Two-loop divergences of scattering amplitudes with massive partons
Andrea Ferroglia, Matthias Neubert, Ben D. Pecjak, Li Lin Yang
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Two-loop divergences of scattering amplitudes with massive partons
Andrea Ferroglia, Matthias Neubert, Ben D. Pecjak, and Li Lin Yang
Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany (Dated: November 11, 2018)We complete the study of two-loop infrared singularities of scattering amplitudes with an arbitrarynumber of massive and massless partons in non-abelian gauge theories. To this end, we calculate theuniversal functions F and f , which completely specify the structure of three-parton correlationsin the soft anomalous-dimension matrix, at two-loop order in closed analytic form. Both functionsare found to be suppressed like O ( m /s ) in the limit of small parton masses, in accordance withmass factorization theorems proposed in the literature. On the other hand, they are unsuppressedand diverge logarithmically near the threshold for pair production of two heavy particles. As anapplication, we calculate the two-loop anomalous-dimension matrix for q ¯ q → t ¯ t near threshold andshow that it is not diagonal in the s -channel singlet-octet basis. I. INTRODUCTION
Recently, much progress has been achieved in the un-derstanding of the infrared (IR) singularities of mass-less scattering amplitudes in non-abelian gauge theories.While factorization proofs guarantee the absence of IRdivergences in inclusive observables [1], in many caseslarge Sudakov logarithms remain after this cancellation.A detailed control over the structure of IR poles in thevirtual corrections to scattering amplitudes is a prerequi-site for the resummation of these logarithms beyond theleading order [2–5]. Catani was the first to predict thesingularities of two-loop scattering amplitudes apart fromthe 1 /ǫ pole term [6], whose general form was only un-derstood much later in [7–9]. In recent work [10], it wasshown that the IR singularities of on-shell amplitudes inmassless QCD can be derived from the ultraviolet (UV)poles of operator matrix elements in soft-collinear effec-tive theory (SCET). They can be subtracted by means ofa multiplicative renormalization factor, whose structureis constrained by the renormalization group. It was pro-posed in this paper that the simplicity of the correspond-ing anomalous-dimension matrix holds not only at one-and two-loop order, but may in fact be an exact result ofperturbation theory. This possibility was raised indepen-dently in [11]. Detailed theoretical arguments supportingthis conjecture were presented in [12], where constraintsderived from soft-collinear factorization, the non-abelianexponentiation theorem, and the behavior of scatteringamplitudes in two-parton collinear limits were studied.It is relevant for many physical applications to gener-alize these results to the case of massive partons. TheIR singularities of one-loop amplitudes containing mas-sive partons were obtained some time ago in [13], butuntil very recently little was known about higher-loopresults. In the limit where the parton masses are smallcompared with the typical momentum transfer amongthe partons, mass logarithms can be predicted based oncollinear factorization theorems [14, 15]. This allows oneto obtain massive amplitudes from massless ones with aminimal amount of calculational effort. A major step to-ward solving the problem of finding the IR divergencesof generic two-loop scattering processes with both mas- sive and massless partons has been taken in [16, 17]. Onefinds that the simplicity of the anomalous-dimension ma-trix observed in the massless case no longer persists inthe presence of massive partons. Important constraintsfrom soft-collinear factorization and two-parton collinearlimits are lost, and only the non-abelian exponentia-tion theorem restricts the allowed color structures in theanomalous-dimension matrix. At two-loop order, two dif-ferent types of three-parton color and momentum cor-relations appear, whose effects can be parameterized interms of two universal, process-independent functions F and f [17]. Apart from some symmetry properties, theprecise form of these functions was left unspecified. Inthis Letter we calculate these functions at two-loop orderand study their properties in some detail. II. ANOMALOUS-DIMENSION MATRIX
We denote by |M n ( ǫ, { p } , { m } ) i a UV-renormalized,on-shell n -parton scattering amplitude with IR singu-larities regularized in d = 4 − ǫ dimensions. Here { p } ≡ { p , . . . , p n } and { m } ≡ { m , . . . , m n } denotethe momenta and masses of the external partons. Theamplitude is a function of the Lorentz invariants s ij ≡ σ ij p i · p j + i p i = m i , where the sign factor σ ij = +1 if the momenta p i and p j are both incom-ing or outgoing, and σ ij = − v i = p i /m i with v i = 1and v i ≥
1. We further define the recoil variables w ij ≡ − σ ij v i · v j − i
0. We use the color-space formal-ism [18], in which n -particle amplitudes are treated as n -dimensional vectors in color space. T i is the color gen-erator associated with the i -th parton and acts as a ma-trix on its color index. The product T i · T j ≡ T ai T aj is summed over a . Generators associated with differentparticles commute, and T i = C i is given in terms ofthe eigenvalue of the quadratic Casimir operator of thecorresponding color representation, i.e., C q = C ¯ q = C F for quarks and C g = C A for gluons. Below, we will la-bel massive partons with capital indices ( I, J, . . . ) andmassless ones with lower-case indices ( i, j, . . . ).It was shown in [10, 12, 17] that the IR poles of suchamplitudes can be removed by a multiplicative renormal-ization factor Z − ( ǫ, { p } , { m } , µ ), which acts as a matrixon the color indices of the partons. More precisely, theproduct Z − |M n i is finite for ǫ → α QCD s used in the calculation of the scatteringamplitude is properly matched onto the coupling α s inthe effective theory, in which the heavy partons are inte-grated out [17]. The relation Z − dd ln µ Z ( ǫ, { p } , { m } , µ ) = − Γ ( { p } , { m } , µ ) (1)links the renormalization factor to a universal anomalous-dimension matrix Γ , which governs the scale dependenceof effective-theory operators built out of collinear SCETfields for the massless partons and soft heavy-quark ef-fective theory fields for the massive ones. For the caseof massless partons, the anomalous dimension has beencalculated at two-loop order in [8, 9] and was found tocontain only two-parton color-dipole correlations. It hasrecently been conjectured that this result may hold toall orders of perturbation theory [10–12]. On the otherhand, when massive partons are involved in the scatteringprocess, then starting at two-loop order correlations in-volving more than two partons appear [16]. At two-looporder, the general structure of the anomalous-dimensionmatrix is [17] Γ = X ( i,j ) T i · T j γ cusp ( α s ) ln µ − s ij + X i γ i ( α s ) − X ( I,J ) T I · T J γ cusp ( β IJ , α s ) + X I γ I ( α s )+ X I,j T I · T j γ cusp ( α s ) ln m I µ − s Ij (2)+ X ( I,J,K ) if abc T aI T bJ T cK F ( β IJ , β JK , β KI )+ X ( I,J ) X k if abc T aI T bJ T ck f (cid:16) β IJ , ln − σ Jk v J · p k − σ Ik v I · p k (cid:17) . The one- and two-parton terms depicted in the first threelines start at one-loop order, while the three-parton termsin the last two lines appear at O ( α s ). The notation( i, j, . . . ) etc. means that the corresponding sum extendsover tuples of distinct parton indices. The cusp angles β IJ are defined viacosh β IJ = − s IJ m I m J = w IJ . (3)They are associated with the hyperbolic angles formedby the time-like Wilson lines of two heavy partons. Thephysically allowed values are w IJ ≥ β IJ ≥
0, or w IJ ≤ − β IJ = iπ − b with real b ≥
0. These possi-bilities correspond to space-like and time-like kinematics,respectively. Since in a three-parton configuration there v v v FIG. 1: Two-loop Feynman graphs (top row) and one-loopcounterterm diagrams (bottom row) contributing to the two-loop coefficient of the renormalization factor Z s . is always at least one pair of partons either incoming oroutgoing, at least one of the w IJ or v I · p k variables mustbe time-like, and hence the functions F and f havenon-zero imaginary parts.The anomalous-dimension coefficients γ cusp ( α s ) and γ i ( α s ) (for i = q, g ) in (2) have been determined to three-loop order in [12] by considering the case of the masslessquark and gluon form factors. Similarly, the coefficients γ I ( α s ) for massive quarks and color-octet partons such asgluinos have been extracted at two-loop order in [17] byanalyzing the anomalous dimension of heavy-light cur-rents in SCET. In addition, the velocity-dependent func-tion γ cusp ( β, α s ) has been derived from the known two-loop anomalous dimension of a current composed of twoheavy quarks moving at different velocity [19, 20].Here we complete the calculation of the two-loopanomalous-dimension matrix by deriving closed analyticexpressions for the universal functions F and f , whichparameterize the three-parton correlations in (2). III. CALCULATION OF F AND f To calculate the function F we compute the two-loop vacuum matrix element of the operator O s = S v S v S v , which consists of three soft Wilson linesalong the directions of the velocities of three massive par-tons, without imposing color conservation. The anoma-lous dimension of this operator contains a three-partonterm given by 6 if abc T a T b T c F ( β , β , β ). Thefunction F follows from the coefficient of the 1 /ǫ polein the bare matrix element of O s . We will then obtain f from a limiting procedure.The operator O s is renormalized multiplicatively, sothat O s Z s is UV finite, where Z s is linked to the anoma-lous dimension in the same way as shown in (1). In orderto calculate the two-loop Z s factor, we have evaluatedthe two-loop non-planar and planar graphs shown in thefirst row of Figure 1, as well as the one-loop countert-erm diagrams depicted in the second row. Contrary toa statement made in [16], we find that F receives con-tributions from all five diagrams, not just from the non-planar graph. The most challenging technical aspect ofthe analysis is the calculation of the diagram involvingthe triple-gluon vertex. We have computed this diagramusing a Mellin-Barnes representation and checked the an-swer numerically using sector decomposition [21]. Wehave also checked that for Euclidean velocities our re-sult for the triple-gluon diagram agrees numerically witha position-space based integral representation derived in[16]. Combining all contributions, we find F ( β , β , β ) = α s π X i,j,k ǫ ijk g ( β ij ) r ( β ki ) , (4)where we have introduced the functions r ( β ) = β coth β ,g ( β ) = coth β (cid:20) β + 2 β ln(1 − e − β ) − Li ( e − β ) + π (cid:21) − β − π . (5)The function f can be obtained from the above resultby writing w = − σ v · p /m , w = − σ v · p /m and taking the limit m → v I · p . In that way,we obtain f (cid:16) β , ln − σ v · p − σ v · p (cid:17) = − α s π g ( β ) ln − σ v · p − σ v · p . (6)Whether a factorization of the three-parton terms intotwo functions depending on only a single cusp angle per-sists at higher orders in α s is an open question.It is interesting to expand the two functions r ( β ) and g ( β ) about the threshold point β = iπ − b with b → + .We find r ( β ) = − iπb + 1 + O ( b ) ,g ( β ) = − π + 2 iπ ln(2 b ) b + (cid:18) π (cid:19) + O ( b ) . (7)Based on the symmetry properties of F and f , it wasconcluded in [16, 17] that these functions vanish when-ever two of the velocities of the massive partons coin-cide. Indeed, this seems to be an obvious consequenceof the fact that F is totally anti-symmetric in its ar-guments, while f is odd in its second argument. Thisreasoning implicitly assumes that the limit of equal ve-locities is non-singular, but is invalidated by the presenceof Coulomb singularities in r ( β ) and g ( β ) near threshold.Consider the limit where two massive partons 1 and 2 areproduced near threshold, with a small relative 3-velocity ~v ≡ ~v − ~v defined in their rest frame. It is thenstraightforward to derive thatlim v → v f = α s π (cid:2) π + 2 iπ ln(2 | ~v | ) (cid:3) cos θ , (8) where θ is the scattering angle formed by the 3-momentaof particles 1 and 3. A similar formula holds for F . Thisresult is anti-symmetric in the parton indices 1 and 2 asrequired, but it does not vanish at threshold.Another interesting limit is that of large recoil, whereall the scalar products w IJ become large in magnitude.In that case, both F and f are suppressed like O (1 /w ),because for large βg ( β ) = 12 w (cid:20) ln (2 w ) − ln(2 w ) + π − (cid:21) + O (cid:16) w (cid:17) . (9)Note that the non-planar contribution from the firstgraph in Figure 1, which was studied in the Euclideanregion in [16], contains the leading-power term F non − planar1 = − α s π ln w w ln w w ln w w + O (cid:16) w (cid:17) (10)and is unsuppressed in this limit. However, this contri-bution cancels against a leading-power term in the planarand counterterm contributions.Using that w IJ = − s IJ / (2 m I m J ), we see that thelarge-recoil limit corresponds to m I m J → s IJ . It follows that the three-parton correlation termsdescribed by F and f vanish like ( m I m J /s IJ ) in thesmall-mass limit. This observation is in accordance witha factorization theorem proposed in [14, 15], which statesthat massive amplitudes in the small-mass limit can beobtained from massless ones by a simple rescaling pre-scription for the massive external legs. IV. ANOMALOUS DIMENSION FOR q ¯ q → t ¯ t NEAR THRESHOLD
As a sample application, we apply our formalism to thecalculation of the two-loop anomalous-dimension matri-ces for top-quark pair production near threshold in the q ¯ q → t ¯ t channel. This matrix (along with the correspond-ing one in the gg → t ¯ t channel) forms the basis for soft-gluon resummation at the next-to-next-to-leading loga-rithmic (NNLL) order. We adopt the s -channel singlet-octet basis, in which the t ¯ t pair is either in a color-singletor color-octet state. For the quark-antiquark annihilationprocess q l ( p ) + ¯ q k ( p ) → t i ( p ) + ¯ t j ( p ), we thus choosethe independent color structures as c = δ ij δ kl and c =( t a ) ij ( t a ) kl . In the threshold limit s = 2 p · p → m t it is convenient to define the quantity β t = p − m t /s ,which is related to the relative 3-velocity ~v t ¯ t between thetop-quark pair in the center-of-mass frame by | ~v t ¯ t | = 2 β t .We find that in the limit β t → Γ q ¯ q = (cid:20) C F γ cusp ( α s ) (cid:18) ln sµ − iπ β t − iπ + 1 (cid:19) + C F γ (2)cusp ( β t ) + 2 γ q ( α s ) + 2 γ Q ( α s ) (cid:21) (cid:18) (cid:19) + N (cid:20) γ cusp ( α s ) (cid:18) iπ β t + iπ − (cid:19) − γ (2)cusp ( β t ) (cid:21)(cid:18) (cid:19) + α s π (cid:2) π + 2 iπ ln(4 β t ) (cid:3) cos θ (cid:18) C F − N (cid:19) + O ( β t ) , (11)where the two-loop expressions for the anomalous dimen-sions γ cusp , γ q , and γ Q can be found in [17], and γ (2)cusp ( β t ) = N α s π (cid:20) iπ β t (cid:18) − π (cid:19) − ζ (cid:21) (12)arises from the threshold expansion of the two-loop coef-ficient of the velocity-dependent cusp anomalous dimen-sion γ cusp ( β, α s ).We stress that, as a consequence of the Coulomb singu-larities, the three-parton correlation term does not vanishnear threshold. Instead, it gives rise to scattering-angledependent, off-diagonal contribution in (11). The off-diagonal terms were omitted in two recent papers [22, 23],where threshold resummation for top-quark pair produc-tion was studied at NNLL order. We leave it to futurework to explore if and how the results obtained by theseauthors need to be modified in light of our findings. V. CONCLUSIONS
The IR divergences of scattering amplitudes in non-abelian gauge theories can be absorbed into a multi-plicative renormalization factor, whose form is deter-mined by an anomalous-dimension matrix in color space.At two-loop order this anomalous-dimension matrix con-tains pieces related to color and momentum correlationsbetween three partons, as long as at least two of them are massive. This information is encoded in two univer-sal functions: F , describing correlations between threemassive partons, and f , describing correlations betweentwo massive and one massless parton. In this Letter wehave calculated these functions at two-loop order. Usingthe exact analytic expressions, we studied the propertiesof the three-parton correlations in the small-mass andthreshold limits. We found that the functions F and f vanish as ( m I m J /s IJ ) in the small-mass limit, in accor-dance with existing factorization theoreoms for massivescattering amplitudes [14, 15]. On the other hand, andcontrary to naive expectations, the two functions do notvanish in the threshold limit, because Coulomb singulari-ties compensate a zero resulting from the anti-symmetryunder exchange of two velocity vectors. This fact hasbeen overlooked in the recent papers [22, 23], where thethree-parton correlations were neglected near threshold.Our results allow for the calculation of the IR polesin an arbitrary on-shell, n -particle scattering amplitudeto two-loop order, where any number of the n partonscan be massive. As an application, we have derived theanomalous-dimension matrix for top-quark pair produc-tion in the q ¯ q → t ¯ t channel. We will explore in futurework to what extent the new off-diagonal entries, arisingfrom three-parton correlation terms, affect the numericalresults for the threshold-resummed t ¯ t production crosssections at the Tevatron and LHC.This Letter completes the study of IR divergences oftwo-loop scattering amplitudes with an arbitrary num-ber of massive and massless external particles, and inarbitrary non-abelian (or abelian) gauge theories withmassless gauge bosons. Details of our calculations willbe presented in a forthcoming article. Acknowledgments:
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