Caso y contexto: un análisis del medio y del mensaje

Let us consider a global variable that measures hadrons everywhere, for instance the thrust ine+_e− _{annihilation, defined as} T = max ~ n P i|~pi·~n| P i|~pi| . (8.20)

The thrust, as all other event shapes, is a measure of the geometrical properties of hadron energy-momentum flow. For instance, for pencil-like events we have T '1, for planar

events T'2/3, while for spherical events T'1/2. This variable has the property of in-

frared and collinear (IRC) safety, i.e. its value does not change after emission of extremely soft particles and/or quasi-collinear splittings. IRC safety is precisely the condition that ensures that observables related to the thrust, i.e. distributions and mean values, can be

computed using the quark-gluon language, in spite of the fact that the variable defini- tion involves hadrons, and the difference between parton and hadron level predictions is suppressed by powers of the hard scaleQ.

The basic quantity we are interested in is Σ(τ), the probability that 1−T < τ, the differential distribution σ−1_{dσ/dτ being the derivative of Σ(τ). Fixed order QCD pre-} dictions are reliable as long as τ is large, but fail as soon as events approach the Born limitτ= 0. In the smallτ region one needs to resum logarithmic enhanced contributions to all orders, and the resulting resummed distribution has the same shape as the data. However, to get on top of the data, one needs to add a further correction that can be inter- preted as the difference between parton and hadron level. This hadronisation correction can be estimated using Monte Carlo (MC) event generators likeherwig[4],pythia[5, 6]

orariadne[7], taking the ratio of the distributions obtained with MC’s before and after

hadronisation, and estimating hadronisation uncertainties using different event generators. This has lead to a successful description of IRC safe event shape distributions and jet rates in e+_e− _{annihilation, giving one of the most accurate measurements of α}

s (see [8] for a

recent review). The validity of this procedure relies strongly on the fact that MC event generators contain the physics that is needed to describe the main features of final-state observables. This statement is in general true for variables whose LL exponentiate, as we shall see in the following.

Consider then a generic final-state variableV({ki}), a function of final-state momenta {ki}, and its rate Σ(v), the probability that V({ki})< v. A generic contribution to Σ(v)

can be represented by the Lund diagram on the left hand side of fig. 8.34. The grey area corresponds to the vetoed region where no real emissions are allowed, and only virtual corrections survive, and one can write in general:

Σ(v) =e−R(v)F(v), (8.21) where R(v) is the exponent representing virtual corrections up to the scale vQ, while real emission outside the vetoed region and the remaining virtual corrections build up the function F(v). The variable V is said to exponentiate if all leading (double) logarithms are contained in the exponentR(v) andF(v) is a pure NLL function, usually denoted by F(R0_{), with R}0_{(v) =−vdR/dv. There are two basic conditions for this to happen, which}

go under the name of recursive infrared and collinear (rIRC) safety conditions [9]:

1. the variable must scale in the same fashion with multiple emissions as with a single emission. Formally, parametrising the momentum ki of each emission in terms of

V(ki),11 the value the variable V would have if only emission ki were present, and

defining V(ki) = ¯vζi, the first rIRC safety condition states that the following limit:

lim ¯ v→0 V(k1(¯vζ1), . . . , kn(¯vζn)) ¯ v (8.22)

has to be finite and non-zero. This ensures that the boundary of the vetoed region in fig. 8.34 does not change substantially whatever is the number of emissions; 2. the variable’s scaling property (8.22) must not be altered by the addition of extra-soft

particles or by quasi-collinear splittings, formally: [ lim ζn+1→0 ,lim ¯ v→0] V(k1(¯vζ1), . . . , kn(¯vζn), kn+1(¯vζn+1)) ¯ v = 0, (8.23)

where the only non-trivial part of the commutator is the one where one takes the limit ζn+1 → 0 after the limit ¯v → 0, the other part being equal to the limit in

11

For simplicity, we will always writeV(k1, . . . , kn) instead of the more correct formV({p˜}, k1, . . . , kn),

eq. (8.22) due to IRC safety. An analogous condition should hold also for collinear splittings. This implies that in fig. 8.34 one can eliminate all emissions in grey, far from the boundary of the vetoed region, without altering the value ofV, and for all emissions with V(ki) ∼V(k1, . . . , kn) one can replace clusters of emissions close in

rapidity with a single emission having the total momentum of the cluster.

ln k_t η /Q ln 1/ ln k_t η /Q

Figure 8.34: The Lund diagram for emissions contributing to a generic final-state observ- able (left) and its simplified version in case of a rIRC safe observable (right).

In the end, for a rIRC safe variable, a generic contribution to Σ(v) can be represented by a Lund diagram like the one on the right hand side of fig. 8.34, where one has a vetoed area, giving rise to the LL exponent R(v), and real emissions contributing to F(R0) at

NLL accuracy are both soft and collinear, well separated in rapidity and confined in a narrow region of width ln 1/ln 1/v close to the boundary of the vetoed area. The fact

that emissions are well separated in rapidity makes it possible to exploit QCD coherence, and consider soft gluons as radiated independently (like in QED) from the hard legs. This simplification of multi-gluon soft matrix elements makes it possible to computeF(R0) with

a MC procedure, where emissions are ordered in V(ki) = ¯vζi, with ζi < ζi−1, V(k1) is fixed at ¯v, and the probability of emission of gluonki collinear to leg ` with rapidity ηi

and azimuthφi (with respect to leg`) is

dP(ki(ζi, ηi, φi, `)) =R0` dηi ∆ηi dφi 2π dζi ζi ζi ζi−1 R0 , X ` R0<sub>`</sub>=R0. (8.24) The function F(R0) can then be computed as the following average:

F(R0) = * lim ¯ v→0 V(k1, . . . , kn) V(k1) −R0+ , (8.25)

where the limit ¯v→0 ensures that the result contains no NNLL contributions. Eq. (8.25)

is an example of application of MC techniques used in parton shower event generators to obtain exact QCD results, and is one of the building blocks of the automated resummation programcaesar [9].

We can now discuss what level of accuracy can be achieved by parton shower event generators. MC parton showers produce emissions in the whole of the phase space (for instance all emissions in the diagram on the left hand side of fig. 8.34), with approximated

matrix elements that are exact in the collinear limit but mistreat the soft large-angle region, both because they do not have full interference terms, and because they are correct only in the large-Nclimit. However, for rIRC safe observables, LL and NLL contributions

are determined only by emissions that are well separated in rapidity and close to the boundary of the vetoed region. MC event generators correctly describe such emissions, so that one expects that they reproduce not only LL, but most NLL contributions to rIRC safe observables.