# Energy helps accuracy: electroweak precision tests at hadron colliders

###### Abstract

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.

^{†}

^{†}preprint: CERN-TH-2016-205

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 ALEPH:2005ab ().

The flaws in this argument are well known to practitioners of Effective Field Theory (EFT). Probing heavy new physics, described by a mass scale , at energies , gives a correction to observables scaling as , for some . For those observables with , hadron colliders benefit from the high CoM energy Domenech:2012ai (); Biekoetter:2014jwa (); Azatov:2015oxa (); Dror:2015nkp (); Butter:2016cvz (); Falkowski:2016cxu (). Is the energy enhancement at hadron colliders sufficient to beat the precision of lepton colliders?

We address this question within the SM EFT Buchmuller:1985jz (); Grzadkowski:2010es (). We study the effect of “universal” new physics Barbieri:2004qk (); Wells:2015uba (); Wells:2015cre () on neutral and charged Drell-Yan (DY) Drell:1970wh () processes: and . Universal theories include scenarios with new heavy vectors that mix with SM ones Salvioni:2009mt (); Contino:2010mh (); delAguila:2010mx (); deBlas:2012qp (); Lizana:2013xla (); Pappadopulo:2014qza (), new electroweak charged particles Cirelli:2005uq (), and electroweak gauge boson compositeness Liu:2016idz (). The effects of universal new physics on the DY process can be parameterized as modifications of electroweak gauge boson propagators and encapsulated in the “oblique parameters” Peskin:1990zt (). At leading order in a derivative expansion they correspond to , , , and Barbieri:2004qk (), which modify the , , and propagators. The effects of and on DY processes do not grow with energy, making it difficult for the LHC to surpass stringent constraints from LEP ALEPH:2005ab (). On the other hand, and , which are generated by the dimension-6 operators of table 1, give rise to effects that grow with energy.

We find that neutral DY has comparable sensitivity to and 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 Chatrchyan:2011nv (); CMS:2011aa (); Aad:2011dm (); Chatrchyan:2014mua (); CMS:2014jea (); Aad:2016naf (); Aad:2016zzw (), Parton Distribution Function (PDF) determination, and NNLO calculations Hamberg:1990np (); Anastasiou:2003yy (); Anastasiou:2003ds (); Melnikov:2006kv (); Catani:2009sm (); Gavin:2010az (); Li:2012wna (). 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 far beyond LEP.
We project the sensitivity of the 13 TeV LHC, and future hadron colliders, and find spectacular reach to probe and .

While we propose to use DY for electroweak precision tests, previous studies have shown DY can probe 4-fermion contact
operators Cirigliano:2012ab (); Aad:2012bsa (); Chatrchyan:2012hda (); Chatrchyan:2013lga (); deBlas:2013qqa (); Aad:2014wca (); Khachatryan:2014tva (); Khachatryan:2014fba (),
the running of electroweak gauge couplings Rainwater:2007qa (); Alves:2014cda (), and quantum effects from superpartners Brensing:2007qm (); Dittmaier:2009cr ().

universal form factor () | contact operator () | |
---|---|---|

W | ||

Y |

EWPT from DY.—
The parameters , , , and modify the SM neutral (, ) and charged () vector boson propagators as ^{1}^{1}1These 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 , , and . This explains the mismatch with Ref. Barbieri:2004qk (), where a different basis is used.

(1) |

where is the four-momentum and , , and are the sine, cosine, and tangent of the Weinberg angle. The parameters and have normalizations that differ from the conventional normalizations Peskin:1990zt () as follows: and . All parameters are constrained at the few per mille level, mainly from precision data collected at LEP Falkowski:2015krw () and from boson mass measurements at the Tevatron Aaltonen:2012bp (); Abazov:2012bv ().

In view of these strong constraints, one might expect that no significant progress is possible at the LHC since DY cross sections, which are the best probes of Eq. (Energy helps accuracy: electroweak precision tests at hadron colliders), are measured with at best a few percent accuracy Chatrchyan:2014mua (); Aad:2016naf (); CMS:2014jea (); Aad:2016zzw (). This expectation is correct for and , which only appear on the pole of the propagator, which is better constrained at LEP. However, and introduce constant terms in the propagator, modifying the cross sections by a factor that grows with energy as . Neutral DY measurements from the 8 TeV LHC CMS:2014jea (); Aad:2016zzw () have already achieved accuracy at a center of mass energy TeV, where this enhancement factor is above . They could thus be already sensitive to values of and as small as , outside the reach of LEP. Moreover, current high-energy measurements are statistics-dominated, the systematic component of the error being as small as . 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, , containing the two dimension-6 operators from the middle column of Table 1. These operators generate the and parameters of Eq. (Energy helps accuracy: electroweak precision tests at hadron colliders). The effects of and on DY are also captured by , which consists of the operators from the right column of Table 1. Here, and are the SU and U currents, and are the corresponding couplings. The current bilinears contain quark-lepton contact operators (a subset of those considered in Ref. deBlas:2013qqa ()) which directly contribute to the DY amplitude with a term that grows with the energy, mimicking the effect of the modified propagators in Eq. (Energy helps accuracy: electroweak precision tests at hadron colliders). The effective Lagrangian is obtained from by field redefinitions, after truncating operators that are higher order in and and with more derivatives. and are physically inequivalent because of this truncation, however they agree in the limits of small and and/or low energy.

Current Limits and Future Prospects.— We compute the tree-level neutral () and charged () DY differential cross sections with the modified propagators of Eq. (Energy helps accuracy: electroweak precision tests at hadron colliders). The differential distribution is integrated in dilepton invariant mass (for neutral DY) and transverse lepton mass (for charged DY) bins and compared with the observations using a test. The value of the cross section in each bin can be written as , , and , are numbers that vary bin-by-bin. The coefficients represent the interference between the SM and the new physics, which is the leading effect in our case. The SM cross section, , is computed at NNLO QCD using FEWZ Melnikov:2006kv (); Catani:2009sm (); Gavin:2010az (); Li:2012wna (); Catani:2007vq (); Gavin:2012sy (). The NNPDF2.3@NNLO PDF Ball:2012cx (); Ball:2013hta (), with , is employed for the central value predictions at 8 and 13 TeV, and to quantify PDF uncertainties. We use NNPDF3.0@NNLO Ball:2014uwa () for 100 TeV projections. The QCD scale and PDF uncertainties are included following Ref. Alves:2014cda (). The photon PDF is not a significant source of uncertainty, because it was recently determined with high precision Manohar:2016nzj ().

Run- limits on and from neutral DY are obtained using the differential cross section measurements performed by ATLAS Aad:2016zzw () and CMS CMS:2014jea (), 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 () 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 . The right panel of Fig. 1 shows the exclusion contours obtained with ATLAS and CMS data in the - plane. The constraint from LEP and from other low-energy measurements Falkowski:2015krw () is displayed as a grey region (marginalizing over and ). Run- limits from neutral DY are already competitive with LEP constraints.

We project neutral/charged DY reach at TeV and at a future TeV collider. We also project the reach of 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 () and uncorrelated () uncertainties. For neutral DY, we use , commensurate with uncertainties achieved in existing 8 TeV measurements. For charged DY we use , consistent with uncertainty attributed to charged DY backgrounds to searches ATLAS:2014wra (); Khachatryan:2014tva (); Aaboud:2016zkn (). We apply the cuts GeV and on leptons, and assume an identification efficiency of () for electrons (muons). For neutral (charged) DY we bin invariant (transverse) mass as in Ref. Alves:2014cda ().

Our TeV results, overlaid with the LEP limit, are shown in Fig. 2 left, for luminosities of , and fb. The projected LHC limits are radically better than present constraints. The expected Run- limit on from charged DY is shown as a dotted green band. The reach far surpasses LEP, even with Run- data. Projections for TeV are shown to the right of Fig. 2 for luminosities of and ab.

In order to delve deeper into our results, Fig. 3 shows how the limit on or changes if only invariant mass (for neutral DY, left panel) or transverse mass (for charged DY, right panel) bins below a certain threshold are included. We learn that our limits mainly rely on measurements below TeV for TeV. The dramatic improvement of reach with is a direct consequence of how the relevant bins scale with , as visible in Fig. 3, leading to an improvement of sensitivity to or that scales as . 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 () or up () with respect to our estimate (i.e., for charged DY) is shown on the right panel of Fig. 3. Similar bounds but for a 100 TeV center of mass collider are shown in Fig 4. In this case the plots show that the bounds mainly rely on invariant mass measurements (transverse mass measurements in the case of charged DY) below 10 TeV.

LEP | ATLAS 8 | CMS 8 | LHC 13 | 100 TeV | ILC | TLEP | CEPC | ILC 500 GeV | |||

luminosity |
|||||||||||

NC |
W | ||||||||||

Y | |||||||||||

CC | W | — | — | — | — | — |

The shape of the limit/reach contours in the - plane can be understood as follows. The interference term in the partonic neutral DY cross section depends on a -independent linear combination of and , when integrated over angles. The orthogonal combination is only constrained when and are large enough for quadratic terms to be relevant. In view of the strong constraint expected on 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 deBlas:2013qqa (). In practice, this does not improve the 8 TeV limits (due to the dominance of the 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 Racco:2015dxa (). 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 or ). The EFT limits become inconsistent if they come from energies above the cutoff. This concept has been addressed in DM EFT searches Racco:2015dxa (); Busoni:2013lha () and electroweak EFT studies Contino:2016jqw (). Depending on whether we consider new physics that directly generates contact interactions (), or modifies the vacuum polarizations (), the maximal cutoff estimate is,

(2) |

The first estimate comes from demanding amplitudes induced by not to exceed the perturbativity bound, the second one from the validity of the derivative expansion, taking into account that is a higher-derivative correction to the (canonically normalized) vector boson kinetic terms. There is no contradiction in the fact that the two pictures have different cutoffs since and are equivalent only if the operators induced by the field redefinition are negligible (as is the case when ).

In order to quantify the impact of the limited EFT validity, Figs. 3 and 4 shows how the reach deteriorates when only data below the cutoff are employed.^{2}^{2}2This is not completely correct in the charged DY case because low transverse mass bins might in principle still receive contributions from reactions that occur at very high center of mass energies, well above the cutoff. These contribution are however negligible for the analysis discussed in this paper. To show this we recalculated the bounds on shown in Table 2 artificially including in the calculation of the New Physics cross section only those events in which the lepton-neutrino invariant mass is below the maximal cutoff at which the derivative expansion breaks down. These new bounds are only weaker than the old ones by a few percent, showing that the contamination mentioned above is numerically irrelevant. 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 panels of Figs. 3 and 4 also show 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 for the SU(2) gauge fields and for the hypercharge. Our results imply TeV from charged DY at 8 TeV and TeV from neutral DY with an LHC luminosity of 300 fb. Our bounds are also applicable to models in which elementary and bosons mix with heavy vector resonances. To discuss the bound in a quantitative way we consider an SU(2) triplet massive vector field, , coupled to the current of the SM. This matter content is described by the following effective Lagrangian:

(3) |

where we define the covariant derivative for as and the current as

(4) |

with running over SM quarks and leptons and being the Higgs boson doublet. Ref. Pappadopulo:2014qza () discusses possible UV realizations of Eq. (3): the vector field can either belong to a weakly coupled UV completion or it can be a composite resonance as those arising in models of Higgs compositeness. Integrating out the vector triplet generates as follows,

(5) |

The model in Eq. (3) is described by two parameters and . In Fig. 5 we show the bounds on the model in the plane coming from , see Eq. (5). We use the current and projected sensitivity of LHC and a 100 TeV Future Circular Collider (FCC) to as extracted from Ref. Thamm:2015zwa ().

Outlook.— In this paper, 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 and . Our results are summarized in Table 2, where we also compare to future lepton colliders.

We conclude by noting that the universal parameters and 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.

###### Acknowledgements.

Acknowledgements.—We thank Daniele Alves, Kyle Cranmer, Jorge de Blas, Roberto Contino, Jamison Galloway, Andy Haas, Alex Pomarol, Francesco Riva, and Jon Walsh for helpful discussions. G. P., R. T. and A. W. thank Javi Serra for collaboration at an early stage of this project. We thank the authors of Ref. Fan:2014axa () for providing the code we used to compute the bounds on and at future lepton colliders. M. F. is supported by the DOE grant DOE-SC0010008. The work of G. P. is partly supported by MINECO under Grant CICYT- FEDER-FPA2014-55613-P, by the Severo Ochoa Excellence Program of MINECO under the grant SO-2012-0234 and by Secretaria d’Universitats i Recerca del Departament dâ’Economia i Coneixement de la Generalitat de Catalunya under Grant 2014 SGR 1450. D. P. is supported by the James Arthur Postdoctoral Fellowship. J. T. R. is supported by the NSF CAREER grant PHY-1554858, and acknowledges the hospitality of the Aspen Center for Physics, supported by NSF grant PHY-1066293. The work of R. T. is supported by the Swiss National Science Foundation under grants CRSII2-160814 and 200020-150060. A. W. acknowledges the MIUR-FIRB grant RBFR12H1MW and the ERC Advanced Grant no. 267985 (DaMeSyFla).## References

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