Renormalization of the radiative jet function

We show how to compute directly the renormalization/evolution of the radiative jet function that appears in the factorization theorems for $B\to \gamma\ell\nu$ and $H\to \gamma\gamma$ through a $b$-quark loop. We point out that, in order to avoid double counting of soft contributions, one should use in the factorization theorems a subtracted radiative jet function, from which soft contributions have been removed. The soft-contribution subtractions are zero-bin subtractions in the terminology of soft-collinear effective theory. We show that they can be factored from the radiative jet function and that the resulting soft-subtraction function gives rise to a nonlocal renormalization of the subtracted radiative jet function. This is a novel instance in which zero-bin subtractions lead to a nonlocality in the renormalization of a subtracted quantity that is not present in the renormalization of the unsubtracted quantity. We demonstrate the use of our formalism by computing the order-$\alpha_s$ evolution kernel for the subtracted radiative jet function. Our result is in agreement with the result that had been inferred previously by making use of the factorization theorem for $B\to \gamma\ell\nu$, but that had been ascribed to the unsubtracted radiative jet function.


I. INTRODUCTION
In the amplitudes for exclusive processes, contributions in which a quark carries a soft momentum appear at subleading power in the ratio of the quark mass to the large momentum transfer in the process. These contributions arise because there is a pinch singularity in the region of soft quark momentum that has a subleading power dependence [1]. They are associated with endpoint singularities in light-cone amplitudes. [2][3][4][5][6][7][8]. 1 In the language of soft-collinear effective theory (SCET) [13][14][15][16][17], the soft-quark contributions occur through a process in which a jet function containing collinear quarks and gluons emits a soft quark via a subleading-power interaction. Such a jet function is called a radiative jet function [18][19][20].
A particular radiative jet function, which is the focus of this paper, enters into the factorization theorem for the exclusive B-meson decay B → γ ν [21] and the factorization theorem for the exclusive decay of the Higgs boson H → γγ through a b-quark loop [22]. 2 In the remainder of this paper, we will refer to this jet function as the radiative jet function.
The renormalization properties of the radiative jet function are an essential ingredient in using these factorization theorems to resum large logarithms of the ratios of m b /µ or m H /m b , where µ is the factorization scale, m b is the b-quark mass, and m H is the Higgs-boson mass.
The radiative jet function has been computed through order α s in Ref. [22], and we have verified this calculation. It has also been computed through order α 2 s in Ref. [25]. The renormalization-group evolution of the radiative jet function in order α s has been inferred from the factorization theorem for B → γ ν, the renormalization-group invariance of the physical amplitude for B → γ ν, and the known renormalization-group evolution kernel of the B-meson light-front distribution, which also appears in the factorization theorem [21]. 3 That analysis has been extended to order α 2 s in Ref. [25]. The renormalization of the radiative jet function that is obtained from these analyses is nonlocal in momentum space in that it involves the convolution of a renormalization factor Z J with the unrenormalized radiative jet function, rather than a simple multiplication. 1 There are also analyses of corrections to inclusive cross sections at subleading power in the inverse of the large momentum transfer. See, for example Refs. [9][10][11][12]. 2 A discussion of the factorization theorem for the decay B → γ ν in the context of the method of regions is given in Ref. [23]. Subleading-power corrections to the decay B → γ ν are discussed in Ref. [24]. 3 This is an application of what is called the consistency condition for the renormalization-group evolution [21].
Although the renormalization properties of the radiative jet function have been known indirectly for almost two decades, a method for computing the nonlocal renormalization factor Z J directly from the definition of the radiative jet function has remained elusive. In the words of Ref. [25], "It is an embarrassment that there is no known method in SCET to derive the anomalous dimensions of the jet functions directly from their operator definitions." In this paper, we present a method to derive the anomalous dimension of the radiative jet function directly from its operator definition. We point out that the radiative jet function contains soft contributions that are already taken into account in the soft functions of the exclusive factorization theorems. These soft contributions must be subtracted from the radiative jet function in order to avoid double counting. Methods for the systematic subtraction of double-counted soft contributions are familiar from the diagrammatic approach to factorization [26,27] and are known in SCET under the name zero-bin subtractions [28].
We call the radiative jet function with the soft contributions subtracted the subtracted radiative jet function. It is the subtracted radiative jet function, rather than the radiative jet function, that should properly appear in the exclusive factorization theorems.
We show that the soft subtractions can be factored from the radiative jet function into a soft-subtraction function by making use of the Grammer-Yennie approximation [29] and the graphical Ward identities that are standard in diagrammatic factorization [30]. The soft-subtraction function gives rise to nonlocal ultraviolet (UV) divergences and accounts for all of the nonlocal contributions in Z J .
In dimensional regularization, soft subtractions (zero-bin subtractions) generally result in scaleless integrals that can be interpreted as being proportional to a difference between UV and infrared (IR) poles. The soft subtractions then have the function of converting IR poles to UV poles. In a fixed-order calculation, if one does not distinguish IR poles from UV poles, then the soft subtractions do not affect the result. As we will see, this is also the case for the soft subtractions of the radiative jet function. However, beyond one-loop order, the UV poles of the soft subtractions are proportional to nonlocal convolutions over light-front momenta, and, so, they lead to nonlocal contributions to Z J . To the best of our knowledge, this is the first time that a nonlocal renormalization owing to the effect of zero-bin subtractions has been observed.
The remainder of this paper is organized as follows. In Sec. II, we present some of the notation and conventions that we use throughout this paper. In Sec. III, we give the operator definition of the radiative jet function and show how to factor the radiative jet function into a convolution of a soft-subtraction function and a subtracted radiative jet function. The renormalization procedure for the subtracted radiative jet function is outlined in Sec. IV. In Sec. V, we record the leading-order expressions for the radiative jet function, the soft-subtraction function, the subtracted radiative jet function, and the softsubtraction renormalization. In Sec. VI, we present the application of our method to the renormalization of the subtracted radiative jet function in order α s . Our result for the orderα s renormalization-group kernel of the subtracted radiative jet function is in agreement with the result in Ref. [21], although, in that work, the renormalization-group kernel is ascribed to the unsubtracted radiative jet function. In Sec. VII, we argue that soft subtractions generally vanish in dimensional regularization if one does not distinguish between UV and IR divergences, and, hence, do not affect existing fixed-order calculations of the radiative jet function. Finally, we summarize and discuss our results in Sec. VIII.

II. NOTATION AND CONVENTIONS
In this section, we establish some of our notation and conventions.
The lowest-order contribution to H → γγ through a b-quark loop is shown in Fig. 1, which establishes our conventions for the momenta of the external particles and the orientation of the internal quark loop. the divergences in the loop integrations.
We make use of the following SCET notations. q s is the soft-quark Dirac spinor, G s = T a G a s , where G a s is the soft-gluon field with color index a and T a is an SU (3) color matrix in the fundamental representation, G n = T a G a n , where G a n is an n-hard-collinear gluon field with color index a, A n is an n-hard-collinear photon field, ξ n is an n-hard-collinear-quark Dirac spinor, which satisfies / nξ n = 0, iD ⊥µ n = i∂ ⊥µ + g s G ⊥µ n + e q A ⊥µ n is a transverse covariant derivative, g s = √ 4πα s is the strong coupling, and e q is the electric charge of the collinear quark.
The scaling of a soft momentum is given by Here Q is the large momentum scale and λ ∼ m b /Q. We follow the convention for the light-front coordinates. The scaling of an n-hard-collinear momentum is given by The soft-quark spinor scales as λ 3 2 , the soft-gluon field scales as λ, the n-hard-collinear spinors scale as λ 1 2 , and the n-hard-collinear gauge fields scale as the n-hard-collinear momentum k µ n in Eq. (7) [17].

III. RADIATIVE JET FUNCTION
A. Operator definition of the radiative jet function In the factorization theorem of Ref. [22], the interactions of virtual hard-collinear quarks and gauge bosons with an outgoing real hard-collinear photon are described by the radiative jet function. The radiative jet function couples to the soft quark by virtue of an interaction between the soft quark and the hard-collinear quarks and gauge fields that first appears at subleading order in the expansion of the SCET Lagrangian. This coupling is given, in the notation of [17], by where H.c. denotes the Hermitian-conjugate contributions. Here, the argument of the softquark field q s is taken to be x − , with x µ − = (n · x) n µ 2 . This argument effects the expansion in momentum space at leading power in λ, in which the collinear subdiagrams depend only on the + component of the soft momentum. (This is the multipole expansion of SCET [17].) W n is a collinear Wilson line, which is defined by where P denotes the path ordering of the exponential. We drop the term involving A n in the remainder of this paper.
In discussing the soft subtraction, we will encounter the soft Wilson line along n which is defined by We take the definition of the radiative jet function J(p 2 ) that is given in Eq. (1.2) of Ref. [25]:J Here, we have made the color indices a and b explicit, and we have definedJ(n · p, p 2 ) to be the complete expression in Eq. (11). γ(k 1 ) denotes a real photon with momentum k 1 = (0, k 1− , 0 ⊥ ) and polarization ε * (k 1 ), and T denotes the time-ordered product. We have inserted a factor i in front of the matrix element in Eq. (11), so that there is a factor i that is associated with each real-photon vertex, whether it arises from the covariant derivative in the matrix element or from the QED interaction Lagrangian. p is defined by and has the interpretation of the soft momentum that is carried by the soft quark in Eq. (8). Note that Throughout this paper, we take the approximation Since k 1 satisfies hard-collinear scaling and satisfies soft scaling, this approximation gives the leading power in λ in the argument p 2 of the radiative jet function. This is the multipole expansion to which we alluded earlier. It follows that This approximation has been invoked in writing p 2 in the arguments ofJ and J. In the remainder of this paper, we suppress the argumentn · p ≈ k 1− inJ.
J(p 2 ) contains contributions in which the real photon attaches to the covariant derivative and contributions in which the real photon attaches to the quark line. Following Ref. [22], we call the former contributionsJ A (p 2 ), and we call the latter contributionsJ G (p 2 ), writinḡ In computing contributions to the radiative jet function in this paper, we do not use the SCET Feynman rules. Instead, we follow a procedure that is equivalent, but more amenable to a graphical analysis. Starting from the Feynman rules for QCD, we insert projectors on the outgoing and incoming ends of the quark lines, respectively, so as to obtain the components of the Dirac spinor that are large in n-hard-collinear scaling and that correspond to the n-hard-collinear spinors ξ n andξ n , respectively.

B. Subtraction and factorization of soft contributions to the radiative jet function
The radiative jet function, as defined in Eq. (11), also contains soft contributions. These must be removed in order to avoid double counting of contributions in the soft function in the factorization theorems. One can factor the soft contributions from the jet function by making use of standard techniques from the diagrammatic methods for proving factorization theorems [31]. We carry out the factorization in the Feynman gauge. However, the resulting expressions are gauge invariant. Our approach in dealing with the soft subtractions is analogous to the one that is given in Sec. 10.8.7 of Ref. [27]. Let us consider first the case of soft gluons that attach to hard-collinear lines, but not to the Wilson lines W n and W † n . At leading order in the scaling parameter λ, the current j µ in the hard-collinear lines to which the soft gluons couple is proportional n µ . It follows that, in the attachment of that soft gluon to a hard-collinear line, its polarization sum g µν can be replaced with k ν n µ /(n·k+iε), where the iε prescription corresponds to a momentum routing in which k flows parallel to the arrow on the quark propagator. Note that the polarization now corresponds to a pure gauge. This is the Grammer-Yennie approximation [29]. Then, one can apply graphical Ward identities to show that sum over the attachments of all such soft gluons to the hard-collinear lines can be replaced with the sum over all attachments of the soft gluons to Wilson lines S n and S † n that attach to the quark line immediately to the outgoing side and immediately to the incoming side, respectively, of the outermost hardcollinear-gluon attachments. Details of this step are given following Eq. (4.3) of Ref. [30].
In SCET, this step can be implemented by making use of field redefinitions [13].
In carrying out this analysis, we omit diagrams that contain n-hard-collinear subdiagrams that are not connected by lines carrying n-hard-collinear momenta to the external photon. These diagrams lead to contributions that are not properly part of the radiative jet function because the disconnected subdiagram does not yield a pinch singularity in the n-hard-collinear momentum region. (At one-loop order, these contributions vanish in di- mensional regularization.) Such diagrams contain quark lines that carry soft momenta and, so, if they were to contribute to the radiative jet function, they would cause the Grammer-Yennie approximation and the factorization of the soft subtractions to fail. We will discuss an example of such a diagram in Sec. VI B.
Next, let us consider the case of soft gluons that attach to the collinear Wilson lines W n and W † n . Here, we use the fact that soft-gluon attachments to the collinear Wilson lines W n and W † n that lie to the outside of all collinear-gluon attachments factor into new soft Wilson lines Sn and S † n , respectively, that lie to the outside of the W n and W † n collinear Wilson lines of the jet function [30]. 4 Soft-gluon attachments to the collinear Wilson lines W n and W † n that lie to the inside of collinear-gluon attachments yield power-suppressed contributions [30]. W n gives rise to Sn, and W † n gives rise to S † n . 5 Note that there is not a one-to-one correspondence betweenS We call the quark line that lies between the covariant derivative and S n in the second diagram in Fig. 2 the n-hard-collinear-soft quark line. Its Feynman rules are obtained from those for the n-hard-collinear quark line by retaining only those contributions that are leading in the soft scaling of the gluon momenta. An n-hard-collinear quark line couples to a soft quark line only through the covariant derivative or W † n in Eq. (8), and at least one n-hard-collinear gluon must attach to the covariant derivative or to W † n in order to produce an n-hard-collinear momentum in the quark line. Therefore, an n-hard-collinear-soft quark line couples to a soft quark line only through the covariant derivative or S † n , and at least one soft gluon must attach to the covariant derivative or to S † n . The blob in the first diagram of Fig. 2 is in the form of a radiative jet function, except that it also contains soft subtractions, which can be implemented by subtracting the Grammer-Yennie form from specific gluon vertices. The blob in the second diagram of Fig. 2 requires some re-arrangement to put it into the form of a radiative jet function (with soft subtractions). One approach is simply to invoke the form of the SCET Lagrangian for nhard-collinear quarks and gluons and their couplings to a soft quark and a real photon to deduce that the blob in the second diagram of Fig. 2 takes the form of a radiative jet function (with soft subtractions) in SCET. Here, it is important that we have definedJ in Eq. (11) in such a way that the vertex for a transverse photon is always ie q γ µ ⊥ , regardless of whether the photon attaches to a quark line or to a covariant derivative. In Appendix A, we sketch how the re-arrangement of the blob in the second diagram of Fig. 2 into the radiative-jet form can be achieved in the diagrammatic approach.
We remind the reader that, owing to the multipole expansion, we keep only the plus components of soft momenta that enterJ sub . Using this fact, we can express the diagrams of Fig. 2, as convolutions over the plus component of momentum: Note that we can also write this convolution in the form where we have made the variable change and we have used p 2 ≈ p + p − [Eq. (15)]. Because bothS andJ sub are Dirac matrices, the order of the factors in the convolution is significant.
The soft-subtraction function is given bȳ whereS In the arguments ofS A andS G , we have suppressed the dependences on the parameter In the equation forS G , the left-hand factor P n arises from the factor P n in the unsubtracted radiative jet functionJ, while the right-hand factor P n arises from applying the identity P n = P 2 n to the factor P n on the left side ofJ sub in Eq. (18). As we have mentioned, at least one soft gluon must attach to the covariant derivative or to S † n inS G . Consequently, one should subtract the non-interacting part, that is, make the replacement At any order in perturbation theory, the soft-subtraction contributions can also be obtained by starting with the jet contributions and retaining only the parts that have leading soft scaling [Eq. (5)]. However, in this form, it is not apparent that the soft subtractions factor from the radiative jet function to yield the form in Eq. (21). As we will see, it is the form in Eq. (21) that leads to the nonlocality in the renormalization of the radiative jet function.
It follows from Eq. (18) thatJ sub is given bȳ The quantityS −1 can be obtained to any order in α s by making use of the expansion ofS in powers of α s , namely,S and solving the equationS iteratively. Specifically, in order α s , Eq. (22) can be written in the form

IV. RENORMALIZATION OF THE RADIATIVE JET FUNCTION
The renormalization ofJ sub arises from two sources: the radiative jet function and the soft function.
The radiative jet functionJ in Eq. (18) is multiplicatively renormalized: where µ is the renormalization scale. That is, the renormalization ofJ is local. This follows from the fact, in order for there to be a nonlocal renormalization forJ, a UV-divergent diagrammatic loop must transfer a plus component of loop momentum k + from one external leg ofJ to the other external leg ofJ. However, a loop that transfers k + in this way does not have a UV-divergent power count because there are too many propagators in the loop.
We will see explicit examples of this in the one-loop calculations in Sec. VI.
The soft-subtraction functionS is nonlocally renormalized: In contrast withJ,S can have nonlocal renormalizations because two of the operators in the definitions ofS in Eq. (21) are separated only in the minus direction. That is, in momentum space, only the plus component of a loop momentum that routes through these operators is constrained. Consequently, a loop momentum k can transfer k + from one external leg of S to the other external leg, and the integration over k ⊥ can still be UV divergent. We will also see explicit examples of this phenomenon in the one-loop calculation in Sec. VI.
Note that we can also write the convolution in the form where we have used the convolution in Eq. (27) is identical to the one in Eq. (18), aside from a trivial rescaling of the argument of Z −1 S with a factor of p 2 and the argument ofS R with a factor p + /k + . Therefore, we use the same notation (⊗) for both convolutions. However, one should keep in mind the rescalings of arguments that are implicit in this notation. It follows that This implies that or, equivalently,J sub where and we have used p 2 ≈ p + p − . Note that, the renormalization factors ZJ and Z −1 S contain only those renormalizations that are associated with the operator matrix element (operator renormalizations) and do not contain the mass and coupling-constant renormalizations of QCD, except for the quark wave-function renormalization.
In order to make contact with the renormalization of the radiative jet function in Ref. [25], we make use of Eq. (31) and the relation between J(p 2 ) andJ(p 2 ) in Eq. (11) to obtain where From the definition ofJ in Eq. (11), it follows that the lowest-order expression forJ is given byJ where J (0) (p 2 ) = 1 is the contribution to J(p 2 ) at leading order in α s . From the definition of the soft-subtraction function in Eq. (21), we find that the lowest-order expressions forS A andS G are given byS Then, making use of the relation in Eq. (18), we find that the lowest-order expression for J sub is the same asJ (0) : Note also that where we have usedS A in the first equality and used k + = (1 − x)p + and p 2 ≈ p + p − in the last equality. We make use of Eq. (36) in computing the order-α s contribution to Z −1 S in Sec. VI.
Finally, we have for the lowest-order contribution to Z −1 VI. RENORMALIZATION OFJJJ sub (p p p 2 ) AT ORDER α s α s α s In this section, we compute the renormalization of the subtracted radiative jet function in order α S .
ZS and ZJ have expansions in powers of α s : Here, 1 represents the lowest-order contribution to ZS [Eq. (37)]. Then, from Eq. (31b) we find that, to order α s in the renormalizations,J sub R is given bȳ where Z (1) Equation (39)   S . Finally, we combine this result with the result for (Z −1 J ) (1) ⊗J sub according to Eq. (39). We evaluate many of the integrals by using contour integration. In these evaluations, we take, for definiteness, p + > 0. This choice corresponds to a time-like argument p 2 > 0 in the radiative jet function because p 2 ≈ p + p − and p − ≈ k 1− is always positive. However, our result for the anomalous dimension of the radiative jet function in the time-like case can be continued analytically to obtain the anomalous dimension in the space-like (p 2 < 0) case.
We carry out the computation in the Feynman gauge.
Feynman diagrams that potentially contribute in order α s to the renormalization of the radiative jet functionJ in the Feynman gauge are shown in Fig. 3. Feynman diagrams that potentially contribute in order α s to the renormalization of the soft subtractions in the Feynman gauge are shown in Fig. 4. In Fig. 4, the blobs representJ sub , which, as we have mentioned, we make explicit in order to keep track of the nonlocal nature of ZS and the Dirac algebra.
In carrying out these computations we make use of the Feynman rules for the Wilson lines, which can be obtained from the definitions in Eqs. (9) and (10). A W n vertex contributes a factor ig s T anµ . A W n propagator contributes a factor i ±n · k + iε .
The plus sign applies when k flows into the Wilson line, and the minus sign applies when k flows out of the Wilson line. In the case of Wn,n is replaced with n. The Feynman rules for W † n and W † n can be obtained from those for W n and Wn, respectively, by taking the Hermitian conjugate and reversing the sign of the momentum in the propagator. The Feynman rules for S n (Sn) are identical to the Feynman rules for Wn (W n ). A. UV divergences ofJJJ sub A (p 2 ) at order α s α s α s Let us discuss the contributions toJ sub A (p 2 ) first. The diagrams that contribute toJ in order α s are shown in Fig. 3. Among the diagrams in Fig. 3, diagrams (a)-(d) are contributions toJ A (p 2 ). Diagram (d), which could potentially yield a nonlocal renormalization, has a UVconvergent power count. This is an example of the general argument that we have given in Sec. IV thatJ has only local renormalizations. Diagram (d) also gives a vanishing contribution, since the collinear gluon is connected to two collinear Wilson lines that are in the same direction, which leads to both a vanishing numerator and a vanishing k + contour integral. Fig. 3 is simply the quark self-energy diagram, and the corresponding amplitude is given by

Diagram (a) in
where C F = (N 2 c − 1)/(2N c ) and N c is the number of colors. There is no soft subtraction in Fig. 4 that corresponds to this contribution. This follows from the fact that, in Eq. (41), (p − k) 2 + iε p 2 − p − k + + iε when k is soft, which implies that there is no pinch that prevents the k − integration contour from being deformed out of the soft region. From the Dirac structure in Eq. (41), we can see that the associated UV divergence corresponds to a quark wave-function renormalization. Extracting the UV divergence and convolving (multiplying) withJ sub , we obtain The subscript UV indicates that the origin of the divergence is UV.
The amplitudes of diagrams (b) and (c) in Fig. 3 are given by We convolve (multiply) the UV-divergent contribution in Eq. (43) withJ sub to obtain Here, the subscript UV indicates that only the UV-divergent part of the expression is to be kept.
As we will explain, the soft subtractions that correspond to diagram (b) in Fig. 3 are given by diagrams (a) and (b) in Fig. 4, and the soft subtractions that correspond to diagram (c) in Fig. 3 are given by diagrams (c) and (d) in Fig. 4. After working out the numerator algebra, we obtain the following contributions: We can see, in the case of the lowest-order expression forJ sub in Eq. We can see as well, in the case of the lowest-order expression forJ sub , that the nonlocal contributions on the right side of Eq. (45) vanish. However, this feature is also special to the lowest-order case.
Note that, if one carries out the integration over k − in Eq. (45) by contour integration, the resulting expression contains a scaleless integral in k ⊥ that yields a factor 1/ UV − 1/ IR . Therefore, if one does not distinguish between UV and IR poles in dimensional regularization, the soft subtraction vanishes. Hence, its role is to convert nonlocal IR poles to nonlocal UV poles.
Owing to the presence of the Wilson-line denominators k − ±iε, the expressions in Eqs. (44) and (45)  and we can make the variable change k → −k to find that Therefore, in the analysis of these contributions, we compute only the combination (−Z (1) We extract the contribution , which gives the UV pole Then, we combine the remaining part of the integrals in (−Z Carrying out the k − contour integration by deforming the contour into the lower halfplane (making the assumption p + > 0) and using p + p − ≈ p 2 , we find that we arrive at After carrying out the k ⊥ integration, we obtain Making the change of variables k + = (1 − x)p + and using p + p − ≈ p 2 , we have where we have used g 2 s = 4πα s , and the plus distribution is defined by Now we can extract the UV divergences of I sub (p 2 ): Using Eq. (39), we find that the total of the UV divergences inJ sub A (p 2 ) is The combined amplitude of diagrams (e) and (f) in Fig. 3 is given by We remind the reader that we retain only terms that contribute in leading power in λ, under the assumption that the momentum = p − k 1 ≈ p 2 n·pn 2 is soft and that the momenta k and k 1 are n-hard collinear.
We extract the UV-divergent contribution from the amplitude in Eq. (57) and convolve it withJ sub to obtain For convenience of computation, we split (−Z (1) J ⊗J sub ) (e+f) into two parts, one with the denominator 1 k − +iε and the other without it: The soft subtractions that correspond to diagrams (e) and (f) of Fig. 3 are given by diagrams (e) and (f) of Fig. 4. Their UV-divergent contributions are Here, we have used Eq. (14) to drop terms that are subleading in the scaling parameter λ and have used the facts that + = p + and P nJ sub =J sub . As in our analysis in Sec. VI A, we can see that, in the case of the lowest-order expression forJ sub in Eq. We can also see that, in the case of the lowest-order expression forJ sub , the nonlocal contributions in Eq. (60) vanish. However, as we have remarked earlier, this feature is special to the lowest-order case.
Owing to the presence of the Wilson-line denominator k − + iε, the expressions in Eqs. (59b) and (60) develop rapidity divergences as k + /k − → ∞. These rapidity divergences cancel in the difference between Eqs. (59b) and (60). Therefore, in order to avoid introducing a rapidity regulator, we compute the expressions in Eqs. (59b) and (60) together. We have organized the expression in the third line of Eq. (60) in such a way as to make the cancellation of the rapidity divergences explicit.
For the jet and soft-subtraction expressions in Eqs. (59) and (60), we perform the k − contour integration first. Carrying out the k − contour integration by deforming the contour into the lower half-plane (making the assumption p + > 0), we obtain Note that the expression for (−Z (1) S ⊗J sub ) (e+f) contains a scaleless integral over k ⊥ that would produce a factor 1/ UV − 1/ IR . That is, the soft subtraction again vanishes in dimensional regularization if one does not distinguish UV and IR poles, and its role is to convert nonlocal IR poles to nonlocal UV poles.
Using Eq. (39), making the changes of variables k 2 ⊥ = tp 2 and k + = (1 − x)p + , and using p + p − ≈ p 2 , we find that the total contribution to (−Z (1) Completing the t integration, we obtain where we have arranged this expression so as to make the cancellation of the rapidity divergences explicit. Finally, extracting the UV divergences, we have C. Renormalization of the subtracted jet functionJJJ sub (p 2 ) at order α s α s α s The total of the UV divergences at order α s is From Eq. (32b), it then follows that Z sub J [(1 − x), p 2 ; µ], up to order α s , is given by where This result is in agreement with Eq. (2.6) of Ref. [25] for the case y = 1. 6 Therefore, our result for the renormalization kernel of the radiative jet function with soft subtractions agrees with the one-loop result that had been inferred from the factorization theorem for B → γ − ν [21]. Note, however, that, in Ref. [21], the renormalization kernel was ascribed toJ, rather than toJ sub .
The result in Eq. (66) is compatible with the analyticity properties ofJ, which is analytic in the p 2 upper half-plane [22]. Hence, although we have derived this result for the time-like case p 2 > 0, it can be continued analytically to the space-like case p 2 < 0. We have also checked the result for the space-like case by explicit calculation.
then the resulting expression is homogeneous in ρ. Therefore, the soft subtractions are scaleless integrals that vanish in dimensional regularization if one does not distinguish between UV and IR divergences. Hence, the soft subtractions do not affect the results of the existing fixed-order calculations of the radiative jet function [22,25].

VIII. SUMMARY AND DISCUSSION
The radiative jet function is a quantity that appears in the factorization theorems for the exclusive processes B → γ ν [21] and H → γγ through a b-quark loop [22]. Its renormalization-group evolution is an essential ingredient in the resummation of logarithms of m b /µ and m H /m b in these processes.
Notwithstanding the importance of these applications, no direct calculation of the renormalization-group kernel for the radiative jet function exists in the literature. Rather, the renormalization-group kernel has been inferred from the factorization theorem for B → γ ν and the known renormalization-group kernel for the B-meson light-front distribution [21].
In this paper, we have argued that the radiative jet function contains, in addition to hard-collinear contributions, soft contributions that must be subtracted in order to avoid double counting of soft contributions that appear in other quantities in the factorization theorems. These soft subtractions are zero-bin subtractions in the language of SCET. We have shown that the radiative jet function can be factored into a convolution over a lightfront momentum of a soft-subtraction function and a subtracted radiative jet function. The subtracted radiative jet function is free of soft divergences, which are contained entirely in the soft-subtraction function. It is the subtracted jet function that should properly appear in the factorization theorems.
The renormalization-group kernel of the subtracted radiative jet function derives from two sources: (1) the renormalization-group kernel of the radiative jet function, which is local and (2) the renormalization-group kernel of the soft-subtraction function, which is nonlocal and leads to the nonlocal contributions in the renormalization-group kernel of the subtracted radiative jet function.
We have shown that the soft-subtraction contributions contain scaleless integrals in dimensional regularization. That is, they are proportional to 1/ UV −1/ IR , and they vanish in calculations in which one does not distinguish UV and IR poles. Hence, existing fixed-order calculations of the radiative jet function are not changed by the soft subtractions, which merely convert IR poles to UV poles. This is the usual role of soft (zero-bin) subtractions in factorization theorems. However, because the UV (and IR) divergences that arise from the soft subtractions are nonlocal beyond one-loop order, the soft subtractions give a nonlocal contribution to the renormalization-group kernel of the subtracted radiative jet function.
To the best of our knowledge, this is a novel phenomenon.
We have illustrated the role of soft subtractions in the renormalization of the subtracted radiative jet function by carrying out a complete calculation of the renormalization-group kernel in order α s . Our result agrees with the renormalization-group kernel that was inferred in Ref. [21]. However, in that work, the renormalization-group kernel was ascribed to the radiative jet function, rather than to the subtracted radiative jet function. The renormalization-group kernel in order α 2 s has also been inferred from the factorization theorem for B → γ ν [25]. It would be interesting to verify that analysis by making use of the methods that we have presented for the direct calculation of the renormalization-group kernel. However, that work is beyond the scope of the present paper.
There is a large class of exclusive processes that proceed at subleading power in the hard-scattering scale [10]. Helicity-flip processes whose amplitudes contain singularities at the endpoints of light-cone distribution amplitudes are included in this class. While factorization theorems have not yet been worked out in detail for most subleading-power processes, it is known that jet functions are a general feature of the factorization theorems [10]. Since collinear functions, including jet functions at subleading power, generally contain soft contributions that must be subtracted in order to avoid double counting, we expect that our methods would be useful in working out the renormalization/evolution of such jet functions.
In particular, there is a jet function that is identical to the one in Eq. (11), except that the external photon state is replaced by an external gluon state. This jet function would be relevant, for example, in the factorization theorem for H → gg through a b-quark loop. We would expect the analysis of soft subtractions in the present paper to go through essentially unchanged in this case because soft-gluon attachments to the external gluon would compensate in the graphical Ward identities for the fact that the external gluon, unlike the external photon, carries color.
The soft function that appears in the factorization theorem for H → γγ [22,33] also has a nonlocal renormalization kernel. It has been conjectured in Ref. [34] that the complete soft sector in the factorization theorem, which includes contributions from the soft-quark function, two radiative jet functions, and a short-distance Wilson coefficient, is a renormalizationgroup invariant. That conjecture leads to a prediction for the renormalization-group kernel of the soft function [13], and that prediction was verified in order α s in Ref. [33]. While the soft-subtraction function that we have defined in the present paper clearly has similarities with the soft function in the factorization theorem, it is not obvious that they would combine to cancel the nonlocal contributions to the renormalization-group kernel for the complete soft sector. It would be interesting to explore systematically the issue of the renormalization-group invariance of the soft sector by making use of the expression for the subtracted radiative jet function that we have presented in this paper.
where the first term in parentheses is the gauge-field part of the covariant derivative and the second term in parentheses is the ordinary-derivative part of the covariant derivative times for γ · j must be kept for gluons with these connections to the quark line.
Finally, we insert a projector P n to the left of the entire expression to project out the components of the quark field that are large under n-hard-collinear scaling. The motivation for this is that we wish to retain only the component of the quark field that corresponds to the effective field theory for the n-hard-collinear sector. The presence of P n insures that the blob cannot contain wrong-collinearity contributions.
At this point, the blob in the second diagram of Fig. 2 is in the form of a radiative jet function (with soft subtractions). [3] M. Beneke, T. Feldmann, and D. Seidel, Systematic approach to exclusive B → V l + l − , V γ