Perturbative T -odd asymmetries in the Drell-Yan process revisited

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I. INTRODUCTION
There is a long history of the study of T -odd asymmetries in QCD hard-scattering processes, starting with the seminal papers [1-6] that showed how T -odd effects may be generated by absorptive parts of QCD loop diagrams.T -odd behavior refers to noninvariance of observables under so-called naive time reversal, that is, under reversal of momenta and spins without interchange of initial and final states.As shown in the early papers, such behavior can occur even in theories that are manifestly invariant under true time reversal.Subsequent work [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] explored T -odd QCD phenomena in a wide range of scattering reactions, among them especially the Drell-Yan process.In Ref. [8] it was proposed to study T -odd asymmetries appearing in the angular distributions of the charged leptons from the decay of W ± bosons produced with high transverse momentum at hadron colliders.The asymmetries manifest themselves as terms proportional to sin ϕ or sin 2ϕ in the lepton distribution, where ϕ is a suitably defined azimuthal angle between the lepton plane and the hadron plane.The T -odd part of the spin-averaged differential cross section for W ± production was expanded in [8] in terms of three structure functions which were computed to lowest order of perturbation theory.The results were later obtained independently in Ref. [10] and extended to the case of longitudinal polarization of one of the initial hadrons in [15,21].
In parallel developments, it was realized that T -odd effects in QCD may also arise in hadronic matrix elements, especially in parton distribution functions (PDFs) [23][24][25][26], where they are associated with correlations among threemomenta, transverse momenta, and polarizations of partons and hadrons and again generate azimuthal-angle dependent terms.This has given rise to an intensive experimental program aiming at the extraction of such T -odd transverse-momentum dependent parton distributions (TMDs) (for a recent review, see Ref. [27]), either in semiinclusive lepton scattering (SIDIS) or via the Drell-Yan process.
The precise connection between T -odd effects in perturbative (collinearly factorized) hard scattering on the one hand and T -odd TMDs on the other has been an area of active research as well.This issue is important both theoretically and for phenomenology, where it is central for the "matching" of resummed calculations based on TMDs to fixed-order perturbation theory.While much progress has been made for leading-twist observables [28][29][30][31][32][33][34][35][36][37][38][39], it was also realized that for many of the azimuthal-angle dependent terms in the Drell-Yan and SIDIS cross sections -both T -odd and T -even -this matching is non-trivial and will involve TMD PDFs at next-to-leading power in the hard scale [40,41].Correspondingly, TMD factorization theorems at next-to-leading power were developed in the literature [42][43][44][45][46].
In the present paper, we advance this area of research by specifically exploring the low-transverse momentum limit of the T -odd terms appearing in the Drell-Yan hard-scattering calculation.The work is carried out in the spirit of Ref. [22] that addressed T -odd effects in SIDIS, but goes well beyond it in terms of calculational techniques.For our purpose, we first perform an independent new analytical calculation of the lowest-order T -odd terms in the Drell-Yan cross section, recovering results at this order from the previous literature [8,10], but also extending them by presenting results for pure Z-boson exchange and γ-Z interference.As one application, we will compare our results to available ATLAS data [47] for the T -odd angular terms taken around the Z resonance.Our main focus, however, is to carefully expand the results for low Q 2 T /Q 2 , where Q T and Q are the boson's transverse momentum and mass, respectively.We do this to first and second power in this ratio, identifying logarithmic behavior as well.As a byproduct we also uncover a novel simple relation between two of the T -odd structure functions valid at leading power in Q 2 T /Q 2 for both partonic channels, q q annihilation and qg Compton scattering.
We hope that our explicit results will be useful in testing TMD factorization at next-to-leading power and ultimately contribute to a better understanding of the matching between the TMD and collinearly factorized regimes.As has been shown in Refs.[48][49][50], at leading power γ/Z interference generates a sin 2ϕ azimuthal dependence in the unpolarized Drell-Yan cross section, entering with the Boer-Mulders function [24], while a term proportional to sin ϕ is not generated.Effects beyond leading power have been investigated in Ref. [51].
In more general terms, TMD factorization theorems make a prediction also for the large-Q T "tail" of the transversemomentum distribution they provide, which may be confronted with the terms generated by the collinearly factorized cross section expanded to low Q T /Q.An important issue is whether there is an overlap region of Q T where the two approaches agree.This decides whether the TMD and collinear contributions to the cross section are manifestations of the same physical origin, or should be regarded as genuinely separate pieces.While such an overlap has been demonstrated in a few important cases, notably the Sivers function [31,32,38], the situation is not clear for next-toleading power observables [40], especially for those that arise only from loop corrections in the collinear-factorization case.In any case, knowledge of both the TMD and the large-Q T (collinear) parts of the cross section is vital for phenomenology, in order to obtain a formalism that encompasses the full range of Q T .We will not address the potential ramifications of our results for TMDs in this paper, but rather view our work as providing a part of a "library" of hard-scattering functions at low transverse momenta.We stress that our techniques for expanding the cross sections for low Q 2 T /Q 2 are completely general and may also be used in a variety of other settings, such as for T -even contributions, collisions of polarized hadrons, and so forth, or perhaps even at the next order in perturbation theory, as available from [18].
Our paper is organized as follows.In Sec.II we present the definition of the structure functions parametrizing the lepton-angular distribution for the Drell-Yan process and the main ingredients for the perturbative calculation of the T -odd contributions.In Sec.III we collect the analytic results, and subsequently in Sec.IV the small Q T expansion is performed.In Sec.V we compare our results with the ATLAS data.Section VI concludes our paper.Some calculational details are collected in the Appendices A-C, and the lengthy results for the small-Q T expansion to next-next-to-leading-power are presented in Supplemental Material [52] to this article.

II. T -ODD STRUCTURE OF THE DRELL-YAN HADRONIC TENSOR
The hadronic tensor W µν for the Drell-Yan process can be written in terms of nine structure functions W i .The most straightforward decomposition of this tensor is obtained by using the helicity formalism proposed in Ref. [53] for reactions with photon exchange and extended in Ref. [10] to the electroweak case.The results of Ref. [10] for the expansion of W µν can be conveniently rewritten using a basis of orthogonal unit vectors T µ = q µ / Q 2 = (1, 0, 0, 0), X µ = (0, 1, 0, 0), Z µ = (0, 0, 0, 1), Y µ = ϵ µναβ T ν Z α X β = (0, 0, 1, 0), proposed in Ref. [53] and constructed from the hadron and virtual-boson momenta.This expansion reads as where q is the momentum of the gauge boson γ, W ± , or Z 0 , with q 2 = Q 2 its Minkowski momentum squared, and ϵ µναβ is the four-dimensional Levi-Civita tensor defined via tr(γ The number of structure functions, 9 = 3 × 3, is determined by the number of possible helicity settings of the gauge boson in the amplitude and its complex conjugate.In the case of the purely weak Drell-Yan reactions or for γ-Z 0 interference we have nine functions, while in the case of the purely electromagnetic Drell-Yan we have only four T -even structure functions.In general, the Drell-Yan hadronic structure functions may be classified as: (a) two transverse functions, the P -even W T and the P -odd W T P ; (b) one longitudinal function W L which is P -even; (c) two transverse-transverse interference (double-spin-flip) functions, the P -even W ∆∆ and the P -odd W ∆∆ P ; (d) four transverse-longitudinal interference (single-spin-flip) functions, the P -even W ∆ , W ∇ and the P -odd W ∆ P , W ∇ P .
The lepton angular distribution dN/dΩ is expanded in terms of the hadronic structure functions as where g i = g i (θ, ϕ) denote the angular coefficients with θ and ϕ being the polar and azimuthal angles of one of the decay leptons in the center-of-mass system (c.m.s.) of the lepton pair.The angle ϕ may be taken to define the orientation of the lepton plane with respect to the hadron plane.In Fig. 1 we show the polar and the azimuthal angles for the Drell-Yan process in the Collins-Soper frame.The six angular coefficients g i (i = T , L, ∆∆, ∆, ∆∆ P , ∆ P ) in (3) are invariant under the P -parity transformation θ → π−θ and ϕ → π+ϕ, while the other three coefficients g i (i = T P , ∇, ∇ P ) change their sign in that case.Therefore, the six partial lepton angular distributions dN i /dΩ (i=T , L, ∆∆, ∆, T P , ∇ P ) are also P invariant, whereas the other three distributions dN i /dΩ (i=∆∆ P , ∆ P , ∇) are P odd and also T odd.As can be seen from Eq. ( 3), the latter distributions are all proportional to either sin ϕ or sin 2ϕ.We note in passing that two other commonly employed, and equivalent, parametrizations of the lepton angular distribution are [10,41,[53][54][55] and + τ sin θ cos ϕ + η cos θ + ξ sin 2 θ sin 2ϕ + ζ sin 2θ sin ϕ + χ sin θ sin ϕ . ( The relations between the three sets of structure functions are recalled in Appendix A. The T -odd structure functions W ∇ , W ∆∆ P , and W ∆ P are generated at O(α 2 s ) in the strong coupling constant α s by the absorptive parts of parton scattering amplitudes.The leading contributions arise from the interference of one-loop and tree-level diagrams.The relevant channels are quark-antiquark annihilation and quark-gluon Compton scattering.Their one-loop diagrams providing an absorptive part for photon exchange in Drell-Yan are shown in Figs. 2 and 3; the diagrams with Z 0 and W ± bosons are generated analogously.The ensuing T -odd effects were first studied in Ref. [8] and later recalculated in [10].Here we will present an independent derivation that will allow us to explore the low-Q T limit of the results.
FIG. 2: One-loop diagrams for q q → gγ that produce an absorptive part.FIG.3: One-loop diagrams for qg → qγ that produce an absorptive part.
For our calculation of the T -odd structure functions we use a convenient orthogonal basis of vectors P, R, K [56], defined by at nonzero Q T .We also introduce the variables relevant in the Q T = 0 limit.The hadronic structure functions W (x 1 , x 2 , ρ 2 ) for the Drell-Yan process with colliding hadrons H 1 and H 2 are related to the parton-level structure functions w ab (x 1 , x 2 , ρ 2 ) by the QCD collinear factorization formula where f i/H (ξ) is the PDF describing the ξ distribution of partons of type i in hadron H.We are suppressing here the scale dependence of the PDFs.We may project onto the parton-level T -odd structure functions in the following way (here we drop parton labels): In the evaluation of the absorptive parts of the one loop diagrams we use the following set of imaginary parts of scalar one-loop integrals [56,57]: where B 0 denotes the two-propagator (bubble) diagram, C 0 the three-propagator (triangle) diagram, and D 0 the four-propagator (box) diagram.We have used dimensional regularization with D = 4 − 2ϵ space-time dimensions; as usual 1/ε = 1/ϵ + log(4π) + γ E with the Euler-Mascheroni constant γ E .We note that in the cases of the electroweak contributions to w ∆∆ P and w ∆ P we have to deal with an odd number of γ 5 matrices in the relevant Dirac traces.We adopt the Larin scheme for treating γ 5 ; details are discussed in Appendix B.

III. ANALYTICAL RESULTS FOR THE PARTONIC T -ODD STRUCTURE FUNCTIONS
In this section we present our analytical results for the parton-level T -odd structure functions w ∆∆ P , w ∆ P , and w ∇ .We first introduce some notation.We use the QCD color factors Specifically, the color factors for q q annihilation and qg scattering are C q q = C F /N c = (N 2 c −1)/(2N 2 c ) = 4/9 and C qg = T F /N c = 1/(2N c ) = 1/6.Furthermore, it is convenient to introduce the coupling factors g q q;i = C q q g Zγ/W EW;i e 2 q α 2 s /(4π) and g qg;i = C qg g Zγ/W EW;i e 2 q α 2 s /(4π), where i = 1, 2. Here, e q is the electric charge of a quark of flavor q.The electroweak couplings g EW;1 and g EW;2 , which incorporate the products of couplings of the gauge bosons (W ± , Z 0 , γ) with quarks and leptons, are given by in the case of electrically neutral gauge bosons (Z 0 , γ) and in the case of the W ± gauge bosons, where In the above expressions, V qq ′ is the relevant element of the Cabibbo-Kabayashi-Maskawa (CKM) matrix and θ W is the Weinberg angle measured to be sin 2 θ W = 0.23121 [58].Furthermore, ) denotes the product of the Breit-Wigner propagator of a weak gauge boson and Q 2 .Its real and imaginary part are given by The masses M G and total widths Γ G of the bosons, taken from the Particle Data Group [58], are M W ± = 80.377±0.012GeV, M Z 0 = 91.1876± 0.0021 GeV, Γ W ± = 2.085 ± 0.042 GeV, and Γ Z 0 = 2.4955 ± 0.0023 GeV.Note that in Eqs. ( 16) and ( 17) the terms proportional to the squares of the Breit-Wigner propagators correspond to the purely weak ZZ and W W contributions to the couplings, while the terms proportional to the real part of the Breit-Wigner As a final ingredient, we note that all one-loop partonic structure functions contain a factor δ (ŝ + t + û − Q 2 )/ŝ arising from phase space and corresponding to the fact that the recoil in the final state consists of a single massless parton.It is convenient to write With this notation in place, we obtain the following partonic T -odd structure functions.For the q q annihilation subprocess, we find wqq For qg scattering we have wqg We note that in the large N c limit the terms proportional to C 1 in the structure functions are suppressed by a factor 1/N 2 c relative to those proportional to C F and C A .The results for g q scattering are obtained from the ones for the qg process by interchange of momenta, p 1 ↔ p 2 , which corresponds to an interchange of Mandelstam variables u ↔ t.
For calculating the hadronic structure functions via the factorization formula (11) and subsequently investigating their small Q T behavior, it is convenient to express our results in terms of the variables one gets for q q annihilation wqq ∆∆ P = − wqq and for qg scattering wqg Here, the functions F 1 and F 2 are defined as: They obey F 1 (1) = F 2 (1) = 0 and F 1 (0) = F 2 (0) = 1.For later reference, we have also given their expansions around z = 1.We now have This factorization is formally valid when Here, we will take the collinear factorization and extrapolate to small values of Q T by formally expanding the result about Q T = 0.This will result in an expansion in powers of ρ 2 which can be matched to TMD results in the region of intermediate Q T for a smooth transition from the TMD to the collinear regime.We will perform the expansion in ρ 2 beyond the leading power, which has been the main focus in the existing literature, to provide information about which higher-power corrections are accounted for in the collinear formalism.

IV. SMALL-QT EXPANSION
In the expansion of hadronic structure functions of the form in Eq. ( 35) we have three contributions.First, there is the direct dependence of the partonic structure function on Q T .Second, the phase space delta function has nontrivial Q T dependence.Third, the variables x 1 , x 2 have implicit Q T dependence.The first type of contribution may be straightforwardly taken into account by simple expansion of the partonic structure functions.The second and third contributions require more discussion.
The phase space delta function in (35) is well known in the literature, see, e.g., Refs.[41,54,59].Its expansion to leading power in ρ 2 = Q 2 T /Q 2 was also given in that reference and reads as Here the "plus" distribution is defined by for a function f that is regular at z = 1.We note in passing that in Ref. [60] a general method for the expansion of distributions was developed, based on Mellin integral techniques.Building on these ideas, we recently formulated [61] an algorithm for the small-Q T expansion of singular functions valid to arbitrary order of ρ 2 and arbitrary number of radiated partons.This will be presented in a separate publication.Here we are only concerned with the expansion of integrals containing the phase space delta function in (36).For a general regular function φ(z 1 , z 2 ) such an integral can be expanded for small Q T including O(ρ 4 , ρ 4 log ρ 2 ) terms in the following way: with Here 1/(1 − z) m +,m−1 is a generalized plus distribution of power m, defined by where f (z) is again a sufficiently regular test function and T m−1 z=1 f (z) denotes the Taylor polynomial of f (z) about z = 1 to order m − 1, A lower integration bound of x instead of zero introduces additional boundary terms of the form where f (z)/(1 − z) m +x,m−1 is the generalized plus distribution defined for an integral starting at a finite lower limit x, i.e.
Comparing Eq. ( 38) with (36) one can see that we reproduce the known leading terms, while the terms of order ρ 2 , ρ 2 log ρ 2 , ρ 4 , and ρ 4 log ρ 2 are new.Substituting the small-Q T expansion of the parton-level structure functions w ab (z 1 , z 2 , ρ 2 ) for the various partonic channels into Eq.( 11) we get for the contributions to the small-Q T expansion of the hadronic structure function were we have abbreviated The expansion coefficients with i = 1, 2, 3 have the structure where denotes a generalized convolution, R i (x 1 , x 2 , L ρ ), P ba,i (z 2 , x 1 , L ρ ), and P ab,i (z 1 , x 2 , L ρ ) are perturbative coefficient functions containing differential operators acting on the PDFs f a/H1 (x 1 ) and f b/H2 (x 2 ).We note that the generalized convolution (47) reverts to the ordinary one, when P(z, y, L ρ ) does not depend on y and L ρ .Details are given in Appendix C. We stress that, as indicated in Eq. ( 44), the functions W i may carry dependence on log ρ 2 , on top of the overall power of ρ that they multiply.However, Eq. (44) is not yet the complete expansion.As mentioned above, we need to take into account that x 1 and x 2 are defined at finite Q T (see Eq. ( 9)) and hence must also be expanded about their respective values at Q T = 0, x 0 1 and x 0 2 in (10).Therefore, we substitute x i = x 0 i 1 + ρ 2 as arguments of the structure functions W i and perform the ρ 2 expansions of the latter.We now present our final result for the full small-Q T expansion of the hadronic structure functions, including the leading-power (LP) term W LP (x 0 1 , x 0 2 , L ρ ), the next-to-leading-power (NLP) term W NLP (x 0 1 , x 0 2 , L ρ ), and the next-next-to-leading-power (NNLP)term W NNLP (x 0 1 , x 0 2 , L ρ ): where Here ∂ m x1 ∂ n x2 W i (x 1 , x 2 , L ρ ) denotes the mth partial derivative with respect to x 1 and the nth partial derivative with respect to x 2 .The calculational techniques for taking these derivatives are discussed in Appendix C.
Explicitly we obtain the following analytical results for the LP contributions W LP;ab J (x 0 1 , x 0 2 , L ρ ) to the T -odd hadronic structure functions (here ab = q q, qg and J = ∆∆ P , ∆ P , ∇): _ FIG.4: Comparison of the full analytical result for the quark-channel contribution to W∆∆ P (black solid line, taken from Eq. ( 30) with expansions to LP (dashed), NLP (dot-dashed), NNLP (blue solid) as given in the Supplemental Material [52]. where and with F i as defined in Eq. (34).Note that the f i are regular functions with f 1 (1) = 1/3, f 2 (1) = 1/2.As shown in (56), there is an interesting relation between the structure functions W LP;ab ∇ (x 0 1 , x 0 2 , L ρ ) and W LP;ab ∆ P (x 0 1 , x 0 2 , L ρ ): valid both for the q q and the qg subprocess at leading power.The NLP and NNLP contributions to the T -odd hadronic structure function are listed in the Supplemental Material to this paper [52].
To illustrate the numerical behavior of these expansions, we consider the q q contribution to the hadronic double-flip structure function, W q q ∆∆ P (x 1 , x 2 ), as an example.In Fig. 4 we compare the full expression without Q T expansion with the LP, NLP, and NNLP results.Here we use the CTEQ 6.1M PDFs of Ref. [62], taken from LHAPDF [63], along with their ManeParse [64] Mathematica implementation.We choose √ s = 8 TeV, Q = 100 GeV, as representative of the kinematics in the ATLAS measurements [47], and the renormalization and factorization scales in the calculations are set to As one can see, the LP piece describes the full result only at low Q T and rapidly departs from it for Q T > 10 GeV or ρ 2 > 0.01.By contrast, already inclusion of the NLP term leads to excellent agreement with the full result out to Q T = 40 GeV (ρ 2 = 0.16), only marginally further improved by the NNLP contribution.In particular, for Q T = 20 GeV, the LP result deviates from the full one by about 20%, whereas at NNLP the relative deviation is only ∼ 0.4%.

V. COMPARISON TO ATLAS DATA
As shown in Eq. ( 4), the Drell-Yan cross section can be expressed in terms of eight angular coefficients A i=0,...,7 .The relations of these coefficients to the hadronic helicity structure functions are recalled in Appendix A. Previous experimental and phenomenological studies mostly focused on the first five A i coefficients [18,47,[65][66][67][68][69][70][71] which are We note that near Q = m Z the contribution by γ-Z 0 interference is suppressed relative to that for pure Z 0 exchange.In the following, we compare our results for the angular coefficients A 5 , A 6 , and A 7 to the ATLAS data.Here we use the full expressions at O(α 2 s ) for the helicity structure functions W ∆∆ P , W ∆ P , and W ∇ .In the denominator of the coefficients, we use the O(α s ) expressions for the transverse and longitudinal structure functions W T and W L (see details in Refs.[41,54,72]).This approach thus consistently gives the leading contribution to A 5 , A 6 , and A 7 , which is of order α s .
To begin with, we investigate the rapidity and Q T dependences of the angular coefficients A 5 , A 6 , A 7 near the Z 0 pole.As in the previous section the calculation is done using √ s = 8 TeV and µ = Q 2 + Q 2 T .Figure 5 shows the rapidity distribution for fixed Q T , while Fig. 6 presents the results as functions of Q T in one of the three rapidity bins accessed by ATLAS.As the plots show, the coefficients are overall small, reaching at most 0.1-0.2%near Q T ∼ m Z or toward larger |y|.This finding does not really come as a surprise: Small values of the T -odd A 5,6,7 coefficients have been predicted in Refs.[8,10] also for W boson production.In the case of Z boson production A 5,6,7 are further suppressed because of the smallness of the corresponding weak couplings, relative to the couplings appearing in W T and W L .
The increase of A 5,6,7 at larger values of rapidity in Fig. 5 -which is consistent with the leading-power relation between W ∇ and W ∆ P we found in Eq. ( 59) -follows the trend observed in Ref. [8] for W boson production in pp collisions at √ s = 540 GeV.Likewise, a similar dependence on rapidity was found for the angular coefficients in Refs. [47,66].At fixed rapidity, the T -odd coefficients are small for small Q T and then increase, peaking when Q T is near the Z mass (see Fig. 6).We also show in the figure the individual contributions by q q and qg scattering, the latter dominating for all kinematics.Among the T -odd structure functions, W ∆∆ P , being symmetric under interchange z 1 ↔ z 2 , has the largest contribution from quark-antiquark annihilation.Figure 7 explores the role played by the pair mass Q for the Q T distribution.We show results for Q = 80 GeV, Q = m Z , and Q = 100 GeV, which span the range of Q used for the ATLAS measurements.As one can see, A 5 and A 6 are rather insensitive to Q, whereas A 7 exhibits a strong dependence, even turning negative at high Q.This suggests that A 7 will be quite sensitive to smearing effects if data are sampled over a sizable range in Q.
We now turn to the actual comparison with the ATLAS data [47].Of their three rapidity bins (|y| < 1, 1 < |y| < 2, and 2 < |y| < 3.5) we only use the two with higher |y| since, as we saw above, the angular coefficients are very small for |y| < 1.We note that ATLAS presents the data in two ways, as an "unregularized" and a "regularized" set.The regularization smoothes the data by correcting for bin migration.This procedure involves the use of Monte-Carlo pseudo data which are at lowest-order accuracy.Details about the data regularization method are presented in Appendixes C and E of Ref. [47].
Figures 8-10 show the comparison.Neither the regularized nor the unregularized data are in particularly good agreement with our theoretical predictions.At best, there is qualitative agreement in that the theoretical results show positive values for all three T -odd angular coefficients, with a similar trend in the data.In particular, for the bin 1 < |y| < 2 one observes a rise of A 5,6,7 with Q T up to about Q T ∼ m Z , exactly as predicted theoretically.Quantitatively, however, the regularized data -which is the set primarily to be used for comparisons -shows overall much higher coefficients than obtained in our calculation.We note that ATLAS used the DYNNLO package [73] to FIG. 6: QT dependence of the angular coefficients A5, A6, and A7 for the rapidity interval 1 < |y| < 2 at Q ∼ mZ and √ s = 8 TeV.We also show the individual q q and qg contributions.obtain theoretical results for A 5,6,7 .DYNNLO predicts values of up to 0.005 for the coefficients A 5,6,7 .However, as stated in [47], the prediction of nonzero values is at the limit of sensitivity of both the theoretical calculation and the data.Hence it seems preliminary to delve into a detailed analysis of the visible discrepancies between our result and the existing data.More robust conclusions regarding the agreement of data and theory for the T -odd structure functions will only become possible after significant improvements of both the experimental results and the theoretical description.A natural question to ask is whether higher-order (say, NLO) QCD corrections to the angular coefficients could lead to a better agreement with the data.In this context it is important to keep in mind that the T -odd coefficients effectively carry an overall factor α s .This immediately means that they will be more susceptible to QCD corrections than ratios of cross sections would normally be.Just to give a simple estimate: For the values of Q T relevant here, varying the scale in α s from Q T /2 to 2Q T easily generates differences of ±15% or more in the calculated coefficients.On top of this, there will be smaller uncertainties associated with the scale dependence and uncertainties of the PDFs.One would thus expect the NLO corrections to A 5,6,7 to be overall non-negligible, even though judging by the sizeable discrepancy in magnitude and shape between the existing data and our result it would come as a surprise if higher-order corrections were to account for the entire difference.We note that NLO corrections could in fact be obtained from Ref. [18].Clearly, a phenomenological study of A 5,6,7 at NLO will be an interesting project for the future.Along with hopefully improved future data it would open the door to careful assessments of the validity of fixed-order perturbation theory for the T -odd angular coefficients.

VI. CONCLUSION
We have performed a detailed analysis of perturbative T -odd effects in charged-and neutral-current Drell-Yan processes, taking into account W ± and Z 0 exchange, as well as γ-Z 0 interference.To this end, we have computed the relevant T -odd structure functions for the q q annihilation and qg Compton channels at order O(α 2 s ), where they become nonvanishing thanks to absorptive contributions to loop amplitudes.While the corresponding results are not new, we have used them in novel ways.Foremost, we have presented a new formalism to expand the results for low transverse momentum, or low ρ = Q T /Q, with the goal of facilitating comparisons to frameworks that analyze T -odd effects in terms of TMDs, especially at nonleading power.Our new formalism is completely general and can in principle be used to obtain expansions to arbitrary order in Q T /Q.As a proof of concept, we have applied it to the T -odd structure functions and expanded them to order O(ρ 4 ).In doing so, we uncovered a new relation between two of the T -odd structure functions, W LP ∇ (x 1 , x 2 ) and W LP ∆ P (x 1 , x 2 ), valid at leading power in the small-Q T expansion.Although in the present paper we have not attempted to connect our results to calculations based on TMD factorization, we think that our paper has much to offer for such comparisons in the future.
We have also presented numerical results for the validity of the expansion in ρ 2 , and we have compared our full results for the T -odd structure functions to available data from the ATLAS experiment.We found that in the present situation it is impossible to draw any quantitative conclusions from this comparison.
In the present paper we have restricted our analysis to the case of the T -odd effects in the Drell-Yan process with unpolarized beams.Extensions to the T -even sector and to polarized scattering will be natural extensions of our work.
Appendix A: Relations among different sets of the structure functions The three sets of structure functions {A i }, {W i } and {λ, µ, ν, . ..} are related as [10,41,[53][54][55] and Appendix B: Treatment of γ 5 matrix in calculation of structure functions In the calculation of the T -odd structure functions we have to deal with the γ 5 matrix in dimensional regularization.We encounter two different cases: (a) Dirac traces with an even number (in practice, two) of γ 5 matrices in case of w ∇ ; (b) Dirac traces with an odd number (in practice, one) of γ 5 matrices in case of w ∆∆ P and w ∆ P .
For case (a), it is permissible even in dimensional regularization to anticommute the two γ 5 matrices toward each other and to use (γ 5 ) 2 = 1.As a result, the contributions of [vector ⊗ vector] and [axial-vector ⊗ axial-vector] couplings to w ∇ are identical.
In case (b) we use techniques established in the literature for the treatment of γ 5 in dimensional regularization.In particular, for the axial-vector spin matrix we use the Larin prescription [74][75][76][77], expressing γ 5 as the product of the four-dimensional Levi-Civita tensor ϵ µναβ and three gamma matrices: Next, in order to evaluate the structure functions w ∆∆ P and w ∆ P , we apply the method proposed in Ref. [77] for the contraction of two Levi-Civita tensors.One Levi-Civita tensor occurs in the definition of the structure functions w ∆∆ P and w ∆ P [see Eqs. ( 12) and ( 13)] and the other appears because of the Larin substitution (B1).In general, the product of two Levi-Civita tensors must be evaluated in terms of the D-dimensional Kronecker tensor δ µ ν in order to preserve Lorentz invariance [77]: For our purposes we need to evaluate two types of contractions [see Eqs. ( 12) and ( 13)], and Using Eq. (B2) we get and partial derivative of the terms with integrals.Let us discuss the treatment of such terms by considering the following integral: The nth derivative of the integral J(x) can be taken using the binomial formula Therefore, we only need to derive an analytical formula for the nth derivative of the integral I(x), where n is an arbitrary natural number.In our derivation we will consider two possible choices for the function P (z): We first consider the simpler case (a).Here, the first-order derivative of I(x) reads as where R ′ (z) = ∂R(z)/∂z.One can prove by induction that the nth derivative of I(x) is given by where For case (b) we recall the definition for the generalized plus distribution when applied to a PDF f (x/z): The x dependence of f (x/z)/(1 − z) m +x,m−1 induces a subtraction of the (m − 1)-th order Taylor polynomial where is the binomial coefficient and f (ℓ) (x) = ∂ ℓ f (x)/∂x ℓ .Note that the partial derivatives δ(1 − z)∂ m−j z f (x/z) with respect to z in Eq. (C7) can be simplified and reduced to derivatives with respect to x as In all cases the derivatives ∂ n I(x)/∂x n can be easily taken by changing the integration variable z → x/ξ and, after some simplifications, returning back to the integration over z.We stress that the choice where R(z) is a regular function, can be reduced to case (b) by carrying out a Taylor expansion of R(z) around z = 1: where R (k) (1) = (∂ k R(z)/∂z k ) z=1 , which results in the cancellation of the respective powers of (1 − z) between the numerator and the denominator of the integrand of I(x).In particular, the distribution P qq (z) defined in Eq. ( 57) contains the term (1 + z 2 )/(1 − z) + , which can be represented as the sum of a regular term corresponding to case (a) and a single distribution 2/(1 − z) + corresponding to case (b), In case (b) the nth derivative of the integral I(x) reads as [1] A. De Rujula, J. M. Kaplan, and E. De Rafael, Nucl.Phys.B35, 365 (1971).
In the following, for convenience, we introduce functions F i (z) with i = 1, 2, 3, 4, 5, 6. Functions F i (z) with i = 3, 4, 5, 6 are obtained from functions F 1 (z) and F 2 (z) by subtraction of the leading and sub-leading terms in an expansion in (1 − z).The main idea of such substracted functions is to make the ratios F i (z)/(1 − z) n regular at z = 1.All functions F i (z) with i = 1, . . ., 6 are defined as They obey the following normalization conditions: It is clear that the ratios are regular at z = 1.In the following, we express the NLP and NNLP contributions to the hadronic structure functions in terms of the functions f i (z).

FIG. 1 :
FIG. 1: Definition of the polar and the azimuthal angles for the Drell-Yan process in the Collins-Soper frame.The hadron plane is depicted in blue, the lepton plane in red.

FIG. 5 :
FIG.5: T -odd angular coefficients A5, A6 and A7 for various transverse momenta QT of the lepton pair, as functions of pair rapidity at Q ∼ mZ and √ s = 8 TeV.

3 A5 2 < 5 Our result Q∼ mZ unregularized Atlas data regularized Atlas data 1 FIG. 8 :1 < |y| < 2 FIG. 9 : 7 1 < |y| < 2 OurA7 2 <
FIG.8: Comparison of our result for A5 to ATLAS data[47] at Q ∼ mZ .The black and red experimental points denote the unregularized and regularized data, respectively, and show their statistical error.The left panel shows the results for 1 < |y| < 2, while the right is for 2 < |y| < 3.5.(The scale on the y axis has been chosen for better visibility of the small values of the coefficient; as a result, some data points fall outside the plot range.The full dataset is shown in the left lower inset in each plot.) (a) P (z) = R(z) is a regular function of the variable z, (b) P (z) = [1/(1 − z) m ] +,m−1 is the generalized plus distribution of power m, defined in Eq. (40) in the main text.