On large angle multiple gluon radiation
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 Eout distribution, but evaluated at a rescaled energy VQ. Comment: 16 pages
arXiv:hep-ph/0303101v1 13 Mar 2003
Preprint typeset in JHEP style. - PAPER VERSION
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
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
3. Analysis and resummation
3.2 Global radiator
4. Multi-gluon correlations in all orders13
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 ), 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
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,
σ(Q) = σBorn(Q) · g?αs(Q2)?,g(0) = 1(Q2≫ m2).
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
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
Σ(Q,V ) ≡
= 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 ) =
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
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
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. ).
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) .
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 . 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
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
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
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 ).
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  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-
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,