Demonstration of the Equivalence of Soft and Zero-Bin Subtractions
Calculations of collinear correlation functions in perturbative QCD and Soft-Collinear Effective Theory (SCET) require a prescription for subtracting soft or zero-bin contributions in order to avoid double counting the contributions from soft modes. At leading order in , where is the SCET expansion parameter, the zero-bin subtractions have been argued to be equivalent to convolution with soft Wilson lines. We give a proof of the factorization of naive collinear Wilson lines that is crucial for the derivation of the equivalence. We then check the equivalence by computing the non-Abelian two-loop mixed collinear-soft contribution to the jet function in the quark form factor. These results demonstrate the equivalence, which can be used to give a nonperturbative definition of the zero-bin subtraction at lowest order in .
In perturbative QCD (pQCD) factorization theorems Collins:1989gx (), a cross section is expressed as a convolution of several distinct functions, each of which captures physics at a given scale. Suppose a process contains a hard scattering characterized by a scale which is much greater than . The factorization theorem typically contains perturbatively calculable hard coefficients, which capture physics at the scale , jet or collinear functions, which describe the propagation of particles in the initial or final state with energies of order but whose invariant mass is typically or , and soft functions which describe low energy quanta emitted in the process.
Soft-Collinear Effective Theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt () is an effective theory that can be used to derive factorization theorems in QCD. In this approach to QCD factorization, QCD is matched onto an effective theory that contains collinear and soft degrees of freedom, whose momentum components in light-cone coordinates scale as
where . Here the light-like vectors are and . This is the power counting of SCET which is relevant for inclusive processes. For exclusive processes, a different power counting is needed, and the appropriate effective theory is SCET Bauer:2002aj (). Below when we refer to SCET, it is implied that we are discussing SCET. Factorization theorems in this approach are obtained by matching QCD onto SCET at the hard scale, , then decoupling the soft and collinear degrees of freedom in the effective theory by a field redefinition Bauer:2001yt (). The matching coefficients in the effective field theory are the hard coefficients of the traditional pQCD factorization theorems. Correlation functions of the collinear and soft fields correspond to the jet functions and soft functions, respectively. SCET is formulated as a systematic expansion in and therefore provides a framework for analyzing power corrections to leading twist pQCD factorization theorems.
In the evaluation of loop or phase space integrals that arise in the perturbative calculation of soft or jet functions, one could impose cutoffs to enforce the constraints of Eq. (Demonstration of the Equivalence of Soft and Zero-Bin Subtractions) so that no virtual soft modes contribute to the calculation of a jet function or vice versa. In practice this would make integrals horribly complicated and one almost always integrates over all momentum space Beneke:1997zp (). This then raises the important question of avoiding double counting of soft contributions in both collinear and soft functions. This problem arises in both pQCD and SCET formulations of the problem. The traditional approach in pQCD has been to argue that jet functions need to be convolved with soft Wilson lines Akhoury:1998gs (); Ji:2004wu (); Collins:1989bt (); Collins:1999dz (); Chen:2006vd (), while in SCET this is implemented by “zero-bin” subtractions Manohar:2006nz ().
An argument for the equivalence (to lowest order in ) of the two formalisms was first given in Ref. Lee:2006nr (). The essence of the argument of Ref. Lee:2006nr () is that the zero-bin mode of the naive collinear field can be decoupled from the purely collinear field by a field redefinition similar to that used to decouple soft modes from collinear modes in SCET. Performing this field redefinition on the naive collinear matrix element, one finds that the naive jet function factorizes into a purely collinear function convolved with a vacuum matrix element of a zero-bin or soft Wilson line. In a previous paper Idilbi:2007ff (), we studied the equivalence of soft and zero-bin subtractions in the quark form factor and in deeply inelastic scattering (DIS) near , where is the Bjorken variable. We emphasized the importance of using an infrared (IR) regulator such as dimensional regularization (DR) which does not take external particles off-shell and therefore spoil the field redefinition that relates the naive and purely collinear functions. We checked the equivalence at one-loop for both the quark form factor and DIS as . We also verified the equivalence of soft and zero-bin subtractions for the two-loop Abelian diagrams contributing to the jet function appearing in the quark form factor.
In this paper we complete the analysis initiated in Ref. Idilbi:2007ff (). Our first objective is to complete the argument for the equivalence of soft and zero-bin subtractions presented in Ref. Lee:2006nr (). A crucial step in the argument is that the Wilson line of the naive collinear gluons factorizes into the product of a Wilson line constructed from the collinear zero-bin and a purely collinear Wilson line. This factorization was assumed in the derivation of Ref. Lee:2006nr () but not derived. In this paper, we provide a derivation of this factorization, valid up to corrections of order . Our second objective is to complete the analysis of zero-bin subtractions in the jet function of the quark form factor Idilbi:2007ff (). In particular, we verify the equivalence of soft and zero-bin subtractions to two-loops in the non-Abelian theory.
The results of this paper demonstrate the equivalence of the soft Wilson line subtraction of pQCD and the zero-bin subtractions in SCET. It is satisfying to understand the relationship between the two approaches to avoiding double counting. The equivalence of soft and zero-bin subtractions may simplify the calculation of higher order loop diagrams in SCET. In addition, the equivalence provides a nonperturbative operator definition of the SCET zero-bin subtraction, which was defined diagrammatically in Ref. Manohar:2006nz ().
The paper is organized as follows. In the next section we review the argument of Ref. Lee:2006nr () for the equivalence and provide the proof required for the factorization of the naive collinear Wilson line. In Section III, we complete the analysis of the two-loop zero-bin contribution to the jet function in the factorization theorem for the quark form factor in non-Abelian gauge theory. The equivalence of the zero-bin subtraction and dividing by the soft Wilson lines requires that mixed collinear and soft zero-bin contributions from certain Feynman diagrams proportional to must add up to zero. This cancellation is verified in this section. Section IV contains our conclusions. In the Appendix we discuss some subtleties in evaluating two-loop zero-bin subtractions that arise in the limit Idilbi:2007ff ().
I Factorization of Collinear Wilson Lines
In this section we briefly review the argument of Ref. Lee:2006nr () and supply a proof of the factorization of the naive collinear Wilson line. Consider the naive collinear matrix element,
Here is the naive collinear field and is the naive collinear Wilson line. By naive we mean that the zero-bin mode has not been removed, either from the definitions of the fields or the SCET Lagrangian. The final state, , contains collinear quanta. For the (unphysical) quark form factor, contains a single collinear quark, while for a physical quantity there will be a sum over infinitely many particles, weighted by a shape variable. We will not specify any properties of since they are not required. Lee and Sterman Lee:2006nr () argue that the zero-bin mode couples to the purely collinear modes in the same way as a soft mode, so the zero-bin modes can be decoupled from purely collinear modes by a similar field redefinition. Let
be the Wilson line constructed from the zero-bin mode of the collinear gauge field. The zero-bin mode is decoupled from other collinear modes by the field redefinition
Though the collinear zero-bin has been decoupled, still contains zero-bin modes of the field . To make the dependence of these modes explicit in what follows, we define (without a prime) to be the zero-bin mode of and from now on use the notation to refer only to the purely collinear contribution, i.e., . The explicit dependence on can be extracted because of the following property of ,
which, as we will see below, holds up to corrections of . Here, is the purely collinear Wilson line which does not contain zero-bin modes, and is another zero-bin Wilson line defined by
We have factorized the final state, , into the product of a state that contains purely collinear degrees of freedom, , and a state containing zero-bin modes only, . We see that the naive collinear matrix element factors into the product of a purely collinear matrix element and a matrix element of zero-bin Wilson lines.
Note that Eq. (8) is not homogeneous in the power counting parameter . The field is , is , and has no definite scaling with since it can act on either the or present in . The factorization of Eq. (5) separates into Wilson lines that are solutions to first order equations that are homogeneous in . To obtain this factorization, we define the function by
Applying the chain rule to , it is straightforward to show that
If we try to write as a power series in , , all terms in the exponential are since is purely collinear. The approximation of Eq. (13) is equivalent to dropping zero-bin momenta relative to purely collinear label momenta in the evaluation of the Wilson line in momentum space.
A physical jet function is not of the form of Eq. (2), but rather the square of such a matrix element with a sum over final states. For example, the jet function in a factorization theorem for an event shape cross section takes the form Lee:2006nr (),
where is the event shape variable and the delta-function ensures that only final states with are summed over. Applying the factorization of the collinear matrix element in Eq. (2) to jet functions, one finds that the naively evaluated jet function can be expressed as a convolution of the purely collinear jet function in Eq. (14) and an eikonal jet function, which is defined in terms of matrix elements of the product of zero-bin Wilson lines, Lee:2006nr (). This convolution can be rendered a simple product in moment space, so one obtains
where , , and are the th-moments of the purely collinear, naive collinear, and eikonal (i.e. soft) jet functions. If the jet function of interest is sufficiently inclusive that the event shape only depends on the total momentum of the final state collinear particles (e.g., the factorization theorem for DIS as ), then one can use translation invariance and completeness to write the jet function as the Fourier transform of a -ordered product of the operators and . The naive and purely collinear jet functions in this case are related by an equation analogous to Eq. (15), where all jet functions are now defined by the corresponding -ordered products. For sufficiently inclusive processes the analog of is the well-known soft function. Therefore the work of Ref. Lee:2006nr () combined with the results of this section provides an effective field theory demonstration of how dividing by the soft function eliminates double counting in pQCD factorization theorems for inclusive processes.
Ii Two-Loop Zero-Bin Subtractions for the Jet Function
Applying the arguments of the last section to the incoming jet function that appears in the SCET factorization theorem for the quark form factor Idilbi:2007ff (), one finds that
Note that, unlike Section I, we are considering a jet function with a particle in the initial rather than final state. This quantity is clearly unphysical, however, it can be used to test the equivalence of soft and zero-bin subtractions since the argument for the factorization of the naively collinear matrix element into purely collinear and zero-bin matrix elements is independent of the initial and final states. The zero-bin subtractions needed for the purely collinear matrix element, , are the same zero-bin subtractions needed for virtual contributions to a physical jet function.
The remainder of this paper is devoted to completing the two-loop check of Eq. (16), initiated in Ref. Idilbi:2007ff (). Specifically, we wish to calculate the two-loop zero-bin subtractions for the left hand side of Eq. (16) and verify that they are reproduced by the right hand side. To the matrix elements on the right hand side of Eq. (16) can be parameterized as:
The and are the contributions to the naive collinear and soft matrix elements, respectively. We have made powers of and the color factors accompanying the diagrams explicit. and denote the contributions from two-loop collinear diagrams proportional to and , respectively. is the contribution from two-loop soft diagrams that are proportional to . The two-loop soft contribution multiplying the term is given by according to the exponentiation theorem. In Ref. Idilbi:2007ff (), we verified that the zero-bin subtractions for the purely collinear matrix element reproduced the terms in Eq. (17) proportional to and . In this section we will check the final term in Eq. (17), which indicates that the two-loop terms proportional to in the zero-bin subtraction and the soft function must be equal.
The two-loop zero-bin contribution to the naive collinear matrix element contains diagrams where all momenta are soft and also mixed collinear-soft zero-bins where one momentum is collinear and the other is soft. Therefore, Eq. (17) implies a nontrivial cancellation among Feynman diagrams with mixed collinear-soft momenta. We will verify this cancellation below. For the remainder of this section we will drop from the subscript on and since this is the only contribution we are concerned with.
The purely collinear contribution to a two-loop integral with integrand is given by Idilbi:2007ff ()
Here , and we use the notation of Ref. Idilbi:2007ff (). The term is the naive collinear contribution. For this contribution, the integrand is evaluated assuming collinear scaling for both momenta, which is denoted by the superscript on and in Eq. (18). The remaining terms in Eq. (18) are the zero-bin contributions which must be subtracted. There is the zero-bin arising when both and are soft, denoted by
The superscript denotes that the momenta are taken to be soft, and indicates that we are taking and to the soft region simultaneously Idilbi:2007ff (). There are also mixed collinear-soft zero-bins that arise when either is collinear and is soft,
or when is collinear and is soft,
The limit is defined by taking soft while is collinear, then taking to be soft. The limit is the same with and interchanged. To see that the limits are in general different, consider what happens to a propagator with momentum in the three limits:
When and are collinear, , , and are all . Taking and to the soft region simultaneously, , , and are all . The propagator in Eq. (II) is in the limit, as opposed to when and are collinear. However, the relative importance of all three terms in the denominator remains the same, so the form of the propagator is unchanged. In the limit, we first take soft while keeping collinear. Then and are but is and and are . The terms in the denominator are dropped. Next, we take soft. The propagator denominator is still , but now and are . The propagator scales as in the , , and limits, but the form of the propagator is different in each case.
Strictly speaking, Eq. (18) is naive since it does not account for the possibility that a mixed collinear-soft zero-bin arises when one linear combination of and becomes soft and the orthogonal linear combination stays collinear. One must check for a zero-bin for every linear combination of loop momenta that appears in a propagator of the Feynman diagram. However, in practice we find that the zero-bin contribution is subleading in unless the gluon propagator connects with the Wilson line. Therefore if we route the momenta so that the momenta of each gluon connected to the Wilson line coincides with one of the loop momenta, then Eq. (18) is sufficient.
Next we turn to the two-loop collinear SCET diagrams that give a contribution proportional to . Two-loop QCD-like graphs are depicted in Fig. 1, while two-loop graphs with the SCET seagull vertex are shown in Fig. 2. We will analyze diagrams in Fig. 1 first, and start by considering the zero-bin from the limit. It is easy to see that this contribution is identical to the contribution of the two-loop soft matrix element proportional to . All calculations are performed in Feynman gauge. The external quark is on-shell and massless. Because the equivalence relies on field redefinitions, we cannot regulate IR divergences by taking external particles off-shell. We will use DR to regulate both UV (ultraviolet) and IR divergences. We denote the momentum of the outermost gluon connected to the Wilson line as . In Figs. 1(a) and (b), the inner gluon connected to the Wilson line has momentum . In Figs. 1(c) and (d), is the momentum of one of the gluons in the quark-gluon vertex correction subgraph. In Fig. 1(e), the blob represents all possible contributions to the gluon self-energy that give rise to a color factor . The loop momentum is the momentum of one of the gluons or ghosts in the self-energy subgraph. Then the propagator denominators of all gluons (or ghosts) in Fig. 1 are , , or . From Eq. (II) we see that these propagators are unchanged in the limit. The coupling of the gluons to the Wilson line is also unaffected by taking the limit. Finally, we need to consider the modification to the coupling of the collinear quark to the soft gluon. In the limit, expanding to lowest order in , it is easy to show that the collinear quark coupling and propagator give the same Feynman rules as the eikonal Wilson line, . No analog of Fig. 1(d) exists in the soft matrix element. In the limit of the collinear matrix element, Fig. 1(d) gives a contribution that is subleading in . The limit of the other diagrams in Fig. 1 are exactly the same as the corresponding two-loop diagrams contributing to the term in the calculation of the soft matrix element, shown in Fig. 3. It is then obvious that
where denotes the sum of the integrands coming from the two-loop diagrams in Fig. 1(a)-(e). Thus the zero-bin subtraction and soft Wilson line subtraction give equivalent results if the remaining zero-bin contributions all vanish. The remaining possible zero-bin contributions are the mixed collinear-soft zero-bins of the diagrams in Fig. 1 and zero-bin contributions from diagrams with the SCET seagull vertex depicted in Fig. 2. Note that the mixed collinear-soft zero-bin subtraction does not vanish for the two-loop Abelian diagrams and is necessary to reproduce the term in Eq. (17).
We next analyze the mixed collinear-soft zero-bin contributions from Fig. 1. With the routing described above it is straightforward to show that the zero-bin subtraction is subleading in unless it comes from the region where is soft and is collinear. There is no mixed collinear-soft zero-bin contribution from Fig. 1(e). The reason is that when is collinear and soft, Fig. 1(e) gives a contribution proportional to
From the naive collinear integrand in Eq. (24) one must subtract the integrand in the limit. However, the integrand in Eq. (24) does not change in the limit, so the integrand when is collinear and is soft vanishes,
The remaining diagrams, Figs. 1(a)-(d), have nonvanishing mixed collinear-soft zero-bin contributions. There is a nontrivial cancellation among the four diagrams that ensures that the two-loop zero-bin subtraction proportional to reproduces the right hand side of Eq.(17). We will describe the evaluation of Fig. 1(b) in some detail and quote our results for the remaining diagrams. Evaluation of Fig. 1(b) gives
When and are taken to be soft and collinear, respectively, we get the naive collinear contributions to ,
The factor of arises because is defined to be the amplitude of the Feynman diagram divided by . The -integral is easily evaluated using Feynman parameters and the result is
The divergences are of IR origin and this is denoted by the subscript on the DR parameter .
Note that the subgraph containing the virtual momentum gives rise to a double IR pole, . In a one-loop diagram, double poles come from regions where the virtual momenta is both soft and collinear. If is purely collinear we do not expect to see a double IR divergence in this subgraph. This arises in the naive collinear graph because we have not excluded the region where is soft. Once we perform the zero-bin subtraction (by removing the region) the final answer must be free from poles. We will see that this is the case below.
Taking the limit of the integrand in Eq. (27) yields
where the last line follows from the change of variables and . The same integral arises in the evaluation of the limit of Fig. 1(c). The evaluation of this integral is subtle and is discussed in the Appendix. The result is
so the mixed collinear-soft zero-bin contribution from Fig. 1(b) is
As expected the poles cancel in the purely collinear loop integral.
We obtain a similar result for Fig. 1(a),
When evaluating Fig. 1(c) in the limit that is soft and is collinear, we find that the -integral can be split into a UV divergent term, an IR divergent term, and a term that is identical to the -integral of Eq. (24). The last term gives a vanishing contribution once the limit of the integrand is subtracted. The remaining terms give
Finally, Fig. 1(d) gives
We see that
so the mixed collinear-soft zero-bin contributions from Figs. 1(a)-(e) add up to zero.
In addition to the mixed collinear-soft zero-bin discussed above there are two-loop SCET diagrams that have no QCD analog. These graphs, given in Fig. 2, involve the seagull interactions with two collinear quarks and two collinear gluons (see Fig. 1 of Ref. Bauer:2000yr ()). The Feynman rule for the SCET seagull interaction is such that if a collinear gluon from the seagull vertex is contracted with the gluon coming from the Wilson line, , the graph vanishes, so the diagrams of Fig. 2(a)-(c) are zero. For Fig. 2(d) we obtain
When is soft and collinear one obtains the naive collinear contribution to :
which by SCET power counting is . The integration over can be performed by combining denominators using the standard Feynman parameterization and after completing the square the resulting integral vanishes by symmetry. Therefore this zero-bin contribution vanishes. It is also easy to check that the zero-bin contributions from both the and regions are subleading in the expansion and can be ignored. This result is essential as Fig. 2(d) has no analog in the soft function diagrams shown in Fig. 3.
In this paper, we completed the two-loop analysis of zero-bin subtractions to the jet function appearing in the quark form factor initiated in Ref. Idilbi:2007ff (). We verified that the zero-bin subtraction is equivalent to dividing by a matrix element of soft Wilson lines to two-loop order. We also supplied a proof of the factorization of the naive collinear Wilson line into a purely collinear Wilson line and a soft Wilson line, valid to . This property is essential for the argument for the equivalence of soft and zero-bin subtractions first presented in Ref. Lee:2006nr (). These arguments imply that the equivalence of soft and zero-bin subtractions should hold to all orders in perturbation theory.
One important consequence of this equivalence (to lowest order in ) is that it provides an operator definition for the zero-bin subtraction procedure that goes beyond perturbation theory. The equivalence may also help to simplify higher order calculations in SCET. For example, in the calculation of the jet function in this paper, we saw that the equivalence led us to anticipate a cancellation between mixed collinear-soft zero-bin subtractions coming from several different SCET diagrams. We expect that the equivalence will simplify SCET calculations of collinear correlation functions, like parton distribution functions, jet functions, and fragmentation functions, since it allows one to work with the naive collinear correlation functions and soft matrix elements, rather than having to remove zero-bins in collinear correlation functions diagram by diagram.
Though the analysis of this paper, as well as Refs. Lee:2006nr () and Idilbi:2007ff (), focuses on avoiding double counting in factorization theorems for inclusive processes, the issue of double counting arises in exclusive processes as well Manohar:2006nz (). It would be interesting to see if one could find an operator definition of the zero-bin subtraction in SCET.
Acknowledgements.This work was supported in part by the Department of Energy under grant numbers DE-FG02-05ER41368, DE-FG02-05ER41376, and DE-AC05-84ER40150. We thank S. Fleming for useful comments.
In this Appendix, we discuss our evaluation of the integral in Eq. (30). First, we use the identity
The parameter is dimensionful and can be made dimensionless by rescaling . The integral is then straightforward to evaluate by completing the square. The result is
The integral is proportional to while the from the integral is clearly IR in origin. Note that we have set a factor of
equal to 1. Anything besides the double poles in the evaluation of the scaleless integral is ambiguous. One source of ambiguity is the freedom to rescale . We chose to rescale so as to obtain a prefactor proportional to , but other rescalings are possible. The other source of ambiguity is due to the ambiguity in expanding any function of when multiplying a factor of .
However, there is even more ambiguity in the result because the coefficient of the double poles actually depends on how one choses to combine denominators using Feynman parameters. To see this we now evaluate Eq. (30) using the identity
The result is then
which differs from Eq. (40) by a factor of 1/2!
Mathematically, there is no inconsistency here, since so the right hand sides of Eq. (40) and Eq. (43) are both equal to zero. Both results are consistent with the well-known result that scaleless integrals vanish in DR. However, if a physical regulator is used, a scaleless integral would be the sum of UV and IR divergent terms. Since the cancellation of UV and IR divergences in physical observables is handled differently in quantum field theory, one often wishes to separate scaleless integrals into their UV and IR divergent parts, even if DR is being used to regulate both. Evidently, for the double poles appearing in the integral of Eq. (30), such a separation is ambiguous.
Faced with this situation, one is forced to invoke a prescription for handling the zero-bin integrals in the limit. One physically motivated prescription was mentioned earlier, that is to fix the overall coefficient so that the poles cancel in the purely collinear integral. This leads to the result of Eq. (40). Another way to resolve the ambiguity is to modify the integral by including a -suppressed correction. For instance, we can replace with with . Now the result of the integral is completely unambiguous and equal to the result of Eq. (40) with the following modification
This prescription unambiguously fixes the coefficient of the mixed pole and is in agreement with the result of Eq. (40). We will use the result of Eq. (40) for the evaluation of the -integral for the zero-bin integral in Eq. (30).
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