The truncation of the standard-model effective field theory, its validity andthe associated uncertainties have been discussed in meetings of the LHC EFT WG.Proposals were made by participants to address these issues. No consensus wasreached and no formal recommendation is therefore put forward at this time.None of the proposals has been approved or validated and further work is neededto establish a prescription. This note aims at summarizing the proposals andpoints of debate.<br

Berger, N.

Brivio, I.

Contino, R.

Dawson, S.

de Blas, J.

Degrande, C.

Durieux, G.

Falkowski, A.

Goertz, F.

Gritsan, A.

Grojean, C.

Lohwasser, K.

Maltoni, F.

Mimasu, K.

Panico, G.

Petrucciani, G.

Riva, F.

Savard, P.

Shepherd, W.

Vryonidou, E.

Wulzer, A.

Zhang, C.

English

MPG.PuRe

CERN-LPCC-2022-01CERN-LHCEFTWG-2021-002November 16, 2022LHC EFT WG note:Truncation, validity, uncertaintiesEditors: Ilaria Brivio,1,2 Sally Dawson,3 Jorge de Blas,4,5 Gauthier Durieux,5 GiovanniPetrucciani,6 Pierre Savard 7Proposal contributors:A. Roberto Contino,8 Adam Falkowski,9 Florian Goertz,10 Christophe Grojean,11 FabioMaltoni,12,13 Giuliano Panico,14 Francesco Riva,15 Andrea Wulzer16B. Céline Degrande,12 Fabio Maltoni,12,13 Ken Mimasu,17 Eleni Vryonidou,18 CenZhang19,20,21C. William Shepherd 22D. Nicolas Berger,23 Andrei V. Gritsan,24 Kristin Lohwasser251 Institut für Theoretische Physik, Universität Heidelberg, Germany2 Physik Institut, University of Zurich, Switzerland3 Department of Physics, Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A.4 CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada,Campus de Fuentenueva, E–18071 Granada, Spain5 Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland6 Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland7 University of Toronto, Toronto, ON M5S 1A7, Canada8 Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy9 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France10 Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany11 DESY, Notkestrasse 85, 22607 Hamburg, Germany12 Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholiquede Louvain, Chemin du Cyclotron, B-1348 Louvain la Neuve, Belgium13 Dipartimento di Fisica e Astronomia, Università di Bologna and INFN, Sezione diBologna, Via Irnerio 46, 40126 Bologna, Italy14 INFN Sezione di Firenze and Dipartimento di Fisica e Astronomia, Università di Firenze,Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy15 Départment de Physique Théorique, Université de Genève, 24 quai Ernest-Ansermet,1211 Genève 4, Switzerland16 Dipartimento di Fisica e Astronomia, Università di Padova, Padova, Italy17 Department of Physics, King’s College London, Strand, WC2R 2LS London, UK18 Department of Physics and Astronomy, University of Manchester, Oxford Road, Manch-ester M13 9PL, United Kingdom1arXiv:2201.04974v2  [hep-ph]  14 Nov 202219 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China20 School of Physical Sciences, University of Chinese Academy of Science, Beijing 100049,China21 Center for High Energy Physics, Peking University, Beijing 100871, China22 Department of Physics and Astronomy, Sam Houston State University, Huntsville TX,USA23 LAPP, Université Savoie Mont Blanc, CNRS/IN2P3, Annecy, France24 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218,USA25 Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UKThe truncation of the standard-model effective field theory (SMEFT), its validity andthe associated uncertainties have been discussed in a dedicated meeting on January 19,2021. Answering a call issued beforehand, three proposals were presented: A, B, and C .A preliminary version of the present note summarizing them was written by the editors,submitted for feedback to the proponents, and presented at the May 3 general meeting.Comments from the wider community were collected in an online document. Experimentalcollaborations provided formal feedback during a second dedicated meeting on June 28.The first version of this summary note was released on January 12, 2022. Proposal Dwas circulated on May 18 and comments collected, ahead of the May 23 general meeting.Extensive discussions took place with the whole community but no consensus emerged.None of the proposals has been approved or validated. No recommendation is thereforeput forward at this time and this note only aims at summarizing the different pointsraised at meetings. Further work is needed to establish a prescription. In particular, thebenchmarking of the different proposals on the working-group fitting exercise has beenproposed and discussed.1 Common groundThere are various points of agreement between proponents of various schemes for dealingwith truncation uncertainties. Most participants agree that:1. most near-future experimental analyses will not aim at probing simultaneously bothdimension-six and dimension-eight operator contributions. The SMEFT truncationof interest is then at the level of dimension-six operators.2. although they only constitute a partial set of 1/Λ4 corrections, the squares of ampli-tudes featuring a single dimension-six operator insertion provide a convenient proxyto estimate 1/Λ4 corrections, as they are well defined and unambiguous. They areindeed gauge invariant and can be translated exactly from one dimension-six operatorbasis to the other. See Appendix A for more detailed statements.3. estimating the relative contributions of dimension-six and dimension-eight opera-tors requires a power counting covering a given range of new-physics scenarios and2depends on its parameters (e.g. mass scales and couplings). Being able to com-pute the dimension-eight dependence of observables is insufficient, as a prescriptiondetermining the relative magnitude of dimension-six and dimension-eight operatorcontributions is still needed.2 ProposalsSpecific points made in each proposal are succinctly summarized under Arabic numbers.Additional considerations highlighting pros and cons are listed under Latin lowercases.Experimental results obtained by following one of the proposals would not be suffi-cient to allow for the a-posteriori application of the other prescription.2.1 Proposals A and BProposal A and B are similar thus discussed together. They advocate:1. providing full multidimensional information on the constrained EFT parameter space,to allow for the proper interpretations in (classes of) new-physics scenarios, andtherefore for the EFT validity assessment.(a) Providing individual and marginalized constraints, on single coefficients and intwo dimensions, is a first but insufficient step. Full likelihoods, or covariancematrices in Gaussian cases, should be published.(b) Correlations between operator coefficients deriving from specific new-physicsassumptions may exclude parameter-space region where linear and quadraticdimension-six truncations diverge significantly and therefore improve the EFTvalidity.2. including squared dimension-six dependences by default and comparing results withthose obtained in the linear SMEFT approximation.(a) The conclusions drawn from this comparison are more qualitative than quanti-tative.In case the two sets of results match each other, one can conclude on the generalvalidity of the dimension-six truncation, as situations in which dimension-eightcontributions would dominate over linear dimension-six ones are likely patho-logical. No additional assumption is required on new physics.When the linear and quadratic results differ significantly, constraints can onlybe applied in scenarios where dimension-eight contributions are generically sup-pressed with respect to quadratic dimension-six ones.(b) The linear-quadratic comparison does not reflect the convergence of the EFTseries when interference contributions suffer (accidental or understood) suppres-sions.(c) The assignment of a truncation uncertainty is not prescribed.3(d) Purely linear fits can be technically more involved, as they formally allow eventrates to turn negative.3. providing experimental results as functions of the maximal energy probed in the dataemployed, introducing where necessary an upper cut (denoted e.g. Ecut orMcut). Dataand prediction are compared in the same phase-space region. This procedure, oftenreferred to as clipping, provides the necessary information to verify the EFT validityfor (classes of) models and enhances the range of scenarios for which a valid EFTinterpretation is possible.(a) For different cut values, different analysis strategies may be required. While ratemeasurements could provide the highest sensitivities at high energies, differen-tial observables may be required to probe the relevant operators at moderateenergies. Upper energy cuts should therefore be considered from the onset inthe analysis design.(b) The reconstruction of the relevant variable to cut on may complicate experimen-tal analyses and result in additional systematic uncertainties (e.g. in final statesfeaturing missing energy). Many EFT analyses (e.g. STXS in the Higgs sector)do however already measure suitable energy variables (e.g. bosons’ transversemomenta, or jet invariant masses) as sensitivity arises from these. Example ofexperimental analyses having adopted such a clipping procedure include dibosonmeasurements by CMS (in Wγ, WZ, ZZ final states).(c) Combining different observables from different processes, each using an uppercut on a different variable, may also raise questions. What variables and cutvalues are compatible in different processes? The study of specific scenarios maybe informative in that regard. Conclusions are expected to be model dependent.(d) Repeating global analyses for several upper cut values would be more costlyboth computationally and in term of personpower.(e) Applying a cut on the EFT signal simulation instead of the data was proposedby experimental collaborations, as described later in section 2.3. As existingrun-2 analyses will not be re-designed, modifying only the signal simulationcould in particular be used to incorporate, into EFT fits, analyses in which nogood energy variable was measured. The proponents A and B however judgethat comparing data in a given phase-space region (without energy cut) withpredictions in a different one (with energy cut) is inconsistent. Further studiescould clarify whether the two approaches are practically equivalent in cases ofinterest.4. assessing, a posteriori (even after the combination of different measurements), therange of models for which the extracted constraints apply, using this Ecut information.The experimental results themselves would therefore not incorporate assumptionsabout new-physics models. This approach allows theorists to interpret the results inthe context of specific (classes of) models.(a) Quantifying missing dimension-eight contributions would require more effort.4A posteriori, one could approximately reproduce the experimental analysis andinclude estimates of dimension-eight contributions in the relevant phase-spaceregion to assess their impact.A priori, one could consider treating linear and quadratic contributions as inde-pendent in experimental analyses and, in interpretations, rescale the quadraticcontribution to estimate dimension-eight effects. The significant increase in thenumber of parameters to be fitted may however not be practical.2.2 Proposal CProposal C advocates:1. using squared dimension-six contributions, which can readily be computed with ex-isting tools, as proxies for missing dimension-eight terms at order 1/Λ4.(a) As the dimension-eight contributions at that order arise from interferences withSM amplitudes, the dimension-eight contributions could have different kine-matic distributions or suffer some accidental suppressions.(b) It is claimed (see online comments) that such contributions violate gauge in-variance, in contradiction with item 2 and Appendix A.2. employing a power-counting rule that would encompass many new-physics models,to estimate dimension-eight contributions from squared dimension-six ones.(a) The line drawn between classes of models that are, and are not, covered by thechosen power-counting rule is somewhat arbitrary.(b) For specific classes of scenarios, this will necessarily be overly conservative.Employing different power-counting rules for different classes of scenarios wouldpermit to quote tighter constraints in the specific cases where they apply.3. considering the known squared dimension-six terms together with the dimension-eightestimates as theoretical uncertainty on the linear dimension-six signal.(a) As the dimension-six squared contributions are known, they may not need tobe included in uncertainties.(b) This uncertainty depends on the SMEFT parameter point and could thereforebe practically difficult to include in analyses.(c) Purely linear fits can be technically more involved, as they formally allow eventrates to turn negative.4. folding these uncertainties directly into experimental analyses.(a) This renders experimental results model dependent, as they then rely on aspecific scaling between dimension-six and dimension-eight operator coefficients.55. multiplying the squared dimension-six terms by the following factor to obtain trun-cation uncertainties:1 +√N8g2SMC6Λ2√1 + 1C26Λ4(1)where N8 is an estimate of the number of contributing dimension-eight operators, gSMis the relevant SM coupling, C6 is a dimensionful dimension-six operator coefficient,and Λ is a scale (such that, if identified with the physical BSM mass scale in atwo-to-two process, one expects C6Λ2 ∼ g2BSM).(a) The classes of models which are covered by this choice is yet to be determined.Without motivation from new-physics models, the various factors may seem adhoc.6. in cases where dimension-eight contributions can be computed, the functional formof dimension-six contributions squared would not need to be used as proxy for thedimension-eight ones and a power counting could be used for dimension-eight (anddimension-six) coefficients directly.2.3 Proposal DThe experimental proposal D suggests to clip the EFT simulation, instead of clipping thedata as prescribed by theorists in item 3 of section 2.1. It stressed that:1. redesigning existing analyses in cases where no energy variable suitable to performdata clipping was measured is difficult (item 3b) and, where the final fitted vari-able is not an energy one (e.g. if multivariate techniques are used to maximise EFTsensitivity), re-running analyses for several energy cut values is costly (item 3d).(a) Presently, many relevant (e.g. unfolded) analyses measure the distribution ofan energy variable from which bins could straightforwardly be removed whenperforming global EFT fits (item 3b).2. clipping the EFT simulation in the tail of an energy variable yields indications aboutthe impact of this kinematic region on extracted EFT parameters. Proper constraintsare derived using the full dataset and simulation, and a comparison with results ob-tained with simulation clipping provides qualitative information about the sensitivityto high energies.(a) Data clipping, by confronting data and simulation in the same phase-spaceregion, additionally allows to derive proper constraints for models with charac-teristic energy scales lower than the energies directly probed in the consideredmeasurement (item 3).3. the EFT simulation can be clipped on any desired variable and at any value, at thecost of a re-computation of the EFT prediction, while clipping the data is only pos-sible on measured variables and at bin boundaries (without re-running the analysis).6(a) There is however no universal best choice of clipping variable. Different modelscan induce energy growths in different kinematic quantities (e.g. s and t channelmediators in the Drell-Yan process, both leading to four-fermion operators).Different operators contributing to the same process could also induce energygrowth in different variables.(b) As for data clipping, the combination of different processes clipped on differentvariables, at different values, is an open question (item 3c). Explicit simplifiedmodels could be examined to inform such choices. One could explore the pos-sibility of using quantile of the SM distribution in each energy variable. Fordata clipping, a public fitting framework with high flexibility in the bins of dataincluded could allow the community to adopt the choices best suited for anyparticular interpretation.4. the clippings of data and EFT simulation are equivalent if performed on a variablethat is actually measured. For data clipping at the reconstructed level and simulationclipping at the truth Monte Carlo level, the two procedures would only differ by finiteresolution effects. If the data does not match the SM expectation, a poor fit wouldbe obtained.(a) On the other hand, clipping the EFT simulation on a variable that is not mea-sured could lead to pathologies arising from the mismatch between the phasespaces of the data and prediction (item 3e). Correlations between the clippedand measured variables could then lead to an overestimation of the strengthof the constraints derived when the data agrees with the SM. In case the datadeviates from the SM, the extracted operator coefficients could also be biasedaway from their true value, in a process- and observable-dependent fashion.Appendix A Well-defined and unambiguous squaresIn the 1/Λ2 expansion of dimension-six amplitudes (S-matrix elements), the zeroth andfirst terms, ASM and A6/Λ2, are separately gauge invariant. The A6/Λ2 term contains alland only 1/Λ2 contributions to the amplitude. The squares of those two terms —|ASM|2,Re{A∗SMA6}/Λ2, |A6|2/Λ4— are thus separately gauge invariant too. More terms do appearin the squared amplitude at order 1/Λ4: from dimension-eight operators, amplitudes withtwo dimension-six operator insertions, or field redefinitions expanded to 1/Λ4 order. Eachof these other subclasses of 1/Λ4 contributions to the squared amplitude is in general notseparately gauge invariant: only the full Re{A∗SMA8}/Λ4 is, where A8/Λ4 collects all andonly 1/Λ4 contributions to the amplitude and is itself separately gauge invariant.As a consequence of the equivalence theorem, dimension-six operators related byclassical equations of motion have identical amplitudes (S-matrix elements) to 1/Λ2 order.Equivalent operator bases can be defined by exploiting this freedom, changing operatornormalizations, or taking linear combinations of them. The linear dimension-six amplitudeA6/Λ2, including all and only 1/Λ2 contributions, can thus be translated exactly from one7dimension-six operator basis to the other by a linear transformation between the two sets ofoperator coefficients {c6} and {c′6}. The same transformation can also be used to translateexactly the square of this linear dimension-six amplitude |A6|2/Λ4 from one dimension-sixbasis to the other.For these two specific reasons, the square of the linear dimension-six amplitude|A6|2/Λ4, where A6/Λ2 contains all and only 1/Λ2 contributions to the amplitude, canbe qualified as well-defined and unambiguous. It can thus for instance be employed as aconvenient proxy for estimating full 1/Λ4 contributions.Acknowledgments: This summary note has been written under the umbrella of theLHC EFT WG and we would like to thank all the participants who contributed to thecorresponding discussions.8

Truncation, validity, uncertainties

http://hdl.handle.net/21.11116/0000-000C-07E9-6

Truncation, validity, uncertainties

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