLs. Explicitly written it reads:

r5() = 1

5! P1

5 + 13! P1

3 NG2+12! P1

2 NG3 + P1 NG4+ NG2 NG3 . (3.29)

Note that the penultimate term in the above equation contains the contribution P1 ( NG2)2/2! (eqs. (3.25) and (3.20)).

The irreducible contribution irr5 is expressed as:

irr5 = Zx1>x2>x3>x4>x5

d 12345 out1 in5

WRVVVR12345 + out2WRRVVR12345 + out3WRVRVR12345 + out4WRVVRR12345 + out2 out3WRRRVR12345+

+ out2 out4WRRVRR12345 + out3 out4WRVRRR12345 + out2 out3 out4WRRRRR12345

, (3.30)

which is neither related to the Sudakov factor nor to NGLs at previous orders, and which contains both C2FC3A and CFC4A terms. The irreducible squared-amplitudes that contribute to irr5 can be classied into two types:

18

proportional only to CFC4A: WRVRRR12345, WRVRVR12345, WRVVRR12345, and WRVVVR12345,

containing both C2FC3A and CFC4A: WRRRRR12345, WRRRVR12345, WRRVRR12345, and WRRVVR12345, which all contain monster terms.

The calculation of irr5 turns out to be trickier and more involved than anticipated. In particular we have not yet been able to simplify (like we did at four-loops) the monster parts of the irreducible amplitudes of the second type above, to forms that can readily be integrated. Other than the monster parts, all remaining terms (either of the rst or second type above) are in fact integrable.

Till the full expression of the irreducible contribution is simplied and integrated, we express the result of irr5 in the following form (based on the pattern seen at four-loops (3.24b) and the pieces found in the integrable amplitudes of eq. (3.30)):

irr5 =

L5

JHEP03(2015)094

5! CFC4A 5

+

CF CA

1

2

(3.31a)

CFC4A 5 + a C2FC3A 2 3 , (3.31b)

where a = 5/ 2 3 0.5244 and the constant coe cients and are yet to be determined.

We discuss the possible values of these constants when we compare our results with those at large Nc in the next section. The form (3.31a) explicitly shows the nite-Nc correction.

The total NGLs contribution at ve-loops then reads:

NG5 = NG2 NG3 + irr5

=

=

L5 5!

L52!3! C2FC3A 2 3 + irr5

=

L5 5!

2 CFC4A 5 (10 a ) C2FC3A 2 3 . (3.32)

The results we obtained up to ve-loops, particularly eq. (3.29), in fact suggest a possible resummation of NGLs into an exponential function of the form:

NG() = exp

L2

2! CFCA 2 +

L3

3! CFC2A 3

L4

4! CFC3A 4

29

8 +

CF CA

1

2

+

L5

+ 5! CFC4A 5

+

CF CA

1

2

+ O( 6s)

. (3.33)

Eq. (3.33) may actually be rewritten in a form analogous to that found in ref. [47] (eqs.(5.10) and (5.11)) for clustering logarithms. To this end we write:

NG() = exp

CF

CA

Xn21n! Sn CA

L

, (3.34)

n

where

S2 = 2 , S3 = 3 , S4 = 4

298 +

CF CA

1

2

, S5 = 5

+

CF CA

1

2

. (3.35)

19

The above-mentioned similarity between clustering logarithms and NGLs emphasises the common (non-global) origin of the two types of logarithms. Moreover, following the pattern in (3.31b), eq. (3.33) may also be recast into the form:

NG() = exp

L2

2! CFCA 2 +

L3

3! CFC2A 3

L4 4!

258 CFC3A 4 +25 C2FC2A 22 +

+

L5 5!

CFC4A 5 + a C2FC3A 2 3 + O( 6s) . (3.36)

The expansion of the above exponential exactly reproduces the terms we have calculated up to ve-loops including all interference terms in the distribution. At each higher order one simply adds a new irreducible NGLs term in the exponent.

In fact, if the pattern deduced in eqs. (3.24a) and (3.24b) persists at higher-loop orders, then one can put forth the following ansatz for the general form of the nth order contribution to the exponent of the resummed NGLs factor:

(1)n1

Lnn! CFCAn1 n

n +

n/2

Xk=2k

CF

CA

k11 2k1

!

, (3.37a)

JHEP03(2015)094

(1)n1

Ln n!

n CFCAn1 n +

n/2

Xk=2k CFk CAnk k nk

, (3.37b)

where n represents the oor function of n, and n, n,

n, and

k are constant coe cients to be determined from integrations. The two formulae presented above for the ansatz are equivalent up to the constant coe cients. The rst form stresses the nite-Nc correction while the second preserves the pattern seen at two-, three- and four-loops (eq. (3.24b)). The above formulae may only be veried once higher-loop orders are carried out explicitly. We hope to perform such calculations in the near future.

We note that in the exponent of (3.33) (and (3.36)) the series in L has alternating signs at each escalating order. To assess the relative size of the three and four-loops corrections to the leading two-loops result, we plot in gure 2 the ratio NG/ exp( NG2) for various truncations of the series in the exponent in eq. (3.33). The leading NGLs coe cient seems to dominate for only relatively small values of L (L [lessorsimilar] 0.15). For larger values the series seems to depart from the leading term in an alternating way (towards larger (smaller) values for odd (even) loop orders). These signicant variations mean that the terms computed thus far are insu cient to capture the full behaviour of the all-orders resummed distribution. We expect, however, that adding few more terms in the exponent may lead to a convergent and more stable behaviour, since one could argue that higher-order terms are suppressed by Ln/n!, while n saturates at 1 as n becomes larger.

To cross-check our results, eqs. (3.33) and (3.36), we compare them, in the next section, to previous calculations at large Nc both at xed order and to all-orders.

20

1.4

3-loops

4-loops

1.3

2)

NG/exp( NG

1.2

1.1

1

JHEP03(2015)094

0.9

0 0.1 0.2 0.3 0.4 0.5

L

Figure 2. Plot of the ratio NG()/ exp( NG2()) in terms of the logarithm L = ( s/) ln(1/).

4 Comparison with large-Nc results

4.1 Comparison with analytical results at large Nc

Having calculated the coe cients of NGLs at nite Nc fully up to four-loops and partially at ve-loops, we can now compare our ndings to those of Schwartz and Zhu [30] obtained through the analytical solution to the BMS equation [26] in the large-Nc limit. To go from the nite-Nc case to the large-Nc approximation we simply invoke the replacement CF CA/2 = Nc/2 (where CA = Nc). This is equivalent to expanding CF to rst order

in colour:

CF = N2c 1

2Nc =

Nc

2 + O

1 Nc

. (4.1)

Then the expansion of our result (3.36) to leading order in colour, i.e., at large Nc, up to

ve-loops is:

NG = 1

2

24(Nc

L)2 + 3

12(Nc

L)3 + 4

34 560(Nc

L)4+

+

2 3

288 +

(2 ) 5 + a 26 3

480

!(Nc L)5 + O (Nc

, (4.2)

where we have written the explicit values of 2 = 2/6 and 4 = 4/90. The full result reported by Schwartz and Zhu (SZ) at large Nc, NGSZ, is [30]:

NGSZ = 1

2

24

L)6

bL2 + 3 12

bL3 + 4 34 560

bL4 +

2 3360 +17 480 5

bL5 + O( 6s) , (4.3)

bL is simply Nc L. The two results are thus identical up to four-loops. Recalling that they were arrived at using di erent approaches, their equality provides a solid cross-check of the correctness of the computed NGLs coe cients (at least up to four-loops). Even though

21

where

we are unable to fully compare our result to that of Schwartz and Zhu at ve-loops, due to the missing values of and , it is ironic to note that the pattern spotted at four-loops, eq. (3.24b), seems to hold true at ve-loops. Whilst the term 5 is apparent in (4.3), the product 2 3 is disguising in the factor 2 3/360. A quick comparison reveals the values:

= 17

2 +

1a , =

2a . (4.4)

Given the above values we expect the nite-Nc result of NGLs up to ve-loops to be expressed as:

NG() = exp

L2

2! CFCA 2 +

L3

3! CFC2A 3

L4 4!

258 CFC3A 4 +25 C2FC2A 22 +

+

172 CFC4A 5 + 2 C2FC3A 2 3 + O( 6s) . (4.5)

We are not, however, claiming to have fully accounted for NGLs at this order since we have not explicitly calculated the coe cients of NGLs with colour factors C2FC3A and CFC4A. We hope that further research on this can help verify the above equation (and eq. (3.37) in general) in which case it may actually be possible to nd a key to the analytical resummation of NGLs both at large and nite Nc.

Our approach additionally has the benet that it sheds light on the possibility of assessing the validity of the large-Nc approximation, by means of judging the impact of neglected nite-Nc corrections. An important note in this regard is that at two- and three-loops, as can be seen by comparing eqs. (3.26) and (4.2) at O(

L2) and O(

L3), there are

no hidden terms buried by the large-Nc approximation, and the nite-Nc result can simply be obtained from the solution of the BMS equation by just restoring the full colour factors through: N2c 2CFCA (at two-loops) and N3c 2 CFC2A (at three-loops). At four-loops

this is not true and in fact there is a hidden correction that is given plainly in (3.24a). We regard this as the rst-order proper nite-Nc correction which introduces new terms that are entirely absent at large Nc. The second-order proper nite-Nc correction occurs at ve-loops and is shown in (3.31a).

4.2 Comparison with all-orders numerical results

In order to verify our resummed formula (3.33), and even (4.5) which includes the ve-loops term, it is instructive to compare it to the all-orders numerical solution of the nite-Nc

Weigert equation [28]. We have not, unfortunately, been able to obtain the output of the MC program, written by Hatta and Ueda [27], for the hemisphere mass distribution.9 We

thus postpone this discussion till the said numerical distribution becomes available. Furthermore, to assess the importance of the missing higher-loop terms in (3.33) we compare it to either the results obtained by the numerical solution of the BMS equation [26] or to the output of the numerical MC program of Dasgupta and Salam (DS) [24]. The latter two numerical solutions are in fact identical within a percent accuracy [26, 30], and we thus restrict ourselves to the DS MC program.

9The hemisphere mass distribution has not yet been coded into the MC program [46].

22

JHEP03(2015)094

L5 5!

large Nc

finite Nc

1

1

0.8

0.8

0.6

0.6

SNG (t)

SNG (t)

3-loops

SNGDS

4-loops

2-loops

0.4

SNGDS

3-loops

4-loops

2-loops

0.4

0.2

0.2

0 0 0.1 0.2 0.3 0.4 0.5

0 0 0.1 0.2 0.3 0.4 0.5

t

t

JHEP03(2015)094

Figure 3. Plot of the NGLs function NG() at large (left) and nite (right) Nc.

Let us introduce the standard evolution parameter t [24], which accounts for the running of the coupling:

t = 1

2

Z

1

eL s(Qx)

dxx =

1 4 0 ln

11 2 0 sL

, (4.6)

where 0 is the one-loop coe cient of the QCD function. At xed order one has t = s L/2 = L/2. Hence substituting L by 2 t into eq. (3.33) up to four-loops we nd:

NG(t) = exp

CFCA23 t2 +43CFC2A 3 t3 4 135

258 CFC3A + C2FC2A t4 + O(t5) . (4.7)

We compare the result (4.7) with the parametrisation for NGLs to all-orders obtained in ref. [24] by tting to the output of the aforementioned DS MC program [24]:

NGDS(t) = exp

CFCA231 + (0.85CAt)21 + (0.86CAt)1.33 t2 . (4.8)

In gure 3 we plot our approximate resummed result (4.7) for various truncations along with the DS resummed factor (4.8) for a range of t [0, 0.5] both at large (left) and nite

(right) Nc. Recall that a value of t = 0.3 corresponds to a value of L = 19 and 108

for s 0.1, which is su cient for phenomenological purposes. Few points to note from

the plots. Firstly, as expected for nite Nc, all curves are shifted up due to the fact that one is using CF = 4/3 1.33 instead of CF = CA/2 = 3/2 = 1.5 (recall that CF is in the

exponent). Secondly, it is striking to observe that the best approximation to the all-orders result for quite a large range of t is the leading two-loops result,10 exp NG2

, for both large and nite-Nc cases. This suggests that the alternating, positive and negative, higher-loop contributions to NG somehow balance out.

Moreover, the main feature of the plots and which has direct link to the purpose of this paper is actually seen at small values of t. We see that the interval of t over which the four-loops result and the all-orders resummed factor overlap is 0 t [lessorsimilar] 0.12. This interval

10This observation was made in refs. [29, 30] too.

23

large Nc

finite Nc

1.2

1.2

3-loops

2-loops

4-loops

3-loops

2-loops

4-loops

SNG / SNG

DS

1

SNG / SNG

DS

1

0.8

0.8

0 0.05 0.1 0.15 0.2 0.25 0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

t

t

JHEP03(2015)094

Figure 4. Plot of the ratio NG()/ NGDS() for both large (left) and nite (right) Nc.

0.8

large Nc

1

SNG (t)

SNGDS 2-loops 3-loops 4-loops

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t

Figure 5. Plot of the NGLs function NG(t) at large Nc for the range 0 t 0.14.

of overlapping is smaller for three-loops, 0 t [lessorsimilar] 0.08, and even smallest for two-loops

0 t [lessorsimilar] 0.05. The latter feature may be seen more clearly in gures 4 and 5. One would

therefore expect that adding more terms in the exponent of (3.33) leads to increasingly larger intervals of overlapping.

A similar observation was also made in ref. [29] for ltering analysis in the case of the ltering parameters nlt = 2 and lt = 0.1, 0.3 (gure 18 of ref. [29]), as well as nlt = 3 and lt = 0.3 (gure 21). However the author of ref. [29], having plotted the expansion of the full ltered Higgs-jet mass distribution including both primary and non-global logarithms in the Cambridge-Aachen jet algorithm [48, 49], ascribed the convergence of the series, as one adds higher-loop terms, to the dominance of the primary series. The author then veried this explanation by plotting the same distribution for higher values of lt (gure

19) where collinear logarithms are expected to be absent and NGLs become of the same order as primary logarithms. Two issues to point out regarding our work compared to

24

that of ref. [29]: rstly, we are plotting purely the NGLs resummed exponential factor and hence the convergence seen in gures 4 and 5 has nothing to do with primary logarithms. Secondly, it is well known [43, 5053] that employing the Cambridge-Aachen jet algorithm not only reduces the size of NGLs but also introduces clustering logarithms that are as important as NGLs. Thus plotting the full distribution, which includes primary, non-global and clustering logarithms, would not tell much about the convergence of the NGLs series.

Moreover, the author of ref. [29] also plotted (gure 24) the pure NGLs resummed factor for the interjet energy ow distribution and concluded that, up to six-loops, the NGLs series seems to be divergent. Recalling that the coe cient of the two-loops NGLs depends on the rapidity gap [25, 43], it is likely that higher-loop NGLs coe cients depend on too. The divergence may thus be due to the presence of the terms. For the hemisphere mass distribution that we have treated in this paper there is no such rapidity gap dependence. A proper answer, however, may only be given once the former interjet energy ow distribution is carefully considered, a task which we hope to perform in coming publications.

Notice nally that the so far discussed NGLs behaviour is in contrast to that seen for clustering logarithms [47] where the whole structure of the all-orders result is mostly captured by the rst few terms in the exponent.

5 Conclusions

In this paper we have considered the calculation of the leading NGLs at nite Nc up to ve-loops for the hemisphere mass distribution in e+e di-jet events. We performed

the calculation by means of integrating the squared amplitudes for the emission of energy-ordered soft gluons in the eikonal approximation, valid at single logarithmic accuracy, over a suitable phase space achieved through a measurement operator. The two and three-loops results were shown to be relatively straightforward to obtain, and were found to be directly related to the Riemann-Zeta function. We noticed that, up to this loop order, nite-Nc corrections are absent. This was a direct consequence of the relatively simple structure of the eikonal amplitudes (up to this order) as well as the combination of real/virtual amplitudes induced by the phase space measurement operator.

Within the same eikonal framework we computed the four-loops contribution to NGLs distribution. The latter turns out to be much harder than the previous two orders and the simple result in terms of a product of a single colour factor and a Zeta function breaks down. This failure originates from a break in the simple structure of the corresponding real-virtual eikonal amplitudes at four-loops order, a phenomenon that was noticed more than two decays ago [32]. Nevertheless, we were able to overcome this complexity, compute NGLs and even spot a new pattern for NGLs at and beyond this order. This pattern helped us to successfully write down the ve-loops contribution to NGLs up to constant coe cients which we extracted from comparisons to previous large-Nc results [30]. We hope to be able to fully compute these constants in the near future. The ve-loops calculation reveals that the NGLs distribution seems to exhibit a pattern of exponentiation. To this end the said

25

JHEP03(2015)094

distribution was cast in an exponential form with full NGLs coe cients and colour factors up to four-loops in the exponent.

Comparisons to large-Nc results obtained by other authors [30] conrmed our ndings, at least in the latter limit. We then took the step forth and compared our exponential function to the all-orders numerical resummed result reported by Dasgupta and Salam [24]. To our surprise, the shape of the all-orders result was best represented by the two-loops approximation for a wide range of the evolution parameter t. In the region of small t, however, adding more terms in the exponent of our resummed result yielded better agreement, then the two-loops result, with the all-orders numerical result. This suggests that more higher-loop contributions are needed for our result to be of any phenomenological signicance (i.e., till the agreement extends to values of t up to 0.20.3). The task of computing these

higher-loop terms might not be impossible after all given that we have developed, over the course of preparing this paper, the machinery for: computing eikonal amplitudes at nite Nc to theoretically any loop order, reducing the dimension of the phase space over which to integrate, and spotting a pattern for NGLs at each order.

Acknowledgments

We would like to thank Prof. Abdelhamid Bouldjedri, head of PRIMALAB research laboratory at Batna University, for making at our service the computer resources of the laboratory. We would also like to thank Mrinal Dasgupta for his comments on the manuscript. This work is supported in part by CNEPRU Research Project D01320130009.

A Angular integrations

In this section we present some denitions and azimuthal/rapidity integrations which have proven useful in our calculations.

Following ref. [30] we dene the following angular functions:

(ij) = cosh( i j) cos(i j) , (A.1a)

hiji =

JHEP03(2015)094

(ij)2 sinh i sinh j . (A.1b)

The basic antennas are expressed as:

w1qq = 2 , w2q1 = e 1+ 2

(12) , w21q =

e 1 2

(12) , w312 =

(12)

(13)(23) . (A.2)

The i-azimuthal averaging over the inverse of the angular function (ij) is given by:

Z

2

di

2

1(ij) = csch| i j| . (A.3)

In the case where a gluon km is constrained within the measured hemisphere region and gluons ki and kj are constrained outside, the km angular integration over wmij yields [30]:

Z

0

0 d m Z

2

dm

2 wmij = ln(1 + hiji) = ln

cosh( i + j) cos(i j)

2 sinh i sinh j , (A.4)

0

26

with i < 0 and j < 0. Furthermore, the azimuthal average over the angle i of the emitter ki yields [30]:

Z

2

di

2

1(ij) ln(1 + hiji) = csch( i j) ln

1 coth i

1 coth j

. (A.5)

With the same conditions ( i < 0, j < 0 and m > 0) we can perform the azimuthal and rapidity integrations for the following antenna functions:

Ajmqq Z

0

0 d m Z

2

dm

2 Ajmqq = 4 ln(1 e2 j) , (A.6a)

Ajmqi Z

JHEP03(2015)094

0

0 d m Z

2

1 coth j

1 coth i

cosh( j + i) cos(j i)

2 sinh j sinh i , (A.6b)

Ajmiq Z

dm

2 Ajmqi = wjqi ln

0

cosh( j + i) cos(j i)

2 sinh j sinh i . (A.6c)

We can also perform further integrations over j and j:

Ajmqq = Z

0

0 d m Z

2

1 coth j

1 coth i

dm

2 Ajmiq = wjiq ln

0

d j Z

2

dj

2 Ajmqq =

2

3 = 2 2 , (A.7a)

Ajmqi = Z

0

0

dj

2 Ajmqi

= ln(1 tanh i) ln((coth i 1) coth i) + 2 Li2

11 tanh i

d j Z

2

0

2 Li2 tanh i , (A.7b)

Ajmiq = Z

0

d j Z

2

dj

2 Ajmiq

0

= ln2 2

2

2 + 2 ln(1 coth i) ln

1 coth i

2 + 2 ln( tanh i) ln(csch(2 i))+

+ 2 Li2 1

1 tanh i

+ 2 Li2 1 tanh i

2 + 2 Li2(1 + tanh i) , (A.7c)

with Li2 the polylogarithm function of order 2. Similarly integrating over the angles of the softest particle kn in the antenna Amnij yields:

Amnij =wmij (ln(1 + himi) + ln(1 + hjmi) ln(1 + hiji)) . (A.8)

At four-loops, the following azimuthal integrations are relevant:

Z

2

1(13)(23)

d3

2 =

coth | 1 3|2 sinh( 1 3) sinh( 2 3)

(A.9)

0

sinh | 1 3|csch| 2 3| cosh( 1 2) cosh( 1 2) cos(1 2)+

+cosh( 1 + 2 2 3) sinh | 1 3|csch| 2 3|

cosh( 1 + 2 2 3) cos(1 2)

+ 1 2 .

27

Then we have:

Z

2

d2 2

Z

d3 2

2

1

(13)(23) ln(1 + h12i) =

coth | 1 3|2 sinh( 1 3) sinh( 2 3)

(A.10)

0

0

sinh | 1 3|csch| 2 3| cosh( 1 2) sinh( 1 2)

ln coth 1 1 coth 2 1 +

+ cosh( 1 + 2 2 3) sinh | 1 3|csch| 2 3|

sinh | 1 + 2 2 3|

ln

csch 1csch 2 sinh2 1 + 2 | 1 + 2 2 3| 2

| 1 + 2 2 3| + 1 2 .

B A note on NGLs- n relation

As a byproduct, we notice from eqs. (2.30) and (3.11) that one may dene the following possibly new logarithmic-integral representation for the Riemann-Zeta function:

s (s) =

(1)s1 (s)

JHEP03(2015)094

Z

+

0 lns1 1 e

d , (B.1)

where (s) = (s 1)! is the Gamma function and the variable s is greater than 1. The

above formula is valid if s is an integer. In the case of non-integer real values one has to take the modulus of the right-hand-side in eq. (B.1). In terms of the polar variables ( , ) the Zeta function admits the integral formula:

s = 1

(s)

Z

0

1

21 c2

lns1

1 c

2 c

dc , (B.2)

where c cos . Notice that the form (B.2) seems to fail (in Mathematica 9) for s > 10.

Moreover, it is interesting to note that the rst non-divergent value of the Zeta function is for s = 2, and so is the rst non-vanishing coe cient of NGLs. If we let Ss denote the

NGLs coe cient at the sth loop order then we can write Ss as the Mellin transform of the

function (e 1)1 [54]. That is:

Ss = s = 1 (s)

Z

+

s1

e 1

d , (B.3)

which is at least true for two and three-loops NGLs coe cients. Recall that the Mellin transform techniques were employed in ref. [55] to compute the rst resummed result for event-shape distributions. Whether there exists a more profound relation between NGLs and the Zeta function (and its related functions such as the polylogarithms) is a subject that requires further investigations.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

28

0

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