ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 4, с. вв0-в74
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
PARTON DYNAMICS, 30 YEARS LATER
© 2008 Yu. L. Dokshitzer*
LPTHE, University Paris VI-VII, CNRS, Paris, France; Petersburg Nuclear Physics Institute, Russian Academy
of Sciences, Gatchina Received March 26, 2007; in final form, August 13, 2007
A review of the basics of the QCD parton evolution picture is given with an emphasis on recent findings that reveal intrinsic beauty, and hint at hidden potential simplicity, of the perturbative quark—gluon dynamics.
PACS: 12.39.-x
1. INTRODUCTION
We are witnessing an explosive progress in analytical and numerical methods and techniques for deriving sophisticated pQCD results, prompted to a large extent by the LHC needs. The permanent fight for increasing the accuracy of pQCD predictions is being fought on two fronts: on the one hand, increasing the as order of the exact matrix element calculations (hard parton cross sections) and, on the other hand, improving perturbative description of space-like (parton distribution functions) and time-like (fragmentation functions) quark—gluon cascades. The first battleground is process specific; the second one is universal and is usually referred to as parton dynamics.
The universal nature of the parton dynamics goes under the name of factorization of collinear ("mass") singularities. Physically, it is due to the fact that quark—gluon multiplication processes happen at much larger space—time distances than the hard interaction itself. It is this separation that makes it possible to describe quark—gluon cascades in terms of independent parton splitting processes. They succeed one another in a cleverly chosen evolution time, t ~ ln Q2, whose flow "counts" basic parton splittings that occur at well-separated, strongly-ordered space—time scales. Perturbative structure of the cross section of a given process p characterised by the hardness scale Q2 can be cast, symbolically, as a product (convolution) of three factors (for a review see [1]):
ahp) (ln Q2) « С(p)[as(t)] ®
(i)
® exp / dTP[as(т)] ® Wh(to), t - ln Q2
E-mail: yuri@lpthe.jusssieu.fr
Here, the functions C[as] (hard cross section; coefficient function) and P[as] (parton evolution; anomalous dimension matrix) are perturbative objects analyzed in terms of the as expansion. The last factor wh embeds nonperturbative information about parton structure of the participating hadron(s) h, be it a target hadron in the initial state (parton distribution) or a hadron triggered in the final state (fragmentation function).
The borderline between perturbative and nonperturbative ingredients in (1) is fictitious; it is set arbitrarily by choosing the launching hardness scale t0 ~ ln Q0. This is, however, not the only arbitrariness present in the representation (1). Namely, beyond the leading approximation (one loop; P « as), the separation between the C and exp(P) factors becomes scheme dependent. Here one talks about factorization scheme dependence. Another negotiable object is the expansion parameter as itself whose definition depends on the ultraviolet renormalization procedure (renormalization scheme dependence). The so-called MS-bar scheme — a precisely prescribed procedure for eliminating ultraviolet divergences, based on the dimensional regularization — won the market as the best suited scheme for carrying out laborious high-order calculations. It is this scheme in which the parton "Hamiltonian" P was recently calculated up to next-to-next-to-leading accuracy, a^, by Moch, Vermaseren, and Vogt in a series of papers [2, 3].
Formally speaking, the physical answer does not depend on the scheme (either factorization or renor-malization) one chooses to construct the expansion. There is a big "if", however, which renders this motto meaningless. It would have been the case, and consolation, if we had hold of the full perturbative expansion for the answer. This goal is not only technically unachievable but, more importantly, it is actually useless. Perturbative expansions in Quantum Field Theory (QFT) are asymptotic
series. This means that starting from some order, n > ncrit = const(p)/as, a series for any observable (p) inevitably goes haywire and ceases to represent the answer. For QED where ncrit ~ 100 this is an academic problem. In QCD, on the contrary, the best hope the perturbative expansion may offer is a reasonable numerical estimate based on the first few orders of the perturbation theory (whose intrinsic uncertainty can often be linked with genuine nonper-turbative effects). This being understood, it becomes legitimate, and mandatory, to play with perturbative series and try to recast a formal as expansion in the most relevant way, the closest to the physics of the problem.
In the beginning of the paper I will remind you of the basics of parton dynamics, of the origin of logarithmically enhanced contributions that lie in the core of the QCD parton picture. The basic one-loop parton Hamiltonian was long known to possess a number of symmetry properties which made the problem look over-restricted and hinted at possible hidden simplicity of the parton dynamics.
We shall discuss in detail the notion of the parton evolution time and how it gets modified due to coherence effects in space-like (DIS) and time-like (e+e- annihilation) processes.
Then, we will have a closer look at an intimate relation between evolution of space- and time-like parton cascades which I believe has not been properly explored. I will argue that paying a deeper respect to this inter-relation should help to better grasp the complicated structure of two- and three-loop anomalous dimensions as they are known today.
2. PARTON DYNAMICS 2.1. Hard QCD Processes and Partons
Hard processes solved the problem of finding out what hadrons are made of. The answer was rather childish but productive: take a hammer and hit hard to see what is it there inside your favorite toy.
We may hit (or heat, if you please) the vacuum as it happens in e+e- — qq — hadrons. Then, one may hit a proton with a sterile (electroweak) probe giving rise to the famous Deep Inelastic lepton—hadron Scattering (DIS): e-p — e- + X. Finally, make two hadrons hit each other hard to produce either a sterile massive object like a pair (the Drell—Yan process), an electroweak vector boson (Z0, W±), or a Higgs, or a direct photon or a hadron with large transverse momentum with respect to the collision axis. Importantly, in all cases it is large momentum transfer which is a measure of the hardness of the process.
Let us turn to DIS as a classical example of a hard process (Fig. 1). Here the momentum q with a large space-like virtuality Q2 = \q2\ is transferred from an incident electron (muon, neutrino) to the target proton, which then breaks up into the final multihadron system. Introducing an invariant energy s = 2(Pq) between the exchange photon (Z0, W±) with 4-momentum q and the proton with momentum P, one writes the invariant mass of the produced hadron system which measures inelasticity of the process as
W2 = (q + P)2 - M2 = q2 + 2(Pq) = s(1 - x),
x = —
q2
2(Pq)
< 1.
The cross section of the process depends on two variables: the hardness q2 and Bjorken x. For the case of elastic lepton—proton scattering one has x = 1 and it is natural to write the cross section as
(2a)
Here, aRuth rc a2/q4 is the standard Rutherford cross section for e.m. scattering off a point charge and Fei stands for elastic proton form factor. For inclusive inelastic cross section one can write an analogous expression by introducing "inelastic proton form factor" which now depends on both the momentum transfer q2 and the inelasticity parameter x:
(lain duRuth
dq2dx dq2
-KM)-
(2b)
What kind of behavior of the form factors (2) could one expect in the Bjorken limit Q2 — oo? Quantum mechanics tells us how the Q2 behavior of the electromagnetic form factor is related to the charge distribution inside a proton:
Fel(Q2) = J d3rp(r) exp {¿Q • r} .
For a point charge p(r) = 53(r), it is obvious that F = 1. On the contrary, for a smooth charge distribution, F(Q2) falls with increasing Q2, the faster the smoother p is. Experimentally, the elastic e-p cross section does decrease with q2 much faster that the Rutherford one (Fel(q2) decays as a large power of q2). Does this imply that p(r) is indeed regular so that there is no well-localized (point-like) charge inside a proton? If it were the case, the inelastic form factor would decay as well in the Bjorken limit: a tiny photon with the characteristic size ~1/Q — — 0 would penetrate through a "smooth" proton like a knife through butter, inducing neither elastic nor inelastic interactions.
However, as was first observed at SLAC in the late sixties, for a fixed x, F^ stays practically constant with q2, that is, the inelastic cross section (with a given inelasticity) is similar to the Rutherford cross section (Bjorken scaling). It looks as if there was a point-like scattering in the guts of it, but in a rather strange way: it results in inelastic break-up dominating over the elastic channel. Quite a paradoxical picture emerged; Feynman—Bjorken partons came to the rescue.
Imagine that it is not the proton itself that is a point-charge-bearer, but some other guys (quark-partons) inside it. If those constituents were tightly bound to each other, the elastic channel would be bigger than, or comparable with, the inelastic one: an excitation of the parton that takes an impact would be transferred, with the help of rigid links between partons, to the proton as a whole, leading to elastic scattering or to the formation of a quasi-elastic finite-mass system (Nn, An, or so), 1 — x < 1.
To match the experimental pattern Fj(q2) < < Fi2l(q2) = O(1) one has instead to view the par-ton ensemble as a loosely bound system of quasifree particles. Only under these circumstances does knocking off one of the partons inevitably lead to deep inelastic breakup, w
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