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Constraint equations for heavy-to-light currents in SCET

Christian M. Arnesen, Joydip Kundu, Iain W. Stewart

TL;DR

This paper constructs a complete NNLO (O(lambda^2)) basis for heavy-to-light currents within SCET and derives comprehensive reparameterization-invariance (RPI) constraints to relate Wilson coefficients across operator orders.By formulating constraint equations for multiple RPI transformations, including projected (STAR/DOLLAR) and SCET/HQET invariances, the authors determine the allowed Dirac structures and express many NNLO coefficients in terms of LO/NLO inputs.The study shows projected RPI on the v_perp=0 surface is complete for these currents up to NNLO and demonstrates the absence of supplementary projected operators, while also addressing boundary-condition independence of the ultrasoft Wilson line Y.In addition to building the operator basis, the work provides tree-level and one-loop conversions between bases and coefficients, enabling cross-checks with established results and facilitating future higher-order extensions.

Abstract

A complete basis for the next-to-next-to leading order heavy-to-light currents in the soft-collinear effective theory is constructed. Reparameterization invariance is imposed by deriving constraint equations. Their solutions give the set of allowed Dirac structures as well as relations between the Wilson coefficients of operators that appear at different orders in the power expansion. The completeness of reparameterization invariance constraints derived on a projected surface is investigated. We also discuss the universality of the ultrasoft Wilson line with boundary conditions.

Constraint equations for heavy-to-light currents in SCET

TL;DR

This paper constructs a complete NNLO (O(lambda^2)) basis for heavy-to-light currents within SCET and derives comprehensive reparameterization-invariance (RPI) constraints to relate Wilson coefficients across operator orders.By formulating constraint equations for multiple RPI transformations, including projected (STAR/DOLLAR) and SCET/HQET invariances, the authors determine the allowed Dirac structures and express many NNLO coefficients in terms of LO/NLO inputs.The study shows projected RPI on the v_perp=0 surface is complete for these currents up to NNLO and demonstrates the absence of supplementary projected operators, while also addressing boundary-condition independence of the ultrasoft Wilson line Y.In addition to building the operator basis, the work provides tree-level and one-loop conversions between bases and coefficients, enabling cross-checks with established results and facilitating future higher-order extensions.

Abstract

A complete basis for the next-to-next-to leading order heavy-to-light currents in the soft-collinear effective theory is constructed. Reparameterization invariance is imposed by deriving constraint equations. Their solutions give the set of allowed Dirac structures as well as relations between the Wilson coefficients of operators that appear at different orders in the power expansion. The completeness of reparameterization invariance constraints derived on a projected surface is investigated. We also discuss the universality of the ultrasoft Wilson line with boundary conditions.

Paper Structure

This paper contains 23 sections, 122 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ingredients from SCET
  3. Degrees of freedom, power counting, gauge invariance, and Wilson lines
  4. Comments on boundary conditions for $Y(x)$
  5. Reparameterization invariance
  6. Completeness of Projected RPI
  7. Heavy-to-Light currents to ${\mathcal{O}}(\lambda^2)$
  8. Current field structures at ${\mathcal{O}}(\lambda^2)$
  9. Constraint equations from reparameterization invariance
  10. RPI-$\star$ at ${\mathcal{O}}(\lambda)$
  11. RPI-$\$$ at ${\mathcal{O}}(\lambda^2)$
  12. SCET RPI-a at ${\mathcal{O}}(\lambda^2)$
  13. SCET RPI-$\star$ at ${\mathcal{O}}(\lambda^2)$
  14. Solutions to the constraint equations
  15. Solutions for scalar and pseudoscalar currents at ${\mathcal{O}}(\lambda^2)$
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Eikonal $i\epsilon$ prescriptions for incoming/outgoing quarks and antiquarks and the result that reproduces this with an ultrasoft Wilson line and sterile quark field.
  • Figure 2: Transformation of operators on and off the $v_\perp=0$ surface. Here ${\mathrm O}_{1,2}$ exist for $v_\perp=0$, while $\overline {\mathrm O}_{3,4,5}$ vanish on the $v_\perp=0$ surface.