Gauge invariance of radiative jet functions in the position-space formulation of SCET

In subleading powers of soft-collinear effective theory (SCET), the Lagrangian contains couplings between soft quarks and hard-collinear quarks. Matrix elements of the hard-collinear parts of these couplings are radiative jet functions. In the position-space formulation of SCET, the Lagrangians are constructed from operators that appear to be gauge invariant. Nevertheless, we find violations of gauge invariance arise in the hard-collinear sector because gauge transformations can shift the momentum of a hard-collinear quark field from the hard-collinear sector to the soft sector, where the hard-collinear fields, by definition, have no support. The violations of gauge invariance are manifested in perturbation theory in the hard-collinear sector through the absence of certain Feynman diagrams that would be present in full QCD. A consequence of the absence of these diagrams is that the radiative jet functions that follow directly from the position-space Lagrangians are not gauge invariant, and we demonstrate this through explicit calculations in lower-order perturbation theory. We obtain gauge-invariant Lagrangians by adding to existing position-space Lagrangians terms that are proportional to the soft-quark equation of motion. These gauge-invariant Lagrangians are valid for nonzero, as well as zero, quark masses. We also remark briefly on the gauge invariance of certain Lagrangians that have been constructed in the label-momentum formulation of SCET.


I. INTRODUCTION
In soft-collinear effective theory (SCET) [1][2][3][4][5], couplings between a quark that carries a soft momentum and a quark that carries a hard-collinear momentum appear in subleading powers of the SCET expansion parameter λ [6][7][8][9].Matrix elements that contain the hardcollinear parts of these couplings are called "radiative jet functions," and they appear in factorization theorems for exclusive processes at subleading power in λ. (See, for example, Refs.[10][11][12].) In general, radiative jet functions are written in terms of operators in which hard-collinear quark and antiquark fields are accompanied by Wilson-line factors, and all derivatives are covariant derivatives.We call such operators "ostensibly gauge-invariant operators."If one replaces the hard-collinear quark and antiquark fields with full QCD fields multiplied by collinear projectors, then the ostensibly gauge-invariant operators are truly gauge invariant.
However, in SCET, the hard-collinear quark and antiquark fields must carry hard-collinear momenta, not soft momenta. 1 This requirement can lead to violations of hard-collinear gauge invariance because hard-collinear gauge transformations multiply quark fields by a phase that, in momentum space, can shift the quark-field momentum from the hard-collinear region to the soft region, where the hard-collinear fields have no support.As we will see, this phenomenon is manifested diagrammatically by the absence of certain Feynman diagrams in the hard-collinear sector that would be present in full QCD.These "missing diagrams" can lead to violations of gauge invariance in the hard-collinear sector.
We carry out an analysis of the Lagrangians that appear in the position-space formulation of SCET in Refs.[6,7], which we refer to as Beneke-Chapovsky-Diehl-Feldmann (BCDF). 2   We demonstrate, through the use of Ward identities, that the BCDF Lagrangians lead to violations of gauge invariance in the hard-collinear sector.The violations occur in order λ 2 , but not in order λ 1 .We find that the violations of gauge invariance can be removed by making use of the soft-quark equation of motion.Therefore, the BCDF Lagrangians lead to gauge-invariant S-matrix elements [13].However, off-shell quantities, such as radiative jet functions can be gauge noninvariant.We use the Lagrangians in Refs.[6,7] directly 1 The requirement that hard-collinear quark and gluon fields carry hard-collinear momenta is essential in working out SCET power counting and in achieving a factorization of the hard-collinear sector of the effective theory from the other sectors at the Lagrangian level.The assumption that the hard-collinear quark and gluon fields carry hard-collinear momenta is used explicitly in Refs.[6,7] in constructing the Lagrangians in those papers. 2The Lagrangians in Eq. (A.1) of Ref. [12] are the BCDF Lagrangians, but expressed in terms of gaugeinvariant building blocks.In this paper, we carry out our analyses in terms of the original BCDF forms of the Lagrangians.
to construct radiative jet functions.That is, we define the radiative jet functions as timeordered matrix elements of the hard-collinear-operator factors in the Lagrangians.We find, through explicit calculations at the lowest nontrivial order in perturbation theory, that the resulting radiative jet functions are not gauge invariant.
We modify the BCDF Lagrangians to obtain gauge-invariant Lagrangians that describe the couplings of a soft quark to a hard-collinear quark by applying the soft-quark equation of motion and by making use of the Bauer-Pirjol-Stewart (BPS) field redefinition in Ref. [4] to factor minus-polarized soft gluons from the hard-collinear subdiagram.We use the concept of missing diagrams to argue that the modified order-λ 2 Lagrangians, as well as the order-λ 1 Lagrangian, are gauge invariant to all orders in perturbation theory.We also demonstrate the gauge invariance by using the modified order-λ 2 Lagrangians to construct radiative jet functions and computing the radiative jet functions in the Feynman gauge and the light-cone gauge at the lowest nontrivial order in perturbation theory.
In the label-momentum formulation of SCET [2], the Lagrangians that describe the interactions of soft quarks with hard-collinear quarks are also constructed from ostensibly gauge-invariant operators [3].We find that the label-momentum Lagrangians in Refs.[8,9] evade the gauge invariance issue that we identify in this paper and that the corresponding operators are truly gauge invariant.
The remainder of this paper is organized as follows.In Sec.II, we establish the notations and conventions that we use throughout this paper.In Sec.III, we present the Lagrangians of Refs.[6,7], discuss the associated Ward identities in lowest-order perturbation theory, and identify the sources of violations of gauge invariance as the "missing diagrams."In Sec.IV, we use the BCDF Lagrangians to compute radiative jet functions in the Feynman gauge and in the light-cone gauge at lowest order in perturbation theory, and we show that these two gauges give different results, verifying the violation of gauge invariance.In Sec.V we modify the BCDF Lagrangians at relative order λ 2 to obtain gauge-invariant Lagrangians that connect a soft quark to a hard-collinear quark.In Sec.VI, we argue that the order-λ 1 Lagrangian and the modified order-λ 2 Lagrangians are gauge invariant to all orders in perturbation theory.We construct radiative jet functions that follow from the modified gauge-invariant Lagrangians in Sec.VII, and we calculate these radiative jet functions in the Feynman gauge and in the light-cone gauge in lowest-order perturbation theory, verifying that the radiative jet functions are invariant with respect to these gauge choices.In Sec.VIII, we observe that certain versions of the label-momentum formulation of SCET evade the gauge-invariance problem that we identify in this paper.Finally, we summarize and discuss our results in Sec.IX.

II. PRELIMINARIES
In this section, we establish the notations and conventions that we use throughout this paper.
We decompose an arbitrary vector in terms of the two lightlike vectors, n and n, as follows: where with n 2 = n2 = 0 and n • n = 2. n and n are the lightlike unit vectors along the z axis: The perpendicular momentum Here, and throughout this paper, we use the notation where r ⊥ is a (D − 2)-dimensional Euclidean vector.
It is convenient to consider the case of a quark with mass m ≫ Λ QCD .Then, a soft momentum r s on the quark line has the scaling behavior where and Q is the hard scale of the process.A collinear momentum r c on the quark line along the n direction has the scaling behavior Since a radiative jet function carries a soft momentum combined with a collinear momentum, the resulting hard-collinear momentum r hc , taken to be along the n direction, has the scaling Note that the hard-collinear and soft momenta have different virtualities in λ: r 2 hc ∼ Q 2 λ 2 , and r 2 s ∼ Q 2 λ4 .The n-hard-collinear Dirac field ψ n can be decomposed into large-and small-component collinear fields by applying collinear projectors P n and P n onto ψ n : where We make use of the following additional notations for SCET fields: q s is a soft-quark field, G µ n is an n-hard-collinear-gluon field, and G µ s is a soft-gluon field.q s , η n and ξ n have scaling of order λ3 , λ 2 and λ, respectively.Each component of the field G µ n has the same scaling as the n-hard-collinear momentum in Eq. ( 8), and each component of the field G µ s has the same scaling as the soft momentum in Eq. (5).g s = √ 4πα s is the strong coupling.
We define the n-hard-collinear Wilson line as and we define the covariant derivatives as

III. WARD IDENTITIES OF THE BCDF SCET LAGRANGIANS A. BCDF Lagrangians
The effective Lagrangians that describe an interaction between a soft quark q s and an n-hard-collinear quark ξ n in the SCET formulation Refs.[6,7] are given by where H.c. denotes the Hermitian conjugate, and the subscripts 1 and 2 indicate the order in λ of these expressions.These Lagrangians generalize slightly those in Refs.[6,7], in that they contain a nonzero quark mass.However, we refer to them as the BCDF Lagrangians.
In deriving these expressions, we have started in full QCD, with a quark with mass m, and we have followed the steps that are given in Refs.[6,7] for the massless case.The detailed derivation of L BCDF 2m is given in Appendix A.
References [6,7] also contain the ξn . . .ξ n , qs . . .q s , and pure gauge-field SCET Lagrangians.The modifications of these Lagrangians for the massive case at orders λ 0 , λ 1 , and λ 2 are also shown in Appendix A. In this paper, we do not use these Lagrangians explicitly.
Instead, we employ an equivalent, but simpler, procedure: we replace the fields ξ n and ξn with hard-collinear Dirac fields, using the expressions in Eq. ( 9); we use full-QCD Feynman rules, with the projectors P n and P n; and we expand to the desired order in λ.
The power counting in λ in Eq. ( 13) follows from the fact that, if we integrate the interaction Lagrangians in Eq. ( 13) over d 4 x, then the integration region is of order λ −4 .In , the factor ix ⊥ρ should be counted as O(λ −1 ) because it scales as the inverse of the transverse momentum that flows into the hard-collinear subdiagram.
Note that, in Eq. ( 13), the soft-quark field q s depends only on x + .That is, the soft-quark field has been multipole expanded in the minus and transverse components of its argument in order to obtain a definite scaling in λ [6].Square brackets indicate that a derivative acts only inside the brackets and that soft fields are evaluated at x + after the derivative is taken.
The factor ix ⊥ρ becomes, in momentum space, a derivative with respect to the transverse component of the soft momentum.In momentum space, the multipole expansion of q s implies that the plus and transverse components of the soft momentum are ultimately set to zero, but only after derivatives with respect to the soft momentum have been taken.
At this stage, minus-polarized soft gluons still attach to the hard-collinear subdiagram.
These attachments can be factored into Wilson lines by making use of the Grammer-Yennie approximation [14], followed by the application of perturbative Ward identities [15].Equivalently, the decoupling can be achieved by making use of the BPS collinear-field redefinitions [4]: where S n is the soft Wilson line, which is defined by The net effect of the BPS field redefinitions is to make the simple replacements qs (x + ) → (13).Note that the arguments of the soft Wilson lines have been multipole expanded to lowest order in λ, and, so, it is still true that only the minus component of the soft momentum enters the momentum-space expressions that derive from the BPS-transformed Lagrangians.

B. Gauge invariance and Ward identities
At first sight, the hard-collinear parts of the interactions in Eq. ( 13) would appear to be gauge invariant with respect to gauge transformation of the hard-collinear fields.We can use Eq.(9a) to replace ξ n with ψ n .Then, if we could replace ψ n with the ordinary Dirac field ψ, the hard-collinear parts of the interactions in Eq. ( 13) would be gauge-invariant full-QCD operators.However, as we will see, the restriction that ψ n (and ξ n ) carry n-hard-collinear momenta implies that certain full-QCD diagrams are missing in hard-collinear functions involving these operators and that, consequently, the hard-collinear functions are not gauge invariant.
In order to investigate the gauge invariance of the O(λ 2 ) SCET Lagrangians in Eq. ( 13 , respectively.The superscripts (0) and ( 1) denote the order-g 0 s and g 1 s contributions of the crossed circle, respectively.
FIG. 3. The Feynman rules for the crossed circles in Fig. 1 for the Lagrangians in Eq. (13c).
V 2b describes the crossed-circle contribution that arises from L BCDF 2b . The superscripts (0) and ( 1) denote the order-g 0 s and order-g 1 s contributions of the crossed circle, respectively.The definition of the operator ∆ ℓ ⊥ ρ is given in Eq. ( 17).
circles can appear only in conjunction with one or more hard-collinear gluons that attach to the Wilson line.Otherwise, hard-collinear momentum would not be conserved at the crossed circles.In deriving these Feynman rules, we make use of the identity and we define the operator which picks up the coefficient of ℓ ρ ⊥ .
The amplitudes for , and L BCDF 2m , which correspond to the diagram in Fig. 1, are respectively.Here, ϵ µ (k) is the polarization of a hard-collinear gluon with momentum k.
Note that, in A 1 , A 2a , and A 2m , p + = k + and p ⊥ = k ⊥ , owing to the multipole expansion of the soft-quark field.In A 2b , p + = k + , but p ⊥ is set equal to k ⊥ only after differentiation with respect to ℓ ⊥ , in accordance with Eq. ( 17).
We can work out the Ward identities for these amplitudes by carrying out a gauge transformation on the gluon field, which, at the lowest order in g s , simply shifts the gluon polarization vector ϵ µ (k) by an amount that is proportional to the gluon momentum k: We drop the constant of proportionality β in subsequent discussions.The gauge term k µ gives the following contributions to the amplitudes A 1 , A 2a , and A 2m in Eq. ( 18): where we have used k ⊥ = p ⊥ , which follows from the multipole expansion of the soft-quark field q s , and The gauge term k µ gives the following contribution to the amplitude A 2b in Eq. ( 18): where we have used k ⊥ = p ⊥ − ℓ ⊥ , owing to the insertion of the ℓ ⊥ into the hard-collinear subdiagram, which follows from the identity in Eq. ( 16).Note that the contributions in which ∆ ℓ ⊥ ρ acts on the ellipsis (the remainder of the diagram) cancel between the first and second terms after the first equality in Eq. (20d).
We see that the order-λ 1 Lagrangian in Eq. ( 13) gives a vanishing contribution to the Ward identities.That is, it is gauge invariant.However, each of the order-λ 2 Lagrangians in Eq. ( 13) produces a nonzero contribution to the Ward identity, i.e., a violation of gauge invariance, and we find that where we have used / nP n = 0. We note that the violations of gauge invariance are proportional to soft-quark free equation of motion.This suggests that we can obtain a gaugeinvariant form of the Lagrangian by discarding terms that are proportional to the soft-quark equation of motion.In Sec.V, we will see that this is the case. 5 complete factorization formula, including both the radiative jet (hard-collinear) function and the soft function, must be gauge invariant because it reproduces full QCD to a given accuracy in λ.Therefore, we expect gauge invariance to be restored if we consider the soft function in conjunction with the radiative jet function.As we have seen, the order-λ 2 contributions to the radiative jet function that violate gauge invariance are proportional to the inverse of the soft-quark propagator − / ℓ − m.At the lowest order in g s , the soft function is just the soft-quark propagator.Hence, soft function is canceled by the gauge-invariance violating contributions.Consequently, as we can see from the result for the radiative jet functions in Secs.IV and VII, all of the poles in the ℓ − complex plane are in the upper half plane.We can then close the ℓ − contour of integration in the lower half plane to obtain a vanishing result for the gauge-invariance-violating contribution.As we have mentioned, in Sec.V, we will use the formal procedure of discarding Lagrangian terms that are proportional to the soft-quark equation of motion in order to implement gauge invariance in the radiative jet function at the integrand level.The phenomenon that we see here, namely, the vanishing of contributions that are proportional to the soft-quark equation of motion, demonstrates the correctness of the formal procedure in lowest-order perturbation theory.

C. Missing diagrams
The violations of gauge invariance that we have noted arise because certain diagrams that would be present in full QCD are missing from the hard-collinear function because they contain a soft-quark propagator.The diagram of this type that appears in order g s is shown on the left side of Fig. 4. Note that, because we are considering diagrams that contain a soft-quark propagator, momentum conservation no longer requires that the orderg 0 s factors from the crossed circles appear in conjunction with hard-collinear gluons that attach to the Wilson line.
The amplitudes for the diagram in Fig. 4 that correspond to the crossed vertices from respectively.Here, in keeping with the multipole expansion of the soft-quark field q s in the Lagrangians in Eq. ( 13), we have discarded terms in the hard-collinear part that are proportional to ℓ ⊥ , except in A 2b,miss .In the case of A 2b,miss , we set ℓ ⊥ to zero only after the derivative in ∆ ℓ ⊥ ρ has been taken.After we carry out the gauge transformation in Eq. ( 19), we obtain the following contributions of the gauge term k µ to Eqs. (22a):  and 4). 6The second term in the square brackets in each of the contributions above cancels the corresponding gauge terms in Eqs.(20b), (20c), and (20d).This cancellation confirms our assertion that the violations of gauge invariance arise because of missing diagrams involving soft-quark propagators.Note that the contribution in Eq. (23a) that arises from the order-λ 1 Lagrangian vanishes, in keeping with the gauge invariance of that Lagrangian.

IV. RADIATIVE JET FUNCTION IN ORDER α s FROM THE BCDF LAGRANGIANS
Now let us test the gauge invariance of radiative jet functions that are derived from the BCDF Lagrangians.From the BCDF Lagrangians at O(λ 2 ) in Eq. ( 13), we can construct the following radiative jet functions: in Eq. ( 13), respectively, while the second factors of the SCET operators, namely, ξn W n , arise from the hard-collinear part of the hard-to-hard-collinear transition operator in the amplitude for a hard-scattering process.The couplings of minus-polarized soft-gluon fields to hard-collinear fields have been removed by making use of the BPS field redefinitions [Eq.( 14)].This has the effect of replacing D in A(ℓ − ) with D n .We have taken the vacuum to Q Q matrix elements of these operators, where Q and Q are massive, on-shell quark states.
These matrix elements are convenient because, for them, the difficulty with gauge invariance appears at the Born level, rather than at the loop level.For definiteness, we take the Q and the Q to be in a spin-triplet, color-singlet state with zero relative momentum between the Q and the Q. (This is an S-wave state.)Then, we can take both the Q and the Q to have momentum p.We can, without loss of generality, choose a frame in which p ⊥ = 0. Note that, because the momentum p is on the mass shell, it satisfies collinear scaling, rather than hard-collinear scaling: Note also that the Ward-identity arguments of Sec.III, which were presented for the case of hard-collinear scaling, are also valid for the case of collinear scaling.
We realize the Q Q color and spin states through the application of the standard spin and color projection operators, whose product is given by where ϵ * is the polarization vector of the external 3 S 2m = 0 (Fig. 2).
Then, a straightforward calculation in the Feynman gauge yields the following Born-level radiative jet functions: where we have kept only the leading nonzero power in λ in the last lines of these expressions.
In the case of B ρ(d),Feynman (ℓ − ), we have used which is valid through the power in λ in which we are interested.

B. Light-cone gauge
Next, let us consider the radiative jet functions in the light-cone gauge, which is defined by the gauge condition from which it follows that the gluon-propagator polarization sum is Here, k is the gluon momentum.In the light-cone gauge, the n-hard-collinear Wilson line [defined in Eq. ( 11)] becomes unity, and so the diagrams involving Wilson lines vanish.
(Equivalently, one can see that the polarization sum in Eq. ( 30 where we have kept only the leading nonzero power of λ in the last line of each expression.
Note that 2m = 0 in Fig. 2. By comparing the light-conegauge results in Eq. ( 31) with the Feynman-gauge results in Eq. ( 27), we conclude that all three radiative jet functions in Eq. ( 24) are not gauge invariant.The difference between the light-cone-gauge and the Feynman-gauge calculations is where we have contracted (−ℓ ρ ⊥ ) into B ρ since an additional factor of (−ℓ ρ ⊥ ) is present in the soft function that is associated with B ρ , relative to the soft functions that are associated with A and M .In the last line of Eq. (32), we have inserted −ℓ + / n 2 / n = 0 in order to obtain the factor − / ℓ − m.As is expected from our Ward-identity analysis, the gauge-variant terms are proportional to the inverse of the soft-quark propagator.

C. Covariant gauge
We can also consider the radiative jet functions in a general covariant gauge, in which the gluon polarization sum is given by where α is the gauge parameter.
In order α s in a general covariant gauge, the sum over all connections of the gluon to the quark line and the W n Wilson line (on the right side of a radiative-jet diagrams) is gauge invariant, independently of the rest of the diagram.Hence, the term in the polarization sum that is proportional to α vanishes in the sum over all connections of the gluon to the quark line and the W n Wilson line, and one obtains the Feynman-gauge result.
However, in order α 2 s the situation is more complicated.We have checked that, in the Abelian case, there is a remnant from the terms that are proportional to α in the polarization sum that is nonzero at the integrand level.This suggests that a gauge dependence exists in general covariant gauges in order α 2 s .

V. GAUGE INVARIANT SCET LAGRANGIANS
In this section, we modify the BCDF Lagrangians to obtain gauge-invariant Lagrangians that describe the coupling of a soft quark to a hard-collinear quark.In order to account for the gauge-violating contribution in Eq. ( 21), we introduce the following subtraction Lagrangian: where, in the second line, we have used / nξ n = / nP n ψ n = 0, and performed an integration by parts for the term that is proportional to in • D s .This integration by parts is valid because the minus component of the soft momentum flows into the hard-collinear parts of the Lagrangian.We remind the reader that the position arguments of the soft fields have been multipole expanded (depend only on the plus component of the coordinate), but only after the derivatives acting on the soft fields have been taken.
The modified Lagrangian that describes the coupling of a soft quark to a hard-collinear quark in order λ 2 is where we omit the position arguments for simplicity.Note that the quark-mass-dependent Lagrangian L BCDF 2m is canceled by the mass term of ∆L 2 in Eq. (34), and so the O(λ 2 ) interactions between a soft quark and a hard-collinear quark are independent of the quark mass.
At this stage, longitudinally polarized soft gluons can still interact with the hard-collinear subdiagram.As we have mentioned previously, such interactions can be factored into Wilson lines by making use of the BPS field redefinitions [4], which are shown in Eq. ( 14).Using the BPS field redefinitions and the identities FIG. 6.The Feynman rules for the crossed circles in Fig. 1 for the Lagrangians in Eq. ( 38).
V mod 2a and V mod 2b represent the crossed-circle contributions that arise from L mod,BPS 2a and L mod,BPS 2b , respectively.The superscripts (0) and ( 1) denote the order-g 0 s and order-g 1 s contributions of the crossed circle, respectively.
we rewrite Eq. ( 35) as follows: where Here, in the first term of L mod,BPS 2a , the covariant derivative (−in • ← − D n ) should be understood as acting only on the Wilson line W † n .The Feynman rules for these modified Lagrangians are given in Fig. 6.Again, we note that, owing to the use of the multipole expansion, the soft transverse momentum ℓ ⊥ should be set to zero, except in the terms involving the operator ∆ ℓ ⊥ ρ [Eq.(17)].For those terms, ℓ ⊥ is set to zero after differentiation with respect to ℓ ⊥ .

A. Gauge invariance and Ward identities of the modified Lagrangians
Let us repeat the gauge-invariance analysis of Sec.III B for the modified O(λ 2 ) SCET Lagrangians in Eq. ( 38).The amplitudes for L mod,BPS 2a and L mod,BPS 2b , which correspond to the diagram in Fig. 1, are respectively.We obtain the Ward identities for these amplitudes by replacing the polarization vector ϵ µ (k) with the factor k µ from the gauge transformation in Eq. ( 19).The results are as follows: = 0, where we have used k ⊥ = p ⊥ for A mod,gauge 2a , owing to the multipole expansion of the softquark field, and k ⊥ = p ⊥ − ℓ ⊥ for A mod,gauge 2b , owing to the insertion of ℓ ⊥ that follows from the identity in Eq. (16).Note that the contributions in which ∆ ℓ ⊥ ρ acts on the ellipsis (the remainder of the diagram) cancel between the first and third terms after the first equality in Eq. (40b).We conclude that the modified Lagrangians in Eq. ( 38  respectively.Here, for A mod 2a,miss , we have performed the multipole expansion to set ℓ ⊥ to 0, and for A mod 2b,miss , we have used We see that the modified Lagrangians in Eq. ( 38) give vanishing contributions in order g s to the missing diagram in Fig. 4 and are, therefore, gauge invariant to this order.

VI. GAUGE INVARIANCE TO ALL ORDERS IN g s
We now argue that the gauge invariances of order-λ 1 Lagrangian in Eq. (13a) and the modified order-λ 2 Lagrangians in Eq. ( 38) hold to all orders in g s .First, we note that the all-orders missing diagrams have exactly one soft-quark propagator and that all collinear gluon attachments are to the collinear-subdiagram side of the soft-quark propagator.These diagrams are of the form that is shown in Fig. 7.A crucial feature of these diagrams is that the crossed-vertex factors are independent of the hard-collinear gluon attachments.That rules in Fig. 6, we obtain We note that A In the light-cone gauge, only the diagram of Fig. 5(d) contributes to the radiative jet function.We find that where we have used the results in Eq. (31), since 2b (Figs. 2,  3, and 6).Note that B mod 2ρ,(d),light-cone (ℓ − ) = 0 because, as can be seen from Fig. 6, the relevant order-g s vertex vanishes.
We conclude that the radiative jet functions are identical in the Feynman gauge and the light-cone gauge: As is expected from our Ward-identity analysis [Eq. ( 40)], A mod is separately gauge invariant, while the sum of B mod 1ρ and B mod 2ρ is gauge invariant.

VIII. GAUGE INVARIANCE IN THE LABEL-MOMENTUM FORMULATION OF SCET
Finally, we mention that certain versions of the label-momentum formulation of SCET [8,9] also contain expressions for the interactions between a soft-quark and a hard-collinear quark that are constructed from ostensibly gauge-invariant operators.The final expressions for the soft-quark-to-hard-collinear-quark Lagrangians are given in Eqs.(33) and (35) of Ref. [8] and Eq.(27) of Ref. [9].They are derived by making use of the soft-quark equation of motion, which, as we have seen, is a crucial ingredient in deriving a gauge-invariant Lagrangian that describes the transitions of a soft quark to a hard-collinear quark.
The soft-quark-to-hard-collinear-quark Lagrangians in Eqs.(33) and (35) of Ref. [8] are proportional to the quantities / M and / B c ⊥ , and the soft-quark-to-hard-collinear-quark Lagrangians in Eq. ( 27) of Ref. [9] are proportional to the quantities and n • M and / B c ⊥ .Since these quantities are commutators of covariant derivatives, the corresponding Lagrangians vanish when the gauge fields are set to zero.That is, these Lagrangians must produce at least one gluon emission if they are to give nonvanishing contributions to amplitudes.
Consequently, the missing diagrams (Fig. 7) receive vanishing contributions from these Lagrangians, and these Lagrangians evade the gauge-invariance issue that we have identified in this paper.That is, the corresponding operators are truly gauge invariant.
As a further check, we have used the Lagrangians in Eqs.(33) and (35) of Ref. [8], to compute radiative jet functions in order α s and found agreement with our results for the radiative jet functions in Sec.VII.Here, it was necessary to sum the contributions from the order-λ operator in Eq. (33) and the / M ⊥ order-λ 2 operator in Eq. ( 35) in order to obtain gauge-invariant results, in accordance with the observation in Ref. [9] that collinear gauge transformations of these Lagrangians mix operators that have different scaling in λ.
We have also used the Lagrangians in Eq. ( 27) of Ref. [9] to compute radiative jet functions in order α s .Again, we have found agreement with our results for the radiative jet functions in Sec.VII.In this computation, we used the Feynman rules that are given in the erratum to Ref. [8].In order to obtain all of the contributions to the radiative jet functions of order λ 2 , it was also necessary, in the case of these Lagrangians, to consider the order-λ correction to the gluon propagator. 7We derived the Feynman rule for this correction to the gluon propagator by applying the field redefinitions in Eq. ( 14) of Ref. [9] to the gauge-fixed gauge-field action.

IX. DISCUSSION
In this paper we have pointed out that the Lagrangians in Refs.[6,7] (BCDF) are not gauge invariant in the hard-collinear sector.This is surprising because these Lagrangians are constructed from operators that are ostensibly gauge invariant: hard-collinear fields are accompanied by Wilson-line factors, and all derivatives are covariant derivatives.The violations of gauge invariance are somewhat subtle.They arise because hard-collinear gauge transformations multiply the quark fields by a phase that, in momentum space, can shift the quark-field momentum from the hard-collinear region to the soft region, where the hardcollinear quark field, by definition, has no support.This phenomenon is manifested in perturbation theory in the hard-collinear sector through the absence of certain diagrams that would be present in full QCD.One consequence of the absence of these diagrams is that, if one uses the BCDF Lagrangians directly to construct radiative jet functions, then the radiative jet functions are not gauge invariant by themselves.
We have demonstrated the violations of gauge invariance by examining the Ward identities for the BCDF Lagrangians and also by computing the radiative jet functions that follow directly from the BCDF Lagrangians at the leading order in g s in the Feynman gauge and in the light-cone gauge.These analyses show that the violations of gauge invariance are, at the leading nontrivial order in g s , proportional to inverse of the soft-quark propagator.
Motivated by the Ward-identity and radiative-jet-function analyses, we have modified the BCDF Lagrangians by adding terms that are proportional to the soft-quark equation of motion.Then, after making use of the BPS field redefinition to factor minus-polarized soft gluons from the hard-collinear subdiagram, we have arrived at gauge-invariant Lagrangians, through order λ 2 , that describe the couplings of a soft quark to a hard-collinear quark.
The fact that the violations of gauge invariance can be removed through the use of a field redefinition that is proportional to the soft-quark equation of motion implies that S-matrix elements of the original BCDF Lagrangians are gauge invariant.
The modified gauge-invariant Lagrangians that we have derived are somewhat more general than the BCDF Lagrangians, in that we have considered the case of a nonzero quark mass.We have demonstrated the gauge invariance of the modified Lagrangians through examination of Ward identities in order g s and through calculations of radiative jet functions in order α s in the Feynman gauge and the light-cone gauge.We have also given a Ward-identity argument to show that the gauge invariance holds to all orders in g s .
In Refs.[11,16], the order-λ 1 Lagrangian in Eq. (13a) was used to construct a radiative jet function that involves a single-photon external state, and that radiative jet function was computed at one and two loops in perturbation theory in the light-cone gauge. 8As we have seen in Sec.VI, the Lagrangian in Eq. (13a) is gauge invariant to all orders in perturbation theory.Consequently, the calculations in Refs.[11,16] are gauge invariant.
We have also examined the label-momentum formulation of SCET in the incarnations that are given in Refs.[8,9].The Lagrangians in these papers that describe the interactions between a soft quark and a hard-collinear quark contain commutators of covariant derivatives.Consequently, these Lagrangians must produce at least one gluon emission if they are to give nonvanishing contributions to amplitudes.It follows that the missing diagrams that correspond to these Lagrangians vanish and that the formulations of SCET in Refs.[8,9] evade the gauge-invariance problem that we have identified in this paper.

Note added
After the present paper appeared on the arXiv, a paper [17] was submitted to the arXiv that addresses the same gauge invariance issue that we address.That paper presents the gauge-invariance issue from an alternative point of view in which the hard-collinear quark

FIG. 1 .
FIG. 1. Feynman diagrams showing a Ward identity for the operators in Eq. (13).The Ward identity is for the case in which a gluon attaches to the Wilson line (W † n ) or to the crossed circle (⊗).The thick, solid line denotes a quark line that has a hard-collinear momentum (−p), and the solid line denotes a quark line that has a soft momentum (−ℓ).The diagram on the right side of the arrow shows the contribution of the term k µ in the gauge transformation in Eq. (19).The short, double lines across a propagator indicate that the propagator has been canceled.
which are represented by the diagram on the right side of Fig. 4. If the quark line with momentum −p is an external line, then the first terms in the square brackets in each of the contributions above vanish on multiplying the quark spinor.Otherwise, they cancel contributions that arise from the gauge transformations for the sum over all attachments of the gluon with momentum k to other parts of the radiative jet function (the ellipses in Figs. 1 where α, β and a, b are Dirac and color indices, respectively.In these expressions, the first factors of the SCET operators arise from the BCDF Lagrangians L BCDF 2a which satisfies ϵ * • p = 0.The O(α s ) diagrams for the radiative jet functions are given in Fig.5.Note that the gluon must connect the soft-quark side of the diagram, which is to the left of the Q Q state, to the hard-quark side of the diagram, which is to the right of the Q Q state, in order for hard-collinear-momentum conservation to be satisfied.The left two diagrams involve the Feynman rule V (0) i (Figs.2 and 3) on their soft-quark sides, and the right two diagrams involve the Feynman rule V(1) i (Figs.2 and 3) on their soft-quark sides with i = 2a, 2b, 2m.

FIG. 5 .
FIG.5.The LO diagrams for the radiative jet functions.Note that the soft momentum ℓ µ flows in through the vertex on the soft-quark side at x and flows out through the vertex on the hard-quark side at 0. The Feynman rules for the crossed circle vertex are given in Figs.2 and 3.
) are separately gauge invariant.B. Missing diagrams for the modified Lagrangians Now let us consider the contributions of the modified Lagrangians to the missing diagram in Fig. 4. The amplitudes for the diagram in Fig. 4 that correspond to the crossed vertices from L mod,BPS 2a and L mod,BPS 2b are A mod 2a,miss = 0, (41a)

FIG. 7 .
FIG.7.The general form of the missing diagrams at all orders in g s .The blob includes gluon self interactions and ghost loops.
is, they are equal to the lowest-order crossed-vertex factors.Furthermore, we have seen in Eqs.(22a) and (41) that each crossed-vertex factor gives a vanishing result when no hardcollinear momentum flows through the crossed vertex.Therefore, the missing diagrams give vanishing contributions, ensuring the gauge invariance of the Lagrangians in Eqs.(13a) and (38).