--- title: Diffractive Parton Distributions in QCD url: https://www.emergentmind.com/topics/diffractive-parton-distributions type: topic --- # Diffractive Parton Distributions in QCD Diffractive parton distributions, usually denoted DPDFs, are the nonperturbative distributions that enter the QCD collinear factorization theorem for diffractive deep-inelastic scattering at fixed \(x_{\!P}\) and \(t\). In processes such as \(ep\to eXp'\), the outgoing proton loses a fraction \(x_{\!P}\) of its momentum but remains intact, or dissociates into a low-mass system \(Y\), and one introduces \(\beta=x/x_{\!P}\) as the longitudinal momentum fraction of the struck parton relative to the diffractive exchange. In phenomenology, DPDFs are commonly represented through proton–vertex factorization or the resolved-pomeron model, while at small \(x\) they can also be derived from operator definitions that reduce to Wilson-line correlators in the shockwave or color-glass-condensate limit [1009.5265] [2510.02254]. ## 1. Kinematics and factorization in diffractive DIS In diffractive DIS one measures the reduced diffractive cross section \[ \sigma_r^{D(4)}(\beta,Q^2,x_{\!P},t) \] related to the diffractive structure functions by \[ \sigma_r^{D(4)} = F_2^{D(4)}(\beta,Q^2,x_{\!P},t) - \frac{y^2}{Y_+}\,F_L^{D(4)}(\beta,Q^2,x_{\!P},t), \qquad Y_+ = 1 + (1-y)^2. \] The kinematic variables most commonly used are \[ x_{\!P} = \frac{(P-P')\cdot q}{P\cdot q} = \frac{x}{\beta},\qquad \beta = \frac{Q^2}{2(P-P')\cdot q} = \frac{x}{x_{\!P}},\qquad t=(P'-P)^2, \] and \[ M_X^2 = (q+P-P')^2. \] These definitions are standard across HERA extractions and later global analyses [1009.5265] [1908.10154]. The hard-scattering factorization theorem for diffractive DIS states that, for fixed \(x_{\!P}\) and \(t\), the diffractive structure functions factorize into the same coefficient functions as in inclusive DIS and diffractive parton distributions: \[ F_{k}^{D(4)}(\beta,Q^2,x_{\!P},t) = \sum_{i=q,\bar q,g} \int_{\beta}^1 \frac{dz}{z}\, C_{k,i}(\beta/z,\alpha_s(Q^2))\,f_i^D(z,x_{\!P};Q^2,t), \qquad k=2,L. \] At leading order, \[ F_2^D(\beta,Q^2,x_{\!P}) = \sum_i e_i^2\,\beta\,f_i^D(\beta,Q^2;x_{\!P}), \] so DPDFs can be read as the diffractive analogues of ordinary PDFs, with the added conditioning on the diffractive proton kinematics [1009.5265]. A recurrent misconception is that the pomeron language is part of the factorization theorem itself. The theorem only requires factorization at fixed \((x_{\!P},t)\); the additional decomposition into flux factors and parton densities in a pomeron is a phenomenological ansatz. This distinction becomes important when comparing Regge-inspired fits with approaches that fit directly at fixed \(x_{\!P}\) [1110.4829]. ## 2. Regge-factorized and resolved-pomeron constructions In the resolved-pomeron model one assumes that the exchanged object can be treated as a pomeron carrying a longitudinal momentum fraction \(\xi\equiv x_{IP}\) of the proton and a squared momentum transfer \(t\). A parton \(i\) in the pomeron then carries a fraction \(\beta\) of the pomeron’s momentum, so that the momentum fraction \(x\) of the parent proton carried by that parton is \[ x=\xi\,\beta. \] The standard collinear diffractive parton distribution of type \(i\) in the proton is then defined by \[ f_i^D(x,\mu^2;\xi,t)\equiv f_{\mathbb P/p}(\xi,t)\;f_{i/\mathbb P}(\beta,\mu^2), \] and, after integrating over \(t\), \[ f_i^D(x,\mu^2)=\int_x^{\xi_{\max}}\frac{d\xi}{\xi}\;f_{\mathbb P/p}(\xi)\;f_{i/\mathbb P}(x/\xi,\mu^2). \] In practice often reggeon exchange is added, \[ f_{i}^{D}(x,\mu^2) = f_{\mathbb P/p}\otimes f_{i/\mathbb P} + f_{\mathbb R/p}\otimes f_{i/\mathbb R}, \] with HERA fits such as H1 2006 Fit A, B, or C providing fluxes and parton densities [1606.06528]. This Regge-factorized form is also the basis of several global DPDF determinations. In one widely used parametrization at \(Q_0^2=1.8\,\mathrm{GeV}^2\), \[ z\,f_q^D(z,Q_0^2) = \alpha_q\,z^{\beta_q}(1-z)^{\gamma_q},\qquad z\,f_g^D(z,Q_0^2) = \alpha_g\,z^{\beta_g}(1-z)^{\gamma_g}, \] with flux factors \[ f_{I P,R/p}(x_{I P},t)=A_{I P,R}\,e^{B_{I P,R}t}\,x_{I P}^{-2\alpha_{I P,R}(t)+1},\qquad \alpha_{I P,R}(t)=\alpha_{I P,R}(0)+\alpha'_{I P,R}t. \] Fits reported that \(\eta_q,\eta_g\) in extended \((1+\eta\sqrt z)\) forms are not constrained and are set to zero [1908.10154] [1802.01363]. Later analyses used closely related but not identical input scales and shapes. The SKMHS23 fits assume proton–vertex factorization, use \[ z\,f_{q/\mathbb P}(z,Q_0^2) =\alpha_q\,z^{\beta_q}(1-z)^{\gamma_q}\,\exp\Big[-\frac{0.01}{1-z}\Big], \] \[ z\,f_{g/\mathbb P}(z,Q_0^2) =\alpha_g\,z^{\beta_g}(1-z)^{\gamma_g}\,\exp\Big[-\frac{0.01}{1-z}\Big], \] at \(Q_0^2=1.69\,\mathrm{GeV}^2\), with a Regge-inspired flux and a subleading Reggeon contribution normalized by \(n_{\mathbb R}\) [2301.10284]. A distinct line of work avoids the Regge theory ansatz altogether. In the fixed-\(x_{\!P}\) method, the singlet and gluon DPDFs are first fitted in each \(x_{\!P}\) bin and the smooth \(x_{\!P}\)-dependence of their parameters is inferred afterward. At \(Q_0^2=2.3\,\mathrm{GeV}^2\) one such parametrization is \[ \beta \Sigma(\beta,Q_0^2) = A_q\,\beta^{B_q}(1-\beta)^{C_q}\exp\!\Big[-\frac{0.01}{1-\beta}\Big],\qquad \beta g(\beta,Q_0^2) = A_g\,\exp\!\Big[-\frac{0.01}{1-\beta}\Big], \] with \(A_q,B_q,C_q,A_g\) promoted to explicit functions of \(x_{\!P}\). This method was presented as being “much closer in relation with the factorization theorem for diffractive hard processes” and as reducing model dependence associated with flux factors [1110.4829] [1205.6356]. ## 3. DGLAP evolution, heavy flavors, and global extractions Once specified at an input scale, DPDFs evolve with the usual DGLAP equations. In NLO form, \[ \frac{d\,f_i^D(\beta,Q^2)}{d\,\ln Q^2} = \frac{\alpha_s(Q^2)}{2\pi} \sum_j \bigl[P_{ij}\otimes f_j^D\bigr](\beta,Q^2), \] with the same splitting functions as in inclusive DIS. Numerical implementations in the literature use standard evolution codes and global fitting tools, including QCDNUM, APFEL, xFitter, Alpos, fastNLO, and NNLOJET, depending on perturbative order and dataset composition [1009.5265] [1802.01363] [2301.10284]. Heavy-flavor contributions have been incorporated in several different schemes. The ZEUS analysis used the Thorne–Roberts general-mass variable-flavour-number scheme with \(m_c=1.35\,\mathrm{GeV}\), \(m_b=4.3\,\mathrm{GeV}\), and \(\alpha_s(M_Z)=0.118\) [1006.3713]. The GKG18 and SKMHS22 analyses used TR GM-VFNS with \(m_c=1.40\,\mathrm{GeV}\), \(m_b=4.75\,\mathrm{GeV}\), while the HK19 and later SKMHS23 analyses adopted FONLL-based GM-VFNS implementations, with HK19 quoting FONLL-B and SKMHS23 using FONLL-A at NLO and FONLL-C at NNLO [1802.01363] [1902.10734] [1908.10154] [2301.10284]. The HERA program progressively constrained both quark and gluon DPDFs. Inclusive diffractive cross sections determine the singlet quark distribution, while diffractive dijet production in DIS is sensitive mainly to the gluon via boson–gluon fusion. The ZEUS SJ fit, which combined inclusive and dijet data, yielded “a well-determined singlet quark distribution, with \(\sim 5\)–\(10\%\) experimental uncertainty across \(0.01<\beta<0.8\)” and “a gluon density of comparable precision, peaking at \(\beta\sim0.1\) and falling towards \(\beta\to1\)” [1009.5265]. Global fits using the combined H1/ZEUS datasets confirmed this picture. GKG18 performed “the first global NLO QCD fit of diffractive PDFs using the xFitter framework,” obtaining \(\chi^2/\mathrm{dof}=322/289=1.11\) in Fit A and \(280/263=1.06\) in Fit B [1802.01363]. The HK19 fracture-function analysis used 499 points after cuts and reported \(\chi^2/\mathrm{d.o.f}=0.89\) at NLO and \(0.91\) at NNLO [1902.10734]. The updated SKMHS23 fits added H1 dijet measurements and found \(\chi^2/\mathrm{d.o.f.}=1.11\) and \(1.10\) for inclusive-only NLO and NNLO fits, improving to \(1.09\) and \(1.07\) when dijets were included [2301.10284]. ## 4. Higher twist, fracture functions, and methodological tensions The treatment of large-\(\beta\) and low-\(Q^2\) data has been a persistent issue in DPDF extractions. One approach imposes kinematic cuts to suppress regions dominated by power corrections. Representative choices include \(M_X>2\,\mathrm{GeV}\), \(\beta\le0.80\), and \(Q^2\ge6.5\,\mathrm{GeV}^2\) or \(Q^2>8.5\,\mathrm{GeV}^2\), depending on the fit [1908.10154] [1902.10734]. A more explicit treatment introduces a phenomenological higher-twist correction. In one NLO analysis, \[ F_2^D(\beta,Q^2) = F_2^{D,LT}(\beta,Q^2)\,[1 + C_{HT}(\beta)/Q^2],\qquad C_{HT}(\beta)=h_0\,\beta^{h_1}(1+h_2\beta), \] with fitted values \[ h_0 = -25.798,\qquad h_1 = 2.536,\qquad h_2 = -1.381. \] This correction was found to be “most significant at large \(\beta\) and low \(Q^2\)” and to improve the fit quality from \(\chi^2/\mathrm{dof} = 390.46/371 = 1.052\) to \(376.08/371 = 1.013\) [1908.10154]. SKMHS22 incorporated a different higher-twist component, namely the twist-4 correction from the longitudinal \(q\bar q\) state in the dipole picture. That analysis stated that “the twist-4 contribution allows to include the high-\(\beta\) region and leads to a better description of the diffractive DIS data sets,” while the inclusion of the subleading Reggeon exchange “significantly improves the description of the diffractive DIS cross-section measurements” [2206.13788]. The fracture-functions formulation provides another angle on the same physics. In that framework, the diffractive distributions are written as proton-to-proton fracture functions \[ {\cal F}_{i/p}^{D}(\beta,Q^2;x_{\mathbb P},t), \] which obey the ordinary homogeneous DGLAP equations after integrating over \(t\) in a narrow range. The HK19-DPDF study presented this as “a QCD framework designed to provide a statistically sound representation of diffractive DIS processes,” with fits performed at NLO and NNLO and uncertainties determined by the Hessian approach with \(T=1\) for \(68\%\) CL [1902.10734]. These different strategies expose an important methodological tension. Regge-factorized fits, fixed-\(x_{\!P}\) fits, and fracture-function fits all implement the same hard-scattering factorization theorem, but they distribute the nonperturbative input differently. This suggests that part of the disagreement among published DPDF sets is attributable not only to data selection and perturbative order, but also to the chosen parameterization of the \(x_{\!P}\) dependence. ## 5. Transverse-momentum-dependent and small-\(x\) formulations The collinear DPDF can be uplifted to a transverse-momentum-dependent object in more than one way. In the \(k_t\)-factorization approach to single-diffractive charm production, the diffractive unintegrated gluon density is generated from the collinear diffractive gluon density through the Kimber–Martin–Ryskin prescription: \[ f_g^D(x,k_t^2,\mu^2) \equiv \frac{\partial}{\partial\ln k_t^2} \bigl[g^D(x,k_t^2)\cdot T_g(k_t^2,\mu^2)\bigr], \] and \[ {\cal F}_g^D(x,k_t^2,\mu^2)\equiv \frac{1}{k_t^2}\,f_g^D(x,k_t^2,\mu^2), \qquad g^D(x,\mu^2)=\int_0^{\mu^2}dk_t^2\,f_g^D(x,k_t^2,\mu^2). \] In that construction the transverse momentum of the pomeron is neglected with respect to the transverse momentum of partons entering the hard process, and the KMR density is frozen below \(k_t<\mu_0\approx1\,\mathrm{GeV}\) [1606.06528]. The corresponding single-diffractive \(c\bar c\) cross section in \(k_t\)-factorization is written as \[ d\sigma^{SD(a)} =\int dx_1\frac{d^2k_{1t}}{\pi}dx_2\frac{d^2k_{2t}}{\pi}\, d\hat\sigma(g^*g^*\to c\bar c)\, {\cal F}_g^D(x_1,k_{1t}^2,\mu^2)\, {\cal F}_g(x_2,k_{2t}^2,\mu^2), \] with an analogous formula for \(SD(b)\). This allows transverse-momentum imbalance and correlation observables that are absent in LO collinear kinematics [1606.06528]. At small \(x\), TMD DPDFs can be defined directly at operator level. For quarks, \[ 2E_{P'}\,\frac{d f_q^D(x,k_\perp;x_{I\!\!P},t)}{d^3P'} = \int\frac{d\xi^-d^2\xi_\perp}{2(2\pi)^6} e^{-ixP^+\xi^- + i k_\perp\cdot\xi_\perp} \langle PS| \bar\psi(\xi)\,{\cal L}_n^\dagger(\xi)\,\gamma^+\,|P'X\rangle \langle P'X| {\cal L}_n(0)\,\psi(0)|PS\rangle, \] and for gluons, \[ 2E_{P'}\,\frac{d f_g^D(x,k_\perp;x_{I\!\!P},t)}{d^3P'} = \int\frac{d\xi^-d^2\xi_\perp}{xP^+(2\pi)^6} e^{-ixP^+\xi^- + i k_\perp\cdot\xi_\perp} \langle PS| F^{+\mu}(\xi)\,{\cal L}_n^\dagger(\xi)\,F_{\mu}{}^{+}(0)\,{\cal L}_n(0)|PS\rangle, \] with future-pointing Wilson lines and diffractive kinematics specified by \(P'=P+\Delta\), \(t=\Delta^2\), \(x_{I\!\!P}\), and \(\beta=x/x_{I\!\!P}\) [2403.19609]. In the shockwave approximation, the same class of distributions reduces to Wilson-line correlators. Quarks pick up \[ V(x_\perp)=P\exp\Big[-i\,g_s\int_{-\infty}^{+\infty}dz^-\,A^+(z^-,x_\perp)\Big], \] gluons pick up \[ U(x_\perp)=P\exp\Big[-i\,g_s\int_{-\infty}^{+\infty}dz^-\,T^cA_c^+(z^-,x_\perp)\Big], \] and the final analytic expressions for coherent quark and gluon diffractive TMDs are given in terms of dipole or quadrupole correlators. Integrating these TMDs over \(k_\perp\) yields closed-form diffractive PDFs \(F_g^D(P,\beta)\) and \(F_q^D(P,\beta)\) in terms of two-dimensional integrals of the same correlators [2510.02254]. The small-\(x\) TMD framework also exposes angular structure. The dipole amplitudes are expanded as \[ {\cal F}_x(q,\Delta)={\cal F}_0(q,\Delta)+2\cos2(\phi_q-\phi_\Delta)\,{\cal F}_\epsilon(q,\Delta), \] \[ {\cal G}_x(q,\Delta)={\cal G}_0(q,\Delta)+2\cos2(\phi_q-\phi_\Delta)\,{\cal G}_\epsilon(q,\Delta), \] leading to elliptic diffractive distributions and a diffractive Sivers function. A striking result is that “the diffractive version of the linearly polarized gluon distribution identically vanishes to leading order” [2403.19609]. ## 6. Phenomenology, validation, and open issues The empirical basis for DPDFs comes primarily from HERA. High-precision inclusive diffractive cross sections from H1 and ZEUS were found to be mutually compatible within measurement uncertainties and to support NLO DGLAP fits with noticeably reduced uncertainties. The gluon density precision improved markedly when diffractive dijet data were included, and the resulting DPDFs described both charm production in diffractive DIS and diffractive dijet photoproduction [1009.5265]. Specific validation tests were particularly influential. In one ZEUS-based study, predictions for the charm contribution \(F_{2}^{D,c\bar c}(\beta,Q^2,x_{\!P})\) at \(x_{\!P}=0.004\) and \(0.02\), \(Q^2=4\) and \(25\,\mathrm{GeV}^2\), agreed within \(\sim10\%\) of the data. The same DPDFs described diffractive dijet photoproduction over \(0.2