Dilepton production in the SMEFT at $\mathcal O(1/\Lambda^4)$

We study the inclusion of $\mathcal O(1/\Lambda^4)$ effects in the Standard Model Effective Field Theory in fits to the current Drell-Yan data at the LHC. Our analysis includes the full set of dimension-6 and dimension-8 operators contributing to the dilepton process, and is performed to next-to-leading-order in the QCD coupling constant at both $\mathcal O(1/\Lambda^2)$ and $\mathcal O(1/\Lambda^4)$. We find that the inclusion of dimension-6 squared terms and certain dimension-8 operators has significant effects on fits to the current data. Neglecting them leads to bounds on dimension-6 operators off by large factors. We find that dimension-8 four-fermion operators can already be probed to the several-TeV level by LHC results, and that their inclusion significantly changes the limits found for dimension-6 operators. We discuss which dimension-8 operators should be included in fits to the LHC data. Only a manageable subset of two-derivative dimension-8 four-fermion operators need to be included at this stage given current LHC uncertainties.


I. INTRODUCTION
The Standard Model (SM) of particle physics has successfully withstood rigorous tests across a wide range of energies, from low-energy nuclear decays to high-energy collisions.
Despite its elegance and successes, the SM is not the final theory of nature, as it does not accommodate neutrino masses [1,2], it does not have a dark matter candidate and it cannot explain the origin of the matter-antimatter asymmetry in the universe [3][4][5][6]. Experiments at the Large Hadron Collider (LHC) are probing the SM at the TeV scale, looking for clues that might lead to solutions to these three outstanding problems and to a better understanding of the mechanism of electroweak symmetry breaking. Despite a few tantalizing hints of discrepancies [7][8][9][10][11], no direct evidence for new particles has so far emerged at the LHC, suggesting that the scale of new physics Λ is larger than the electroweak scale.
A powerful theoretical framework for investigating indirect signatures of heavy new physics is the SM Effective Field Theory (SMEFT). The SMEFT is formed by augmenting the SM Lagrangian with higher-dimensional operators consistent with the SM gauge symmetries and formed only from SM fields. The higher-dimensional operators in the SMEFT are suppressed by appropriate powers of a characteristic energy scale Λ below which heavy new fields are integrated out. Complete, non-redundant bases for the dimension-6 [12][13][14] and dimension-8 operators [15,16] have been constructed. Odd-dimensional operators violate lepton-number and will not be considered here. It is an ongoing effort to analyze the numerous available data within the SMEFT framework, primarily in partial analyses of individual SMEFT sectors . Recent work has been devoted to performing a global, simultaneous fit of all data available [38][39][40][41][42][43][44][45][46][47][48][49], and to study the interplay between SMEFT fits and the extraction of parton distributions from data [48,50].
Most of these global fits have focused on the truncation of the SMEFT expansion to dimension-6 operators at O(1/Λ 2 ). An issue that must be addressed with such an approach is the sensitivity of fits to O(1/Λ 4 ) effects from dimension-8 operators and the square of dimension-6 terms. Intuitively their effects should be suppressed, but since many measurements at the LHC probe high energies this assumption must be tested. Furthermore, dimension-8 effects sometimes represent the leading SMEFT contributions in models with certain approximate symmetries [51,52], and their identification will be crucial to determine the UV model responsible for deviations from the SM in LHC and low-energy data 2 [53][54][55]. Previous analyses of the impact of dimension-8 operators can be found in the literature [56][57][58][59][60][61][62][63][64], and there is a growing body of work devoted to study the constraints from basic principle of quantum field theory, such as unitarity and analyticity, on the allowed space of dimension-8 coefficients [53][54][55][65][66][67][68][69][70] Our goal in this work is to explore the sensitivity of high-energy Drell-Yan production of lepton pairs to dimension-8 effects in the SMEFT. Since it is calculated to high precision in the SM and is measured with residual experimental uncertainties approaching the percent level, it is an ideal channel in which to search for such effects. In previous work we have shown that a category of dimension-8 operators induce novel angular dependences not generated by QCD, and which can be potentially measured at the LHC [61]. In this work we study the impact of all sources of 1/Λ 4 effects, which can arise from either genuine dimension-8 operators or from the square of dimension-6 effects, on existing experimental measurements. We include the SM next-to-next-to-leading order (NNLO) QCD and NLO electroweak corrections in the next-to-leading logarithmic approximation, as well as NLO QCD corrections to the 1/Λ 2 and 1/Λ 4 terms. We note that higher-order electroweak corrections to the (1/Λ 2 ) terms have recently been calculated [71]. We summarize below the main messages of our analysis.
• Effects quadratic in the dimension-6 Wilson coefficients have a significant impact on fits of the current data. For dimension-6 four-fermion operators that interfere with the SM, including 1/Λ 4 effects can shift bounds on the Wilson coefficients by factors of 2-3 depending on the operator.
• Genuine dimension-8 operators can be strongly constrained by existing Drell-Yan measurements at the LHC. For example, existing high-precision measurements of the Drell-Yan invariant mass distribution up to 1.5 TeV can probe dimension-8 operator scales approaching Λ ≈ 4 TeV in the case of two-derivative operators of the form ∂ µ (ψγ ν ψ) ∂ µ (χγ ν χ), with ψ (χ) a lepton (quark) field.
• Dimension-6 scalar and tensor semileptonic four-fermion operators, which contribute to the cross section at O(1/Λ 4 ), are currently probed at the same level as four-fermion operators that interfere with SM.
• The inclusion of dimension-8 operators in the fit can significantly change the allowed regions of dimension-6 Wilson coefficients.
In the light of these results, we advocate for the inclusion of the dimension-6 squared contributions and of the most relevant dimension-8 operators in the analysis of LHC Drell-Yan data. At a minimum the two-derivative four-fermion operators that give contributions to the cross section scaling as O(s 2 /Λ 4 ) in the high-energy limit should be included in fits to the current data to avoid misleading bounds. The number of such operators at dimension-8 is only O (10), so that the complexity of including the full dimension-8 operator set is avoided in this setup.
The paper is organized as follows. In Section II we provide the definition of the dimension-6 and -8 operators relevant for dilepton production. In Section III we sketch the calculation of the cross section and analyze the directions in parameter space that can be probed by the Drell-Yan process. In Section IV, we study the numerical impact of O(1/Λ 4 ) effects. In Sections V and VI we perform a fit to the dilepton invariant mass distribution, measured at the center-of-mass energy of 8 TeV [72]. To assess the sensitivity of existing data to O(1/Λ 4 ) effects, we first perform a single coupling analysis, in which only one operator coefficient is turned on at the new physics scale Λ. We then study to which extent dimension-8 effects can cancel dimension-6 contributions by performing a multiple parameter fit. We conclude in Section VII.

II. OPERATOR BASIS
The SMEFT Lagrangian contains the most general set of operators that are invariant under the Lorentz group, the gauge group SU (3) c ×SU (2) L ×U (1) Y , and that have the same field content as the SM, with the Higgs boson belonging to an SU (2) L doublet. SMEFT operators are organized according to their canonical dimension, with operators of higher dimension suppressed by higher powers of the new physics scale Λ. The rapid advance of Hilbert series methods [73] has allowed the derivation of the complete SMEFT Lagrangian up to dimension nine [15,16,[74][75][76]. In this work we are concerned with O(1/Λ 4 ) contributions to the Drell-Yan cross section, which arise from the square of dimension-6 operators and from the interference of dimension-8 operators with the SM. Here we list the relevant dimension-6 and dimension-8 operators that we consider. We first establish our notation. The left-handed quarks and leptons and the scalar field while the right-handed quarks, u R and d R , and charged leptons, e R , are singlets under GeV is the scalar vacuum expectation value (vev), h is the physical Higgs field and U (x) is a unitary matrix that encodes the Goldstone bosons. We will denote bỹ ϕ the combinationφ = iτ 2 ϕ * . The gauge interactions are determined by the covariant derivative where B µ , W I µ and G a µ are the U (1) Y , SU (2) L and SU (3) c gauge fields, respectively, and g, g , and g s are their gauge couplings. y denotes the field hypercharge assignment, given explicitly by Furthermore, τ I /2 and t a are the SU (2) L and SU (3) c generators, in the representation of the field on which the derivative acts.

A. Dimension six operators
The dimension-6 SMEFT Lagrangian was constructed in Refs. [12,14], and the operators that give the most important contributions to Drell-Yan can be organized into three different classes: • ψ 2 Xϕ contains dipole couplings to the U (1) Y , SU (2) L and SU (3) c gauge bosons. Here we focus on weak dipoles, which contribute at tree level:

5
• ψ 2 ϕ 2 D contains corrections to the W and Z boson couplings to fermions where The right-handed charged-current operator C Hud contributes to pp → ν, but does not induce corrections to + − production, at LO in electroweak interactions.
• L ψ 4 includes four-fermion operators. The most relevant for Drell-Yan are semileptonic four-fermion operators, Hq , C He , C Hu , C Hd and the seven four-fermion operators C (1,3) q , C eu , C ed , C u , C d and C qe interfere with the SM, and thus give corrections to the Drell-Yan cross section at O(1/Λ 2 ). The operators in the first class shift the value of Z and W boson couplings to quarks and leptons by O(v 2 /Λ 2 ) with respect to the SM expectation, and thus give rise to cross sections that have the same energy behavior as the SM. Operators in these classes can be sensitively probed by electroweak precision data at the Z-pole [77]. 6 In addition, they give important contributions to diboson production or Higgs production in association with W /Z, where they induce corrections that grow with energy [37,78].
Four-fermion operators, on the other hand, induce contributions that grow with energy and scale as O(s/Λ 2 ). If we neglect small quark and lepton Yukawas, the dipole operators in Eq.
(5) do not interfere with the SM and they thus contribute to the cross section at where the power of s arises from the additional derivative in the dipole interaction with respect to the SM. Finally, the scalar and tensor operators C edq , C The Wilson coefficients of the operators in Eqs. (5), (6) and (7) are in principle matrices in flavor space. Here, for simplicity and to avoid stringent constraints from flavor physics, we choose them to be universal in both quark and lepton flavor.

B. Dimension eight operators
At dimension eight, we only consider operators that can interfere with the Standard Model. We can split the dimension-8 operators into four categories: As in the dimension-6 section we do not explicitly list those operators that shift the electroweak couplings at O(1/Λ 4 ).
• The two derivative operators are These operators interfere with the SM to generate a O(s 2 /Λ 4 ) correction to the cross section. The other class of two-derivative operators has the form[79] where the notation (µν) denotes symmetrization over the indices µ and ν These operators give rise to interesting angular distributions, which we considered in Ref. [61]. The interference with the SM, however, vanishes once we integrate over the lepton angle cos θ. While the cuts on the leptons transverse momenta and rapidities prevent an exact cancellation, these operators cannot be efficiently probed with the distributions we study in this paper.
• There are several semileptonic operators with two Higgses: 2 q 2 H 2 does not interfere with the SM. The net effect of the other operators is to provide an independent coefficient in the 12 channels e i u j , e i d j , ν L u j , ν L d j with i, j ∈ {L, R}. The contributions of the operators in Eq. (12) to each flavor and helicity channel are given in Eq. (A5).
• The next class we consider are fermion bilinear operators with three derivatives. They 8 give rise to vertices of the formf γ µ f ∂ 2 Z µ : Only six linear combinations contribute to Drell-Yan. We introduce the coupling to left-handed electrons The other linear combinations are C (1) • The final corrections we consider are fermion bilinear operators with a single derivative.
These give rise to momentum independent corrections to the Z-boson vertices. We can write the relevant operators as

+C
(1) 9 This class of operators introduces seven independent Wilson coefficients. Since they give rise only to momentum-independent vertex corrections they are difficult to disentangle from similar effects at dimension-6, and are typically much smaller than other shifts of the cross section induced by SMEFT operators.
The corrections to the cross section from the operators in Eqs. (12) and (13)

III. CALCULATION OF THE CROSS SECTION
The Drell-Yan cross section has the structure where we collectively denote by C (6) and C (8) the coefficients of dimension-6 and dimension-8 operators, respectively. As we already mentioned, the chiral structure of the SM implies that only a limited number of interference terms exist. Similarly, if we neglect small lepton and quark Yukawas, the interference terms between different dimension-6 operators, b (6) ij , are limited to interference between the purely left-handed operators C (1) q and C (3) q , between the SU (2) L and U (1) Y dipole operators, and between scalar and tensor operators. The latter vanishes when integrating over the angular variables, leaving some small residual effects due to the cuts on the lepton transverse momenta and rapidities.
A detailed discussion of the SMEFT vertices that enter the Drell-Yan cross section is given in Appendix A. We summarize here the main points of this discussion. At the O(1/Λ 4 ) level, three distinct contributions to the Drell-Yan cross section are possible.
• The most important class of corrections scales as O(s 2 /Λ 4 ). Examples of operators that lead to this dependence are momentum-dependent four-fermion operators at dimension-8, and scalar/tensor dimension-6 four-fermion operators. These terms have a large effect on the cross section, and our numerical fits in the next section show that they cannot be neglected in fits to the current data.
• The second type of correction scales as O(v 2 s/Λ 4 ). Dimension-6 dipole operators and momentum-dependent Z-vertex corrections at dimension-8 lead to this behavior. The impact of the these terms on fits to the Drell-Yan data is smaller than those in the previous category. but not explicitly considered here.
We compute a (6) , a (8) and b (6) at NLO in QCD. A first O(α s ) effect arises from the renormalization group evolution of the Wilson coefficients of SMEFT operators. The dimension-6 corrections to the Z and W boson couplings, the vector-like semileptonic operators and the dimension-8 operators in Eqs. (9), (12), (13) and (15) do not run at one loop in QCD.
For the dipole, scalar and tensor operators, we evolve the coefficients from the scale µ 0 , chosen to be close to the new physics scale Λ, to the renormalization scale µ R , using twoloop anomalous dimensions [80][81][82]; see Appendix B and Ref. [29] for more details. The renormalization of the two-derivatives operator C (2) 2 q 2 D 2 , and similar operators with different quark and lepton chiralities, has been studied in the context of higher-twist operators, and it is known to three-loops [83]. The second effect arises from QCD virtual and real emissions.
For the dimension-6 and dimension-8 operators with the same chiral structure as the SM, these corrections are identical to QCD corrections to SM amplitudes. For dipole, scalar, and tensor operators we use the calculation of Ref. [29], while we compute the corrections , we include all corrections proportional to operators that contribute at the Born level, and that thus generate contributions enhanced in the soft and collinear limits. We do not include corrections from dimension-8 operators with gluons, such as While these operators induce small corrections to the dilepton invariant mass distribution, and we can safely neglect them, they might play a more prominent role in studies of the dilepton transverse momentum distribution.
In the invariant mass bins we consider, NLO QCD corrections increase the SM cross section by about 20%, reaching 26% in the highest bin. SMEFT cross sections receive contributions of similar size, about 30% for scalar, tensor and dipole operators.  becomes approximately 25% and 50% of the linear piece in the two highest invariant mass bins. The quadratic and linear terms become comparable in the highest invariant mass bin, m ∈ [1.0, 1.5] TeV, for Λ = 3 TeV, which we also expect to be in the range of validity of the EFT. For C u , b (6) is even more important, being, in the highest bin, larger than the linear term at Λ = 4 TeV and 70% of the linear term at Λ = 5 TeV. The values of the coefficients a (6) /Λ 2 and b (6) /Λ 4 induced by the dimension-6 SMEFT operators that can interfere with the SM are given in Tables II and III in Appendix C for Λ = 4 TeV. In the highest invariant mass bin, m ∈ [1.0, 1.5] TeV, the quadratic term ranges from half to two times a (6) . We thus expect quadratic contributions to have a significant impact on the coefficient fits.
In tensor coefficients are stronger than on vector operators. The dipole operators also do not interfere with the SM. In their case, however, the correction to the SM cross section grows as s/Λ 2 compared to s 2 /Λ 4 for four-fermion operators. For Λ = 4 TeV, the dimensionless dipole coefficients C f W and C f B need to be larger than four-fermion coefficients by about a factor of ten to cause comparable corrections to the cross section. The corrections to dσ/dm induced by dipole, scalar and tensor operators, in the binning of Ref. [72], are given in Tables IV, V and VI. Fig. 4 we show the corrections induced by four dimension-8 operators that couple right-handed quarks and electrons: the two-derivative operator C e 2 u 2 D 2 , the correction to the dimension-6 coupling C e 2 u 2 H 2 , and two derivative couplings of the Z boson to quark and leptons, C e 2 H 2 D 3 and C u 2 H 2 D 3 . We see that the derivative operator induces sizable  corrections. For such a low scale, of course, a SMEFT analysis of the data in Ref. [72] is not justified. We note that this conclusion is valid only when assuming that the underlying UV completion giving rise to the SMEFT is weakly coupled. The dimensionless Wilson coefficients generically behave as C i ∼ g 2 U V , where g U V represents a coupling constant of the UV model, and we have assumed that the SMEFT operators are generated at tree-level. If we assume that the UV completion is strongly coupled we can have g U V ≈ 4π. In the case of the Z-boson form factor operators being generated by strongly coupled UV physics we would instead arrive at an effective scale of Λ ≈ 1 TeV. The corrections to dσ/dm induced by dimension-8 operators are given in Tables VII and VIII.

V. SINGLE COUPLING ANALYSIS
We now extract bounds on SMEFT coefficients from the results of Ref. [72], which We choose the UV scale Λ = 4 TeV, which is above the highest invariant mass bin studied in the experimental analysis, as a reference scale. We calculate the SM cross section at nextto-next-to leading order (N 2 LO) in QCD using the N -jettiness subtraction method [84,85] as implemented in MCFM [86] and include next-to-leading-logarithmic (NLL) electroweak corrections [87,88], which become important in the high invariant mass bins. The theoretical uncertainties in the SM arise from the parton distributions (PDFs), from missing higher order corrections and from uncertainties in the SM parameters. We estimate PDF uncertainties by using the 100 members of the NNPDF31 nnlo as 0118 PDF set [89]. The PDF error ranges between less than 1% and 2.8%. PDF uncertainties between different bins are strongly correlated. We estimate the theoretical error from missing higher-order corrections by separately varying the renormalization and factorization scales in the range m /2 ≤ µ R,F ≤ 2m subject to the constraint 1/2 ≤ µ R /µ F ≤ 2. To provide a conservative uncertainty estimate we vary the scales in the NLO cross section. The scale uncertainty estimated in this way ranges from 1.2% to 3.1% in the highest invariant mass bin. We assume that the scale uncertainty is uncorrelated between the experimental bins. The SMEFTinduced corrections are calculated at NLO in the QCD coupling constant. We have assumed no underlying hierarchy regarding the dimension-6 and dimension-8 coefficients, and rely instead upon the experimental data to determine their allowed ranges. Figure 5 shows the comparison between the SM prediction and the measurement of Ref. [72]. We see that there is in general a very good agreement. For m > 300 GeV the data lie below the SM expectation by about one sigma. Taking into account the experimental and theoretical correlations, for the SM cross section we find a χ 2 per degree of freedom (dof) of 11.7/12 = 1.05.
The bounds from turning on only a single coefficient at a time are shown in Figs. 6-10.
We begin by discussing the bounds on dimension-6 four-fermion coefficients in Fig. 6. We We proceed to discuss next the bounds on the dipole, scalar and tensor couplings shown in Fig. 7. As explained previously these enter the cross section quadratically, so the bounds are symmetric around zero. The χ 2 per dof is slightly over unity, indicating a reasonable fit to the data. Since these operators always increase the SM cross section and no destructive LeQu , C LedQ and the dipole couplings C f B , C f W , with f ∈ {u, d} to be proportional to the identity in the quark mass basis [91].
In this case, the strongest limit on scalar operators arise from the ratios which scale as m 2 e /m 2 µ in the SM, but are not suppressed in the presence of pseudoscalar operators. Assuming flavor universality, and, in addition, that the couplings are real, one The limits on Λ can be weakened by one order of magnitude assuming quark flavor diagonal rather than flavor universal couplings [92].
For the scalar and tensor couplings, the best constraints come nuclear beta decays, R π and radiative pion decays [18,[92][93][94]. In this case one finds [93,94] − 0.6 (4 TeV) 2 < at 95% CL. These bounds are very close to those showed in Fig. 7. Therefore, while the linear combination constrained by R π and R K is out of the LHC reach, we can conclude that for the other two linear combinations of chiral-breaking scalar and tensor coefficients there is a strong interplay between low-and high-energy searches, as already pointed out in Refs. [29,[93][94][95]. Similar conclusions apply to the real part of flavor-diagonal dipole operators.
Electric dipole moments put strong constraints on the imaginary part of the coefficients of flavor-diagonal chiral-breaking operators, so that for these the LHC is never competitive.
Finally we proceed to discuss the dimension-8 operator bounds in Figs. 8, 9, and 10. The first plot discusses operators of the form L ψ 4 D 2 that give momentum-dependent four-fermion corrections that scale as O(s 2 /Λ 4 ), the second gives momentum-independent four-fermion corrections from L ψ 4 H 2 that scale as O(v 2 s/Λ 4 ), and the third gives momentum-dependent   fb −1 of 13 TeV data [96][97][98][99][100][101][102]. While these datasets play an important role in further constraining the SMEFT expansion, they are published with less detailed error information, and the extraction of reliable bounds requires a detailed detector simulation not available to theorists. We note that the search for contact interactions in Ref. [101] considers a subset of the dimension-6 operators that we included, and uses a signal region of 2 < m < 6 TeV.
The uncertainties on the background in this region are quite significant, and the limits on the new physics scale Λ, once converted into the conventions of Eq. (7), are about Λ ∼ 7 TeV, stronger by only a factor of approximately 1.5 compared to our analysis. This again highlights the importance that precise data at all energies can have on SMEFT analyses.

VI. MULTIPLE COUPLINGS SCENARIOS
We study in this section the impact of turning on several dimension-6 and dimension-8 SMEFT operators at the same time. Since angular information can in principle disentangle operators with different helicities [29], we consider one specific helicity channel, with righthanded u quarks and right-handed electrons. The dimension-6 operator that contributes in this channel is C eu . At dimension eight, we turn on C e 2 u 2 H 2 , the derivative operator C e 2 u 2 D 2 , and one momentum-dependent correction to the Z-vertex, C (1) u 2 H 2 D 3 also contributes to the helicity channel with right-handed quarks and lefthanded electrons). With these four couplings, we find a best fit χ 2 /dof = 5.8/8, indicating that multiple couplings do not significantly improve the fits. The 95% CL limits on the four SMEFT operators we considered are shown in Table I obtained by marginalizing over the remaining three couplings. In this case, the correlation matrix of the four couplings (C eu , C e 2 u 2 D 2 , C e 2 u 2 H 2 , C (1) indicating strong correlations between dimension-6 and dimension-8 four-fermion operators.
The rightmost column show marginalized bounds, but with the coefficients of dimension-8 operators allowed to vary between ±256, so that the effective scale of the operators does not go below 1 TeV.
From Table I, we see that the bounds on the dimension-6 operator C eu can be weakened by turning on dimension-8 operators with arbitrary coefficients. In the case that all couplings are allowed to vary freely without enforcing consistency of the EFT expansion for all nonzero couplings, the bounds on C eu are weakened by more than an order of magnitude. Even if we constrain the dimension-8 operators to have coefficients compatible with the EFT expansion, the derivative operator C e 2 u 2 D 2 plays an important role, and weakens the bounds on C eu by a factor of 2 compared to the single coupling analysis truncated at O(1/Λ 2 ). We conclude that fits to the Drell-Yan data that truncate to dimension-6 operators only can be misleading by a significant amount. Wilson coefficients becoming as important as the linear effects far below the UV scale Λ.
Energy-dependent dimension-8 four-fermion operators whose effects scale as s 2 /Λ 4 in the high-energy limit also become nearly as large as the dimension-6 terms for s Λ.
To illustrate the impact of these findings we perform fits to the ATLAS high-mass data from Ref. [72]. Although the inclusion of the full dimension-8 corrections in the SMEFT may at first appear to be a daunting task, our results show that only the subset of dimension-8 operators consisting of two-derivative four-fermion interaction must be included. Other categories of corrections, including energy-independent dimension-8 four-fermion operators whose effects scale as v 2 s/Λ 4 and Z-boson vertex corrections, can be neglected given the current data precision. We believe that our findings provide a solid foundation for future analyses of LHC Drell-Yan measurements within the SMEFT framework. To this goal, on the experimental side it will be important to have access to more differential distributions, including rapidity and angular distributions, at high invariant mass. These measurements will allow to disentangle the flavor and helicity structure of dimension-6 and dimension-8 operators, reducing the degeneracies that affect the dilepton invariant mass distribution. On the theoretical side, a consistent dimension-8 fit will need to include the "positivity" constraints that can be inferred by fundamental principles of quantum field theory [53][54][55][65][66][67][68][69][70]. While the most naive elastic positivity constraints do not apply to the operators in the class L ψ 4 D 2 that are most relevant to Drell-Yan [67], a more detailed analysis of elastic positivity and extremal positivity bounds is necessary [67,68]. In order to discuss exactly which combinations of Wilson coefficients can be probed in the Drell-Yan process, it is helpful to introduce general parameterizations for the gauge boson-fermion vertices and for the four-fermion interactions. We begin with the four-fermion interactions. We parameterize the four-fermion interaction as (quark momenta incoming, lepton momenta outgoing): As explained previously we have neglected operators of the typeq L γ We note that this parameterization holds for both up and down type quarks. The coefficients S (6) , T (6) , and the V (8) all begin contributing to the cross section at O(1/Λ 4 ). The V (6) begin at O(1/Λ 2 ). They have the following expansion: It is straightforward to use the operators listed in the previous section to determine these vertex factors in terms of Wilson coefiicients. We begin with the scalar and tensor couplings defined in L ψ 4 : All six scalar and tensor Wilson coefficients can in principle be determined since they appear in different vertex factors, and lead to different angular dependences in the cross section.
The matrix element squared for these interactions, which only appears interfered with itself, scales as O(s 2 /Λ 4 ).
We next consider the V (8) vertex factors which come from L ψ 4 D 2 . These can be written 27 in terms of Wilson coefficients as All seven Wilson coefficients appear in separate vertex factors. We note that these contributions lead to different energy dependences than the V 6 vertex correction factors defined in Eq. (A1), and the effects of these two classes of operators can in principle be disentangled in the Drell-Yan process. Contributions from these vertex factors scale as O(s 2 /Λ 4 ) in the high-energy limit.
We now proceed to the momentum-independent four-fermion operators that first contribute at O(1/Λ 2 ). The first terms in their expansion take the forms 28 The O(1/Λ 4 ) terms in the expansion come from L ψ 4 H 2 , and take the form We now proceed to parameterize the corrections to the photon and Z-boson vertices. We express these vertices as P L,R are the standard left and right-handed projection operators, and Q i is the electric charge of fermion i. The normal SM couplings receive corrections in SMEFT, and must be expanded in κ = 1/Λ 2 :ē =ê + κē 1 + κ 2ē 2 , We have shown as well the expansion of the momentum-independent Z-boson vertex factors.
We note that the dipole corrections, as well as the momentum-dependent vertex corrections factors W Z , contribute first at O(κ 2 ). The expansion of the input parameters in κ has been studied to all orders in the SMEFT [63,103].
We can write all of these variables in terms of the input parameters and Wilson coefficients. We begin with the dipole terms: The O(κ 0 ) weak mixing angle and Higgs vev are defined in the (G F , M W , M Z ) scheme used here asŝ Since the dipole terms first contribute at O(κ 2 ) we have replaced the couplings with their O(κ 0 ) values. The following six Wilson coefficients contribute to the dipole terms: C eW , C eB , C uW , C uB , C dW , C dB .
These corrections scale as O(v 2 s/Λ 4 ) in the high-energy limit.
We now present the momentum-dependent vertex corrections. These can be written as where C (e) H 2 D 3 was defined in Eq. (14). The O(κ 0 ) Z-coupling iŝ Since these corrections contribute first at O(κ 2 ) we can use the leading-order expressions for the Z-boson coupling. We see that these couplings depend on the following six combinations of Wilson coefficients: There are 14 Wilson coefficients in total, so multiple flat directions appear in the parameter space. We enumerate the eight directions that cannot be probed in neutral-current Drell-Yan below: The first linear combination is in principle accessible in pp → νν or Z → νν. Probing the remaining combinations requires processes with multiple Higgs and gauge bosons. The W Z terms induce corrections to the Drell-Yan cross section that scale as O(v 2 s/Λ 2 ) in the high-energy limit.
We now proceed to study the momentum-independent Z-boson vertex factors. The O(κ 0 ) pieces are given by The O(κ 1 ) pieces contain two distinct contributions: the expansion of the couplingsĝ Z andŝ 2 W , and the explicit dimension-6 vertices. The explicit vertices can be obtained from Ref. [15]: These vertices are dependent on the following six Wilson coefficient combinations: Hq , C He , C Hu , C Hd , C (1) Hq .
There are seven Wilson coefficients total. The combination C vertex factors. The explicit vertex corrections can be found in Ref. [15]: We comment here briefly on the number of Wilson coefficients that enter our calculation.
After accounting for the redefinitions of the input parameters, a total of 28 dimension-6 Wilson coefficients and 54 dimension-8 Wilson coefficients enter our result in the flavoruniversal limit assumed here (we note that many enter only in linear combinations, and cannot independently be probed). The number of contributing dimension-8 coefficients is reduced by the fact that since these couplings must interfere with the SM amplitude, the contributions from all scalar and tensor four-fermion operators, as well as all dipole operators, vanish in the massless fermion limit. This removes approximately 20 additional Wilson coefficients that would appear if fermion masses were not neglected.

33
Appendix B: Renormalization group evolution of SMEFT coefficients Most operators we consider are built out of quark vector and axial currents, which do not run in QCD [104]. This is the case for C (1,3) Hq , C Hd , C Hu and C Hud in Eq. (6), C (1,3) q , C eu , C ed , C u , C d and C qe in Eq. (7) and all the operators in Eq. (12) and (15). The additional derivatives in the operators in Eq. (9) do not affect the renormalization of these operators under QCD. The operators C (1,2,3,4) Eq. (13) have a covariant derivative acting on the quark field, which could in principle affect the renormalization of these operators. In the combinations that contribute to Drell-Yan, however, the covariant derivative can be moved on the weak bosons and Higgs fields, so that again these operators do not renormalize in QCD. For example, we can write where the . . . denote terms with more Higgs and weak gauge boson fields, implying that where the two loop anomalous dimensions are [80][81][82] γ (0) Here C F = 4/3, N C = 3 and n f = 5 is the number of light flavors. The limits in Figure 7 are on scalar and tensor coefficients defined at the arbitrary scale µ 0 = 1 TeV. They can be translated into limits at other scales by using Eq. (B3).

34
Finally, two-derivative operators in the same class as C (2) 2 q 2 D 2 behave under QCD like twist-two operators. Their anomalous dimension is known to three loops [83,105,106], and, at one loop: