Factorization Violation and Scale Invariance

Factorization violating effects in hadron scattering are due mainly to spectator-spectator interactions. While it is known that these interactions cancel in inclusive cross sections, like for the Drell-Yan process, not much is known about for what classes of observables factorization is violated. We show that for pure Glauber ladder graphs, all amplitude-level factorization violating effects completely cancel at cross section level for any single-scale observable (such as hadronic transverse energy or beam thrust). This result disproves previous claims that these pure Glauber graphs are factorization-violating. Our proof exploits scale invariance of two-to-two scattering amplitudes in an essential way. The leading factorization-violating effects therefore come from graphs with at least one soft gluon, involving the Lipatov vertex off of the Glauber ladders. This implies that real soft radiation must be involved in factorization-violation, shedding light on the connection between factorization-violation and the underlying event.


Introduction
Factorization is essential to the predictive power of perturbative QCD at hadron colliders. The essential point of factorization is that it lets us separate the dynamics of the proton from the dynamics of the scattering that produces hard radiation. Unfortunately, factorization is known not to hold universally. There is both theoretical evidence for non-factorization through effects like super-leading logarithms, and experimental evidence, as calculations performed assuming factorization can have significant deviations from experiment. In order to continue to push the precision of predictions for the Large Hadron Collider and other machines, it will be essential to understand more about how and when factorization violation occurs. The hope is that with better understanding we might be able to either choose observables for which factorization violation is minimal or find some universality in factorization-violating effects.
Factorization has been shown to hold rigorously in perturbation theory in certain circumstances. For example, it holds at the amplitude level for processes with only final-state radiation. It also holds at the amplitude level for processes with only one colored particle in the initial state (like deep-inelastic scattering at the parton level). In addition, it holds for any process with a fixed number of external particles. These results were reviewed and clarified recently in [1][2][3]. Factorization is known to be violated both at the amplitude and cross section level for processes with two colored initial state particles and at least one final-state particle collinear to an initial-state one [4][5][6]. In such situations, while the amplitude can still be written as a splitting amplitude times the amplitude with the collinear pair replaced by their mother particle, the splitting amplitude necessarily depends on the quantum numbers of non-collinear particles. [4].
Even if factorization is violated at the amplitude level, it may still cancel when amplitudes are squared and integrated over phase space. The celebrated example of this phenomenon is the Collins-Soper-Sterman (CSS) proof of factorization for Drell-Yan [7]. There, when all relevant processes describing the hadro-production of a lepton pair are included, the cross section still can be written as the convolution of parton-distribution functions and a hard scattering kernel.
In recent years, there has been renewed interest in extending the CSS argument to other processes, and understanding its failure. Partly this has been motivated by the advent of jet substructure techniques [8][9][10][11][12], where predictions of observables like jet mass or beam thrust are apparently more sensitive to factorization-violating effects than traditional kinematic observables, like the jet p T spectrum. Partly it has been driven by theory developments that give new handles on factorization-violating effects. In particular, Soft-Collinear Effective Theory (SCET) [13][14][15][16][17][18] has allowed for higher precision jet substructure calculations. Recently, Rothstein and Stewart [19] have explained how to account for Glauber effects within the SCET framework (see also [20][21][22][23]).
In this current paper, we attempt to shed some light on when factorization occurs by combining the SCET Glauber picture with observations by CSS and others. In particular, it has been argued in Ref. [19,24] that factorization can be violated by diagrams with pure Glauber gluon exchange, but not involving any real gluon radiation. We revisit these arguments and show that in fact factorization is not violated by the pure Glauber graphs. Our calculation amounts to showing that after summing over all possible cuts, contribution from pure Glauber graphs vanish, Here, the vertical dashed lines denote possible cuts and p µ 3 and p µ 4 are the momenta of the two outgoing spectators generated by the cuts.
The cancellation we find implies that the differential cross-section for any single-scale observable, like hadronic transverse energy, or beam thrust, gets no contribution from pure-Glauber graphs. The leading factorization-violating effects therefore must involve diagrams with the Lipatov vertex [25], that is, diagrams where soft gluons are exchanged between Glauber rungs. Our argument does not apply to doubly-differential observables, like those studied in [26].
We begin in Section 2 with a review of some results about factorization at the amplitude and cross section level. This section can be skipped by an informed reader, but may be useful to a reader who finds the literature on factorization marginally impenetrable. Section 3 presents our main new result, that the contribution of pure Glauber ladder graphs to cross section exactly vanishes for any single-scale observable. To not disrupt the logic, we include in this section only summaries of the calculations, with details relegated to Appendices B through E. While elements of our calculation are similar to the summation of Glauber ladder graphs into a phase in position space for forward scattering calculations, we work instead in momentum space where a choice of variables respecting scale invariance can be made. Section 4 discusses graphs beyond the Glauber ladders, necessarily involving soft gluons and the Lipatov vertex. For these graphs, scale-invariance is violated by quantum effects and so factorization can be violated. A brief summary and conclusions are in Section 5.

Factorization and Factorization Violation
We begin with a review of some known results about factorization. There are no new results in this section. The goal of the section is merely to clarify what we mean by factorization, and what is known, using relatively clear and precise language.
Almost all the literature about factorization refers to statements about soft and collinear divergences in perturbative QCD. A clean way to describe perturbative factorization was developed in [1][2][3]6]. Consider an initial state |Z and a final state X| each of which is made up of some quarks and gluons with various momenta. We can group those momenta into sectors either collinear to a set of directions n µ j or soft. Each sector has an associated scale. For example, the scale λ j associated with the n j -collinear sector may be defined so that each momentum p j in the n j collinear sector has n j ·p j ≤ λ j p 0 j . For the soft sector a parameter λ s can be defined so that the energy of each soft momentum k µ has k 0 ≤ λ s Q where Q is a hard scale, such as the center-of-mass energy, or the energy of some jet. Thus we can write X| = X s | X 1 | · · · X N | and |Z = |Z s |Z N +1 · · · Z M . For any such decomposition in which no initial-state direction is collinear to a final-state direction, amplitudes factorize: In this relation ∼ = means the two sides agree at leading power in all of the λ j and λ s , including all infrared divergences (soft and collinear divergences). Another way to say this is that the Wilson coefficient C(S ij ), defined as the ratio of the left-side amplitude to the factorized ampliutde on the right, is finite as all the λ go to zero. Here, the matrix elements are written as amplitudes of operators in scalar QED, for simplicity; the same factorization formula holds in QCD but the color and spin notation is more cumbersome. The W j and Y j are Wilson lines. All operators in this factorization formula are written in terms of the fields of an ordinary quantum field theory (scalar QED or QCD) -no effective field theory interactions are required. This factorization is closely related to factorization in SCET where there are soft fields and collinear fields for each direction with leading-power interactions. One important implication of factorization is that it relates amplitudes with different external states. For example, the Wilson coefficient C(S ij ) depends only on hard scales S ij = (P µ i +P µ j ) 2 , with P µ j the net momenta in the n j -collinear sector. Thus the factorization formula relates processes with different soft and collinear particles.
An application of factorization is to prove the universality of collinear splittings. Treating amplitudes as vectors in color space, a splitting amplitude is defined as the ratio between an amplitude |M with n − m particles to an amplitude |M with n particles. For m = 1 the relationship can be written as |M(p 1 , · · · , p n ) ∼ = Sp(p 1 , p 2 ; p 3 , · · · , p n ) · |M(P, p 3 , · · · p n ) Here M and M are amplitudes like in Eq. (2). The momentum P µ on the right is a single particle momentum that splits into p µ 1 and p µ 2 : P µ ∼ = p µ 1 + p µ 2 . Factorization in Eq. (2) implies that and therefore The key point is that the splitting amplitude is universal -it only depends on the the fields in the direction collinear to the splitting. Now consider a situation where some outgoing particles are collinear to some incoming ones, such as in forward scattering. Then factorization at the amplitude level does not hold and Eq. (2) is invalid. Instead, we can write something similar by combining all the operators into one: To evaluate the right-hand side one needs to use the SCET Lagrangian which has collinear fields, soft fields, and Glauber interactions The labels j and s have become quantum numbers labelling the sectors. Thus interactions between different collinear sectors are forbidden by j and s superselection rules (the terms in each L j or L s only involve fields with the same quantum numbers). The denominator factors on the right-hand-side of Eq. (2) are replaced by a diagram-level zero-bin subtraction procedure to remove the soft/collinear overlap. With this understanding, when L G ij can be ignored then Eq. (2) and Eq. (6) are equivalent.
The terms L G ij in the SCET Lagrangian are the "Glauber interactions", introduced in [19] (see also [22,23]). They contain interactions among collinear particles in the i and j sectors, as well between these collinear directions and soft particles. The Glauber interactions are non-local, involving explicit factors of 1 The diagrams involving Glauber exchange to lowest order in g s are the same as those arising by taking the Glauber-scaling limit of diagrams in full QCD. For higher-order diagrams there is not a simple method-of-regions correspondence with QCD. Instead the all-orders form of these terms is fixed by symmetry arguments (reparameterization invariance) and direct matching calculations. In addition to adding these new interactions, SCET with L G ij requires a new zero-bin subtraction: the Glauber limit of soft and collinear graphs must be subtracted diagram-by-diagram so as not to double count. Finally, there is an implicit non-analytic rapidity regulator required to define the theory. Some understanding of why the regulator must be non-analytic was discussed in [6], but understanding the regulator in more detail (e.g. what properties it must have, what is regulator-independent, etc.) is an open area of research.
As the Glauber interactions involve different collinear sectors, they violate factorization. It is nevertheless a non-trivial check that the Glauber interactions can reproduce known factorization-violating effects in full QCD. One such check was performed in [6] where the generalized splitting function (i.e. one that depends on multiple directions) was calculated using the SCET formalism, finding a result in agreement with full QCD. Other checks, such as reproducing the Glauber phase in forward scattering or the BFKL equation, were discussed in [19].
For factorization to hold in hadronic collisions, we would like to be able to separate the dynamics of the proton from the hard scattering. Let's start with a situation where factorization does hold, deep-inelastic scattering (DIS). In pictures, factorization implies n 2 n 1 n 1 The first diagram on the right-hand side further factorizes, into soft and collinear (not shown). The n 1 and n 2 labels indicate to which directions the various lines are collinear. The red shallow curled lines are meant to indicate soft gluons interacting between the two collinear sectors. In words, the dynamics of the initial state, collinear to the n µ 1 direction, factorizes from the hard scattering. In equations where p 1 µ = ξp µ 1 is the momentum participating in the hard partonic scattering. Does this imply that factorization holds in terms of parton distribution functions for the proton? Not necessarily. Indeed, factorization of DIS at the amplitude level is neither a necessary condition (factorization could hold at the cross section level) or a sufficient condition (we are just working in perturbation theory here, so we cannot say anything about non-perturbative physics). Nevertheless it is suggestive of factorization and a good start.
For Drell-Yan, the analogous factorization does not hold The problem originates with the soft sector, but affects purely collinear emissions too through virtual effects. There is a quick way to see what the problem is with soft radiation. Soft radiation factorizes when it is sensitive to only the net color charge going in a particular direction. This is similar to Gauss's law in electromagnetism -at large distances only the net charge in a region matters to the leading approximation. But what is the net charge when there are incoming and outgoing particles in the same sector? For DIS, we can use a trick and move all the color of the n 1 -collinear sector to a Wilson line in the scattering operator. By color conservation, we know the net color that the outgoing n 2 jet sees is negative of the net n 1 color. For Drell-Yan, this trick does not work. Indeed, the reason the minimal number of colored particles needed for factorization breaking is four is that with three or fewer, color conservation can be used to ensure that the scattering is only sensitive to the net color. This argument can be found in [4] and is explained in depth in [6].
The obstruction to factorization can be traced to the invalidity of the eikonal approximation in describing soft radiation. Soft radiation refers to regions of real or virtual phase space in which all the components of a gluon's momentum are small compared to the energy scale Q of the particles emitting the soft radiation, |k µ | Q. For two momenta p µ and q µ we can always write The eikonal or Grammer-Yennie approximation amounts to dropping the second term on the right with respect to the first. When the eikonal approximation can be used, soft-collinear factorization holds [1,2]. For p µ hard (p 0 ∼ Q) and k µ soft (k 0 Q), it seems like the second term can always be dropped since The problem is that it is possible for fixed. In light-cone coordinate the scaling of such mode can be written as k ∼ Q(λ 2 , λ 2 , λ). This kind of scaling toward k µ = 0 is known as Glauber scaling and can foil factorization. An intuitive way to understand the obstruction from Glauber scaling is to contrast the soft limit k µ → 0 with the limit Q → ∞. If Q decouples, so we can take Q → ∞ to describe soft radiation, then the sources emitting soft radiation can be treated as scale-invariant Wilson lines Y n . Soft-collinear factorization is based on being able to use this scale-invariant limit. Glauber scaling involves Q in an essential way, so it obstructs scale-invariance.
To prove that the eikonal limit can be used, one must show that the region of soft momenta described by Glauber scaling is contained in scaleless integrations over soft momenta. In CSS language, this happens when there is no pinch in the Glauber region. In the language of SCET, it is when contributions form the Glauber Lagrangian (L G ij in Eq. (7)) are not exactly canceled by the Glauber-soft and Glauber-collinear zero-bin subtractions [19]. The relation between these two ideas was further explored in [6].
So factorization does not hold in Drell-Yan at the amplitude level, as in Eq. (10), because the Glauber region is not contained in the soft region. Graphs for Drell-Yan have a pinch in the Glauber region. What CSS showed was that despite the non-factorization at the amplitude level, factorization still holds for Drell-Yan as long as the observable is inclusive over all QCD radiation. In pictures To prove this, CSS showed that for soft gluons with momenta k µ i that interact with collinear momenta p µ j , the transverse components of the soft momenta can be neglected. That is, we can replace Once we know the result is unaffected by dropping transverse components then there is only one way to scale k µ → 0; there is no Glauber region and the eikonal approximation k 2 k·p → 0 is always justified around k 2 = 0. In more detail, CSS's argument used old-fashioned perturbation theory. They showed that the same cross section results with and without neglecting k ⊥ and k + by summing over all possible cuts in spectator-spectator graphs (13) as well as in active-spectator graphs. In addition, their argument exploited observations about the polarizations of the gluons coupling to the collinear sector. While there is no doubt that the CSS proof is correct, it gives little guidance as to what we might learn about situations where factorization is violated.

Factorization violation
Factorization holds for the Drell-Yan process where only the lepton momenta are measured. A typical Drell-Yan observable is the • Lepton transverse momentum q T : defined in the Drell-Yan process as the transverse momentum of the lepton pair q T = | p 1,⊥ + p 2,⊥ | where p µ 1 and p µ 2 are the lepton momenta.
By momentum conservation, q T is also equal to the net transverse momentum of all the hadronic final state particles.
For other observables, results about factorization are murkier. Two other variables we will discuss are • Hadronic transverse energy E T . Assuming all measured particles are massless, transverse energy is the scalar sum of the particles' transverse momenta E T = j | p jT |.
• Beam thrust. This is a hadronic event shape observable. For a process involving vector boson production (pp → V + X), beam thrust defined in hadronic center-ofmass frame is where Q and Y V are the vector-boson mass and rapidity and p jT and Y j are the transverse momenta and rapidities of the other final state particles [27,28].
q T was shown to factorize by CSS. For the other observables no rigorous results are known. The general lore is that factorization violation shows up in event generator simulations as sensitivity of an observable to the underlying event. [19,24,26,[29][30][31].
Consider for example beam thrust. In the original papers [27,28,32], beam thrust was thought to factorize. A rough argument along the lines of CSS about why the Glauber contributions should cancel was presented in [27]. Unfortunately, beam thrust is extremely sensitive to models of the underlying event; turning underlying event on or off in simulations has an order-one effect on the beam thrust distribution, completely destroying predictivity of the theoretical calculation [33]. Beam thrust was revisited by Gaunt [24] who argued that the pure Glauber contribution is factorization-violating. Similar conclusions were reached in [19].
Let us briefly review Gaunt's argument. He first considered summing cuts of a diagram with a single Glauber exchange. There are two possible cuts. (14) The two cuts contribute to the cross section with opposite signs and the sum of them vanishes, independent of the observable. Next, he looked at diagrams with two Glauber gluons exchanged. There are three cuts through such graphs: (15) Gaunt argued that for the observables E T and beam thrust, the Glauber effects do not cancel when summing over these cuts and there is a factorization-violating effect. Gaunt showed that for a single Glauber exchange diagram, one can achieve the cancellation by relabelling the transverse components of integrated momentum. However, for double Glauber exchange, Gaunt argued that no change of variables should exist to achieve the cancellation.
A similar argument to Gaunt's can be found in Stewart and Rothstein's treatise on Glauber gluons in SCET. These authors argued that when all the ladder graphs are summed, the result is a Glauber phase exp(iφ( b ⊥ )) with b ⊥ the impact parameter conjugate to the relative transverse momentum of the two spectators ∆ p ⊥ . When one calculates the cross section there is an integral over this b ⊥ as well as the b ⊥ for the complex-conjugate amplitude. The argument is that these phases do not cancel unless the observable is independent of ∆ p ⊥ . Therefore factorization should be violated by Glauber ladder diagrams for any observable other than q T .
While both Gaunt and Rothstein/Stewart provide strong arguments, their arguments rely on assuming no unusual cancellations can happen. Their choices of variables are certainly suggestive that it would take a miracle for cancellation to happen. But miracles do happen when there are symmetries. In this paper, we show that a choice of variables that respects the scale invariance of the problem makes a general cancelation manifest.
Our main result is that the sum of all the ladder graphs is zero when integrated over the kinematic variables other than a single infrared-safe observable: The variables z andz that respect the symmetry of the scattering will be discussed in the next section. The leading factorization violating effect comes from graphs with soft gluons exchanged between the Glauber rungs: (17) These graphs involve the Lipatov vertex -the 3-gluon vertex connected 2 Glauber gluons to a soft gluons [25]. The Lipatov vertex is currently known to one loop in QCD [34,35]. The Lipatov vertex is embedded in the Glauber operator in SCET in Eq. (7).

Glauber ladder graphs
In this section, we prove that summing over all Glauber ladder graphs, factorization is preserved. We start with the 2-loop result which demonstrates all the essential features of our argument. We then discuss the all-orders result.

2-loop Glauber cancellation
Consider the amplitude for Drell-Yan production from quark initial states with two outgoing gluons: At up to 2-loop order order, the diagrams that can produce factorization violating effects are In these diagrams, the red dotted exchanges are Glauber gluons. The Feynman rules for these gluons can be found in [14] using SCET, or more simply by power expanding the amplitude in QCD using Glauber scaling: | k ⊥ | k + , k − . Glauber scaling makes the propagators for a momentum k µ depend only on k ⊥ , i.e.
Since the propagators only involve transverse momentum, they are more like contact interactions than propagators but we draw them as extended lines to exhibit their origin from diagrams in QCD.
The leading order matrix element is We assume the outgoing gluons are collinear to the incoming quarks, so the tree-level matrix element is given at leading power by the product of splitting amplitudes: This leads to a simplified form for the matrix N µν in Eq. (20): By inserting the Glauber potential operators and integrating out the light-cone components of the Glauber momenta, the one-loop and two-loop Glauber ladder graphs reduce to the following integrals in the d = 2 − 2 transverse plane, Here s ij = (p i + p j ) 2 , and we use the convention that all particle momenta are incoming; outgoing momenta are obtained by crossing p → −p. We have computed the amplitudes M 0 , M G1 and M G2 in d dimensions as a series in = 4−d 2 , using some master integrals from [36]. Details are given in the appendices and will be summarized here.
In computing the matrix-element squared, we find that up to 2-loop order the IR divergences exactly cancel. This cancellation is non-trivial, but also expected from general results. Of the 8 possible helicity combinations, only 2 are independent. If both collinear quark/gluon pairs have the same helicity (or if both pairs have opposite helicity), such that h 3 = ±h 1 , h 4 = ±h 2 , then we find (see Appendix D) If one collinear quark/gluon pair has the same helicity and the other pair has the opposite helicity, such that h 3 = ±h 1 , h 4 = ∓h 2 , then where the color sum is implicit in |M| 2 . The 2-loop and tree-level color factors are c 2G = with z 13 ≡n · (p 1 + p 3 ) n · p 1 < 1, and z 24 ≡ n · (p 2 + p 4 ) n · p 2 < 1. and These results are expressed in terms of the dimensionful variable q T = | q ⊥ | and two dimensionless variables u and v. These are defined as At fixed q T , the collinear limits are u → 0, v → 1 or v → 0, u → 1.
To proceed, let's consider how to integrate over d 2 p 3,⊥ , or equivalently u and v. Since q ⊥ ≡ p 3,⊥ + p 4,⊥ , the three vectors form a triangle in the transverse plane. We can rotate this triangle so that q ⊥ is conveniently oriented along the real axis, and rescale out q T , leading to a simpler triangle defined by one point z in the complex plane The relation between u, v and z is then Using z andz facilitates integrating over d 2 p 3,⊥ . Explicitly, The phase-space integrals then become regular two-dimensional integrals over conformal coordinates.
Explicitly, for |M| 2 ++ , the relevant integral is (see Appendix E) For |M| 2 +− , the relevant integral is Intriguingly both integrals give the same result. This result is non-zero, and not power suppressed as q T → 0 so it would seem to indicate factorization violation. If this were the end of the story, we would have found a non-zero factorization-violating contribution to any observable, even q T for which factorization is proven to hold. In fact, there is another piece contributing to the cross section. The term of order 1 in |M| 2 has the form . Due to the factor 1 uv in |M 0 | 2 in Eq. (27), the first term in braces is singular when integrated over 4-dimensional phase space. Thus in d dimensions it will give a −1 factor which cancels the 1 prefactor giving an O( 0 ) contribution. The remaining terms in the braces are integrable over phase space and thus do not contribute as → 0. The singular term is independent of helicity, so we leave the helicity labels implicit.
Performing the integral over the singular piece in d dimensions, we find When added to the contributions from O( 0 ) in |M| 2 , Eqs. (33) and (34)- (35), we see that the net contribution is zero: and a rescaled transverse momentum p µ 3,⊥ so that Working with q T and p 3,⊥ as the independent variables, Eq. (41) can be written as the sum of the following two integrals canceling each other at O( 0 ), Then we can change variables from q T to E T easily. For 2 → 2 scattering Thus, noting that that the Jacobian ∂ ln q T ∂ ln E T = 1, The integration over z andz in the first integral is unaffected by the change of variables. The second integral is changed by a factor of (|z| + |1 − z|) 6 . However, the behavior of the integrand in the singular regions z → 0 and z → 1 is unchanged. This new factor only contributes at order O( ). Therefore the cancellation between the two integrals still holds at O( 0 ). The key point is that the inclusive integration over z andz is observable independent. A more detailed discussion of the change of variables is given in Appendix A. For another example, consider beam thrust. For 2 → 2 scattering, where q µ = (q + , q − , q T ) is the total momentum of the lepton pair. Changing from q T to τ B results in an expression similar to Eq. (46). The integration around the singular region again only contributes new terms that start at O( ), and thus there is no factorization-violating effect from the 2-loop Glauber ladder graphs. The same argument holds for any infrared-safe single-scale transverse observable X. Any such observable must be expressible as ln X = a ln q T + g(z,z) for some a > 0 and some function g(z,z) that is regular as z → 0 and z → 1. If g(z,z) were not regular in the collinear limits, of if a ≤ 0, then the observable cannot be infrared safe. Because g(z,z) is regular in collinear limits, the same inclusive integrals over z can be done without generating singularities, and thus an observable X does not exhibit factorizationviolating effects at this order.

All-orders Glauber cancellation
We showed that at 2-loop order, the Glauber ladder diagrams alone do not violate factorization for any single scale observable. Now we will show that the cancellation we found persists to all orders. By direct calculation, we find that to all orders in perturbation theory, the sum of Glauber ladder diagrams has the form (49) where the · · · are terms that do not contribute to the cross section. The combination appears naturally and the phase is A key property of the expression in Eq. (49) is that all of the infrared divergences are contained in the phase. This is shown in Appendix B. Another property is that once this phase is factored out, the order 0 term denoted f reg is integrable over the collinear regions. This decomposition and the calculation of φ g ( α s ) are given in Appendix C. The expression for f reg at 2-loops, which appeared in Eqs. (25) and (26), is computed in Appendix D.
The leading non-integrable piece we call f sing . It can be extracted from a general formula we derive for the singular part of M G . At order α n s , this singular part is Expanding e −iφg( αs) M n sing gives f sing (α; ) = (3ζ 3 + 9 2 ζ 4 2 +· · · )α 2 +(ζ 4 −16ζ 5 2 +· · · )iα 3 +(−5ζ 5 +· · · )α 4 +O(α 5 ) (53) The first term in this expansion was used in Eq. (36). Since all of the IR divergences cancel in |M| 2 , the cross section has the form dσ d ln q T Glauber ladders = dzdzF reg (z,z) + 1 In Section 3.1 we showed that at two loops both of these terms are separately non-zero and only their sum vanishes. For observable q T , this must also be true to all orders as a result of CSS poof of factorization for q T . Now say we change variables from ln q T to ln E T , using Eq. (45). The first integral in Eq. (54) is unaffected since the integrand depends only on the conformal coordinates. Changing q T into E T also has no effect on the singular behavior of the second integral, since E T → q T in the collinear limits. Thus both integrals give the same result at O( 0 ), and there is no factorization-violating effect from Glauber ladders for E T , to all orders. The same argument holds for any infrared-safe single-scale observable X. In this section we will focus on the potentially-factorization-violating contributions from graphs with soft gluon exchanges between the Glauber ladders. Similar graphs have also been briefly discussed in Ref. [24].

Non-ladder potentially-factorization-violating contributions to the cross section involve soft emissions from the Glauber legs. For example, virtual or real contributions could look like
For graphs like these, there is neither proof nor expectation that the IR divergences should resum into a phase. Instead, we expect there to be IR divergences in |M| 2 that cancel only when the phase space integral is done. Let us denote the cross section from all graphs with Glaubers as σ G . Quite generally, we can write with q T the only dimensionful variable and the u, v, w, · · · variables dimensionless. For an IR or rapidity divergent contribution, another scale µ or ν can appear. This scale can only appear logarithmically, so the q T dependence must be logarithmic and we can write a series expansion (57) Some of these integrals may be IR divergent. We know however, that the final result must be IR finite, so the IR divergences must cancel among the various contributions. Moreover, we know by q T -factorization that the sum of all of these contributions is exactly zero. Since each term multiplies a different power of ln q T , each one separately must integrate to zero. Now consider the cross section for E T rather than q T . For 2 → 2 scattering, . For graphs with more final-state gluons, we would have a more complicated function but still a linear relation between ln q T and ln E T . So let us write Other single scale variables, like beam thrust, will have a similarly linear relation with a different function g. Then The terms on the first line vanish by q T factorization, but the terms on the second may not.

Observable dependence
Let us now put a little more detail into Eq. (59). We discuss two types of observables for which factorization may not hold 1. Non-global observables that do not include the collinear region, so that the real soft gluon phase space is not fully integrated over.
2. Global observables that are rapidity-independent in the collinear region.
The first category includes non-global observables only sensitive to soft emissions within some region of the detector, but inclusive over the collinear particles along the beam directions. An example of such an observable is the mass of the hardest jet. The second category includes global hadronic event shapes, like E T or transverse thrust To understand the factorization breaking effects due to the these two types of measurements, let us look at diagrams with one soft gluon as an example. The leading order real-emission diagram has a single soft gluon coming off of a Glauber line: This diagram contributes to the cross section at order g 6 s |M 0 | 2 . Virtual diagrams that contribute at this same order are the square of 1-loop graphs with a Glauber gluon and a single soft loop: There are also 2-loop graphs with two Glauber gluons relevant at this order: These contribute at order g 6 s |M 0 | 2 as well as through interference with the tree-level graphs M 0 . Note that there are no graphs where a soft gluon connects to a collinear line -collinear fields only interact with ultrasoft gluons at leading power First, let's consider observables in class 1, that are inclusive over the beam. Let's call the observable X and the measurement function on the emitted radiation f X ( k). For these observables we know that if we were inclusive over everything, all the Glauber graphs would exactly cancel, like for q T . So the uncanceled part is only due to the real emission in the measured region of phase space. Thus we only need to consider Lipatov diagrams, like M 1 R with real soft emissions. The leading factorization violating effect can then be computed by where Ω is the area of phase space of the observable. In the limit that the angular region A = ∆y∆φ in phase space is small, the real emission amplitude will be independent of rapidity y and azimuthal angle φ. Then we expect the factorization violating effect to be proportional to the area and integrand to only depend on the transverse momentum of the emitted gluon. Thus, where γ Lip is given by Holding k ⊥ fixed, the phase-space integral d 2 q T d 2 p 3,⊥ γ Lip (q T , p 3,⊥ , k ⊥ ) is IR finite. The k ⊥ → 0 limit is not allowed by the measurement function. Therefore Eq. (64) will give us a positive finite number proportional to the area A and to α 4 s . Now let's proceed with the second type of observable, which measures particles emitted in all rapidity regimes. In particular, we will assume that the observable is independent of the rapidity of soft or collinear particles but only sensitive to their transverse momenta. The measurement function acting on a two-body and three-body final state can be writen as where z, w are complex dimensionless variables defined by By infrared safety, f X (z,z) and f X (z,z, w,w) have the following properties, Unlike the non-global observables, here all the diagrams contribute to dσ dX . Since the diagrams have entangled virtual and real IR divergences in different regimes, it is hard to see directly whether dσ dX is non-vanishing. The way we will deal with it by adding and subtracting a subtraction term. We do this by defining a new measurement function f that is not sensitive to soft radiation collinear to the spectators so that f satisfies and for 2-body final states f X = f X . By adding and subtracting the cross section with f X , the cross section for f X can be written as the sum of two terms: where σ denotes the cross section computed with f X . The first term only acts non-trivially on wide-angle soft real emission diagrams, where the divergence as k ⊥ goes to zero is cured by the measurement function. To compute the second term, we can first integrate inclusively over soft momentum k, after which soft real and virtual divergences cancel. Then the integrand contains only finite integrable functions in two-body phase space. In the following we will show in detail that either term in Eq. (71) could be non-vanishing. Let's start with the first term. To better understand the behavior of this integral, especially how it depends on the measurement function, we need to study the scaling behavior of the squared amplitude. Since the measurement is independent of the soft gluon rapidity, the integral generates a rapidity divergence. Regulating this divergence with a scale ν gives a rapidity log term that breaks scale invariance. Thus, we we can no longer prove the cancellation using scale invariance as we did for Glauber ladder diagrams., even for a single-scale observable. In order to show the failure of cancellation, we only need to keep track of the rapidity logs. Expanding the matrix element, we see Note that the k z integral in Eq. (72) leads to rapidity divergence. We have used the rapidity regulator of Ref. [37,38] to regularize it, which leads to the divergent 2/η term in Eq. (72). The rapidity scale ν in Eq. (72) is similar to the µ in dimensional regularization. This determines the scaling behavior of the squared amplitude where we use γ Lip and M Lip to denote the dimensionless version of γ Lip and M Lip by dividing with appropriate power of q T . The second term in Eq. (73) respects scale invariance and is therefore insensitive to the difference in measurement functions. The first term in Eq. (73) is of the form anticipated in Eq. (57). It will give us a non-vanishing integral Now let's move onto the second term in Eq. (71) . In order to compute dσ dX , let us take the Lipatov diagram and fully integrate over the real soft momentum k, then add it to the virtual diagrams. Doing so allows us to write down a squared amplitude |M V | 2 inc , which corresponds to the sum of all four-loop cut diagrams with fixed q T and p 3,⊥ , Then the cross section becomes a two-body phase-space integral over Focusing on the divergent terms, we are able to determine the scaling behaviour of |M V | 2 inc , which takes the following form where Γ inc and γ inc are determined by integrals in transverse dimensions over the soft and Glauber momenta. Again, the cross section with measurement f X can be written as its difference with measurement q T , Putting the pieces together Generically we expect the integrals above not to vanish. The ν dependence here should cancel with contributions from Glauber diagrams with one additional collinear gluon, and ν 2 should be replaced by a hard scale related to q + q − . It would certainly be interesting to see whether the explicit forms of these integrals can help determine some observables for which the integrals do vanish, and factorization violation is then postponed to higher order. So far our discussion is restricted to diagrams with soft emissions from the Glauber line. At leading order, real-emission diagrams can also have a single soft gluon coming off of an active collinear parton. Example tree-level and 1-loop diagrams are A factorization-violating effect can come from interference of M 0 Ra with two-loop realemission diagrams M 2 R : and from interference between M 1 Ra and the one-loop diagram M 1 R in Eq. (60). These interference terms contribute to the cross section at order g 6 s |M 0 | 2 . Both M 1 Ra (q, p 3 , k) and M 2 R (q, p 3 , k) contain IR divergences from the Glauber loop labeled in the diagram. With the presence of Lipatov vertex, the color generators of the two Glauber operator insertions in M 2 R do not commute, therefore IR divergences do not cancel at amplitude-squared level. The interference term M 1 Ra (M 1 R ) * + M 0 Ra (M 2 R ) * could therefore contain q T -dependent IR poles, generating factorization-violating effects.
The same issue appears in M R at higher loop order, for example, though diagarms Since color generators to the left of the Lipatov vertex do not commute to the right, IR divergences in real-emission diagrams do not cancel in the squared amplitudes |M R | 2 N-Glauber or (M R M * Ra ) N-Glauber , for N > 3. Understanding the structure of the IR divergences of the active/spectator soft-emission diagrams terms could be important to understanding factorization-violation in more detail and should be an interesting area for future research.

Summary and Conclusion
The main result of this paper is that for any single scale observable, factorization is not violated due to processes involving only Glauber exchange between spectator quarks. Previous studies, working in position space [19] or using more standard kinematic variables [24], suggested that the sum of all of these graphs would not vanish for a generic observable without some magical cancellation. In our proof, we show that the magical cancellation does happen, and is clearest using a natural set of conformal coordinates z andz. With these coordinates, the integral over phase space for spectator emission is completely inclusive over z for all infrared-safe single-scale observables, and hence the Glauber cancellation for q T implies the Glauber cancellation for all such observables.
To gain more insight into the cancellation, we computed the complete contribution of Glauber ladder graphs at next-to-leading order. In order to see the cancellation explicitly, we needed the terms up to order 0 at 2 loops and the terms up to order at 1-loop. These order terms are needed because a 1 phase space divergence appears when integrating over the spectator transverse momentum. We demonstrated a non-trivial cancellation between the O( 0 ) IR-finite 2-loop contribution and the 1 × 1-loop contribution, confirming our general result.
To demonstrate the cancellation to all orders, we showed that all of the IR divergences in the ladder diagrams exponentiate into a phase. After the phase is pulled out, the O( 0 ) terms are integrable over phase space while the O( ) terms are not. Thus at each order in perturbation theory, a non-trivial cancellation between finite contributions and phasespace singular contributions will occur. While we have focused on hadronic observables associated with a hard Drell-Yan scattering process in this paper, the method we use is general and can also be applied to more complicated processes. For example, it has been shown that dijet production at hadron colliders violate factorization if the dijets are backto-back and have small total transverse momentum [39]. It would be interesting to see whether or how factorization is violated through explicit calculation along the line of this work. Another interesting process to consider using our formalism is tt production at small transverse momentum [40][41][42], where it has been shown that final state interactions of top quarks and spectators leads to factorization violation [43].
One corollary of our proof is that that the leading order factorization-violating effects must involve soft radiation, through the Lipatov vertex associated with the Glauber lines. The graphs involving such soft radiation can have IR divergences that do not exponentiate and/or rapidity divergences. Such divergences, when regulated, generate a scale in the amplitude which prohibits the application of our scale-invariance argument for observable independence. We briefly studied the forms that these higher order terms can have. Diagramatically these contributions are similar to the factorization violation in Regge fac-torization [19,44]. It would be interesting to evaluate these diagrams explicitly, and see if there is any universality in the perturbative factorization violating terms. With the explicit expression for the factorization-violating contribution, it would also be interesting to search for hadronic observables that can delay factorization-violation to higher order or that avoid factorization-violating effects all together.
where α = 2 + 2n with n Glauber loops. The difference of the two parameterizations is O( ) or above. Since after cancelling the infrared divergent phase factor in Eq. (51), individual pure Glauber graphs can give at most finite corrections, these O( ) terms can be safely neglected.
The equality in Eq. (84) does not hold when the squared amplitude develops anomalous dimension, either from an incomplete cancellation of soft radiation, or from a rapidity divergence. These two cases are discussed in Sec. 4.

B Exponentiation of IR divergences
Consider diagram M Gn with n Glauber gluons exchanged: The diagram with n Glauber gluons is related to the one with n − 1 Glauber gluons through the following recurrence relation where α s (µ) ≡ 4πg 2 s (4πµ 2 ) . The integral in Eq. (94) can be divergent due to the region around ⊥ = 0. To remove this divergence, let us define an amplitude M G,R ≡ ∞ n=0 M n G,R recursively through the following equations: and for n ≥ 1,

C Singular region to all-orders
In this appendix, we examine the behavior of amplitude in the phase-space-singular regime, when either p 3 is aligned with p 1 or when p 4 is aligned with p 2 . These singular regions correspond to u = 0, v = 1 and u = 1, v = 0. Only the leading power terms in this expansion generate phase-space singularities, so we denote the leading-power approximation to the amplitude as M sing . The remainder is integrable, i.e. there exists some δ > 0 such that Thus we need to evaluate the singular part in d dimensions and the remainder can be evaluated in 4 dimensions. M G is determined by master integrals in d = 2 − 2 Euclidean space with three mass terms: p 2 3,⊥ , ( q ⊥ − p 3,⊥ ) 2 , q 2 T . In the singular limit where one of the masses becomes small, M sing is determined by single-scale integrals which we can compute to all-loop order. For example, in the limit where u = where C 0 and B 0 are one-loop scalar triangle and bubble integrals d = 2 − 2 dimension. M sing in the first few orders of α s are The general formula of M sing at n-loop order is, and the all-orders phase factor φ g is Note that the 1 term in φ g agrees with Eq. (98). The key point represented in the expression in Eq. 109 is that once the phase is factored out the singular terms start at order . + δ h 3 + δ h 4 + ln u ln v + z(1 −z) Split(z ij ) encodes the energy dependence of the splitting amplitude, where z ij is defined as the energy fraction of the daughter fermion in q → qg time-like splitting, see Eqs. (28) and (29). Except for an overall factor of q −4 T in |M 0 | 2 , the squared amplitude depends only on dimensionless parameters u and v, up to terms suppressed by . This is a consequence of the scale symmetry of ladder integrals in d = 2 transverse plane, as well as the cancellation of IR singularities.
The integral over the radial distance can be written as The integral over σ from 1 to ∞ can be inverted into an integral from 0 to 1, so that Therefore