Gao-Liang Zhou,Zheng-Xin Yan,Xin Zhang and Feng Li

1 College of Science,Xi’an University of Science and Technology,Xi’an 710054,China

2 College of Science,Nanjing University of Posts and Telecommunications,Nanjing 210023,China

Abstract Effects of Glauber gluons,which cause the elastic scattering process between different jets,are studied in the frame of soft-collinear effective theory.Glauber modes are added into the action before being integrated out,which is helpful in studies on the Glauber couplings of collinear and soft particles.It is proved that the final state interactions cancel out for processes inclusive enough.So are interactions with the light cone coordinates x+ and x- greater than those of the hard collision.The eikonalization of Glauber couplings of active particles and absorption of these couplings into soft and collinear Wilson lines are discussed,which is related to the loop level definitions of Glauber gluons here.The active-spectator coherence is proved to be harmless to the inclusive summation of spectator-spectator and spectator-soft Glauber exchanges.Based on this result,spectator-spectator and spectator-soft Glauber exchanges cancel out in the processes considered here.Graphic aspects of the cancellation are also discussed to explain relations between the graphic and operator level cancellation of Glauber gluons.

Keywords:soft-collinear effective theory,QCD factorization,Glauber gluons

1.Introduction

Soft-collinear factorization is crucial in connecting perturbative QCD calculations with experimental data of high energy hadron processes.Such factorization is proved in many typical processes in the frame of perturbative QCD,(see,for example,[1–8]).Compared to these results,the factorization in soft-collinear effective theory(SCET)[9–12]provides us with a new perspective to understand the QCD factorization,which seems more intuitive.Collinear,soft and ultrasoft particles are described by effective modes in SCET.Especially,the hard collisions between collinear particles are described by various effective operators in SCET.In the original SCET Lagrangian,ultrasoft gluons decouple from collinear fields after a unitary transformation and collinearsoft couplings are absorbed into soft Wilson lines.Thus the soft-collinear factorization holds at the Lagrangian level in the original SCET.

Despite great advantages,the original SCET did not deal with Glauber gluons properly.Glauber gluons,which take space-like momenta,are responsible for elastic scattering processes between different jets.Results in[13]displayed how Glauber gluons break the QCD factorization in processes involving two initial hadrons like the Drell–Yan process.For Drell-Yan process,leading pinch singularities in the Glauber region cancel out according to the unitarity[2–4].One may then deform the integral path of loop momenta to avoid the Glauber region.After the deformation,couplings between Glauber gluons and collinear particles eikonalize.That is,Glauber gluons behave like soft or collinear gluons after the deformation.In summary,the soft-collinear factorization of the Drell–Yan process is not violated by Glauber gluons[2–4].However,the factorization can be violated by Glauber gluons for processes that are not inclusive enough(see,for example,[14–18]).In these processes,cancellation of leading pinch singularities in the Glauber region is hindered by Glauber couplings of the detected final particles

Although leading pinch singular singularities in the Glauber region may cancel out,the Glauber gluon effects are visible in processes in which the factorization works.Since the contour deformation to avoid the Glauber region relies on explicit processes,whether the collinear and soft Wilson lines appearing in parton(distribution and fragmentation)functions and soft factors are past-pointing or future-pointing is process dependent[19].In other words,Glauber gluons affect the directions of various Wilson lines in the factorization.Such effects are more obvious in transverse momentum-dependent objects like the Sivers function[20–22],although they are not concerned here.

Glauber gluons in SCET are more subtle.In[23–27],Glauber gluon fields are added into the SCET Lagrangian to describe jets in dense QCD matter.The effective theory is termed SCETG.Glauber gluons in SCETGbehave like a QCD background and do not cause scattering between different jets directly.This is different from the situation one confronts in usual soft-collinear factorization.In[28–30],Glauber gluons are integrated out and effective operators that describe elastic scattering effects between collinear particles are introduced into the effective action of SCET.These operators are nonlocal,in which the decoupling of ultrasoft gluons from collinear fields is no longer manifest.Matching the coefficients of these operators relies on a suitable subtraction scheme[28,31]to avoid double counting in loop integrals and a systematic scheme[32,33]to regularize the rapidity divergences.These effective operators may violate the factorization theorem in SCET as they cause coherence between different jets.In[30],the authors discuss the properties of these operators.

In[30],ladder diagrams like those shown in figure 1 were discussed.Explicitly,the ladder diagrams of spectatorspectator Glauber exchanges like those shown in figure 2 cancel out for processes inclusive enough as discussed in[30].In addition,the ladder-like spectator-active and active–active Glauber exchanges can be absorbed into collinear and Wilson lines according to results in[30].

These discussions are necessary for proofs of the factorization theorem in SCET.However,they are not enough as these discussions are restricted to ladder-like diagrams.More discussions on these topics are necessary.This is the primary motivation of our paper.

Glauber gluons should be viewed as modes different from collinear and soft gluons no matter whether they are integrated out or not.The question is how to distinguish them from collinear and soft gluons in loop integrals.In other words,how to avoid the double counting in loop integrals?Let us start from the coupling between a gluon q and a pluscollinear particle k.If q is soft or ultrasoft or minus-collinear,then the coupling between q and k eikonalize at a leading power of λ.Hence we can define the non-eikonalized part of the coupling as Glaube coupling of k even if q locates in the non-Glauber region without affecting the leading power results.On the other hand,the eikonalized part of the coupling can be absorbed into soft or ultrasoft or collinear Wilson lines even if q locates in the Glauber region.Thus we always define the non-eikonalized part as Glaube coupling of k and absorb the eikonalized part into soft or ultrasoft or collinear Wilson lines.For a Glauber gluon q exchanged between pluscollinear and minus-collinear particles,one has

One can also neglect l+(l-)in couplings between l and pluscollinear(minus-collinear)particles.These approximations are helpful to describe the Glauber couplings in loop integrals.Let us consider the coupling between a gluon q and a plus-collinear particle k further.If q is soft or minus-collinear,then the coupling between q and k eikonalize.The eikonalized part of the coupling can be absorbed into collinear or soft Wilson lines and does not affect the factorization even if q locates in the Glauber region.The non-eikonalized part should be power suppressed in the minus-collinear or soft(ultrasoft)region of q.Hence the rapidity divergences in this definition can be controlled through the regulator presented in[30,32,33],which reads

We find it convenient to introduce Glauber gluon fields into the SCET action before integrating them out.This is helpful to determine the power counting for couplings involving Glauber gluons.Especially,it helps us to see the origins of leading power effective operators in[30].We should mention that the power counting for various modes may depend on explicit gauge conditions in perturbative calculations as shown in[28–30].Compared to the covariant gauge in[28–30],it is more convenient to work in the Feynman gauge for issues considered here.There are superleading powers in practical diagrams in the Feynman gauge.However,such superleading powers cancel out in physical observable according to the Ward identity[6,7].This is confirmed by the power counting result presented in this paper.Thus superleading powers in the Feynman gauge do not disturb us.

The subtraction of eikonalized couplings from the definition of Glauber gluon modes is important in the treatment of elastic scattering processes.The eikonalized part of couplings involving collinear particles should be viewed as soft and collinear and ultrasoft interactions of these collinear particles even if there are gluons with Glauber momenta.Especially,the Glauber couplings of active particles eikonalize and these couplings should be absorbed into the definition of collinear or soft gluons.Besides,the subtraction does not affect spectator-spectator type Glauber gluons in the dimensional regularization scheme after the approximation(1)as dimensionless integrals vanish in the scheme.

The paper is organized as follows.In section 2,we add Glauber gluons into the SCET action.The power counting for couplings involving Glauber gluons is also presented in this section.In section 3,we discuss how to understand the cancellation of ladder-like Glauber exchanges between spectators presented in[30].These discussions can be viewed as operator level interpretation of such cancellation.The cancellation of final state interactions for processes inclusive enough is also proved in this section,which is crucial in our general discussions.In section 4,we consider the Glauber couplings of active particles.We prove the eikonalization of these couplings and explain why these couplings are equivalent to zero bins of soft and collinear couplings of active particles.The absorption of spectator-active Glauber exchanges(including non-ladder-like cases)into collinear Wilson lines and active–soft and active–active Glauber exchanges(including non-ladder-like cases)into collinear Wilson lines is the direct result of discussions in this section.In section 5,we prove the cross section level cancellation of spectator-spectator Glauber exchanges for processes inclusive enough.We exclude the influence of spectator-active coherence by proving that the coherence should occur before spectatorspectator Glauber exchanges.Hence the summation over final spectators without affecting spectator-active coherence is enough for the cancellation of these Glauber exchanges.After the cancelation,Glauber modes can be absorbed into collinear and soft Wilson lines which are graph independent.In section 6,we explain how our operator skills are related to the graphic cancellation of spectator–spectator Glauber exchanges in[2–4].Our conclusions and some discussions are presented in section 7.In appendix A,we discuss the evolution operator of the effective theory.In appendix B,we discuss how to define spectators and collinear particles.The reparameterization invariance of the effective theory is discussed in appendix C.

2.Glauber gluons in SCET

In this section,we introduce Glauber gluon fields into the SCET action and study Glauber interactions.These discussions are helpful for studies of Glauber effects in hadronic processes at leading power.Glauber gluons take space-like momenta and cause elastic scattering between collinear particles.For example,we consider a gluon with momentum scales as

2.1.Power counting for couplings between Glauber gluons and other particles

In this section,we consider Glauber interactions.We first consider couplings between Glauber gluons and ultrasoft gluons.According to table 1,the power counting for ultrasoft particles and Glauber gluons reads λ3In[9,10],the authors work in the covariant gauge,

where ξ is the gauge parameter.The power counting for the field AnreadsIt is required that 1-ξ is not too small to get this result.While working in the Feynman gauge,there are superleading power terms involving n·An,pin SCET Lagrangian.This does

not disturb us as n·An,pandappear in pairs in practical diagrams.In fact,if there is a collinear gluon polarized as n·An,p,then its other end should polarize asin the Feynman gauge according to the gμνtensor in its propagator.The power counting for such a pair reads λ2according to table 1,which is equivalent to that of the covariant gauge in[9,10].

,λ

2

and

.The integral volumes of these couplings scale as ∫d

4

x ～Q

-4

λ

-4-b

.In these couplings,there are at least two Glauber gluons and one ultrasoft gluon.The power counting for the combination of these fields reads λ

4+b

.There is an additional gluon field or momentum operator in these couplings according to Lorentz covariance.The power counting for the additional gluon feild reads

That of the momentum operator reads λ or λ

2

.Thus the infrared power counting for these couplings reads λ

r

,where

That is to say,the infrared power counting for these couplings reads λ or higher.For the couplings between Glauber gluons and ultrasoft fermions,the situation is similar.The integral volumes scale as ∫d4x ～Q-4λ-4-b.In these couplings,there are at least one ultrasoft fermion and one Glauber gluon and one fermion with momentum scales asThe power counting for these fields readsrespectively. Momenta of these fields are quite small and thereare no energy scales that may produce the minus power of λ.Thus the infrared power counting for these couplings readsλ32orhigher.and the power counting for them in the Feynman gauge,where.

Table 1.Relevant modes in the propagation of particles collinear to and the power counting for them in the Feynman gauge,where.

Table 1.Relevant modes in the propagation of particles collinear to and the power counting for them in the Feynman gauge,where.

Modes Fields Momenta scales⊥n p n p p,, n(· ·)Infrared power counting Collinear quarks ξn Q(λ2,1,λ)λ Collinear gluons μAn Q(λ2,1,λ)λ Soft quarks ψs Q(λ,λ,λ)λ3/2 Soft gluons μ As Q(λ,λ,λ)λ Ultrasoft quarks ψu(yù)s Q(λ2,λ2,λ2)λ3 Ultrasoft gluons μ Aus Q(λ2,λ2,λ2)λ2 Glauber gluons μ AnG Q(λ2,λb,λ)(b=1,2)λ +1 b 2

2.2.Leading power action including Glauber gluons

We consider the leading power SCET action with Glauber gluon fields in this section.There are two kinds of SCET Lagrangian in literature,SCETIand SCETII,which are suitable for studies on different objects.We do not distinguish them here.For simplicity,we neglect the couplings involving ultrasoft particles and the couplings between Glauber gluons without soft gluons at first.

We start from a Glauber gluon AnGwhich couples to ncollinear particles.The power counting for such coupling readsaccording to the results in table 2.The other end of the Glauber gluon may couple to ultrasoft particles,Glauber gluons,soft particles or particles collinear to other directions.If the Glauber gluon couple to particles collinear to other directions at that end,then the power counting for that coupling reads.The final power counting for couplings at two ends of the Glauber gluon reads λ0.4There may be additional powers of λ in practical diagrams even if these diagrams involve only leading order couplings as shown in[30].This does not disturb us here as we are only concerned with the power counting for effective iterations in this paper.

If the other end of the Glauber gluon involves soft particles,then the power counting for that coupling readsor λ.For the former case,the final power counting for couplings at two ends of the Glauber gluon reads λb-1≥λ0.For the latter case,in which b=1,the other end of AnGinvolves a Glauber gluon of the typeIf the other end of thecouple to particles collinear tothen the final power counting for these couplings readsIf the other end ofcouples to soft particles,then we can repeat the procedure and get the same result.In conclusion,the final power counting for these couplings reads λ0if there are couplings between Glauber gluons and soft particles.

If a Glauber gluon is exchanged between soft particles,then the power counting for the combination of couplings at the two ends of the diagram reads,λb≥λ1.We notice that the minus powers of λ can only be produced by couplings between glauber gluons and collinear particles.Thus the combination of couplings involving Glauber gluons is power suppressed except for those couplings between Glauber gluons and collinear particles are involved.

We then consider couplings involving ultrasoft particles and couplings between Glauber gluons without soft gluons.The power counting for the couplings between Glauber gluons and ultrasoft particles reads λ1or λ3/2.The power counting for couplings between Glauber gluons without soft gluons reads5One should not be confused with the possible fractional power in this coupling.There are other powers of λ in diagrams involving this coupling.For example,one considers the coupling between three Glauber gluons of the type AnG.Two of them connect to a fermion collinear to nμ and one of them collinear toμ .At leading power,two of them are of the type ·AGand one of them is of the type n·AG in the coupling between these three Glauber gluons.According to Lorentz covariance,there is a momentum term of the type n·k ～λ2 instead of the type k⊥～λ,which is produced by the couplingAccording to the above discussions,the combination of other couplings involving Glauber gluons does not produce minus powers of λ.Thus couplings involving ultrasoft particles and couplings between Glauber gluons without soft gluons are power suppressed.

In Summary,(1)Glauber gluons are exchanged between collinear particles and soft particles or between collinear particles at leading power;(2)there may be intermediate couplings between Glauber gluons and soft particles in these exchanges at leading power;(3)couplings between Glauber gluons and ultrasoft particles are power suppressed;(4)couplings between Glauber gluons without soft gluons are power suppressed.This is compatible with the results in[30].

The leading power effective action can then be written as6The perpendicular components decouple from other fields at the leading power of λ.We simply disregard them here.7We should mention that there is overlap between and ,which should be subtracted from the effective action.The subtraction is not displayed here for simplicity.

Some power suppressed terms are added into the action to maintain the BRST covariance as discussed in appendix A.We see that diagrams involving AnG(k)rely on n·k through various propagators and vertexes involving AnG(k)are independent of n·k at leading power.This is crucial in our following discussions.

Table 2.The infrared power counting for couplings involving Glauber gluons,where.

Table 2.The infrared power counting for couplings involving Glauber gluons,where.

Couplings Fields Power counting Glauber gluons and ultrsoft gluons(AnG,Aus)λ or higher Glauber gluons and ultrsoft fermions(AnG,Aus)λ32 or higher Glauber gluons(AnG)λb2 or higher Glauber gluons and soft gluons(AnG,AnG ,As)λ or higher Glauber gluons and soft gluons(AnG,As)λb2 or higher Glauber gluons and soft fermions(AnG,ψs)λb2 or higher Glauber gluons and collinear fermions(AnG,ξn)λb-1 2 or higher Glauber gluons and collinear gluons(AnG,An)λb-12 or higher Glauber gluons and collinear fermions(AnG,ξn)λ -1 b 2 or higher Glauber gluons and collinear gluons(AnG,An)λ -1 b2 or higher

Before the end of this section,we briefly discuss the Glauber gluon states.Although the momenta square of Glauber gluon modes is of order Q2λ2like those of collinear and soft particles,there are not time derivative terms of Glauber gluon modes in the leading power action(7).Hence Glauber gluons should be viewed as constraints not canonical variables in the effective theory.In other words,Glauber gluon fields do not correspond to quantum physical states.On the other hand,one may try to solve the constraint,which is equivalent to integrating out Glauber gluon fields.We show the motion equation of Glauber gluon fields in appendix A.While referring to Glauber gluon states,we always mean the states corresponding to the field configuration of the solution of Glauber gluons.In other words,the relevant quantum states in the soft-collinear factorization are not changed by Glauber gluon fields.

3.Elastic scattering effects in hadron collisions

We do not consider quantities dependent on q⊥in this paper.Thus one can integrate out q⊥in the following discussion.

3.1.Elastic scattering effects without interactions between spectators and active particles

In this section,we start from spectator–spectator interactions without the active-spectator coherence8Effects of the active-spectator coherence are discussed in sections 5.2–5.4..We show an operator level explanation for the spectator–spectator cancellations discussed in[30].Our explanation based on the unitarity is quite general and helpful to understand what happens in inclusive processes.Given that the active-spectator coherence is neglected,such cancelation can be easily extended to non-Glauber interactions.

Let us start from the diagrams shown in figure 2.These diagrams can be written as,

where Iacis the active part of Ieffand Ispis the spectator part of Ieff9Given that the active-spectator coherence is neglected,such decomposition is feasible.Otherwise,there should be active-spectator coherence terms..The interactions between spectators and active particles have been dropped in the above decomposition.We have

where the summation is made over all possible final particles with fixed total momentum and color and angular momentum.Thus the spectator–spectator interactions cancel out in processes inclusive enough given that the active-spectator coherence haven been dropped as in figure 3.

3.2.Cancellation of final state interactions

In this section,we take into account the active-spectator coherence and prove the cancellation of final state interactions in the process is inclusive enough.Such general cancellation originates from the unitarity.

For diagrams with the active-spectator coherence,like those shown in figure 4 and their conjugations,the situation is more complicated.Let us consider the quantity,

where H(q)represents the detected lepton or jet pair with total momenta q and the summation is made over all possible final states.Generally,there is active-spectator coherence in U(t1,t2)and the factorization(23)does not simply work.

Fortunately,the contractions betweenO()and other fields occur only if the time coordinates of fields inO(x)are smaller than those of other fields.Otherwise,the contraction does not contribute toH.We can writeH as

where the order between operatorsO anddoes not affect the result.

and have

The summation in the above quantity is made over all possible final states.The completeness of the final states X10X represents all possible other states.Hence H(q)X represents all possible states that contain the detected lepton or jet pair.H(q)X forms the invariant subspace of QCD evolution unless one considers more hard processes after the hard collision(so that the jet pair may vanish).Effects of such hard processes are power suppressed in according to the physical picture of pinch singular surfaces[6,34].and the unitarity of U(t1,t2)hint that

We notice that

We have,

Hence final state interactions cancel out inThis is compatible with the results in[2–4].

Especially,the effects of couplings between Glauber gluons and final active particles cancel out inH as these couplings occur after the hard collision.

4.Glauber gluons coupling to active particles

4.1.Eikonal approximation in couplings between collinear particles and soft(ultrasoft)gluons

In this section,we briefly explain the eikonal approximation in soft and ultrasoft couplings of collinear particles.11Couplings between collinear particles and soft(ultrasoft)fermions are power suppressed[9,10].Let us consider the coupling between a plus-collinear particle p and a soft or ultrasoft gluon q.The power counting for p and q reads

We have12

That is to say,the absorption of soft and ultrasoft gluons into light-like Wilson lines is the direct result of the eikonal approximation.13We do not distinguish the Wilson lines of soft and ultrasoft gluons here as it is irrelevant to the main result in this section.

The approximation(40)also works in couplings between plus-collinear particles and minus-collinear gluons.Although the eikonal approximations(39)and(40)seem rather diagrammatic,it is necessary in loop level definition of effective modes considered here.14In fact,the eikonalization approximation is crucial for the higher order definition of spectator and active in[30].It is also the origin of various Wilson lines in SCET[11].Hence discussions on the approximations are necessary to examine whether Glauber interactions can be absorbed into collinear and soft Wilson lines or not.

However,the Glauber couplings of collinear particles are more subtle.Taking the coupling between a plus-collinear particle p and a Glauber gluon l as an example,the power counting for p and l reads

We have

and the eikonal approximation does not simply work here.

The mermaid kissed his high, smooth forehead, and stroked back his wet hair; he seemed to her like the marble statue in her little garden, and she kissed him again, and wished20 that he might live

4.2.Eikonalization of couplings between active particles and Glauber gluons in H(P,-P,q+,q-)

In this section,we consider the Glauber couplings of active particles inand prove the eikonalization of these couplings.We should emphasize that such eikonalization is independent of the other ends of the Glauber gluons.Hence we consider couplings between Glauber gluons and an arbitrary active particle with the other ends of the Glauber gluons discussed in sections 4.3–4.5.

The distinction between spectators and active particles is discussed in appendix B.According to results in appendix B,the plus-collinear(minus-collinear)active particles are defined as:(1)plus-collinear(minus-collinear)particles coupling to the hard vertex directly;(2)plus-collinear(minus-collinear)particles for which there is a path made up of plus-collinear(minus-collinear)particles and the plus momenta(minus momenta)of them flow into the hard vertex through the path.We discuss these two cases one by one.

We start from active particles which participate in the hard collision directly.Without loss of generality,we consider Glauber couplings of a plus-collinear active particle k

Let us consider couplings between k and some Glauber gluons l1,…,lnas shown in figure 5.

At the leading power of λ and η,we have

One can easily verify that

Hence

at leading power of λ and η.That is to say,the Glauber of couplings k eikonalize at leading power of λ and η.

We then consider active particles which do not couple to the hard vertex directly.We consider a plus-collinear active particlek′ .′+kis restricted to flow into the hard vertex.We consider Glauber couplings of active particles along the flow of the plus momenta.k′ may couple to some Glauber gluons l1,…,lnand other particles q1,…,qm.The momentum of the active particle becomesk′+l1+ … +ln+q1+ …+qm.Finally one meets the hard vertex,which is independent of l1,…,ln.At the leading power of λ and η,these couplings rely on l1,…,lnthough the term

For the Glauber couplings of minus-collinear active particles,we have similar results.In conclusion,Glauber couplings of active particles eikonalize at the leading power of λ and η in.

4.3.Glauber gluons exchanged between active and soft particles

We consider active-soft Glauber exchanges in this section.These Glauber gluons can be absorbed into soft Wilson lines according to discussions here.

Without loss of generality,we consider the Glauber gluons exchanged between plus-collinear active particles and soft particles.To specify our discussions,let us consider the Glauber gluons(l1,…,ln)exchanged between a plus-collinear active particle k and a soft particle ks.

The couplings between k and Glauber gluons read(49)

at leading power of λ and η,where D=4-2ε.We compare the result with the case that k couple to soft gluons q1,…,qn.After the eikonal approximation,the couplings between q1,…,qnand k read

at leading power of λ and η.Comparing(51)with(52),we see that Glauber gluons behave like soft gluons while coupling to k.

l1,…,lncouple to soft particles kson the other ends.While coupling to soft particles,Glauber gluons behave like soft gluons as demonstrated in the effective action(7).In fact,the Glauber region can be viewed as the subregion of the soft region in loop integrals once the Glauber couplings eikonalize.Hence the couplings between Glauber gluons and kscan be absorbed into soft couplings of ksby absorbing the Glauber region into the soft region in loop integrals.In summary,the active-soft type Glauber gluons behave like soft gluons on both ends.

We also notice that the couplings between Glauber gluons and ultrasoft particles are power suppressed according to table 2.So are the couplings between soft gluons and ultrasoft particles.Hence Glauber gluons behave like soft gluons at the leading power of λ while coupling to ultrasoft particles.

In addition,the propagators of ljbehave like those of qjin the special momenta region.

According to the above facts,we see that Glauber gluons exchanged between ksand k behave like soft gluons and can be absorbed into soft Wilson along +-direction.For other active-soft Glauber exchanges,we have similar results.In other words,active-soft Glauber exchanges can be absorbed into zero bins of soft Wilson lines.15The Wilson lines should be past pointing according to the poles’location ofin(49).One may also see this by simply realizing that final interactions cancel out in as discussed in section 3.2.

4.4.Glauber gluons exchanged between spectators and active particles

We consider active-spectator Glauber exchanges in this section.These Glauber gluons are absorbed into collinear Wilson lines according to discussions here.

Without loss of generality,we consider the Glauber gluons exchanged between a minus-collinear spectatorand a plus-collinear active particle k(k+＞0 as plus momenta flow from plus-collinear particles to the hard vertex).While coupling to spectators,these Glauber gluons behave like the collinear gluonswithOn the other hand,couplings between Glauber and ultrasoft gluons are power suppressed according to the power counting results in table 2.Hence,we can use the eikonal approximation in couplings between Glauber and ultrasoft gluons without affecting the leading power results.So are couplings between these Glauber gluons and other Glauber gluons.16Couplings between Glauber and soft gluons are discussed in section 4.4.In other words,these Glauber gluons behave like collinear gluonswithwhile coupling to spectators and ultrasoft and Glauber gluons.

On the other end,these Glauber gluons couple to pluscollinear active particles.According to the results of section 4.2,such couplings eikonalize at the leading power of λ and η.The couplings between these Glauber gluons and kread(49)

at leading power of λ and η,where D=4-2ε.On the other hand,one may consider the couplings between k and minuscollinear gluons.We denote the momenta of these collinear gluons asThe couplings between these collinear gluons and k can be written as

at leading power of λ and η,where we have made use of the eikonal approximation in couplings between k andWe see that the couplings between k and l1,…lnbehave like those between k and.

In summary,Glauber gluons exchanged betweenand k behave like collinear gluonswithon both ends.The propagators of these Glauber gluons behave like those of the collinear gluonsHence the Glauber gluons exchanged betweenand k can be absorbed into minus-collinear Wilson lines by extending the collinear region to include the Glauber region in the loop integrals.17The Wilson lines should be past pointing as final interactions cancel out in as discussed in section 3.2

For general active-spectator Glauber exchanges,we have similar results.In conclusion,active-spectator Glauber exchanges can be absorbed into collinear Wilson lines.

4.5.Glauber gluons exchanged between active particles

We consider active–active Glauber exchanges in this section.They can be absorbed into soft Wilson lines for ladder diagrams[30].In this section,we extend the result to the general situation.

For general active–active Glauber exchanges,we have similar results.Hence active–active Glauber exchanges can be absorbed into soft Wilson lines by extending the soft region to include the Glauber region in loop integrals.18These Wilson lines should be past pointing as final interactions cancel out in .

5.Cancellation of spectator–spectator and spectator-soft Glauber exchanges in H(P,-P,q+,q-)

In this section,we prove the cancellation of the spectator–spectator and the spectator-soft Glauber exchanges in19Soft–soft Glauber exchanges are power stressed and can be absorbed into soft exchanges between soft particles..Calculations in[30]show the cancellation of the spectator–spectator type Glauber exchanges in ladder diagrams.According to our discussions in section 3.1,ladder diagrams of Glauber gluons exchanged between spectators in H can be understood as perturbative series of the object

In general cases,the active-spectator coherence may obstruct the summation over all possible final spectators.For example,one may consider the diagram shown in figure 6.

The gluon exchanged between spectators and active particles should not be collinear in the first two diagrams.Otherwise the two diagrams do not contribute to the process considered here.In other words,the summation over Glauber interactions of spectators is hampered by the active-spectator coherence in figure 6.One should deal with the active-spectator coherence carefully to get the cancellation of spectatorspectator and spectator-soft Glauber exchanges.

If the Glauber interactions occur after the spectator-active and the soft-active interactions like the first diagram in figure 7 then the summation over all states after the spectatoractive and the soft-active interactions is inclusive enough for the spectator–spectator and the spectator-soft Glauber exchanges.

The cancellation(58)can be extended into this case even if there are spectator-active and soft-active interactions.It seems important for us to exclude spectator-active and softactive interactions after spectator–spectator and spectator-soft Glauber exchanges like those in figure 6.However,instated of the time evolution of collinear and soft states,we find it convenient to consider the evolution of these states along a nearly light-like direction.This is displayed explicitly in the following sections.

According to the discussions in section 4,Glauber gluons coupling to active particles should be absorbed into zero bins of collinear or soft gluons.Hence we view Glauber interactions of active particles as collinear or soft interactions of these particles.While referring to Glauber interactions,we always mean the spectator–spectator and the spectator-soft Glauber exchanges in the following texts.

5.1.

5.2.Couplings involving Glauber gluons A+G in H(P,,q+,q-)

In this section,we consider the Glauber gluons A+Gexchanged between plus-collinear and other particles.Weprove the canc

ellation of the Glauber couplings at vertexes ofwhich the coordinatesare smaller than those of somevertexes free from A+G(except for the hard vertex).

To distinguish vertexes involving A+Gfrom others,we denote the coordinates of couplings between A+Gand pluscollinear particles as yiand those between A+Gand minus-collinear and soft particles asand those free from A+Gas zi(i=1,…).Without loss of generality,we consider aGlauber gluon l1exchanged between the two vertexes y1andcan be neglected at the vertex.21represents the plus momenta of plus-collinear particles and -ls and -pˉ represent the minus momenta of soft andminus-collinear particles.

The propagator of l1is independent ofat therelies leading power of λ and η.As a result,onandonly through the term

and the integrals over-l1and+

l1read

The vertex y1and the hard vertex are connected throughplus-collinear particles.For a plus-collinear particle withmomenta ξP+k(ξ ～O(1),kμQλ for plus-collinear exter-nal lines),the propagator reads

where N(P)represents possible numerators in the propagators.

We have

on the left side of the final cut and

on the right side of the final cut,where we have made use ofthe fact

5.3.Cancellation of Glauber gluon A+G

Considering that

where

for interactions involving the Glauber gluons A+G23While considering the next leading power results,one has .as dis-cussed in section 5.2,we have

on the left side of the final cut and

5.4.Cancellation of Glauber gluons A-G

6.Graphic cancellation of Glauber gluons

6.1. evolution of plus-collinear particles

where the term

contributes to the momenta conservation δ-function of thehard subprocess and can be dropped here.

Without loss of generality,we define l1as the momentumof a plus-collinear internal line connecting to y and have

Considering that

for plus-collinear particles and Glauber gluons A+G24Couplings between collinear and soft(ultrasoft)particles eikonalize.Hence the coordinates x- of plus-collinear particles remain unchanged inthese couplings.Be compatible with our discussions for couplings betweenGlauber gluons here, and x+-order of plus-collinear states areequivalent to each other even if one considers soft(ultrasoft)interactions ofthese states.,wehave

That is to say,the couplings involving a plus-collinear par-ticle l1(except for hard interactions)can be calculated throughthe Feynman diagram skill except that one should make thesubstitution

for the δ-function of momenta conservation.

We then integrate out the minus momenta of plus-colli-near internal lines.For an arbitrary plus-collinear internal linek,H relies on l-through the propagator

and the two vertexes

where qiand qjrepresent the momenta of Glauber and soft(ultrasoft)gluons.We take l+＞0 as plus momenta of plus-collinear particle flow from the initial particle P to the hardvertex or final cut.We can then integrate out l-by taking the residue of poles located in the upper half plane.After thisoperation,we have

We repeat this procedure and get terms shaped like

on the left side of the final cut and

on the right side of the final cut,where qi(qj)represents themomenta of Glauber and soft(ultrasoft)gluons and li(lj)repre-sents the momenta of plus-collinear internal lines and ki(kj)represents the momenta of on-shell plus-collinear particles.Thesummation of li(lj)is made over states with the coordinatesbetween two given vertexes.The summation of qi(qj)andki(kj)is made over states connecting to vertexes with the coor-dinatessmaller than those of li(lj).We should mentionthat li(lj)may rely on transverse momenta of external lines andGlauber and soft(ultrasoft)gluons through the δ-function oftransverse momenta conservation.

Compared to the light-cone perturbative series in[4],terms of the type(99)and(100)correspond to contributionsof states with the coordinates x+(or the coordinatessmaller than that of the hard collision.Interactions with thecoordinatesgreater than that of the hard collisioncancel out as demonstrated in section 5.1.

6.2.evolution of minus-collinear and soft particles

We then consider couplings free from plus-collinear particles.Considering that

for soft and ultrasoft particles and

for Glauber gluons,we have25Couplings between soft and ultrasoft particles are power suppressed andcan be neglected.So are couplings between Glauber gluons and ultrasoftparticles and those between Glauber gluons.

6.3.Glauber graphs in H(P,,q+,q-)

For general diagrams,one may consider the flow of theplus momenta of plus-collinear particles.According to thediscussions in section 6.1,the plus momenta should flowfrom these collinear particles to the hard vertex or the finalcut.We consider the couplings involving these collinearparticles through the order of the flow26According to the physical picture in[6,34],such order should be definiteon pinch si ngular surfaces..We can define themomenta of these particles so that the plus momenta ofthese particles are positive.We then integrate out the minusmomenta of these collinear particles by taking the residuesof poles of the vertex from which the plus momenta of thecollinear particles flow out of.If we meet Glauber cou-plings before non-Glauber couplings in some diagrams thenthere are some Glauber gluons of which all poles of theminus momenta(no less than two)locate in the upper halfplane as shown in the example figure 9 and the formula(112)27These poles originate from the propagators of plus collinear particles coupling to Glauber gluons and vertexes which plus momenta of these particles flow into..Hence these diagrams vanish after the cancellationof interactions with the coordinatesgreater than thatof the hard collision.

For Glauber couplings of minus-collinear and softparticles,one may consider the flow of the minus momentaof these particles.According to the discussions insection 6.2,we can consider these couplings through theorder of the flow and meet the hard vertex or the final cutfinally.We define the momenta of these particles so that theminus mom

ent a of these particles are positive.Be similar tothe Glauber couplings of plus-collinear particles,if we meetGlauber couplings before non-Glauber couplings in somediagrams then all poles of the plus momenta(no less thantwo)of some Glauber gluons locate in the upper half plane.These diagrams do not contribute to.

In summary,the active-spectator coherence with thecoordinatesgreater than those of Glauber interac-tions cancel out inAs a result,the sum-is inclusiveenough for spectator–spectator and spectator-soft Glauberinteractions involving A+Geven if there are detected finalstates.This is crucial in the cancellation of these Glauberinteractions.mation over final states in

6.4.Cancellation of spectator-spectator and spectator-soft Glauber subgraphs

After these integrals,the Glauber exchange subprocess relieson the minus momenta of initial plus-collinear spectators onlythrough their propagators or wave function.In other words,the whole process(including the Glauber and non-Glaubersubprocess)relies on the minus momenta of the spectatorsonly through the vertex at which the spectators are producedand propagators or wave functions of the spectators.28For example,one may check the p--terms in(115).Wethen integrate out the minus momenta of the spectators bytaking the poles of the propagators(if the spectators are off-shell)or using the on-shell condition(if the spectators are on-shell).Obviously,these two results are equivalent to eachother.For Glauber couplings of other particles,we havesimilar results.That is,the cancellation of spectator–spectatorand spectator-soft Glauber gluons is irrelevant to the off-shellness of initial particles of the Glauber subprocess at theleading power of λ and η.

7.Conclusions and discussions

summation of spectator-soft and spectator-spectator Glauberexchanges.Fortunately,these interactions cancel out forprocesses considered here as discussed in section 5.2.Intui-tively,spectator-soft and spectator-spectator exchange ofGlauber gluons A+Gand A-Gshould occur at the vertexeswith the coordinates x+and x-equivalent to those of the hardcollision according to the classical trajectories of collinearparticles29Pinch singular surfaces are related to classical trajectories as discussedin[34].and locality of the Glauber propagators in the x+and x-directions.Therefore one should not be surprised tosee such a cancellation.

Considering that Glauber couplings may change the transverse momenta of final particles,the summation over these final states,especially the integration over transverse momenta of these final states,is necessary for the cancellation of Glauber exchanges.While considered observable is not inclusive enough,the final states may not form the invariant subspace of the evolution induced by Glauber couplings and the Glauber cancellations induced by the unitarity of the evolution may break up.That is to say,our proofs of Glauber cancellation only work for quantities inclusive enough.For electromagnetic processes like the Drell–Yan process,the result is stronger.In these processes,there are no QCD interactions between the lepton pair l+l-and undetected states X.Thus QCD interactions between final states do not change the total momentum of the lepton pair at the lowest order of electromagnetic interactions.Hence the final state interactions cancel out in these processes even if q⊥is not integrated out.

According to our discussions,the operator level factorization theorem should be tightly related to the graphic aspects of the theorem.However,the operator level factorization differs from the graphic factorization through the aspects:(1)instead of explicit perturbative calculations,we get the operator level cancellation of final state interactions through the unitarity of the time evolution operator;(2)instead of light cone perturbative series,we exclude the active-spectator coherence with x+and x-greater than those of spectator–spectator and spectator-soft Glauber exchanges through the(operator level)evolution property of collinear particles;(3)instead of light cone perturbative series,we get the cancellation of spectator-spectator and spectator-soft Glauber exchanges through the operator level unitarity.

Acknowledgments

The work of G L Zhou is supported by The National Nature Science Foundation of China under Grant No.11805151 and The Scientific Research Foundation for the Doctoral Program of Xi’an University of Science and Technology under Grant No.6310116055 and The Scientific Fostering Foundation of Xi’an University of Science and Technology under Grant No.201709.The work of Z X Yan is supported by The Department of Shanxi Province Natural Science Foundation of China under Grant No.2015JM1027.The work of F Li is supported by China Postdoctoral Foundation under Grant No.2015M581824,20161001 and International Postdoctoral Exchange Fellowship Program between JUELICH and OCPC.

Appendix A.Time evolution operator of the effective theory

We discuss the time evolution operator of the effective theory here.Generally speaking,the time evolution operator in the interaction picture reads

where H0and H are the free and full Hamiltonian of the theory.However,the Hamiltonian formulation of non-Abelian gauge theory is quite nontrivial(see e.g.[35,36]),not to mention the difficulties caused by Glauber gluon fields in(7),which should be viewed as constraints.According to the effective action(7),one has

where collinear and soft momenta are not displayed explicitly for simplicity.

Solving the constraints corresponding to Glauber gluon fields is equivalent to integrating out Glauber gluon fields in the effective theory.Let us consider the solution of n·An,G

We substitute the solution into couplings between n-collinear quarks and the Glauber gluons An,Gand have

While neglecting the Glauber gluon terms in the solution of n·AnG,we obtain the scattering processes induced by the exchange of a Glauber gluon.Some examples of these scattering processes are shown in figure A1

At leading of power λ,figure A1 reads

For the Glauber gluon terms in equation(A4),one may obtain their contribution order by order and see elastic processes induced by the exchange of more sequential Glauber gluons.We show some examples of such elastic scattering in figure A2.

At leading of power λ,figure A2 reads

One can verify that equations(A5)and(A6)are equivalent to the full QCD results at the leading power of λ.

For couplings between Glauber and n-collinear gluons,we have similar results.Although Glauber gluons are not integrated out here,one should keep in mind that they are constraints and do not correspond to any quantum states.

Instead of the Hamiltonian formula,we would like to discuss the time evolution operator in the frame of the path integral.Let us start from the time evolution operator of an ordinary gauge theory.According to the relation between the Hamiltonian formulation and path integral method,one has

for arbitrary physical states∣Aphy〉and∣Bphy〉,whereC is a constant inde pendent of the fields and the states∣Aphy〉and∣Bphy〉.The matrix element(A7)can be extended to arbitrarystates(including non-physical states)30According to the discussions in[37],physical states should be annihilated by the BRST-charge.

and the mixture between Glauber and nonphysical polarizedgluons vanishes at the leading power of λ31The power counting for couplings between soft and Glauber gluons reads or higher.If there is additional suppression of order or higher,then the mixture between soft and Glauber gluons affects the cancellation of nonphysical polarized soft gluons at order λb.,where

Hence the mixture is power suppressed unless the contractionbetween non-physical polarized gluons and superleadingvertexes occurs in the process. The superleading interactions(order) originate from couplings between the Glaubercomponents n·AnGwith b=1 and n-collinear particles.Hence such superleading vertexes can contract with theGlauber components n·AnGnotAccording to thepower counting results of nonphysical polarized gluons in(A18) and (A19), such contraction (order λ2b-1or higher)vanishes at the leading power of λ.

Appendix B.Active particles and spectators in hard collisions

We consider the definition of active particles and spectators inthis appendix.Considering that such a definition relies onspecific processes and graphs,we should consider the defi-nition at the graphic level.That is to say,whether a collinearparticle should be viewed as an active particle or a spectatordepends on its behaviour in a specific diagram.We shouldconsider a suitable definition to facilitate the disposal ofGlauber couplings of collinear particles

Literally,active particles should be the collinear particlesthat participate in the hard subprocess32At the leading power of λ,soft and ultrasoft particles and Glauber gluons are free from the hard subprocessand spectators shouldbe those free from the subprocess.On the hand,there may becollinear particles that do not participate in the hard sub-process directly but can not be cut by the final cuts like k1infigure B1.

Glauber couplings of these particles are not removed bythe summation of final spectators.One may verify thatGlauber couplings of k1eikonalize.It seems that k1should bedefined as active particles even if it does not participate in thehard subprocess directly.

Without loss of generality,let us consider plus-collinearactive particles and spectators.Trajectories of these particlesoriginate from the initial plus-collinear hadron and end at thefinal cuts or the hard vertex.If the plus momentum of a plus-collinear particle k1flows into the hard vertex finally,then k1can not survive after the hard collision.In other words,k1cannot be cut by the final cuts even if it does not participate in thehard subprocess directly.For example,one may consider theparticle k1in figure B1.+k1flow into the hard vertex throughthe path k1→k2→hard vertex.

According to the above discussions,if there is a pathmade up of plus-collinear particles k1,…,kn33Obviously,P+(kj)＞0(1 ≤j ≤n)as the reflux of plus momenta,which contradicts classical trajectories of plus-collinear particles,is absent on theleading pinch singular surfaces.flows into thehard vertex through the path k1→…→kn→hard vertexthen kjcan not be cut by the final cuts and they should bedefined as active particles.One may verify the eikonalizationof Glauber couplings of kj(1 ≤j ≤n).Let us consider thepropagator of kj,which reads

where N(j)represents terms independent of Glauber momentain the propagator and l1,…lj-1represent the momenta ofGlauber gluons coupling to k1,…,kj-1.34kj may couple to other non-Glauber particles(except for k1,…kn).We define the momenta of these particles as variables independent of Glauber gluons coupling to kj.According to theform of the propagators of kj(1 ≤J ≤n),poles of l1,…,ln-1are all located in the lower half plane.Hence one can take theeikonal approximation in Glauber couplings of kjwithout changing the leading power result. This is compatible withdiscussions in section 4.2.

Hence l-is not pinched in the Glauber region at the leadingpower of λ.35In fact,figure B2 cancels out in summation over final states as discussed in section 5.2.This is compatible with discussions insection 4.2.

For a plus-collinear particle p1free from the hard vertex,if there are no paths through whichflow into the hardvertex then p1flows into the final cuts.There should be a pathp1→…pn→final cuts.The propagator of pj(1 ≤j ≤n-1)reads

and the final cut on pnreads

It is interesting to consider a path including non-collinearparticles.For example,we consider a plus-collinear particle p1and a path p1→…→l →…→pn→hard vertex,where l issoft or ultrasoft or Glauber.Considering that the trajectory of ldiffers from those of collinear particles,this path should not beviewed as an effective path through which plus momenta ofactive particles flow into the hard vertex.In fact,Glauber cou-plings of p1cancel out if l is soft or ultrasoft according todiscussions in section 5.2 since there are non-Glauber interac-tions after Glauber coupling of p1.In the case that l is Glauber,whether Glauber coupling of p1eikonalize depends on whetherthere is a path made up of plus-collinear particles through which

+p1flow into the hard vertex.In both cases,one should inves-tigate other possible paths made up of plus-collinear particlesthrough which+

p1flow into the hard vertex.According to our definition,plus momenta of plus-col-linear spectators do not flow into the hard vertex or they flowinto the hard vertex through paths including non-collinearparticles.In the former case,spectators may survive after thehard collision and the spectators can be cut by the final cuts.That is,inclusive summation over spectators is possible inthis case.In the latter case,Glauber couplings of spectatorscancel out if the paths include soft or ultrasoft particles asdiscussed in section 5.2.If the paths do not contain soft orultrasoft particles and plus momenta of spectators flow intothe hard vertex through Glauber gluons and active particles,we can then deal with Glauber couplings of these spectatorsaccording to the skills in sections 4.1 and 5.

For minus-collinear particles,we have a similardefinition.

In summary,we define plus-collinear(minus-collinear)active particles as:36The two cases may overlap.(1)the plus-collinear(minus-collinear)particles coupling to particles collinear to other directions orhard modes directly;(2)the plus-collinear(minus-collinear)particles of which the plus(minus)momenta flow into thehard vertex through a path made up of collinear particles.Other plus-collinear(minus-collinear)particles are defined asspectators37Certainly,a high loop level definition of collinear particles relies on how they couple to other particles and a suitable subtraction scheme which is beyond the scope of this paper..The definition of active particles is compatiblewith discussions in section 4.2.And further,we can deal withGlauber exchanges between spectators and other particlesaccording to skills in sections 4.1 and 5.?

We consider the reparameterization invariance in thisappendix.For simplicity,we neglect ghost fields in thisappendix.In original SCET,the reparameterization invarianceof the n-collinear sector arises from two types of ambi-guity[38,39]:

for ultrasoft particles.Therefore the type-1 transformations leave the action(7)invariant at the leading power of λ.

The type-2 transformations are more subtle.The transformations of relevant operators under the reparameterization-2 are exhibited in table C1[38,39]38We define nμ andμso thatn · =1,which is different from that in[38,39]..

Appendix C.Reparameterization invariance of the effective theory

Table C1.Transformations of relevant operators under the type-2 reparameterization.