One-loop matching of QCD currents to power-suppressed two-jet operators

Technical Summary: One-loop matching of QCD currents to power-suppressed two-jet operators

Problem and Context
The paper addresses the need to extend theoretical predictions for hard processes beyond the leading-power approximation in the high-energy expansion. While factorization in Soft-Collinear Effective Theory (SCET) successfully separates short- and long-distance dynamics, systematic inclusion of subleading contributions is increasingly required to match experimental precision in event shapes and deep-inelastic scattering (DIS) in the Bjorken-$x \to 1$ limit.

Specifically, the authors focus on the matching of QCD quark-antiquark currents onto a basis of two-jet operators in SCET. While leading-power matching coefficients are known to high orders, and subleading power ($O(\lambda)$) coefficients have been computed for specific cases (heavy-light currents, back-to-back jets), the one-loop matching coefficients for operators suppressed at second order in the power expansion ($O(\lambda^2)$) were previously unknown. Here, $\lambda = p_\perp/Q$ represents the ratio of transverse momentum to the hard scale. These $O(\lambda^2)$ corrections are particularly relevant for processes where azimuthal dependence is not resolved, as they contribute to the next-to-leading power (NLP) term alongside double insertions of $O(\lambda)$ operators.

Methodology
The authors perform a one-loop matching calculation in perturbative QCD. The methodology involves:

1. Operator Basis Construction: Defining a complete set of gauge-invariant SCET operators up to $O(\lambda^2)$. These include two-particle operators (involving collinear and anti-collinear quark fields) and three-particle operators (involving an additional gluon field). The basis includes operators with transverse derivatives and additional gluon building blocks ($A_{c\perp}$). The authors utilize dimensional regularization ($d=4-2\epsilon$) and identify evanescent operators that vanish in $d=4$ but are necessary for consistent renormalization.
2. Matching Procedure: The matching coefficients are determined by evaluating the defining equation for the QCD current in on-shell matrix elements.
   • For $A$-type operators (no extra gluon), the matrix element $\langle \bar{q}q | \dots | 0 \rangle$ is used.
   • For $B$-type operators (one extra gluon), the matrix element $\langle g \bar{q} q | \dots | 0 \rangle$ is used.
   • The calculation involves expanding QCD amplitudes in the power counting parameter $\lambda$ while retaining only the "hard" region of loop integrals. Special care is taken to handle diagrams where internal propagators are collinear but contribute to the local short-distance coefficients at $O(\lambda^2)$ via a decomposition of the propagator or by applying equations of motion.
3. Renormalization: The authors verify their results by checking the renormalization group evolution. They demonstrate that the calculated bare coefficients, when convoluted with the known renormalization constants ($Z$-factors) for subleading-power operators, yield finite, renormalized coefficients. This serves as a consistency check against known anomalous dimensions.
4. Endpoint Factorization: The behavior of the coefficient functions is analyzed in the limit where partonic momentum fractions approach zero (endpoint region). The authors show that these coefficients satisfy factorization relations that ensure the cancellation of endpoint singularities in the complete factorization formula.

Key Contributions and Results

• First Calculation of $O(\lambda^2)$ Matching: The paper provides the first computation of one-loop matching coefficients for QCD currents onto $O(\lambda^2)$ suppressed two-jet operators in SCET.
• Explicit Coefficients: The authors derive explicit expressions for the short-distance coefficients of:
  • Six two-particle operators ($J_{A1,A1}$ and $J_{A0,A2}$ types).
  • Twelve three-particle operators ($J_{A1,B1}$ and $J_{A0,B2}$ types).
  • The results are presented in momentum space as functions of the momentum fraction $x$ and the hard scale $Q$.
• Relations via Reparameterization Invariance (RPI): The authors demonstrate that many of the calculated coefficients are not independent but are related to the leading-power coefficient $C^{(A0)}$ and the $O(\lambda)$ coefficient $C^{(B1)}$ via RPI. This reduces the number of independent calculations required.
• Evanescent Operators: The paper details the treatment of evanescent operators (those vanishing in $d=4$) and how their inclusion affects the definition of the physical operator basis and the corresponding Wilson coefficients.
• Consistency Checks:
  • The results are shown to be consistent with the known anomalous dimensions of subleading-power operators, confirming that the divergent parts of the matching coefficients correctly cancel the UV divergences of the SCET operators.
  • The endpoint factorization of the coefficients is explicitly derived, validating the consistency of the factorization theorem for power corrections.

Significance
The paper claims that these results constitute a necessary step in the ongoing program to compute power corrections and sum power-suppressed logarithmically enhanced terms for event shapes and DIS. By providing the missing one-loop matching coefficients for $O(\lambda^2)$ operators, the work enables more precise theoretical predictions for two-jet processes. The demonstration of endpoint factorization relations is highlighted as crucial for the consistent cancellation of singularities in the complete factorization formula, ensuring the validity of the effective theory description in the kinematic limits where power corrections become significant. The results are extended to (pseudo)scalar currents in the appendices, broadening their applicability to various phenomenological settings.