Factorization and transverse phase-space parton distributions
CCERN-TH-2021-019
Factorization and transverse phase-space parton distributions
Bin Wu
1, 2, 3, ∗ Theoretical Physics Department, CERNCH-1211 Geneva 23, Switzerland ∗ Instituto Galego de F´ısica de Altas Enerx´ıas IGFAE,Universidade de Santiago de Compostela, E-15782 Galicia-Spain LIP, Av. Prof. Gama Pinto, 2, P-1649-003 Lisboa, Portugal
We first revisit impact-parameter dependent collisions of ultra-relativistic particlesin quantum field theory. Two conditions sufficient for defining an impact-parameterdependent cross section are given, which could be violated in proton-proton collisions.By imposing these conditions, a general formula for the impact-parameter dependentcross section is derived. Then, using soft-collinear effective theory, we derive a factor-ization formula for the impact-parameter dependent cross section for inclusive hardprocesses with only colorless final-state products in hadron and nuclear collisions. Itentails defining thickness beam functions, which are Fourier transforms of transversephase-space parton distribution functions. By modelling non-perturbative modes inthickness beam functions of large nuclei in heavy-ion collisions, the factorization for-mula confirms the cross section in the Glauber model for hard processes. Besides, thefactorization formula is verified up to one loop in perturbative QCD for the inclusiveDrell-Yan process in quark-antiquark collisions at a finite impact parameter.
Keywords: factorization; impact-parameter dependent collisions; SCET; Glauber model ∗ [email protected] a r X i v : . [ h e p - ph ] F e b I. INTRODUCTION
In hadron and nuclear collisions, cross sections for hard processes are always a blend ofshort-distance and long-distance physics. QCD factorization allows one to systematicallystudy incoherence between effects at various distance scales [1]. The predictive power ofperturbative QCD for hadron collider physics relies on validity of factorization theoremsand universality of quantities encoding long-distance physics. Factorization also provides asystematic approach to resum large logarithms of ratios between different scales to all ordersin perturbation theory [2].Factorization for the Drell-Yan cross section in hadron collisions has been extensivelystudied in the literature. Its validity has been proved within the context of perturba-tive QCD [3–6]. It has been alternatively studied using the soft-collinear effective theory(SCET) [7–11] in Refs. [10, 12]. Such an effective field theory approach facilitates explor-ing factorization and resummation in other similar processes, such as inclusive Higgs bosonproduction [13].Long-distance physics in all the processes mentioned above manifests itself either inparton distribution functions (PDFs) [14, 15], transverse-momentum-dependent (TMD)PDFs [15] or the beam functions [16] in SCET. They are all defined as matrix elementsof gauge invariant operators sandwiched between some momentum eigenstate of collidinghadrons. And they do not possess any information on the spatial distribution of quarks andgluons inside the hadrons, needed for a holistic snapshot of quantum phases-space partondistributions [17].The spatial distribution of partons in a hadron, say, a proton, could be revealed bystudying impact-parameter dependent collisions, as have been extensively studied in nucleus-nucleus (AA) collisions. The impact parameter in AA/heavy-ion collisions can be determinedvia centrality measurements by using the Glauber model [18]. In this model, the cross sectionfor the Drell-Yan production of vector bosons reduces to that in binary nucleon-nucleoncollisions, which is consistent with recent measurements at the LHC within experimentaluncertainties [19–23]. In order to unambiguously define the impact parameter of proton-proton (pp) collisions, one needs first to clarify conceptual difference between small collidingsystems like pp collisions and large colliding systems like AA collisions.The discovery of collectivity in pp collisions [24–29], however, blurs the boundary between
FIG. 1. The quantum picture of impact-parameter dependent collisions. This figure combinesthe amplitude and the conjugate amplitude for the hard process in Eq. (1) into a cut diagram.The transverse momenta of particles A and B are known to be around some design values withan uncertainty ∆ p T . Accordingly, their transverse locations x A and x B = x A + b can only bedetermined with an accuracy ∆ x T limited by the uncertainty principle. This accounts for thedisplacement of their transverse positions across the cut. The product C with momentum p µC iscreated at some hard scattering vertex around X , initiated by two partons respectively collinearto A and B . Different collinear partons can communicate with one another via exchange of softgluons. The misalignment of the hard vertex in the transverse plane across the cut, that is, itstransverse coherence length, is of order 1 /p C,T . large and small colliding systems [30]. Concepts based on (classical) collision geometry andthe impact parameter in heavy-ion collisions have been frequently employed to interpretcollectivity in pp collisions in many theoretical discussions without scrutiny (see Ref. [31]for a recent review). Nowadays, redefining the boundary of physical concepts respectivelyapplicable to pp, pA and AA collisions is one of the main focuses in high-energy nuclearphysics [30, 31], which demands a unified theoretical approach in QCD to treat all thesecollisions on the same footing.The sole purpose of this paper is to lay the groundwork for a unified description ofhard processes in impact-parameter dependent pp, pA and AA collisions based on QCDfactorization. As depicted in Fig. 1, we restrict ourselves to a generic inclusive hard processin hadron and nuclear collisions at an impact parameter b : A + B → C + anything else (1)with A and B either hadrons or nuclei, and C some colorless final-state object, such aselectroweak gauge bosons and Higgs bosons. Below, we motivate and outline how we proceedto derive the factorized cross section for such a hard process.In Sec. II, we revisit the fuzzy quantum picture of impact-parameter dependent collisions.The motivation for this section is two-fold: First, in a typical pp collision at the LHC, theproton size R , the impact parameter b and even the proton transverse spatial dispersion ∆ x T could be comparable with one another. In this case, a quantum description of the collisionbecomes more appropriate. Second, in heavy-ion collisions, the Glauber model [18] hasbeen broadly employed not only in theoretical calculations but also in experimental studiesof collision geometry from centrality measurements. Behind the Glauber model underlies aclassical picture of the impact parameter, which can even trace back to Rutherford’s seminaldiscovery in Ref. [33] that helped usher in the quantum age. Giving this model a justificationin QCD entails a quantum picture instead. However, the formula for the impact-parameterdependent cross section has not been derived in quantum scattering theory [34, 35]. Thissection is aimed to fill this gap.We consider a collision of two ultra-relativistic particles A and B at the impact parameter b . In collider physics, all we know is that their momenta are determined to be around somedesign values P µi with i = A and B . From this information, we do know that the statesof the particles can be described by some wave packets in momentum space, with theirmomentum width constrained by experimental uncertainties. Accordingly, the particles arelocalized in position space, described by some well-defined spatial wave packets. We aim todefine the impact-parameter dependent cross section for the collision, which is intrinsic tothe colliding particles and should be independent of the beam particles’ wave packets.We find that such an impact-parameter dependent cross section can be defined, as givenin Eq. (27), if the following two conditions are fulfilled. Condition i) is the high-energylimit: | P iz | (cid:29) P iT , ∆ p T , ∆ p z with ∆ p T and ∆ p z respectively the particle’s transverse andlongitudinal momentum dispersions. It is needed to identify the longitudinal momenta of At the LHC, the crossing angle at the interaction point θ c can be measured with an accuracy ∆ θ c < µ rad [32]. For a E = 7 TeV proton, the uncertainty principle dictates ∆ x T ≥ E ∆ θ c > . the wave packets in the amplitude and the conjugate amplitude. Condition ii) is to requirethat the fuzziness in particles’ transverse positions should be smaller than b : b (cid:29) ∆ x T . Itis needed to identify the transverse momenta of the wave packets in the amplitude and theconjugate amplitude. Therefore, they are both necessary for integrating out the wave packetsin order to define the cross section and for unambiguously defining the impact parameter.Otherwise, the probability for producing any final states in the collision explicitly dependson the wave packets, which are not guaranteed to be the same in different experiments. And,one ends up measuring a quantity which differs across experiments.With the general formula for the impact-parameter dependent cross section derived inSec. II, one can calculate it for the hard process in Eq. (1) using perturbative QCD. Ingeneral, one needs to deal with the long-distance behavior in perturbative series. We appealto QCD factorization to factor out long-distance physics in Sec. III.In Sec. III A, using SCET we first give a detailed derivation of the factorized cross sectionfor the process in Eq. (1). The main features of this factorization formula can be understoodbased on the following heuristic argument:Imagine that besides the hard scale Q we measure the transverse momentum p C,T (cid:29) Λ QCD of the product C . As illustrated in Fig. 1, C is produced at some hard scatteringvertex around X with a transverse coherent length | x | ∼ /p C,T (cid:29) t h ∼ /Q . It effectively picks out two partons located around X respectively from A and B with a transverse spatial accuracy of the order of | x | . The distributions of these partonsin the colliding particles are described by a new type of PDFs, which are referred to as thickness beam functions in this paper.The thickness beam functions are expected to be universal as a consequence of thehard-collinear and soft-collinear factorization. Generic hard processes all involve radiationcollinear to the two beam directions n µA and n µB . The coherent time of a n i -collinear partonwith momentum p µn i is given by t n i = 1 n i · p n i ∼ λ Q (cid:29) t h ∼ Q (2)where the expansion parameter λ , as a ratio between a soft momentum scale determinedby some specific observable to be measured and the hard scale Q , is assumed to be muchsmaller than unity. That is, the beam collinear radiation takes place too early to interferewith the hard scattering. Different beam collinear partons may communicate, via exchangeof soft gluons, with each other or with final-state colored objects since soft gluons have acoherent time t s (cid:29) t h : t s ∼ λQ , or 1 λ Q . (3)As long as soft gluons cannot resolve the substructure of collinear splittings, there is factor-ization between soft and beam collinear partons. And the thickness beam functions shouldbe universal to all the hard processes in which these two types of factorization hold true.Because their formation time is much longer than t h , the hard scattering occurs toorapidly to radiate soft gluons. That is, soft gluons are detached from the hard scatteringvertex. As a consequence, for the process in Eq. (1), soft radiation only couples to the totalcolor charges of collinear partons from A or B and organizes itself into soft Wilson lines.They act on the vacuum state to define another module of the factorization formula: thesoft function. Since it is the vacuum state that is involved in its definition, the soft functionis independent of X and does not carry any information on the impact parameter.Based on the above argument one can expect that the factorization formula, as given inEq. (44), schematically takes the following form: dσ AB d b dy C d p C ∼ (cid:88) j,k (cid:90) d X T j/A ( X − x A ) ⊗ T k/B ( X − x B ) ⊗ H j,k → C ⊗ S (4)with the impact parameter b = x B − x A , X denoting the transverse position of the hardscattering and x i being the (average) transverse location of particle i , as shown in Fig.1. Here, the hard function H j,k → C can be calculated from the partonic process j, k → C + anything else and S is the soft function. They are the same as for conventional crosssections in pp collisions [10, 12, 13]. The thickness beam functions T j/i are related to thequantum transverse phase-phase PDFs via the Fourier transform. That is, they encodethe information on the distribution of parton j carrying a momentum fraction z in bothtransverse momentum and position spaces inside particle i . A detailed discussion about theproperties of T j/i is presented in Sec. III B.In Sec. III C, we study the connection between the factorization formula in Eq. (4) andthe cross section in the Glauber model in heavy-ion collisions. We find that T j/i reducesto a product of the thickness function in the Glauber model and the corresponding beamfunction, as the Fourier transform of the TMD PDF, when the incoming nucleus is treatedas an assembly of uncorrelated nucleons. This gives the success of the Glauber model [36]a QCD justification for such hard processes. On the other hand, the factorization formulaallows one to explore refined details about the parton distributions in heavy nuclei. It canbe used to optimize the potential of the LHC and HL-LHC, with increased accuracy, insystematically studying cold nuclear effects [37–39], which are absent in the aforementionedmodelling. Nuclear modifications of PDFs had been first revealed in deep inelastic scatteringby the European Muon Collaboration [40] and have been favored by recent measurementsin PbPb collisions at the LHC [41]. The factorization formula is important for investigatingthe LHC/HL-LHC’s potential to pin down such effects since the hard and soft functions canbe calculated at high accuracy in perturbative QCD.Using the factorization formula in Eq. (4), one can potentially measure the transversephase-space parton distributions in protons through inclusive hard processes in impact-parameter dependent pp collisions, which have not been experimentally explored. On theother hand, given the fact that the aforementioned two conditions needed to define theimpact-parameter dependent collisions are not always fulfilled, caution is needed when theimpact-parameter dependent cross section is studied in pp collisions. A brief discussion onthis issue is given in Sec. III D.At this point, we have two ways to calculate the impact-parameter dependent crosssection by either using the general formula in Sec. II or the factorization formula in Sec.III. This provides a way to verify factorization order by order in perturbation theory: first,calculate the cross section in perturbative QCD using the general formula; then, expandthe perturbative QCD results at small λ or, equivalently, large Q ; and finally compare theleading-order results in Q to those given by the factorization formula.In Sec. IV, a verification of the factorization formula is carried out for the inclusiveDrell-Yan process q ¯ q → γ ∗ . This process involves two scales: the photon virtuality Q andthe impact parameter b , which set the expansion parameter λ = 1 / ( bQ ). We only focuson the physically interesting case with λ (cid:28)
1, that is, the impact parameter being muchlarger than 1 /Q . At leading order in λ , the factorization formula is confirmed up to oneloop in perturbative QCD. We also calculate the one-loop spatial quark distribution in afast moving quark, which carries information complementary to the corresponding TMDPDF. The probability to find a quark carrying a momentum fraction z at a transversedistance r from the original transverse location of the incoming quark is found to be inverselyproportional to r .Detailed derivations and calculations backing up the above summary can be found in theensuing sections. And the interested reader is invited to vet their details. II. THE IMPACT-PARAMETER DEPENDENT CROSS SECTION
In this section we revisit the concepts of the impact parameter and the impact-parameterdependent cross section in quantum field theory (QFT). These two concepts are well-definedin classical physics, as exemplified by the original derivation of the Rutherford scatteringformula [33]: the deflection angle of a charged particle scattering off a Coulomb potential isgiven by a unique function of the impact parameter. In contrast, in the textbook derivationof the conventional cross section (see, e.g., Chapter 4 of [35]), there is no unique relationbetween the deflection angle and the impact parameter. Yet, Rutherford’s formula can beeasily reproduced. Below, we redo the derivation of the formula for the cross section in QFTin order to restore its impact-parameter dependence.Consider a collision of two ultra-relativistic particles A and B respectively from twocounter-moving beams. Before the collision, the momentum of particle i with i = A or B isaccelerated to be around some design value (cid:126)P i with an uncertainty ∆ (cid:126)p i . So, all we know isthat its wave packet peaks about (cid:126)P i with a momentum width equal to or smaller than ∆ (cid:126)p i ,which can be generically written in the form | φ i (cid:105) = (cid:90) d (cid:126)p (2 π ) e − i p · x i (cid:112) E p φ i ( (cid:126)p ) | (cid:126)p (cid:105) , (5)where φ i is normalized, that is,1 = (cid:104) φ i | φ i (cid:105) = (cid:90) d (cid:126)p (2 π ) | φ i ( (cid:126)p ) | , (6)the phase factor in front of φ i ( (cid:126)p ) accounts for the spatial translation in the transverse planeand x i is the transverse position vector of particle i , to be determined later. Here andbelow, we denote two-dimensional vectors in the transverse plane by bold letters and three-dimensional vectors by letters with an arrow overhead.The impact parameter of the collision is defined as b ≡ x B − x A , (7) We only focus on unpolarized collisions and ignore the wave packet’s spin dependence here. once x A and x B are given. However, the particles are, at best, known to locate somewherewith an uncertainty limited by the uncertainty principle. In order to quantitatively define x i , one needs the particle’s spatial wave packet, which is given by˜ φ i ( x − x i , z ) = (cid:104) (cid:126)x | φ i (cid:105) ≡ (cid:90) d (cid:126)p (2 π ) e i(cid:126)p · (cid:126)x − i p · x i φ i ( (cid:126)p ) (8)with the position eigenstate [42] | (cid:126)x (cid:105) ≡ (cid:90) d (cid:126)p (2 π ) e − i(cid:126)p · (cid:126)x (cid:112) E p | (cid:126)p (cid:105) . (9)As long as (cid:126)x is not measured with a resolution better than the particle’s de Broglie wave-length, | ˜ φ i | admits interpretation as the probability density to find particle i at (cid:126)x . Given ˜ φ i , x i can be chosen to be the average transverse position vector of particle i . Obviously, thereare alternative choices for x i , such as the center of mass or the transverse peak location of | ˜ φ i | . Below, we show that such an ambiguity in defining b ≡ | b | is negligible when one isallowed to define an impact-parameter dependent cross section.In principle, one can predict the probability for producing any final state |{ p f }(cid:105) in thecollision at the impact parameter b according to P b ( φ A , φ B → { p f } ) = (cid:104) φ A φ B | ˆ S † |{ p f }(cid:105)(cid:104){ p f }| ˆ S | φ A φ B (cid:105) . (10)Since we are only interested in the cross section, we can replace the S -matrix element by (cid:104){ p f }| ˆ S | p A , p B (cid:105) → (2 π ) δ (4) ( p A + p B − (cid:88) p f ) iM ( p A , p B → { p f } ) . (11)Then, plugging the wave packets in Eq. (5) into Eq. (10) and using one of the delta functionsfrom the above replacement to integrate out ¯ p Az , ¯ p Bz and ¯p B yields P b ( φ A , φ B → { p f } ) = (cid:90) p A , ¯p A , p B ,p Az ,p Bz e i b · ( p A − ¯p A ) × φ A ( (cid:126)p A ) φ ∗ A ( (cid:126) ¯ p A ) φ B ( (cid:126)p B ) φ ∗ B ( (cid:126) ¯ p B ) σ ( p A , p B → { p f } ← ¯ p A , ¯ p B ) , (12)where the measure (cid:82) n (cid:81) j =1 dp j π is denoted by (cid:82) p , ··· ,p n for brevity, the off-diagonal cross-sectionis defined as σ ( p A , p B → { p f } ← ¯ p A , ¯ p B ) ≡ (2 π ) δ (4) ( p A + p B − (cid:80) p f ) (cid:112) E p A E ¯ p A E p B E ¯ p B | ¯ v Az − ¯ v Bz |× M ( p A , p B → { p f } ) M ∗ (¯ p A , ¯ p B → { p f } ) , (13)0 ¯p B = p A + p B − ¯p A , and the longitudinal momenta ¯ p Az and ¯ p Bz are solutions to E p A + E p B = E ¯ p A + E ¯ p B , ¯ p Az + ¯ p Bz = p Az + p Bz . (14)The impact-parameter dependent probability P b generally depends on the wave packets.However, the beam particles’ wave packets are not measured in collider physics. Moreover,they are not guaranteed to be the same in different experiments. The cross section, on theother hand, is intrinsic to the colliding particles and, therefore, should be independent ofthe wave packets in order to allow comparison across experiments. Below, we show that thefollowing two conditionsi) | P iz | (cid:29) | P i | , ∆ p T , ∆ p z ; ii) | b | (cid:29) ∆ x T (15)are sufficient for defining the impact-parameter dependent cross section from P b . Here, ∆ x T ,∆ p T and ∆ p z are respectively the transverse spatial, transverse and longitudinal momentumdispersions of the colliding particles.Condition i) is the high-energy limit in which both particles are moving predominantlyalong the beam ( ± z) directions. In this limit, the solutions to Eq. (14) are given by¯ p Az − p Az = p Bz − ¯ p Bz ≈ | p A | − | ¯p A | P Az − | p B | − | ¯p B | P Bz (16)for P Az > P Bz <
0. Since these terms are small , we will drop them and take¯ p Az = p Az , ¯ p Bz = p Bz . (17)Following [43], we define transverse Wigner functions W i ( X , P ) ≡ (cid:90) d χ (2 π ) (cid:90) dze i P· χ ˜ φ i (cid:16) X − χ , z (cid:17) ˜ φ ∗ i (cid:16) X + χ , z (cid:17) . (18)In terms of W i , P b can be expressed as P b ( φ A , φ B → { p f } ) = (cid:90) d X A d P A d X B d P B (cid:90) d q (2 π ) e i q · ( b + X B − X A ) × W A ( X A , P A ) W B ( X B , P B ) σ ( p A , p B → { p f } ← ¯ p A , ¯ p B ) (19)with p A = P A + q , p B = P B − q , ¯ p A = P A − q , ¯ p B = P B + q . (20) For example, estimated from crossing angles at the LHC [32], one has | ¯ p iz − p iz | / | P iz | < − . P iz . For collider phenomenology, one only needs to keep leading-order terms in ∆ /P iz with ∆ = P i , ∆ p T or ∆ p z , which is equivalent to replacing P i by and p iz by their designvalues P iz in the off-diagonal cross section defined by Eq. (13) . | q | ∼ /b could also bemuch smaller than | P iz | . However, as it will become clear in the following sections, we donot expand the amplitude squared in the off-diagonal cross section about | q | /P iz = 0 here.We only do so when it is needed in detailed calculations, as exemplified in Sec. IV. As aresult, we have σ ( p A , p B → { p f } ← ¯ p A , ¯ p B ) = 12 s M ( p A , p B → { p f } ) M ∗ (¯ p A , ¯ p B → { p f } ) × (2 π ) δ (4) ( p A + p B − (cid:88) p f ) , (21)where the Mandelstam variable s = ¯ n A · P A ¯ n B · P B , (22)the incoming momenta are given by p µA = ¯ n A · P A n µA q µT − q T n A · P A ¯ n µA , p µB = ¯ n B · P B n µB − q µT − q T n B · P B ¯ n µB , ¯ p µA = ¯ n A · P A n µA − q µT − q T n A · P A ¯ n µA , ¯ p µB = ¯ n B · P B n µB q µT − q T n B · P B ¯ n µB , (23)with masses being neglected, and for any four-vector V µ we define V µT = g T µν V ν = (0 , V x , V y , . (24)Note that one has V T = −| V | . Here, the light-like vectors are chosen to be n µA = ¯ n µB ≡ (1 , , , , n µB = ¯ n µA ≡ (1 , , , − , (25)and the transverse metric is defined as g µνT = g µν − n µA n νB + n νA n µB . (26)As a result, the integrand on the right-hand side of Eq. (19) depends on P i only throughthe transverse Wigner functions. In order to integrate out the wave packets and get unity, Another justification of this approximation is that modern detectors typically have limited resolutionwhich can not resolve such a small variation of P i [35]. X i from its phase factor. Condition ii) is sufficient to justify such anapproximation. If one goes back to Eq. (12), this, equivalently, means that the differencebetween p i and ¯ p i is negligible compared to their average. Obviously, this is also thecondition needed to eliminate the ambiguity in defining the impact parameter in Eq. (7)due to alternative choices of x i .At the end, one can integrate out the transverse Wigner functions in Eq. (19) and obtainthe impact-parameter dependent differential cross section for producing any observable Odσd b dO = (cid:90) d q (2 π ) e i q · b (cid:90) (cid:89) f (cid:2) d Γ p f (cid:3) δ ( O − O ( { p f } )) × s M ( p A , p B → { p f } ) M ∗ (¯ p A , ¯ p B → { p f } )(2 π ) δ (4) ( p A + p B − (cid:88) p f ) , (27)where O ( { p f } ) defines the observable O as a function of the final-state momenta { p f } , theincoming momenta are given in Eq. (23) and the phase-space measure in d -dimensionalspacetime for a particle of mass m (cid:90) d Γ p ≡ (cid:90) d d p (2 π ) d (2 π ) δ ( p − m ) θ ( p ) = (cid:90) dyd d − p π ) d − (28)with y the particle’s rapidity. III. INCLUSIVE HARD PROCESSES IN HADRON AND NUCLEARCOLLISIONS
Based on the heuristic argument outlined in the introduction, one can expect that a newtype of PDFs, which describe the parton distributions in transverse phase space, can beuniversally defined for inclusive hard processes in impact-parameter dependent hadron andnuclear collisions. In this section, we justify this argument using SCET [7–11].
A. Factorization for inclusive hard processes with colorless final states
In this subsection, we derive a factorized form of Eq. (27) for the process in Eq. (1). Inhadron and nuclear collisions, there are additional length scales that one needs to consider:the sizes R i of the colliding particles. Accordingly, the impact-parameter dependent crosssection for hadron and nuclear collisions is defined in the range: R A + R B (cid:38) b (cid:29) ∆ x T . We are only interested in strong interactions and hence ignore the cases with b (cid:29) R A + R B . q T ∼ /b ∼ R i in addition to perturbativemodes with p f (cid:38) Λ QCD that contribute to the observable O ( { p f } ) to be measured. Sincethese non-perturbative modes are collinear to the colliding hadrons or nuclei, we take themas submodes of the corresponding beam collinear modes.
1. Basics of SCET
We first briefly review the elements of SCET that are relevant to our discussion. Withthe modes of q T ∼ /b taken as submodes of the corresponding beam collinear modes, allthe relevant infrared degrees of freedom for the process under study are the same as thosefor the conventional cross section in Refs. [10, 12, 13]: n i -collinear: p µn i ∼ Q ( λ , , λ ) n i ¯ n i , soft: p µs ∼ λ Q for SCET I λQ for SCET II , (29)where λ (cid:28) V µ V µ = ¯ n µi n i · V + n µi n i · V + V µT ≡ ( n i · V, ¯ n i · V, V ) n i ¯ n i (30)in terms of a pair of light-like vectors n i and ¯ n i with n i · ¯ n i = 2.The S -matrix element at leading order in λ can be expressed generically in the form (cid:104) p C , { p X }| ˆ S | φ A φ B (cid:105) = (cid:90) d xe ip C · x (cid:104){ p X }| i ˆ M ( x ) | φ A φ B (cid:105) (31)with { p X } standing for momenta of unmeasured infrared partons and the amplitude operatorˆ M being a convolution of relevant SCET operators and corresponding Wilson coefficients.Let us combine everything but the collinear and soft fields in the SCET operators with theWilson coefficients and denote their sum by the coefficient C . In this way, irrespective of thespecies of the product C , ˆ M can be always written in the form i ˆ M ( x ) = (cid:90) dt A dt B C a A a B α A α B ( (cid:15), t A , t B )[ S n A φ n A ( x + t A ¯ n A )] α A a A [ S n B φ n B ( x + t B ¯ n B )] α B a B , (32)where the coefficient C is to be determined by the matching procedure after the species of theproduct C being specified, a i and α i are respectively the color and Lorentz/spinor indices,4 S n i is the soft Wilson line along the collinear direction n i and the collinear building blocks φ n i stand for [44] χ n i ( x ) = W † n i ( x ) /n i / ¯ n i ψ n i ( x ) , ¯ χ n i ( x ) , B µn i T = 1 g s W † n i ( x ) iD µn i T W n i ( x ) , (33)respectively for n i -collinear quarks, antiquarks or gluons with D µn i T ≡ ∂ µT − ig s A µn i T and W n i the n i -collinear Wilson line.Both the hard-collinear factorization and the soft-collinear factorization (at leading orderin λ ) are implemented through ˆ M , which are independent of the initial states of the collidingparticles. Soft radiation decouples from the hard scattering encoded in the coefficient C ,which has been proved for certain processes based on infrared power counting in perturbativeQCD [1, 2]. Soft gluons can couple to collinear partons only through their unphysicalpolarizations n i · A s , which gives rise to the soft Wilson lines in ˆ M . Soft and collinear fieldsdecouple in the SCET Lagrangian for both SCET I (after decoupling transformation [9]) andSCET II . The soft Wilson lines in coordinate space take the form S n i ( x ) = P e ig s (cid:82) −∞ dtn i · A s ( tn i + x ) for φ n i + (incoming particles)¯ P e − ig s (cid:82) ∞ dtn i · A s ( tn i + x ) for φ n i − (outgoing antiparticles) (34)where φ n i ± respectively stand for the positive and negative energy parts of φ n i and A s ≡ A as T a with T a being SU ( N c ) generators in the corresponding color representation. Thecollinear Wilson lines take the form W n i ( x ) = P e ig s (cid:82) −∞ dt ¯ n i · A ni ( t ¯ n i + x ) or ¯ P e − ig s (cid:82) ∞ dt ¯ n i · A ni ( t ¯ n i + x ) . (35)The inclusion of collinear Wilson lines in the collinear building blocks is mandated by thecollinear gauge invariance [9].
2. Derivation of the factorization formula
With the S -matrix element given in Eq. (31), we are ready to derive the factorized crosssection for the process in Eq. (1). We impose the two conditions in Eq. (15) and identify This is equivalent to the approximation ( p c + p s ) ≈ ¯ n · p c n · p s in all propagator denominators withinternal momenta given by a sum of n -collinear ( p c ) and soft ( p s ) momenta. The justification of thisapproximation in perturbative QCD is rather technical event for the Drell-Yan process [6]. P b with the cross section at the outset: dσd b dy C d p C = 12(2 π ) (cid:90) d Xd x e − ip C · x × (cid:88) { p X } (cid:104) φ A φ B | ˆ M † (cid:16) X + x (cid:17) |{ p X }(cid:105)(cid:104){ p X }| ˆ M (cid:16) X − x (cid:17) | φ A φ B (cid:105) . (36)Here, for definiteness we measure the rapidity y C and the transverse momentum p C of theobject C .The following derivation is pretty much the same as that in the previous section, exceptthat we keep X unintegrated. It tells us the transverse location of the hard scattering vertexas shown in Fig. 1. One can first integrate out X and X z in Eq. (36) by using themomentum operator and then the longitudinal momenta associated with the wave packetsin the conjugate amplitude. After that, one identifies the longitudinal momenta in theamplitude and the conjugate amplitude, as in Eq. (17). And, in terms of the transverseWigner functions in Eq. (18), the cross section can be expressed as dσd b dy C d p C = (cid:90) d X (cid:89) i = A,B (cid:20)(cid:90) d X i d P i d q i (2 π ) e i q i · ( X − x i − X i ) W i ( X i , P i ) (cid:21) × dσdy C d p C ( p A , p B → p C ← ¯ p A , ¯ p B ) , (37)where in terms of ˆ M , the off-diagonal cross section in Eq. (21) takes the form dσdy C d p C ( p A , p B → p C ← ¯ p A , ¯ p B ) = 12(2 π ) s (cid:90) d xe − ip C · x × (cid:88) { p X } (cid:104) ¯ p A ¯ p B | ˆ M † (cid:16) x (cid:17) |{ p X }(cid:105)(cid:104){ p X }| ˆ M (cid:16) − x (cid:17) | p A p B (cid:105) , (38)and the incoming momenta are given by p µA = ¯ n A · P A n µA q µA − q A n A · P A ¯ n µA , p µB = ¯ n B · P B n µB q µB − q B n B · P B ¯ n µB , ¯ p µA = ¯ n A · P A n µA − q µA − q A n A · P A ¯ n µA , ¯ p µB = ¯ n B · P B n µB − q µB − q B n B · P B ¯ n µB , (39)with q µi = p µi − ¯ p µi orthogonal to n i and ¯ n i . Finally, the two conditions in Eq. (15) allow oneto integrate out the transverse Wigner functions and one has dσd b dy C d p C = (cid:90) d X (cid:90) q A , q B e i q A · ( X − x A )+ i q B · ( X − x B ) dσdy C d p C ( p A , p B → p C ← ¯ p A , ¯ p B ) . (40)6Since the collinear and soft modes decouple, the right-hand side of the above equationcan be further written in a factorized form. The coefficient C in ˆ M is independent of theinitial states of the colliding particles, which is combined into the hard function togetherwith its complex conjugate. The soft Wilson lines only act on the vacuum state to definethe soft function. As a result, the soft function is independent of X , as shown below. Thatis, the soft and hard functions are the same as for the conventional cross section [10, 12, 13].The collinear fields act on the incoming states to give a new type of PDFs, which arereferred to as thickness beam functions in this paper. The n i -collinear sector in Eq. (40)takes the form T α (cid:48) αa (cid:48) a ( r i , n i · x, x ) = (cid:90) d q (2 π ) e i q · r i × (cid:68) ¯ n i · P i , − q (cid:12)(cid:12)(cid:12) [ φ † n i ] α (cid:48) a (cid:48) (cid:16) n i · x , x (cid:17) [ φ n i ] αa (cid:16) − n i · x , − x (cid:17) (cid:12)(cid:12)(cid:12) ¯ n i · P i , q (cid:69) (41)with r i ≡ X − x i and the spin of particle i being implicitly averaged over. Given someprojector P α (cid:48) α , we define the corresponding thickness beam function T j/i as T α (cid:48) αa (cid:48) a ( r i , n i · x, x ) → (cid:90) dzz P α (cid:48) α ( z ¯ n i · P i ) d c i δ a (cid:48) a T j/i ( r i , z, x ) e i ni · x z ¯ n i · P i (42)with d c i the dimension of the color representation of φ n i and j the parton species corre-sponding to φ n i . The most common projectors for unpolarized collisions are given by thespin/polarization average, which take the form P ¯ α i α i ( k ) = ( /k ) ¯ α i α i for quarks/antiquarks d − ( − g ¯ α i α i T ) for gluons (43)with d the spacetime dimension.Inserting the expression of ˆ M in Eq. (32) into Eq. (40) and making the replacement inEq. (42) eventually yields the factorization formula: dσ AB d b dy C d p C = 14 πs (cid:88) j,k (cid:90) d X (cid:90) d x e i p C · x (cid:90) dz A z A dz B z B T j/A ( X , z A , x ) T k/B ( X − b , z B , x ) × (cid:90) (cid:89) f [ d Γ p f ] (cid:89) i = A,B δ ( z i ¯ n i · P i − ¯ n i · p C − (cid:88) ¯ n i · p f ) × H ¯ a A ¯ a B a A a B ( z A P A , z B P B → p C , { p f } ) S ¯ a A ¯ a B a A a B ( x ) , (44)7where we have taken x A = 0 and x B = b , P µi = ¯ n i · P i n µi , the soft function is defined by S ¯ a A ¯ a B a A a B ( x ) ≡ (cid:104) | ¯ T [ S † a (cid:48) B ¯ a B n B ( x + ) S † a (cid:48) A ¯ a A n A ( x + )] T [ S a A a (cid:48) A n A ( x − ) S a B a (cid:48) B n B ( x − )] | (cid:105) = (cid:104) | ¯ T [ S † a (cid:48) B ¯ a B n B ( x ) S † a (cid:48) A ¯ a A n A ( x )] T [ S a A a (cid:48) A n A (0) S a B a (cid:48) B n B (0)] | (cid:105) (45)with x ± ≡ X ± x /
2, and the hard function, given by the partonic process j ( z A P A ) + k ( z B P B ) → C ( p C ) + anything else( { p f } ), takes the form H ¯ a A ¯ a B a A a B ≡ P ¯ α A α A d c A P ¯ α B α B d c B ˜ C ∗ ¯ a A ¯ a B ¯ α A ¯ α B ˜ C a A a B α A α B (46)with ˜ C given by˜ C ( (cid:15), z A ¯ n A · P A , z B ¯ n B · P B ) = (cid:90) dt A dt B e i ( t A z A ¯ n A · P A + t B z B ¯ n B · P B ) C ( (cid:15), t A , t B ) . (47)For the spin-averaged projectors in Eq. (43), the hard function H a A a B a A a B = | M | (48)with | M | the square of the amplitude averaged over initial-state colors, spins or polariza-tions [45]. B. Thickness beam functions and transverse phase-space parton distributions
The thickness beam functions are universal. Their emergence only relies on the hard-collinear and soft-collinear factorization. Therefore, they should universally show up in allinclusive hard processes in hadron and nuclear collisions as long as the processes admit ofthese two types of factorization. And hard processes with colorless final states like that inEq. (1) provide the cleanest way to measure the thickness beam functions. The discussionin the previous subsection can be easily generalized to deep inelastic scattering in electron-proton and electron-ion collisions. Therefore, they could also be measured using the futureElectron-Ion Collider [46] once the impact-parameter dependence of the collisions can bedetermined experimentally.The thickness beam functions are related to transverse phase-space PDFs (TPS PDFs) via the Fourier transform with respect to x : f j/i ( r , z, p ) = (cid:90) d x e i p · x T j/i ( r , z, x ) . (49) TPS PDFs have a corresponding definition in QCD, in which the collinear building blocks are replaced bythe corresponding fields and the collinear Wilson lines are combined into gauge links. Subtlety, however,could arise from the difference between the product of collinear Wilson lines and the chosen gauge linksas for the beam functions [16] and TMD PDFs [15]. T j/i in Eq.(42), which generally takes the form T j/i ( r , z, x ) = (cid:90) d q (2 π ) e i q · r (cid:90) dt π e − izt ¯ n · P × (cid:68) ¯ n · P, − q (cid:12)(cid:12)(cid:12) [ φ † n ] α (cid:48) a (cid:18) t ¯ n x T (cid:19) Γ α (cid:48) α [ φ n ] αa (cid:18) − t ¯ n − x T (cid:19) (cid:12)(cid:12)(cid:12) ¯ n · P, q (cid:69) (50)with Γ α (cid:48) α to be determined after P α (cid:48) α are chosen. By using the momentum operator, onecan write T j/i ( r , z, x ) = (cid:88) m (cid:90) d q (2 π ) (cid:68) ¯ n · P, − q | [ φ † n ] α (cid:48) a ( r ) | m (cid:69) Γ α (cid:48) α (cid:68) m | [ φ n ] αa ( r ) | ¯ n · P, q (cid:69) × e i p m · x δ (¯ n · p m − (1 − z )¯ n · P ) . (51)In the first line, the right-hand side of this equation can be interpreted as the ”probability” of measuring a collinear field of type j at r inside particle i and producing a final state | m (cid:105) with total momentum p µm . Accordingly, the corresponding TPS PDF is f j/i ( r , z, p ) = (cid:88) m (cid:90) d q (2 π ) (cid:68) ¯ n · P, − q | [ φ † n ] α (cid:48) a ( r ) | m (cid:69) Γ α (cid:48) α (cid:68) m | [ φ n ] αa ( r ) | ¯ n · P, q (cid:69) × δ (2) ( p + p m ) δ (¯ n · p m − (1 − z )¯ n · P ) , (52)which, after being coarse-grained, is the probability of finding a collinear parton at r carryinga momentum fraction z and transverse momentum p .In the aforementioned coarse-grained sense, thickness beam functions and TPS PDFsadmit interpretation respectively as the beam functions [16] and TMD PDFs [15] at r sinceone typically has | r | (cid:29) | x | for hard processes. For inclusive hard processes with | x | ∼ /Q , x can be dropped from the thickness beam functions as a result of multipole expansion.And T j/i ( r , z, ) can be viewed as the corresponding conventional parton distribution func-tions [14, 15] at r , that is, the transverse spatial PDFs. In this case, the off-diagonal ma-trix element needed for defining T j/i ( r , z, ) is the same as generalized parton distributions(GPDs) [47] with the incoming momenta differing only in the transverse plane. Like the Wigner function in quantum mechanics [43], we don’t expect that this term is always positive.It can be literally interpreted as a probability only after being coarse-grained, say, measured at r with arelatively large uncertainty. T q/i ( r , z, x ) = (cid:90) d q (2 π ) e i q · r (cid:90) dt π e − izt ¯ n · P × (cid:68) ¯ n · P, − q (cid:12)(cid:12)(cid:12) ¯ χ n (cid:18) t ¯ n x T (cid:19) / ¯ n χ n (cid:18) − t ¯ n − x T (cid:19) (cid:12)(cid:12)(cid:12) ¯ n · P, q (cid:69) , (53)for antiquarks, T ¯ q/i ( r , z, x ) = (cid:90) d q (2 π ) e i q · r (cid:90) dt π e − izt ¯ n · P × (cid:68) ¯ n · P, − q (cid:12)(cid:12)(cid:12) Tr (cid:20) / ¯ n χ n (cid:18) t ¯ n x T (cid:19) ¯ χ n (cid:18) − t ¯ n − x T (cid:19)(cid:21) (cid:12)(cid:12)(cid:12) ¯ n · P, q (cid:69) = (cid:90) d q (2 π ) e i q · r (cid:90) dt π e − izt ¯ n · P × ( − ) (cid:68) ¯ n · P, − q (cid:12)(cid:12)(cid:12) ¯ χ n (cid:18) − t ¯ n − x T (cid:19) / ¯ n χ n (cid:18) t ¯ n x T (cid:19) (cid:12)(cid:12)(cid:12) ¯ n · P, q (cid:69) , (54)and for gluons T g/i ( r , z, x ) = z ¯ n · P ( − g T α (cid:48) α ) (cid:90) d q (2 π ) e i q · r (cid:90) dt π e − izt ¯ n · P × (cid:68) ¯ n · P, − q (cid:12)(cid:12)(cid:12) B aα (cid:48) nT (cid:18) t ¯ n x T (cid:19) B aαnT (cid:18) − t ¯ n − x T (cid:19) (cid:12)(cid:12)(cid:12) ¯ n · P, q (cid:69) . (55)For gluons, there are also other possible projectors as combinations of g µνT , x µT and r µT . Wewill not exhaust all the possible forms of the projectors in this paper, whose relevance willdepend on specific processes under consideration. C. The Glauber model for hard processes in heavy-ion collisions
In heavy-ion collisions, one has | r i | ∼ | b | ∼ R i (cid:29) / Λ QCD (cid:29) | x | , /Q. (56)That is, r i -dependence of thickness beam functions lies deep in the non-perturbative regime.In principle, one could also quantitatively study such non-perturbative degrees of freedomusing the effective field theory approach. We, instead, content ourselves with connecting ourfactorization formula in Eq. (44) to the cross section in the Glauber model [18] by modellinglarge nuclei.0In heavy-ion collisions, the two conditions in Eq. (15) are easily fulfilled. In principle, theimpact parameter in Eq. (7) can be determined better than 1 fm as long as one is not aimedat high accuracy in the determination of the beam particles’ transverse momenta and keeps∆ p T (cid:38) . Since the quantum fuzziness is much smaller than the impact parameter,the classical concept of collision geometry used in the Glauber model is indeed a reasonableapproximation to the underlying quantum picture.Let us approximate the thickness beam functions by appealing to a commonly used modelfor heavy nuclei as in the Glauber model [18]. We also consider the difference betweenprotons and neutrons [48]. Large nuclei are known to be loosely bound with the bindingenergy per nucleon ∆ E ≈ t = ¯ n · P i m i E ≈ γ fm in the lab frame.Such a time scale is much longer than any other time scales in the problem. Therefore,nucleons in the nuclei are to be treated as free particles with a normalized distribution ˆ ρ i proportional to that of electric charges [49]. Accordingly, the probability to find a nucleonper unit transverse area around r i is given byˆ T i ( r i ) ≡ (cid:90) dz ˆ ρ i ( r i , z ) . (57)And, the thickness beam functions of nucleus i , which is made of Z i protons and N i neutrons,can be replaced by T j/i ( r i , z, x ) → T i ( r i ) (cid:20) Z i Z i + N i B j/p ( z, x ) + N i Z i + N i B j/n ( z, x ) (cid:21) , (58)where the nuclear thickness function is defined as [18] T i ( r i ) ≡ ( Z i + N i ) ˆ T i ( r i ) , (59) B j/p and B j/n are the beam functions for protons and neutrons, respectively. That is, in theGlauber model, the thickness beam functions are products of the thickness functions andthe beam functions.Inserting Eq. (58) into Eq. (44) gives the cross section for hard processes in the Glaubermodel dσ AB d b dy C d p C = (cid:90) d X T A ( X ) T B ( X − b ) dσ nn dy C d p C , (60) In this paper, we are not concerned about how to determine the impact parameter of a collision experi-mentally but only about what is speakable and unspeakable about the impact parameter in the quantumpicture. σ nn given by the factorization formula forcolliding two nucleon beams with neutron-to-proton ratios respectively equal to N i /Z i . Thatis, the nuclear modification factor R AA ≡ dσ AB d b dy C d p C T AB ( b ) dσ nn dy C d p C = 1 , (61)with T AB ( b ) ≡ (cid:90) d X T A ( X ) T B ( X − b ) . (62) D. Impact-parameter dependent pp collisions
Collective phenomena among produced soft particles have been studied in pp colli-sions [24–29], which are presumptively related to collision geometry according to some mod-els [31]. Inclusive hard processes in impact-parameter dependent pp collisions are worthbeing explored experimentally as well. Based on the factorization formula in Eq. (44), suchhard processes can be potentially used to measure transverse phase-space parton distribu-tions inside protons.Caution is, however, needed when one studies impact-parameter dependent pp collisions.For hard processes, one has | r i | ∼ | b | ∼ R i ∼ / Λ QCD (cid:29) | x | , /Q. (63)Therefore, like heavy-ion collisions, T j/i ( r , z, x ) can be viewed as the distribution of parton j with a transverse size ∼ | x | , i.e., the beam function B j/i ( z, x ), located at r inside proton i . On the other hand, in contrast to heavy-ion collisions, the two conditions in Eq. (15) arenot always fulfilled in pp collisions. As discussed in Sec. II, in order to measure universalquantities across experiments, one needs to maintain∆ p T ≥ x T (cid:29) Λ QCD ≈
100 MeV . (64)∆ p T , however, can not be too large on modern colliders. For example, the crossing angle atthe interaction point θ C ∼ µ rad at the LHC [32], which is one of the crucial parameters In experiments, one measures R AA with both its numerator and denominator averaged over a rangeof impact parameter ∆ b corresponding to some centrality bin. Since ∆ b (cid:29) / | p C | , one can identify theaverage impact-parameter dependent cross section with the average number of hard processes per collision. θ C can be determined with an accuracy ∆ θ C = 10 µ rad, whichgives us ∆ p T ≤
70 MeV (65)for E = 7 TeV proton beams. If the measurements like those in Refs. [24–29] are subjectto a similar constraint (with lower beam energies), the required theoretical calculationsalways involve the wave packets of colliding protons, as shown in Eqs. (19) and (37), which,however, are not measured . IV. THE IMPACT-PARAMETER DEPENDENT DRELL-YAN PROCESS IN q ¯ q COLLISIONS
The modes associated with q T ∼ /b are non-perturbative in hadron and nuclear colli-sions. In order to make a model-independent verification of the factorization formula in Eq.(44), in this section we study impact-parameter dependent q ¯ q collisions with b (cid:46) / Λ QCD .Since the factorization formula is valid at all orders in α s and at leading order in λ , its va-lidity in such collisions can be verified order by order in perturbation theory by comparingto the results of the general formula in Eq. (27). For this task, the factorization formula isrequired to reproduce the perturbative QCD results, expanded to leading order in λ . A. The impact-parameter dependent Drell-Yan cross section
We calculate the impact-parameter dependent cross section for the Drell-Yan process q ¯ q → γ ∗ + anything else (66)with the virtuality of the photon p C = Q at the impact parameter b (cid:46) / Λ QCD . Thequark and antiquark are taken as massless onshell particles. In this case, there are onlytwo scales: Q and 1 /b , which set the expansion parameter λ = 1 / ( bQ ) (cid:28)
1. Like theconventional cross section, the impact-parameter dependent cross section is not infrared safebecause of insufficient average over the initial states. Its singularities are to be regularizedby dimensional regularization with d = 4 − (cid:15) . When Condition ii) in Eq. (15) is violated, the factorization formula in Eq. (44) is still valid but thedefinition of thickness beam functions will depend on the protons’ wave packets. O and then singles out one ofthe final-state particles as γ ∗ to obtain the general formula for the total cross section dσ q ¯ q d b = πs (cid:90) d q (2 π ) e i q · b (cid:90) (cid:89) f (cid:2) d Γ p f (cid:3) δ ( p C − Q ) × M ( p A , p B → p C , { p f } ) M ∗ (¯ p A , ¯ p B → p C , { p f } ) (67)with p C = p A + p B − (cid:80) p f and the incoming momenta given by Eq. (23). For unpolarizedcollisions, it can be written in the following compact form dσ q ¯ q d b = πs (cid:90) d q (2 π ) e i q · b (cid:90) (cid:89) f (cid:2) d Γ p f (cid:3) δ ( p C − Q ) | M | ( p A , p B → p C , { p f } ← ¯ p A , ¯ p B ) (68)with | M | ( p A , p B → p C , { p f } ← ¯ p A , ¯ p B ) being the off-diagonal amplitude squared in the lastline of Eq. (67) averaged over initial-state spins.For the factorization formula, one needs to integrate over y C and p C in Eq. (44). Thiscan be easily done by using Eq. (28) to introduce δ ( p C − Q ) and then integrate out thefour-momentum p µC instead. We only consider the production threshold: Q ∼ s . In thiscase, one can replace δ ( p C − Q ) by δ (¯ n A · p C ¯ n B · p C − Q ) and the p C -integral gives δ (2) ( x ).Then, integrating out x gives dσ q ¯ q d b = πs (cid:88) j,k (cid:90) dz A z A dz B z B (cid:90) d X T j/q ( X , z A ) T k/ ¯ q ( X − b , z B ) (cid:90) (cid:89) f [ d Γ p f ] × H ( z A P A , z B P B → p C , { p f } ) δ (¯ n A · p C ¯ n B · p C − Q ) (69)with ¯ n i · p C = z i ¯ n i · P i − (cid:80) ¯ n i · p f . Here, we have used the fact that the soft function becomesunity at x = 0 and the hard function H is equal to | M | for the partonic process j, k → C + anything else, as given in Eq. (48). We define T j/i ( r , z ) ≡ T j/i ( r , z, ) , (70)which are referred to as transverse spatial PDFs. In heavy-ion collisions, the above equationgives the factorization formula for the inclusive Drell-Yan production of γ ∗ , W ± and Z witha single hard scale Q , valid for all orders in α s . The interested reader is referred to Refs.[37, 48] for fixed-order calculations and phenomenological studies. Below, we only focuson the verification of the factorization formula by a detailed calculation for q ¯ q collisions atleading order in λ .4 B. Factorization at the Born level
At zeroth order in α s , the off-diagonal amplitude squared is given by | M (0) | ( p A , p B → p C ← ¯ p A , ¯ p B ) = p A p B p C p A _ p B _ = − e q Tr (cid:104) v s (¯ p B )¯ v s ( p B ) γ µ u s (cid:48) ( p A )¯ u s (cid:48) (¯ p A ) γ µ (cid:105) (71)with e q the electric charge of the quark and antiquark. Expand it around λ = 0, and atleading order one has | M (0) | ( p A , p B → p C ← ¯ p A , ¯ p B ) = e q (1 − (cid:15) ) s, (72)where s = ¯ n A · P A ¯ n B · P B and we have used the following identities (cid:88) s u s ( p )¯ u s ( p (cid:48) ) = /p + m (cid:112) m + E ) (cid:0) γ (cid:1) /p (cid:48) + m (cid:112) m + E (cid:48) ) , (cid:88) s v s ( p )¯ v s ( p (cid:48) ) = − /p + m (cid:112) m + E ) (cid:0) − γ (cid:1) − /p (cid:48) + m (cid:112) m + E (cid:48) ) . (73)Inserting Eq. (72) into Eq. (68) and expanding the δ function in Q as well gives dσ (0) d b = π e q δ ( s − Q ) δ (2) ( b ) . (75)That is, the hard scattering is initiated by the quark and antiquark only when they passvery close to one another at a distance ∼ /Q , which defines ”point-like” in our calculation.Now, let us calculate the factorized Born cross section using Eq. (69). At zeroth order in α s , the transverse spatial quark/antiquark distributions, according to Eqs. (53) and (54),take the form T (0) q/q ( r , z ) = T (0)¯ q/ ¯ q ( r , z ) = δ (1 − z ) δ (2) ( r ) , (76)and the hard function is given by H (0) ( z A P A , z B P B → p C ) = | M (0) | ( z A P A , z B P B → p C ) = e q (1 − (cid:15) ) z A z B s. (77)Plugging them into Eq. (69) gives the same result as Eq. (75), hence confirming the validityof factorization at the Born level. As an exercise to illustrate the validity of such an expansion, one can work out Q (cid:90) d q (2 π ) e i q · b δ ( | q | − Q x ) = Q π J ( Qb √ x ) → δ (2) ( b ) δ ( x ) as Q → ∞ , (74)by using test functions in both b and x . C. Factorization at one loop At O ( α s ), the off-diagonal amplitude squared for the general formula in Eq. (68) containsboth virtual and real diagrams. Instead of evaluating them exactly, we constrict ourselvesto showing that expanding them to leading order in λ yields the one-loop factorized resultgiven by Eq. (69).The virtual diagrams include the quark/antiquark self-energies and the one-loop quark-photon vertex function. In dimensional regularization, the quark/antiquark self-energiesvanish due to the cancellation between ultraviolet (UV) and infrared (IR) poles for masslessonshell particles. Therefore, one only needs to include the one-loop quark-photon vertexfunction, which contains both UV and IR divergences. Since its counterterm cancels withthat for quark/antiquark self-energies, the UV pole is, effectively, converted into an IR poleand the singularities in the virtual diagrams are of IR origin. Using the Laudau equation [50],one can see that there are three potentially IR divergent regions in the phase space of thevirtual gluon: 1) n A -collinear region; 2) n B -collinear region; and 3) the soft region. Theseregions produce a double IR pole in (cid:15) . The one-loop quark-photon vertex function is wellknown, which only depends on 2 p A · p B (see, e.g., Ref. [51]). Its explicit form is not relevantfor our discussion here, and we write its contribution to the total cross section in the followingcompact form dσ (1) r d b = πs (cid:90) d q (2 π ) e i q · b | M (1) v | ( p A , p B → p C ← ¯ p A , ¯ p B ) × δ (( p A + p B ) − Q ) (78)with | M (1) v | the spin-averaged amplitude squared which contains virtual-gluon contributions.The real amplitude includes two diagrams: iM (1) r ( p A , p B → p C , k ) ≡ p A p B k p C p A p B kp C + . (79)Accordingly, the real correction in perturbative QCD takes the form dσ (1) r d b = πs (cid:90) d q (2 π ) e i q · b (cid:90) d Γ k | M (1) r | ( p A , p B → p C , k ← ¯ p A , ¯ p B ) × δ (( p A + p B − k ) − Q ) . (80)6It also contains collinear and soft divergences.We use the method of regions (see Ref. [45] for an introduction) to expand the real (Eq.(78)) and virtual (Eq. (80)) integrals in each relevant region of the gluon momentum k µ inorder to verify the factorization formula at one loop. The relevant regions include:1. The hard region: k µ ∼ Q In this region, upon expanding the off-diagonal amplitude squared, at leading order in λ it equals the conventional amplitude squared with p i and ¯ p i both replaced by their designvalues P i . As a result, the expansion of the virtual and real integrals in this region gives dσ (1) h d b = δ (2) ( b ) σ (1) ( P A , P B → p C ) (81)with σ (1) the conventional one-loop cross section for q ¯ q → γ ∗ , which can be found in, e.g.,Ref. [51]. The double poles in virtual and real contributions cancel out but the collineardivergences do not cancel, which produces the 1 /(cid:15) pole in the final result of σ (1) . If oneinserts into the factorization formula the zeroth-order transverse spatial PDFs in Eq. (76)and the spin-averaged virtual and real amplitude squared with incoming momenta equal to P i as the one-loop hard function, one evidently reproduces the same result as Eq. (81).2. Two collinear regions: k µ ∼ Q ( λ , , λ ) n i ¯ n i Expanded in these regions, the virtual diagrams become scaleless integrals and, hence, van-ish. As a result, one only needs to consider the real diagrams as shown in Eq. (79).Let us take for example the n A -collinear region in which p µA ∼ ¯ p µA ∼ k µ ∼ ( λ , , λ ) n A ¯ n A , p µB ∼ ¯ p µB ∼ (1 , λ , λ ) n A ¯ n A . (82)With the above scaling in mind, expanding the δ function in Eq. (80) gives δ ( z A s − Q ) (83)with (1 − z A ) ≡ ¯ n A · k/ ¯ n A · P A . Then, expand the real amplitude M (1) r in λ as well. Aftersome algebra, we have | M r,n A | ( p A , p B → p C , k ← ¯ p A , ¯ p B ) = e q (1 − (cid:15) ) s g s C F N qq ( z ) [4 | k | − (1 − z ) | q | ] (cid:12)(cid:12) k + (1 − z ) q (cid:12)(cid:12) (cid:12)(cid:12) k − (1 − z ) q (cid:12)(cid:12) (84)with N qq ( z ) = (cid:0) z (cid:1) − (cid:15) (1 − z ) . (85)7In this way, one can get the leading-order contribution from the n A -collinear region. In orderto verify the factorization formula, we need to show that dσ (1) r,n A d b = πs (cid:90) d q (2 π ) e i q · b (cid:90) d Γ k | M r,n A | ( p A , p B → p C , k ← ¯ p A , ¯ p B ) δ ( z A s − Q )= πs (cid:90) dz A z A T (1) q/q ( b , z A ) H (0) ( z A P A , P B → p C ) δ ( z A s − Q ) (86)with the zeroth-order hard function H (0) given in Eq. (77) and T (1) q/q the one-loop transversespatial quark distribution function to be calculated below. The physical meaning of thisequation is quite obvious: the quark, recoiling against a radiated gluon, approaches to theantiquark located at b and then they annihilate into a virtual photon with virtuality Q .Let us confirm Eq. (86) using the factorization formula in Eq. (69). Here, we need tocalculate the one-loop transverse spatial quark distribution function in the incoming quark.According to its definition in Eq. (53), one has, in ¯ n · A = 0 lightcone gauge, T (1) q/q ( r , z ) = (cid:90) d q (2 π ) e i q · r (cid:90) d − (cid:15) k (2 π ) − (cid:15) M π (1 − z )¯ n · P (87)with M = p pk _ = g s C F
12 Tr (cid:104) u sp ¯ u s ¯ p /(cid:15) ∗ λ ( k ) (cid:0) / ¯ p − /k (cid:1) / ¯ n (cid:0) /p − /k (cid:1) /(cid:15) λ ( k ) (cid:105) ( p − k ) (¯ p − k ) . (88)Here, the gluon polarization sum in the lightcone gauge is given by (cid:88) λ (cid:15) µλ ( k ) (cid:15) ∗ νλ ( k ) = − g µν + ¯ n µ k ν + ¯ n ν k µ ¯ n · k . (89)The incoming quark momenta in the amplitude ( p ≡ P + q T /
2) and the conjugate amplitude(¯ p ≡ P − q T /
2) are both onshell: P µ ± q µT ≡ − q T n · P ¯ n µ n · P n µ ± q µT . (90)The gluon momentum k µ can be decomposed as k µ = − k T (1 − z )¯ n · P ¯ n µ − z )¯ n · P n µ k µT . (91)8Accordingly, the denominators in Eq. (88) are given by (cid:16) P ± q T − k (cid:17) = (cid:0) k T ∓ (1 − z ) q T (cid:1) − z ∼ λ . (92)One only needs to keep terms of O ( λ − ) in M . After expanding it according to thescaling in Eq. (82), one can easily obtain M = g s C F ¯ n · P N qq ( z ) [4 | k | − (1 − z ) | q | ]2 (cid:12)(cid:12) k + (1 − z ) q (cid:12)(cid:12) (cid:12)(cid:12) k − (1 − z ) q (cid:12)(cid:12) (93)with N qq ( z ) given in Eq. (85). The equality of Eq. (86) is confirmed by plugging M in theabove equation (via Eq. (87)) and the zeroth-order hard function in Eq. (77) into the lastline of Eq. (86).The one-loop transverse spatial quark distribution in a fast-moving quark can be calcu-lated analytically. One combines the denominators on the right-hand side of Eq. (93) byintroducing a Feynman parameter x : x (cid:12)(cid:12)(cid:12) k + (1 − z ) q (cid:12)(cid:12)(cid:12) + (1 − x ) (cid:12)(cid:12)(cid:12) k − (1 − z ) q (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˜k (cid:12)(cid:12)(cid:12) + ∆ (94)with ˜k = k −
12 (1 − x )(1 − z ) q , ∆ = x (1 − x )(1 − z ) | q | . (95)Then, by changing variables to ˜k , one can easily integrate out k in Eq. (87) and obtain T (1) q/q ( r , z ) = α s C F π (cid:90) dx (cid:90) d q (2 π ) e i q · r N qq ( z )(1 − (cid:15) )Γ( (cid:15) ) (cid:16) e γE µ x (1 − x ) | q | (cid:17) (cid:15) (1 − z ) (cid:15) = α s C F π Γ( (cid:15) )Γ (1 − (cid:15) )Γ(1 − (cid:15) ) N qq ( z )(1 − z ) (cid:15) (cid:90) d q (2 π ) e i q · r (cid:18) e γ E µ | q | (cid:19) (cid:15) = α s C F π r cos( π(cid:15) )Γ (1 − (cid:15) )Γ (cid:18) (cid:15) + 12 (cid:19) N qq ( z )(1 − z ) (cid:15) (cid:0) e γ E µ r (cid:1) (cid:15) , (96)where we have used the integral in Eq. (A6) and replaced the bared coupling g s by g s π = α s (cid:18) µ e γ E π (cid:19) (cid:15) (97)with µ the MS renormalization scale and γ E the Euler constant. At the end, by using1(1 − z ) (cid:15) = − (cid:15) δ (1 − z ) + 1(1 − z ) + , (98)9we have T (1) q/q ( r , z ) = α s C F π r (cid:20) − (cid:18) (cid:15) + L T (cid:19) δ (1 − z ) + 1 + z (1 − z ) + (cid:21) (99)with L T ≡ (cid:16) rµ e − γ E (cid:17) . (100)The singularity in (cid:15) arises from the fact that the virtual-gluon radiation can only contributeto the prefactor in front of δ ( r ) and there is no real-virtual cancellation at a nonzero trans-verse distance r .Similarly, one can also identify the contribution from the transverse spatial antiquarkdistribution function of the incoming ¯ q by expanding the integrand of Eq. (80) in the n B -collinear region. One hence confirms the contributions from one-loop transverse spatialPDFs in the factorization formula.3. The soft region: k µ ∼ λQ As a consistency check, we show that the contribution from the soft region vanishes. Weexpand both the diagram for the one-loop spatial quark distribution function in Eq. (88)and the virtual and real integrals for the general formula given respectively by Eqs. (78) and(80) in this region. The former corresponds to the zero-bin contribution to the spatial PDFswhile the latter corresponds to the one-loop correction to the soft function. In both cases,one ends up with scaleless integrals, which vanish in dimensional regularization. Therefore,the soft region is indeed irrelevant.In summary, we have verified the validity of the factorization formula at one loop: thecorrection from the one-loop hard function is given by the expansion of the virtual and realintegrals for the general formula in the hard region, as given in Eq. (81); the correction fromone-loop transverse spatial PDFs is equivalent to expanding these virtual and real integralsrespectively in the two collinear regions, as given in Eq. (86) for the quark collinear region;and the correction from the soft function vanishes.
ACKNOWLEDGEMENTS
The author would like to thank F. Gelis, Y. V. Kovchegov, J. G. Milhano and U. A.Wiedemann for reading through the manuscript and informative comments. The author is0 CC C ReIm x FIG. 2. Integration contour for I x . The two branch cuts extend respectively from i + to i ∞ andfrom i − to − i ∞ . also grateful to Y.-T. Chien, R. Rahn, S. S. van Velzen, D. Y. Shao and W. J. Waalewijn forilluminating discussions on aspects of SCET. This work has received support from Xuntade Galicia (Centro singular de investigaci´on de Galicia accreditation 2019-2022 and TalentAttraction Program), from the European Union ERDF and from the Spanish Research StateAgency by Mara de Maeztu Units of Excellence program MDM-2016-0692. The author isalso indebted to CERN TH department for continued support when this work was finalized. Appendix A: An integral
In this appendix, we evaluate the integral I = (cid:90) d q (2 π ) e i q · r ( | q | ) (cid:15) . (A1)One can choose r to align with the positive x -axis and split the integral into two pieces: I = 1(2 π ) r − (cid:15) I x I y . (A2)Here, I x ≡ (cid:90) dx ( x ) − (cid:15) e ix with x = q x r = (cid:90) C dx [( x − i + )( x − i − )] − (cid:15) e ix (A3)1with the contour C running from −∞ to + ∞ . This integral is well-defined only for therange < Re (cid:15) <
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