Soft-Collinear Effective Theory: BRST Formulation

We provide a systematic BRST formalism for the soft-collinear effective theory describing interactions of soft and collinear degrees of freedom in the presence of a hard interaction. In particular, we develop full BRST symmetry transformation for SCET theory. We further extend the BRST formulation by making the transformation field dependent. This establishes a mapping between several SCET actions consistently when defined in different gauge conditions. In fact, a definite structure of gauge-fixed actions corresponding to any particular gauge condition can be generated for SCET theory using our formulation.


Introduction
Effective field theories are used to separate the the contributions associated with different scales, a high-energy and a low-energy scales, of Quantum Chromodynamics (QCD). Over the past two decades, soft-collinear effective field theory (SCET) [1][2][3] has become amongst one of the important theories describing low-energy effective field theories of the Standard Model. In QCD, the low-energy part is nonperturbative in particular. In order to derive the factorization theorems and to perform the resummation of Sudakov logarithms, SCET provides an alternative to the traditional diagrammatic techniques [4]. SCET has been applied to a large variety of processes, from B-meson decays to jet production at the Large Hadron Collider (LHC). In Ref. [3], the factorization of soft and ultrasoft gluons from collinear particles is shown at the level of operators.
In order to describe jet-like events of QCD in SCET, it is convenient to write fields in either collinear, anti-collinear or soft (low energetic) modes with the help of the light cone unit vectors satisfying n 2 =n 2 = 0 and n ·n = 2. A momentum in the light-cone basis is represented as where ⊥ components are orthogonal to both collinear unit vector n and anti-collinear unit vector n.
The gauge symmetry structure in SCET is richer than the QCD as the former involves more than one distinct gluon fields. Therefore, the idea of background fields is required to give well defined meaning to several distinct gluon fields [5]. Based on momentum regions, SCET is categorized in two formulations: SCET I and SCET II. SCET I and SCET II scale soft sector of the theory differently. For instance, in SCET I all the momentum components of the soft fields are scaled similar to the small component of the collinear fields, while in SCET-II the momentum components of soft fields are scaled like the transverse component of the collinear fields.
The celebrated Becchi-Rouet-Stora-Tyutin (BRST) formulation is a comparatively rigorous mathematical scheme [6][7][8] which provides a powerful technique to quantise gauge field theories. The range of applicability of BRST formulation further enhanced by extending it, where the anti-commuting transformation parameter is made finite and field-dependent [9]. The finite field-dependent BRST transformations have been discussed successfully in many field theoretic systems with gauge symmetries and have been found many applications [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Although BRST formulation simplifies the renormalizability greatly and helps to show unitarity of many theories, the implementation of this approach in SCET is quite cumbersome task. Thus even though a full field theoretic description for hadronic processes is developed, the BRST formulation for SCET is not studied so far. This provides us with an opportunity to bridge this gap.
In this paper, we consider a gauge invariant SCET I action which admits different sets of gauge invariance in different momentum regions. We develop two sets of BRST symmetries which leave the Faddeev-Popov actions for collinear and ultrasoft sectors, separately. Moreover, we formulate an extended version of BRST symmetries by making the transformation parameter field dependent. We call such transformation field-dependent BRST (FDBRST) transformation. In contrast to the standard case, this eventually leads to a non-trivial Jacobian for functional measure in the expression of transition amplitude. This Jacobian extends the BRST-exact parts of the action. We show that for some appropriate choices of field-dependent parameters an exact form of gauge-fixed action corresponding to different gauge condition can be generated through FDBRST.
The plan of the paper is as following. In section 2, we construct fermionic rigid collinear and ultrasoft BRST transformations. These symmetry transformations are further generalized by making he transformation parameters field dependent in a traditional way in section 3. Moreover, we implement such FDBRST transformations with appropriately constructed transformation parameters to the generating functional. We summarize the outcome of this formulation with their significance in the section 4.

SCET I action and BRST symmetry
In the Ref. [25], it has been shown that the leading-order SCET collinear quark action should satisfy following requirement: (a) it should yield proper spin structure of the collinear propagator, (b) it should have both collinear quarks and collinear antiquarks, (c) it should interact with both collinear gluons and ultrasoft gluons, (d) and it should lead to the correct low-order propagator for different situations. These requirements allow us to write down the effective leading-order SCET action. Further by splitting the fermion field into big and small components using the usual projectors ( / n/ n 4 and / n/ n 4 ) and eliminating (using the equations of motion) the small components, one can write first the leading-order collinear quark action with collinear modes in n direction as [25] where P µ is a label operator which provides a definite power counting for derivatives and the collinear covariant derivatives are defined as Even in the presence of ultrasoft fields, one can write collinear quark action equivalent to (2.1) as where spinor components ϕn are subleading in the collinear limit and In order to write the collinear gluon action, ultrasoft gauge field A µ us is treated as a background field with respect to collinear gauge field A n µ . In this way, the QCD gluon action leads to the leading-order collinear gluon action in a covariant gauge as follows [25] where τ is a gauge fixing parameter for collinear gluon and The lowest-order Faddeev-Popov action for ultrasoft quarks and ultrasoft gluons is a covariant gauge can be written by [25] S us = d 4 x ψ us i / D us ψ us − Tr 1 2 G us µν G µν us + τ us (∂ µ A µ us ) 2 + 2c us ∂ µ D µ us c us , where τ us is a gauge fixing parameter for ultrasoft gluon and iD µ us = i∂ µ + A µ us .
The complete Faddeev-Popov effective action for a single set of quark and gluon collinear modes in the n direction, and quark and gluon ultrasoft modes in a covariant gauge is given by S scet = S nξ + S ng + S us . (2.8) We construct the following collinear and ultrasoft BRST transformations, (a) collinear BRST: (b) ultrasoft BRST:

General setup
To construct FDBRST we define a generic notation for the BRST transformations in Eqs. (2.9) and (2.10) for a collective field (having both collinear and ultrasoft fields) φ(x) as follows: where s b φ is a Slavnov variation and Λ is an global infinitesimal anticommuting parameter. Following the standard procedure [9], a field-dependent BRST transformation is constructed via interpolation of a continuous parameter κ (0 ≤ κ ≤ 1) as: where Θ ′ [φ(κ)] is an infinitesimal field-dependent parameter. In contrast to standard BRST transformation, this field-dependent transformation is not the symmetry of the path integral measure and amounts a precise Jacobian in the generating functional. This Jacobian contribution can be expressed as exponential of some functional of local fields and modifies the BRST exact part of the action [9]. The Jacobian of functional measure is given by [26,27] This Jacobian therefore extrapolates the action (within functional integration) of the SCET theory (2.11) as follows: This modified expression due to FDBRST does not amount any changes in the physical content of the theory but rather simplifies various issues in a dramatic way. In the next subsection we are going to demonstrate this.

Collinear FDBRST transformation
Following above methodology, we construct the infinitesimal collinear FDBRST transformations as where Θ ′ n is an infinitesimal collinear field-dependent transformation parameter. This parameter can be chosen arbitrarily provided that must be nilpotent in nature. In the next section we will construct appropriate Θ ′ n to show how the generating functionals corresponding to various effective actions in different gauges are related.

Ultrasoft FDBRST transformation
In the similar fashion the infinitesimal ultrasoft FDBRST transformations are derived as where Θ ′ us is an arbitrary infinitesimal ultrasoft field-dependent transformation parameter.

Implementation of FDBRST transformation
In this subsection, we assign some specific values for the field dependent parameters Θ ′ n and Θ ′ us and calculate Jacobians of functional measures under respective FDBRST transformations. In this regard, we first choose the parameter of collinear FDBRST transformation Θ ′ n as Here, we try to emphasize that we utilize an appropriate BRST transformation for the antighost fields according to gauge conditions. On the other hand the field dependent parameter Θ ′ us for ultrasoft BRST is chosen as where final effective action is defined by These are nothing but the leading-order collinear gluon action with gauge-fixing condition f 1 [A nµ , A µ us ] = 0 and leading-order action for ultrasoft quarks and ultrasoft gluons with gaugefixing condition f 2 [A µ us ] = 0. Thus, FDBRST transformation upon implementation on generating functional changes the gauge-fixing and ghost sectors of collinear and ultrasoft gluon actions. This result will be very useful in handling the Feynman processes of the theory. Since calculations of different Green's functions depend on the choice of gauge-fixing condition and for some particular choices the calculations are simplified greatly, the structure of Faddeev-Popov action corresponding to that gauge can be achieved easily from this FDBRST formulation.

Applications and conclusions
In this paper, we have considered an effective theory in light-cone coordinates which describes the interactions of soft and collinear modes in the presence of a hard interaction. By eliminating the small components after decomposition of fermion field, we have written a gauge invariant SCET I action which admits different sets of gauge invariance in different momentum regions. In order to quantize correctly, we need to extend classical action by adding suitable terms which break the local gauge invariance. Such gauge variant terms attribute ghost terms in the generating functional of the theory. We have developed two independent sets of BRST symmetries which leave the Faddeev-Popov actions for collinear and ultrasoft sectors invariant separately. These BRST transformations may help to write the counter terms to make the theory renormalizable. Furthermore, we have extended these sets of BRST symmetries by making the transformation parameter field dependent. The difference of these extended symmetries to the usual one lies to the fact that these are not symmetry of the functional measure and, in contrast to the usual one, eventually lead to a local Jacobian. On the physical ground, this Jacobian do not modify the theory as all the changes attributed to the BRST-exact parts of the action. We have shown that for some specific choices of field-dependent parameters the exact expressions for various gauge-fixed actions can suitably be derived. These results are of particular importance for the theoretical estimation of decay processes. This is because the certain diagram calculations get simplified greatly in some particular gauge choices. For instance, it has been shown that by extending SCET formulation to the class of singular gauges, a new Wilson line, the T Wilson line, has to be invoked as a basic SCET building block [28,29]. It is shown there that study in non-covariant gauges extend the range of applicability of SCET. The transition from one gauge to another in SCET can easily be done through our approach.