Two-Loop DGLAP Splitting Functions from Light Cone Perturbation Theory

TL;DR

This paper develops a two-loop LCPT framework to compute the nonsinglet NLO DGLAP splitting function $P^{-,(1)}$, showing that a positive $k^+$ regulator yields unambiguous results while transverse loops are handled by dimensional regularization. By expressing PDFs as operator expectations in a dressed quark state and renormalizing to introduce a factorization scale $$, they relate UV poles to collinear evolution and extract the NLO kernel from the $1/$ terms, including a careful treatment of one-loop subdivergences and endpoint contributions. A detailed two-loop diagram (B4) illustrates the computation, and an automated approach to spinor algebra, tensor reduction, and master integrals is presented; the final result for $P^{-,(1)}(x)$ matches the covariant Curci–Furmanski–Petronzio expression. The work demonstrates the viability of higher-order LCPT calculations, highlights a transparent renormalization perspective for PDFs in this framework, and points to a systematic path toward automation and applications in high-energy QCD and CGC physics. Overall, the paper establishes LCPT as a rigorous, interpretable, and scalable approach for precision parton dynamics at two loops and beyond.

Abstract

We perform a two-loop calculation in Light Cone Perturbation Theory (LCPT) to evaluate the next-to-leading order nonsinglet splitting function. Our calculation demonstrates the methodology and feasibility of performing higher order calculations in LCPT. Since in Hamiltonian perturbation theory the longitudinal $k^+$ momentum is always positive, poles in $1/k^+$ can be regularized by a simple cutoff which cancels in physical results, without any associated ambiguities. For transverse momentum integrals we use dimensional regularization. Developing methods for loop calculations in LCPT paves the way for a systematical, automatizable procedure for precision calculations in this framework with a transparent physical partonic interpretation. This can provide a standard framework in higher order calculations in the gluon saturation regime of QCD.

Figures (4)

• Figure 1: Leading order diagram contributing to the quark-quark splitting function, built from the square of $\psi^{q\rightarrow qg}_{(0)}$. Here, the dashed lines represent the intermediate states (and the corresponding energy denominators), the curvy line in the middle separates the wave function from its conjugate, and the cross represents that the quark is measured, i.e. it carries the momentum fraction $x$. Here $h$ and $h'$ are quark helicities, $\lambda$ is the gluon polarization, and $\alpha, \beta$ and $a$ are color indices in the fundamental and adjoint representations, respectively. The three-momenta are $\vec{p}=(p^+,\mathbf{0})$, $\vec{k}=(xp^+,\mathbf{k})$, and $\vec{q}=\vec{p}-\vec{q}=((1-x)p^+,-\mathbf{k})$. We have also indicated that the incoming quark is set off-shell with $p^-=-\mu^2_0/(2p^+)$.
• Figure 2: Diagrammatic visualization of terms arising from the expectation value of the fermion number operator in the $qq\bar{q}$ state, as given in Eq. \ref{['eq: quark in quark PDF expansion NLO']}. The different diagrams arise by contracting the tree-level LCWFs with different time and momentum orderings, with the quark carrying the momentum fraction $x$ tagged by a cross. The first and third diagrams are equal, giving a factor of two, while the fourth and sixth diagrams, where either the sea quark or sea antiquark is measured, are equal and cancel upon subtraction. Here we have not drawn diagrams with instantaneous interactions and conjugate diagrams, as well as the the multiplicative coefficients arising from summing and contracting the LCWFs.
• Figure 3: Diagrams contributing to the NLO non-singlet splitting function. Conjugate diagrams are not drawn explicitly, but should be included for each non-symmetric diagram. Lines with a small horizontal bar depict instantaneous interactions.
• Figure 4: Diagram B4, with kinematics and energy denominators. The left hand side of the cut depicts $\psi_{(0)}^{q\rightarrow qgg'}$, while the right hand side is $\left(\psi_{(0)}^{q\rightarrow qg'g}\right)^\dagger$. The gluon carrying momentum $\vec{l}\,'=\vec{p}-\vec{k}-\vec{l}$ is tagged with a prime in the notation for the wave functions and energy denominators.