Abstract
In soft-collinear effective theory, we analyze the structure of rapidity divergence due to the collinear and soft modes residing in disparate phase spaces. The idea of an effective theory is applied to a system of collinear modes with large rapidity and soft modes with small rapidity. The large-rapidity (collinear) modes are integrated out to obtain the effective theory for the small-rapidity (soft) modes. The full SCET with the collinear and soft modes should be matched onto the soft theory at the rapidity boundary, and the matching procedure becomes exactly the zero-bin subtraction. The large-rapidity region is out of reach for the soft mode, which results in the rapidity divergence. The rapidity divergence in the collinear sector comes from the zero-bin subtraction, which ensures the cancellation of the rapidity divergences from the soft and collinear sectors. In order to treat the rapidity divergence, we construct the rapidity regulators consistently for all the modes. They are generalized by assigning independent rapidity scales for different collinear directions. The soft regulator incorporates the correct directional dependence when the innate collinear directions are not back-to-back, which is discussed in the N-jet operator. As an application, we consider the Sudakov form factor for the back-to-back collinear current and the soft-collinear current, where the soft rapidity regulator for a soft quark is developed. We extend the analysis to the boosted heavy quark sector and exploit the delicacy with the presence of the heavy quark mass. We present the resummed results of large logarithms in the form factors for various currents with the light and the heavy quarks, employing the renormalization group evolution on the renormalization and the rapidity scales.
Highlights
Effective field theories enable us to understand important physics by extracting relevant ingredients and disregarding the unnecessary remainder
The structure of the paper is as follows: in section 2, we discuss the idea of applying an effective theory to a system with the collinear and soft modes, and explain that the rapidity divergence in the collinear part comes from the matching procedure that corresponds to the zero-bin subtraction
We have presented a new perspective regarding the origin of the rapidity divergence in Soft-collinear effective theory (SCET) and its consistent treatment
Summary
Effective field theories enable us to understand important physics by extracting relevant ingredients and disregarding the unnecessary remainder. The soft-collinear factorization with the zero-bin subtraction is identified as the matching of the full SCET onto the soft part, and the collinear part can be considered as the matching coefficient describing the large-rapidity region. Since the origin of the rapidity divergence is soft dynamics, the divergence in the heavy quark sector comes from the zero-bin subtraction. This is another new observation we make. The structure of the paper is as follows: in section 2, we discuss the idea of applying an effective theory to a system with the collinear and soft modes, and explain that the rapidity divergence in the collinear part comes from the matching procedure that corresponds to the zero-bin subtraction. In appendix B, we show the details of the calculations for extracting the rapidity divergences in the boosted heavy quark sector
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