Energy helps accuracy: electroweak precision tests at hadron colliders

We show that high energy measurements of Drell-Yan at the LHC can serve as electroweak precision tests. Dimension-6 operators, from the Standard Model Effective Field Theory, modify the high energy behavior of electroweak gauge boson propagators. Existing measurements of the dilepton invariant mass spectrum, from neutral current Drell-Yan at 8 TeV, have comparable sensitivity to LEP. We propose measuring the transverse mass spectrum of charged current Drell-Yan, which can surpass LEP already with 8 TeV data. The 13 TeV LHC will elevate electroweak tests to a new precision frontier.

Introduction.-Hadron colliders are often viewed as "discovery machines." They have limited precision, due to their messy QCD environments, but their high Center of Mass (CoM) energies allow them to directly produce new, heavy, particles. Hadron colliders are often contrasted with less energetic lepton colliders, which can reach high precision to indirectly probe new heavy physics, as exemplified by LEP, which tested the electroweak sector of the Standard Model (SM) with unprecedented per-mill accuracy [1].
The flaws in this argument are well known to practitioners of Effective Field Theory (EFT). Probing heavy new physics, described by a mass scale M , at energies E M , gives a correction to observables scaling as (E/M ) n , for some n ≥ 0. For those observables with n > 0, hadron colliders benefit from the high CoM energy [2][3][4][5][6][7]. Is the energy enhancement at hadron colliders sufficient to beat the precision of lepton colliders?
We address this question by studying the effect of "universal" new physics [8] on neutral and charged Drell-Yan (DY) [9] processes: pp → + − and pp → ν. Universal theories include scenarios with new heavy vectors that mix with SM ones [10][11][12][13][14][15], new electroweak charged particles [16], and electroweak gauge boson compositeness [17]. The effects of universal new physics on DY process can be parameterized as modifications of electroweak gauge boson propagators and encapsulated in the "oblique parameters" [18]. At leading order in a derivative expansion they correspond toŜ,T, W, and Y [8], which modify the γ, Z, and W propagators. The effects ofŜ andT on DY processes do not grow with energy, making it difficult for the LHC to surpass stringent constraints from LEP [1]. On the other hand, W and Y, which are generated by the dimension-6 operators of table I, give rise to effects that grow with energy.
We find that neutral DY has comparable sensitivity to W and Y as LEP, already at 8 TeV. This sensitivity follows from the growth in energy, as well as the percent-level precision achieved by LHC experiments [19][20][21][22][23][24][25], Parton Distribution Function (PDF) determination, and NNLO calculations [26][27][28][29][30][31][32]. We propose that the LHC can carry out similar measurements in charged DY (using the transverse mass spectrum), which with current data is sensitive to W far beyond LEP. We project the sensitivity of the 13 TeV LHC, and future hadron colliders, and find spectacular reach to probe W and Y.
While we propose to use DY for electroweak precision tests, previous studies have shown DY can probe 4-fermion contact operators [33][34][35][36][37], the running of electroweak gauge couplings [38,39], and quantum effects from superpartners [40,41].   Fit to CMS [23] and ATLAS [25] dilepton invariant mass distributions measured at 8 TeV. Left: comparison of data and SM prediction. The error bars include the fractional experimental uncertainties, while the thickness of the SM predictions include uncertainties from PDF and scale variation. The smaller error bars in the ATLAS plot show the systematic uncertainties. We also show how the central value of the theoretical prediction changes when W varies within its 95% CL range. Right: 95% CL constraints in the W-Y plane. gators as 1 where q is the four-momentum and s, c, and t are the sine, cosine, and tangent of the Weinberg angle. All 4 parameters are constrained at the few per-mill level, mainly from precision data collected at LEP [42]. In view of these strong constraints, one might expect that no progress is possible at the LHC since DY cross sections, which are the best probes of Eq. (1), are measured with at best a few percent accuracy [22][23][24][25]. This expectation is correct forŜ andT, which only appear on the pole of the propagator, which is better constrained at LEP. However, W and Y introduce constant terms in the propagator, modifying the cross sections by a factor that grows with energy as q 2 /m 2 W . Neutral DY measurements from the 8 TeV LHC [23,25] have already achieved 10% accuracy at a center of mass energy q ∼ 1 TeV, 1 These modified propagators encapsulate all new physics effects because they are written in the field basis where the vector boson interactions with fermions are identical to those of the SM, once expressed in terms of the input parameters αem, G F , and m Z . This explains the mismatch with Ref. [8], where a different basis is used.
where this enhancement factor is above 100. They could thus be already sensitive to values of W and Y as small as 10 −3 , outside the reach of LEP. Moreover, current high-energy measurements are statistics-dominated, the systematic component of the error being as small as 2%. Big improvements are thus possible at 13 TeV thanks to higher energy and luminosity.
The electroweak gauge boson propagators are modified by an effective Lagrangian, L, containing the two dimension-6 operators from the middle column of Table I. These operators generate the W and Y parameters of Eq. (1). The effects of W and Y on DY are also captured by L , which consists of the operators from the right column of Table I. Here, J L and J Y are the SU(2) L and U(1) Y currents, and g 1,2 are the corresponding couplings. The current bilinears contain quark-lepton contact operators (a subset of those considered in Ref. [35]) which directly contribute to the DY amplitude with a term that grows with the energy, mimicking the effect of the modified propagators in Eq. (1). The effective Lagrangian L is obtained from L by field redefinitions, after truncating operators that are higher order in W and Y and with more derivatives. L and L are physically inequivalent because of this truncation, however they agree in the limits of small W and Y and/or low energy.
Current Limits and Future Prospects.-We compute the tree-level neutral (pp → l + l − ) and charged (pp → lν) DY differential cross sections with the modified propagators of Eq. (1). The differential distribution is integrated in dilepton invariant mass (for neutral DY) LEP I-II Projected 95% CL exclusions in the W-Y plane. Left: exclusion from neutral (purple) and charged (green) DY from LHC measurements at various luminosities and energies, compared to LEP bounds (gray). Right: projected reach from a 100 TeV collider (notice the change of scale). and transverse lepton mass (for charged DY) bins and compared with the observations using a χ 2 test. The value of the cross section in each bin can be written as σ = σ SM (1 + p a p C p + pq b pq C p C q ), C = {W, Y}, and a p , b pq are numbers that vary bin-by-bin. The coefficients a p represent the interference between the SM and the new physics, which is the leading effect in our case. The SM cross section, σ SM , is computed at NNLO QCD using FEWZ [29-32, 43, 44]. The NNPDF2.3@NNLO PDF [45,46], with α s = 0.119, is employed for the central value predictions at 8 and 13 TeV, and to quantify PDF uncertainties. We use NNPDF3.0@NNLO [47] for 100 TeV projections. The QCD scale and PDF uncertainties are included following Ref. [39]. The photon PDF is not a significant source of uncertainty, because it was recently determined with high precision [48].
Run-1 limits on W and Y from neutral DY are obtained using the differential cross section measurements performed by ATLAS [25] and CMS [23], including the full correlation matrix of experimental uncertainties. The left panel of Fig. 1 shows the comparison of the ATLAS and CMS measurements with our theoretical predictions for the cross section in each bin in the SM (W = Y = 0) hypothesis. Theoretical uncertainties from PDF and scale uncertainty are displayed as a shaded band, while the black error bars represent experimental uncertainties. Our predictions reproduce observations, under the SM hypothesis, over the whole invariant mass range. We also notice that statistical errors are by far dominant at high mass, the theoretical and systematical uncertainties being one order of magnitude smaller, around 2%. The right panel of Fig. 1 shows the 95% exclusion contours obtained with ATLAS and CMS data in the W-Y plane. The constraint from LEP and from other lowenergy measurements [42] is displayed as a grey region (marginalizing overŜ andT). Run-1 limits from neutral DY are already competitive with LEP constraints.
We project neutral/charged DY reach at 13 TeV and at a future 100 TeV collider. We also project the reach of 8 TeV for charged DY (differential cross section measurements are presently unavailable at high transverse mass). In order to estimate experimental uncertainties, we include fully correlated (δ c ) and uncorrelated (δ uc ) uncertainties. For neutral DY, we use δ c = δ uc = 2%, commensurate with uncertainties achieved in existing 8 TeV measurements. For charged DY we use δ c = δ uc = 5%, consistent with uncertainty attributed to charged DY backgrounds to W searches [49][50][51]. We apply the cuts p T > 25 GeV and |η | < 2.5 on leptons, and assume an identification efficiency of 65% (80%) for electrons (muons). For neutral (charged) DY we bin invariant (transverse) mass as in Ref. [39].
Our 13 TeV results, overlaid with the LEP limit, are shown in Fig. 2 left, for luminosities of 100, 300, and 3000 fb −1 . The projected LHC limits are radically better than present constraints. The expected Run-1 limit on W from charged DY is shown as a dotted green band. The reach far surpasses LEP, even with Run-1 data. Projections for 100 TeV are shown to the right of Fig. 2 for luminosities of 3 and 10 ab −1 .
In order to delve deeper into our results, Fig. 3 shows how the limit on W or Y changes if only invariant mass (for neutral DY, left panel) or transverse mass (for charged DY, right panel) bins below a certain threshold Λ cut are included. We learn that our limits mainly rely on measurements below 1 (2) TeV for √ s = 8 (13) TeV. The dramatic improvement of reach with √ s is a direct consequence of how the relevant bins scale with √ s, as visible in Fig. 3, leading to an improvement of sensitivity to W or Y that scales as q 2 /m 2 W ∝ s. By highlighting the relevant bins, Fig. 3 illustrates the ranges of invariant/transverse mass where percent-level experimental systematics will be important. The effect of varying the systematic uncertainties down (2%) or up (10%) with respect to our estimate (i.e., 5% for charged DY) is shown on the right panel of Fig. 3.
The shape of the limit/reach contours in the W-Y plane can be understood as follows. The interference term in the partonic neutral DY cross section depends on a q 2 -independent linear combination of W and Y, when integrated over angles. The orthogonal combination is only constrained when W and Y are large enough for quadratic terms to be relevant. In view of the strong constraint expected on W from charged DY, this flat direction is irrelevant in practice. However, we note that the flat direction can in principle be constrained with neutral DY only, using angular information such as the energy dependence of forward-backward asymmetries [35]. In practice, this does not improve the 8 TeV limits (due to the dominance of the q L q R → l − L l + R amplitude), but may be significant at higher energies/luminosities. We leave a full study of the power of angular distributions to future work.
Beyond EFT's.-When using EFTs to describe high energy processes, one has to keep in mind that an EFT provides an accurate description of the underlying new physics only at energies below the new physics scale. The latter scale is the EFT cutoff and it should be regarded as a free parameter of the EFT [55]. A related concept is that of "maximal cutoff", which is the maximal new physics scale that can produce an EFT operator of a given magnitude (e.g., a given value of W or Y). The EFT limits become inconsistent if they come from energies above the cutoff. This concept has been addressed in DM EFT searches [55,56] and electroweak EFT studies [57]. Depending on whether we consider new physics that directly generates contact interactions (L ), or modifies the vacuum polarizations (L), the maximal cutoff estimate is, The first estimate comes from demanding 2 → 2 amplitudes induced by L not to exceed the 16π 2 perturbativity bound, the second one from the validity of the derivative expansion, taking into account that L is a higherderivative correction to the (canonically normalized) vector boson kinetic terms. There is no contradiction in the fact that the two pictures have different cutoffs since L and L are equivalent only if the d > 6 operators induced by the field redefinition are negligible (as is the case when q < Λ). In order to quantify the impact of the limited EFT validity, Fig. 3 shows how the reach deteriorates when only data below the cutoff are employed. 2 If the resulting curve stays below the maximal cutoff lines corresponding to Eq. (2), as in our case, the EFT limit is self-consistent. The right panel of Fig. 3 also shows how lowering the systematic uncertainties moves the limit curve far from the maximal cutoff line. This allows to test EFTs with below maximal cutoffs.
Our results can be applied to various new physics scenarios. Higher derivative corrections to the SM gauge boson kinetic terms directly test their compositeness above a scale Λ 2 ≈ m W / √ W for the SU(2) gauge fields and  [42], marginalizing overŜ and T. Bounds from Z-peak ILC [52] and TLEP [53] are from Ref. [39]. Bounds from off-peak measurements of e + e − → e + e − at lepton colliders are extracted from [54].
Y for the hypercharge. Our results imply Λ 2 4 TeV from charged DY at 8 TeV and (Λ 2 , Λ 1 ) (6.5, 5) TeV from neutral DY with an LHC luminosity of 300 fb −1 . Our bounds are also applicable to composite Higgs with partial compositeness, in which elementary W and B bosons mix with composite vector resonances. Following the notation of Ref. [15], and using the results of Ref. [58], we find that charged DY measurements presented in this paper can surpass direct searches of heavy vector triplets W /Z for 3.5 TeV < m W < 4 TeV and g V ∼ g 2 at 8 TeV and for 6.5 TeV < m W < 10 TeV and g V 2g 2 with a luminosity of 300 fb −1 at the LHC. Outlook.-In this letter, we have demonstrated that hadron colliders can be used to perform electroweak precision tests, and in particular that the LHC is now surpassing LEP in sensitivity to the universal parameters W and Y. Our results are summarized in Table II, where we also compare to future lepton colliders.
We conclude by noting that the universal parameters W and Y are just two examples from the class of operators of the SM EFT whose effects grow with energy. The LHC, and future hadron colliders, therefore have great potential to perform precision tests, because high center of mass energy compensates limited accuracy. We advocate exploration of a broad program of precision tests at hadron colliders, where SM measurements can be leveraged as indirect probes of new physics that is too heavy to produce directly.