Threshold resummation for pair production of coloured heavy (s)particles at hadron colliders
aa r X i v : . [ h e p - ph ] O c t TTK-10-45IPPP/10/61DCPT/10/122FR-PHENO-2010-026SFB/CPP-10-65arXiv:1007.5414 [hep-ph]September 28, 2010
Threshold resummation for pair production ofcoloured heavy (s)particles at hadron colliders
M. Beneke a , P. Falgari b , C. Schwinn ca Institute f¨ur Theoretische Teilchenphysik und Kosmologie,D–52056 Aachen, Germany b IPPP, Department of Physics, University of Durham,Durham DH1 3LE, England c Albert-Ludwigs Universit¨at Freiburg, Physikalisches Institut,D-79104 Freiburg, Germany
Abstract
We derive a factorization formula for the production of pairs of heavy colouredparticles in hadronic collisions near the production threshold that establishes fac-torization of soft and Coulomb effects. This forms the basis for a combined re-summation of Coulomb and soft corrections, including the non-trivial interferenceof the two effects. We develop a resummation formalism valid at NNLL accuracyusing the momentum-space approach to soft gluon resummation. We present nu-merical results for the NLL resummed squark-antisquark production cross sectionat the LHC and Tevatron, including also the contribution of squark-antisquarkbound states below threshold. The total correction on top of the next-to-leadingorder approximation is found to be sizeable, and amounts to (4–20)% in the squarkmass region 200 GeV – 3 TeV at the 14 TeV LHC. The scale dependence of thetotal cross section is also reduced.
Introduction
In perturbative calculations of partonic cross sections at hadron colliders there often ariseterms that are kinematically enhanced in certain regions in phase space. In the Drell-Yanprocess, for instance, Sudakov logarithms of the form [ α s ln (1 − z )] n arise, where z = Q / ˆ s ,and Q and ˆ s represent the invariant mass squared of the lepton pair and the partoniccentre-of-mass (cms) energy, respectively. In the threshold region z →
1, these terms arelarge, and spoil the convergence of the perturbative expansion in the QCD coupling α s .If the dominant contribution to the hadronic cross sections originates from the partonicthreshold region, which is certainly the case when Q approaches the cms energy s , theselogarithms need to be resummed to all orders in perturbation theory to attain a reliabletheoretical description. This was accomplished in [1, 2] for the inclusive Drell-Yan crosssection by solving evolution equations in Mellin space. In production processes of pairsof heavy coloured particles such as top quarks or coloured particles in extensions of thestandard model, e.g. squarks and gluinos in supersymmetric extensions, the partonic crosssection contains terms of the form ( α s log β ) n (“threshold logarithms”), where β = (1 − M / ˆ s ) / is the heavy particle velocity, and ( α s /β ) n (“Coulomb singularities”), which arealso enhanced near the partonic threshold ˆ s ≈ M . Resummation of threshold logarithmsfor heavy-particle and di-jet production has also been implemented in Mellin space [3–9]up to now, and has been used for improved predictions of the top-pair production crosssection at hadron colliders [10–16]. Recently the resummation of threshold logarithmsfor production processes of supersymmetric coloured particles [17–20] and colour octetscalars [21,22] has been studied as well. The resummation of multiple exchanges of Coulombgluons and bound-state effects has been studied for the total top-antitop cross section [10,23] and the invariant mass distribution of top quarks and gluinos [24–26].The theoretical basis for resummation is a factorization of the partonic cross section ˆ σ in the partonic threshold region into hard and soft contributions of the schematic formˆ σ = H ⊗ S (1.1)with a hard function H and a soft function S both of which are matrices in colour space.If the invariant mass Q = ( p + p ) of the heavy-particle pair H ( p ) H ′ ( p ) is held fixed,the “partonic threshold” is then defined more generally as the kinematical region where Q is close to the partonic centre-of-mass energy ˆ s . Arguments from perturbative QCD [27],the properties of Wilson lines [28, 29], or effective theories [30–33] can then be used todemonstrate exponentiation of the enhanced contributions by solving evolution equationsfor the functions S and H . Traditionally this resummation is performed in Mellin spacewhich requires a numerical transformation of the resummed result back to momentumspace. A resummation method directly in momentum space was proposed in [31] and hasbeen applied subsequently e.g. to Drell-Yan or Higgs production [33–35] and recently tothe top-quark invariant mass distribution [36].In this paper we consider the production of a pair of heavy coloured particles H, H ′ with masses m H and m H ′ , respectively, in the collision of hadrons N and N , N ( P ) N ( P ) → H ( p ) H ′ ( p ) + X (1.2)1nd concentrate on the resummation of the total cross section. In this case, after integrationover the invariant mass of HH ′ , the enhanced logarithms appear whenˆ s ≈ ( m H + m H ′ ) . (1.3)In this limit, the factorization of soft gluons is complicated by the fact that the non-relativistic energy of the heavy particles is of the same order as the momenta of the softgluons, in contrast to the assumptions made in the derivation of the usual factorizationformula (1.1). We show that in this kinematical regime the partonic cross sections factorizesinto three contributions, ˆ σ = H ⊗ W ⊗ J , (1.4)where H is determined by hard fluctuations, W by soft fluctuations and J accounts for thepropagation of the heavy-particle pair including Coulomb-gluon exchange. The short-hand W ⊗ J includes a convolution which accounts for the energy loss of the heavy particles dueto soft emissions (the precise form of the factorization formula is given in (2.63) below).The main points of the present approach are as follows: • Our approach is largely model-independent and highlights the universal features ofsoft-gluon and Coulomb resummation: the precise nature of the heavy particles andthe physics model enter the factorization formula (1.4) only in the hard function H ; the soft function and the Coulomb function depend only on the colour and (inthe latter case) the spin quantum numbers of the heavy-particle pair. The hardfunction can be obtained directly from the HH ′ production amplitude, expandednear threshold, without any need to perform a full cross section calculation. • Eq. (1.4) generalizes previous results of the form (1.1) by factorizing the Coulombeffects in addition to those from soft gluons. Multiple exchange of Coulomb gluonsassociated to corrections ∼ α ns /β m can be resummed in the function J using methodsfamiliar from non-relativistic QCD (NRQCD). The presence of the Coulomb function J leads to a more complicated colour structure of the soft functions W comparedto previous treatments based on a factorization of the form (1.1). The factorizationformula (1.4) allows a combined resummation of soft and Coulomb gluons and justifiesearlier treatments [11, 18, 24, 25] where the factorization of Coulomb from soft gluonswas put in as an assumption. • The kinematical structure of the soft function W simplifies in the threshold re-gion (1.3) compared to the more general situation considered in [7]. This has al-lowed us to construct a basis in colour space that diagonalizes the soft functionsrelevant for hadron collider pair production processes of heavy particles in arbitraryrepresentations of the colour gauge group to all orders of perturbation theory [37].The diagonal colour bases for resummation of threshold logarithms for all produc-tion processes of pairs of coloured supersymmetric particles at hadron colliders, i.e.squark-antisquark, squark-squark, squark-gluino and gluino-gluino production have2een provided in [37], extending explicit one-loop results for soft anomalous dimen-sions at threshold [7, 17, 18, 20]. This result greatly simplifies threshold resumma-tions at NNLL accuracy and has allowed us to extract the two-loop soft anomalousdimension for arbitrary colour representations [37] from results of [38–40], in agree-ment with an independent two-loop study for top-pair production [41]. The morecomplicated colour structures in the two-loop soft anomalous dimension for massiveparticles with generic kinematics [42–44] have been shown not to contribute to theNNLO total production cross section of a heavy-particle pair produced in an S -waveat threshold [45]. • We use the method of [31, 32] to perform the resummation of threshold logarithmslog β directly in momentum space solving evolution equations for the functions H and W . The resummation of the total cross section in the threshold region has beenperformed previously in Mellin space [9, 11]. Our derivation gives a field theoreti-cal definition of the quantities appearing in that approach and the resummation inmomentum space allows an analytical treatment, since no numerical inverse Mellintransform is necessary. The relation between the Mellin-space and momentum-spaceformalisms has been discussed in [31–33, 37]. • The formalism includes the case of heavy particles with sizeable decay widths. Tofirst approximation finite-width effects enter only in the Coulomb function and canbe included by a shift of the non-relativistic energy E → E + i Γ as familiar fromtop-quark pair production at electron-positron colliders. A systematic descriptionof finite width effects beyond leading order can be achieved in the effective theoryapproach [46–49].In this paper we focus on establishing the factorization of soft and Coulomb gluons anddemonstrate the new features arising in a combined resummation by presenting numericalresults for the example of squark-antisquark production. While we set up a method suitablefor resummation at NNLL accuracy, in our initial application we restrict ourselves toNLL accuracy since the required colour separated hard production coefficients for squark-antisquark production are currently unknown. We also do not include finite squark decaywidths which are small in typical supersymmetry (SUSY) scenarios and would introduce adependence on the SUSY parameters but, as mentioned above, could easily be included inour approach. Note that, since in the present work we are concerned with the total crosssection, there are no sizable finite-width corrections as long as Γ ≪ M , unlike the case ofthe invariant mass distribution near threshold.The paper is organized as follows. In section 2 we derive the factorization formula (1.4)using effective field theory methods. We compare to previous treatments of combined softand Coulomb corrections and extend the diagrammatic argument of [45] on the absence ofcorrections from subleading couplings of soft gluons to non-relativistic particles at NNLLto all orders in the strong coupling using the effective theory approach. In section 3 wediscuss the process-independent ingredients in the factorization formula, the soft function W and the Coulomb function J , collect their explicit results and perform the summation of3hreshold logarithms using evolution equations in momentum space. We also give a simpleprescription for how to obtain the hard-function H , the only process-dependent ingredientin the resummation formalism, from a standard fixed-order calculation. In section 4 weperform the combined soft-Coulomb resummation at NLL accuracy for squark-antisquarkproduction and present numerical results for the Tevatron and the LHC. A number ofadditional results concerning parton densities, the Coulomb potential in various colourrepresentations and the resummed and expanded cross section are collected in the appendix.
In this section we derive a factorization formula of the form (1.4) for the process (1.2) nearthe partonic threshold. The heavy particles are assumed to transform under representations R and R ′ of the colour gauge group SU (3) C . The inclusive production cross section isdescribed by the factorization formula σ = X p,p ′ Z dx dx f p/N ( x , µ ) f p ′ /N ( x , µ ) ˆ σ pp ′ ( x x s, µ ) , (2.1)where ˆ σ pp ′ are the hard-scattering cross sections for the partonic subprocesses p ( k ) p ′ ( k ) → H ( p ) H ′ ( p ) + X (2.2)and pp ′ ∈ { qq, q ¯ q, ¯ q ¯ q, gg, gq, g ¯ q } . As usual the nucleon masses are neglected and partonmomenta k , are related to the incoming nucleon momenta by k = x P , k = x P . (2.3)The partonic cms energy is given byˆ s ≡ ( k + k ) = x x s. (2.4)At hadron colliders ˆ s is not a fixed quantity but one may argue that the steep rise of theparton distribution functions with decreasing x leads to an enhanced contribution to thetotal production cross section from the partonic threshold region z ≡ M ˆ s ∼ , (2.5) M = ( m H + m H ′ ) / M /s = zx x & .
2. Therefore weexpect the resummation of threshold logarithms to be relevant for particles with masses Preliminary results have appeared in [50, 51]. & . , . ,
10 TeV) and M &
450 GeV at theTevatron which are at the border of the estimated reach for coloured SUSY particles at theLHC [52,53] and the exclusion limits obtained at the Tevatron [54,55]. However, experiencewith the NLO corrections to top-quark production suggests that also for smaller massesthere is a sizable contribution to the total cross section from those terms in the partoniccross section that are enhanced for z → β = r − M ˆ s = √ − z, (2.6)which for equal masses also corresponds to the non-relativistic velocity of the heavy parti-cles in the partonic cms. The aim of this paper is to perform a resummation of thresholdlogarithms log β and Coulomb corrections 1 /β . In order to discuss the systematics of thecombined resummations of the two corrections we count both α s ln β and α s /β as quanti-ties of order one and introduce a parametric representation of the expansion of the crosssection of the form [37]ˆ σ pp ′ ∝ ˆ σ (0) pp ′ X k =0 (cid:18) α s β (cid:19) k exp h ln β g ( α s ln β ) | {z } (LL) + g ( α s ln β ) | {z } (NLL) + α s g ( α s ln β ) | {z } (NNLL) + . . . i × (cid:8) α s , β (NNLL); α s , α s β, β (NNNLL); . . . (cid:9) . (2.7)With this counting, the resummed cross section at LL accuracy includes all terms of or-der 1 /β k × α n + ks ln n β relative to the Born cross section near threshold. Next-to-leadingsummation includes in addition all terms of order α s ln β ; α s (cid:8) /β × ln β, ln β (cid:9) ; . . . , whilefurthermore all terms α s ; α s (cid:8) /β, ln , β (cid:9) ; . . . are included in NNLL approximation. In the threshold region, the cross section of the process (2.2) receives contributionsfrom hard, potential, soft and two collinear momentum regions defined byhard ( h ) : k ∼ M potential ( p ) : k ∼ M λ, ~k ∼ M √ λ soft ( s ) : k ∼ ~k ∼ M λn -collinear ( c ) : k − ∼ M, k + ∼ M λ, k ⊥ ∼ M √ λ ¯ n -collinear (¯ c ) : k + ∼ M, k − ∼ M λ, k ⊥ ∼ M √ λ , (2.8) The NNLL terms odd in β at order α s and α s , α s β ln , β ; α s β ln , β , vanish due to rotationalinvariance for the total cross section when the heavy-particle pair is dominantly produced in an S -wavestate at tree level [37, 45]. See also below in section 2.5. λ ∼ − z = β ≪
1. For the collinear momentum we employ thelight-cone decomposition p µ = p − n µ + p + n µ + p µ ⊥ (2.9)with two light-like vectors n and ¯ n satisfying n · ¯ n = 2. Due to the threshold kinematics( k + k ) ∼ M , no collinear modes can appear in the final state so that the particlesdenoted by X in (1.2) are entirely given by soft modes.In order to achieve the separation of the effects corresponding to the different re-gions (2.8) it is useful to describe the heavy-pair production process in the frameworkof effective field theories. The relevant effective Lagrangian contains elements of non-relativistic QCD (NRQCD), describing the interaction of potential and soft modes, andsoft-collinear effective theory (SCET) [57–60] describing the interaction of soft and collinearmodes. Hard modes with virtuality of order M are integrated out and are not part ofthe effective Lagrangian. This effective theory is similar to that used in [48, 49, 61] todescribe W -pair production near threshold in e + e − collisions and the construction of theeffective Lagrangian proceeds along similar lines. As in the discussion of the Drell-Yanprocess in [33] the following considerations will be adequate to establish factorization ofthe partonic cross section; the discussion of the hadronic cross section requires to includeadditional modes with a scaling given by powers of Λ QCD /M [32]. We also note the exis-tence of an additional region with scaling k ∼ M √ λ , usually referred to as “soft” in theNRQCD literature. It is related to potential forces between the heavy particles, and willnot be relevant for the following discussion of the leading Coulomb corrections.In the EFT framework, the factorization formula is derived by a series of steps involvingfirst a matching calculation from QCD (or another “full theory” such as the MSSM) tothe EFT followed by field redefinitions that decouple soft gluons from collinear [58, 62] andpotential [63] fields. These will be described in the remainder of this section. We alsoshow that, up to NNLL accuracy, corrections from subleading interactions to the effectiveLagrangians not included in the derivation of the factorization formula either vanish or canbe straightforwardly incorporated.
The first step in the derivation of the factorization formula consists of representing thescattering amplitude for the process (1.2) in terms of effective theory matrix elements ofproduction operators O ( ℓ ) { a ; α } corresponding to the partonic sub-processes p ( k ) p ′ ( k ) → H ( p ) H ′ ( p ) X , multiplied by coefficient functions that contain the hard part of the ampli-tude. After this step the partonic scattering amplitude is given by A ( pp ′ → HH ′ X ) = X ℓ C ( ℓ ) { a ; α } ( µ ) h HH ′ X |O ( ℓ ) { a ; α } ( µ ) | pp ′ i EFT . (2.10) The corresponding discussion of factorization of electromagnetic effects and resummation for the W -pair production cross section at e + e − colliders has been given in [64]. → p ( k ) p ′ ( k ) → H ( p ) H ′ ( p ). Near the partonic threshold, kinematics forbids additional massive or en-ergetic particles in the production operator, while additional soft fields connected directlyto the hard process imply additional highly off-shell propagators and therefore lead to op-erators power-suppressed in λ ∼ − z relative to the leading four-particle operators. Therequired four-particle operators take the form O (0) { a ; α } ( µ ) = h φ c ; a α φ ¯ c ; a α ψ † a α ψ ′ † a α i ( µ ) . (2.11)The form of the production operators is thus similar to those describing the productionof non-relativistic W -pairs in e + e − collisions discussed in [48, 49]. Several remarks on thenotation are in order:(1) The fields ψ † and ψ ′ † are non-relativistic fields which create the heavy particles H and H ′ with momenta p µ = m H w µ + ˜ p µ , p µ = m H ′ w µ + ˜ p µ (2.12)with the cms velocity w µ ( w = 1) and small residual momenta ˜ p / in the potential region(2.8). In the partonic cms frame w = (1 , ~ p = − ˜ p ≡ ˜ p for the spatial componentsof ˜ p , . The fields φ c ( φ ¯ c ) are collinear (anticollinear) fields that destroy the initial statepartons p and p ′ with momenta k µ ≈ M n µ , k µ ≈ M ¯ n µ , (2.13)respectively. We provide more details on the definition of these fields and their effectiveLagrangians in the following subsection.(2) Greek indices α i denote the spin (or Lorentz group representation) index of thefield. Where convenient we use the multi-index notation { α } = α α α α . Repeatedmulti-indices are summed by summing over all four components. A similar conventionis used for colour indices, which we denote by Latin letters { a } = a . . . a . We employcreation fields for the heavy final state particles and destruction fields for the initial state.If, for example, H ′ is the antiparticle of H , then this convention implies that ψ ′ transformsin the complex conjugate SU(3) representation of ψ . Similarly, an initial-state antiquarktransforms in the antifundamental representation. In effective-theory calculations it isconventional to decompose operators into a complete basis in spin and colour, makingoperators and coefficient functions “scalars”. Here we prefer to work with operators andcoefficient functions that carry open spin and colour indices, see (2.11). This turns out to We do not have to consider operators where the φ c destroy particles different from p and p ′ . Thesewould arise from a splitting of a collinear gluon into a collinear and a soft quark or of a collinear quarkinto a soft quark and a collinear gluon, but these interactions are present only in the power-suppressedpart of the SCET Lagrangian [59].
7e convenient for the spin indices, since the soft gluon contributions are spin-independent,as well as the Coulomb contributions needed at NLL accuracy. Thus, it is more direct toperform the spin summations on the square of the amplitude as in standard unpolarizedcross section calculations instead of decomposing the amplitude. At NNLL accuracy thespin-dependence of the potential corrections has to be taken into account in order toobtain all terms of the order α s ln β . For this purpose a spin decomposition of the potentialfunction is introduced in section 2.4. As concerns colour, given the representations r, r ′ and R, R ′ of the initial and final state particles, we can introduce a set of independent colourmatrices c ( i ) that form an orthonormal basis of invariant tensors in the representation r ⊗ r ′ ⊗ ¯ R ⊗ ¯ R ′ . The colour structure of the Wilson coefficients can be decomposed in thisbasis according to C ( ℓ ) { a ; α } = X i c ( i ) { a } C ( ℓ,i ) { α } (2.14)This allows us to write C ( ℓ ) { a ; α } ( µ ) O ( ℓ ) { a ; α } ( µ ) = X i C ( ℓ,i ) { α } ( µ ) h c ( i ) { a } O ( ℓ ) { a ; α } ( µ ) i ≡ X i C ( ℓ,i ) { α } ( µ ) O ( ℓ,i ) { α } ( µ ) (2.15)in (2.10). For example, in a 3 + ¯3 → q ¯ q → t ¯ t andsquark-antisquark production), a convenient colour basis is [7] c (1) { a } = 1 N c δ a a δ a a c (2) { a } = 2 √ D A T ba a T ba a , (2.16)with D A = N c −
1, expressing colour conservation and the fact that the initial and finalstates can be in a colour-singlet or a colour-octet state. Suitable colour bases for thedifferent colour scattering processes have been constructed in [37] where also more detailson the properties of the basis tensors can be found.(3) In (2.10) the argument µ denotes the factorization scale dependence of the operatorsand hard functions. We drop this in the following, but note that exploiting the invarianceof the physical amplitude under changes of the factorization scale will be a key elementin deriving the resummation formula. The superscript ℓ stands for higher order terms inthe expansion of the amplitude, which correspond to operators of the form (2.11) withderivatives acting on the fields, as discussed below.(4) Fields without space-time arguments are at x = 0. In general, the SCET repre-sentation of a scattering amplitude involves a convolution of the Wilson coefficient withcollinear fields evaluated at different positions along the light ray [59], instead of a prod-uct with a local bilinear operator φ c ; a α φ ¯ c ; a α as in (2.10). However, since here thereis only one collinear field of a given type in the operator, translation invariance may beused to rewrite the operator in a local form [59, 65]. Applying the translation operatorson the initial state turns the convolution integrals with the coefficient function into theFourier-transform of the position-space coefficient functions to momentum-space coefficient8unctions. These then depend on the large momentum components of the partons, andare the natural objects to work with. The hard functions C ( ℓ ) { a ; α } in (2.10) refer to themomentum-space quantities.Despite the somewhat complicated formalism and notation, there is a straightforwardgeneral prescription for the computation of the matching coefficients C ( ℓ ) { a ; α } in the expres-sion (2.10). One simply calculates the renormalized scattering amplitude for the par-tonic process p ( k ) p ′ ( k ) → H ( p ) H ′ ( p ) without any averages or sums over colour andspin states, and expands it in ˜ p , defined by (2.12) around the production threshold( k + k ) = ( m H + m H ′ ) . Beyond tree level, the matching coefficients can be extracteddirectly by expanding the loop integrand under the assumption that all loop momenta arehard [66] and by performing the loop integration of the expanded integrand in dimensionalregularization. As a special case, the leading term in the kinematical expansion is foundby calculating the full amplitude directly at threshold. In this case the loop corrections tothe effective theory matrix element evaluated directly at threshold vanish since they aregiven by scaleless integrals. Therefore only tree diagrams have to be considered on theeffective theory side of the matching calculation and the hard function is determined from A ( pp ′ → HH ′ ) | ˆ s =4 M = X i C (0) { a ; α } h HH ′ |O (0) { a ; α } | pp ′ i EFT tree . (2.17)Since the matrix element on the right-hand side is simply a product of spinors (polarizationvectors) and the operator renormalization factor, the MS scheme coefficient function C (0) { a ; α } equals the scattering amplitude at threshold with the spinors (polarization vectors) strippedoff, and all 1 /ǫ poles set to zero. The Born-level matching required for squark-antisquarkproduction is discussed explicitly in section 4.The leading term in the kinematical expansion describes the production of HH ′ in an S -wave state. The next term is related to the terms linear in ˜ p , , more precisely theircomponents orthogonal to w µ , denoted by the symbol ⊤ , since the other component is ofhigher order in the potential region. The P -wave production operators corresponding tothis term are of the form (2.11) with the replacement ψ † a α ψ ′ † a α → ψ † a α (cid:18) − i ←→ D µ ⊤ (cid:19) ψ ′ † a α ≡ − i (cid:16) ψ † a α [ −→ D µ ⊤ ψ ′ † ] a α − [ ψ † ←− D µ ⊤ ] a α ψ ′ † a α (cid:17) (2.18)in the non-relativistic part of the operator, with the covariant derivative D µ in the appro-priate colour representation. The series in ℓ in (2.10) accounts for the threshold expansionin powers of λ . In the following, we will mostly consider only the leading term ℓ = 0. Inparticular, the factorization formula we derive is valid only for the case that the Born crosssection is dominated by S -wave production. 9 .2 Effective Lagrangians The propagation and interactions of the non-relativistic fields ψ and ψ ′ are described by thepotential NRQCD Lagrangian [67–70]. That is, we assume that the modes with momentum k ∼ M √ λ and potential gluons have already been integrated out. At leading order, inthe partonic cms frame, the effective Lagrangian, including the decay widths of the heavyparticles, is given by L PNRQCD = ψ † iD s + ~∂ m H + i Γ H ! ψ + ψ ′ † iD s + ~∂ m H ′ + i Γ H ′ ! ψ ′ + Z d ~r h ψ † T ( R ) a ψ i ( ~r ) (cid:16) α s r (cid:17) h ψ ′ † T ( R ′ ) a ψ ′ i (0) . (2.19)Here T ( R ) are the SU (3) generators in the appropriate representation of the heavy par-ticle. In this convention ψ ′ transforms in the antifundamental representation, if H ′ is anantiparticle in the fundamental representation. This deviates from the standard NRQCDnotation for quark-antiquark production that employs a field χ that creates an antiquarkand transforms in the fundamental representation. Our notation is convenient since itaccommodates the case where the heavy particles produced are not particle-antiparticlepairs. This convention leads to a different sign of the potential term compared to theconventional formulation.The only relevant interactions in (2.19) are the exchange of Coulomb (potential) glu-ons, rewritten as an instantaneous, but spatially non-local operator in the second line, andan interaction with the zero component (that is, w · A s , in a general frame) of the softgluon field in the soft covariant derivative iD s ψ ( x ) = ( i∂ + g s T ( R ) a A a ( x )) ψ ( x ), and theanalogous expression for ψ ′ with the generator in the representation R ′ . Using the canoni-cally normalized NRQCD Lagrangian (2.19), one has to take into account a normalizationfactor √ E H ≡ (2 m H (1 + ~ ˜ p /m H )) / = √ m H + O ( λ ) in the definition of the state | H i (similarly for | H ′ i ) on the NRQCD side of the matching relation. The NRQCD Lagrangian (2.19) is invariant under soft gauge transformations in theappropriate representation: ψ ( x ) → U ( R ) s ( x ) ψ ( x ) , ψ ′ ( x ) → U ( R ′ ) s ( x ) ψ ′ ( x ) . (2.20)The soft gauge transformation depends only on x since soft fields have to be multipoleexpanded when multiplied with potential fields in order to maintain a uniform powercounting [70–72]. The invariance of the Coulomb potential for arbitrary representationsfollows from the property U ( R ) † T ( R ) a U ( R ) = U (8) ab T ( R ) b (2.21) For Γ H = 0. See [47] for the general case. U ( R ) is a SU (3) transformation in the representation R and “8” denotes the adjoint.Therefore under a soft gauge transformation: (cid:2) ψ † T ( R ) a ψ (cid:3) ( x + ~r ) ⇒ (cid:2) ψ † T ( R ) b ψ (cid:3) ( x + ~r ) U (8) s,ab ( x ) (2.22)and analogously for the ψ ′ fields. The gauge invariance of the Coulomb potential termfollows then from the fact that the transformations in the adjoint representation are realso that U (8) ab U (8) ac = ( U (8) † U (8) ) bc = δ bc . For this is essential that only the time componententers the soft gauge transformation since the ψ fields and the ψ ′ fields in the Coulombpotential are defined at different points in space but at the same time. The propagation and interactions of quark and gluon collinear modes with large momentumproportional to n µ is described by the SCET Lagrangian [57]. At leading order, in theposition-space formalism of SCET [59, 60] it is given by L c = ¯ ξ c (cid:18) in · D + iD/ ⊥ c i ¯ n · D c iD/ ⊥ c (cid:19) ¯ n/ ξ c −
12 tr (cid:0) F µνc F cµν (cid:1) . (2.23)Here ξ c denotes the n -collinear quark field, which satisfies n/ξ c = 0 and the projectionidentity ( n/ ¯ n// ξ c = ξ c . The covariant derivatives are defined as iD c = i∂ + g s A c with A c the matrix-valued gluon field in the fundamental representation. The covariant derivative D without subscript, however, contains both the collinear and soft gluon field. The quantity F cµν is the field-strength tensor built from the collinear gauge field in the usual way, exceptthat n − · D rather than n − · D c appears [60]. The collinear and soft fields are evaluatedat x , but in products of soft and collinear fields the soft fields are evaluated at x µ + =(¯ n · x/ n µ ≡ x − n µ , according to the multipole expansion. In this notation x − is a scalar,while x µ + is a vector. The ¯ n -collinear fields are described by an identical Lagrangian withthe roles of n and ¯ n interchanged. The corresponding quark field satisfies ¯ n/ξ ¯ c = 0 and(¯ n/n// ξ ¯ c = ξ ¯ c . The two collinear sectors couple only via soft gluon interactions.It is convenient to express the collinear part of the operators (2.11) in terms of fieldsthat are invariant under the collinear gauge transformation as defined in [60]. Let W c bethe collinear Wilson line in the ¯ n direction, W c ( x ) = P exp (cid:20) ig s Z −∞ dt ¯ n · A c ( x + ¯ nt ) (cid:21) . (2.24)Then φ c in (2.11) is given by the combinations W † c ξ c ( x ) , ¯ ξ c W c ( x ) (2.25)for the quark and antiquark initial state, respectively. Note that our conventions implythat the conjugate fields φ † c are ξ † c W c = ¯ ξ c γ W c for the quark and W † c γ ξ c for the antiquark11nitial state. For the gluon initial state we use the following definition that is invariantunder the collinear gauge transformations [73]: A ⊥ µc ( x ) = g − s ( W † c [ iD µ ⊥ c W c ])( x ) = Z −∞ ds ¯ n ν (cid:0) W † c F νµ ⊥ c W c (cid:1) ( x + s ¯ n ) . (2.26)The anticollinear fields are defined similarly with ¯ n replaced by n . These collinear operatorstransform under the soft gauge symmetry of the SCET Lagrangian [60] according to( W † c ξ c )( x ) → U (3) s ( x − )( W † c ξ c )( x ) A ac ( x ) T a → U (3) s ( x − ) A ac ( x ) T a U (3) † s ( x − ) = U (8) ab ( x − ) A bc ( x ) T a , (2.27)where U ( R ) s ( x − ) is a soft gauge transformation in the representation R at the point x − n µ . Since the collinear fields φ c , anticollinear fields φ ¯ c and the potential fields ψ † in the produc-tion operators (2.11) interact with each other only via exchange of soft gluons, an essentialstep in deriving a factorization of the hadronic cross section is the decoupling of soft glu-ons from the collinear and potential degrees of freedom. In SCET it is well known thatthe soft gluons can be decoupled from the collinear fields at leading power by performingfield redefinitions involving Wilson lines [58]. Here we will show that an analogous trans-formation also decouples the soft gluons from the potential fields at leading order in thenon-relativistic expansion. As a result, the production operators factorize into products ofnon-interacting collinear, anticollinear, potential and soft contributions.In order to decouple the soft gluons from the collinear and anticollinear fields, therequired redefinitions are familiar from the derivation of factorization formulas in SCET [58,62] ξ c ( x ) = S (3) n ( x − ) ξ (0) c ( x )¯ ξ c ( x ) = ¯ ξ (0) c ( x ) S (3) † n ( x − ) = ¯ ξ (0) c ( x ) S (¯3) Tn ( x − ) A a,µc ( x ) = S (8) n,ab ( x − ) A (0) b,µc (2.28)with the soft Wilson lines S ( R ) n for a particle in the representation R of SU (3) given by S ( R ) n ( x ) = P exp (cid:20) ig s Z −∞ dt n · A cs ( x + nt ) T ( R ) c (cid:21) (2.29)and similarly for particles collinear to ¯ n . Consistent with our treatment of incoming an-tiparticles as particles in the complex conjugate representations, we have rewritten thetransformation of the conjugate collinear spinor as a transformation in the antifundamen-tal representation with the generators T (¯3) = − T T .12hese transformations induce analogous transformations on the collinear fields φ c ∈{ W † c ξ c , ¯ ξ c W c , A ⊥ c } entering the production operators: φ c ; aα ( x ) = S ( r ) n,ab ( x − ) φ (0) c ; bα ( x ) (2.30)as follows from the transformation W c ( x ) → S (3) n ( x − ) W (0) c ( x ) S (3) † n ( x − ) (2.31)of the collinear Wilson line. For the incoming parton in the anticollinear direction weexchange n by ¯ n , x − by x + in the argument of the Wilson lines, and the representation r by r ′ . In the expressions with the superscript (0) all collinear fields are replaced by thedecoupled fields A (0) c and ξ (0) c and the soft gluon fields are set to zero.Turning to the non-relativistic sector, we first note that from the diagrammatic per-spective the coupling between the non-relativistic and soft fields is non-trivial, since thenon-relativistic energy of the heavy fields and the energy of soft gluons is of the same order M λ ∼ M β . Thus, soft gluons can attach to the non-relativistic lines, to and in betweenthe Coulomb ladder rungs, leaving potential lines in the potential region. However, a redef-inition of the non-relativistic fields by a time-like Wilson line decouples the soft gluons alsofrom the potential fields at leading order in the non-relativistic expansion. Namely, theinteraction with soft gluons in the PNRQCD Lagrangian given in (2.19) can be eliminatedthrough ψ a ( x ) = S ( R ) w,ab ( x ) ψ (0) b ( x ) (2.32) ψ † a ( x ) = ψ (0) † b ( x ) S ( R ) † w,ba ( x ) , (2.33)where the soft Wilson lines S ( R ) w are defined as S ( R ) w ( x ) = P exp (cid:20) − ig s Z ∞ dt w · A cs ( x + wt ) T ( R ) c (cid:21) (2.34) S ( R ) † w ( x ) = P exp (cid:20) ig s Z ∞ dt w · A cs ( x + wt ) T ( R ) c (cid:21) (2.35)and analogously for the primed field transforming under the representation R ′ . This trans-formation eliminates the interaction contained in the soft covariant derivative D s in (2.19),since S ( R ) † w iD s S ( R ) w = i∂ , (2.36) Note that in contrast to (2.30) we do not use a field redefinition of the non-relativistic annihilationfields ψ with Wilson lines extending from −∞ to x but (2.32), since the former definition would obscurethe decoupling of the soft gluons in the PNRQCD Lagrangian. As discussed in [74], the two different formsof the redefinition are equivalent if one takes an appropriate phase for external non-relativistic in-statesinto account. Since we encounter only external outgoing non-relativistic particles, this subtlety is irrelevantin the present context. U are the Wilson lines S w [37], and since the S w in the transformations (2.32), (2.33) only depends on the timecoordinate, the soft gluon Wilson lines drop out from the Coulomb potential term ex-pressed in terms of the redefined fields. In other words, in terms of the redefined fields thePNRQCD Lagrangian takes exactly the same form as (2.19) except that D s → ∂ , so thatthe Lagrangian is independent of the soft-gluon field. At the next order in the velocityexpansion, the PNRQCD Lagrangian contains an interaction involving the gauge invariantchromoelectric operator ~x · ~E s ( t,
0) [67, 75] that is not removed by the above transforma-tion so the decoupling of soft and potential modes is only valid at leading order in thenon-relativistic expansion. We demonstrate in section 2.5 that the corrections to the totalcross section from a single insertion of this operator vanish to all orders.Assembling the different contributions shows that the operators (2.11) for S -wave pro-duction ( ℓ = 0) factorize into an expression of the form O (0 ,i ) { a ; α } ( x, µ ) = S ( i ) { j } ( x ) h φ (0) c ; j α φ (0)¯ c ; j α ψ (0) † j α ψ ′ (0) † j α i ( x, µ ) , (2.37)with the universal soft contribution S ( i ) { j } ( x ) = c ( i ) { a } S n,a j ( x − ) S ¯ n,a j ( x + ) S † w,j a ( x ) S † w,j a ( x ) . (2.38)Here and in the following we suppress the representation labels on the soft Wilson lines, andtake it as understood that the Wilson lines always are in the representations appropriate forthe respective particles. The decoupled fields ψ (0) † , φ (0) c and φ (0)¯ c in (2.37) do not interactwith the soft Wilson lines and with each other at leading power in SCET and at leadingorder in the non-relativistic expansion.Near the partonic threshold only soft radiation can occur in the final state so the state | X i is free of collinear radiation. Therefore the Fock space is a direct product of potential,soft, collinear and anticollinear contributions, | pp ′ i = | p i c ⊗ | p ′ i ¯ c ⊗ | i s ⊗ | i p , | HH ′ X i = | i c ⊗ | i ¯ c ⊗ | X i s ⊗ | HH ′ i p , (2.39)and we obtain the scattering amplitude in the factorized form: A ( pp ′ → HH ′ X ) = X i C (0 ,i ) { α } ( µ ) h X |S ( i ) { j } (0) | i h HH ′ | ψ (0) † j α ψ ′ (0) † j α | i× h | φ (0) c ; j α | p i h | φ (0)¯ c ; j α | p ′ i . (2.40)As remarked above, soft-potential and soft-collinear factorization is only valid at leadingorder in the non-relativistic and collinear expansion. Similarly for a P -wave productionoperator (2.18) the field redefinitions (2.33) do not eliminate the soft gluons in the covariantderivative. In order to achieve factorization of soft gluons in a kinematical regime where relativistic correctionsbecome important one can describe the heavy fields by two copies of heavy quark effective theory withdifferent velocities w and w , see also [36]. Performing separate field redefinitions for H and H ′ withWilson lines defined in terms of the two velocities decouples the soft gluons up to corrections of the order1 /m H , /m H ′ . .4 Factorization formula for the cross section near threshold We now show that the total partonic cross section for the production of a heavy-particlepair near the production threshold factorizes into hard, potential and soft contributionsas sketched in (1.4). To this end, we consider the cross section for the production of aheavy-particle pair near threshold in an S -wave from the partonic process (2.2) for on-shell, massless initial state partons p and p ′ with zero transverse momentum. In this caseonly the hard, soft, collinear and anticollinear modes (2.8) contribute to the cross sectionsince no smaller scale related to Λ Q CD , parton masses or off-shellness is introduced andthe modes included in the effective theory introduced in section 2.1 are sufficient. We cantherefore use the expression for the amplitude (2.40) in terms of decoupled collinear andpotential fields and soft Wilson lines to compute the cross section as usual by squaring thescattering amplitude, averaging over initial state and summing over final state polarizationsand colours, and integrating over the final-state phase space: σ pp ′ ( s ) = 12 s Z d Φ d Φ d Φ X X pol X colour |A ( pp ′ → HH ′ X ) | (2 π ) δ (4) ( p + p + P X − P ) (2.41)Here P is the total incoming momentum and P X is the total momentum of the hadronicstate X . We suppress the normalization factors for the initial-state averages. We do not aimhere at a proof of the factorization of the hadronic cross section, that would also require thetreatment of modes related to the QCD scale, as in the case of deep inelastic scattering for x → O ( β ) corrections, subleading soft interactions inthe effective Lagrangian that are not removed by the decoupling transformations couldcontribute to the cross section starting from NNLL. At fixed-order NNLO accuracy it wasshown in [45] that these corrections do not, in fact, contribute to the total cross section andthe only corrections not generated by the leading order effective Lagrangian are the one-loop hard function and NLO Coulomb and non-Coulomb potentials. These considerationswill be elaborated on in section 2.5 and extended to all orders in the coupling expansion.Therefore the derivation of the simple factorization of the form (1.4) given in the followingholds at NNLL accuracy for production processes dominated by S -wave production.As for the hadronic cross section (2.1), standard QCD factorization implies that thecross section for the parton initial states factorizes according to σ pp ′ ( s ) = X ˜ p, ˜ p ′ Z dx dx f p/ ˜ p ( x , µ ) f p ′ / ˜ p ′ ( x , µ ) ˆ σ ˜ p ˜ p ′ ( x x s, µ ) , (2.42)where the parton distribution functions of parton ˜ p in parton p , f p/ ˜ p ( x, µ ), contain thecollinear modes. We will consider this cross section near the production threshold, s ≈ M , and show that the short-distance cross section ˆ σ ˜ p ˜ p ′ itself factorizes into hard, soft andpotential contributions as sketched in (1.4). At threshold, the PDFs in (2.42) necessarilyappear in the limit x →
1, where the flavour off-diagonal PDFs are suppressed, so it issufficient to consider the case p = ˜ p , p ′ = ˜ p ′ . This is consistent with the fact that, atleading power, we only have to consider EFT matrix elements in (2.10) where the collinear(anticollinear) field φ c ( φ ¯ c ) annihilates the state | p i ( | p ′ i ). According to QCD factorization,ˆ σ ˜ p ˜ p ′ is independent of the initial states p , p ′ so it is identical to the hard-scattering crosssection to be used in the hadronic cross section (2.1). Therefore the factorized form ofthe short-distance cross section (1.4) derived in this way can be used in the hadroniccross section in the partonic threshold region where x x s ∼ M . As discussed above,this region is expected to give a sizable contribution to the total cross section for heavy-particle masses 4 M ∼ . s . For values of x , outside the partonic threshold region,the factorized formula is not necessarily a good approximation to the cross section andthe resummed cross section should be matched to the fixed-order expansion as discussedfurther in section 3.6.We now proceed with the derivation of the factorization formula which follows similarsteps as that for the Drell-Yan process in [33]. The new ingredients here are the presenceof the potential fields and additional Wilson lines for the heavy coloured particles, result-ing in a more involved colour and spin structure. After inserting the factorized matrixelement (2.40) into the standard expression for the cross section (2.41), we use an integralrepresentation of the δ function,(2 π ) δ (4) ( p + p + P X − P ) = Z d z e − iz · ( p + p + P X − P ) . (2.43)Rewriting the exponential in terms of the momentum operator acting on the externalstates, we can translate the collinear, potential and soft operators in the conjugate matrixelement to z , e.g. e iz · k h p ( k ) | φ (0) † c (0) | i = h p ( k ) | e iz · ˆ P φ (0) † c (0) e − iz · ˆ P | i = h p ( k ) | φ (0) † c ( z ) | i .We then obtain for the cross section: σ pp ′ ( s ) = 12 s X pol X colour X i,i ′ C (0 ,i ) { α } C (0 ,i ′ ) ∗{ β } Z d z e − iMw · z J β β α α k k j j ( z ) × h p | φ (0) † c ; k β ( z ) | i h p ′ | φ (0) † ¯ c ; k β ( z ) | i h | φ (0) c ; j α (0) | p i h | φ (0)¯ c ; j α (0) | p ′ i× Z d Φ X h |S ( i ′ ) ∗{ k } ( z ) | X i h X |S ( i ) { j } (0) | i . (2.44)Here we introduced the definition of the potential function in terms of the PNRQCD matrixelements: J { α }{ a } ( z ) = Z d Φ d Φ X pol , colour h | [ ψ ′ (0) a α ψ (0) a α ]( z ) | HH ′ i h HH ′ | [ ψ (0) † a α ψ ′ (0) † a α ](0) | i . (2.45)The factor e − iMw · z = e − i ( m H + m H ′ ) w · z arises because, by definition, the non-relativistic fieldsdepend only on the residual momentum so that ψ ( z ) = e iz · ˆ P ψ (0) e − iz · ˆ P = e − iz · ˜ p ψ (0). We16an introduce a colour basis for the potential function J by introducing projectors P R α onto the irreducible representations R α appearing in the decomposition of the final-staterepresentations, R ⊗ R ′ = P R α R α : J { α }{ a } = X R α P R α { a } J { α } R α . (2.46)A systematic procedure for the construction of the projectors for arbitrary representationsout of Clebsch-Gordan coefficients for the coupling R ⊗ R ′ → R α is reviewed in [37] wherethe projectors needed for all production processes of pairs of coloured SUSY particles arecollected and further properties of the projectors can be found.Since the leading order PNRQCD Lagrangian is spin independent, the spin structureof the leading order potential function (resumming multiple Coulomb exchange) is trivial: J { α } R α ( x ) = δ α α δ α α J R α ( x ) . (2.47)However, as mentioned above, spin-dependent non-Coulomb potentials become relevantstarting from NNLL accuracy in the combined counting in α s ln β and α s /β defined in (2.7).Therefore we introduce a decomposition of the potential function into different spin states: J { α } R α ( x ) = s + s ′ X S = | s − s ′ | Π { α } S J SR α ( x ) . (2.48)Here s ( s ′ ) is the spin of the heavy particle P ( P ′ ). The projectors Π S project on the HH ′ state with total spin S . They satisfy the completeness relation s + s ′ X S = | s − s ′ | Π S { α } = δ α α δ α α . (2.49)As an example, for two spin particles, the projectors on singlet and triplet states read inthe cms frame: Π { α } = 12 δ α α δ α α , Π { α } = 12 σ iα α σ iα α . (2.50)The explicit expression for the function J is given in section 3.3.In order to bring (2.44) into the form of the standard factorization formula (2.42) wehave to identify the collinear matrix elements with the PDFs which requires to combine theproducts of the matrix elements of the collinear fields into a single expectation value (thesame discussion applies for the anticollinear fields). This can be achieved by recalling thatdue to the threshold kinematics collinear radiation from the initial state is inhibited. Wecan therefore formally sum over the states of the collinear final state Fock space | C i since17he only contribution comes from the vacuum. After subsequent use of the completenessrelation we can simplify the product of collinear matrix elements to h p | φ (0) † c ( z ) | i h | φ (0) c (0) | p i ⇒ X C Z d Φ C h p | φ (0) † c ( z ) | C i h C | φ (0) c (0) | p i = h p | φ (0) † c ( z ) φ (0) c (0) | p i . (2.51)The same result for the collinear matrix element is obtained in the standard collinearfactorization away from threshold where the final state contains collinear radiation and thesum over the collinear Fock space is present from the beginning. We therefore define h p ( P ) | φ (0) † c ; k β ( z ) φ (0) c ; j α (0) | p ( P ) i | avg. = δ k j Z dx x e ix ( z · P ) N pα β ( x P ) f p/ ˜ p ( x , µ ) , (2.52)where ˜ p is the parton annihilated by the field φ and the average refers to colour andpolarization. We show in appendix A that the so defined functions f p/ ˜ p ( x , µ ) coincide withthe standard definition of the quark, antiquark and gluon parton distribution functions.Since the momentum of the external state is P µ = ¯ n · P n µ /
2, the matrix element dependson z only through z + = n · z ∝ z · P since z · P = ( n · z )(¯ n · P ) /
2, which has been usedto represent it as a one-dimensional Fourier transform. This also allows us to replace z µ by ( n · z ) ¯ n µ / χ α of the SCET fields N pαβ ( x P ) = 1 n s D r X λ χ λα ( x P ) χ λ ∗ β ( x P ) , (2.53)where the sum extends over the physical polarization states. D r denotes the dimension ofthe SU(3) colour representation of the initial state parton p and the number of spin-degreesof freedom is n s = 2 for all relevant cases. Note that the polarization wave functions areevaluated for a particle with momentum x P , which corresponds to the standard rule forcalculating parton scattering cross sections. At the same time the factors 1 /x (and 1 /x for the other parton) from the definition (2.52) combine with the prefactor 1 / (2 s ) in (2.44)to yield the standard flux factor 1 / (2ˆ s ) for partonic scattering.The soft matrix elements in the cross section (2.44) can be collected in the soft function ˆ W R α ii ′ ( z, µ ) = X X Z d Φ X h |S ( i ′ ) ∗{ j j k k } ( z ) | X i P R α { k } h X |S ( i ) { j j k k } (0) | i , (2.54)which is a matrix in the colour basis given by the c ( i ) { a } . Since the collinear matrix elementsare diagonal in colour and due to the colour decomposition of the potential function (2.46),the initial-state colour indices j , j are summed over and the final state indices are pro-jected on the irreducible representations R α . Since we are concerned with the total crosssection, the soft matrix elements can be combined in a correlation function by summing18ver the hadronic final state and using the completeness relation X X Z d Φ X | X i h X | = I (2.55)of the soft Hilbert space. The soft function can then be written asˆ W R α ii ′ ( z, µ ) = P R α { k } c ( i ) { a } ˆ W { k }{ ab } ( z, µ ) c ( i ′ ) ∗{ b } (2.56)with the correlation function of soft Wilson linesˆ W { k }{ a,b } ( z, µ ) = h | T[ S w,b k S w,b k S † ¯ n,jb S † n,ib ]( z )T[ S n,a i S ¯ n,a j S † w,k a S † w,k a ](0) | i . (2.57)In combining the soft matrix elements into a correlation function we have introducedtime- and anti-time-ordering symbols as discussed in [33]. In evaluating this correlationfunction, a soft gluon propagator connecting fields in the time-ordered and anti-time-ordered products is given by a cut propagator, therefore this prescription reproduces theusual rules for the real soft corrections. A colour basis that diagonalizes the soft matrices W R α ii ′ to all orders in perturbation theory has been constructed in [37] as reviewed insection 3.1 below.Inserting these definitions, the cross section becomes a convolution of the PDFs withthe potential and soft functions: σ pp ′ = Z dx dx f p/ ˜ p ( x , µ ) f p ′ / ˜ p ′ ( x , µ ) X i,i ′ s + s ′ X S = | s − s ′ | H Sii ′ ( µ ) × Z d q (2 π ) J SR α ( q ) Z d z e i ( x P + x P − Mw − q ) · z ˆ W R α ii ′ ( z, µ ) , (2.58)where we have introduced the Fourier transform of the potential function J SR α ( q ) = Z d z e iq · z J SR α ( z ) (2.59)and the spin and colour dependent hard functions defined in terms of the short-distancecoefficients averaged over the initial state spins and projected on the spin state of theheavy-particle system: H Sii ′ ( µ ) = 12ˆ s h N pα β ( x P ) N p ′ α β ( x P ) C (0 ,i ) { α } ( µ )Π β β α α S C (0 ,i ′ ) ∗{ β } ( µ ) i = 12ˆ s D r D r ′ X λ λ ( C (0 ,i ) { α } ( µ ) χ λ α χ λ α )Π β β α α S ( C (0 ,i ′ ) { β } ( µ ) χ λ β χ λ β ) ∗ . (2.60)In the second line we have inserted the prefactors (2.53) (omitting the momentum argu-ment of the polarization wave functions). Summing over the spin states and using the In (2.60) we could replace ˆ s by 4 M to make the hard function energy-independent. However, in thenumerical evaluation it is trivial to keep the kinematic factor 1 / (2ˆ s ) and this is the convention we adoptin Section 4. H ii ′ ( µ ) = s + s ′ X S = | s − s ′ | H Sii ′ ( µ ) = 12ˆ s h N pα β ( x P ) N p ′ α β ( x P ) C (0 ,i ) α α γδ ( µ ) C (0 ,i ′ ) ∗ β β γδ ( µ ) i (2.61)A simple prescription for the computation of the hard functions directly from the scatteringamplitudes will be given in section 3.2.Comparing (2.58) to the factorization formula (2.42) we obtain the factorized expressionfor the short-distance cross section by stripping off the integrals with the parton distribu-tions functions. The remaining expression can be further simplified in the threshold region.For notational simplicity, we will perform these manipulations in the partonic cms framewhere w = (1 , ~ q integral by introducing the new integration variable q ′ = ( x P + x P − M w − q ) = ( √ ˆ s − M ) w − q and obtainˆ σ pp ′ (ˆ s, µ ) = X i,i ′ s + s ′ X S = | s − s ′ | H Sii ′ ( µ ) Z d q ′ (2 π ) J SR α ( Ew − q ′ ) Z d z e iq ′ · z ˆ W R α ii ′ ( z, µ ) . (2.62)Here we defined the partonic centre-of-mass energy measured from threshold, E = √ ˆ s − M = M (1 − z ) + O ( λ ). The function ˆ W contains soft fields that, by definition, varysignificantly only on distances z ∼ /λ . Hence, only soft momenta ( q ′ ∼ λ ) contribute tothe q ′ integration. On the other hand the function J , which is defined in terms of potentialfields alone, is a function of q = ( √ ˆ s − M ) w − q ′ . In the partonic centre of mass frame wehave by assumption x P + x P = √ ˆ sw = p + p + k s = 2 M w + k ′ s with soft momenta k s , k ′ s , therefore q is also soft. Since potential fields with scalings ( k , ~k ) ∼ ( λ, √ λ ) candepend on soft momenta only through their small time-like components, J is a function of q = √ ˆ s − M − q ′ alone. This enables us to perform the ~q integration in (2.62) whichsets ~z = 0 in the argument of the soft function. We then obtain the final expression forthe factorized short-distance cross sections:ˆ σ pp ′ (ˆ s, µ ) = s + s ′ X S = | s − s ′ | X i,i ′ H Sii ′ ( µ ) Z dω X R α J SR α ( E − ω W R α ii ′ ( ω, µ ) . (2.63)Here we defined the Fourier transform of the soft function W R α ii ′ ( ω, µ ) = Z dz π e iωz / ˆ W R α ii ′ ( z , ~ , µ ) . (2.64)The formula (2.42), with the partonic cross section (2.63), establishes the factorization intocollinear (the parton distribution functions), potential (the function J ) and soft (the softfunction W ) contributions for heavy particles produced in an S -wave, and constitutes themain theoretical result of this work. Some further comments on the structure and validityrange of the factorization formula as well as a comparison to previous results are given insection 2.5 below. The simplification of the colour sum over i, i ′ due to the existence of adiagonal basis [37] is reviewed in section 3.1.20 .5 Some comments on the factorization formula Let us briefly comment on the gauge invariance of the ingredients in the factorizationformula. The hard function is defined in terms of on-shell fixed-order scattering amplitudesprojected on a given colour and spin state. The gauge invariance of the hard functionthen follows from the gauge invariance of the on-shell scattering amplitude and the linearindependence of the elements of the colour and spin bases. The effective-theory Lagrangianis invariant under the collinear gauge transformations of SCET [60] and the soft gaugetransformations (2.27) and (2.20). Collinear gauge invariance is built into the formalismsince the operators are constructed from the invariant fields φ c . The invariance of thecomponents of the soft function (2.64) under soft gauge transformations follows since theelements of the colour basis are invariant tensors satisfying c ( i ) { a } = U ( R ) † a b U ( R ′ ) † a b c ( i ) { b } U ( r ) b a U ( r ′ ) b a . (2.65)Note that for this it is essential that only the soft function with vanishing spatial argument, W R α ii ′ ( x , ~ , µ ), enters the final factorization formula since for the soft function at arbitrary x the gauge transformations of the collinear, anticollinear and potential fields appear withdifferent arguments x − , x + and x . Finally the collinear matrix elements and the potentialfunction are defined in terms of the decoupled fields that transform trivially under softgauge transformations. In our derivation of the factorization formula we relied on field redefinitions that decouplethe soft gluons from the leading SCET and PNRQCD Lagrangians. The subleading effec-tive Lagrangians include higher-order potentials, but also interactions of the soft gluons tothe potential fields via the ~x · ~E interaction mentioned above and analogous couplings to thecollinear fields. These terms cannot be eliminated using the decoupling transformations.However, being sub-leading, these interactions can be treated as perturbations in β and,since the soft gluons decouple from the leading effective Lagrangian, the cross section canbe written to all orders in the non-relativistic and SCET expansions in the schematic formˆ σ = X a H a [ W a ⊗ J a ] , (2.66)with higher-order hard, soft and potential functions labeled by the index a . At NNLLaccuracy, several effects could be relevant that are not included in the leading term a = 0in (2.66) considered so far in this work. We now discuss these effects and show that theyeither do not contribute at NNLL or are incorporated in a straightforward way withoutspoiling the factorization.It is worth mentioning at this point that in standard situations of soft gluon resum-mation not requiring the consideration of Coulomb singularities, the expansion in powers21f 1 − z and α s can be considered separately. Thus the discussion of power-suppressedinteractions is not necessary to any order in α s as long as one drops subleading terms in1 − z . In joint soft-gluon and Coulomb resummations the two expansions are linked bythe counting α s /β ∼
1, so that O ( β ) suppressed terms that would normally be referred toas power corrections now appear at the same NNLL order as α s terms, see (2.7). This isa complication characteristic of perturbative non-relativistic systems that have kinematicsingularities which are stronger than logarithmic. Hard effects.
Higher-order production operators such as the P -wave operator (2.18) canappear at O ( β ), which according to (2.7) corresponds to a NNLL effect. These can includeinteractions with soft gluons through a spatial covariant derivative that are not removedby the decoupling transformation. However, since there is no interference between S - and P -wave production for the total cross section, these corrections are at least O ( β ), beyondNNLL accuracy. Potential effects.
Beyond the leading-order Coulomb potential in (2.19), subleading effectslead to further potential interactions of the form Z d ~r (cid:2) ψ † a ψ b (cid:3) ( ~r ) δV abcd ( r ) (cid:2) ψ ′ † c ψ ′ d (cid:3) (0) . (2.67)The running of the strength of the Coulomb potential causes a NLL correction beginningwith α s /β × log β , which is accounted for by evaluating the coupling in the leading-orderCoulomb potential at a scale of order m red β , where m red denotes the reduced mass of theheavy-particle system, or by an explicit logarithmic correction to δV abcd ( r ). Genuine loopcorrections to the colour Coulomb potential lead to the substitution α s (1 /r ) r → α s (1 /r ) r (cid:16) a α s π + . . . (cid:17) (2.68)in the Lagrangian (2.19). Only the one-loop correction ˆ a is needed at NNLL, and con-tributes terms beginning with α s /β . In addition there exist the spin-dependent and inde-pendent non-Coulomb potentials of form α s /r and (summarily) α s /r , see e.g. [76], leadingto NNLL terms beginning with α s log β . Here the logarithm arises from non-relativisticfactorization and is related to the fact that the short-distance functions H Sii ′ ( µ ) have poten-tial infrared divergences in addition to the familiar soft and collinear divergences. Theseadditional potential terms in the PNRQCD Lagrangian do not involve soft gluon fields andthe decoupling transformation can be applied as for the leading potential. Therefore theseterms can be included in the evaluation of the potential function J , as was done in [45] tocompute all log β and 1 /β enhanced terms of the t ¯ t -production cross section at O ( α s ). Subleading potential-soft interactions.
We next discuss possible contributions to subleadingterms in (2.66) arising from the higher-order couplings of soft gluons to collinear andpotential fields that are not decoupled by the field redefinition. In a diagrammatic language,these arise from the interference of subleading soft gluons coupling to the initial or finalstate with potential gluon exchange, that could contribute NNLL terms beginning with22 s /β × α s β log , β . As shown in [45] using power-counting and rotational invariance, thefixed-order NNLO corrections of this form vanish for the total cross section. We will nowuse the effective theory language to show that the NNLL corrections arising from thesubleading SCET and PNRQCD Lagrangians vanish to all orders in the strong coupling,so that the leading factorization formula (2.63) does not require modification at NNLL.We begin with the O ( β ) suppressed interactions in the subleading PNRQCD La-grangian [67, 70, 75], L (1)PNRQCD = − g s h ψ † ( x ) ~x · ~E s ( x , ~ ψ ( x ) + ψ ′† ( x ) ~x · ~E s ( x , ~ ψ ′ ( x ) i ≡ − g s ~x · ~E as ( x , ~ J a ( x ) (2.69)with the bilinear product of potential fields J a = ψ † T ( R ) a ψ + ψ ′† T ( R ′ ) a ψ ′ . (2.70)We treat the chromoelectric vertex perturbatively, i.e. we consider contributions to thecross section where one of the matrix elements contains an insertion of the vertex ~x · ~E and evaluate these matrix elements with the LO PNRQCD Lagrangian. The first-ordercorrection to (2.10) from the interaction (2.69) is given by∆ A (1) ( pp ′ → HH ′ X ) = C (0) { a ; α } ( µ ) Z d x h HH ′ X | T[ L (1)PNRQCD ( x ) O (0) { a ; α } ( µ )] | pp ′ i EFT . (2.71)The expectation value is evaluated with the leading-order effective-theory Lagrangian, sothe soft, collinear and potential fields decouple after the field redefinition. Under thisredefinition, the chromoelectric interaction is transformed into ψ † ( x ) ~x · ~E s ( x , ~ ψ ( x ) = ψ † (0) ( x ) ~x · ~ E s ( x , ~ ψ (0) ( x ) (2.72)with E s = S † w E s S w . Independent of the colour representation of the heavy particles, due tothe identity (2.21), which also holds for the Wilson lines, the components of the redefinedchromoelectric fields are given by ~ E as ( x ) = S (8) w,ab ~E bs ( x ) . (2.73)In analogy to (2.40), the effective theory matrix element factorizes and the correctionto the scattering amplitude can be written as∆ A (1) ( pp ′ → HH ′ X ) = X i C (0 ,i ) { α } ( µ ) h | φ (0) c ; j α | p i h | φ (0)¯ c ; j α | p ′ i× Z d x ~x h X | T[ ~ E as ( x , ~ S ( i ) { j } (0)] | i h HH ′ | T[ J a (0) ( x ) ψ (0) † j α ψ ′ (0) † j α ] | i . (2.74)Repeating the steps leading to the factorization formula (2.62) and the subsequent discus-sion, we obtain an analogous formula with the replacement23 dq (2 π ) J R α ( E − q ) Z dz e iq z ˆ W R α ii ′ ( z , µ ) ⇒ Z dq (2 π ) Z d k (2 π ) J a (1) R α ( E − q , k ) Z dz e iq z Z d x e − ik · x ~x · ~ ˆ W a,R α (1) ii ′ ( z , x , µ ) + c.c . (2.75)Here we have defined the subleading soft function with an insertion of the chromoelectricfield according to ~ ˆ W a,R α (1) ii ′ ( z , x , µ ) = P R α { k } h | T[ S ( i ′ ) ∗{ ijk k } ( z )]T[ ~ E as ( x ) S ( i ) { ijk k } (0)] | i (2.76)as well as a subleading potential function with an insertion of the bilinear (2.70): J a (1) { α } R α ( q, k ) = P R α { a } Z d z e iq · z Z d x e ik · x h | [ ψ (0) α a ψ ′ (0) α a ]( z )T[ J a (0) ( x )[ ψ (0) † α a ψ ′ (0) † α a ](0)] | i . (2.77)Since the matrix elements are evaluated with the spin-independent leading order PNRQCDLagrangian, the spin dependence simplifies in analogy to (2.47) as was used in (2.75). Inthe terms denoted by “c.c.” in (2.75) the operator E s is inserted in the anti-time-orderedproduct in the subleading soft function and the potential function is defined with theinsertion of J (0) in an anti-time-ordered product with the annihilation fields.Because the subleading soft function does not depend on ~x , we can perform the integralover the spatial components of x in (2.75) and obtain the expression Z d k (2 π ) δ (3) ( ~k ) ∂∂~k J a (1) R α ( E − q , k ) = 0 , (2.78)which vanishes since (in the partonic cms frame) there is no three-momentum left that theintegral over J (1) can depend on. Hence we conclude that the correction to the cross sectiondue to a single insertion of the NLO potential Lagrangian vanishes to all orders in the strongcoupling constant. Similar to the diagrammatic argument in [45], an essential ingredient inthe argument was the multipole expansion of soft fields when multiplying potential fields.Because of this, only ~E ( x ) appears in the subleading PNRQCD interaction and, as aconsequence, the subleading soft function depends only on x which allowed to perform thesimplification in the last step. The second ingredient is rotational invariance in combinationwith the independence of the potential function on any external three-momentum. Subleading collinear-soft interactions.
The effects potentially relevant at NNLL arise fromthe effective Lagrangians at the next-to-leading order in the kinematic expansion. For theSCET Lagrangian of collinear quarks, the relevant interactions are given by [59, 60] L (1) ξ = ¯ ξ (cid:0) x µ ⊥ n ν W c gF s µν W † c (cid:1) ¯ n ξ , (2.79)for n -collinear modes and similar terms involving transverse derivatives or factors of x ⊥ for the couplings to collinear gluons, and an interaction L (1) ξq = ¯ ξ iD/ ⊥ c W c q s + ¯ q s W † c iD/ ⊥ c ξ (2.80)24nvolving soft quarks but only collinear gluons. As argued already in [45], these verticeswith transverse derivatives or x µ ⊥ cannot contribute to the total cross section since theinitial-state momenta can be chosen to have zero transverse momentum. Then loop in-tegrals with transverse momentum factors vanish by rotational invariance in the planetransverse to the beam axis. Eq. (2.80) contains a vertex that describes the collinear split-ting of a quark into a gluon and a soft quark, which is also an O ( β ) term. However, anon-vanishing contribution to the squared amplitude requires two such vertices to contractthe soft quark field, hence there is again no contribution at NNLL. Note that insertionsof power-suppressed interaction Lagrangians lead to collinear matrix elements that do nottake the form of the standard parton densities. At this point it appears that one mustdistinguish the collinear expansion in powers of k ⊥ ∼ M √ λ ∼ M β from the standardcollinear expansion in k ⊥ ∼ Λ QCD . However, as mentioned above, this issue is not relevantfor NLL and NNLL resummations.To summarize, the factorization formula (2.63) is valid at NNLL accuracy providedsubleading Coulomb and non-Coulomb potentials are included in the computation of thepotential function and the hard and soft functions are computed at NLO.
The result (2.63) can be compared to the analogous formula (1.1) given in [7] for thepair production of relativistic coloured particles with different four-velocity vectors w , w ,where potential degrees of freedom do not appear. In this case the potential function isabsent in the factorization formula and the soft function is given by (taking into accountour convention for the antiparticles):ˆ S { ab } ( z ) = h | T[ S † n,ib S † ¯ n,jb S w ,b l S w ,b k ]( z )T[ S ¯ n,a j S n,a i S † w ,ka S † w ,la ](0) | i . (2.81)In our case the extraction of the potential function J leads to a more complicated colourstructure of the soft function W , but to a simpler kinematical structure due to the equalfour-velocities of the heavy particles. Using the completeness relation of the projectors andthe definition of the components of the soft function (2.56), the limit of (2.81) for the caseof equal four velocities, w = w = w , is formally related to the sum of the soft functionsfor the different final-state representations: S ii ′ | w = w = X R α W R α ii ′ . (2.82)The resummation coefficients used in the Mellin space approach to threshold resummationhave been calculated at the one-loop level for various colour representations [7, 17, 20] fromthe threshold limit of the soft function S . This amounts to taking the threshold limit afterextracting the 1 /ǫ poles of the eikonal diagrams contributing to S . Applying this approachat the two-loop level [42] one encounters three-parton colour correlations proportional tolog β that arise from an interference of soft and potential divergences, and lead to off-diagonal contributions in the colour basis constructed in [37] that diagonalizes the soft25unctions W R α . In contrast, the effective theory approach constructs an expansion in β before taking the ǫ → β -dependent divergences are attributedto higher-dimensional soft and potential functions due to the subleading ~x · ~E vertex inthe PNRQCD Lagrangian and subleading potentials, while the leading soft function (2.56)is diagonal to all orders. In order to resum all terms that are enhanced by powers oflog β and inverse powers of β at a given accuracy one should use the formula (2.66),while (1.1) is appropriate for partonic thresholds where the relative velocities of the finalstate particles are relativistic, as considered e.g. in the recent resummation of the invariantmass distribution of top-quark pairs in [36]. The enhancement due to single-Coulomb exchange has been included in the Mellin momentspace approach to resummation as part of the hard function [11], which implicitly assumesthe factorization of soft and potential gluons. We now briefly discuss the relation of ourfactorization formula (2.63) to the Mellin space formalism and show that the convolutionof the soft and potential function factorizes in Mellin space in the large N limit, whichjustifies the earlier treatment. We recall that the Mellin moments of the cross section inthe variable z = 4 M / ˆ s are defined as:ˆ σ ( N, µ ) = Z dz z N − ˆ σ (ˆ s, µ ) . (2.83)In order to Mellin-transform the factorized cross section (2.63), we approximate E = √ ˆ s − M ≈ M (1 − z ) and make the transformation of variables ω = 2 M (1 − w ). For stableparticles and neglecting bound-state contributions, the potential function is non-vanishingonly for E − ω/ ≈ M ( w − z ) > σ pp ′ (ˆ s, µ ) ≈ M X S,i,i ′ H Sii ′ Z z dww X R α J SR α ( M (1 − zw )) W R α ii ′ (2 M (1 − w ) , µ ) . (2.84)Here we have used z < w . E − ω/ ≈ M ( w − z ) ≈ M (1 − z/w ) and introduceda factor w ≈
1. Since a convolution of the form R dww f ( w ) g ( z/w ) factorizes under a Mellintransform, we conclude that, up to power suppressed terms, our factorization formulaimplies multiplicative soft-Coulomb factorization of the cross section in Mellin space:ˆ σ pp ′ ( N, µ ) ≈ X S,i,i ′ H Sii ′ X R α J SR α ( N ) S R α ii ′ ( N, µ ) , (2.85)where the soft function in Mellin space is given by S R α ii ′ ( N, µ ) = 2 M Z dz z N − W R α ii ′ (2 M (1 − z ) , µ ) = Z ∞ dωe − N M ω W R α ii ′ ( ω, µ ) + O ( N − ) , (2.86)26nd J SR α ( N ) = Z dz z N − J SR α ( M (1 − z )) . (2.87)This shows that previous treatments that put the one-loop Coulomb corrections into thehard function [11] can be extended to include the higher-order Coulomb corrections as well. The combination of soft and Coulomb effects for the invariant mass distribution of top-quark and gluino pairs near threshold has been performed in several approximations as-suming soft-Coulomb factorization [24–26]. We now show that a factorization formula forthe differential cross section dσ/dM HH ′ with M HH ′ = ( p + p ) , valid for M HH ′ near 2 M ,can be derived from our main result (2.63).We first introduce the parton luminosity L pp ′ ( τ, µ ) = Z dx dx δ ( x x − τ ) f p/N ( x , µ ) f p ′ /N ( x , µ ) (2.88)to express the hadron scattering cross section as σ N N → HH ′ X = X p,p ′ = q, ¯ q,g Z M /s dτ L pp ′ ( τ, µ ) ˆ σ pp ′ ( τ s, µ ) , (2.89)with ˆ σ pp ′ ( τ s, µ ) given by (2.63). Next we observe that the argument E − ω/ √ τ s − M − ω/ J SR α in that equation corresponds to the non-relativisticenergy of the HH ′ pair in the partonic cms frame. In this frame the three momentum ofthe pair is soft and therefore makes a negligible contribution to the invariant mass, so therelation M HH ′ = 2 M + E − ω O ( M β ) = √ τ s − ω/ O ( M β ) (2.90)applies. We now change variables from ω to M HH ′ in (2.63) and interchange the τ and(implicit) ω integration in (2.89) using Z M /s dτ Z E dω = 2 Z √ s M dM HH ′ Z M HH ′ /s dτ (2.91)and the fact that for stable particles, and neglecting bound-state contributions, the Coulombfunction has support only for positive values of its argument, which fixes the upper limit2 E . This results in dσ N N → HH ′ X dM HH ′ = X p,p ′ = q, ¯ q,g Z M HH ′ /s dτ L pp ′ ( τ, µ ) d ˆ σ pp ′ ( τ s, µ ) dM HH ′ = X p,p ′ = q, ¯ q,g s + s ′ X S = | s − s ′ | X i,i ′ H Sii ′ ( µ ) X R α J SR α ( M HH ′ − M )27 Z M HH ′ /s dτ L pp ′ ( τ, µ ) W R α ii ′ (2( √ τ s − M HH ′ ) , µ ) (2.92)with d ˆ σ pp ′ (ˆ s, µ ) dM HH ′ = s + s ′ X S = | s − s ′ | X i,i ′ H Sii ′ ( µ ) X R α J SR α ( M HH ′ − M ) W R α ii ′ (2( √ ˆ s − M HH ′ ) , µ ) , (2.93)which is the desired result. This shows that Coulomb and soft effects factorize multi-plicatively in the invariant mass distribution near threshold, as assumed in [24–26]. Notethat in the hadronic cross section only the soft function is averaged with the parton lu-minosity. Eqs. (2.92), (2.93) apply unmodified to bound-state contributions, in whichcase the invariant mass distribution extends below the nominal threshold 2 M , and unsta-ble particles, where in addition the Coulomb function is evaluated at complex argument M HH ′ − M + i (Γ H + Γ H ′ ) / The factorization formula (2.63) provides the basis for resummation of soft and Coulombgluon effects in the soft function W and the potential function J , respectively. The softgluon resummation is performed by solving evolution equations for the soft and hard func-tions [27–31], while the resummation of potential effects can be performed using resultsobtained for top-quark production in electron-positron collisions [76–78]. In this section weprovide the explicit results for the resummed soft and potential functions. In section 3.1we recall the colour basis constructed in [37] that diagonalizes the soft function to all or-ders in the strong coupling constant and quote the result for the one-loop soft function.In section 3.2 we relate the hard function to the colour- and spin decomposed Born crosssection and the colour decomposed one-loop amplitudes at threshold to demonstrate thatthe implementation of the resummation formula requires only a standard calculation infixed-order perturbation theory. In section 3.3 we provide the leading order potential func-tion summing up multiple Coulomb gluon exchange, as required for the NLL resummationperformed in section 4. In section 3.4 we obtain evolution equations for the short-distancecoefficients C and the soft function W that allow to resum soft-gluon effects. The anoma-lous dimensions required up to NNLL have already been collected in [37]. The final resultfor the resummed cross section using the momentum-space formalism of [31–33] is pre-sented in section 3.5 with the explicit expression of the resummation exponent up to NLLrelegated to appendix C. Expansions of the resummed result to order α s , as required forthe matching to a fixed-order calculation, are given in appendix D. In the factorization formula (2.63), the hard- and soft function are matrices in colour spacein the basis of the tensors c ( i ) introduced in (2.14). The soft function can be diagonalized28o all orders in the strong coupling for an arbitrary heavy coloured particle productionprocess using the colour basis constructed in [37]. In this basis, the components of the softfunction (2.56) can be expressed in terms of the soft function for a single heavy final stateparticle in the irreducible colour representation R α . This gives a precise meaning to thepicture of soft gluon radiation resolving only the total colour charge of a heavy-particlepair at rest [11]. We review this construction here and quote the result of the one-loop softfunction.It is useful to perform a decomposition of the product of the representations of theinitial state and final state particles into irreducible representations: r ⊗ r ′ = X α r α , R ⊗ R ′ = X R α R α . (3.1)It is intuitively clear that a final state pair in an irreducible colour representation R α canonly be produced from an initial state system in an equivalent representation. In order toimplement this picture formally, one forms pairs P i = ( r α , R β ) of equivalent initial and finalstate representations r α and R β , treating multiple occurrences of equivalent representationsin the decomposition as distinct, e.g. in the case of a symmetric or antisymmetric couplingof 8 ⊗ →
8. It has been shown in [37] that a basis respecting colour conservation (2.65)can always been chosen by forming for every pair P i of equivalent representations theassociated basis element c ( i ) { a } = 1 p dim( r α ) C r α αa a C R β ∗ αa a , (3.2)where the C R α αa a are Clebsch-Gordan coefficients implementing a unitary basis transfor-mation from the basis vectors of the tensor product space R ⊗ R ′ to basis vectors ofthe irreducible representations R α , and analogously for the initial state representations.The Clebsch-Gordan coefficients, basis elements and projectors for all squark and gluinoproduction processes at hadron colliders have been provided in appendix B of [37]. Forsquark-antisquark production from quark-antiquark annihilation, the allowed pairs of rep-resentations are P i ∈ { (1 , , (8 , } (3.3)and the basis has been given already in (2.16). For the gluon fusion channel the allowedpairs of representations are P i ∈ { (1 , , (8 S , , (8 R , } (3.4)and the basis is given in (4.13) below.Using properties of Wilson lines and the Clebsch-Gordan coefficients it was shown in [37]that the components of the soft function (2.56) in the basis (3.2) can be obtained from thesoft function for the production of a single particle in the representation R α ˆ W R α { aα,bβ } ( z, µ ) ≡ h | T[ S R α w,βκ S † ¯ n,jb S † n,ib ]( z )T[ S n,a i S ¯ n,a j S R α † w,κα ](0) | i (3.5)29y contracting with appropriate Clebsch-Gordan coefficients W R α ii ′ ( ω, µ ) = 1 p dim( r α )dim( r α ′ ) δ R α R β ′ δ R α R β C r α { aα } W R α { aα,bβ } ( ω, µ ) C r ′ α ∗{ bβ } . (3.6)As indicated, the elements are non-vanishing only if the irreducible representation R α isidentical to both final state representations R β and R β ′ in the pairs P i = ( r α , R β ) and P i ′ = ( r α ′ , R β ′ ) that define the tensors c ( i ) , c ( i ′ ) . This is intuitively plausible since weproject on a specific final state representation so only production from initial states in anequivalent representation is possible. This shows that the soft function is automaticallyblock diagonal in the basis (3.2) where off-diagonal elements are only possible if severalinitial state representations r α are equivalent. For initial state quarks, antiquarks andgluons this only happens for two gluons in the initial state that can be combined intoa symmetric and antisymmetric octet, see (3.4). Using Bose symmetry it is furthermorepossible to show that the symmetric and antisymmetric octet production channels do notinterfere so the soft function is diagonal, i.e. W R α ii ′ ( ω, µ ) = W R α i ( ω, µ ) δ ii ′ δ R α R β . (3.7)The one-loop term in the loop expansion of the soft function, W R α i ( ω, µ ) = ∞ X n =0 (cid:18) α s ( µ )4 π (cid:19) n W ( n ) R α i ( ω, µ ) , (3.8)was calculated in [37] and depends only on the quadratic Casimir operators C R of therepresentations of the initial state particles and the final state pair, in agreement withcalculations for specific colour representations [7, 17, 20]. The result in position space canbe written in terms of the variable L = 2 ln (cid:0) iz µe γE (cid:1) and is given byˆ W (1) R α i ( L, µ ) = ( C r + C r ′ ) (cid:18) ǫ + 2 ǫ L + L + π (cid:19) + 2 C R α (cid:18) ǫ + L + 2 (cid:19) , (3.9)where the normalization is such that ˆ W (0) R α i ( L, µ ) = 1. For a final state singlet and aquark-antiquark initial state this agrees with the Drell-Yan Wilson line [29], for a colouroctet final state the result agrees with [21]. The Fourier transform of this result entersthe factorization formula (2.63) and has been computed in [37]. Since in the momentumspace formalism the solution of the renormalization group equation for the soft functionis obtained using the Laplace transform of the momentum space soft function that can beobtained from the position space result by a simple replacement rule, the momentum spaceexpression is not explicitly needed in this paper.
In order to simplify the application of the factorization formula (2.63) we show how tobypass the matching to the effective theory used in our derivation and express the hard30unction (2.60) directly in terms of hard-scattering amplitudes for the production of theheavy-particle pair. The potential and soft functions are universal functions, dependingonly on the colour quantum numbers. The only process dependent input for NLL resumma-tion are then the colour-separated leading-order production cross sections for the processof interest. For NNLL resummation, in addition the tree cross sections for individual spinstates of the heavy-particle pair and the colour separated NLO amplitudes at thresholdare required. We work in the colour basis that diagonalizes the soft function and denotethe diagonal elements of the hard function by H Si .We recall that the S -wave matching coefficients C (0 ,i ) { α } are simply given by the scatteringamplitudes at threshold, with polarization vectors removed and projected on a specificcolour channel according to A { α } = P i c ( i ) { α } A ( i ) . Thus the scattering amplitude for a givencolour configuration and fixed helicities and spins λ i can be written as A ( i ) ( p λ p ′ λ → H λ H ′ λ ) = 2 √ m H m H ′ C (0 ,i ) { α } ( µ ) χ λ α χ λ α ξ λ ∗ α ξ λ ∗ α , (3.10)where we have accounted for the normalization factors √ m H in the definition of the non-relativistic states. Here the ξ are the polarization spinors or vectors of the non-relativisticparticles satisfying the completeness relation X λ ξ λα ξ λ ∗ β = δ αβ . (3.11)Computing the spin averaged partonic tree cross section for the production of a heavy-particle pair in a fixed colour state using the expression of the amplitude in the effectivetheory (3.10) we obtainˆ σ (0 ,i ) pp ′ (ˆ s ) = 12ˆ s λ / (ˆ s, m H , m H ′ )8 π ˆ s D r D r ′ X pol |A ( i )Born ( pp ′ → HH ′ ) | ≈ ˆ s → M ( m H m H ′ ) / M β π ˆ s D r D r ′ X λ i ( C (0 ,i ) { α } ( µ ) χ λ α χ λ α ) δ α β δ α β ( C (0 ,i ) { β } ( µ ) χ λ β χ λ β ) ∗ . (3.12)In the second line we have approximated the prefactor using λ (ˆ s, m H , m H ′ ) = (ˆ s − M )(ˆ s − ( m H − m H ′ ) ) ≈ (ˆ s − M )4 m H m H ′ and the C (0 ,i ) { α } ( µ ) should be computed in the treeapproximation. Comparing to the definition of the hard function (2.60), we obtain asimple relation between the leading order hard functions and the spin averaged total crosssection for a given colour channel:ˆ σ (0 ,i ) pp ′ (ˆ s ) ≈ ˆ s → M ( m H m H ′ ) / M β π H (0) ii . (3.13)The tree-level hard functions can therefore be obtained from the threshold limit of theBorn cross sections in a specific colour channel. With the aim of extending the validity ofthe resummed expressions one may consider defining an improved hard function by using31ull tree-level cross section instead of its threshold limit. In this way, some higher orderterms in β are included, though not systematically.At NNLL spin-dependent hard functions are required in the tree approximation, whichcan be obtained by a formula analogous to (3.13) from the cross section for a fixed colourand spin channel. In the framework of a standard computation of the leading order crosssection, the projection on the final-state spin can be performed introducing scatteringamplitudes for a fixed final state spin S that can be obtained from the helicity amplitudesfor the production of the HH ′ pair according to A ( i ) ( p λ p ′ λ → ( HH ′ ) S,λ ) = X λ λ N S ( λ | λ λ ) A ( i ) ( p λ p ′ λ → H λ H ′ λ ) . (3.14)The N S ( λ | λ λ ) are Clebsch-Gordan coefficients that can be found for the case of spinone-half particles, e.g. in [79].Furthermore, at NNLL accuracy also the one-loop hard function is required. Since thespin-dependence of the potential function is already an O ( α s ) effect, one may use the spinsummed one-loop hard functions H (1) i at NNLL without formal loss of accuracy. The one-loop hard function is given by the interference of the Born and the one-loop amplitudesfor a given colour channel, evaluated directly at threshold: H (1) i ( µ ) = 12ˆ s h N pα β N p ′ α β C (0 ,i )1 − loop α α γδ ( µ ) C (0 ,i )tree ∗ β β γδ ( µ )) i = 18 m H m H ′ ˆ s D r D r ′ X pol (cid:16) A ( i ) ∗ Born ( pp ′ → HH ′ ) A ( i )NLO ( pp ′ → HH ′ ) (cid:17) . (3.15)Here A NLO is the UV-renormalized one-loop amplitude evaluated directly at thresholdwith IR singularities regularized dimensionally and subtracted in the MS-scheme. Thisis therefore the only process independent input required for NNLL resummations and isfar simpler to compute than the full NLO cross section. Alternatively, the hard functioncan be extracted from the threshold expansion of the NLO partonic cross sections in eachcolour channel [45]. These are available for t ¯ t production at hadron colliders [80] but notyet for the production of squarks or gluinos. We now discuss the relation of the potential function J to the Coulomb Green functionfamiliar from PNRQCD. The momentum-space potential function (2.59) can be written interms of the correlation function J { α }{ a } ( q ) = Z d z e iq · z Z d Φ d Φ X pol,colour h | [ ψ ′ (0) a α ψ (0) a α ]( z ) | HH ′ i h HH ′ | [ ψ (0) † a α ψ ′ (0) † a α ](0) | i = Z d z e iq · z h | [ ψ ′ (0) a α ψ (0) a α ]( z )[ ψ (0) † a α ψ ′ (0) † a α ](0) | i , (3.16)32here the matrix elements are evaluated with the leading order PNRQCD Lagrangian. In the second line we used that in this approximation the soft gluon fields are decoupledfrom the fields ψ (0) so we can replace the sum over the two-particle Hilbert space by asum over the full Hilbert space and use the completeness relation. The definition of J given here is sufficient for S -wave production of the heavy particles; for P -wave productionalso expectation values of fields with derivatives have to be considered. As explained insection 2.5.2, effects of the ~x · ~E vertex in the sub-leading PNRQCD Lagrangian are notincluded in the calculation of the potential function J at higher orders but lead to additionalcontributions to the factorization formula with generalized soft and potential functions, tobe calculated with the leading effective Lagrangians. As shown in section 2.5.2 these termsare only relevant for the total cross section beyond the NNLL order.Defining a tensor product notation for the decoupled potential fields,( ψ ⊗ ψ ′ ) a a ( t, ~r, ~R ) = ψ (0) a ( t, ~R + ~r ) ψ ′ (0) a ( t, ~R − ~r ) , (3.17)with the relative coordinates r and the cms coordinates R and using the projectors ontothe irreducible representations R α introduced in (2.46), we can perform the colour decom-position ( ψ ⊗ ψ ′ ) a a = X R α P R α { a } ( ψ ⊗ ψ ′ ) a a ≡ X R α ( ψ ⊗ ψ ′ ) R α a a . (3.18)This is analogous to the singlet-octet decomposition discussed in [67] (see also [81], inparticular eq. (48)). With this notation, the colour structure of the Coulomb potentialsimplifies to (cid:2) ψ (0) † T ( R ) a ψ (0) (cid:3) h ψ ′ (0) † T ( R ′ ) a ψ ′ (0) i = X R α ,R β ( ψ ′ † ⊗ ψ † ) b b P R β { b } T ( R ) bb a T ( R ′ ) bb a P R β { a } ( ψ ⊗ ψ ′ ) a a = X R α ( ψ ⊗ ψ ′ ) R α † D R α ( ψ ⊗ ψ ′ ) R α . (3.19)Here the coefficients of the Coulomb potential are defined by the relation T ( R ) ba c T ( R ′ ) ba c P R α c c a a = D R α P R α { a } (3.20)and we have used the projection property P R α a a b b P R β b b c c = δ R α R β P R α a a c c . (3.21)Using the simplification of the Coulomb potential, the leading order PNRQCD La-grangian for the decoupled fields can be written in the tensor-product notation as a sum As discussed in section 2.5, to reach NNLL accuracy, the LO Lagrangian is to be supplemented byNLO Coulomb and the leading non-Coulomb potentials. L PNRQCD = X R α n ( ψ ⊗ ψ ′ ) R α † i∂ s + ~∂ r m red + ~∂ R M ) ! ( ψ ⊗ ψ ′ ) R α + Z d ~r (cid:2) ( ψ ⊗ ψ ′ ) R α † ( t, − ~r/ , (cid:3) (cid:18) α s D R α r (cid:19) (cid:2) ( ψ ⊗ ψ ′ ) R α ( t, ~r/ , (cid:3)o , (3.22)with the reduced mass m red = m H m H ′ / ( m H + m H ′ ). The kinetic term can be written interms of the tensor field as shown here after a projection on the two-particle sector of thetheory [67, 81].For top-antitop and squark-antisquark production the irreducible representations aregiven by 3 ⊗ ¯3 = 1 ⊕ D = − C F = − N C − N C (attractive) ,D = − (cid:20) C F − C A (cid:21) = 12 N C (repulsive) . (3.23)The explicit values of the coefficients for all remaining representations relevant for theproduction of coloured SUSY particles are collected in appendix B.Since the Lagrangian is now diagonal in colour and spin, the leading-order CoulombGreen function is of the form J { α }{ a } ( q ) = X R α P R α { a } δ α α δ α α J R α ( q ) (3.24)with the correlation function J R α ( q ) = Z d z e iq · z h | [ ψ ′ (0) ψ (0) ]( z )[ ψ (0) † ψ ′ (0) † ](0) | i . (3.25)In this expression the fields carry no colour and spin indices any more and the correlationfunctions are to be calculated with the term in (3.22) corresponding to the representation R α with Coulomb potential α s D R α /r . Since the annihilation fields ψ annihilate the vacuum,we can replace ( ψψ ′ )( z )( ψ † ψ ′ † )(0) = [( ψψ ′ )( z ) , ( ψ † ψ ′ † )(0)] in the vacuum expectation valueand express the correlation function in terms of the imaginary part of a time orderedproduct J R α ( q ) = Z d z e iq · z h | T[ ψ ′ (0) ψ (0) ]( z )[ ψ (0) † ψ ′ (0) † ](0) | i = 2 Im G R α (0)C (0 , E ) . (3.26)Here we introduced the zero-distance Coulomb Green function of the Schr¨odinger operator − ~ ∇ / (2 m red ) − ( − D R α ) α s /r , i.e. the Green function G R α (0)C ( ~r , ~r ; E ) evaluated at ~r = ~r = 0. 34sing the representation of the Green function given in [82], the MS-subtracted zero-distance Coulomb Green function including all-order gluon exchange, reads as follows [70]: G R α (0)C (0 , E ) = − (2 m red ) π (r − E m red + ( − D R α ) α s (cid:20)
12 ln (cid:18) − m red Eµ (cid:19) −
12 + γ E + ψ (cid:18) − ( − D R α ) α s p − E/ (2 m red ) (cid:19)(cid:21)) . (3.27)Here γ E is the Euler-Mascheroni constant and one should apply the prescription E → E + iδ for stable heavy particles and E → E + i (Γ H + Γ H ′ ) / E the potential function evaluates to the Sommerfeld factor J R α ( E ) = (2 m r ed ) πD R α α s π (cid:18) e πD Rα α s q m red E − (cid:19) − E > . (3.28)If the potential is attractive, D R α <
0, there is a sum of bound states below thresholdgiven by J bound R α ( E ) = 2 ∞ X n =1 δ ( E − E n ) (cid:18) m red ( − D R α ) α s n (cid:19) E < E n = − m red α s D R α n . (3.30)A series representation of the imaginary part of the Coulomb Green function for finitewidths and arbitrary energies can be found in [23].The above results suffice for resummation with NLL accuracy as performed in section 4.For NNLL accuracy in the counting (2.7) one has to resum in addition all α s × ( α s /β ) n corrections as well as the non-relativistic logarithms of the form α s ln β , α s ln β, . . . . Thefixed-order NNLO corrections of this form have been obtained in [45]. An analyticalresult resumming all α s × ( α s /β ) n terms was obtained in [76] and is quoted e.g. in [25].Resummation of the non-relativistic logarithms requires the generalization of results fortop-quark pair production obtained e.g. in [83, 84] to arbitrary colour representations. In the momentum-space approach to threshold resummation one calculates the short-distance coefficients C ( i ) ( µ ) at a hard scale µ h ∼ M and the soft function W ( ω, µ ) ata soft scale of the order of µ s ∼ M β . Threshold logarithms log β are resummed by usingevolution equations to evolve both functions to an intermediate factorization scale µ f . Inthis subsection we will provide these evolution equations in the colour basis (3.2) that diag-onalizes the soft function. In this case the structure of the evolution equations is similar tothose in the Drell-Yan process [33, 35] and the complications of matrix-valued anomalous35imensions depending on the kinematics [7, 36] are avoided. The resummed cross sectionobtained from solving the equations is given in 3.5.The evolution equation of the hard coefficients has been obtained in [37] from the resultsof [38] for the IR structure of general massive QCD amplitudes: dd ln µ C ( i ) { α } ( µ ) = (cid:18) γ cusp (cid:20) ( C r + C r ′ ) (cid:18) ln (cid:18) M µ (cid:19) − iπ (cid:19) + iπC R α (cid:21) + γ Vi (cid:19) C ( i ) { α } ( µ ) . (3.31)Here Casimir scaling was used to write the cusp anomalous dimension for a massless partonin the representation r in the form Γ r cusp = C r γ cusp which holds at least up to the three-looporder. At least up to the two-loop level the anomalous dimension γ Vi can be written interms of single-particle anomalous dimensions: γ Vi = γ r + γ r ′ + γ R α H,s . (3.32)The anomalous dimension γ R α H,s is related to a massive particle in the final state representa-tion R α in the pair P i = ( r ′ α , R α ) defining the colour basis element c ( i ) with index i . Whilethe one- and two-loop anomalous-dimension coefficients γ r of massless quarks and gluonshave been available for some time, the two-loop results for the heavy particle soft anoma-lous dimension γ R α H,s have become available only recently [37, 41]. The one-loop coefficientsof the cusp and soft anomalous dimensions are simply γ (0)cusp = 4 and γ R α (0) H,s = − C R α .The anomalous dimensions γ r related to the light partons do not obey Casimir scalingalready at one loop. The explicit one- and two-loop results for all remaining anomalousdimensions appearing in this section are available in the literature and have been collectedin [37]. We observe that, as noted for the production of a colour octet final state particlein [21], the imaginary part in the evolution equation (3.31) can not simply absorbed in theargument of the logarithm by the continuation M → − ( M + i
0) which complicates theresummation of “ π -enhanced” terms compared to colour singlet final states as in Higgsproduction [34, 35]. We will not consider such a resummation of constant terms here. Theevolution equation of the hard-functions is obtained from (3.31) as dd ln µ H Si ( µ ) = (cid:18) γ cusp ( C r + C r ′ ) ln (cid:18) M µ (cid:19) + 2 γ Vi (cid:19) H Si ( µ ) . (3.33)As discussed in [37] and section 2.5.2, starting from the two-loop order there are IRdivergent contributions to the short-distance coefficients that are related to UV divergencesof insertions of the non-Coulomb PNRQCD potentials in the extended factorization for-mula (2.66). These divergences cause additional scale dependence, not included in (3.33),which is cancelled by a non-trivial scale-dependence of the potential function including thenon-Coulomb potential insertions. The factorization scale dependence in the separationof H and J is related to additional non-relativistic log β terms of the NNLL order. Sincein this paper we do not consider the resummation of these non-relativistic logarithms, inorder to obtain the evolution equation of the soft function, these contributions to the scale36ependence of the hard function have to be dropped as done in (3.33). Beyond NNLL fur-ther complications can arise from the structure of the extended factorization formula (2.66)including terms with higher-dimensional soft functions, but the discussion of resummationbeyond NNLL is beyond the scope of this paper.The evolution equation of the soft function can be obtained from that of the hardfunction using the factorization scale independence of the hadronic cross section. Consis-tent with our discussion above, we consider the potential function to be scale independentwhich is appropriate for the NLO potential function quoted in [25]. Scale invariance of thetotal cross section and the known factorization scale dependence of the PDFs implies theevolution equation of the partonic cross section dd ln µ ˆ σ pp ′ ( z, µ ) = − X ˜ p, ˜ p ′ Z z dxx (cid:0) P p/ ˜ p ( x ) + P p ′ / ˜ p ′ ( x ) (cid:1) ˆ σ ˜ p ˜ p ′ ( z/x, µ ) , (3.34)where P p/ ˜ p ( x ) are the Altarelli-Parisi splitting functions for the splitting of a parton p intoa parton ˜ p . Consistent with the derivation of the factorization formula in section 2 fromthe on-shell scattering process pp ′ → HH ′ X at production threshold, we use the x → p in the colour representation r , P p/ ˜ p ( x ) = (cid:18) r cusp ( α s ) 1[1 − x ] + + 2 γ φ,r ( α s ) δ (1 − x ) (cid:19) δ p ˜ p + O (1 − x ) . (3.35)As discussed in section 2.5.2, subleading collinear terms could potentially be enhancedby the Coulomb singularity, so care in dropping higher order corrections in equations likethis is required. In the present case, however, the corrections to (3.35) are of the order1 − z = β so they they are not relevant at NNLL accuracy. We recall the property of theplus distribution, Z z dx f ( x ) (cid:20) − x ) (cid:21) + = Z z dx f ( x ) − f (1)(1 − x ) − Z z dx f (1)(1 − x ) . (3.36)Inserting the factorized form of the partonic cross section into (3.34) and making useof the evolution equation of the short-distance coefficients (3.31) leads to an evolutionequation of the soft function. Using the relations ˆ s ≈ M , (1 − z/x ) ≈ (1 − z ) − (1 − x )valid in the x → z → ω = 4 M (1 − x ), theresulting equation can be written in the same form as the equation found for the Drell-Yanprocess [33]: dd ln µ W R α i ( ω, µ ) = − (cid:20)(cid:16) Γ r cusp + Γ r ′ cusp (cid:17) ln (cid:18) ωµ (cid:19) + 2 γ R α W,i (cid:21) W R α i ( ω, µ ) − (cid:16) Γ r cusp + Γ r ′ cusp (cid:17) Z ω dω ′ W R α i ( ω ′ , µ ) − W R α i ( ω, µ ) ω − ω ′ , (3.37)with the anomalous-dimension coefficient γ R α W,i = γ Vi + γ φ,r + γ φ,r ′ . (3.38)37hese results hold for the colour basis (3.2) that diagonalizes the soft function to allorders [37], where only the diagonal elements W i = W ii of the soft function have to beconsidered. Analogously to the anomalous dimension γ Vi of the hard function (3.32), atleast up to the two-loop level the anomalous-dimension coefficient of the soft function (3.38)can be written in terms of separate single-particle contributions γ R α W,i = γ R α H,s + γ rs + γ r ′ s (3.39)with γ rs = γ r + γ φ,r . (3.40)The anomalous-dimension coefficients γ rs vanish at one-loop level so that γ R α (0) W,i = γ R α (0) H,s = − C R α . Taking the Fourier transform of the evolution equation of the position space softfunction given in (3.30) of [37] one obtains the momentum-space evolution equation quotedin (3.37). In order to obtain this result, the logarithmic term multiplied by Γ cusp has to betreated with care, for instance by writing ln x = lim ǫ → ǫ (1 − x − ǫ ) (see also [28]).The terms proportional to the cusp anomalous dimensions in the evolution equationsare related to the resummation of double logarithms ln β , while the coefficients γ Vi and γ R α W,i are related to single logarithms. For LL summation one needs the one-loop cuspanomalous dimension while all other quantities enter at LO. For an NLL resummationthe required ingredients are the two-loop cusp anomalous dimensions and the one-loopanomalous dimensions γ Vi and γ R α W,i . For a NNLL resummation one needs in addition tothe three-loop cusp anomalous dimensions and the two-loop anomalous dimensions, theone-loop soft function (3.9) and the one-loop hard function (3.15).
As mentioned in the beginning of section 3.4, the resummation of soft-gluon corrections inthe approach of [31–33] is performed by calculating the hard and soft functions at scales µ h and µ s that minimize the radiative corrections to these quantities and subsequently usingthe renormalization group equations (3.33) and (3.37) to evolve to a common scale µ f to compute the partonic cross section (2.63) and perform the convolution with the PDFsevaluated at the same scale (see figure 1).Since we have determined a colour basis that diagonalizes the soft function to all ordersin perturbation theory for all hadron collider processes of interest, the evolution equationshave the same form as those for the Drell-Yan process [33]. The resummed hard functionsolving the evolution equation (3.33) is given by H Si ( µ ) = exp[4 S ( µ h , µ ) − a Vi ( µ h , µ )] (cid:18) M µ h (cid:19) − a Γ ( µ h ,µ ) H Si ( µ h ) , (3.41)with the functions S , a Vi and a Γ defined as [33] S ( µ h , µ ) = − Z α s ( µ ) α s ( µ h ) dα s Γ r cusp ( α s ) + Γ r ′ cusp ( α s )2 β ( α s ) Z α s α s ( µ h ) dα ′ s β ( α ′ s ) , s µ f f ( µ f ) f ( µ f ) H ( µ f ) J R α W R α ( ω, µ f ) µ hW R α ii ′ ( ω, µ s ) H ( µ h ) Figure 1: Sketch of the resummation of soft gluon corrections using RGEs. a Γ ( µ h , µ ) = − Z α s ( µ ) α s ( µ h ) dα s Γ r cusp ( α s ) + Γ r ′ cusp ( α s )2 β ( α s ) ,a Vi ( µ h , µ ) = − Z α s ( µ ) α s ( µ h ) dα s γ Vi ( α s ) β ( α s ) . (3.42)Here α s ( µ ) represents the QCD coupling in the MS scheme and β ( α s ) the corresponding β -function. Explicit results for the functions β , a Γ and the Sudakov exponent S up to theNLL order as needed for section 4 are collected in appendix C; expressions up to the N LLorder can be found in [33].The evolution equation of the soft function (3.37) can be solved in momentum space [31,32] by introducing the Laplace-transform with respect to the variable s = 1 / ( e γ E µe ρ/ ),˜ s R α i ( ρ, µ ) = Z ∞ − dωe − sω W R α i ( ω, µ ) , (3.43)where we have defined the MS-renormalized soft function W R α i . In practice, it is notnecessary to perform the Laplace transform explicitly, since the soft function in positionspace ˆ W R α i ( z , µ ) depends on the arguments solely through the combination [28, 29] iz µe γ E ≡ e L/ . (3.44)It is then easy to see that the function ˜ s R α i ( ρ ) is obtained by simply replacing L → − ρ inthe MS-renormalized result for the soft function in position space. From the results of thetree-level and one-loop soft function in the diagonal basis (3.2) quoted in (3.9) we obtain,discarding the 1 /ǫ poles due to the MS subtraction,˜ s (0) R α i ( ρ, µ ) = 1 , (3.45)˜ s (1) R α i ( ρ, µ ) = ( C r + C r ′ ) (cid:18) ρ + π (cid:19) − C R α ( ρ − . (3.46)39he resummed soft function can be written in terms of the Laplace transform with theargument ρ replaced by the derivative with respect to an auxiliary variable η that is set to η = 2 a Γ ( µ s , µ ) after performing the derivatives: W R α , res i ( ω, µ ) = exp[ − S ( µ s , µ ) + 2 a R α W,i ( µ s , µ )]˜ s R α i ( ∂ η , µ s ) 1 ω (cid:18) ωµ s (cid:19) η θ ( ω ) e − γ E η Γ(2 η ) . (3.47)Here the functions a R α W,i are defined in analogy to the functions (3.42) with the obviousreplacement of the anomalous dimensions γ Vi → γ R α W,i . For η < µ f weobtain the result for the resummed cross sectionˆ σ res pp ′ (ˆ s, µ f ) = s + s ′ X S = | s − s ′ | X i H Si ( µ h ) U i ( M, µ h , µ s , µ f ) (cid:18) Mµ s (cid:19) − η × ˜ s R α i ( ∂ η , µ s ) e − γ E η Γ(2 η ) Z ∞ dω J SR α ( E − ω ) ω (cid:18) ωµ s (cid:19) η = s + s ′ X S = | s − s ′ | X i H Si ( µ h ) U i ( M, µ h , µ s , µ f ) × Z ∞ dω J SR α ( E − ω ) ω (cid:16) ω M (cid:17) η ˜ s R α i (cid:18) (cid:18) ωµ s (cid:19) + ∂ η , µ s (cid:19) e − γ E η Γ(2 η ) , (3.48)where have defined the evolution function U i ( M, µ h , µ f , µ s ) = (cid:18) M µ h (cid:19) − a Γ ( µ h ,µ s ) (cid:18) µ h µ s (cid:19) η × exp h S ( µ h , µ f ) − S ( µ s , µ f )) − a Vi ( µ h , µ s ) + 2 a φ,r ( µ s , µ f ) + 2 a φ,r ′ ( µ s , µ f ) i (3.49)and now η = 2 a Γ ( µ s , µ f ). The sum over the final state representations R α in the factor-ization formula (2.63) has disappeared in the colour basis (3.2) since there is a uniquefinal state representation for each term in the sum over i (see the expression (3.7) for thediagonal soft function). The evolution function U i has been simplified using the identity a Γ ( µ h , µ f ) + a Γ ( µ f , µ s ) = a Γ ( µ h , µ s ) . (3.50)It could be further simplified using the identity [32] S ( µ h , µ f ) − S ( µ s , µ f ) = S ( µ h , µ s ) − a Γ ( µ s , µ f ) ln µ h µ s (3.51)40o obtain U i ( M, µ h , µ f , µ s ) = (cid:18) M µ h (cid:19) − a Γ ( µ h ,µ s ) × exp h S ( µ h , µ s ) − a Vi ( µ h , µ s ) + 2 a φ,r ( µ s , µ f ) + 2 a φ,r ′ ( µ s , µ f ) i , (3.52)which makes it clear that the µ f dependence is related to the parton density functions.Eq. (3.51) is formally satisfied at whatever N k LL accuracy employed, but not strictly validin an expansion in the strong coupling since S and a Γ are evaluated at different orders. Inthe numerical evaluation we do not use this simplification but use (3.49).A similar convolution of the resummed Coulomb Green function as in (3.48) was en-countered in [61] where the mixed Coulomb and one-loop soft corrections to W -pair pro-duction were considered. The formula (3.48) generalizes the result in [61] to QCD andby resumming soft-gluons to all orders. An analogous formula was already used in [64] toresum QED corrections of the form ln(Γ W /M W ) for W -pair production in electron-positroncollisions.At the NLL order only the trivial, LO O ( α s ) soft function, the spin-independent LOpotential function and the leading-order hard functions are required so that the resummedcross section is given byˆ σ NLL pp ′ (ˆ s, µ f ) = X i H (0) i ( µ h ) U i ( M, µ h , µ s , µ f ) e − γ E η Γ(2 η ) Z ∞ dω J R α ( E − ω ) ω (cid:16) ω M (cid:17) η . (3.53)As mentioned before, for η < ω (cid:0) ω M (cid:1) η should be understood in the distri-bution sense [32]. Neglecting heavy particle decay and bound-state effects, the potentialfunction J R α vanishes for E < ω integral is defined as the star-distribution [32] Z E dω f ( ω ) (cid:20) ω (cid:16) ω M (cid:17) η (cid:21) ∗ = Z E dω f ( ω ) − f (0) − ωf ′ (0) ω (cid:16) ω M (cid:17) η + (cid:20) f (0)2 η + 2 E η + 1 f ′ (0) (cid:21) (cid:18) EM (cid:19) η , (3.54)where the double subtraction is sufficient to render the integral convergent for η > − η = 0 and η = − show up explicitly in the secondline but are cancelled by the overall prefactor 1 / Γ(2 η ) in (3.48) and (3.53).Note that the all-order solution (3.48) to the RGEs does not depend on the hard and softscales µ h and µ s , but truncating the perturbative expansion at a finite order in α s introducesa residual dependence on these scales, which is of higher order in α s . As pointed outin [32, 33] the standard resummation formula in Mellin space following [1, 2] corresponds tothe implicit scale choices µ h = µ f and µ s = M/N for the N -th Mellin moment. The explicitappearance of these scales in the momentum-space formalism allows to use the residualscale dependence of the resummed cross section to estimate the remaining uncertaintiesfrom uncalculated higher order corrections. Our choices for these scales and the estimatesfor the remaining scale uncertainties are discussed in section 4.2.41 .6 Matching to the fixed-order NLO calculation For the implementation of the NLL resummed cross section, we match to the fixed-orderNLO calculation valid outside the threshold region. The matched result for the partoniccross section is then given byˆ σ matched pp ′ (ˆ s ) = h ˆ σ NLL pp ′ (ˆ s ) − ˆ σ NLL(1) pp ′ (ˆ s ) i θ (Λ − [ √ ˆ s − M ]) + ˆ σ NLO pp ′ (ˆ s ) ≡ ∆ˆ σ NLL pp ′ (ˆ s, Λ) + ˆ σ NLO pp ′ (ˆ s ) , (3.55)where ˆ σ NLO pp ′ (ˆ s ) is the fixed-order NLO cross section obtained in standard perturbationtheory and ˆ σ NLL(1) pp ′ is the resummed cross section expanded to NLO. The result is givenexplicitly in (D.1) in appendix D. Since the resummed NLL cross section is expected tobe a good approximation to the total cross section only near the threshold, we allowed fora cutoff E = √ ˆ s − M <
Λ to switch off the resummation outside the threshold region, aswas done in [14]. In the calculation of squark-antisquark production we choose, however,not to introduce this cut-off as discussed in section 4.2.The total hadronic cross section at NLL is then obtained by convoluting (3.55) withthe parton luminosity, σ matched N N → HH ′ X ( s ) = X p,p ′ = q, ¯ q,g Z M /s dτ L pp ′ ( τ, µ f ) ˆ σ matched pp ′ ( sτ, µ f ) , (3.56)where the parton luminosity is defined in terms of the PDFs in (2.88). In this section we use the results of section 3 to perform the NLL resummation for squark-antisquark production in proton-proton and proton-antiproton collisions. According tothe systematics introduced in (2.7), NLL accuracy requires the combined resummation ofCoulomb gluons and soft gluons. Therefore our results extend previous ones on squark-antisquark production at higher order [17–20], that treated soft and Coulomb resummationsseparately or used a fixed-order expansion. In section 4.1 we perform the matching to theeffective theory, and compare the threshold approximation of the LO and NLO cross sec-tions to the full results. In 4.2 we discuss our choices for the hard, soft and Coulomb scalesappearing in the resummed cross section and present predictions for squark-antisquarkproduction at the Tevatron and the LHC at cms energies of 7 ,
10 and 14 TeV. We alsocompare to results from soft gluon resummation in the Mellin-moment approach [20].
In the following we perform the matching from the MSSM to an effective theory containingonly (anti-)collinear partons and non-relativistic squarks and antisquarks, and derive the42ard functions for the partonic subprocesses. As discussed in section 3.2, the hard functionsrequired for NLL resummation can be inferred from the leading-order cross sections for thecolour-singlet and octet production channels [18, 19]. Therefore the explicit matching tothe EFT is not strictly necessary, but is included here in some detail in order to provide anillustration for the somewhat abstract discussion in section 2. We also use the general resultfor the threshold expansion at NLO (D.3) to reproduce the known threshold behaviour ofthe NLO cross section [85].We consider squark-antisquark production at hadron colliders, which at leading orderproceeds through quark-antiquark and gluon-induced subprocesses: q i ( k )¯ q j ( k ) → ˜ q σ k ( p )˜ q σ l ( p ) g ( k ) g ( k ) → ˜ q σ i ( p )˜ q σ j ( p ) , (4.1)where i, j, k, l denote the quark/squark flavours and σ , = L/R label the scalar partnersof the left/right-handed quarks. The relevant Feynman diagrams are shown in figure 2.We follow the setup of [85] and take all squarks to be mass degenerate and do not considertop squarks. The first restriction is done for simplicity (and was also used in other recentworks on higher-order soft-gluon effects [17–20]) and is not essential in our formalism, sincethe results in section 3 include the case of different final state masses. The application totop squarks would require an extension of our framework, since in the quark-antiquarkproduction channel they are produced in a P -wave [86]. Resummation for top squarks wasperformed recently in [87] assuming that the resummation formalism for S -wave productioncan be applied to P -wave dominated processes. Since, in the present work, we are concernedwith the total cross section, there are no sizable corrections from the finite squark widthΓ as long as Γ ≪ M unlike the case of the invariant mass distribution near threshold.Contributions to the cross section from below the nominal production threshold will beincluded through the bound-state contribution to the Coulomb Green function. From arecent study of threshold effects in gluino production [26] we expect additional finite-widtheffects at most of the order of these bound state corrections.To obtain the leading-order production operators we evaluate the tree-level scatteringamplitudes for the processes (4.1) at the production threshold, ( k + k ) = 4 m q , anddetermine the short-distance coefficients from the matching condition (2.17) in order tocompute the hard functions (2.60). The tree-level scattering amplitude for quark-antiquark induced squark-antisquark pro-duction at threshold is dominated by t -channel gluino exchange, if quark and squark (andantiquark and antisquark) flavours are identical. Near threshold the full-theory tree-levelscattering amplitude for the quark-antiquark channel is i A (0) { a } ( q i ¯ q j → ˜ q kL ˜ q lR ) = − ig s m ˜ g m q + m g δ ik δ jl T ba a T ba a ¯ v ( m ˜ q ¯ n ) P L u ( m ˜ q n ) . (4.2)43 ˜ q ˜ g ˜ q ¯ q , q ˜ q ¯ q g ˜ q g ˜ qg g ˜ q , g ˜ q ˜ qg ˜ q , g ˜ q ˜ qg ˜ q , g ˜ qg ˜ q Figure 2: Leading-order Feynman diagrams for the quark-antiquark (top) and gluon-fusion(bottom) induced production of squark-antisquark pairs.Here we use the usual projectors P R/L = (1 ± γ ) /
2. There are analogous expressions withleft and right labels exchanged.In the effective theory we introduce fields ψ i ( ψ ′ i ) that annihilate a left squark (rightantisquark) of flavour i . The quark fields are given by the SCET fields (2.25). We canthen match the amplitude onto the effective theory according to the condition (2.17), withthe production operator given by (no sum over flavours is implied) O (0) { a ; α } = (cid:2) ( ¯ ξ j,c ; α W c ) a ( W † c ξ i,c ; α ) a (cid:3) (cid:16) ψ † i ; a ψ ′ † j ; a (cid:17) . (4.3)Here ξ i,c and ξ i, ¯ c are the SCET fields describing collinear and anticollinear quarks withflavour i . From the matching condition (2.17) we can read off the matching coefficient,taking the non-relativistic normalization factor 2 m ˜ q into account: C q ¯ q, { a ; α } = − g s m ˜ g m ˜ q ( m q + m g ) T ba a T ba a ( P L ) α α . (4.4)We now introduce the decomposition of the coefficients into a colour basis (2.14), butleave the spin indices open. For the case at hand, the colour basis has been given alreadyin (2.16). We find for the matching coefficients for the two colour channels: C (1) q ¯ q, { α } = ( − C F ) 4 πα s m ˜ g m ˜ q ( m q + m g ) ( P L ) α α ,C (2) q ¯ q, { α } = r C F N C πα s m ˜ g m ˜ q ( m q + m g ) ( P L ) α α . (4.5)From the definition (2.60) and the polarization sum (A.7) we obtain the diagonal elements44f the hard function , H (0) ii = 18 m q h (0) i (cid:18) πα s m ˜ g m ˜ q ( m q + m g ) (cid:19) m q N c tr (cid:20) n/ n/ P L (cid:21) = 2 h (0) i (cid:18) πα s m ˜ g N c m ˜ q ( m q + m g ) (cid:19) , (4.6)with colour factors h (0)1 = C F , h (0)2 = C F N c (4.7)for the two colour channels. From (3.13) we obtain the total partonic Born cross sectionat threshold ˆ σ (0) q i ¯ q j → ˜ q iL ˜ q jR | ˆ s =4 m ˜ q = C F β πN c (cid:18) πα s m ˜ g ( m q + m g ) (cid:19) . (4.8)Adding the same expression with left and right squark labels exchanged reproduces theresult given e.g. in [85]. In figure 3 we compare the partonic cross section at threshold tothe full partonic LO cross section [85], for a squark mass of 1 TeV at the LHC with 14 TeVcms energy. We plot the integrand of the convolution in the formula for the hadronic crosssection (2.89), i.e. the product of the partonic cross section and the parton luminosity, asa function of β , taking the Jacobian ∂τ∂β into account: dσ (0) q ¯ q → ˜ q L ˜ q R dβ = 8 βm q s (1 − β ) L q ¯ q ( β, µ f )ˆ σ (0) q ¯ q → ˜ q L ˜ q R . (4.9)In the plot we use the MSTW2008LO PDFs [88] with µ f = m ˜ q . It can be seen that thethreshold approximation is applicable for β . .
3, while the main contribution to the crosssection comes from the region β ≈ .
6. The results shown here are for a gluino-squarkmass ratio of 1 .
25. For other gluino masses the discrepancies in the region β > . α s . The resulting expression can be written in theform ∆ σ (1) pp ′ = σ (0) pp ′ α s π f (1) pp ′ + O ( β ), where the scaling functions f (1) pp ′ for an arbitrary colourchannel are given explicitly in (D.3). Summing up the two colour channels we obtain thethreshold expansion of the scaling function for the quark-antiquark channel f (1) q ¯ q = π ( N c − N c β + 8 C F (cid:20) ln (cid:18) m ˜ q β µ (cid:19) + 8 − π (cid:21) − N c − N c ln (cid:18) m ˜ q β µ (cid:19) + 12 N c + h (1) q ¯ q ( µ ) + O ( β ) . (4.10)The logarithmically enhanced terms and the Coulomb correction agree with the resultsof [85]. The one-loop hard coefficient h (1) is currently unknown in analytical form. In theright-hand plot in figure 3 we plot the full NLO corrections to the partonic cross section For simplicity, in (4.6) and (4.15) we set 1 / (2ˆ s ) ∼ / (8 m q ). However, in our numerical implementationof the resummed cross section (3.53) this prefactor is kept unexpanded. .2 0.4 0.6 0.8 1.0 Β Σ H L d Β @ pb D Β DΣ NLO d Β @ pb D Figure 3: Partonic cross sections for q i ¯ q j → ˜ q k ¯˜ l q , multiplied with the quark-antiquarkluminosity and summed over quark and squark flavours, for m ˜ q = 1 TeV and m ˜ g =1 .
25 TeV at the 14 TeV LHC. Left panel: exact leading order result (black/solid) andthreshold approximation (blue/dashed). Right panel: exact NLO result (black/solid) andthe approximation based on (4.10) with h (1) q ¯ q ( µ ) = 0, using the threshold approximation ofthe tree (red/dot-dashed) and the full tree (blue/dashed).obtained from the parameterization given in [19] (solid/black) to the threshold approxima-tion (4.10) with the constant h (1) q ¯ q set to zero. In the red/dot-dashed curve we multiply thecorrection (4.10) with the threshold approximation of the tree, in the blue/dashed curve weuse the full tree cross section. It can be seen that the NLO corrections are peaked closerto threshold than the tree cross section, as might be expected. Multiplying the scalingfunction by the full tree leads to an improved agreement with the full NLO result. Overallthe approximation of the full NLO result by the threshold expansion is quite good, in thesense that integrating the approximated partonic cross section captures the bulk part ofthe exact result. For the gluon induced process the s -channel diagram is again P -wave suppressed, so thedominant contribution comes from the t - and u -channel diagrams and the quartic vertex.These diagrams give i A (0) { a } ( gg → ˜ q iR ˜ q jR ) = ig s δ ij { T a , T a } a a ( ǫ µ g ⊥ µν ǫ ν ) , (4.11)with the transverse metric g ⊥ µν = g µν − ( n µ ¯ n ν + ¯ n µ n ν ). This form of the scatteringamplitude at threshold is reproduced by the matrix element of the effective theory operator(no sum over flavours is implied) O (0) { µ ; a } = A ⊥ c ; a µ A ⊥ ¯ c ; a µ ψ † i ; a ψ ′ † i ; a . (4.12)46ere it was used that the polarization vector corresponding to a free SCET gluon fields A ⊥ is given by ǫ µ ⊥ = g ⊥ µν ǫ ν . This can be seen from (2.26) using a Fourier transformationafter setting the strong coupling to zero, so that W c = 1 and the non-abelian terms in thefield strength vanish.To introduce the colour basis, note that the gluon-gluon system admits the decomposi-tion 8 ⊗ ⊕ s ⊕ a ⊕ ⊕ ⊕
27. The squark-antisquark pair is again either in a singletor octet state, but this time the latter can be produced by two different initial colour-octetstates. The colour basis obtained for this case from the prescription (3.2) agrees with theone given in [7], c (1) { a } = 1 √ N c D A δ a a δ a a c (2) { a } = 1 √ D A B F d ba a T ba a c (3) { a } = r N c D A F ba a T ba a , (4.13)where d abc are the usual symmetric invariant tensors of SU (3) and F ba a = if a ba . We alsodefined the coefficients D A = N c − B F = N C − N C = . Thus, the matching coefficientsfor the three colour channels are (including the non-relativistic normalization 2 m ˜ q ) C (0 , gg, { µ } = p C F πα s m ˜ q g ⊥ µ µ ,C (0 , gg, { µ } = p D A B F πα s m ˜ q g ⊥ µ µ ,C (0 , gg, { µ } = 0 . (4.14)The diagonal elements of the hard function are given by H (0) ii = 18 m q h (0) i (cid:18) πα s m ˜ q (cid:19) D A g ⊥ µµ = (cid:18) πα s D A m q (cid:19) h (0) i (4.15)with the colour factors h (0)1 = 2 C F , h (0)2 = 2 D A B F , h (0)3 = 0 . (4.16)Using P i h (0) i = C F ( N c −
2) gives for the total partonic cross section at thresholdˆ σ (0) g ¯ g → ˜ q iL ˜ q iR | ˆ s =4 m ˜ q = πα s β m q N c − N c ( N c − . (4.17)Adding the same expression for left and right squark labels exchanged again reproducesthe result of [85]. In figure 4 we compare the partonic cross section at threshold to thefull partonic LO cross section [85], for a squark mass of 1 TeV at the LHC with 14 TeV47 .0 0.2 0.4 0.6 0.8 1.0 Β Σ H L d Β @ pb D Β DΣ NLO d Β @ pb D Figure 4: Partonic cross sections for gg → ˜ q ¯˜ q multiplied with the gluon-gluon luminosityfor m ˜ q = 1 TeV at the 14 TeV LHC. Left panel: exact leading-order result (black/solid)and threshold approximation (blue/dashed). Right panel: exact NLO result (black/solid)and the approximation (4.18) with h (1) q ¯ q ( µ ) = 0, using the threshold approximation of thetree (red/dot-dashed) and the full tree (blue/dashed).cms energy. As for the quark-antiquark induced subprocess we plot the product of thepartonic cross section with the parton luminosity and the Jacobian ∂τ∂β . Again the thresholdapproximation is adequate up to β ∼ . β ≈ . f (1) gg = π ( N c + 2) N c ( N c −
2) 1 β + 8 N c (cid:20) ln (cid:18) m ˜ q β µ (cid:19) + 8 − π (cid:21) − N c N c − N c − (cid:18) m ˜ q β µ (cid:19) + 12 N c N c − N c − h (1) gg ( µ ) + O ( β ) . (4.18)The β -dependent terms agree with [85]. Again, the hard corrections h (1) gg are currently notavailable analytically. In the right plot of figure 4 we compare the full NLO correctionin the parameterization of [19] (black/solid) to the threshold approximation of the NLOcorrection (4.18) with h (1) gg = 0, using the threshold approximation of the tree (red/dot-dashed) or the full tree cross section (blue/dashed) as pre-factor. In this case, the thresholdapproximation gives a better agreement with the full NLO result than for the quark-antiquark initial state. Using the full tree further improves the agreement with the full NLOcorrection in the full β -range. Once again the integrated threshold approximation providesa very good approximation to the exact NLO correction, especially when multiplying (4.18)by the full Born cross section. This can be compared to the Born cross section itself, forwhich the leading term in the β -expansion is typically a less reliable approximation.48 .2 Results for squark-antisquark production at the LHC andTevatron We are now ready to perform the combined soft and Coulomb NLL resummation for squark-antisquark production. We first discuss our choice for the soft, hard and Coulomb scalesentering the resummed cross section, and then present numerical results for the LHC andthe Tevatron. We also compare to the results of a soft gluon resummation in Mellin spacepresented in [20].
Evaluating the NLL resummed cross section (3.53) requires a choice for the soft, hard andCoulomb scales. As can be seen from the Sudakov logarithm in the evolution equation ofthe hard function (3.33), the natural scale of the hard corrections is of the order µ h ∼ M .Therefore we will choose the default value of the hard scale to be ˜ µ h = 2 m ˜ q . Analogously,the form of the approximate NLO corrections (4.10) and (4.18) implies that the scale elim-inating large logarithms in the soft corrections to the partonic cross section is given by µ s ∼ E ≈ M β . However, it was argued in [32, 33] that this choice leads to similarproblems as the inversion of Mellin-space resummation formulae for the partonic cross sec-tion [10], i.e. an ill-defined convolution with the parton luminosity. Here we follow [33] andchoose a soft scale such that it minimizes the relative fixed-order one-loop soft correctionto the hadronic cross section. More precisely, we vary the scale in the PDFs and the softcorrection and determine the value ˜ µ s that minimizes the relative soft corrections:0 = dd ˜ µ s X p,p ′ Z m q /s dτ L pp ′ ( τ, ˜ µ s ) ˆ σ (1) pp ′ , soft ( τ s, ˜ µ s ) σ (0) N N ( s, ˜ µ s ) . (4.19)Here the fixed-order NLO soft correction ˆ σ (1) pp ′ soft is obtained from (4.10) and (4.18), settingthe Coulomb correction and the hard corrections h i to zero, and we divide by the leading-order hadronic cross section. This procedure results in values˜ µ s = 123 −
455 GeV for m ˜ q = 200 − √ s = 14 TeV. The fact that the soft scale is over half of the massfor light squarks, but considerably smaller for larger masses, indicates that the thresholdresummation is more important for heavier squarks, as expected. We can approximatelyfit the numerical results for the soft scale by a function of the form proposed in [33],˜ µ LHC s ≈ m ˜ q (1 − ρ ) √ . ρ , (4.21)with ρ = 4 m q /s , which is accurate to better than 5% for m ˜ q >
450 GeV. We find that fordifferent cms energies √ s the ratio ˜ µ LHC s /m ˜ q is to a good approximation a function of ρ only, that is, for a squark of mass 500 GeV at 7 TeV cms energy the ratio is approximately49 .6 0.8 1 1.2 1.5 2 Μ s (cid:144) ΜŽ s DΣ NLL @ Μ s D(cid:144) DΣ NLL @ ΜŽ s D m q Ž = m q Ž = m q Ž =
500 GeV Μ h (cid:144) ΜŽ h DΣ NLL @ Μ h D(cid:144) DΣ NLL @ ΜŽ h D m q Ž = m q Ž = m q Ž =
500 GeV
Figure 5: Dependence of the NLL corrections at the 14 TeV LHC on the soft scale (leftpanel) and the hard scale (right panel), normalized to the corrections at the default scales˜ µ s determined from (4.19) and ˜ µ h = 2 m ˜ q . Black (solid): m ˜ q = 500 GeV, blue (dashed): m ˜ q = 1 TeV, red (dot-dashed): m ˜ q = 2 TeVthe same as for a 1 TeV squark at 14 TeV. In our numerical NLL results, however, we usethe numerical values for the soft scale obtained directly by the condition (4.19). At theTevatron the values for the soft scale can be fitted by the function˜ µ TeV s ≈ m ˜ q (1 − ρ ) √ . . ρ , (4.22)and representative values for the default soft scale are˜ µ TeV s (200GeV) = 84GeV , ˜ µ TeV s (400GeV) = 86GeV , ˜ µ TeV s (600GeV) = 65GeV . (4.23)Note that the soft scale begins to decrease above a certain value of ρ .Figure 5 shows the dependence of the NLL corrections to the hadronic cross sectionat the LHC with 14 TeV on the soft and hard scales, varied around the default values ˜ µ s and ˜ µ h = 2 m ˜ q . Here by “NLL corrections” we mean the convolution of the expression∆ˆ σ NLL pp ′ (ˆ s, Λ) in (3.55) with the parton luminosity, i.e. the corrections on top of the exactNLO cross section due to the higher-order soft-gluon and Coulomb terms (bound-statecorrections are not included in these curves, and we set the cutoff Λ to the maximumpossible value). In the left panel, the hard scale is fixed to the default value µ h = ˜ µ h andwe vary the soft scale in the interval 0 . µ s . . . µ s . Analogously, in the right panel the softscale is fixed to the default value and we vary the hard scale in the interval m ˜ q . . . m ˜ q .The ambiguity due to the choice of the soft scale becomes smaller for larger masses, inagreement with the expectation that soft-gluon resummation is better justified for largermasses, where the contribution from the threshold region to the total cross section is moreimportant.We now address the choice of the scale used for the strong coupling constant in theCoulomb Green function. The only scale dependence of the imaginary part of the leading-50 .6 0.8 1 1.2 1.5 2 Μ C (cid:144) ΜŽ C DΣ NLL @ Μ C D(cid:144) DΣ NLL @ ΜŽ C D m q Ž = m q Ž = m q Ž =
500 GeV
Figure 6: Dependence of the NLL corrections at the 14 TeV LHC on the Coulomb scale,normalized to the corrections at the default scale ˜ µ C = max { m q β, µ B } . Black (solid): m ˜ q = 500 GeV, blue (dashed): m ˜ q = 1 TeV, red (dot-dashed): m ˜ q = 2 TeVorder Coulomb Green function (3.27) enters through the coupling constant of the leading-order Coulomb potential, so the potential function is separately renormalization-groupinvariant at NLL, and the coupling constant can be evaluated at a scale µ C differentfrom the factorization scale. Since the necessary truncation of the perturbative seriesfor the running coupling constant introduces a residual higher-order scale dependence,the scale µ C should be chosen such that higher-order corrections are minimized. Sincethe Coulomb corrections are related to the exchange of potential gluons with momentumtransfer | ~k | ∼ M √ λ , a natural scale choice µ C is expected to be of the order µ C ∼ M √ λ = M β ∼ M α s , (4.24)where we have used the NRQCD counting β ∼ α s . Indeed, as mentioned in section 2.5.2,the effect of the scale-dependent strength of the Coulomb potential can be incorporatedusing the choice µ C = 2 m ˜ q β . The choice µ C ∼ O ( M β ) is in fact required to sum correctlyall NLL terms, since µ C ∼ M would miss terms such as α s /β × ln β , which arise in partfrom the small virtuality of Coulomb gluons. However, very small values of β are integratedover in the convolution of the partonic cross section with the PDFs, and with the choice µ C ∝ β the strong coupling α s ( µ C ) would hit the Landau pole. On the other hand, therelevant scale for bound-state effects is set by the inverse Bohr radius of the first HH ′ bound state, 1 /r B = C F m ˜ q α s / ≡ µ B /
2. Therefore, as our default scale choice for theCoulomb Green function we use ˜ µ C = max { m ˜ q β, µ B } , (4.25)where we solve the equation µ B = C F m ˜ q α s ( µ B ) iteratively. The Bohr scale was used inrecent studies of threshold effects in top-pair or gluino production [24–26], while in therecent calculation [18] of squark-antisquark and gluino production the Coulomb scale wasset equal to the factorization scale. The dependence of the NLL corrections at the LHC51ith 14 TeV cms energy on the Coulomb scale (with all other scales fixed to their defaultvalues) is shown in Figure 6, where we vary the Coulomb scale in the interval 0 . µ C . . . µ C .The scale dependence is of a similar magnitude as for the soft and hard scales, and againimproves for larger masses. In the results given below, we estimate the uncertainty dueto the scale choices by varying all scales µ f , µ s , µ h and µ C from one-half to twice theirdefault value and add the uncertainties in quadrature. Having fixed the default scale values, we present numerical results for the combined NLLresummation of soft and Coulomb effects obtained by inserting the potential function (3.28)into the cross section (3.53). We use the MSTW2008 PDF set [88] and the associated valueof the strong coupling. The results for the fixed-order NLO cross section are obtained usingthe program Prospino , which is based on the calculation of [85]. In contrast to the defaultsetting of
Prospino , we include contributions from initial-state bottom quarks using b -quark PDFs. Unless stated otherwise, we use the squark-gluino mass ratio of m ˜ g = 1 . m ˜ q .As discussed in section 4.1, we consider the squarks to be mass-degenerate and do notinclude top squark final states. Our default value for the factorization scale is ˜ µ f = m ˜ q .In table 1 we compare the Prospino
NLO predictions obtained using the MSTW08and the CTEQ6.6M [89] PDF set for the Tevatron and the LHC at 14 TeV. While wefind good agreement for small squark masses, the differences between the two sets grow toover 10% for large masses. Since we expect very similar effects for the results includingNLL resummation, we present our predictions using the MSTW PDFs below, but theuncertainties due to differences between current PDF sets at large x should be taken intoaccount in interpreting our results.In figure 7 we show the NLL corrections to the partonic cross sections multiplied by theparton luminosities and the Jacobian ∂τ∂β . It can be seen that the dominant contributionsarise from the threshold region β ≤ .
2, and that there is only a small difference betweenusing the hard function obtained from the tree at threshold, as determined in section 4.1,or from the full tree, as mentioned in section 3.2. Thus, in the results presented below weuse the hard functions determined from the tree at threshold, and set the cutoff Λ in thematching formula (2.17) to the maximal value, so that the resummed correction is appliedin the full phase space. Another plausible choice is to switch off the NLL corrections atthe point where they turn negative [14]. In the example shown in figure 7 this occursat β ≈ .
4, where the validity of the threshold approximation is not immediately clear.Since the integrated contributions from the region β > . β andthat other choices might be required for other processes. There are some differences to preliminary results presented in [50, 51] due to a different choice of hardscale ( µ h = m ˜ q ) in the NLL results, and because the qg initiated subprocess was not included in the NLOcross section. ( p ¯ p → ˜ q ¯˜ q )(pb), √ s = 1 .
96 TeV m ˜ q [GeV] NLO MSTW
NLO
CTEQ
200 11 . +1 . − . . +1 . − .
300 6 . +1 . − . × − . +1 . − . × −
400 4 . +0 . − . × − . +0 . − . × −
500 2 . +0 . − . × − . +0 . − . × − σ ( pp → ˜ q ¯˜ q )(pb), √ s = 14 TeV m ˜ q [GeV] NLO MSTW
NLO
CTEQ
500 14 . +1 . − . . +1 . − . . +0 . − . × − . +0 . − . × − . +0 . − . × − . +0 . − . × − . +1 . − . × − . +0 . − . × − Table 1: NLO results for the squark-antisquark production cross section at the Teva-tron and the LHC at √ s = 14 TeV for m ˜ g = 1 . m ˜ q obtained with Prospino , usingMSTW08NLO and CTEQ6.6M PDFs. The error estimate is obtained by varying thefactorization scale in the interval m ˜ q / ≤ µ f ≤ m ˜ q Β- - DΣ NLL d Β @ pb D Β- - DΣ NLL d Β @ pb D Figure 7: NLL corrections to the partonic cross sections, multiplied with the parton lu-minosity, for q i ¯ q j → ˜ q k ¯˜ q l (left) and gg → ˜ q k ¯˜ q l at the 14 TeV LHC for m ˜ q = 1 TeV and m ˜ g = 1 .
25 TeV, using the threshold approximation of the tree (red, dot-dashed) and thefull tree (blue/dashed). 53
00 1000 1500 2000 2500 3000 m q Ž @ GeV D DΣ (cid:144) Σ NLO H % L NLL s + h NLL s + h + CNLL noBS
NLL
200 300 400 500 600 m q Ž @ GeV D DΣ (cid:144) Σ NLO H % L NLL s + h NLL s + h + CNLL noBS
NLL
Figure 8: NLL corrections to the squark-antisquark cross section at the LHC with √ s =14 TeV (above) and the Tevatron (below) relative to the NLO cross sections in variousapproximations. For explanations see text.In figure 8 we plot the K -factor relative to the NLO corrections,∆ K NLL = σ NLL − σ NLO σ NLO , (4.26)for the √ s = 14 TeV LHC and for the Tevatron, as a function of the squark mass, andconsider various approximations to our NLL result to study the effect of the Coulomb andsoft corrections separately. The different curves in the figure are defined as follows: NLL (black, solid): Results for the full NLL soft and Coulomb resummation using theresummed cross section (3.53) in the matching formula (3.55), including bound-stateeffects obtained from the potential function below threshold (3.30).
NLL noBS (green, dashed): Bound-state effects are omitted from the full NLL result.54 LL s+h (blue, dashed): NLL resummation without Coulomb corrections, i.e. the crosssection (3.53) with the trivial potential function J (0) ( E ) = m q π q Em ˜ q . NLL s+h +C (red, dot-dashed): The pure Coulomb corrections, obtained by using thetrivial soft function W (0) = δ ( ω ) and the resummed potential function (3.28) withoutbound state effects, are added to the NLL s + h approximation.It can be seen that the soft-Coulomb interference effects, given by the difference of theNLL noBS and NLL s+h +C curves, are sizable, in particular for large squark masses. Mostof this effect is due to the interference of the first Coulomb correction with the resummedsoft corrections; for instance at the 14 TeV LHC the relative correction at m ˜ q = 2 TeV inthis approximation is ∆ K NLL = 9 .
0% as opposed to ∆ K NLL = 9 .
7% in the full NLL softplus Coulomb resummation. The NLL s+h corrections are of a similar magnitude as the softNLL corrections obtained in [17, 18, 20] from resummation in Mellin space. A quantitativecomparison to these previous results will be performed below. The Coulomb correctionsare larger than those observed in [18]. This can be traced to the choice of the Coulombscale, which was taken to be µ C = µ f in that reference, while we use the smaller scale (4.25)required to sum all NLL terms, as discussed above.In tables 2 and 3 we compare the NLL resummed results pertaining to p ¯ p collisions atthe Tevatron cms energy √ s = 1 .
96 TeV and pp collisions at LHC with √ s = 7 ,
10 and14 TeV to the LO and NLO results obtained from
Prospino . For the LO predictions wehave used the MSTW2008LO set of PDFs, for the NLO and resummed predictions theMSTW2008NLO set. One observes a sizable reduction in the scale dependence for largersquark masses. This improvement can also be seen in figure 9. In the left-hand plots thescale dependence of the NLO cross section, the full NLL and the NLL s+h correction areplotted as a function of the squark mass for the Tevatron and the 14 TeV LHC. For theNLL results, the soft, hard, Coulomb scales and the factorization scale are varied and theuncertainties added in quadrature. Clearly, a significant reduction of the scale dependencerequires the inclusion of soft-Coulomb interference. In the right-hand plots in figure 9we plot the LO, NLO and NLL cross section as a function of the factorization scale for m ˜ q = 1 TeV at the 14 TeV LHC and for m ˜ q = 400 GeV at the Tevatron. One observes astabilization of the result with respect to variations of the factorization scale in going fromthe NLO to the NLL approximation if the factorization scale is varied in the usual interval0 . m ˜ q ≤ µ f ≤ m ˜ q , even after taking the variation of the soft, hard and Coulomb scales(shown as the green band) into account.In table 4 we compare to the NLL result of [20] obtained from threshold resummationin Mellin space. In that reference no Coulomb resummation was considered. In order tocompare to their result we use the approximation NLL s+h defined above, but in additionset the hard scale to µ h = µ f . As can be seen from the comparison, this approximation,denoted by NLL s , is in good agreement with the result from [20] at the 14 TeV LHC,while the full NLL corrections including Coulomb effects are considerably larger at highermasses. At the Tevatron the NLL s approximation results in smaller corrections than theones obtained in the Mellin space approach, while again the Coulomb corrections and55 ( p ¯ p → ˜ q ¯˜ q )(pb), √ s = 1 .
96 TeV m ˜ q [GeV] LO NLO NLL ∆ K NLL
200 9 . +4 . − . . +1 . − . . +1 . − . . . +2 . − . × − . +1 . − . × − . +0 . − . × − . . +1 . − . × − . +0 . − . × − . +0 . − . × − . . +1 . − . × − . +0 . − . × − . +0 . − . × − . . +4 . − . × − . +2 . − . × − . +0 . − . × − . σ ( pp → ˜ q ¯˜ q )(pb), √ s = 7 TeV m ˜ q [GeV] LO NLO NLL ∆ K NLL
200 1 . +0 . − . × . +0 . − . × . +0 . − . × . . +1 . − . . +0 . − . . +0 . − . . . +1 . − . × − . +0 . − . × − . +0 . − . × − . +1 . − . × − . +0 . − . × − . +0 . − . × − . . +2 . − . × − . +0 . − . × − . +0 . − . × − . . +3 . − . × − . +1 . − . × − . +0 . − . × − . σ ( pp → ˜ q ¯˜ q )(pb), √ s = 10 TeV m ˜ q [GeV] LO NLO NLL ∆ K NLL
200 4 . +1 . − . × . +0 . − . × . +0 . − . × . . +0 . − . × . +0 . − . × . +0 . − . × . . +0 . − . . +0 . − . . +0 . − . . . +0 . − . × − . +0 . − . × − . +0 . − . × − . . +1 . − . × − . +0 . − . × − . +0 . − . × − . . +0 . − . × − . +0 . − . × − . +0 . − . × − . . +0 . − . × − . +0 . − . × − . +0 . − . × − . √ s = 7 TeV and 10 TeV for m ˜ g = 1 . m ˜ q at leading order, next-to-leadingorder (NLO) and NLO plus resummed soft and Coulomb corrections (NLL). The errorestimates of the LO and NLO results are obtained by varying the factorization scale in theinterval m ˜ q / ≤ µ f ≤ m ˜ q , the error estimate of the NLL result is obtained by varying µ i ∈ { µ f , µ h , µ s , µ C } in the interval ˜ µ i / ≤ µ i ≤ ˜2 µ i around the default values and addingthe variations of the cross section in quadrature.56 ( pp → ˜ q ¯˜ q )(pb), √ s = 14 TeV m ˜ q [GeV] LO NLO NLL ∆ K NLL
200 0 . +0 . − . × . +0 . − . × . +0 . − . × . . +1 . − . × . +0 . − . × . +0 . − . × . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . . +0 . − . × − . +0 . − . × − . +0 . − . × − . . +0 . − . × − . +0 . − . × − . +0 . − . × − . . +3 . − . × − . +1 . − . × − . +0 . − . × − . . +3 . − . × − . +1 . − . × − . +0 . − . × − . . +2 . − . × − . +1 . − . × − . +0 . − . × − . √ s = 14 TeV, setup asin table 2. σ ( pp → ˜ q ¯˜ q )(pb), √ s = 14 TeV m ˜ q [GeV] NLO NLL Mellin (ref. [20]) NLL s NLL200 1 . × . × (1%) 1 . × (1%) 1 . × (3 . . × . × (1 . . × (1 . . × (4 . . × − . × − (1 . . × − (1 . . × − (5 . . × − . × − (3 . . × − (3 . . × − (11%)3000 7 . × − . × − (6 . . × − (5 . . × − (21%) σ ( p ¯ p → ˜ q ¯˜ q )(pb), √ s = 1 .
96 TeV m ˜ q [GeV] NLO NLL Mellin (ref. [20]) NLL s NLL200 1 . × . × (1 . . × (1%) 1 . × (6 . . × − . × − (2 . . × − (1 . . × − (10%)400 4 . × − . × − (4 . . × − (2 . . × − (17%)500 2 . × − . × − (7 . . × − (4 . . × − (27%)600 9 . × − . × − (11)% 10 . × − (7 . . × − (42%)Table 4: Comparison to the results of [20] for the LHC at 14 TeV (above) and the Tevatron(below) for m ˜ g = m ˜ q . In brackets we quote the corrections ∆ K NLL relative to the NLOresult. 57
00 1000 1500 2000 m q Ž @ GeV D Σ X H Μ L(cid:144) Σ X H Μ L NLONLL s + h NLL Μ f m q Ž Σ @ pb D NLLNLOLO
300 400 500 600 m q Ž @ GeV D Σ X H Μ L(cid:144) Σ X H Μ L NLO
NLL s + h NLL Μ f m q Ž Σ @ pb D NLLNLOLO
Figure 9: Left: Scale dependence of the NLO, NLL and NLL s+h approximations at theLHC with √ s = 14 TeV (top) and the Tevatron with √ s = 1 .
96 TeV (bottom). Thecorrections are normalized to the value at the default scales. Right: Dependence of theLO, NLO and NLL result on the factorization scale, for m ˜ q = 1 TeV at the 14 TeV LHC(top) and for m ˜ q = 400 GeV at the Tevatron (bottom). The green band indicates theuncertainty on the NLL result from the choice of the soft, hard and Coulomb scales.mixed soft/gluon corrections are sizable (note that, due to statistical fluctuations, some ofthe NLO results from Prospino differ from the numbers quoted in [20] in the last digit).
The production of pairs of massive coloured particles is an integral part of hadron colliderphysics programmes, since such particles, the top quark for certain, may be produced inlarge numbers. Precise calculations of the production cross sections are hampered by po-tentially large quantum corrections originating from the suppression of soft gluon emissionand the exchange of Coulomb gluons near the partonic production threshold, especially forlarge masses of the produced particles. The techniques for summing soft gluon effects to58ll orders in perturbation theory have been available for some time, and have been usedextensively to improve Drell-Yan production and related processes. Likewise, the summa-tion of Coulomb effects is well established in the context of calculations of top quark pairproduction near threshold in e + e − collisions. But the interplay of both effects as relevantto hadronic pair production had not been studied to all orders up to now.Joint soft-gluon and Coulomb resummation is a non-trivial issue, since the energy ofsoft gluons is of the same order as the kinetic energy of the heavy particles produced.Soft-gluon lines may therefore connect to the heavy-particle propagators in between theCoulomb ladders, as well as to the Coulomb gluons itself, impeding the standard factoriza-tion arguments that assume either no couplings to the final state, or to energetic particles.In this paper we employed a combination of soft-collinear and non-relativistic effective fieldtheory techniques to show that, despite these apparent complications, soft gluons can bedecoupled from the non-relativistic final state. Their effect is then contained in a morecomplicated soft function including two time-like Wilson lines for the final state particles,and in a convolution in energy of the soft function with the final state Coulomb function,since the latter depends on the energy but not the momentum taken away by the radiatedgluons. Our main result is therefore a new factorization formula for the partonic cross sec-tion of heavy-particle pair production near threshold in hadron collisions given by (2.63),and the corresponding equation (2.93) for the invariant mass distribution, together withthe resummation formula (3.48) for the partonic cross section.The result is naturally formulated in momentum space. When expressed in Mellinmoment space, the factorization is multiplicative in each separate colour channel, and givenby the product of a hard, soft and Coulomb (or potential) function. This justifies previoustreatments of Coulomb effects at the one-loop order and extends them to all orders. Theresummation of Coulomb effects to all orders is in fact formally required even at the leading-logarithmic level, though numerically the two- and higher-loop Coulomb correction is rathersmall. Starting from the NNLL level the combined soft-gluon and Coulomb resummationmight have involved subleading soft gluon couplings from the effective Lagrangians (beyondthe standard eikonal approximation in the collinear sector), making the resummation muchmore difficult. We applied symmetry considerations to show that these couplings do notcontribute at NNLL. A new effect at this order is, however, that the potential functionacquires factorization-scale dependence that cancels with the hard function, resulting inadditional threshold logarithms on top of the ones from soft gluon emission. Eq. (2.63) isthus valid at NNLL, but beyond that order further, more complicated soft functions aremost likely required.Factorization and resummation are largely model-independent. The details of the hardproduction mechanism enter only in the momentum-independent short-distance coeffi-cients. We exemplified our theoretical approach by considering squark-antiquark produc-tion in pp and p ¯ p collisions at LHC and Tevatron energies with NLL accuracy. We find thatthe effect of resummation on top of the fixed-order NLO cross section is around (12–27)%for squark masses that give cross sections at the femtobarn level, and increases with massas expected. The scale uncertainty is considerably reduced after resummation suggestinga reduction of the remaining theoretical error from which experimental limits or determi-59ations of squark masses would benefit. We find resummation effects larger than previousestimates in the literature, since we normalize the Coulomb function at the scale M β tosum all NLL terms, and include the interference terms between soft gluon and Coulombexchange, as well as squark-antisquark bound-state production in the total cross section.A more extensive phenomenological analysis of sparticle production covering all final statesat the NNLL level is therefore of interest.
Acknowledgements
We thank U. Langenfeld and S. Moch for providing interpolating functions of squark pro-duction cross sections for performing numerical checks, G. Watt for discussions on theMSTW PDFs, J.R. Andersen for computing advice, and S. Klein for helpful discussions.M.B. thanks the CERN theory group for its hospitality, while part of this work was done.The work of M.B. is supported in part by the DFG Sonderforschungsbereich/Transregio 9“Computergest¨utzte Theoretische Teilchenphysik”; the work of P.F. by the grant “PremioMorelli-Rotary 2009” of the Rotary Club Bergamo.
A Relation of collinear matrix elements to PDFs
In this appendix we show that collinear matrix elements of the form h p | φ c ( x ) φ c (0) | p i canbe expressed in terms of the parton distribution functions (PDFs) and a polarization sumof the external wavefunctions, as stated in (2.52). We recall the operator definitions of thePDFs of quarks and gluons in full QCD [90]: f q/N ( x , µ ) = 12 π Z dt e − ix t ¯ n · P h N | [ ¯ ψ ( t ¯ n ) ¯ n/ W ¯ n ( t ¯ n, ψ (0) | N i , (A.1) f g/N ( x , µ ) = 12 πx (¯ n · P ) Z dt e − ix t ¯ n · P h N | (¯ n µ F µν ( t ¯ n ) W ¯ n (¯ nt, n ρ F νρ (0)) | N i , (A.2)where ψ is the quark field in QCD, F µν the full gluon field strength and an average overthe physical polarization states is understood. W ¯ n ( x + ,
0) denotes a Wilson line with thefull gluon field ¯ n · A extending from 0 to x + , W ¯ n ( x + ,
0) = P exp (cid:20) ig s Z x + dt ¯ n · A (¯ nt ) (cid:21) . (A.3)The PDFs in SCET have been introduced in [62] and are collected in [91] in the SCETconventions used here. The purpose of the present discussion is to fix our conventions andidentify all the numerical prefactors.In order to relate the collinear matrix elements to the PDFs, we decompose the collinearand anticollinear matrix elements into a basis of spin structures Γ i , so that (suppressingcolour indices) h p | φ † c ; β φ c ; α | p i = X i ¯Γ iαβ h p | φ † c Γ i φ c | p i (A.4)60ith tr[¯Γ i Γ i ] = 1. We now discuss the cases of quarks and gluons separately. For n -collinearquarks the fields are given by φ c = W † c ξ c . The identities n/ξ c = 0 and ( n/ ¯ n// ξ c = ξ c satisfied by the collinear spinors imply [62] that the independent spin structures for thedecomposition of a matrix element h p | ¯ ξ c Γ ξ c | p i can be taken as Γ i ∈ { ¯ n/, ¯ n/γ , ¯ n/γ µ ⊥ } , up tonormalization. Since we sum over the initial state parton spin, only the ¯ n/ term contributes.In colour space the product of ξ † and ξ decomposes into a singlet and an octet componentbut the sum over the initial state parton colours projects on the colour singlet. We thereforeobtain for the collinear quark matrix element h p ( P ) | φ (0) † c ; kβ ( z ) φ (0) c ; iα (0) | p ( P ) i| avg. = 12 N c δ ki (cid:18) n/ γ (cid:19) αβ h p | [ ¯ ξ (0) c W (0) c ]( z ) ¯ n/ W (0) † c ξ (0) c ](0) | p i| avg. = N qαβ ( P ) δ kj Z dx e ix ( z · P ) f q/p ( x , µ ) . (A.5)In the last step we have identified the quark PDF defined in terms of the SCET fields f q/p ( x , µ ) = 12 π Z dt e − ix t ¯ n · P h p | [ ¯ ξ (0) c W (0) c ]( t ¯ n ) ¯ n/ W (0) † c ξ (0) c ](0) | p i| avg. , (A.6)which is consistent with the definition (2.52) in the main text, since for quarks the spin-dependent normalization factor, identified as a polarization sum of collinear quark spinorswith momentum P µ = (¯ n · P ) / n µ , N qαβ ( P ) = ¯ n · P N c (cid:18) n/ γ (cid:19) αβ = 12 N c X λ u λα ( P ) u λ ∗ β ( P ) , (A.7)satisfies N q ( x P ) = x N q ( P ). (The factor γ arises because of the definition of theconjugate quark and antiquark fields as discussed below (2.25).) Eq. (A.6) is of the sameform as the PDF in full QCD (A.1) for external quark states, apart from the fact thatthe collinear quarks and Wilson lines appear instead of the full QCD fields. Since we areconsidering external massless on-shell partons with vanishing transverse momentum, thecollinear region is the only one contributing to the PDFs. As the SCET Lagrangian for asingle collinear direction is equivalent to full QCD [59], one can replace the collinear fieldsin (A.6) by the full QCD fields and identify the PDF with (A.1). For a matrix elementwith collinear antiquarks we obtain an analogous expression with the antiquark PDF f ¯ q/p ( x , µ ) = 12 π Z dt e − ix t ¯ n · P h p | tr (cid:26) ¯ n/ W (0) † c ξ (0) c ]( t ¯ n )[ ¯ ξ (0) c W (0) c ](0) (cid:27) | p i| spin avg. (A.8)and the normalization factor N ¯ qαβ ( P ) = ¯ n · P N c (cid:18) γ n/ (cid:19) βα = 12 N c X λ ¯ u λ ∗ β ( P )¯ u λα ( P ) . (A.9)61or the matrix element of n -collinear gluons, the only available transverse symmetrictensor is g ⊥ µν so that [62, 91] h p |A ⊥ c ; kµ ( z + ) A ⊥ c ; iν (0) | p i| avg. = 12( N c − g ⊥ µν δ ki h p |A ⊥ µc,j ( z + ) A ⊥ c ; jµ (0) | p i| avg. = N gµν ( P ) δ ki Z dx x f g/p ( x , µ ) e ix ( z · P ) (A.10)with the gluon PDF in SCET f g/p ( x , µ ) = − x (¯ n · P )2 π Z dt e − ix t ¯ n · P h p |A ⊥ µc ( t ¯ n ) A ⊥ c,µ (0) | p i| avg. , (A.11)which is consistent with the definition (2.52) in the main text. The normalization factor is N gµν ( P ) = − g ⊥ µν N c −
1) = 12( N c − X λ ǫ λµ ( P ) ǫ λ ∗ ν ( P ) . (A.12)For the last identity we use the polarization sum of the gluon polarization vectors X λ ǫ λµ ( P ) ǫ λ ∗ ν ( P ) = − g µν + q µ P ν + P µ q ν P · q = − g µν + q µ n ν + n µ q ν n · q (A.13)and choose the arbitrary light-like vector q proportional to ¯ n .The equivalence of the SCET and QCD definitions of the gluon PDF is establishedmost easily in light-cone gauge ¯ n · A = 0. In this gauge the SCET gluon operators (2.26)reduce to the collinear gluon fields: A ⊥ c = A ⊥ c , while the QCD definition (A.2) becomes: f g/p ( x , µ ) = − πx (¯ n · P ) Z dt e − ix t ¯ n · P h p | (¯ n · ∂A ν ( t ¯ n )(¯ n · ∂A ν (0)) | p i| avg. . (A.14)Upon integration by parts and Fourier transformation one sees that this agrees with theSCET definition. B Coulomb potential for gluinos and squarks
Here we collect the coefficients D R α of the Coulomb potential relevant for the productionof coloured SUSY particles defined by the relation (3.20). This relation can be solved forarbitrary representations in terms of the quadratic Casimir operators of the representationsof the two heavy particles and the final state system D R α = 12 ( C R α − C R − C R ′ ) , (B.1)62s can be deduced from the generator of the product representation T ( R ⊗ R ′ ) = T ( R ) ⊗ ( R ′ ) + ( R ) ⊗ T ( R ′ ) and by projecting the identity T ( R ) b ⊗ T ( R ′ ) b = 12 h T ( R ⊗ R ′ ) b T ( R ⊗ R ′ ) b − ( C R + C R ′ ) ( R ) ⊗ ( R ′ ) i (B.2)on the irreducible representation. The explicit results for gluino pair production are longknown [92] while all the remaining cases have recently been collected in [93]. We providethese results here for completeness. Note that our different sign convention implies thatnegative values of D R α correspond to an attractive Coulomb force, positive values to arepulsive one. Squark-antisquark production.
The coefficients for squark-antisquark production have beenquoted already in (3.23): D = − C F = − N C − N C = − , D = C A − C F = 12 N C = 16 . (B.3) Squark-squark production.
In squark-squark production qq → ˜ q ˜ q the final state system iseither in the ¯3 or 6 representation and the coefficients of the Coulomb potential are D ¯3 = − (cid:18) N C (cid:19) = − , D = 12 (cid:18) − N C (cid:19) = 13 . (B.4) Gluino-squark production.
For gluino-squark production qg → ˜ q ˜ g the final-state represen-tations appear in the decomposition 3 ⊗ D = − N c − , D ¯6 = − , D = + 12 . (B.5) Gluino-pair production.
For gluino pair production the final state representations appearin the decomposition 8 ⊗ ⊕ s ⊕ a ⊕ ⊕ ⊕
27 with the coefficients of the Coulombpotential: D = − , D S = D A = − , (B.6) D = 0 , D = 1 . (B.7)Note that only the singlet and octet states can be produced from a quark-antiquark initialstate while all the states appear in gluon fusion. C Evolution functions
For convenience we quote here the explicit expressions of the resummation functions S , a Vi and a Γ given in (3.42) up to NLL. The expressions for the corresponding functions for63he Drell-Yan process up to N LL order have been given in [33]. Up to NLL the relevantexpressions read S ( µ h , µ s ) = Γ (0) ,r cusp + Γ (0) r ′ cusp β (cid:20) πα s ( µ h ) (cid:18) − α s ( µ h ) α s ( µ s ) − ln α s ( µ s ) α s ( µ h ) (cid:19) + Γ (1) ,r cusp + Γ (1) r ′ cusp Γ (0) ,r cusp + Γ (0) r ′ cusp − β β ! (cid:18) − α s ( µ s ) α s ( µ h ) + ln α s ( µ s ) α s ( µ h ) (cid:19) + β β ln α s ( µ s ) α s ( µ h ) , (C.1) a Γ ( µ h , µ ) = Γ (0) ,r cusp + Γ (0) r ′ cusp β ln α s ( µ ) α s ( µ h ) , (C.2) a Vi ( µ h , µ ) = γ (0) ,Vi β ln α s ( µ ) α s ( µ h ) . (C.3)and an analogous expression for a ψ,r . Here we have introduced the perturbative expansionsof the beta function and the anomalous dimensions: β ( α s ) = − α s X n =0 β n (cid:16) α s π (cid:17) n +1 , (C.4) γ ( α s ) = X n =0 γ ( n ) (cid:16) α s π (cid:17) n +1 , (C.5)with β = 113 C A − n f , β = 343 C A − C A n f − C F n f . (C.6)Explicit expressions for the anomalous dimensions required in the present work can befound, e.g. in [37]. D Fixed-order expansions
The expression for the resummed cross section (3.48) can be expanded to a fixed order α ns in the strong coupling, providing an approximation to the full O ( α ns ) QCD calculation.According to the counting (2.7), the expansion of the NLL resummed cross section to order α s compared to the leading order cross section is accurate up to terms of the order α s × β .This approximation to the NLO cross section is obtained from (3.53) by expanding theevolution function U i to O ( α s ), performing the convolution of the leading order Coulomb-Green function for arbitrary η , identifying η = 2 a R α Γ ( µ s , µ f ) and expanding up to O ( α s ).Inserting the explicit results for the one-loop cusp and soft anomalous dimensions, theresult reads f NLL(1) pp ′ ( i ) = − π D R α β + ( C r + C r ′ ) (cid:20) (cid:18) ln (cid:18) Eµ f (cid:19) − ln (cid:18) Eµ s (cid:19)(cid:19) + ln (cid:18) M µ h (cid:19) − ln M µ f ! (cid:21) − C R α + 4 ( C r + C r ′ )) ln (cid:18) µ s µ f (cid:19)
64 2 ( γ φ,r (0) + γ φ,r ′ (0) + 2 C R α − β ) ln (cid:18) µ h µ f (cid:19) + O (1) . (D.1)We have introduced the expansion of the cross section in the strong couplingˆ σ ( i ) pp ′ ( β, µ f ) = ˆ σ (0 ,i ) pp ′ ( α s ( µ f )4 π f (1) pp ′ ( i ) + (cid:18) α s ( µ f )4 π (cid:19) f (2) pp ′ ( i ) + O ( α s ) ) . (D.2)The term involving 2 β in the last line of (D.1) appears, since we express ˆ σ (0 ,i ) pp ′ ∝ α s in terms of the strong coupling at scale µ f , while in the factorization formula the hardfunction involves the scale µ h . The expansion (D.1) is used in (3.55) to match the NLLresummed prediction for squark-antisquark production to the fixed-order NLO calculation.At the order considered here, one can further approximate E = M β + O ( β ) to write theresult in a more familiar form.The NLL approximation to the NLO cross section (D.1) can be improved to includein addition all terms of order α s × β by inserting the Laplace transform of the one-loopsoft function (3.9) as well as the hard function in (3.48). Performing the derivatives withrespect to η and expanding to O ( α s ) afterwards we obtain for the NLO corrections to thecross section in the colour-channel i [45]: f (1) pp ′ ( i ) = − π D R α β + 4 ( C r + C r ′ ) (cid:20) ln (cid:18) Eµ f (cid:19) + 8 − π (cid:21) − C R α + 4 ( C r + C r ′ )) ln (cid:18) Eµ f (cid:19) + 12 C R α + h (1) i ( µ f ) + O ( β ) . (D.3)We note that the dependence on the soft scale has canceled between the one-loop soft func-tion and the expansion of the evolution function. Similarly the evolution equation (3.33)implies that the dependence on the hard scale cancels between h (1) i ( µ h ) ≡ H (1) i ( µ h ) /H (0) i and the evolution function. This can also be checked explicitly for the case of top-pairproduction where the one-loop hard functions can be obtained from [80]. In contrast, theNLL result (D.1) contains residual dependence on the soft and hard scale that is formallyof higher order if the scales are chosen of the order µ s ∼ E ∼ M β and µ h ∼ M . Weobserve that the ln 8 constant terms in the NLO result (D.3) can be reproduced exactlywith µ s = 8 E .Expanding the NNLL resummed cross section in the same way results in an approx-imation to the NNLO cross sections f (2) pp that is accurate up to terms α s β . The resultsfor the production of heavy particles of arbitrary spin and colour and the application totop-pair production have already been presented in [45] and will not be repeated here. References [1] G. Sterman,
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