[Paper Review] Generalized threshold resummation for semi-inclusive e+e- annihilation

N. A. Lo Presti, A. Vogt+1 more|arXiv (Cornell University)|Feb 23, 2012

Cryptography and Data Security17 references3 citations

TL;DR

This paper presents a generalized threshold resummation framework for semi-inclusive e+e− annihilation (SIA), extending fixed-order results and dimensional regularization techniques to resum all-order large-𝑥 double logarithms in timelike splitting functions and fragmentation functions. The method achieves next-to-next-to-leading logarithmic (NNLL) accuracy for off-diagonal timelike splitting functions and coefficient functions, with new results confirmed by comparison to independent DIS calculations.

ABSTRACT

Recently methods have been developed to extend the resummation of large-x double logarithms in inclusive deep-inelastic scattering (DIS) to terms not addressed by the soft-gluon exponentiation. Here we briefly outline our approach based on fixed-order results, the general large-x structure in dimensional regularization and the all-order factorization of mass singularities, which is directly applicable also to semi-inclusive e+e- annihilation (SIA). We then present some main results for the corresponding timelike splitting functions and transverse and longitudinal fragmentation functions. The close relation between DIS and SIA facilitates the determination of additional third-order results for the latter function which is fully known only at the next-to-leading order. Therefore all above quantities can be resummed at next-to-next-to-leading logarithmic accuracy.

Motivation & Objective

• To extend threshold resummation beyond soft-gluon exponentiation in semi-inclusive e+e− annihilation (SIA), addressing double logarithmic terms not captured by standard methods.
• To generalize the approach used in deep-inelastic scattering (DIS) to the timelike case, enabling all-order resummation of large-𝑥 logarithms in SIA.
• To determine the next-to-next-to-leading logarithmic (NNLL) coefficients for off-diagonal timelike splitting functions and coefficient functions in SIA, including transverse and longitudinal fragmentation functions.
• To confirm consistency with existing DIS results through cross-checking physical kernels, particularly for the $F_L^T$ and $F_T^T$ fragmentation functions.
• To lay the groundwork for extending resummation to higher-order terms in the (1−x) expansion and to high-energy (small-x) logarithms in future work.

Proposed method

• The method employs fixed-order results in perturbative QCD, combined with the general large-𝑥 structure in dimensional regularization (D=4−2ε) to extract the full ε-expansion of SIA fragmentation functions.
• It uses the all-order factorization of mass singularities and the relation between spacelike and timelike splitting functions via Mellin convolution to derive the timelike evolution kernels.
• The approach relies on the expansion of the unfactorized SIA fragmentation functions $\widehat{F}_{a,k}^T$ in powers of $\epsilon$, where coefficients contain increasing powers of $\ln(1-x)$.
• The resummation is achieved by matching the $\epsilon^{-n}$ and $\epsilon^{-n+1}$ terms in the $\epsilon$-expansion to the fixed-order splitting functions $P_{m}$ and beta function coefficients $\beta_m$.
• The method enables the determination of NNLL coefficients for $P_{ij}^T$, $C_{T,g}^T$, $C_{L,g}^T$, $C_{\phi,q}^T$, and $C_{L,q}^T$ using lower-order fixed-order information.
• Cross-verification with DIS results is performed by comparing physical kernels, confirming consistency for $F_L^T$ and $F_T^T$ up to the NNLL level.

Experimental results

Research questions

• RQ1 Can the generalized threshold resummation method used in deep-inelastic scattering (DIS) be extended to semi-inclusive e+e− annihilation (SIA) for timelike splitting functions?
• RQ2 What are the NNLL coefficients for the off-diagonal timelike splitting functions $P_{ij}^T$ and the corresponding coefficient functions in SIA?
• RQ3 How do the resummed large-𝑥 double logarithms in SIA compare quantitatively to fixed-order results, particularly at $Q^2 \simeq M_Z^2$?
• RQ4 Can the method be used to determine higher-order terms in the (1−x) expansion beyond the leading double logarithms?
• RQ5 Why is the method not directly applicable to Drell-Yan or Higgs production in pp collisions, and what alternative tools are needed?

Key findings

• The paper achieves next-to-next-to-leading logarithmic (NNLL) resummation for all off-diagonal timelike splitting functions $P_{ij}^T$ and the corresponding coefficient functions in SIA, including $C_{T,g}^T$, $C_{L,g}^T$, $C_{\phi,q}^T$, and $C_{L,q}^T$.
• The NNLL coefficient for the $F_L^T$ fragmentation function is confirmed by comparison with independent DIS results, with a specific coefficient expression involving $C_F$, $C_A$, $\zeta_2$, and $n_f$ terms.
• The numerical effect of resummed double logarithms on the splitting functions in Mellin-$N$ space is less than 1% for $N \lesssim 20$ at $Q^2 \simeq M_Z^2$, indicating small corrections beyond fourth order.
• For coefficient functions, contributions from order $\alpha_s^5$ are non-negligible, and N³LL corrections are expected to be significant, suggesting that higher-order terms are phenomenologically relevant.
• The method confirms that all large-𝑥 double logarithms in SIA are fixed by lower-order fixed-order information, with NℓLL coefficients determined by NℓLO results.
• The approach is extendable to small-𝑥 double logarithms, with NNLL results for $x^{-1}$ terms in timelike splitting functions already derived, and $x^0$ terms for DIS quantities in preparation.