# Parton distribution functions (definition)

Post-publication activity

Curator: John Collins

Central to much phenomenology of high-energy scattering with hadron beams or targets is the concept of a parton density in quantum chromodynamics (QCD). This article explains the definition of parton densities (also called parton distribution functions). It includes the use of the $$\overline{\mbox{MS}}$$ renormalization prescription, which provides the most commonly used scheme for measured parton densities.

## Factorization, and the need for parton densities

Parton densities are used in factorization theorems, where a particular hard-scattering cross section is a convolution product of one (or two) parton densities (or similar quantities) and a perturbatively calculable hard scattering. The importance of factorization theorems is that they unlock a lot of predictive power of QCD. By far the best developed method for predicting scattering cross sections is weak-coupling perturbation theory. However, typical scattering reactions involve phenomena on a wide range of distance or momentum scales. In this situation, unadorned perturbation theory is useless, because its coefficients contain large logarithms of ratios of the scales relevant for a process.

In factorization properties for appropriate processes, like deep inelastic scattering or hadron-hadron scattering to jets, each factor involves phenomena on roughly a single scale. The renormalization group can be used in each factor separately to give the QCD coupling its value appropriate for the factor's typical scale. Now QCD is asymptotically free. That is, its effective coupling is weak for short-distance quantities, which are therefore perturbatively calculable to a useful accuracy.

In contrast, the long-distance factors are non-perturbative; these are the parton densities and similar quantities, where the relevant scale is that of the size of a hadron (roughly $$10^{-15}$$ m). Although our ability to predict parton densities from QCD is very limited, they are universal between different reactions. Thus they can be measured from a limited set of reactions at a limited set of energies, and then used in factorization theorems for the same reaction at different energies and for different reactions. Herein lies much of the available predictive power of QCD.

In fact, parton-density universality is modified in QCD, since the parton densities depend on the momentum scale for which they are being used. There is an equation, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation, for the scale dependence. Further predictions arise from the perturbative calculability of the kernels of the DGLAP equation.

### Factorization formula

A typical factorization theorem is for a structure function for deeply inelastic scattering (DIS) of leptons on hadrons ($$l+P\to l'+X$$). The DIS cross section is written in terms of scalar structure functions like $$F_1(x,Q)\ ,$$ which are conventionally written in terms of standard kinematic variables $$x$$ and $$Q$$ — see QCD evolution equations for parton densities. The factorization theorem for $$F_1$$ has the form \tag{1} \begin{align} F_1(x,Q) = {}& \sum_j \int_{x}^{1} \frac{ d\xi }{ \xi } \hat{F}_{1j}(x/\xi,Q/\mu;\alpha_s(\mu)) f_{j/H}(\xi;\mu) + \mbox{power-suppressed correction (p.s.c.)}. \end{align} This gives a factorized form for $$F_1$$ valid up to corrections that are suppressed by a power of $$Q$$ when $$Q$$ is large enough. The quantity $$f_{j/H}$$ is the density of partons of flavor $$j$$ in the target particle $$H\ .$$ In QCD the parton index can have the values $$g$$ (for gluon), or one of the flavors of quark or antiquark ($$u\ ,$$ $$\bar{u}\ ,$$ $$d\ ,$$ $$\bar{d}\ ,$$ etc). The argument $$\xi$$ is the fractional momentum of the parton relative to the target, in a sense to be defined below using light-front coordinates. The second argument $$\mu$$ is the energy scale at which the parton density is defined; we will identify it with the renormalization scale used in $$\overline{\mbox{MS}}$$ renormalization. Finally $$\hat{F}_{1j}$$ is a short-distance coefficient function.

Predictions are made by computing the coefficient function in low-order perturbation theory. To make this effective, $$\mu$$ should be taken to be of the order of the large momentum scale $$Q\ .$$ There are then no large logarithms (provided $$x$$ is not close to $$0$$ or $$1$$), while $$\alpha_s(Q)$$ is small because of the asymptotic freedom of QCD.

A coefficient function like $$\hat{F}_{1j}(x/\xi)$$ can be conceptualized as a structure function for DIS on a parton target of flavor $$j$$ and momentum $$\xi P\ ,$$ where $$P\$$ is the target momentum. But it is defined with subtractions so that it only concerns the short-distance part of DIS.

Positivity of the final-state energy implies that the parton density is zero for $$\xi$$ above unity, while the coefficient function is zero for $$\xi<x\ .$$ Thus the limits on the $$\xi$$ integral are $$x$$ and $$1\ .$$ However, the parton densities and coefficient functions may be generalized functions with singularities at the endpoints. So to make the integrals mathematically correct, the integral should be extended slightly beyond the kinematic endpoints.

### Scale dependence

The scale dependence of the parton densities is governed by the DGLAP equation: $\tag{2} \frac{ d }{ d\ln\mu } f_{j/H}(\xi;\mu) = 2 \sum_{j'} \int_{\xi}^{1} \frac{ dz }{z} P_{jj'}(z;\alpha_s(\mu)) f_{j'/H}(\xi/z;\mu),$ with the overall factor 2 being used so that the kernel $$P\$$ has the normalization convention given by the Particle Data Group — see Nakamura et al. (2010) and PDG structure-function review; other conventions exist. Within the $$\overline{\mbox{MS}}$$ approach, the DGLAP equation is the renormalization-group (RG) equation for the parton densities. The kernel $$P$$ can be obtained from the UV renormalization coefficients of the parton densities — see Eq. (24) below.

Note, however, that the original derivation by Altarelli and Parisi (1977) of the DGLAP equation used a somewhat different interpretation.

The overall structure of factorization and its application can be summarized by writing the factorization equation and the DGLAP equation in a convolution notation: $\tag{3} F_1(Q) = \hat{F}_1 \otimes f + \mbox{p.s.c.},$ $\tag{4} \frac{ df(\mu) }{ d\ln\mu } = P(\alpha_s(\mu)) \otimes f(\mu) .$ The solution has the form $\tag{5} F_1(Q) = \hat{F}_1(Q) \otimes \exp\left[ \int_{Q_0}^Q P(\alpha_s(\mu)) \frac{d\mu}{\mu} \right] \otimes f(Q_0) + \mbox{p.s.c.}$ Here the structure function at a large scale $$Q$$ is expressed in terms of the parton densities at a fixed scale $$Q_0\ .$$ Algebra on the kernel $$P\$$ is in the sense of convolution. Provided that $$Q_0$$ is not too small, the DGLAP kernel as well as the coefficient function are perturbatively computable. Hence they can be predicted from first principles in QCD to a useful accuracy.

## Light-front quantization

To motivate the definition of a parton density in full QCD, let us start with a candidate definition that arises from elementary parton-model considerations by way of light-front quantization.

In Figure 1 is depicted a DIS collision in the center-of-mass frame. The process is $$l + H(P) \longrightarrow l' + X\ ,$$ where the notation $$X$$ indicates an inclusive cross section, summed over hadronic final states. The proton, coming from the left, has high energy, and is highly time-dilated and Lorentz-contracted. The electron, from the right, undergoes a wide angle scattering with large momentum transfer $$Q$$ over a short distance scale of order $$1/Q\ .$$

The parton model arose before QCD from the suggestion that the electron scattering occurs off a quasi-point-like constituent in the proton, and that the only relevant scales are the short distance scale of the electron scattering and the long distance scale of self-interactions in the time dilated hadron. The constituents are called partons, a term that is now effectively a collective name for quarks and gluons. Then the cross section is the product of a short-distance cross section and the density of partons. Since all but the longitudinal momentum of the parton can be neglected in the hard scattering, the density is the single-particle density of the parton as a function of its large longitudinal momentum.

To implement this idea in quantum field theory, one uses light-front quantization. We first define light-front coordinates for a general vector $$V$$ by $\tag{6} V = (V^+, V^-, \boldsymbol{V}_{\text{T}}) = \left( \frac{V^0+V^z}{\sqrt2}, \frac{V^0-V^z}{\sqrt2}, V^x, V^y \right),$ in a frame where the target hadron is moving in the $$+z$$ direction. Next let $$|H\rangle$$ be a state of a hadron moving with momentum $$P=(P^+,M^2/(2P^+),\boldsymbol{0}_{\text{T}})\ .$$ In light-front quantization one constructs creation and annihilation operators, $$b_{k,\alpha,j}^{\dagger}$$ and $$b_{k,\alpha,j}\ ,$$ for a parton of flavor $$j$$ with momentum fraction $$\xi=k^+/P^+\ ,$$ transverse momentum $$\boldsymbol{k}_{\text{T}}\ ,$$ and light-front helicity $$\alpha\ .$$ Then the parton density is $\tag{7} f_{j/H}(\xi) = \frac{1}{ 2\xi (2\pi)^3 } \sum_\alpha \int d^2\boldsymbol{k}_{\text{T}} \frac{ \langle H | b_{k,\alpha,j}^{\dagger} b_{k,\alpha,j} | H \rangle } { \langle H|H\rangle } .$ Since momentum eigenstates have infinite normalization, there is an implicit limit operation in Eq. (7)$|H\rangle$ is initially a normalizable wave-packet state, and then the limit of a momentum eigenstate is taken. In addition, the vacuum expectation value of the number operator is to be subtracted.

Expressing the operators in terms of quark fields gives $\tag{8} f_{j/H}(\xi) = \int \frac{ dw^- }{2\pi } \, e^{-i\xi P^+w^-} \langle P| \overline{\psi}_j(0,w^-,\boldsymbol{0}_{\text{T}}) \frac{\gamma^+}{2} \psi_j(0) |P\rangle_{\text{c}} .$ The target state is now an exact momentum eigenstate, $$|P\rangle\ ,$$ defined to have the Lorentz-invariant normalization $\tag{9} \langle P' | P \rangle = (2\pi)^32P^+ \, \delta(P^+-{P'}^+) \, \delta^{(2)}(\boldsymbol{P}_{\text{T}}-\boldsymbol{P}_{\text{T}}').$ In Eq. (8), the subscript "c" denotes that the vacuum expectation value of the operator is subtracted. In Feynman graph terms, this means that we retain only graphs in which the quark fields are connected to the target state. The parton density uses fields on the light-like line depicted in Figure 2. Figure 2: The space-time location of the fields in a parton density is along a light-like line (marked in red). The shaded region represents the approximate location of the target hadron in its rest frame.

For calculation of Feynman graphs, we write the above definition in terms of a momentum-space Green function of quark fields between the target state: $\tag{10} f_{j/h}(\xi) \, = \, \mathrm{Tr} \frac{\gamma^+}{2} \int \frac{ dk^- d^2\boldsymbol{k}_{\text{T}} }{ (2\pi)^4 } \mbox{Graph below}$

## Parton densities in QCD

The above definition is exactly correct, appropriate, and valid, but only in a model QFT where the parton model is correct. In QCD it must be modified: First a UV regulator is applied to the theory to cut off all UV divergences. Bare parton densities are then defined by a gauge-invariant modification of Eq. (8). Then UV renormalization factors are applied so that the resulting renormalized parton densities remain finite when the regulator is removed. As a UV regulator, we will use dimensional regularization with $$n=4-2\epsilon$$ space-time dimensions.

### Gauge-invariant bare parton densities

The definition of a bare quark density in QCD is $\tag{11} f_{(0)\,j/h}(\xi) = \int \frac{ dw^- }{ 2\pi } \, e^{-i\xi P^+w^-} \langle P| \overline{\psi}^{(0)}_j(0,w^-,\boldsymbol{0}_{\text{T}}) W(w^-,0) \frac{\gamma^+}{2} \psi^{(0)}_j(0) |P\rangle_{\text{c}} ,$ where the Wilson-line factor is $\tag{12} W(w^-,0) = P\!\left[ e^{-ig_0 \int_0^{w^-} dy^- A^+_{(0)\alpha}(0,y^-,\boldsymbol{0}_{\text{T}}) t_\alpha } \right] ,$ a path-ordered exponential of the gluon field along the line joining the quark and antiquark fields. Here $$g_0$$ is the bare gauge coupling of QCD, and $$t_\alpha$$ are the generating matrices of the SU(3) gauge group of QCD, normalized to $$\mathrm{Tr} t_\alpha t_\beta = \frac12 \delta_{\alpha\beta}\ .$$ As indicated by the sub- or superscripts $$(0)\ ,$$ the fields are the bare fields of QCD, i.e., the ones with standard canonical (anti)commutation relations.

Bare gauge-invariant antiquark densities are defined with the roles of the quark and antiquark fields exchanged, and with a corresponding hermitian conjugation of the Wilson line factor: $\tag{13} f_{(0)\,\bar{\jmath}/h}(\xi) = \int \frac{ dw^- }{2\pi } \, e^{-i\xi P^+w^-} \mathrm{Tr} \frac{\gamma^+}{2} \langle P| \psi_j^{(0)}(0,w^-,\boldsymbol{0}_{\text{T}}) \overline{\psi}^{(0)}_j(0) W(w^-,0)^\dagger{} |P\rangle_{\text{c}} .$ The bare gluon density is defined by $\tag{14} f_{(0)\,g}(\xi) = \sum_{j,\alpha} \int \frac{ dw^- }{2\pi \xi P^{+}} e^{-i\xi P^{+}w^{-}} \langle P | G_{(0)\,\alpha}^{+j}(0,w^{-},\boldsymbol{0}_{\text{T}}) W_A(w^-,0)_{\alpha\beta} G_{(0)\,\beta}^{+j}(0) |P\rangle_{\text{c}} .$ Here, $$G_{(0)\,\alpha}^{\mu\nu}$$ is the bare field-strength tensor, $$G^\alpha_{(0) \, \mu\nu} = \partial_\mu A^\alpha_{(0) \, \nu} - \partial_\nu A^\alpha_{(0) \, \mu} - g_0 f_{\alpha\beta\gamma}A^\beta_{(0) \, \mu} A^\gamma_{(0) \, \nu} \ ,$$ and the subscript $$A$$ on $$W_A$$ denotes that it is a Wilson line in the adjoint representation of SU(3) rather than the fundamental representation. The extra factor $$1/\xi P^+$$ in (14) arises when one takes the gluonic version of (8) and converts it to be expressed in terms of the gluonic field strength tensor.

### Feynman rules for parton densities

Perturbative calculations of parton densities are made by a dimensionally regulated Feynman-graph expansion of Eq. (11), i.e., $\tag{15} f_{j/h}(\xi) \, = \, \mathrm{Tr}_{\text{Dirac,color}} \frac{\gamma^+}{2} \int \frac{ dk^- d^{2-2\epsilon}\boldsymbol{k}_{\text{T}} }{ (2\pi)^{4-2\epsilon} } \mbox{Graph below}$

together with the corresponding equation for the gluon density. This formula shows an integral over external $$k^-$$ and $$\boldsymbol{k}_{\text{T}}$$ and a trace over Dirac and color indices. The external plus-component of momentum at the top is fixed at $$k^+=\xi P^+\ .$$ The vertical line corresponds to a on-shell final state, with normal Feynman rules on its left and hermitian-conjugated Feynman rules on the right. The double line indicates the Wilson line.

To be able to use standard Feynman rules, we need a time-ordered product of operators on the left of the final-state cut, and anti-time-ordered operators on the right. To make this work for the Wilson line, it is converted to a form in which it goes to infinity from the quark field and comes back to the antiquark field: $\tag{16} W(w^-,0) = W(+\infty, w^-)^\dagger{} W(+\infty, 0).$ Then path ordering in $$W(+\infty, 0)$$ and $$W(+\infty, w^-)^\dagger$$ corresponds to the appropriate time-ordering on the left of the final-state cut and to anti-time-ordering on the right.

The resulting Feynman rules for the Wilson-line elements are shown in Figure 3. The factors of $$Z_3^{1/2}$$ arise from writing the rules for using the renormalized gluon field. The rules for attaching the gluon in the gluon density are given in Figure 4. Figure 3: Feynman rules for Wilson lines in parton densities. Here, $$n^\mu=\delta^\mu_-=(0,1,\boldsymbol{0}_{\text{T}})\ .$$ In the Wilson line for a gluon pdf, the generating matrix for the adjoint representation was used$(T_{\alpha})_{kj} = if_{k\alpha j}\ .$ The indices $$\alpha$$ and $$\beta$$ are for color, $$\mu$$ is the Lorentz index of the gluon, and the $$j$$ index is summed over as in Eq. (14).

### UV divergences and renormalization

UV divergences (in the limit $$n\to4$$) arise not only from field strength renormalization, but also from integrating up to infinite $$\boldsymbol{k}_{\text{T}}$$ and $$k^-$$ in Eq. (11). The simplest examples are given by the upper loops in graphs of the structure shown in Figure 5(a) and (b). Figure 5: (a) and (b) Examples of graphs with UV divergences for quark density. (c) Structure of UV divergences of parton densities: The UV divergences are from integrals to infinity of $$k_{\text{T}}$$ and $$k^-$$ in subgraphs of the form of the upper subgraph labeled "UV".

The general UV divergence is associated with the diagrammatic decomposition of Figure 5(c). Finite renormalized parton densities are obtained by convoluting the bare parton densities with suitable regulator-dependent renormalization factors, in the following form. $\tag{17} f_{j/H}(\xi) = \sum_{j'} \int_{\xi-}^{1+} \frac{ dz }{z} \, Z_{jj'}(z,g,\epsilon) \, f_{(0)\,j'/H}(\xi/z),$ with a sum over parton flavors. Implicitly, the bare densities depend on the regulator, and a limit $$\epsilon\to0$$ is to be applied to get the renormalized parton densities for the physical limit of QCD.

### Renormalization

The $$\overline{\mbox{MS}}$$ scheme of renormalization is defined by requiring that, order-by-order in perturbation theory, the renormalization factor is written as pure pole terms at $$\epsilon=0$$ with a slight modification: $\tag{18} Z_{jj'}(z,g,\epsilon) = \sum_{n=0}^\infty \left( \frac{ g^2 }{ 16\pi^2 } \right)^n Z^{[n]}_{jj'}(z,\epsilon)$ with the coefficients required to have the form $\tag{19} Z^{[n]}_{jj'}(z,\epsilon) = \sum_{l=1}^n Z_{n,l;jj'}(z) S_\epsilon\frac{1}{\epsilon^l},$ where the $$\overline{\mbox{MS}}$$ scheme is defined by the choice that $\tag{20} S_\epsilon = \frac{ (4\pi)^\epsilon }{ \Gamma(1-\epsilon) }.$ (Some definitions of the $$\overline{\mbox{MS}}$$ scheme in the literature are formulated somewhat differently. But all these different definitions can be shown to give the same results for renormalized quantities at $$\epsilon=0\ .$$)

In the $$\overline{\mbox{MS}}$$ scheme the evolution of the coupling obeys $\tag{21} \frac{ d\alpha_s/4\pi }{ d\ln\mu } = -2\epsilon \frac{\alpha_s}{4\pi} + S_\epsilon^{-1} 2\beta(\alpha_sS_\epsilon/(4\pi)),$ where the only $$\epsilon$$ dependence is in the $$-\epsilon \alpha_s$$ term and in the explicit factors of $$S_\epsilon\ .$$ It is convenient to write this equation for the evolution of the natural expansion parameter $\tag{22} \frac{\alpha_s}{4\pi} = \frac{g^2}{16\pi^2}.$

The detailed dynamics of the evolution of the coupling is in the $$\beta$$ function in (21), which has the value $\tag{23} \beta(\alpha_s/4\pi) = - \left( 11 - \frac{2}{3} n_f \right) \frac{\alpha_s^2}{16\pi^2} + O(\alpha_s^3).$

### DGLAP equations

Bare quantities are RG invariant. We can therefore derive an RG equation of the form of (2); this is called the DGLAP equation. Its evolution kernel $$P_{jj'}$$ is obtainable from the renormalization factor for the parton densities: \tag{24} \begin{align} \sum_{j'} \int \frac{ dz' }{z'} P_{jj'}(z',g,\epsilon) Z_{j'k}(z/z',g,\epsilon) & = \frac12 \, \frac{ d }{ d\ln\mu } Z_{jk}(z,g,\epsilon) \\ & = \frac12 \, \frac{ dg(\mu) }{ d\ln\mu } \frac{ \partial }{ \partial g} Z_{jk}(z,g,\epsilon), \end{align} i.e., essentially $\tag{25} P = \frac12 \, \frac{ d }{ d\ln\mu } \ln Z .$ In this equation the logarithm of $$Z$$ is in an algebra in which multiplication is interpreted in the sense of convolutions on $$z\ ,$$ and in the sense of matrices on the partonic indices, as on the left-hand-side of Eq. (24). Since the kernel gives a derivative of the finite renormalized parton densities, the kernel is itself finite, despite its being calculated from the renormalization factor, with its UV divergences. The scale dependence of the coupling $$dg(\mu)/d\ln\mu$$ can be written in terms of the RG beta-function of QCD.

### Perturbative calculations

Parton densities are expectation values of certain operators in hadronic states, and as such are non-perturbative. But the renormalization factors (and the consequent DGLAP kernels) are independent of which hadronic state $$|H\rangle$$ is used. So perturbative calculations for the DGLAP kernel, etc, are conveniently done by replacing $$|H\rangle$$ by an on-shell state of a single parton of flavor $$j'\ ,$$ and performing strictly finite-order perturbative calculations of $$f_{(0)j/j'}(\xi)$$ from the Feynman rules given later. Then we arrange the renormalization factor to cancel the UV divergences.

In more detail, we generalize Eq. (18) to define the coefficients of the perturbation expansion of the bare and renormalized parton densities. The $$n$$-loop expansion of the renormalization equation Eq. (17) gives $\tag{26} f^{[n]}_{j/k}(\xi) = \sum_{n'=0}^n \sum_{j'} \int \frac{ dz }{ {z} } \, Z^{[n']}_{jj'}(z,g,\epsilon) \, f^{[n-n']}_{(0)\,j'/k}(\xi/z),$ where we now use a partonic target $$k\ .$$ The lowest-order terms are trivial: $\tag{27} Z^{}_{jj'}(z) = \delta_{jj'} \delta(z-1), \quad f^{}_{j/j'}(\xi) = f^{}_{(0)\,j/j'}(\xi) = \delta(\xi-1)\delta_{jj'} .$ Then the one-loop renormalized parton density is \tag{28} \begin{align} f^{}_{j/k}(\xi) & = (Z_{2j}^{-1}f)^{}_{(0)j/k}(\xi) + (Z_{2j}Z)^{}_{jk}(\xi,g,\epsilon) \\ & = (Z_{2j}^{-1}f)^{}_{(0)j/k}(\xi) + Z_{2j}^{} \delta_{jk} \delta(\xi-1) + Z^{}_{jk}(\xi,g,\epsilon). \end{align} This is the one-loop bare density plus the one-loop term in the renormalization factor. A slight complication is to multiply the bare parton density by $$Z_{2j}^{-1}$$ which is the inverse of the field renormalization factor for parton $$j\ .$$ Thus the first term on the right of Eq. (28) is computed from a modified version of formulae like Eq. (11) for bare densities, with the modification that the bare quark field is replaced by the renormalized quark field.

The point of this last manipulation is that UV divergences in self-energy and vertex subgraphs are canceled by counterterms in the QCD Lagrangian. Then, with the use of renormalized fields, the only remaining UV divergences are associated with integration over momenta flowing through the vertices for the operators defining the parton density. But to obtain the DGLAP kernels it is necessary to transform the results back to give the renormalization factor for the parton densities relative to the parton densities written in terms of bare fields; this is because the derivation of the DGLAP kernels from the renormalization factors uses the renormalization-group invariance of the bare parton densities.

## The IR point-of-view

A common approach to presenting perturbative QCD and factorization (e.g., pp. 118–128 of Dissertori et al., 2003) is to assert (without proof) that cross sections like that for DIS are given as convolutions of bare parton densities with unsubtracted massless partonic cross sections. In a massless theory, the partonic cross sections have collinear divergences, which can be factored out and absorbed into a redefinition of the parton densities. The use of an $$\overline{\mbox{MS}}$$ scheme therefore appears to be associated with removal of mass divergences, i.e., divergences that appear when masses are set to zero.

Fundamentally this is a mistaken view. This is most easily demonstrated by considering a model QFT in which all the fields have non-zero masses. The bare parton densities, as defined above, demonstrably have no collinear or soft divergences; the only divergences are pure UV divergences. Then conventional renormalization is an appropriate tool for defining finite renormalized parton densities. Even in actual QCD, we have color confinement, so that divergences due to the masslessness of the gluon are cut off (non-perturbatively) by a confinement radius.

Nevertheless the IR view is popular and gives correct results for hard scattering. The link between the IR and UV views is given by the methods of Smirnov (2002) for extracting the asymptotic behavior of Feynman graphs. The connection of the two views can be shown as follows.

We start with the factorization theorem for some process (e.g., DIS) on a hadron target: $\tag{29} \sigma_H = \hat{\sigma} \otimes f_H + \mbox{p.s.c.}$ Here, $$\sigma_H$$ is the cross section on the hadron target $$H\ ,$$ $$f_H$$ is the array of $$\overline{\mbox{MS}}$$-renormalized parton densities on the same target, and $$\hat{\sigma}$$ is the set of hard-scattering coefficients (which have incoming partons). The same theorem on an on-shell massless partonic target gives: $\tag{30} \sigma_p = \hat{\sigma} \otimes f_p,$ where the subscript $$p$$ denotes the partonic target. The hard scattering $$\hat{\sigma}$$ is the same no matter what the target. Since the partons are massless, the power corrections (a power of mass divided by $$Q$$) are now zero. To avoid collinear divergences caused by the on-shell massless initial state, factorization on a partonic target must be applied in a regulated theory, with space-time dimension $$4-2\epsilon>4\ .$$

The renormalization equations of the parton densities on hadron and partonic targets are: $\tag{31} f_H = Z \otimes f^{(0)}_H, \qquad f_p = Z \otimes f^{(0)}_p,$ with the same renormalization coefficient $$Z\ .$$ Hence \tag{32} \begin{align} \sigma_p & = \hat{\sigma} \otimes Z \otimes f^{(0)}_p \\ & = \hat{\sigma} \otimes Z. \end{align} The last line follows since, when all masses are zero, loop corrections to a bare parton density in a parton are zero. This is because they give scale-free integrals over transverse momenta, e.g., $\tag{33} \int d^{2-2\epsilon}\boldsymbol{k}_{\text{T}} k_{\text{T}}^\alpha = 0.$ That such integrals are zero is a theorem of dimensional regularization: UV divergences are exactly equal and opposite to certain corresponding collinear divergences. Thus $$f^{(0)}_p$$ equals its lowest-order value, as given in Eq. (27).

From Eqs. (29), (31), and (32), it follows that \tag{34} \begin{align} \sigma_H &= \hat{\sigma} \otimes f_H + \mbox{p.s.c.} \\ &= \sigma_p \otimes Z^{-1} \otimes f_H + \mbox{p.s.c.} \\ &= \sigma_p \otimes f^{(0)}_H + \mbox{p.s.c.} \end{align} That is, the hadronic cross section is the unsubtracted massless on-shell partonic cross section convoluted with the bare parton density. Collinear divergences in the partonic cross section are removed by the factor $$Z^{-1}$$ shown in the second line, and this is compensated by the change from bare to renormalized parton densities.

This result is physically misleading, since it suggests that there are actual collinear divergences that are removed by absorbing them into a redefinition of the parton densities. But these collinear divergences are not present in the actual theory, as already observed. The only true divergences to be concerned with are the UV divergences in bare parton densities, which are to be canceled by renormalization. It is the use of Eq. (33) that relates the true UV divergences to numerically equal and opposite (unphysical) collinear divergences in the unsubtracted massless partonic cross section.

## Further developments

The above account of parton densities leads rather naturally to certain generalizations, of which a brief account is given in this section.

### Polarized parton densities

The parton densities defined above correspond to simple number densities for partons. (But the need for renormalization implies that the number-density interpretation is not strictly correct.) The definitions can be generalized to take account of possible polarization states for the partons and hadrons. This leads to the definition of what are called polarized parton densities. For the details, see for example Sec. 6.3.6 of Collins (2011).

### Dealing with heavy quarks

Modifications to the factorization property and the associated definitions of parton densities are needed when there are heavy quarks. These are quarks whose masses are significantly larger than the scale $$\Lambda$$ of non-perturbative phenomena, which is a few hundred MeV. The currently known heavy quarks are the $$c\ ,$$ $$b\ ,$$ and $$t\ ,$$ and the modified definitions of parton densities imply that the pure $$\overline{\mbox{MS}}$$ definitions are not exactly the ones used in practice.

The physics issues are illustrated by DIS with $$Q$$ around 5 GeV. This value is in the perturbative domain, i.e., $$\alpha_s(Q) \ll 1\ ,$$ but is comparable to the mass of the $$b$$ quark. For light quarks, a useful approximation to DIS is the lowest-order hard scattering on the light quark, e.g., $$\gamma^*+u \to u$$ (multiplied by the $$u$$-quark density). But for the $$b$$ quark, the appropriate basic approximation uses next-to-leading order production by scattering off a (light) gluon. $$\gamma^*+g \to b + \bar{b}\ .$$

For a review of methods for dealing with heavy quarks in factorization see Thorne and Tung (2008). Of the schemes described there, one that keeps the definition of parton densities close to the $$\overline{\mbox{MS}}$$ definition is that of Aivazis, Collins, Olness, and Tung (ACOT), where the parton densities are defined using the renormalization scheme of Collins, Wilczek, and Zee (1978).

This definition is a composite of several schemes labeled by the number $$N_{\rm act}$$ of "active quarks". The active quarks are the $$N_{\rm act}$$ lightest in the theory, and the remaining $$N_{\rm tot}-N_{\rm act}$$ are called inactive. The gluons are always considered active, since they have zero mass in the QCD Lagrangian. The UV divergences in diagrams that only involve active quarks are renormalized by the $$\overline{\mbox{MS}}$$ method. But the subtraction for a graph with heavy quark loop is defined by zero momentum subtraction.

In this scheme, the DGLAP and renormalization-group equations when there are $$N_{\rm act}$$ active flavors are exactly those obtained with pure $$\overline{\mbox{MS}}$$ renormalization when the fields for the inactive quarks are deleted from QCD. Perturbative calculations can be made of the matching conditions between parton densities in the subschemes with different numbers of active quarks. The density of a heavy quark of mass $$m_q$$ is suppressed by a power $$\Lambda/m_q$$ when a subscheme is used in which the quark is inactive.

### Transverse-momentum-dependent (TMD) parton densities

It is natural to examine the entity obtained by deleting the $$k_T$$ integral in the initial candidate definition (7) of a parton density. This quantity, notated $$f_{j/H}(\xi,k_T)$$, is a number density in the space of $$\xi$$ and $$k_T$$, and is therefore called a transverse-momentum-dependent (TMD) parton density or an unintegrated parton density. TMD densities are useful, even essential, to a treatment of hard processes that are sensitive to partonic transverse momentum. A classic example is the Drell-Yan process of the production of lepton pairs of high invariant mass in a hadron-hadron collision, e.g., $$H_A+H_B \to \mu^++\mu^-+X$$.

However, a correct and useful definition of a TMD parton density is quite non-trivial to construct in QCD. See Soper (1979) and Collins and Soper (1982) for some of the original work on appropriate definitions of TMD parton densities in QCD. A modernized definition is presented in Ch. 13 of Collins (2011). For some of the complications that work on TMD functions has encountered in the SCET framework, see for example Mantry and Petriello (2010) and Becher and Neubert (2010).