Consistent treatment of rapidity divergence in soft-collinear effective theory

In soft-collinear effective theory, we analyze the structure of rapidity divergence which emerges because the collinear and soft modes reside in disparate phase spaces. The idea of an effective theory is applied to a system of collinear modes with large rapidity and soft modes with small rapidity. The large-rapidity (collinear) modes are integrated out to obtain the effective theory for the small-rapidity (soft) modes. The full SCET with the collinear and soft modes should be matched onto the soft theory at the rapidity boundary, and the matching procedure is exactly the zero-bin subtraction. The large-rapidity region is out of reach for the soft mode, which results in the rapidity divergence. And the divergence in the collinear sector comes from the zero-bin subtraction, which ensures the cancellation of the rapidity divergences from the soft and collinear sectors. In order to treat the rapidity divergence, we construct the rapidity regulators consistently for all the modes. They are generalized by assigning independent rapidity scales for different collinear directions. The soft regulator incorporates the correct directional dependence when the innate collinear directions are not back-to-back, which is discussed in the $N$-jet operator. As an application, we consider the Sudakov form factor for the back-to-back collinear current and the soft-collinear current, where the soft rapidity regulator for a soft quark is developed. We present how to resum large logarithms in the Sudakov form factor, employing the renormalization group evolution on the renormalization and the rapidity scales.


I. INTRODUCTION
Effective field theories enable us to understand important physics by extracting relevant ingredients and disregarding the unnecessary remainder.Soft-collinear effective theory (SCET) [1-3] is an effective theory for QCD, which is appropriate for high-energy processes with energetic particles immersed in the background of soft particles.We pick up the collinear and soft modes to describe high-energy processes, and all the other modes are either integrated out or neglected.The degrees of freedom in the effective theory are classified by the phase spaces they reside in.
Since there are various modes in different phase spaces in SCET, it possesses an additional divergence known as the rapidity divergence [7,8] as well as the traditional ultraviolet (UV) and infrared (IR) divergences.A boundary in the phase space is introduced to separate collinear and soft modes, and it is the origin of the divergence by constraining these modes in different phase spaces.In full QCD, there is no rapidity divergence because all the phase space is taken into account without any constraint.Therefore in SCET there may be rapidity divergence in each sector, but when we sum over all the contributions from different sectors, there should be no rapidity divergence.This is a good consistency check for the effective theory.In this respect, the rapidity divergence seems to be an artifact in SCET by dissecting the phase space, but it gives a richer structure of the effective theory and we can obtain deeper understanding of underlying physics.
Here we consider a system of the collinear and soft modes with the same offshellness, in which they are distinguished by their rapidities.The soft modes have small rapidity, while the collinear modes have large rapidity.For the factorization into the soft and collinear parts in SCET, the central idea is to apply the conventional effective theory approach that is widely used for separating long-and short-distance physics.
We first construct an effective theory for the soft mode out of the full SCET.The full SCET contains both collinear and soft modes, while the soft theory contains only soft modes with small rapidity.By requiring that physics be the same near the rapidity boundary, the full SCET with both modes is matched onto the soft theory, and produces the Wilson coefficient.The Wilson coefficient is obtained by subtracting the contribution of the soft theory from that of the full SCET.In the matching near the boundary, the contribution of the soft theory to be subtracted is exactly the zero-bin contribution in SCET.Therefore the collinear contribution with the zero-bin subtraction [4] can be interpreted as the Wilson coefficients for the soft theory.
Note that the soft mode cannot resolve the large rapidity region.And we expect the rapidity divergence as the soft momentum approaches the rapidity boundary.Suppose that the collinear mode behind the boundary is n-collinear.Then the rapidity divergence in the soft sector arises as the momentum component n • k → ∞ (and n • k → 0).On the other hand, the naive collinear contribution before the zero-bin subtraction has no corresponding rapidity divergence since n • k is bounded from above.However the collinear part after the matching, which is exactly the zero-bin subtraction, contains the rapidity divergence with the same origin as the soft part.Therefore the rapidity divergences in the collinear and the soft sectors have the opposite sign, which ensures the cancellation of the rapidity divergence when both are combined.
The naive collinear part contains another type of the rapidity divergence as n • k → 0.
But that region is shared with the soft part and this type of rapidity divergence is cancelled in the matching.And the true rapidity divergence with n • k → ∞ in the collinear sector is recovered by the zero-bin subtraction, similar to the pullup mechanism [5,6].As a result, the soft-collinear factorization with the zero-bin subtraction is identified as the matching of the full SCET onto the soft part, and the collinear part can be considered as the matching coefficient describing the large-rapidity region.
The main issue of this paper is to implement this idea for the consistent treatment of the rapidity divergence.The first step is to establish a proper method for regulating the rapidity divergences in the collinear and the soft sectors.We prescribe the rapidity regulators in both sectors from the same origin.
In addition to the conventional regularization method [7,8], there have been many suggestions to regulate the rapidity divergence, such as the use of the Wilson lines off the lightcone [9], the δ-regulator [10,11], the analytic regulator [12], the exponential regulator [13], and the pure rapidity regulator [14], to name a few.The gauge invariance and the consistency in power counting to all orders have also been recent issues in selecting appropriate rapidity regulators [8,14,15].
The construction of the rapidity regulators is interwoven between the collinear and the soft sectors, and let us first look at the Wilson lines in SCET.The emission of collinear or soft gluons from collinear, energetic particles is eikonalized and exponentiated to all orders to yield the collinear and soft Wilson lines by integrating out large offshell modes.The collinear and the soft Wilson lines in the lightlike n direction are written as [3] W where A n (A s ) is the collinear (soft) gauge field.The lightlike vectors n and n satisfy n 2 = n 2 = 0, n • n = 2, and n • P (n • P) is the operator extracting the incoming momentum component in the n (n) direction.
Note that the nature of the eikonalization is different in both cases.For the collinear Wilson line W n , we consider the emission of the n-collinear gluons from the counterpart, that is, from the n-collinear field in a back-to-back current or from the heavy quark in a heavyto-light current.Whatever the sources are, when the intermediate states are integrated out, and the leading terms are taken, we obtain the collinear Wilson line which depends on n • P.
On the other hand, the soft Wilson line S n is obtained by the emission of soft gluons from an n-collinear field, and the intermediate states are integrated out.Note that the source of the gluon emission is different.Therefore when we consider the rapidity divergences in the soft and the collinear sectors, we should compare the n-collinear gluons from the counterpart in the collinear sector, and also the soft gluons from the same source as in the collinear sector, and take the n-collinear limit.Simply put, the rapidity matching does not happen between W n and S n , but happens between W n and S n for the back-to-back current.
Let us take an example of the back-to-back current ξ nW nS † nγ µ S n W † n ξ n , which will be generalized later, and consider a soft gluon emitted from the soft Wilson line, not from S n , but from S † n.In order to consider the matching with the n-collinear sector, in which the n-collinear gluon is emitted from the n-collinear field, we take the limit in which the component n • k of the soft momentum becomes large, compared to other components with k 2 fixed.And it is taken to infinity in the soft sector because the large scale is beyond reach for the soft particles.Therefore the region n • k → ∞ is where the possible rapidity divergence occurs and we apply the rapidity regulator to extract it.We can choose the rapidity regulator of the form (ν/n • k) η , where ν is the rapidity scale introduced and the rapidity divergence appears as poles in η.In order to be consistent, we also choose the same rapidity regulator in the n-collinear sector since the modification of the rapidity region due to the regulator should be the same in the overlapping region.
We can also include collinear currents which are not back-to-back, or even the N -jet operator in which there are N well-separated collinear directions.We emphasize that the same rapidity regulator should be employed both in the collinear and soft sectors.When the rapidity divergence in one collinear sector is to be matched, the soft gluons emitted from other collinear directions are selected and the collinear limit in the given collinear direction is taken.In this process, the directional dependence in the soft sector is correctly produced.Furthermore, we can assign a different rapidity scales to each collinear direction if there is a hierarchy of scales in different directions.The total contribution is free of every rapidity divergence associated with each collinear direction.The rapidity regulator for the soft-collinear current can be also consistently constructed from this method.
When the collinear and the soft particles have different offshellness, the collinear and the soft particles are distinguished by their offshellness, not by their rapidity.The effective theory in this situation is called SCET I .There is no rapidity divergence in each sector because collinear and (u)soft particles do not overlap.In practice, the rapidity divergences in the virtual correction and in the real emission cancel.But in the Sudakov form factor or in the N -jet operator, in which there is no contribution from real emissions, the rapidity divergence can be present in each sector.When the offshellness of the collinear and the soft particles is of the same magnitude, they should be distinguished by their rapidities and it is described by SCET II .
The structure of the paper is as follows: In Section II, we discuss the idea of applying an effective theory to a system with the collinear and soft modes, and explain that the Wilson coefficients in matching corresponds to the zero-bin subtraction.And we show how to set up the rapidity regulator with the zero-bin subtraction in the collinear sector.The soft rapidity regulator is defined by employing the same principle for the collinear sector, but with the appropriate expression for the soft Wilson line.In Section III, the Sudakov form factor is analyzed for the back-to-back current.We suggest how to implement the rapidity regulator in the N -jet operator in Section IV.In Section V, we consider the Sudakov form factor for the soft-collinear current, which is compared to the result by boosting the back-to-back current.We conclude and describe the outlook in Section VI.In Appendix, the resummation of the large logarithms in the Sudakov form factor is performed using the renormalization group (RG) equation with respect to the renormalization and rapidity scales.The evolutions are described explicitly for the two currents, showing that they are equal to next-to-leading logarithmic (NLL) accuracy, and are independent of the paths in the RG evolution.The issue here is to see the consistency in the RG evolutions with a single rapidity scale versus two rapidity scales in the soft-collinear and the back-to-back current.

II. RAPIDITY DIVERGENCE AND THE ZERO-BIN SUBTRACTION
We start with the matrix element of the back-to-back collinear current in SCET where p and p are the on-shell momenta of the collinear quarks in the n-and n-directions respectively.The soft Wilson lines Sn (S n) are present by redefining the collinear fields, which results in the decoupling of the soft and the collinear interactions.The notations follow the convention used in Ref. [16], depending on the originating collinear particles.In higher-order contributions, the UV, IR, and rapidity divergences are produced, in which the UV and the IR divergences are controlled by dimensional regularization.If we introduce a nonzero gluon mass M , the IR divergence is regulated by the mass.However, a new regulator is needed to regularize the rapidity divergence.
The widely-used rapidity regulator has been suggested in Refs.[7,8], by modifying the original collinear and soft Wilson lines as Here the rapidity scale ν is introduced and the rapidity divergence appears as poles in η.
As described in Introduction, we use the rapidity regulator of the form (ν/n • k) η for the n-collinear sector, which is the same as the prescription in Eq. ( 3) for W n .However, here we construct the soft rapidity regulator which is the same as the collinear rapidity regulator, but applied to S n.Therefore the soft rapidity regulator is different from that in Eq. ( 3), and needs further explanation.

A. Effective theory approach to treating the rapidity divergence
We begin with the general argument in treating the rapidity divergence.In SCET II where the collinear and the soft modes have the same offshellness, we distinguish these modes by their rapidities.In radiative corrections, there appears the integral of the form which can be separated into soft and collinear integrals as where p + is a hard momentum, and µ L is a soft scale.Here Λ is some arbitrary scale which separates the soft and collinear regions.
In the spirit of the zero-bin contribution, the integral I, by rearranging the second term, can be written as The first term in the parenthesis can be interpreted as the naive collinear contribution, and the second term as the zero-bin contribution.We clearly see that double counting is avoided in the collinear part.And if there is any divergence as k + → 0, it is removed.It guarantees the factorization to secure the independence of the collinear sector as stressed in Ref. [15].
The cutoff Λ appears only in the soft integrals, including the zero-bin contribution.And it can be taken to infinity as far as the soft modes are concerned.Then with the rapidity regulator, Eq. ( 6) can be expressed as 1 The soft and collinear parts in Eq. ( 7) are given as The rapidity divergence and the scale dependence in the soft and collinear sectors cancel when they are combined.But the evolutions with respect to the rapidity scale ν to µ L for the soft sector and to p + for the collinear sector are necessary for the resummation of the large logarithms in p + /µ L [7,8].
Note that the separation into the soft and collinear parts in Eq. ( 7) is similar to separating the long-and short-distance physics in effective theories.The effective theory at lower energy is matched at the cutoff scale to the full theory, yielding the Wilson coefficients.The same mechanism applies to Eq. (7).Consider all the modes with the same offshellness, and there is a cutoff rapidity which distinguishes the soft modes with small rapidity and the collinear modes with large rapidity.We match the two regions near the cutoff, which is accomplished by the zero-bin subtraction.It yields the Wilson coefficients, which corresponds to the collinear contribution in this case, when the contributions from the large rapidity (collinear) region and the small rapidity (soft) region are matched at the boundary.
As a consequence, the rapidity divergence arises entirely due to the fact that the soft part cannot describe large-rapidity physics.And the collinear part contains the divergence through matching onto the soft theory with small rapidity, i.e., the zero-bin subtraction.
There is always one-to-one correspondence for the rapidity divergences between the soft and the collinear sectors.

B. Collinear contribution and the zero-bin subtraction
The matrix element in Eq. ( 2) is factorized into the n-, n-collinear and the soft parts.
Let us first consider the n-collinear contribution at one loop.The corresponding Feynman diagram is shown in Fig. 1 (a).The naive collinear contribution is given by 1 Some readers may wonder if the pole 1/η can (mathematically) regularize the pole at k + = 0 as well in the zero-bin contribution in Eq. (7).But the purpose of the η-regulator is to capture the divergence as k + → ∞, meaning that we take η to be slightly positive, i.e., η = +0.It does not regularize the divergence for k + = 0, and we may introduce another η = −0 to regulate the divergence as k + → 0 as in pure dimensional regularization.However, the divergence for k + = 0 cancels and we simply put the regulator only for k + → ∞. (a) H r S m t l y L S 1 D X e u B F j 8 b r 4 J n 7 I g r 7 5 u / 0 4 9 k 2 Y N / c 1 r r 2 0 g r 9 K 6 p t j I / 7 4 a t + + P l 1 7 + R 9 f X 3 t o y P 0 D L 1 A I X q D T t A p G q I R w i h d O e y 8 6 5 x 2 P q 3 Q 7 a 0 6 5 j F q P J 0 v f w E V u g G A < / l a t e x i t > k < l a t e x i t s h a 1 _ b a s e 6 4 = " A 4 t m f c 6 2 7 F A K A o j j W h a 1 / Y o e P 0 4 = " n < l a t e x i t s h a 1 _ b a s e 6 4 = " x P j B C z n 2 6 K F + 9 l W n W K j r a y Z S 3 H r S m t l y L S 1 D X e u B F j 8 b r 4 J n 7 I g r 7 5 u / 0 4 9 k 2 Y N / c 1 r r 2 0 g r 9 K 6 p t j I / 7 4 a t + + P l 1 7 + R 9 f X 3 t o y P 0 D L 1 A I X q D T t A p G q I R w i h F P 9 E v 9 L t z 1 H n X O e 1 8 W q O 7 O 3 X M E 9 R 4 O l / + A h 8 R A b M = < / l a t e x i t > W n < l a t e x i t s h a 1 _ b a s e 6 4 = " M P u A 9 E U V N a l S L W E a 9 w r m s r m J j a s = "     where µ 2 MS = (µ 2 e γ E /4π) in the MS scheme and we employ the rapidity regulator in Eq. ( 3).The massless fermion is on shell (p 2 = 0), and the nonzero gluon mass M is inserted as an IR regulator, or the real gauge boson mass in electroweak processes.Performing the contour integral on n • k, we obtain where p + ≡ n • p is the largest component of the external momentum p, and x = k + /p + .
In Eq. ( 11), the pole in η comes from the region k + → 0. The collinear gluon momentum k + has an upper limit p + , and there is no rapidity divergence in the region k + → ∞ in this naive collinear amplitude by itself.However, as we have explained in Introduction, it is cancelled when we subtract the zero-bin contribution because the soft sector shares the same phase space.Due to the zero-bin subtraction, the rapidity divergence in the collinear sector is pulled up to the divergence for k + → ∞.
The zero-bin contribution is obtained from Eq. ( 11) by taking the limits k + p + and x → 0 in the integrand, and the upper limit in the integral of x to infinity.It is given by Then the legitimate collinear contribution is obtained by subtracting the zero-bin contribution, Eq. ( 12), from the naive collinear contribution, Eq. (11).It is given as In the second line, we divide the integration region of the zero-bin contribution into x ∈ [0, 1] and x ∈ [1, ∞].Then the integral for x ∈ [0, 1] is combined with the naive collinear contribution Ma .Note that there is no pole in η for the integral with x ∈ [0, 1], hence we can put η = 0.The η-regulator is employed in the second integral, where k + (or x) goes to infinity.Finally we have shown that the correct rapidity divergence in the collinear sector is captured through the zero-bin subtraction.

C. Rapidity regulators for the soft sector
We have to find a consistent rapidity regulator for the soft sector that conforms to the regulator in the collinear sector.Here we extend the collinear current to q n Γq n , where the lightcone directions n and n are not necessarily back-to-back, but n • n ∼ O(1).Let us consider the configuration in which a collinear or a soft gluon is emitted, which is shown in  lines W n and S n is determined by the power counting of the collinear and soft momenta.
The collinear momentum and the collinear gauge field scale as Q(1, λ, λ 2 ), while the soft momentum and the soft gauge field scale as Q(λ, λ, λ).In order to take a consistent rapidity regulator in the collinear and the soft sectors, we choose the configuration of Fig. 2 (b) and take the n-collinear limit.
The soft Wilson line S n is written as where n • P returns incoming momentum of the soft gluon with p if we take the n-collinear limit of the soft momentum k, it becomes k µ ≈ (n • k)n µ /2.In this Therefore the soft rapidity regulator to capture the divergence The important point in taking this limit is to express the original regulator (ν/n • k) η in terms of n •k, with which the soft Wilson line S n is expressed.As a consequence, we suggest that the soft rapidity regulator for Fig. 2 (b) is given by because it corresponds to the regulator (ν/n • k) η in the limit n • k → ∞.Accordingly, the soft Wilson line is modified as For S n , we switch n and n .These soft Wilson lines appear in the collinear current q n Γq n .
Consider an N -jet operator with one n-collinear operator, and the remaining (N − 1) , in which we are interested in the rapidity divergence associated with the n direction.For each n i direction, we can modify S n i using different rapidity regulators with η and ν in the form It properly captures the rapidity divergence in the n direction when a soft gluon is radiated from the n i -collinear sector to the n direction in the limit n • k → ∞.Note that each separate rapidity scale η i can be assigned to each n i direction, and the corresponding rapidity divergences are cancelled when the collinear and soft contributions are added.We will discuss the N -jet operator in more detail in Section IV.

III. SUDAKOV FORM FACTOR FROM BACK-TO-BACK CURRENT
A. Soft one-loop contribution for the back-to-back collinear current We now return to the back-to-back collinear current in Eq. (2), and consider its soft contribution at one loop, which is depicted in Fig. 1 (c).Due to the presence of S † n and S n in the current, the soft contribution contains the factor It provides two types of rapidity divergences as n sector, and the rapidity divergence as n The rapidity divergence is not regulated by the dimensional regularization because it appears irrespective of the UV (k 2 ⊥ → ∞), or the IR (k 2 ⊥ → 0) limits.From Eq. ( 17), the soft Wilson lines S † n and S n with the rapidity regulator are written as with n•n = 2.The iε-prescription is provided explicitly according to Ref. [16].We introduce the rapidity regulators for each collinear direction.The rapidity regulator with η − in Eq. (20) regulates the divergence in the n direction, and the one with η + in Eq. ( 21) for the divergence in the n direction, with the corresponding two rapidity scales, ν ∓ .
The soft one-loop contribution, before the regulator is inserted, is given as where we first perform the contour integral on k 0 , with the relation The rapidity regulators do not affect the pole structures for the contour integral, and are dropped for the moment.Now we assign the rapidity regulators according to Eqs. ( 20) and ( 21).As can be seen in Fig. 3 (a), the green regions in the phase space, where the rapidity divergences arise, are well separated.Therefore, for practical purposes, it is convenient to divide the phase space by the line k + = k − in the k + -k − plane.Then, we can employ the regulator from S n only in the region for k + > k − , while we employ the regulator from S n only in the region for k + < k − because the omitted regulators produce no rapidity divergence.As a result, the rapidity regulator at one loop can be written as2 It amounts to specifying independent rapidity scales for different collinear directions.If we set η = η + = η − and ν = ν + = ν − , we obtain the same result using the regulator proposed in Refs.[7,8], in which the soft regulator is written as As Dividing the full phase space by the line k + = k − , the soft rapidity regulator can be written as which reduces to R S with ν + = ν − = ν at leading order, neglecting subleading corrections.
Applying Eq. ( 24) to Eq. ( 22), the soft contribution can be written as where the phase space A (B) is the region with [See Fig. 3 (a).]The contribution from the region A is given by where we require that η go to zero faster than n with n > 0. The contribution from the region B, M B S , is obtained from M A S by switching (η + , ν + ) ↔ (η − , ν − ).The complete soft contribution at one loop is given as It is consistent with the result in Ref. [8] with a single η.

B. Factorization of the Sudakov form factor
The n-collinear contribution at one loop in Fig. 1 (a) is given by Eq. ( 13).Combining it with the field strength renormalization and the residue we obtain the complete contribution to the n-collinear sector at one loop as Replacing (p + , η + , ν + ) with (p − , η − , ν − ), the result for the n-collinear sector is given by (2), the Sudakov form factor is factorized as where The advantage of introducing multiple rapidity scales ν ± in Eq. ( 33) is that we can systematically deal with various cases with a hierarchy of scales between p + and p − .For example, if p + p − M , the range of the evolution in ν − is smaller than the range of ν + .
It is very interesting to consider the limit p + p − ∼ M , in which we can directly describe the soft-collinear current3 from Eq. (33).Identifying ν − ∼ M in C n and S nn in Eq. ( 33), we can combine the two functions into a new soft function to describe the soft sector.We refer to Section IV for more details.
C. On-shell regularization with a massless gluon We can also employ pure dimensional regularization with a massless gluon, in which the UV and IR divergences are expressed as poles in UV and IR respectively.The UV and IR divergences are separated, and the problematic mixed divergence such as (1/ UV ) • (1/ IR ) does not appear.
Compared to Eq. ( 13) with nonzero M for the n-collinear sector, the result with M = 0 is given by Here the integration over k 2 ⊥ in pure dimensional regularization is expressed as The rapidity regulator in the integral over x ∈ [0, 1] is not needed since there is no rapidity divergence.With the self-energy contribution where the residue is given by R (1) ξ = α s C F /(4π IR ), we obtain the n-collinear contribution at one loop as Similarly, the n-collinear contribution is given as The soft virtual contribution with a massless gluon, yet without the rapidity regulator, in Fig. 1 (c) is given as Applying the rapidity regulator in Eq. ( 24), we divide the soft phase space into the regions To compute the contribution from the region A, it is useful to consider the phase space in (k + , |k ⊥ |) in Fig. 3 (b).
The contribution from the region A in Fig. 3 (b) can be written as , where we divide the integration region for k 2 ⊥ into [Λ 2 , ∞] and [0, Λ 2 ] in order to separate the UV and IR divergences.The dependence on the arbitrary scale Λ 2 cancels at the end of calculation.The two terms in Eq. (39) are labelled as M A1 S and M A2 S , and are given by Combining these two contributions, we have The contribution from the region B can be obtained from Eq. (42) by switching (η + , ν + ) → (η − , ν − ).Finally the soft contribution at one loop using the pure on-shell dimensional regularization is given by The total contributions from Eqs. (36), (37) and ( 43) are free of the rapidity scales and the IR divergence of the full theory is reproduced.With Q 2 = p + p − , they are given by

D. Soft contributions to timelike processes
So far we have considered the back-to-back collinear current with the spacelike momentum transfer.For the current with the timelike momentum transfer as in Drell-Yan (DY) process, the current in SCET is given by where p (p ) is the incoming n-(n-)collinear momentum.Compared to Eq. (2), S † n is replaced by S † n [16].The soft Wilson line S † n from the n-collinear antiquark is given by The matrix element of the full-theory current is schematically factorized as where Q 2 = 2p • p = p + p − , and u n and v n are the spinors for the n-collinear quark and n-collinear antiquark respectively.The hard coefficient H DY depends on −Q 2 , in contrast to +Q 2 for a spacelike process, and its anomalous dimension for H DY at one loop is given by The minus sign in the logarithm in γ H,DY also shows up in V µ DY,SCET , and it appears specifically in the soft function S DY since C n and C n are the same.Fig. 4 shows the different paths of the soft Wilson lines for S DY with respect to the spacelike process, which generates the relative minus sign in the logarithm.The amplitude for S DY = 0|S † n S n |0 at one loop is written as

S †
n < l a t e x i t s h a 1 _ b a s e 6 4 = " r x U J I 5 q x 4 M p d X 1 4 R V U B r 6 x T / 0 c Y 9 x X u 9 P 3 H T c B 9 + a 3 1 3 a U V h 1 d U 1 z g + P I h f H M Q f X g 6 O 3 j T X 1 w 5 6 i B 6 j p y h G r 9 A R e o d G a I w w q t B P 9 A v 9 7 j 3 r j X o n v S 8 r 9 N p W 4 / M A t V o v + w t w / g f Q < / l a t e x i t >

+1
< l a t e x i t s h a 1 _ b a s e 6 4 = " L + L P S e 6 H v a J 4 7 i X u H 6 U m 0 o P e u 9 7 H 3 a Y 3 u 7 j Q x j 1 D r 6 U 3 / A t 6 n A 8 Q = < / l a t e x i t > n < l a t e x i t s h a 1 _ b a s e 6 4 = " w Z q 2 6 F u q n H r S m t l y L S 1 D X e u B F j 8 b r 4 J n 7 I g r 7 5 u / 0 4 9 k 2 Y N / c 1 r r 2 0 g r 9 K 6 p t j I / 7 4 a t + + P l 1 7 + R 9 f X 3 t o y P 0 D L 1 A I X q D T t A p G q I R w i h H r S m t l y L S 1 D X e u B F j 8 b r 4 J n 7 I g r 7 5 u / 0 4 9 k 2 Y N / c 1 r r 2 0 g r 9 K 6 p t j I / 7 4 a t + + P l 1 7 + R 9 f X 3 t o y P 0 D L 1 A I X q D T t A p G q I R w i h where I DY is given by by setting n µ = (1, 0, 0, 1) and n µ = (1, 0, 0, −1), where k Encircling the contour in the lower-half plane, the two poles at k 0 = |k| − i and k 0 = k z − i contribute to the integral.The result can be written as where I 0 is the residue from the pole k 0 = |k| − i and I T is from the pole k 0 = k z − i .I 0 is given as Note that the contribution I 0 (k) in Eq. ( 49) is the same as MS in Eq. (38) (or Eq. ( 22) with M 2 = 0).So I 0 (k), or MS is common to both the spacelike and the timelike processes.
The residue I T from the pole k 0 = k z − i is present for the timelike process only, and it is given as And the contribution from I T in Eq. ( 49) is given by There is no rapidity divergence here, and the factor −iπ gives a negative sign in the argument of the logarithm in M S .
As a result, applying the rapidity regulator in Eq. ( 24) to M DY S , we obtain the soft where N {p} is the N -jet amplitude at tree level with the contributions from the external on-shell spinors and polarization vectors.F N ({p}) is the form factor that can be expanded in powers of α s .
The form factor F N ({p}) can be factorized as where H N is the hard matching coefficient.The string {2σ ij p i • p j } represents all the possible combinations of the hard momentum transfers with different i and j (i, j = 1, • • • , N ).When both p i and p j are all incoming or all outgoing, σ ij = −1, and otherwise σ ij = +1.The corresponding string {σ ij n i •n j /2} appears in S N .Since all the external partons are on-shell, the momentum can be written as Let us consider the one-loop contribution to F N ({p}) in SCET.The n i -collinear contribution at one loop for Fig. 5 (a) can be obtained from Eq. (37), and is given by where γi = 3C F for a quark or an antiquark, and β 0 for a gluon.T i is the color charge of the i-th collinear particle, and T 2 i = T a i • T a i is C F for a quark or an antiquark and C A for a gluon.
Let us now consider the one-loop calculation for S N .For a soft gluon exchange between S n i and S n j in Fig. 5 (b) 4 , the amplitude contains the factor The rapidity divergence can occur both in the n i and n j directions.To clarify this, we can apply the same reasoning employed in the back-to-back current.The rapidity divergence in the n i direction arises when n i • k → 0 and n j • k ≈ (n i • n j /2) n i • k → ∞, and the rapidity divergence in the n j direction arises when n j • k → 0 and As discussed in Section II C, we can introduce the rapidity regulators in the soft Wilson lines as in Eq. ( 17).For the rapidity divergence in the n i direction, the rapidity regulator is inserted in S n j , and the regulator for the divergence in the n j direction, it is inserted in Then the contribution for the soft gluon exchange between S n i and S n j is given by It is written to make the expression look symmetric, and the simultaneous appearance of the two regulators may look confusing.But in extracting the rapidity divergence in the n i direction, the η j regulator can be dropped because there is no pole in η j , and vice versa.
We can directly compute Eq. ( 62) by decomposing the momentum vector in the n i -n j basis.Interestingly, there is another convenient way to recycle the result for the back-toback current.Let us boost the reference frame in order that two lightcone vectors n i and n j become back-to-back [23].With the lightcone vectors n µ i = (1, ni ) and n µ j = (1, nj ), we find that the boost is obtained by the velocity β = (n i + nj )/2.The lightcone vectors in the boosted frame are given as where the Lorentz factor γ is given by In Eq. ( 63), the boosted lightcone vectors are not normalized, but can be normalized by s q G a C P l 0 v o O S 3 8 i A 3 c h n C a w F M F 4 P r e b o 9 d W + F Y w 0 1 k y 9 3 L h y l X n v k 1 a c 9 L F 2 z x v o C A P T Q 2 T r P B 1 7 K S J w l q / 3 A C X H U K 0 I 6 T N p r t c G O q K 7 I R c H 7 0 k K 2 n u f 4 S S 1 n g I b e Z t Y 9 f z B 2 l X P u 1 E C Q p X S q 9 o W C 7 O M w f w k l c s 8 5 V p q 9 B 7 Q y o W M F W h h o U X X U + o O 2 h t b b k W l 1 B e 6 V H g P Z 6 s n V P / R x j 3 F e 7 0 w 6 d t w H 3 5 r f X d p R W H V 1 T X m J w c x y + P 4 4 + v B q d v m + t r H z w G T 8 F z E I P X 4 B S 8 B y M w B g g Y 8 B P 8 A r 9 7 w 9 6 k N + u l a 3 R 3 p / F 5 C F q t t / g L N O I K d g = = < / l a t e x i t > p 0 k < l a t e x i t s h a 1 _ b a s e 6 4 = " R T 8 t 6 p V X w y n q S I M 2 V / Q s T 1 N 7 N i B K 4 1 k s e W 5 K D u d C + 5 p z / 0 i a F S T 9 M y 0 z I w q C g q 4 n S g g U m D 9 y 2 g y R T S A 1 b W g O o y u x a A 3 o B C q i x x e n 3 I 4 H f a c 4 5 i K S M U q 6 r S T g t y 0 F Y R S / c 7 0 3 V I Z i H 1 F C b 4 r S y g 1 0 q B e Y l 4 K q R F P e V 0 U 1 Q O a r 8 n D F W b p h n o s R v R d 3 E y s u A D k F r / Q + I E T a S C F A K l t 0 k s M 6 y J t p I v q j c g M q d P S 9 c S q t J V N L z J 3 P r T x G G V b 9 q N A o g V 7 C H C f W F M B R T 8 t 6 p V X w y n q S I M 2 V / Q s T 1 N 7 N i B K 4 1 k s e W 5 K D u d C + 5 p z / 0 i a F S T 9 M y 0 z I w q C g q 4 n S g g U m D 9 y 2 g y R T S A 1 b W g O o y u x a A 3 o B C q i x x e n 3 I 4 H f a c 4 5 i K S M U q 6 r S T g t y 0 F Y R S / c 7 0 3 V I Z i H 1 F C b 4 r S y g 1 0 q B e Y l 4 K q R F P e V 0 U 1 Q O a r 8 n D F W b p h n o s R v R d 3 E y s u A D k F r / Q + I E T a S C F A K l t 0 k s M 6 y J t p I v q j c g M q d P S 9 c S q t J V N L z J 3 P r T 2 A + R + U p Y i N Z 2 V m y c H J q 2 1 X X v k 1 W 5 U E X b / O i g b w 6 N D 2 M 4 t T 1 s V M m B r V + t Q a u O o R s Z 5 g 1 m + 5 y f q p r s p N y d f S i O G e J + x B y V u M + t J 6 where C n and S q are the collinear and the soft functions for the soft-collinear current.The one-loop result for C n is given in Eq. ( 31) or (36).
Let us consider the rapidity divergence in S q including a soft quark [Fig.6  As a result, we suggest the rapidity regulator for the soft quark sector at one loop as where p µ = En µ , E is the energy of the soft quark and n µ is the lightcone vector for the soft massless quark.The last limit shows that, in the soft quark sector, we pick up the rapidity divergence consistently as in the n-collinear sector when n • k → ∞.
At higher orders, it is complicated to set up a consistent rapidity regulator in the soft quark sector especially with the n-collinear regulator in Eq. (3).That is because multiple soft gluon radiations from the soft quark are not eikonalized.However, as discussed in Section II C, the origin of the rapidity divergence from the collinear and soft gluon radiations is the same.Therefore, once we set up the n-collinear rapidity regulator by modifying W n or the phase space, we can trace the corresponding factor in the multiple soft gluon radiations from the soft quark.
Before employing the regulator, the contribution to S q in Fig. 6 (a) with the pure on-shell dimensional regularization at one loop can be written as Here the momentum k µ is decomposed as where k µ ⊥ is perpendicular to n µ and n µ .And k 2 is given by In obtaining the first term in Eq. ( 78), the integration measure is written as and the contour integral in the complex n • k plane is performed.
In Eq. ( 78), the rapidity divergence appears only in the first term.Applying the rapidity regulator, it can be written as In order to separate different types of divergences clearly, the integration region for n • k is divided into 0 < n • k < ∆ and n • k > ∆, where ∆ is an arbitary soft energy scale.Then Eq. ( 78) can be written in terms of the three parts as The rapidity divergence arises only in Eq. (84), and the rapidity regulator is inserted here.
In Eqs.(82) and Eq.(83), they have IR and UV divergences respectively, and there is no need for the rapidity regulator.
Combining all the results, we obtain equation.On the other hand, in the soft-collinear current, there is only one rapidity scale associated with a single collinear direction.However, the evolution of the Sudakov form factor is the same, and independent of the factorization scales.It is also independent of the order of evolution with respect to the renormalization scale and the rapidity scale.Because it is technical, the detail is deferred to Appendix.

VI. CONCLUSIONS
We have presented a new perspective regarding the origin of the rapidity divergence in SCET and its consistent treatment.It is based on applying the effective theory to a system with a hierarchy of rapidities, that is, a system with collinear particles with large rapidity and soft particles with small rapidity, but with the same offshellness.The effective theory with soft modes is obtained by integrating out the collinear modes with large rapidity.The Wilson coefficient through the matching is obtained by subtracting the low-rapidity physics from the full theory, and can be interpreted as the zero-bin subtraction for the collinear sector.In the naive collinear calculation, the rapidity divergence occurs only in the lowrapidity region, but it is cancelled by the zero-bin subtraction converting the divergence from small rapidity to large rapidity.It has the opposite sign of the rapidity divergence in the soft sector, guaranteeing the cancellation of the rapidity divergence in the total contribution.
The main point in extracting rapidity divergence in each sector is that we have to trace the same configurations both in the collinear and in the soft sectors.For example, the ncollinear Wilson line W n is obtained by the emission of the n-collinear gluons from the other part of the current, say, the n-collinear sector from the back-to-back current.Therefore the corresponding soft sector in the matching, or the zero-bin subtraction, should come from the emission of soft gluons from the n-collinear sector, that is S n, not S n which is built by the emission of the soft gluons from the n-collinear sector.
We employ the same rapidity regulator of the form (ν/n • k) η both in the soft sector and the collinear sector to regulate the rapidity divergence at large rapidity.When the current is not back-to-back, or when there is a soft quark involved, the same rapidity regulator is employed, but we use the appropriate expressions conforming to the corresponding soft Wilson lines.In this process, the directional dependence enters in the soft rapidity regulator and it is essential in extracting Lorentz invariants in the full theory when combined with the energy dependence from the collinear sector.Furthermore, unless the correct directional dependence is incorporated, an additional UV divergence is induced because the directional dependence can appear in the coefficients of the 1/ UV pole.With our prescription, the directional dependence is correctly implemented without any problem.
By extending the treatment of the rapidity divergence, we can associate independent rapidity scales for each collinear direction.Since physics should be independent of the rapidity divergence in any collinear direction, its cancellation in the combination of the soft and the collinear contributions gives a severe constraint on the structure of the effective theory.On the practical side, when the factorized collinear and soft parts share the same size of the rapidity scale, they can be combined to be a single function.It is illustrated in obtaining the soft function of the Sudakov form factor from the soft-collinear current, which is also confirmed from explicit calculations.This can be applied to various physical processes, in which the rapidity scales can be varied depending on physics so that part of the factorized parts can be combined.
It is important to include the evolution with respect to the rapidity scale in SCET II because it yields the correct resummation of the large logarithms when the rapidity divergence is involved.Therefore the understanding of the rapidity divergence is essential, and we have explained the origin of the rapidity divergence, proposed how to extract it consistently in each sector.Our future plan is to apply this prescription systematically to various processes, with a variety of physical observables.
7 1 w I s e n 6 6 C Z + 6 L K O 2 b v 9 O j k 0 3 A v r m t 9 e 2 l F f p X V N c 4 P d g P 3 + 6 H x + 8 G h x + b 6 2 u H P C c v y W s S k v f k k H w h I z I m l C D 5 S X 6 R 3 7 1 n v c P e 5 9 7 R C r 2 1 1 c Q 8 J a 2 n d / I X I b s B T w = = < / l a t e x i t > p k < l a t e x i t s h a 1 _ b a s e 6 4 = " M h O O r H p 4 7 w o 9 t 1 I C K B U l C E H j r l E = " > A A A F m n i c d Z T P b 9 M w F M e 9 s c I o P 7 b B e s / 2 e 4 1 g y q s 1 w + G d r + 9 Z O 7 / a d 3 b v 9 e / c f P H y 0 t / / 4 T O e F w m S M c 5 a r i x g 0 u y K g o y b e i b m P l Z S A O I d b 6 H x A T 2 E g i Q C l Y d p P A O s u a a C P 5 o n I D U e 7 0 e e F S W k 0 r S b m C W o 9 v d O / H L 4 C F w = = < / l a t e x i t > S † n < l a t e x i t s h a 1 _ b a s e 6 4 = " / K n p w c v U m x 8 y 8 o + a 2 K 2 D 0 i 8 X O l w = " > A A A F r n i c d Z T P b 9 M w F M e 9 s c I o P 7 a B O H 1 j c n I c v z y O P 7 4 a n L 5 t r q 9 9 8 B g 8 B c 9 B D F 6 D U / A e j M A Y I G D A T / A L / O 4 N e 5 P e r J e u 0 d 2 d x u c h a L X e 4 i 8 2 F g p 6 < / l a t e x i t > S n < l a t e x i t s h a 1 _ b a s e 6 4 = " p U m 0 Q G C H I 9 0 W v G g e X B T M 7 3 a 8 C 5 w FIG.1.Feynman diagrams for the back-to-back collinear current at one loop.

Fig. 2 .
Fig. 2. In Fig. 2 (a), a collinear gluon from the n -collinear quark q n is emitted in the n direction.It produces the collinear Wilson line W n at first order.The same configuration is exhibited in Fig. 2 (b) except that a soft gluon is emitted, producing the soft Wilson line S n at first order.The momentum of the soft gluon scales as (n • k, k ⊥ , n • k) = Q(λ, λ, λ) with a large scale Q.But the soft sector is an exact copy of QCD where Q is taken to infinity.Therefore when the soft momentum is in the corner of phase space with n • k → ∞, n • k → 0, it approaches the n-collinear momentum.It is the region where the rapidity divergence occurs.It means that the soft rapidity regulator associated with the n-collinear sector should be implemented in S n , not S n .The collinear rapidity regulator for W n in Eq. (3) for Fig. 2 (a) is given by (ν/n • k) η .And the rapidity divergence shows up as poles of 1/η with n • k → ∞.It will be consistent to use the same rapidity regulator for the soft part as (ν/n • k) η .However, the form of the Wilson (a) (b) FIG. 2. (a) Collinear and (b) soft gluon emissions from the n -collinear sector in the current qn Γq n , which yield the leading contributions to W n and S n respectively.If n • k s → ∞ and n • k s → 0 in the soft phase space, k s becomes the soft version of the n-collinear momentum k n , and the rapidity divergence arises when we separate the soft and collinear gluons in SCET.
leading order, where p is the n -collinear momentum.When the internal fermion is integrated out in Fig.2 (b), we obtain gn • A s /n • k.On the other hand,

FIG. 3 .
FIG. 3. Structure of the phase space for the soft gluon in the back-to-back current.The rapidity divergence in the n (n) direction arises in the green region with η + (η − ).(a) the phase space in the k + -k − plane with nonzero gluon mass M .The (red) curve k + k − = M 2 is the IR cutoff.(b) the phase space in k + -|k ⊥ | plane with massless gluons.The IR divergence arises at |k ⊥ | = 0.
the momentum transfer to the current.To next-to-leading order (NLO) in α s , C n,n = 1 + M n,n and S nn = 1 + M S , where M n,n,S are the renormalized functions from Eqs. (31), (32), and (29).Each rapidity scale dependence in C n and C n is cancelled by the soft function S nn .But the evolution of ν + from p + to M and that of ν − from p − to M are needed to resum the large logarithms of Q/M .
t e x i t s h a 1 _ b a s e 6 4 = " p U m 0 Q G C H I 9 0 W v G g e X B T M 7 3 a 8 C 5 w = " > A A A F n X i c d Z T d b t M w F M e 9 s c I o X x t c c k 9 B P 9 Q r 8 7 D z p v O u 8 7 H 9 b o 7 k 4 d c w 8 1 n s 7 4 L 2 / a A 4 U = < / l a t e x i t > t 0 g s I 6 y J t p I u b C + I 9 K f y c B d C K c J I k U w n s /t + u i 1 F b 4 R z H S W z L 0 8 c + W q c 9 8 m r T n s 4 m 2 e N 1 C Q h 6 a G S T b z d e y k i U G t X 6 y B i w 4 h 2 h H S Z t N d L g x 1 S X Z C r o 5 e k p U s 9 z 9 C y W o 8 h N b z t r G r + Y O 0 K 5 9 2 q g S D p d J L F p a L 8 8 w B v O R V k f n K t F X w 3 s D E G a Q q 1 I j w o u s p c w e t r S 1 W 4 g L k p R 4 F 3 u P j l X P q / w j j v s K d v v + 4 C b g v v 7 W + u 7 T i 8 I r q G s e H B / G L g / j D y 8 H R m + b 6 2 k W P 0 B P 0 D M X o F T p C 7 9 A I j R F G F v 1 E v 9 D v 3 m H v t J f 2 Y I V u b z U + D 1 Cr 9 b 7 + B a W 9 C 4 Y = < / l a t e x i t > S n < l a t e x i t s h a 1 _ b a s e 6 4 = " p U m 0 Q G C H I 9 0 W v G g e X B T M 7 3 a 8 C 5 w = " > A A A F n X i c d Z T d b t M w F M e 9 s c I o X x t c c k 9 B P 9 Q r 8 7 D z p v O u 8 7 H 9 b o 7 k 4 d c w 8 1 n s 7 4 L 2 / a A 4 U = < / l a t e x i t > S † n < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 W S 1 NO V L l 3 M B a q C P s F 8 J s Y a 3 M u o = " > A A A F q H i c d Z T P b 9 M w F M c 9 W G G U H 9 v g y C V Q I S E O o x l I c J z g A r f y o 9 t Q U 6 I X x + 2 i 2 Y 6 x n a L K 8 4 G / g y v 8 U f w 3 2 G n W N Q 4 4 i v X i7 + c 9 2 + 8 5 z g Q t l B 4 O / 2 x d u 7 7 d u 3 F z 5 1 b / 9 p 2 7 9 3 b 3 9 u 8 f q 7 K S m I x x S U t 5 m o

r 4 I
X 7 I k r 7 5 u / 0 w 5 d t w L 6 5 r f X t p R X 6V 1 T X O D s + C l 8 d h Z 9 f D 0 5 P m u t r H z 1 G T 9 F z F K I 3 6 B S 9 R y M 0 R h h d o J / o F / r d C 3 r v e h 9 7 n 9 b o 7 k 4 T 8 w i 1 n t 7 0 L 9 Q z A 8 I = < / l a t e x i t > 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " d Y f I z X + S W 2 X a 9 N W z G l 9 2 O J i D j y Y = " > A A A F n n i c d Z T f b 9 M w E M e 9 s c I o v z Z 4 5 C V Q I f H C a A Y S e 5 y E h O A B U R B d K z V V d X G c z p r t GN s p i r I 8 8 C f w C n 8 Z / w 1 2 m n W N A 6 l q X e 7 7 u b N 9 5 z i W j G o z H P 7 Z 2 b 2 x 1 7 t 5 a / 9 2 / 8 7 d e / c f H B w + P N N Z r j A Z 4 4 x l a h q D r t G N s p i r I 8 8 C f w C n 8 Z / w 1 2 m n W N A 6 l q X e 7 7 u b N 9 5 z i W j G o z H P 7 Z 2 b 2 x 1 7 t 5 a / 9 2 / 8 7 d e / c f H B w + P N N Z r j A Z 4 4 x l a h q D 9 o P e u 9 7 H 3 a Y 3 u 7 j Q x j 1 D r 6 U 3 / A t 6 n A 8 Q = < / l a t e x i t > 1 < l a t e x i t s h a 1 _ b a s e 64 = " d Y f I z X + S W 2 X a 9 N W z G l 9 2 O J i D j y Y = " > A A A F n n i c d Z T f b 9 M w E M e 9 s c I o v z Z 4 5 C V Q I f H C a A Y S e 5 y E h O A B U R B d K z V V d X G c z pr t G N s p i r I 8 8 C f w C n 8 Z / w 1 2 m n W N A 6 l q X e 7 7 u b N 9 5 z i W j G o z H P 7 Z 2 b 2 x 1 7 t 5 a / 9 2 / 8 7 d e / c f H B w + P N N Z r j A Z 4 4 x l a h q D

w 0 7 6 G
0 o l 2 z J g 9 a G 1 t s R I X o K 7 1 w I s e n 6 6 C Z+ 6 L K O 2 b v 9 O j k 0 3 A v r m t 9 e 2 l F f p X V N c 4 P d g P 3 + 6 H x + 8 G h x + b 6 2 u H P C c v y W s S k v f k k H w h I z I m l C D 5 S X 6 R 3 7 1 n v c P e 5 9 7 R C r 2 1 1 c Q 8 J a 2 n d / I X F 1 M B T Q = = < / l a t e x i t > n < l a t e x i t s h a 1 _ b a s e 6 4 = " S 3 f R 1 7 Y h E i U Z / x k k 1 E D 1 + W n a m 2 w = " > A A A F m X i c d Z T P b 9 M w F M e 9 s c I o v z Y 4 7 h K o k D i V Z i D B c c B l c C q D dp O a q n p x n N a a 7 R j b K a q y H L h z h f + N / w Y 7 z b r G g V S 1 X t 7 3 8 5 7 t 9 x z H k l F t B o M / O 7 u 3 9 j q 3 7 + z f 7 d 6 7 / + D h o 4 P D x 2 O d 5 Q q T E c 5 Y p i 5 i 0

FIG. 4 .
FIG. 4. Soft Feynman diagrams for the virtual contributions to (a) the spacelike process and (b) the timelike (DY) process.Here the double line represents the path for a given soft Wilson line.
l e W C N + A W 3 o y 3 w U 6 z r n E g V a 2 T 8 / + d Y / s c x 7 G k m T b 9 / p + d 3 W t 7 n e s 3 9 m 9 2 b 9 2 + c / f e w e H 9 k R a 5 w m l e W C N + A W 3 o y 3 w U 6 z r n E g V a 2 T 8 / + d Y / s c x 7 G k m T b 9 / p + d 3 W t 7 n e s 3 9 m 9 2 b 9 2 + c / f e w e H 9 k R a 5 w m e s / 2 e 4 1 g y q s 1 w + G d r + 9 Z O 7 / a d 3 b v 9 e / c f P H y 0 t / / 4 T O e F w m S M c 5 a r i x g 0 e s / 2 e 4 1 g y q s 1 w + G d r + 9 Z O 7 / a d 3 b v 9 e / c f P H y 0 t / / 4 T O e F w m S M c 5 a r i x g 0 e s / 2 e 4 1 g y q s 1 w + G d j 8 9 Z W 7 / a d 7 b v 9 e / c f P H y 0 s / v 4 R O e F w m S M c 5 a r s x g 0 g 9 b W 5 k t x D u p a D 7 z o 8 c k y e O a + i N K + + T s 9 + r I O 2 D e 3 t b 6 9 t E L / i u o a J / t 7 4 Z u 9 8 P P b w c F h c 3 1 t o 6 f o B X q F Q v Q O H a C P a I T G C K M M / U S / 0 O / e s 9 5 h 7 6 j 3 a Y l u b j Q x T 1 D r 6 R 3 / B X / I A i o = < / l a t e x i t > O N < l a t e x i t s h a 1 _ b a s e 6 4 = " L 3 A Z b 7 Y O 1 + P l v 8 D m k D j 4 K p J s Q g s = " > A A A F n 3 i c d Z T P b 9 M w F M e 9 s c I o v z Y 4 c s m o k D i N Z i D B c Y I D c B k F 0 a 6 o q a o X x 2 m j 2 Y 6 x n a L K y 4 F / g S v 8 Y / w 3 2 G n W N Q 6 k q v X y v p / 3 b L / n O B Y 0 U 7 r f / 7 O z e 2 O v c / P W / u 3 u n b v 3 7 j 8 4 O H w 4 U n k h M R n i n O Z y H I M i N O N k q D N N y V h I A i y m 5 D y + e O v 0 8 y W R K s v 5 F 7 0 S Z M p g z r M 0 w 6 C t a x w x b D 6 W s 7 P Z Q a 9 / 3 K + e o G 2 E t d F D 9 T O Y H e 7 9 i J I c F 4 x w j S k o e 9 C a 2 n I t L k F e 6 Y E X P R y t g 2 f u i z D 2 z d / p h 8 / b g H 1 z W + v a S y v 0 r 6 i 2 M T o 5 D l 8 c h 5 9 e 9 k 7 f 1 N f X P n q M n q B n K E S v 0 C l 6 j w Z o i D C i 6 C f 6 h X 5 3 j j r v O m e d w R r d 3 a l j H q H G 0 / n 6 F 0 X F B E U = < / l a t e x i t > O N < l a t e x i t s h a 1 _ b a s e 6 4 = " L 3 A Z b 7 Y O 1 + P l v 8 D m k D j 4 K p J s Q g s = " > A A A F n 3 i c d Z T P b 9 M w F M e 9 s c I o v z Y 4 c s m o k D i N Z i D B c Y I D c B k F 0 a 6 o q a o X x 2 m j 2 Y 6 x n a L K y 4 F / g S v 8 Y / w 3 2 G n W N Q 6 k q v X y v p / 3 b L / n O B Y 0 U 7 r f / 7 O z e 2 O v c / P W / u 3 u n b v 3 7 j 8 4 O H w 4 U n k h M R n i n O Z y H I M i N O N k q D N N y V h I A i y m 5 D y + e O v 0 8 y W R K s v 5 F 7 0 S Z M p g z r M 0 w 6 C t a x w x b D 6 W s 7 P Z Q a 9 / 3 K + e o G 2 E t d F D 9 T O Y H e 7 9 i J I c F 4 x w j S k o

FIG. 5 .
FIG. 5. Feynman diagrams for the N -jet operator in SCET at one loop.(a) the n i -collinear gluon exchange, (b) the soft gluon exchange between the soft Wilson lines S n i and S n j .

p 0 <
l a t e x i t s h a 1 _ b a s e 6 4 = " k L v 4 l t x n z w B I d w / 3 1 f B f q H v o 2 S M = " > A A A F m X i c d Z T P b 9 M w F M e 9 s c I o P 7 b B c Z d A h e A 0 m o E E J z T E Z X A q g 3 a T m q p 6 c Z z W m u 0 Y 2 y m q s h y 4 c 4 X / j f 8 G O 8 2 6 x o F U t V 7 e 9 / O e 7 f c c x 5 J R b f r 9 P 1 v b t 3 Y 6 t + / s 3 u 3 e u / / g 4 d 7 + w a 2 D d / p x / P N g H 7 5 r b W t Z d W 6 F 9 R b W N 0 f B S + O g o / v + 6 d v K u v r 1 1 0 i J 6 i F y h E b 9 A J O k U D N E Q Y p e g n + o V + d w 4 7 7 z u n n U 8 r d H u r j n m M G k / n y 1 8 U h g F 8 < / l a t e x i t > p < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 p 5 t w o y v p / 3 b L / n O J Y s 0 2 Y 4 / L N 1 6 / Z 2 7 8 7 d n X v 9 + w 8 e P n q 8 u / f k V O e F o j i m O c v V e Q w a W S Z w b D L D 8 F w q B B 4 z P I s v P z n 9 b I F K Z 7 n 4 a p Y S p x z m I k s z C s a 6 j u V s d z D c H 9 H T N m D 1 p b W 6 z E B a h r P f C i x 6 e r 4 J n 7 I k r 7 5 u / 0 6G Q T s G 9 u a 3 1 7 a Y X + F d U 1 T g / 2 w 7 f 7 4 f G 7 w e H H 5 v r a I c / J S / K a h O Q 9 O S R f y I i M C S V I f p J f 5 H f v W e + w 9 7 l 3 t E J v b T U x T 0 n r 6 Z 3 8 B S C H A U s = < / l a t e x i t > S † n < l a t e x i t s h a 1 _ b a s e 6 4 = " m X 7 r V l X Q / j L l / m n p v y x k 7 u g T o W I = " > A A A F r n i c d Z T P b 9 M w F M e 9 s c I o P 7 a B O H E J V E i c R j O Q 4 I Q m c Y F b + d F u U l M i x 3 F a a 7 Z j b K e o 8 n z g T + E K f x H / D Xa a d Y 0 D j m K 9 + P t 5 z / Z 7 j j N B i d L D 4 Z + d 3 R t 7 v Z u 3 9 m / 3 7 9 y 9 d / / g 8 O j B R J W V R H i M S l r K 8 w w q T A n H Y 0 0 0 x e d C Y s g y i s + y i 3 d e P 1 t i q z v b d 4 5 j y a g 2 / f 6 f n d 1 r e 5 3 r N / Z v d m / d v n P 3 3 s H h / Z H O c o X J E G c s U + c x a M K o I E N D D S P n U h H g M S N n 8 e L U 6 W d L o j T N x F e z k m T C Y S Z o S j E Y 5 5 L P X i y m B 7 3 + U b 9 6 g r Y R 1 k Y P 1 c 9 g e r j 3 I 0 o y n H M i D G a g 9 T j s S z M p Q B m K G S m 7 U a 6 J B L y A G R l b U w A n e l J U i y 2 D p 9 a T B G m m 7 F + Y 9 7 j z q n H Y + d j 6 t 0 d 2 d O u Y B a j y d 4 V 9 + H A I o < / l a t e x i t > p 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " k L v 4 l t x n z w B I d w / 3 1 f B f q H v o 2 S M = " > A A A F m X i c d Z T P b 9 M w F M e 9 s c I o P 7 b B c Z d A h e A 0 m o E E J z T E Z X A q g 3 a T m q p 6 c Z z W m u 0 Y 2 y m q s h y 4 c 4 X / j f 8 G O 8 2 6 x o F U t V 7 e 9 / O e 7 f c c x 5 J R b f r 9 P 1 v b t 3 Y 6 t + / s 3 u 3 e u / / g 4 d 7 + w a s h a 1 _ b a s e 6 4 = " x p S T z s e h w b 4 G g m i 7 1 6 e r 2 D p z 4 o Q = " > A A A F m n i c d Z T P b 9 M w F M e 9 s c I o P 7 b B H W b L r L + a m u y U 7 K 5 d G L 4 p w l 7 k P I W Y 3 7 0 G r e N n Y z v 1 d 2 7 c p O t W S w 0 G b B / H Y J E V t A 5 K L g s e t M W w U X D U y e w 0 z 7 G p F O t C N l 9 q C 1 t f l S n I O 6 1 g M v e n y y D J 6 5 L 6 K 0 b / 5 O j 7 6 u A / b N b a 1 v L 6 3 Q v 6 K 6 x s n B f v h 6 P / z y Z n D 4 v r m + t t E T 9 B y 9 R C F 6 i w 7 R J z R C Y 4 R R h n 6 i X + h 3 7 2 n v Q + + o 9 3 m J b m 4 0 M Y 9 R 6 + k d / w W 9 C g I y < / l a t e x i t > A s < l a t e x i t s h a 1 _ b a s e 6 4 = " b m d f Y a l a E h 8 M w d b T 1 R M 2 Y q f L v A 4 = " > A A A F m n i c d Z T P b 9 M w F M e 9 s c I o P 7 b B E Q 6 B C o n T a A Y S n N C A C x O X A u s 2 q a m q F 8 f J r N m O s Z 2 i K s u B P 4 A r / G 3 8 N 9 h p 1 j U O p K r 1 8 r 6 f 9 2 y / 5 z i W j G o z H P 7 Z 2 L y x 1 b t 5 a / t 2 / 8 7 d e / d 3 d v c e n O i 8 U J i M c c 5 y d R a D J o w K M j b U M H H W b L r L + a m u y E 7 K 5 d G L 4 p w l 7 k P I W Y 3 7 0 G r e N n Y 9 v 1 d 2 7 c p O t W S w 0 G b B / H Y J E V t A 5 K L g s e t M W w U X D U y e w 0 z 7 G p F O t C N l 9 q C 1 t f l S n I O 6 0 g M v e n y y D J 6 5 L 6 K 0 b / 5 O j 7 6 s A / b N b a 1 3 b L / n O J Y s 0 2 Y 4 / L N 1 6 / Z 2 7 8 7 d n X v 9 + w 8 e P n q 8 u / f k V O e F o j i m O c v V e Q w a W S Z w b D L D 8 F w q B B 4 z P I s v P z n 9 b I F K Z 7 n 4 a p Y S p x z m I k s z C s a 6 j i 9 n u 4 P h / r B + g q 4 R N s a A N M 9 o t r f 9 I 0 p y 8 p P 8 I r 9 7 z 3 q H v c + 9 o x V 6 a 6 u J e U p a T + / k L w a D A U Y = < / l a t e x i t > k < l a t e x i t s h a 1 _ b a s e 6 4 = " H g Uu p A c B C 1 0 d T f G V Y l K v Z J 0 p L q s = " > A A A F m H i c d Z T P b 9 M w F M c 9 W G G U X x v c 4 B K o k D i N Z i D B C U 3 i A L t 1 E 9 0 m N V X 1 4 r x 0 0 W z H 2 E 5 R l e X A m S v 8 c f w 3 2 G n W N Q 6 k q v X y v p / 3 b L / n O J Y s 0 2 Y 4 / L N 1 6 / Z 2 7 8 7 d n X v 9 + w 8 e P n q 8 u / f k V O e F o j i m O c v V e Q w aW S Z w b D L D 8 F w q B B 4 z P I s v P z n 9 b I F K Z 7 n 4 a p Y S p x z m I k s z C s a 6 j i 9 n u 4 P h / r B + g q 4 R N s a A N M 9 o t r f 9 I 0 p y W n A U h j L Q e h I O p Z m W o E

FIG. 6 .
FIG. 6.(a) Feynman diagram of a soft gluon exchange from a soft quark, (b) In the limit when the momentum of the soft gluon from the soft quark becomes collinear to the n direction, the rapidity divergence arises.
(a)].It arises when a soft gluon from the soft quark becomes n-collinear and its momentum k reaches n • k → ∞ (and n • k → 0), as shown in Fig. 6 (b).[See Fig. 2 for comparison.]In order to be consistent with the n-collinear sector, the rapidity regulator should be (ν + /n • k) η + as n • k goes to infinity.On the other hand, since the propagator in Fig. 6 (b) is proportional to 1/p • k, the rapidity regulator in the form ν + /(p • k) is desired.