An effective field theory for forward scattering and factorization violation

Abstract

Starting with QCD, we derive an effective field theory description for forward scattering and factorization violation as part of the soft-collinear effective field theory (SCET) for high energy scattering. These phenomena are mediated by long distance Glauber gluon exchanges, which are static in time, localized in the longitudinal distance, and act as a kernel for forward scattering where |t| s. In hard scattering, Glauber gluons can induce corrections which invalidate factorization. With SCET, Glauber exchange graphs can be calculated explicitly, and are distinct from graphs involving soft, collinear, or ultrasoft gluons. We derive a complete basis of operators which describe the leading power effects of Glauber exchange. Key ingredients include regulating light-cone rapidity singularities and subtractions which prevent double counting. Our results include a novel all orders gauge invariant pure glue soft operator which appears between two collinear rapidity sectors. The 1-gluon Feynman rule for the soft operator coincides with the Lipatov vertex, but it also contributes to emissions with ≥ 2 soft gluons. Our Glauber operator basis is derived using tree level and one-loop matching calculations from full QCD to both SCET II and SCET I . The one-loop amplitude's rapidity renormalization involves mixing of color octet operators and yields gluon Reggeization at the amplitude level. The rapidity renormalization group equation for the leading soft and collinear functions in the forward scattering cross section are each given by the BFKL equation. Various properties of Glauber gluon exchange in the context of both forward scattering and hard scattering factorization are described. For example, we derive an explicit rule for when eikonalization is valid, and provide a direct connection to the picture of multiple Wilson lines crossing a shockwave. In hard scattering operators Glauber subtractions for soft and collinear loop diagrams ensure that we are not sensitive to the directions for soft and collinear Wilson lines. Conversely, certain Glauber interactions can be absorbed into these soft and collinear Wilson lines by taking them to be in specific directions. We also discuss criteria for factorization violation.

A complete basis of helicity operators for subleading factorization

“…The soft and collinear dynamics are entirely described by the Lagrangian of the effective theory, and the subleading power SCET Lagrangian is also required at the same order in the power expansion. At leading power, the BPS 1 This assumes that leading power Lagrangian interactions that can couple soft and collinear modes through Glauber exchange operators [79] that involve 1/P 2 ⊥ potential can be ignored at the active parton level. It is known that this is the case for the full e + e − →2-jet event shapes at leading power and for inclusive Drell-Yan at leading power [3], and that this is not the case for spectator effects and O(α 4 s ) perturbative corrections in certain Drell-Yan event shapes [80,81].…”

“…The only exception to this is the leading power Glauber Lagrangian [79] which couples together soft and collinear fields, and which will violate factorization if it can not be shown to give canceling contributions or that it is irrelevant. Power suppressed Lagrangians have been analyzed in the literature [82][83][84][85][86][87], and the SCET I Lagrangian is currently known to O(λ 2 ) [87] (excluding power suppressed Glauber exchange operators).…”

“…The Lagrangians L (i) hard contain the hard scattering operators O (i) , whose structure is determined by the matching process, as described in section 5. The leading power Glauber Lagrangian [79], L (0) G , describes leading power interactions between soft and collinear modes in the form of potentials. It breaks factorization unless it can be shown to cancel out or absorbed into other interactions such as Wilson line directions.…”

A subleading operator basis and matching for gg → H

“…1 The dependence on the underlying hard 1 By referring to active-parton factorization we imply that this formula ignores contributions from proton spectator interactions [22] that occur through the Glauber Lagrangian of ref. [23]. There are also perturbative corrections at O(α 4 s ) that are described by a single function Bgg in place of BgBg [23,24].…”

“…[23]. There are also perturbative corrections at O(α 4 s ) that are described by a single function Bgg in place of BgBg [23,24].…”