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arXiv:hep-ph/0303101v1 13 Mar 2003

Preprint typeset in JHEP style. - PAPER VERSION

Bicocca–FT–03–6

LPTHE-03-10

hep-ph/0303101

February 2003

On large angle multiple gluon radiation

Yu.L. Dokshitzera∗and G. Marchesinia,b

aLPTHE, Universit´ es Paris 6 et 7, CNRS UMR 7589, Paris, France

bDipartimento di Fisica, Universit` a di Milano-Bicocca and INFN, Sezione di

Milano, Italy

Abstract: Jet shape observables which involve measurements restricted to a part

of phase space are sensitive to multiplication of soft gluon with large relative angles

and give rise to specific single logarithmically enhanced (SL) terms (non-global logs).

We consider associated distributions in two variables which combine measurement of

a jet shape V in the whole phase space (global) and that of the transverse energy flow

away from the jet direction, Eout(non-global). We show that associated distributions

factorize into the global distribution in V and a factor that takes into account SL

contributions from multi-gluon “hedgehog” configurations in all orders. The latter

is the same that describes the single-variable Eoutdistribution, but evaluated at a

rescaled energy V Q.

Keywords: QCD, Jets, LEP HERA and SLC Physics, Hadronic Colliders.

∗On leave from St. Petersburg Nuclear Institute, Gatchina, St. Petersburg 188350, Russia

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Contents

1. Introduction1

2. Observables6

3. Analysis and resummation

3.1Mellin factorization

3.2Global radiator

3.3 Non-global radiator

7

7

9

9

4. Multi-gluon correlations in all orders 13

5. Conclusions 15

1. Introduction

In a theory with dimensionless coupling one would naively expect cross sections to

scale in a simple manner at very large Q2, Q2≫ m2, with m a generic mass scale.

However, in QCD this is not true for two reasons. Firstly, due to UV divergences, the

effective interaction strength does vary with scale (as in any quantum field theory

with dimensionless coupling). As a result perturbative (PT) corrections to cross sec-

tions slowly decrease with Q2as powers of αs∝ 1/ln(Q2/Λ2

generally speaking, collinear and infrared divergences (collinear 2-parton splittings,

soft gluon radiation). As well known (Bloch-Nordsieck [1]), in “good” inclusive ob-

servables which do not include observation (fixing the momentum) of a single hadron,

either in the initial or final state, the logarithmic collinear and infra-red divergences

cancel.

The classical examples are the total cross section of e+e−annihilation into

hadrons, τ → hadrons decay width. Here the collinear and infra-red divergences,

present in both real and virtual corrections, cancel completely in the (unrestricted)

sum.Since here we don’t have dimensional parameters other than Q2(in the

m2/Q2→ 0 limit), the total cross section is given by the simple Born expression

modulo calculable αn

QCD). Secondly, there are,

s(Q2) corrections.

σ(Q) = σBorn(Q) · g?αs(Q2)?,g(0) = 1(Q2≫ m2).

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Moving to less inclusive measurements one faces different situations. The first

case involves fixing (measuring) momentum of a hadron, e.g. that of the initial pro-

ton in DIS, DY, ...(structure functions, SF) or of a final hadron (fragmentation

function). Then soft divergences still cancel but collinear ones do not, making such

observables not calculable at the parton level.

to be universal and, given a proper technical treatment, can be factored out as

non-perturbative (NP) inputs. What remains under control then is only the Q2-

dependence (scaling violation pattern).

Collinear divergences in the final state may be avoided altogether if one looks at

energy flows rather than individual hadrons. More generally, one can study a family

of the so-called collinear and infra-red safe (CIS) jet shapes V ,

These effects, however, turn out

V =

?

i

v(ki), (1.1)

where the sum runs over all particles (hadrons) in the final state, v(k) being a con-

tribution of a single particle with 4-momentum k. Being (by construction) linear in

particle momenta, such observables are also free from collinear (and soft) divergences.

However, here the cancellation of real and virtual effects is not complete and

leaves trace in the PT-calculable distributions over V . Indeed, taking the value of a

generic jet shape observable V ≪ 1 we squeeze the phase space thus inhibiting real

parton production and multiplication. Since the virtual PT radiative contributions

remain unrestricted, the divergences do cancel but produce finite but logarithmically

enhanced leftovers:

Σ(Q,V ) ≡

?V

0

dVdσ(V )

σtotdV

= f?αs(Q2),lnV?.(1.2)

Each gluon emission brings in at most two logarithms (one of collinear, an-

other of infra-red origin). These leading contributions are due to multiple soft gluon

bremsstrahlung off the primary hard partons that form the underlying event, which

can be looked upon as independent gluon radiation and can be easily treated. A

clever reshuffling of PT series, based on universal nature of soft and collinear radia-

tion (factorization) results in the exponentiated answer in the form

lnΣ(Q,V ) =

∞

?

n=1

αn

s(Q2)?Anlnn+1V + BnlnnV + ···?.(1.3)

In what follows we will discuss only these two series of term and neglect subleading

small corrections of the order of αs, α2

double logarithmic (DL) and Bnas single logarithmic (SL).

The whole leading series An actually originates from the effect of running of

the coupling in the basic one gluon radiation term, αsln2V , n = 1. Indeed, the

slnV , .... The An series is referred to as

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expansion in αs(Q2) is pretty artificial since in reality it is a wide range of scales,

(V Q)2≪ k2≪ Q2, at which the coupling αs(k2) actually enters:

A1αs(Q2)ln2V =⇒ A1

?Q2

(V Q)2

dk2

k2

t

t

αs(k2

t)lnQ

kt

=

∞

?

n=1

An· αn

s(Q2)lnn+1V, (1.4)

where ktis the gluon transverse momentum and the logarithmic factor is due to the

soft enhancement. As a result, the coefficients Anare straightforward to obtain. It

is important to realise that what makes the coupling run in Minkowskian observ-

ables is collinear gluon splittings (into gluons with energies of the same order, “hard

splitting”, and q¯ q pairs) and the inclusive CIS nature of the observable (see, e.g. [2]).

Let us remark in passing that the scales below (V Q)2with k2

zero are actually present as well. They, however, do not contribute at the PT level

but give rise to the NP power suppressed corrections (see [2,3]).

The subleading series Bnstart from the first SL correction αslnV (n = 1) (of

either collinear or soft origin). In higher orders, n ≥ 2, taking care of SL terms

involves a careful treatment of αs(physical scheme), of its running argument, as well

as precise fixing of the scales of the leading logarithmic terms in (1.3).

In the case of the so-called “global” observables, that is measurements in which

the observable (1.1) accumulates contributions from final state particles in the whole

phase space, all SL contributions are generated by a careful treatment of gluon

bremsstrahlung off the primary partons (Sudakov exponentiation). Obviously, there

exists also gluon multiplication and, in particular, gluon bremsstrahlung off sec-

ondary partons. As already stated, collinear “hard” gluon decays make the coupling

run. As for soft gluon bremsstrahlung off secondary gluons, it does not affect global

observables at the DL+SL accuracy (1.3) (the first correction being O(α2

Global observables have been intensively studied in the literature both perturba-

tively [4, 5] (DL+SL as in (1.3)) and with account of the first leading NP power

correction O(1/Q) [3].

However, as was recently noticed by Dasgupta and Salam, certain jet shape

measurements turn out to be sensitive to soft gluon multiplication effects at the SL

level starting from n = 2. These observables involve measurements restricted to a part

of phase space and were correspondingly dubbed non-global [6]. The simplest example

is given by particle transverse energy flow Eoutin e+e−annihilation, measured in an

angular (pseudorapidity) region Cout which is away from the jet direction (thrust

axis) by a finite angle θ0:

trunning down to

slnV )).

Σout(Q,Eout),Eout=

?

i∈Cout

kti. (1.5)

It is a CIS observable. Moreover, there is obviously no collinear enhancement (An≡

0). The leading effect is SL – exponentiation of independent large angle soft gluon

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emission off the primary jets. Apart from it, however, additional contributions of

the same order emerge here due to soft gluon-gluon correlations, which were absent

in global observables.

Imagine a system of gluons whose energies are strongly ordered:

kℓ≪ kℓ−1≪ ... ≪ k1≪ p ≃ Q/2.

Then only the hardest of the gluons belonging to Coutwould contribute to the observ-

able while the contributions of all other (much softer) gluons would cancel against

corresponding virtual corrections. This means that it suffices to consider multi-gluon

systems with the softest offspring kn∈ Coutbeing the only one to be measured, while

all harder companions do not contribute to the observable, ki∈ Cin, i < n, where

Cinis the complementary angular region, close to the jets.

Such configurations of n ≥ 2 gluons contribute to the integrated distribution

(1.5) at the SL level as (αsln(Q/Eout))n. This is the origin of the Dasgupta–Salam

discovery.

These specific subleading contributions are not easy to analyse analytically order

by order. In spite of the fact that in the strong energy ordered kinematics all (n+2)!

amplitudes are known and given by soft insertion rules, to give a compact answer to

the square of the n-gluon matrix element is possible only in the large-Nclimit (see [7]).

Moreover, angles between gluons are of the order one (hedgehog configurations), so

that the phase space integrations can be handled only numerically. As we shall

show below, the all-order resummation of these contributions can be carried out in

the large-Nclimit. This leads to an evolution equation [8] which has a highly non-

linear structure. Therefore, its solution can be given in an analytic form only in an

academic high-Q limit (αsln(Q/Eout) ≫ 1).

Recall that non-linear evolution equations for generating functionals describing

multi-parton ensembles have a long history. In the leading collinear approximation

they, in particular, form the basis for Monte Carlo generators and for the standard

theoretical jet studies (jet rates, in-jet particle multiplicities and spectra, etc.) [7,9].

When large angle soft gluon radiation becomes important (as, e.g., in inter-jet particle

flows – the bread for the so-called string effect), in the collinear approximation (that

is when other secondary gluons stay quasi-parallel to the primary hard parton) the

QCD coherence is at work. As a result, a single soft gluon emission at large angles

is very simple and is determined by the total colour charge of the jet.

However, when the large-angle hedgehogs are considered, the new evolution equa-

tion for such systems has an essentially different (and more complicated) structure

which, as mentioned above, can be practically handled only in the large-Ncapprox-

imation.

It is important to stress that most of the actual experimental measurements are

subject to non-global effects. Experiments often (if not always) involve phase space

restriction. For example,

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• the very first CIS jet cross section invented by Sterman and Weinberg back in

1979 [10] provides a perfect example of a non-global observable;

• in hadron–hadron interactions accompanying hadron production is studied in a

limited rapidity range (in particular, the famous pedestal distributions in hard

hadron–hadron collisions);

• direct photon studies necessarily involve photon isolation criteria;

• a family of QCD string/drag effects deals with particle flows in restricted inter-

jet angular regions;

• profiles of a separate jet (rather than studying shape variables of the whole

event). For example, characteristics of the current fragmentation jet in DIS

which is based in a one-hemisphere particle selection.

All these and many other similar measurements contain non-global PT corrections.

With exception of the current jet in DIS (see [11]) and the case study of the Eoutdis-

tribution in e+e−[8,12,13], the non-global effects have not been studied theoretically

even at level of the first (αslog)2correction.

In spite of being formally subleading, these effects may significantly modify the

PT predictions for the non-global observables, as was shown by the case study of the

Eoutdistribution in e+e−by Dasgupta and Salam [12].

Berger, Kucs and Sterman [14] have recently formulated a programme of how

to avoid non-global logarithms in a measurement of transverse energy flow away

from jets (Eout). They suggested to squeeze the jets and thus suppress multi-gluon

hedgehogs in the Cin region. In e+e−this amounts to introducing the associated

distribution in two variables, Eout and V (for example, V = 1 − T with T the

thrust), and considering the region V ≪ 1 which selects 2-jet-like configurations,

Σ2ng(Q,V,Eout).

They treated the region in which the two characteristic scales of the problem are

comparable, V Q/Eout= O(1), so that

αslnV Q

Eout

(1.6)

amounts to a negligible correction O(αs). The authors stated in the Conclusions

to [14] that in this approximation their “formalism is sensitive mainly to radiation

stemming directly from primary hard scattering”.

In this paper we consider a general case1of Eoutbeing potentially much smaller

than V Q. We show that, having extracted the global DL and SL enhanced term

1We take V to be a global observable, though one can equally well restrict V to Cinas was done

in [14].

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in lnV , the remaining SL corrections (αslnV )nand (αsln(Q/Eout))nconspire to

produce the powers in (1.6).

We analyse these non-global logarithms in all orders and demonstrate that the

associated distributions factorize as

Σ2ng(Q,V,Eout) = Σ(Q,V ) · Σout(V Q,Eout). (1.7)

Here Σ(Q,V ) is the standard global distribution (1.2) and Σout(Q,Eout) is the SL

distribution in Eout(at the same total energy Q) which takes into account contri-

butions from multi-gluon hedgehog configurations in all orders and contains the full

dependence on the geometry of the measurement (θ0). This is the main result of the

paper.

We will also discuss the asymptotic behaviour which according to (1.7) reduces

to that of the simplest non-global distribution Σoutwhich had been studied in [8].

Let us stress that one needs to keep under the best possible control effects in-

duced by radiation of relatively soft gluons not merely for the sake of improving PT

predictions. What makes such studies even more important and interesting is the

fact that the physics of small transverse momentum gluons is that of confinement.2

2. Observables

In this paper we will consider e+e−annihilation into hadrons and define a cone

around the thrust axis and denote by Coutand Cinthe regions outside and inside the

solid cone (inter and intra jet regions).

We start by considering the first factor in (1.7) that is the (integrated) global

distribution Σ(Q,V ) defined as

Σ(Q,V ) =

?

n

?

dσn

σtotΘ

?

V −

?

i∈Ctot

v(ki)

?

,(2.1)

with the sum extended to the full phase space Ctot= Cin+ Cout. Here dσnand σtot

are the production and total cross sections. The probing function v(k), linear in the

particle momentum k, can be represented as

v(k) =kt

Q· h(η), (2.2)

with ktthe transverse momentum with respect to the thrust axis and η the pseu-

dorapidity. To get the distribution in thrust, T, we have to take h(η) = e−|η|and

the set V = 1−T. For small 1 − T, summing over final particles we may neglect

contributions from the primary hard partons (quarks) since h(η) vanishes at large

|η|, and restrict ourselves to to considering only secondary soft gluons.

2In market terms, a $106problem [15]

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The distribution in broadening B, for example, corresponds to h(η) = 1, V = 2B.

In this case, even for B ≪ 1, one needs to include the contributions from the recoiling

primary quarks.

Now we define analogously the non-global distribution (the second factor on the

r.h.s. of (1.7)):

Σout(Q,Eout) =

?

n

?

dσn

σtotΘ

?

Eout−

?

i∈Cout

kti

?

, (2.3a)

where we have chosen to measure the transverse energy flow accumulated in the

angular region Coutaway from the jets.

Finally, as was suggested by Berger, Kucs and Sterman [14], we introduce the

shape observable distribution in two variables (the associated distribution)

Σ2ng(Q,V,Eout) =

?

n

?

dσn

σtotΘ

?

V −

?

i∈Ctot

v(ki)

?

·Θ

?

Eout−

?

i∈Cout

kti

?

.(2.3b)

3. Analysis and resummation

We are interested in the soft region

V,Eout≪ Q. (3.1)

3.1 Mellin factorization

First we recall how the resummation with the SL accuracy is done for global ob-

servables. To this end one has to factorize the theta-function in (2.1) by the Mellin

transform:

Σ(Q,V )=

?

?

dν

2πiνeν V ˜Σ(Q,ν),

?

i∈Ctot

˜Σ(Q,ν) =

n

dσn

σtot

?

u(ki),u(k) = e−ν v(k).

(3.2)

As has been mentioned above, in the thrust case the contributions of the primary

partons can be dropped (since v(k) vanishes at large |η|).

In the case of broadening the recoiling primary partons do contribute. This

causes certain complication. This recoil effects can be taken care of by expressing

the momenta of recoiling quarks in terms of those of the secondary partons via 3-

momentum conservation. Thus, also in the case of broadening one can restrict the

product in (3.2) to the secondary soft partons, provided one factorizes also the mo-

mentum conservation constraints by the Fourier transform. This amounts to incor-

porating into˜Σ an additional Fourier variable associated with transverse momentum

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conservation. Peculiarities of B as a variable employed to squeeze the jets, can be

analyzed along the lines of [5]. This leads to the known modification of the global

distribution but will not affect the dependence on Eoutin (1.7).

In what follows to simplify the discussion we neglect this complication and con-

centrate on the thrust case.

For the out-distribution we have

Σout(Q,Eout) =

?

?

dνout

2πiνout

?

eνoutEout/Q˜Σout(Q,νout),

˜Σout(Q,νout) =

n

dσn

σtot

?

i∈Ctot

uout(ki),

uout(k) = θin+ θoute−νoutkt/Q= 1 − θout

?1 − e−νoutkt/Q?.

(3.3)

Here θin, θoutare the support functions for k in the regions Cinand Cout, respectively:

θin= ϑ(|η| − η0),θout = ϑ(η0− |η|), (3.4)

with η0the pseudorapidity corresponding to the opening (half-)angle of the cone,

η0 = −lntanθ0

2.

Note that the fact the probing function uout(k) = 1 for k ∈ Cin means that this

parton does not contribute to the measurement.

Finally, for the associated distribution we introduce a double Mellin transform:

Σ2ng(Q,V,Eout) =

?

?

dν

2πiν

?

dνout

2πiνout

dσn

σtot

i∈Ctot

eνV +νoutEout/Q˜Σ2ng(Q,ν,νout),

˜Σ2ng(Q,ν,νout) =

n

?

u2ng(ki).

(3.5)

The new source function is given by the expression

u2ng(k) = e−ν v(k)?θin+ θoute−νoutkt/Q?= e−ν v(k)−θoute−ν v(k)?1 − e−νoutkt/Q?. (3.6)

It corresponds to measuring V everywhere and Eoutonly in Cout.

In the Mellin space all the distributions become exponents of the so-called radi-

ators:

˜Σ(Q,ν) = e−R(Q,ν),

˜Σout(Q,νout) = e−Rout(Q,νout),

˜Σ2ng(Q,ν,νout) = e−R2ng(Q,ν,νout).

(3.7)

(3.8)

(3.9)

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3.2 Global radiator

For the global distribution the answer is simple and the radiator is known to be given

by the SL-improved Sudakov expression

R(Q,ν) = 2CF

?Q

0

dk2

k2

t

t

αs(k2

π

t)

?ηmax

0

dη?1−e−ν v(k)?,ηmax= lnQe−3

4

kt

. (3.10)

In the definition of ηmaxwe have incorporated a SL correction to the “hard” piece of

the quark splitting function

αP(α) =1 + (1 − α)2

2

= 1 − α

?

1 −α

2

?

,α =kt

Qeη.

It is straightforward to verify then that the virtual SL contribution due to large ra-

pidities, α ∼ 1, can be embodied into rescaling of the upper limit of the η integration

as follows:

?1

The two-loop radiator (3.10) (with the coupling αsin the physical “bremsstrahlung”

scheme [16]) has single-logarithmic accuracy. This means that the neglected correc-

tion is O(α2

kt/Q

dαP(α) =⇒

?ηmax

0

dη.

slnν) which translates into α2

slog(Q/V ) (see, e.g. [5]).

3.3 Non-global radiator

The radiators for both out and associated non-global distributions, (3.8) and (3.9),

have the structure

RA= R(1)

Here R(1)

term due to correlated multi-gluon radiation. (The latter is absent (negligible within

the SL accuracy) in the case of a global observable.)

A+ R(c)

A,A = out, 2ng. (3.11)

Ais the standard Sudakov one-gluon emission contribution and R(c)

Ais the

Sudakov piece.

The Sudakov piece R(1)

A,

R(1)

A= 2CF

?

d2kt

πk2

t

αs(k2

π

t)

?ηmax

0

dη[1 − uA(k)],

(3.12)

is given by the expression (3.10) with proper probing functions:

[1 − uout(k)] = θout

[1 − u2ng(k)] = [1 − u(k)] + θoute−νv(k)?1 − e−νoutkt/Q?

= [1 − u(k)] + e−νv(k)[1 − uout(k)].

?1 − e−νoutkt/Q?, (3.13a)

(3.13b)

We compute first R(1)

out. Substituting (3.13a) into (3.12) we obtain

R(1)

out(Q,νout) = 2CF

?Q

0

dk2

k2

t

t

αs(k2

π

t)

?η0

0

dη?1−e−νoutkt/Q?, (3.14)

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with the rapidity integral restricted to the Cout region. Here we have used that

η0∼ 1 < ηmax. This expression does not have a collinear singularity and therefore is

a SL function. Therefore we can approximate [1−uout] in (3.14) by a theta-function

ϑ(kt−Q/νout) and evaluate the inverse Mellin integral by simply substituting Q/Eout

for νout. We arrive at

R(1)

out(Q,νout) =⇒

4CF

CA

η0· ∆(Q,Eout) ≡ R(1)

out(Q,Eout), (3.15)

where we have introduced a convenient SL function

∆(Q,E) = CA

?Q

E

dkt

kt

αs(k2

π

t)

. (3.16)

Similarly, using (3.13b) in (3.12) gives the expression for the Sudakov piece of the

associated distribution radiator:

R(1)

2ng(Q,ν,νout) = R(1)(Q,ν) + δR(1)(Q,ν,νout). (3.17)

Here the first term reconstructs the global distribution Σ(Q,V ) in (1.7) and the

addition contribution δR(1)to the Sudakov radiator reads

δR(1)

2ng(Q,ν,νout) = 2CF

?Qdk2

t

k2

t

αs(k2

π

t)

?η0

0

dηe−ν v(k)?1 − e−νoutkt/Q?. (3.18)

This expression is identical to that for the out-case, (3.14), apart from the additional

factor exp(−ν v(k)). Since (3.18) is a subleading SL correction, we can treat this

factor, within our accuracy, as simply proving an additional restriction upon the

transverse momentum. As long as η = O(1), we have v(k) ∼ ktand thus

e−v(k)ν

=⇒ϑ(Q − νkt).

In V -space this translates into the condition

kt ? V Q ≪ Q.

Then the integral for δR(1)becomes the same as that for R(1)

outwith Q replaced by V Q:

δR(1)

2ng(Q,ν,νout) =⇒4CF

CA

η0· ∆(V Q,Eout) = R(1)

out(V Q,Eout).(3.19)

We conclude at the level of the Sudakov contributions to the radiators the factorized

answer (1.7) holds.

Now we have to analyse the specific correlation contribution R(c)in (3.11). This

is also a SL function whose PT expansion starts at the level of (αslog)2.

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2-Loop correlation.

At two loops R(c)is given by the integral

R(c)

A

=

?

dω2(k1,k2) [UA(k1,k2) − uA(k1)uA(k2)] ,(3.20)

where dω2is the distribution for the correlated emission of two secondary partons off

the primary q¯ q system. Here U(k) is an “inclusive” source function for the parent

massive gluon. As was shown in [17], to properly reconstruct the two-loop Sudakov

expression with the running physical coupling, this source has to be defined in terms

of the single parton source function uA(k) = uA(kt,η) as

UA(k1,k2) = uA(

?

k2

t+ m2,η);k = k1+ k2, m2= k2.(3.21)

Here η is the rapidity of the parent,

?

k2

t+ m2e±η= kt1e±η1+ kt2e±η2.

The distribution dω2is singular at m2= 0 when the two partons become collinear

or one of them is much softer than the other. In particular, in the collinear limit

the secondary partons obviously belong to the same angular region (Cin or Cout)

and then, as can be easily seen from (3.20), the combination of the sources in the

square brackets vanishes thus regularizing the singularity. The same is true for the

soft (energy ordered) two-gluon configuration, provided the gluons are in the same

angular region. This results in a negligible subleading contribution O(α2

As we will see shortly, the only relevant logarithmically enhanced contribution

emerges in the case when the two secondary partons belong to the complimentary

angular regions, Cinand Cout, so that the sources u(k1) and u(k2) become different

and the cancellation gets broken. This is the configuration that gives rise to the

non-global SL corrections, as was found by Dasgupta and Salam [6].

Given that there are no collinear logarithms, we can simplify (3.20) by taking

strongly ordered parton energies,

slog).

kt2≪ kt1≪ Q.(3.22)

In this configuration the correlated 2-gluon distribution dω2in (3.20) reads [18]

dω2(k1,k2) = 2CFCA

?αs

π

?2

2?

i=1

?dωi

ωi

dΩi

4π

?

· C(Ω1,Ω2),(3.23)

The function C depends on parton angles and is given by the expression

C(n1,n2) =

(n¯ n)

(nn1)(n1n2)(n2¯ n)+

= wn¯ n(n1)[wnn1(n2) + wn1¯ n(n2) − wn¯ n(n2)],

(n¯ n)

(nn2)(n2n1)(n1¯ n)−

(n¯ n)

(nn1)(n1¯ n)

(n¯ n)

(nn2)(n2¯ n)

(3.24)

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where

wab(c) ≡

(ab)

(ac)(cb),(nanb) ≡ 1 − cosΘab

and n, ¯ n stand for the direction 4-vectors of the primary partons and ni, i = 1,2

of the secondary gluons. The angular distribution (3.24) becomes singular when the

gluon momenta?k1and?k2are parallel.

First we observe that in the soft limit (3.20) we can put m2= 0 in the source U

to approximate

[UA(k1,k2) − uA(k1)uA(k2)] ≃ uA(k1) [1 − uA(k2)].(3.25)

Let us first analyse the more complicated case of the associated distribution. Using

the explicit expression for the source (3.6) we obtain two potential contributions:

k1∈ Cin, k2∈ Cout:

e−νv(k1)?1 − e−νv(k2)−νoutk2t?=⇒ min{Eout,V Q} < k2t< kt1< V Q;

k1∈ Cout, k2∈ Cin:

e−νv(k1)−νoutkt1?1 − e−νv(k2)?=⇒ V Q < k2t< kt1< min{Eout,V Q}.

We see that the only contributing region is that in (3.26a) where a harder gluon is

close to the jet axis while a softer one contributes to the out-of-jet kt–flow, provided

Eout≪ V Q. Therefore in (3.20) we can substitute

(3.26a)

(3.26b)

[U2ng(k1,k2) − u2ng(k1)u2ng(k2)] ⇒ θin(k1)θout(k2)ϑ(V Q − k1t)ϑ(k2t− Eout) (3.27)

and we get at two loops

R(c)

2ng=⇒

CF

CA

∆2(V Q,Eout) · F(η0).(3.28)

Here F(η0) is given by the angular integral

F(η0) =

?

Cin

d2Ω1

4π

?

Cout

d2Ω2

4π

C(n1,n2) =π2

6− Li2

?e−4η0?. (3.29)

The same conclusion holds for the out-distribution:

[Uout(k1,k2) − uout(k1)uout(k2)] ⇒ θin(k1)θout(k2)ϑ(Q − k1t)ϑ(k2t− Eout). (3.30)

Comparing this with (3.27) we see that this case reduces to setting V = 1 in (3.26),

and we obtain

R(c)

CA

The result (3.28) is just the same expression that we derived for the away from jet

distribution, except that what was Q there is now replaced by V Q.

out=⇒

CF

∆2(Q,Eout) · F(η0). (3.31)

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This completes the proof of (1.7) with account of the 2-loop gluon correlations.

So we can write the integrated non-global distributions (2.3) with the SL accu-

racy as

Σout(Q,Eout) = e−R(1)

Σ2ng(Q,V,Eout) = e−R(1)

out(Q,Eout)e−R(c)

out(Q,Eout),(3.32a)

out(V Q,Eout)e−R(c)

2ng(Q,V,Eout)· Σ(Q,V ), (3.32b)

where R(1)

of the 2-gluon correlation (O(∆2))

outwas defined in (3.15). We have shown in this section that with account

R(c)

2ng(Q,V,Eout) = R(c)

out(V Q,Eout). (3.33)

In the following section we show that (3.33) hold at all orders in ∆.

4. Multi-gluon correlations in all orders

Since there is no collinear singularities in R(c), all SL contributions arise from en-

sembles of soft gluons with strongly ordered energies – the leading logarithmic soft

approximation. As a result of the strong energy ordering, it is only the hardest

of these soft gluons in the angular region Coutthat contributes to Eout (since the

contribution of softer gluons are negligible in the leading soft approximation).

Therefore, to collect these specific SL contributions to the non-global distribu-

tions (both the out- and associated distributions) it suffices to analyse multi-gluon

systems with many gluons in Cinand a single one in Cout. Let us denote the mo-

mentum of this gluon by q. Then, among the gluons ki∈ Cinit suffices to consider

only those that are harder then q, since only such gluons may affect the radiation

of q. Softer ones, ki≪ q, don’t contribute to the measurement due to real-virtual

cancellations (except as power-suppressed corrections).

With account of multi-gluon effects the non-global correlation function R(c)

comes much more involved. At the same time, the structure of its dependence on Q

and V remains the same as in the 2-loop case. Indeed, the logarithmic integration

in the energy of the hardest gluon k1in the Cinregion is limited from above either

by a pure phase space restriction k10< Q for the case of R(c)

k10< V Q in the case of the associated distribution R(c)

the only place where the dependence on Q (QV ) enters, we automatically derive the

relation (3.33). Therefore it suffices to consider R(c)

the subscript and study

Abe-

outor, alternatively, by

2ng. Since this upper limit is

out(Q,Eout). Hereafter we suppress

R(c)(Q,Eout) ≡ R(c)

out(Q,Eout) = R(c)

2ng(Q,V = 1,Eout).

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Generalisation.

as follows. We take k ∈ Cinand q ∈ Cout, q ≪ k, and write (¯ αs≡ CAαs/π)

For the sake of generalisation the 2-loop result can be rephrased

R(c)(Q,Eout) =

?Q

× {rnnk(k0,Eout) + rnk¯ n(k0,Eout) − rn¯ n(k0,Eout)},

Eout

dk0

k0

?

Cin

dΩk

4π

¯ αswn¯ n(nk)

(4.1)

where rabis the 1-loop radiator for the out-distribution generated by a colorless dipole

ab with a,b ∈ Cin:

rab(Q,Eout) =2CF

CA

?Q

Eout

dq0

q0

?

Cout

dΩq

4π

¯ αswab(nq).(4.2)

In the large-Nclimit the structure of soft multi-gluon multiplication can be described

in terms of the iterative procedure (see [7]):

Wab(1,...,m) = wab(ℓ)Waℓ(1,...,ℓ−1)Wℓb(ℓ+1,...,m),Wab(1) = wab(1), (4.3)

where ℓ is one of m soft gluons emitted by the colour singlet dipole (ab). We choose

ℓ to be the hardest of the secondary gluons. Following this path we arrive at the

following generalisation:

R(c)

ab(Q,Eout) =

Q

?

Eout

dk0

k0

?

Cin

dΩk

4π

¯ αswab(nk)

?

1 −Zank(k0,Eout)Znkb(k0,Eout)

Zab(k0,Eout)

?

,

(4.4a)

where

−lnZab(E,Eout) = rab(E,Eout) + R(c)

ab(E,Eout).(4.4b)

In (4.4) Zabis the generating function describing the gluon cascade which originates

from the ab-dipole. We remark that the coupled equations (4.4) describing the large-

angle (hedgehog) gluon configurations have a highly non-linear structure.

Once the equation is solved, we substitute the directions of the primary partons

n¯ n for ab to obtain

R(c)

out(Q,Eout) = R(c)

n¯ n(Q,Eout).(4.5)

Recall that to obtain the all-order correlation function R(c)

ciated distribution we simply have to substitute Q → V Q in (4.5).

2ng(Q,V,Eout) for the asso-

The asymptotic regime.

αsln(Q/Eout) ≫ 1 (αsln(V Q/Eout) ≫ 1) was analysed in [8]. For the partons a and

b inside the same Cincone and θab≪ 1

The behaviour of the answer in the asymptotic limit

Zab≃ h

?θ2

ab

2θ2

c

?

,θc≃ λ(η0)e−c

2∆,c ≃ 2.5

(4.6)

14