# QCD evolution equations for parton densities

Post-publication activity

Curator: Guido Altarelli

QCD Evolution Equations for Parton Densities valid in the theory of the strong interactions (Quantum Chromodynamics or QCD), determine the rate of change of parton densities (probability densities to find a quark or a gluon in the proton) when the energy scale chosen for their definition is varied.

## Introduction

At a fundamental level the strong interactions, which bind the nucleons (i.e. protons and neutrons) in the nuclei and also the quarks in the nucleons, are described in terms of quarks and gluons (denoted as “partons”) in a gauge theory based on the “colour” charges (QCD) Ellis et al (2003), Dissertori et al (2003). Altarelli (1982), Altarelli (2008). There are 8 gluons (one for each colour charge) and 3 colours of quarks. The quark triplets exist in 6 kinds, called “flavours” (the u, c, t “up” quarks of charge +2/3, in units of the proton charge, and d, s, b the “down” quarks of charge -1/3). The up and (separately) the down quark flavours are all identical except for their mass and all of them have the same QCD interactions.

Two very important properties of QCD are confinement and asymptotic freedom. Confinement is the property that no isolated coloured charge can exist as a free particle but only colour singlet particles can be isolated. In particular, a proton is made up of 3 quarks, uud (2 up’s and 1 down), in a neutral colour configuration. All observed “hadrons” (particles with strong interactions) are made up either of 3 quarks (baryons) or of a quark-antiquark pair (mesons), all singlets under colour. Confinement is due to the fact that the potential between two colour charges, for example a quark and an antiquark, has a Coulomb-like part at short distances but a linearly rising term at long distances. The linearly rising term in the potential makes it energetically impossible to separate the two colour charges. If a pair is created at one space-time point, for example in e+e- annihilation, and then the quark and the antiquark start moving away from each other in the center of mass frame, it soon becomes energetically favourable to create additional pairs, smoothly distributed between the two leading charges, which neutralise colour and allow the final state to be reorganised into two jets of colourless hadrons, that communicate in the central region by a number of "wee" hadrons with small energy. It is just like the familiar example of a broken magnet: if one tries to isolate a magnetic pole by stretching a dipole, the magnet breaks down and new poles appear at the breaking points.

Asymptotic freedom is the property that the QCD coupling becomes weak at high energies, due to quantum corrections, so that the theory becomes perturbative in this regime (that is, the theoretical predictions can be expressed as an expansion in powers of the coupling limited to the first few terms). In the QCD theory there are no fundamental parameters with the dimension of mass except for the quark masses. At energies large enough that masses can be neglected (in a unit system where c=1, energy and mass are measured in the same units, because of $$E \sim ~mc^2$$), naively one would expect that dimensionless measurable quantities would become “scale invariant”, namely independent of the absolute scale of energy, and only functions of energy ratios (“scaling variables”).

## Hard processes

A “hard” process is a process that: i) occurs at high energies; ii) for it all energy variables are large and of the same order; we denote by Q the common energy scale; iii) it is “infrared safe”, that is it is well defined in the limit of vanishing quark masses and free of infrared singularities. For hard processes any measurable quantity can only depend on Q and on a number of scaling variables $$x_i \ ,$$ i.e. the large Q limit should naively be scale invariant (we should have “scaling”). In reality scale invariance is broken in QCD by quantum corrections: even starting with vanishing quark masses the procedure of quantisation and renormalisation of the theory necessarily introduces a scale of mass $$\Lambda_{QCD}$$ (denoted simply as $$\Lambda$$ in the following). Scaling is broken, but the scaling violations are only logarithmic and computable.

The simplest hard processes are those where no hadrons are present in the initial state and the final state is totally inclusive (that is, the sum over all possible hadronic final states is taken). Examples are e+e- $$\rightarrow$$ hadrons, Z$$\rightarrow$$ hadrons (where Z is the weak neutral current gauge boson) and $$\tau \rightarrow \nu_\tau +$$ hadrons. For these processes the scaling violations only enter through the “running coupling” $$\alpha_s(Q)$$ which is the expansion parameter of the perturbative series. In terms of gs, the QCD gauge coupling (for the $$q \bar q g$$ or $$ggg$$ vertices, where q and g are the quark and gluon fields, respectively), the quantity $$\alpha_s$$ is given by $$\alpha_s = \frac{g_s^2}{4\pi} \ .$$ Here gs and $$\alpha_s$$ are renormalised couplings defined at some scale $$\mu \ .$$ Due to the quantum corrections (described by Feynman diagrams with loops) the effective coupling becomes scale dependent$\alpha_s(Q)$ with, in first approximation: $\tag{1} \alpha_s(Q) \sim \frac{\alpha_s}{1+b \alpha_s t}=\frac{1}{b\log{Q^2/\Lambda^2}}$

where $\tag{2} t = \ln{Q^2/\mu^2}, ~~~~~~~~b=\frac{33-2n_f}{12\pi}$

with $$n_f$$ being the number of quark flavours with mass m < Q. The constant b is the first coefficient of the QCD beta function. Note that for $$Q=\mu \ ,$$ we have t=0 and $$\alpha_s(\mu)=\alpha_s \ .$$ Actually one can trade the dimensional scale $$\mu$$ with the scale $$\Lambda$$ defined, in lowest approximation, as shown in eq. (1). If for the simplest hard processes with no hadrons in the initial state we take a dimensionless quantity $$F \ ,$$ like, for example, the cross section divided by the lowest order (Born) cross section $$F = \sigma/ \sigma_{Born}$$ then we would expect in general that $$F=F[t,\alpha_s] \ .$$ Actually a very important result that can be proven in QCD is that $$F$$ can depend on the scale Q only through the running coupling according to $$F =F[0,\alpha_s(Q)] \ ,$$ that is $$F$$ can be given in terms of an expansion in $$\alpha_s(Q) \ :$$ $\tag{3} F= F[0,\alpha_s(Q)]= 1+c_1 \alpha_s(Q) + c_2 \alpha_s(Q)^2 + \dots$

where the $$c_i$$ are calculable constants (once the renormalised coupling has been precisely defined).

## Deep inelastic scattering

At the next level of complication one has hard processes with one (and only one) hadron in the initial state, like in deep inelastic scattering (DIS): l+N $$\rightarrow$$ l’ + X, where l, l’ are leptons, i.e. either an electron or a muon or a neutrino, N is a hadron, like a proton or a nucleus, and X is summed over all possible hadronic final states.

For DIS shown in Fig.1, we have, in the lab system where the nucleon of mass m is at rest: $\tag{4} Q^2~=~-q^2~=~-(q.q) ~=~2(k.k')~=~4EE'\sin^2{\theta/2};~~~~~~~~m\nu~=~(p.q);~~~~~~~~x~=~\frac{Q^2}{2m\nu}$

where k and k' are the initial and final lepton 4-momenta (q=k-k') and the product (a.b) of two Lorentz 4-vectors $$a^\mu=(a^0, a^i), b^\mu=(b^0, b^i)$$ is given by $$(a.b)=a^0b^0-a^ib^i \ .$$ In this case the virtual momentum q of the gauge boson is spacelike. x is known as the Bjorken variable. DIS processes have played and still play a very important role for our understanding of QCD and of nucleon structure. In the '60's the demise of hadrons from the status of fundamental particles to that of bound states of constituent quarks was the breakthrough that made possible the construction of a renormalisable field theory for strong interactions. In fact, the presence of an unlimited number of hadron species, many of them with large spin values, presented an obvious dead-end for a manageable field theory. The evidence for constituent quarks emerged clearly from the systematics of hadron spectroscopy. But, at the beginning, confinement that forbids the observation of free quarks was a clear obstacle towards the acceptance of quarks as real constituents and not just as fictitious entities describing some purely mathematical pattern. The early measurements at SLAC (Stanford Linear Accelerator Centre) of DIS dispelled all doubts: the observation of Bjorken scaling and the success of the "naive" (not so much after all) parton model of Feynman established quarks as the basic fields for describing the nucleon structure (parton quarks).

The set of DIS processes actually provides us with a rich laboratory for theory and experiment. There are several structure functions that can be studied, $$F_i(x,Q^2)\ ,$$ each a function of two variables. This is true separately for different beams and targets and different polarizations. Depending on the charges of l and l' we can have neutral currents ($$\gamma\ ,$$ Z) or charged currents in the l-l' channel (Fig. 1). The cross-section $$\sigma\sim L^{\mu \nu}W_{\mu \nu}$$ is given in terms of the product of a leptonic ($$L^{\mu \nu}$$) and a hadronic ($$W_{\mu \nu}$$) tensor. While $$L^{\mu \nu}$$ is relatively simple and easily obtained from the lowest order electroweak (EW) vertex plus QED radiative corrections, the complicated strong interaction dynamics is contained in $$W_{\mu \nu}\ .$$ The latter is proportional to the Fourier transform of the forward matrix element between the nucleon target states of the product of two EW currents: $\tag{5} W_{\mu \nu}~=~\int{~dx~\exp{iqx}~ }$

Note that for processes like e+e- $$\rightarrow$$ hadrons the matrix element between vacuum states of a similar product of currents applies. At large $$Q^2$$ the Fourier integral is dominated by the region of the light cone $$x^2 \sim 0\ .$$ Near the light cone an operator product expansion for the relevant product of currents can be written down. The naive parton model limit corresponds to the expansion in free field theory, while the QCD improved parton model follows in the interacting theory. Structure functions are defined starting from the general form of $$W_{\mu \nu}$$ given Lorentz invariance and current conservation. For example, for EW currents between unpolarized nucleons we have: $\tag{6} W_{\mu \nu}~=~(-g_{\mu \nu}~+~\frac{q_{\mu}q_{\nu}}{q^2})~W_1(\nu,Q^2)~+~(p_{\mu}~-~\frac{m \nu}{q^2}q_{\mu})(p_{\nu}~-~\frac{m \nu}{q^2}q_{\nu})~\frac{W_2(\nu,Q^2)}{m^2}~-~$

$~-~\frac{i}{2m^2}\epsilon_{\mu \nu \lambda \rho}p^{\lambda}q^{\rho}~W_3(\nu,Q^2)$ $$W_3$$ arises from VA interference and is absent for pure vector currents. In the limit $$Q^2>>m^2,~x$$ fixed, the structure functions obey approximate Bjorken scaling which in reality is broken by logarithmic corrections that can be computed in QCD: $\tag{7} mW_1(\nu,Q^2)\rightarrow F_1(x)$

$\nu W_{2,3}(\nu,Q^2)\rightarrow F_{2,3}(x)$ For example, the $$\gamma$$ -N cross-section is given by ($$W_i~=~W_i(Q^2,\nu)$$): $\tag{8} \frac{d\sigma^{\gamma}}{dQ^2d\nu}~=~\frac{4\pi\alpha^2E'}{Q^4E}\cdot [2\sin^2{\frac{\theta}{2}}W_1~+~\cos^2{\frac{\theta}{2}}W_2]$

In the scaling limit the longitudinal and transverse cross sections are given by: $\tag{9} \sigma_L\sim\frac{1}{s}[\frac{F_2(x)}{2x}~-~F_1(x)]$

$\sigma_{RH,LH}\sim \frac{1}{s}[F_1(x)~\pm~F_3(x)]$ $\tag{10} \sigma_T=\sigma_{RH}~+~\sigma_{LH}$

where L, RH, LH (longitudinal, right-handed, left-handed) refer to the helicity 0, 1, -1, respectively, of the exchanged gauge vector boson and T stands for transverse.

## The QCD improved parton model

In the language of Bjorken and Feynman the virtual $$\gamma$$ (or, in general, any gauge boson) sees the quark partons inside the nucleon target as quasi-free, because their (Lorentz dilated) QCD interaction time is much longer than $$\tau_{\gamma}\sim 1/Q\ ,$$ the duration of the virtual photon interaction. As a result, the structure functions are proportional to the density of partons with fraction x of the nucleon momentum, weighted with the squared charge. The quark charges were derived from the data on the electron and neutrino structure functions: $\tag{11} F_{ep}=4/9u(x)~+~1/9d(x)~+~.....~;~~~~~~F_{en}~=~4/9d(x)~+~1/9u(x)~+~....$

$F_{\nu p}=F_{\bar{\nu}n}~=~2d(x)~+~.....~;~~~~~~~~~~~~~~F_{\nu n}~=~F_{\bar{\nu}p}~=~2u(x)~+~.....$ where $$F\sim 2F_1\sim F_2/x$$ and u(x), d(x) are the parton number densities in the proton (with fraction x of the proton longitudinal momentum), which, in the scaling limit, do not depend on $$Q^2\ .$$ The normalisation of the structure functions and the parton densities are such that the charge relations hold: $\tag{12} \int_0^1[u(x)-\bar u(x)]dx=2,~~~\int_0^1[d(x)-\bar d(x)]dx=1,~~~\int_0^1[s(x)-\bar s(x)]dx=0$

Also it was proven by experiment that at values of $$Q^2$$ of a few GeV$$^2\ ,$$ in the scaling region, about half of the nucleon momentum, given by the momentum sum rule: $\tag{13} \int_0^1[\sum_i(q_i(x)+\bar{q}_i(x))~+~g(x)]xdx~=~1$

is carried by neutral partons (gluons). These results are maintained in QCD at the leading order and allow to confirm the quantum numbers of the quarks. Also, the observation that $$R~=~\sigma_L/\sigma_T\rightarrow 0$$ implies that the charged partons have spin 1/2.

In QCD there are calculable log scaling violations induced by $$\alpha_s(Q)\ .$$ The parton rules just introduced can be summarised in the formula: $\tag{14} F(x,t)~=~\int_x^1dy\frac{q_0(y)}{y}\sigma_{point}(\frac{x}{y},\alpha_s(Q))~+~o(\frac{1}{Q^2})$

Before QCD corrections $$\sigma_{point}=e^2\delta(x/y-1)$$ and $$F=e^2q_0(x)$$ (here we denote by e the charge of the quark in units of the proton charge, i.e. e=2/3 for the u quark). QCD modifies

$$\sigma_{point}$$ at order $$\alpha_s$$ via the diagrams of Fig.2.

Note that the integral is from x to 1, because the energy can only be lost by radiation before interacting with the photon (which eventually has to find a fraction x). From a direct computation of the diagrams one obtains a result of the following form: $\tag{15} \sigma_{point}(z,\alpha_s(Q))~\simeq ~e^2[\delta (z-1)~+~\frac{\alpha_s}{2\pi}(t\cdot P(z)~+~f(z))]$

For y>x the correction arises from diagrams with real gluon emission. The log arises from the virtual quark propagator. Actually the log should be read as $$\log{Q^2/m^2}$$ because in the massless limit a genuine mass singularity appears. But in correspondence to the initial quark we have the (bare) quark density $$q_0(y)$$ that appears in the convolution integral. This is a non perturbative quantity that is determined by the nucleon wave function. So we can factorize the mass singularity in a redefinition of the quark density: we replace $$q_0(y)\rightarrow q(y,t)~=~q_0(y)~+~\Delta q(y,t)$$ with: $\tag{16} \Delta q(x,t)~=~\frac{\alpha_s}{2\pi}t\int_x^1dy\frac{q_0(y)}{y}\cdot P(\frac{x}{y})$

Here the factor of t is a bit symbolic: it stands for $$\log{Q^2/km^2}\ ,$$ with k a constant, and what we exactly put below $$Q^2$$ depends on the definition of the renormalised quark density, which also fixes the exact form of the finite term f(z) in eq.~ (15).

## The QCD evolution equations

The effective parton density q(y,t) that we have defined is now scale dependent. In terms of this scale dependent density we have the following relations, where we have also replaced the fixed coupling with the running coupling, as it can be proven to be appropriate: $\tag{17} F(x,t)=\int_x^1dy\frac{q(y,t)}{y}e^2[\delta (\frac{x}{y}-1)~+~\frac{\alpha_s(Q)}{2\pi}f(\frac{x}{y}))]~=~e^2q(x,t)~+~o(\alpha_s(Q))$

$\frac{d}{dt}q(x,t)=\frac{\alpha_s(Q)}{2\pi}\int_x^1dy\frac{q(y,t)}{y}\cdot P(\frac{x}{y})~+~o(\alpha_s(Q)^2)$

We see that in lowest order we reproduce the naive parton model formulae for the structure functions in terms of effective parton densities that are scale dependent. The evolution equations for the parton densities are written down in terms of kernels (the "splitting functions") that can be expanded in powers of the running coupling. At leading order, we can interpret the evolution equation by saying that the variation of the quark density at x is given by the convolution of the quark density at y times the probability of emitting a gluon with fraction x/y of the quark momentum.

It is interesting that the integro-differential QCD evolution equation for densities can be transformed into an infinite set of ordinary differential equations for Mellin moments Gross and Wilczek (1973), Politzer (1973). The moment $$f_n$$ of a density f(x) is defined as: $\tag{18} f_n~=~\int_0^1dxx^{n-1}f(x)$

By taking moments of both sides of the second of eqs. (17) one finds, with a simple interchange of the integration order, the simpler equation for the n-th moment: $\tag{19} \frac{d}{dt}q_n(t)~=~\frac{\alpha_s(Q)}{2\pi}\cdot P_n \cdot q_n(t)$

The solution is: $\tag{20} q_n(t)=[\frac{\alpha_s}{\alpha_s(Q)}]^{\frac{P_n}{2\pi b}}~q_n(0)$

Moments correspond to local operators appearing in the light cone operator product expansion. The evolution of $$q_n(t)$$ was originally derived by renormalization group techniques applied to the operator product expansion.

Up to this point we have implicitly restricted our attention to non-singlet (under the flavour group) structure functions. The $$Q^2$$ evolution equations become non diagonal as soon as we take into account the presence of gluons in the

target. In fact the quark which is seen by the photon can be generated by a gluon in the target (Fig.3).

The quark evolution equation becomes: $\tag{21} \frac{d}{dt}q_i(x,t)~=~\frac{\alpha_s(Q)}{2\pi}[q_i\otimes P_{qq}]~+~\frac{\alpha_s(Q)}{2\pi}[g\otimes P_{qg}]$

where we introduced the shorthand notation: $\tag{22} [q\otimes P]~=~[P\otimes q]~=~\int_x^1dy\frac{q(y,t)}{y}\cdot P(\frac{x}{y})$

(it is easy to check that the convolution, like an ordinary product, is commutative). At leading order, the interpretation of eq. (21) is simply that the variation of the quark density is due to the convolution of the quark density at a higher energy times the probability of finding a quark in a quark (with the right energy fraction) plus the gluon density at a higher energy times the probability of finding a quark (of the given flavour i) in a gluon.

The evolution equation for the gluon density, needed to close the system, can be obtained by suitably extending the same line of reasoning to a gedanken probe sensitive to colour charges, for example a virtual gluon. The resulting equation is of the form: $\tag{23} \frac{d}{dt}g(x,t)~=~\frac{\alpha_s(Q)}{2\pi}[\sum_i (q_i+\bar q_i)\otimes P_{gq}]~+~\frac{\alpha_s(Q)}{2\pi}[g\otimes P_{gg}]$

Eqs. (20), (22) are the QCD evolution equations for parton densities Gribov and Lipatov (1972), Altarelli and Parisi (1977), see also Dokshitzer (1977), also called DGLAP or GLAP or AP equations. The explicit form of the splitting functions in lowest order can be directly derived from the QCD vertices so that they are a property of the theory and do not depend on the particular process the parton density is taking part in Altarelli and Parisi (1977) and is given by: $\tag{24} P_{qq}=\frac{4}{3}[\frac{1+x^2}{(1-x)_+}~+~\frac{3}{2}\delta (1-x)]~+~o(\alpha_s)$

$P_{gq}=\frac{4}{3}\frac{1+(1-x)^2}{x}~+~o(\alpha_s)$ $P_{qg}=\frac{1}{2}[x^2+(1-x)^2]~+~o(\alpha_s)$ $P_{gg}=6[\frac{x}{(1-x)_+}~+~\frac{1-x}{x}~+~x(1-x)]~+~\frac{33-2n_f}{6}\delta (1-x)~+~o(\alpha_s)$ For a generic non singular weight function f(x) the "+" distribution is defined as: $\tag{25} \int_0^1\frac{f(x)}{(1-x)_+}dx~=~\int_0^1\frac{f(x)-f(1)}{1-x}dx$

The $$\delta(1-x)$$ terms arise from the virtual corrections to the lowest order tree diagrams. Their coefficient can be simply obtained by imposing the validity of charge and momentum sum rules. In fact, from the request that the charge sum rules in eq.~ (12) are not affected by the $$Q^2$$ dependence one derives that $\tag{26} \int_0^1P_{qq}(x)dx~=~0$

which can be used to fix the coefficient of the $$\delta(1-x)$$ terms of $$P_{qq}\ .$$ Similarly, by taking the t-derivative of the momentum sum rule in eq.~ (13) and imposing its vanishing for generic $$q_i$$ and g, one obtains: $\tag{27} \int_0^1[P_{qq}(x)~+~P_{gq}(x)]xdx~=~0,~~~~~~\int_0^1[2n_fP_{qg}(x)~+~P_{gg}(x)]xdx~=~0.$

At higher orders the DGLAP evolution equations are easily generalised but the calculation of the splitting functions rapidly becomes very complicated. For many years the splitting functions were only completely known at the next-to-leading accuracy$\alpha_s P~\sim~\alpha_s P_1~+~\alpha_s^2 P_2~+...\ .$ Then in recent years the next-next-to-leading results $$P_3$$ have been first derived in analytic form for the first few moments and, then the full analytic calculation, a really monumental work, was completed in 2004 by Moch, Vermaseren and Vogt (2004) and Vogt, Moch and Vermaseren (2004). Beyond leading order, a precise definition of parton densities should be specified. Once the definition of parton densities is fixed, the coefficients that relate the different structure functions to the parton densities at each fixed order can be computed. Similarly the higher order splitting functions also depend, to some extent, from the definition of parton densities, and a consistent set of coefficients and splitting functions must be used at each order.

At present DIS still remains very important for quantitative studies and tests of QCD. The theory of scaling violations for totally inclusive DIS structure functions, based on the light cone operator expansion or diagrammatic techniques and on renormalisation group methods, that we have just summarised, is crystal clear and the predicted $$Q^2$$ dependence can be tested at each value of x. The scaling violations are clearly observed by experiment and their pattern is very well reproduced by QCD fits at NLO. These fits provide an impressive confirmation of a quantitative QCD prediction. The measurement of quark and gluon densities in the nucleon, as functions of x at some reference value of $$Q^2$$ is performed in DIS processes. At the same time one measures $$\alpha_s(Q^2)$$ and the DIS values of the running coupling can be compared with those obtained from other processes.

The parton densities defined and measured in DIS are instrumental to compute hard processes initiated by two colliding hadrons via the Factorisation Theorem (FT) which is based on diagrammatic techniques. Suppose you have a hadronic process of the form $$h_1+h_2 \rightarrow$$ X+all where $$h_i$$ are hadrons and X is some triggering particle or pair of particles which specify the large scale $$Q^2$$ relevant for the process, in general somewhat, but not much, smaller than s, the total c.o.m. squared mass. For example, in pp or $$p\bar p$$ collisions, X can be a W or a Z or a virtual photon with large $$Q^2 \ ,$$ or a jet at large transverse momentum $$p_T \ ,$$ or a pair of heavy quark-antiquark of invariant mass M. By "all" we mean a totally inclusive collection of final state hadrons. The FT states that for the total cross-section or some other sufficiently inclusive distribution we can write, apart from power suppressed corrections, the expression: $\tag{26:label exists!} \sigma(s,\tau)~=~\sum_{AB}\int dx_1dx_2 p_{1A}(x_1,Q^2)p_{2B}(x_2,Q^2)\sigma_{AB}(\alpha_s(Q), x_1x_2s,\tau)$

Here $$\tau=Q^2/s$$ is a scaling variable, $$p_{iC}$$ are the densities for a parton of type C inside the hadron $$h_i \ ,$$ $$\sigma_{AB}$$ is the partonic cross-section for parton-A + parton-B $$\rightarrow$$ X + all', computable in perturbation theory. Here by all’ we denote a totally inclusive collection of quarks and gluons. This result is based on the fact that the mass singularities that are associated with the initial legs are of universal nature, so that one can reproduce the same modified parton densities, by absorbing these singularities into the bare parton densities, as in DIS. Once the parton densities and $$\alpha_s$$ are known from other measurements, the prediction of the rate for a given hard process is obtained with not much ambiguity (e.g from scale dependence or hadronisation effects). The QCD evolution equations are essential in order to evolve the measured parton densities from one scale Q to a different one. The next-to–the-leading calculation of the partonic cross-section is needed in order to correctly specify the scale and in general the definition of the parton densities and of the running coupling in the leading term. The residual scale and renormalisation scheme dependence is often the most important source of theoretical error.