# Interpreting mass measurements in the SMEFT.

###### Abstract

Measurements of the mass () provide an important consistency check of the Standard Model (SM) and constrain the possibility of physics beyond the SM. Precision measurements of at hadron colliders are inferred from kinematic distributions of transverse variables. We examine how this inference is modified when considering the presence of physics beyond the SM expressed in terms of local contact operators. We show that Tevatron measurements of using transverse variables are transparent and applicable as consistent constraints in the Standard Model Effective Field Theory (SMEFT) with small measurement bias. This means that the leading challenge to interpreting these measurements in the SMEFT is the pure theoretical uncertainty in how these measurements are mapped to Lagrangian parameters. We stress the need to avoid using naive combinations of Tevatron and LEPII measurements of without the introduction of any SMEFT theoretical error to avoid implicit UV assumptions.

## I. Introduction.

The lack of any statistically significant deviation from the Standard Model (SM) to date, argues that a mass gap is present between the scale of new physics – – and the electroweak scale . Such a mass gap, if limited to can also be consistent with expectations of UV physics motivated by naturalness concerns for the Higgs mass (). Broad classes of new physics scenarios consistent with this assumption can be studied efficiently using Effective Field Theory (EFT) methods to analyse data limited to energies . This includes future data gathered on the “Higgs pole” where .

It is of interest to examine possible future deviations in such measurements in a consistent theoretical framework that also incorporates lower energy experimental data gathered on and near the and poles. This theoretical framework has come to be known as the Standard Model Effective Field Theory (SMEFT). Near pole data can be directly incorporated in the SMEFT Grinstein:1991cd (); Han:2004az (); Pomarol:2013zra (); Ciuchini:2014dea (); Ellis:2014jta (); Petrov:2015jea (); Berthier:2015oma (); Berthier:2015gja (); David:2015waa () by interpreting the well known LEP electroweak pseudo-observables ALEPH:2005ab () in terms of constraints on higher dimensional operators. Many operators at dimension six can contribute to these observables, and further measurements are required to constrain all the operators that can contribute to such LEP data without flat directions in the SMEFT parameter space. Measurements of the boson’s decay width and mass () are a particularly important further source of constraints on the SMEFT in this sense. Note that we use bar superscripts to denote the canonically normalized parameters in the SMEFT, and hat superscripts to denote parameters derived from the measurements of our input parameter set at tree-level in the SM. Measurements of these parameters are more challenging to incorporate consistently in global studies, as the leptonic decay of the blocks precise direct experimental reconstruction of a mass peak due to the presence of final state missing energy. This challenge is overcome with the use of transverse kinematic variables, where a Jacobian peak in the corresponding distributions are exploited to extract vanNeerven:1982mz (); Arnison:1983rp (); Banner:1983jy (); Smith:1983aa (); Barger:1983wf (). Modern fits to reported using this technique with Tevatron data obtain the values in Table 1.

Determinations of at LEPII use two distinct methodologies. In one of these, scans of the differential cross section in the threshold region () exploit the rise in the cross section proportional to the velocity of the bosons, given by , to extract . This is done for the processes , with at LEPII, and this approach is relatively insensitive to some corrections Hagiwara:1987pb () that could come about in the SMEFT. Such corrections include modifications of the couplings of the and to the initial and final state fermions, and possible anomalous Triple Gauge Coupling (TGC) parameters. The presence of these corrections is a related concern when interpreting the results of the second measurement employed at LEPII, where is extracted via kinematic fits to the W pair invariant mass distribution away from the threshold region. This issue is related to the issue we study in detail at the Tevatron, and we return to this point in Section VI. We stress that all the experimental approaches reported in Table 1 are appropriate when consistency testing the SM, and can be combined as in Ref. Agashe:2014kda (); Schael:2013ita (); Group:2012gb () under the SM assumption.

This paper is aimed at the consistent interpretation of measurements in a different field theory than the SM, the SMEFT. When all dimension six operators are allowed to be present with arbitrary Wilson coefficients, interpretation of the measurements is modified. This modification comes about by an impact on the measurement itself, and in the mapping of the experimental results to different Lagrangian parameters. In this paper, we focus on the first question. We examine whether a bias on the measured arises due to SMEFT corrections and thus whether an additional theoretical error of this form should be included in fits of SMEFT parameters due to such a bias. After describing the modifications of parameters in the SMEFT in Section II, the potential bias is analysed in detail for the Tevatron measurement in Sections III-V. In Section VI we comment on the LEPII measurements, whereas Section VII is reserved for conclusions

Result | Value | Ref. |

DØ | 80.375 0.023 | Abazov:2012bv () |

CDF | 80.387 0.019 | Aaltonen:2012bp () |

Tev. Comb. | 80.387 0.016 | Agashe:2014kda () |

LEP threshold | 80.42 0.20 0.03 | Schael:2013ita () |

LEP direct | 80.375 0.025 0.022 | Schael:2013ita () |

LEP. Comb. | 80.376 0.033 | Schael:2013ita () |

Global Comb. | 80.385 0.015 | Group:2012gb () |

## II. The mass and Width in the SMEFT.

In the SM, at tree level, the input parameter set fix where

(1) |

The shift in the pole mass in the symmetric version of the SMEFT is given by Berthier:2015oma ()

(2) |

where . We use a to indicate a shift in a quantity due to the complete set of corrections present at leading order in the power counting in the SMEFT. We use the Warsaw basis of dimension six operators in the SMEFT Grzadkowski:2010es () that defines the Wilson coefficients . The cut off scale has been absorbed into the definition of the Wilson coefficients so that the mass dimension of these parameters is . The value of can be predicted in the SM with complete one loop Denner:1990tx () and even full two loop corrections Awramik:2003rn (). The full one loop corrections compared to the Born approximation to are a Beenakker:1996kt () correction. The size of this correction depends on how the tree level value of is related to the input observables. Absorbing universal radiative corrections into the parameters defining the width in an improved Born approximation reduces the size of the remaining perturbative corrections at one loop to Rosner:1993rj (); Beenakker:1996kt (); Denner:1990tx (). The effect of still neglected higher order perturbative terms is then expected to be a loop factor smaller than this variation.

The expression for the shift of in terms of the input parameter and the derived value in the SMEFT is given by

(3) |

where . Here and we are considering the massless fermion limit. Being conservative, so long as it is clear that neglected SMEFT corrections can have a non negligible impact on the theory error of an extracted value of at the Tevatron. This condition corresponds to a bound on the Wilson coefficients and the cut off scale of the form , which are the cases of interest motivated by the hierarchy problem.

## III. Spectra for Extractions of at the Tevatron.

We illustrate the effect of generalizing these measurements into the SMEFT following the analytic methods of Ref.vanNeerven:1982mz (). This is sufficient for our purposes as detector resolution effects can only be approximated without direct access to the experimental data, and are substantial.

The value of is extracted from Tevatron data using kinematic templates for distributions in the variables . The latter is found to have a small effect on the fit Abazov:2012bv (), so we neglect this spectrum. We define the transverse mass to be , with and the momenta and angle between the leptons in the perpendicular plane to the collision axis. We agree with the result in Ref.vanNeerven:1982mz () for for the effective kinematic spectra once the partonic production mechanism is factorized out, consistent with a narrow width expansion, and integrated over Parton Distribution Functions (PDF’s), generating an effective for . Note however the correction to the last term in the numerator of the function (derived from the Jacobian), correcting a typo in Ref.vanNeerven:1982mz (). Here , and . is the transverse momentum of the boson present due to the effects of PDFs. We find the following results for

(4) |

Still following Ref.vanNeerven:1982mz (), we have introduced a normalized Breit-Wigner resonance, consistent with a narrow width normalization in the limit for the unstable boson. The angular dependence and normalization of the partonic is given by

(5) |

where is the sum of the branching ratios to the specific lepton final states included in the analysis, and is a PDF dependent production cross section factor, that can be taken to be constant within , since the main SMEFT dependence of the production cross section has been included, by pulling out of this production cross section. We follow Ref. Berthier:2015oma () absorbing the SMEFT shifts of the input parameters into the redefined -coupling . We have also included the initial state partonic factor of and note that in our numerical simulations this leading factor of is replaced with consistent with the narrow width approximation factorization of the process. The angular dependence here is defined in the un-boosted rest frame w.r.t a axis along the direction, with the electron and neutrino momentum decomposed as

(6) | |||||

(7) |

We express in terms of

(8) | ||||

(9) |

where, having denoted , we have also related to the lepton transverse momenta. A similar calculation gives the result for the spectrum

(10) | |||||

(11) |

We find the phase space in this case to be

## IV. Variation in Extractions of .

Kinematic template fits to extract can be impacted by the presence of local contact operators in the SMEFT as follows. The pole mass is shifted compared to the expected value in the SM, as given in Eq. (2). This is how the constraint on measurements is intended to impact the fit constraint space of the SMEFT, at leading order in . The value of the width is modified, see Eq. (3). This shift is assumed to vanish when SM extractions of are performed and gives a theoretical error when interpreting these measurements in the SMEFT.

The overall normalization of both of the spectra is modified with a shift

(13) |

due to the SMEFT shift of the normalization of Eq. (5). This normalization effect is the correction that is not directly related to the -dependent normalization of the Breit-Wigner function.

In a flavour symmetric scenario, we can relate the CKM matrix in the SMEFT to the CKM matrix in the SM in a straightforward fashion as . The production spectra are the direct sum over all partonic quarks when considering hadronic collisions. Incorporating PDF’s, the partonic differential cross sections are convoluted and varied over the PDF’s in their uncertainty band in the experimental analyses. The flavour dependence of is expected to be subdominant to this SM variation. However, the remaining flavour universal variation due to is neglected in SM analyses, which leads to a further theoretical error.

Both and stem from shifts of the -couplings, and are therefore correlated. We decompose

(14) |

where captures all the correlation, and corresponds to directions in Wilson-coefficient-space where the overall normalization of the spectra is unchanged. In the limit we find

(15) |

where the proportionality factor is obtained by expressing in a basis of the space spanned by that includes as a basis vector. The relation is modified in a non-flavour-symmetric scenario, but the decomposition in Eq.(14) is still possible.

## V. Numerics.

To estimate these effects, we generate fit templates for each spectra varying in steps of 1.25 MeV around the central value of GeV Agashe:2014kda (). In generating these, we employ the tree level SM relation , fixed so that GeV Agashe:2014kda (). We extract the -distribution from the -distribution of Fig.(52) in D0:2013jba (), which is well approximated by the distribution

(16) |

with mode GeV and GeV. The follows the same distribution at a momentum scale lower by a factor of privateD0 ().

The -distribution is robust to variations of and we found it sufficient to set GeV when generating the -templates. The -distribution is very dependent on , and the result in Eq.(10) was averaged over the distribution in Eq.(16) when generating -templates. Detector effects are significant and we approximate them as a convolution of the calculated spectra with a Gaussian resolution function , taking . We choose the parameters to match the spectra published in Ref. Abazov:2012bv (); D0:2013jba (); Aaltonen:2012bp (); Aaltonen:2013vwa () both with and without detector effects. This is an approximation to the true acceptance, which depends on the detailed kinematics and not just the variable . However, it can reproduce the spectra of the above references reasonably well as shown in Fig. (3).

We choose an overall normalization to match the event counts, and thereby statistical significance found in the experiments, by matching the peak-height of our spectra to the results in Ref. D0:2013jba (); Aaltonen:2013vwa (), after approximating detector effects (thereby we eliminate the need to know and the integrated luminosity).

Using the same methodology, we have generated “data samples”, keeping but including the SMEFT variations discussed above. We obtain an estimated by a binned log-likelihood fit of the simulated data samples to the template set, using the Poisson likelihood function

where is the expected event count in bin cf. the template, and is the data event count. We then investigate the measurement bias as a function of the included SMEFT shift. We parametrize the shifts in and as described in Eqs.(14)-(15), and vary both by , consistent with TeV. The results are shown in Fig. 4. We also show the goodness-of-fit parameters corresponding to the best fit, for each of the SMEFT shifted data samples. In order to obtain the plots, we have done the analysis for many different instantiations of Poissonian counting noise, and averaged the results. Only yields a substantial bias on , but that this bias can be on the order of magnitude of and still yield reasonable values. Note that for the plots shown for the spectra respectively, so that a shift in the goodness of fit test is respectively yielding a bias . However, this shift is due to the normalization change of the data, compared to the templates, as can be seen by the effect of not introducing a significant bias. As the experimental collaborations float the overall normalization as a free parameter, this effect will not be present in the experimental result. Therefore, the measurement results are expected to reproduce with no significant bias, when interpreted in the SMEFT. This is our main conclusion.

## VI. LEPII mass extractions.

At threshold we find the leading order (in ) SMEFT correction to the Born cross section approximation of is

(17) |

Note the lack of s channel contributions due to possibly anomalous TGC parameters in the threshold limit. These corrections first appear at order , as is well known (see the review Beenakker:1994vn ()). We agree with the discussion in Ref.Hagiwara:1987pb () that threshold extractions can be relatively insensitive to normalization corrections, by fitting the shape of the cross section rise (in ) with a free normalization factor to account for the corrections in the first line. Further, only using specific final states in such extractions, the remaining corrections of the form can be corrected for. Unfortunately, the legacy LEP fit results on using threshold data assumes the SM, and combines final states under the SM assumption. As such, a further theoretical error must be assigned when using this data for the neglect of SMEFT corrections. The requirement to use data away from threshold to reduce statistical errors in a manner that makes the measurement competitive with Tevatron results introduces further SMEFT corrections due to anomalous TGC parameters. It is inconsistent to utilize the same data set for extractions of assuming vanishing anomalous TGC parameters in template fits at LEP, and then simultaneously use the same data with the extracted value of to constrain anomalous TGC parameters in the SMEFT. Considering the robustness of the Tevatron extractions, away from threshold LEP data should be reserved for direct constraints on anomalous TGC parameters. We discuss this issue in more detail in a companion paper fitpaper ().

## VII. Conclusions.

In this paper we have examined if considering the SMEFT generalisation of the SM introduces a measurement bias in reported values of the mass in Tevatron data.
Such a bias would have to be incorporated as an extra component of the theoretical error assigned in using these measurements, and was potentially dominant over any pure theoretical error
associated with mapping these measurements to a bound on the Lagrangian parameters given by Eqn. 2. We have found^{1}^{1}1To our surprise. that the effect of the SMEFT modification of the
measurement introduces a negligible bias in the extracted value of the mass. Our results show that mapping this very precise measurement to the SMEFT Lagrangian consistently (i.e. using Eqn. 2, a leading order result in the non-perturbative and perturbative expansion, to map to the SMEFT Lagrangian), is the dominant
issue in interpreting these results in the SMEFT. We stress that our results on the measurement bias are only the expected order of magnitude, however our estimate matches well with the estimate
given in Ref.Aaltonen:2013iut () on the correlation between errors in and assuming the SM. Two parameter extractions of , not assuming the SM, with the simultaneous reporting of a correlation matrix (expected to have small off diagonal entries troels ()) using template fits to transverse variables
is very well motivated in the SMEFT and can be robustly interpreted. We encourage the experimental collaborations to also perform such an analysis on the legacy Tevatron data used to extract .

Acknowledgements M.T. acknowledges generous support by the Villum Fonden and the Discovery center. We thank L. Berthier for collaboration on the companion paper fitpaper () and comments on the manuscript. We thank members of the Tevatron mass working group for helpful correspondence, and in particular Rafael Lopez de Sa. We also thank Andre Tinoco Mendes and Troels Petersen for very useful comments.

## References

- (1) L. Berthier, M Bjørn and M. Trott, arXiv:1606.06693
- (2) B. Grinstein and M. B. Wise, Phys. Lett. B 265, 326 (1991).
- (3) Z. Han and W. Skiba, Phys. Rev. D 71, 075009 (2005)
- (4) A. Pomarol and F. Riva, JHEP 1401, 151 (2014)
- (5) M. Ciuchini et al. arXiv:1410.6940 [hep-ph].
- (6) J. Ellis, V. Sanz and T. You, JHEP 1503, 157 (2015)
- (7) A. A. Petrov et al. Phys. Rev. D 91, no. 7, 073001 (2015)
- (8) L. Berthier and M. Trott, JHEP 1505, 024 (2015)
- (9) L. Berthier and M. Trott, JHEP 1602, 069 (2016)
- (10) A. David and G. Passarino, Rev. Phys. 1, 13 (2016)
- (11) S. Schael et al. Phys. Rept. 427, 257 (2006)
- (12) W. L. van Neerven et al. NIKHEF-H/82-20.
- (13) G. Arnison et al. [UA1 Collaboration], Phys. Lett. B 122, 103 (1983).
- (14) M. Banner et al. [UA2 Collaboration], Phys. Lett. B 122, 476 (1983).
- (15) J. Smith et al. Phys. Rev. Lett. 50, 1738 (1983).
- (16) V. D. Barger et al. Z. Phys. C 21, 99 (1983).
- (17) V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 108, 151804 (2012)
- (18) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 108, 151803 (2012)
- (19) K. A. Olive et al. [PDG Collaboration], Chin. Phys. C 38, 090001 (2014).
- (20) S. Schael et al. [LEP Electroweak Collaborations], Phys. Rept. 532, 119 (2013)
- (21) T. E. W. Group [CDF and D0 Collaborations], arXiv:1204.0042 [hep-ex].
- (22) K. Hagiwara and D. Zeppenfeld, Phys. Lett. B 196, 97 (1987).
- (23) B. Grzadkowski et al. JHEP 1010, 085 (2010)
- (24) A. Denner and T. Sack, Z. Phys. C 46, 653 (1990).
- (25) M. Awramik et al. Phys. Rev. D 69, 053006 (2004)
- (26) W. Beenakker et al., hep-ph/9602351.
- (27) J. L. Rosner, M. P. Worah and T. Takeuchi, Phys. Rev. D 49, 1363 (1994)
- (28) V. M. Abazov et al. [D0 Collaboration], Phys. Rev. D 89, no. 1, 012005 (2014)
- (29) T. A. Aaltonen et al. [CDF Collaboration], Phys. Rev. D 89, no. 7, 072003 (2014)
- (30) Private comm. with members of the D coll.
- (31) W. Beenakker and A. Denner, Int. J. Mod. Phys. A 9, 4837 (1994).
- (32) W. J. Stirling, Nucl. Phys. B 456, 3 (1995)
- (33) T. A. Aaltonen et al. [CDF and D0 Collaborations], Phys. Rev. D 88, no. 5, 052018 (2013)
- (34) T. Peterson, Private communication.