Abstract
We derive an operator based factorization theorem for the energy-energy correlation (EEC) observable in the back-to-back region, allowing the cross section to be written as a convolution of hard, jet and soft functions. We prove the equivalence of the soft functions for the EEC and color singlet transverse-momentum resummation to all-loop order, and give their analytic result to three-loops. Large logarithms appearing in the perturbative expansion of the EEC can be resummed to all orders using renormalization group evolution. We give analytic results for all required anomalous dimensions to three-loop order, providing the first example of a transverse-momentum (recoil) sensitive e+e− event shape whose anomalous dimensions are known at this order. The EEC can now be computed to next-to-next-to-next-to-leading logarithm matched to next-to-next-to-leading order, making it a prime candidate for precision QCD studies and extractions of the strong coupling constant. We anticipate that our factorization theorem will also be crucial for understanding non-perturbative power corrections for the EEC, and their relationship to those appearing in other observables.
Highlights
Using a recently introduced rapidity regulator [40], which allows both the regulator and the measurement function to be described by spacetime shifts of the Wilson lines, we prove that the soft function is invariant under the crossing of the Wilson lines, allowing us to use the recently derived results for qT soft function to derive the anomalous dimension and soft function for the energy correlation (EEC)
In this paper we have presented an analytic result for the three-loop soft function for the EEC observable in the back-to-back region
This result was derived from a new factorization theorem describing the leading power dynamics in the back-to-back region, whose soft function is identical to the case of qT for color singlet production up to the direction of the Wilson lines
Summary
We discuss in detail the kinematics of the EEC observable in the back-toback region, χ → π in eq (1.1). Contributions to the observable arise only from correlations between collinear partons in different collinear sectors It is a simple geometric exercise to relate their perpendicular momentum to z, as relevant for the EEC. Where k⊥h ,s is the total transverse momentum of soft final-state hadrons relative to thrust axis, and k⊥h ,i(j) is the tranverse momentum of a collinear hadron relative to its respective jet axis, defined as the direction with largest energy flow. 1 − z is related to the vector sum of the transverse-momentum in the different sectors, and in particular, the only property of the soft radiation that is measured is the total transverse-momentum This is much simpler than other recoil sensitive e+e− observables such as broadening, where it is the scalar sum of the transverse momentum that is measured, making the measurement function extremely complicated for configurations with multiple emissions
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