EFTfitter---A tool for interpreting measurements in the context of effective field theories
Nuno Castro, Johannes Erdmann, Cornelius Grunwald, Kevin Kröninger, Nils-Arne Rosien
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
EFT fitter —A tool for interpreting measurements in the contextof effective field theories
Nuno Castro , , Johannes Erdmann , Cornelius Grunwald , KevinKr¨oninger , Nils-Arne Rosien Laborat´orio de Instrumenta¸c˜ao e F´ısica Experimental de Part´ıculas, Departamento de F´ısica, Universidade do Minho,4710-057 Braga, Portugal Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universidade do Porto, 4169-007 Porto, Portugal Lehrstuhl f¨ur Experimentelle Physik IV, TU Dortmund, Otto-Hahn-Straße 4a, 44227 Dortmund, Germany II. Physikalisches Institut, Universit¨at G¨ottingen, Friedrich-Hund-Platz 1, 37077 G¨ottingen, GermanyReceived: date / Accepted: date
Abstract
Over the past years, the interpretation ofmeasurements in the context of effective field theo-ries has attracted much attention in the field of par-ticle physics. We present a tool for interpreting sets ofmeasurements in such models using a Bayesian ansatzby calculating the posterior probabilities of the corre-sponding free parameters numerically. An example isgiven, in which top-quark measurements are used toconstrain anomalous couplings at the
W tb -vertex.
Keywords
Effective field theory · Combination ofmeasurements · Bayesian inference · Uncertaintypropagation
With the recent start of the Run-2 of the LHC, searchesfor physics beyond the Standard Model (BSM) willreach unprecedented sensitivity. The LHC’s increasedcentre-of-mass energy opens a new kinematic regimeand enhances the direct production cross section ofheavy, yet unknown, particles—if they exist. It is a goodtime for bump hunters.On the other hand, it is not obvious that the massscale of such new particles is anywhere near the energywhich can be reached by current, or even future, accel-erators. However, in contrast to the direct production ofheavy particles, their impact on observables accessibleat current collider experiments can be probed indirectlyin the context of effective field theories. Such theoriesextend the Standard Model (SM) Lagrangian by termsallowed in quantum field theory and which share thegauge symmetries of the SM. These terms contain oneor several effective operators and corresponding coeffi-cients, often referred to as Wilson coefficients, which de- fine the individual strength of these operators. Depend-ing on the type of operator, the additional terms in theLagrangian can have an impact on different observables,which, in turn, can be compared to a set of correspond-ing measurements. Comparisons of SM predictions andobservations can be used to constrain the Wilson coef-ficients by propagation of uncertainty. The strategy toindirectly infer on the parameters of a physics model,may it be an effective model or a full model, has provento be successful in a variety of applications, e.g. in thefield of flavor physics [1], super-symmetry [2], or elec-troweak precision measurements [3].This paper describes a generic tool, the EFT fitter ,for performing such interpretations in the context ofuser-defined physics models and formulating them interms of Bayesian reasoning. Emphasis is placed on thestatistical treatment of the combination of measure-ments correlated by their uncertainties as well as onoften overlooked issues in interpretations, such as thenecessity to consider model-specific efficiency and ac-ceptance corrections for the measurements, or the pres-ence of physical constraints on observables and parame-ters. An example is given for the case of an effective fieldtheory in the top-quark sector, which is an active fieldof research. This example is motivated by the wealth ofexperimental data delivered by the Tevatron and LHCexperiments and the increasing precision of measure-ments involving top quarks. Also from a theoretical per-spective, the interpretation of top-quark measurementsis attractive: recent calculations, e.g. predictions of thecross section of top-quark pair production [4,5,6], reachnext-to-next-to-leading order (NNLO) precision in per-turbative QCD. A historical example for the interpre-tation of experimental data in the context of top quarks a r X i v : . [ h e p - e x ] M a y is the successful prediction of the top-quark mass fromelectroweak precision measurements, see, e.g., Ref. [7].This paper is organised as follows. Section 2 de-scribes the statistical procedure of combining severalmeasurements, while their interpretation is discussedin Section 3. The numerical implementation of theEFT fitter is introduced in Section 4, and an examplefor interpretations in the field of top-quark physics isgiven in Section 5. The paper is concluded in Section 6. In Bayesian reasoning, inference of the free parameters λ of a model M is based on the posterior probabilityof those parameters given a data set x , p ( λ | x ). It iscalculated using the equation of Bayes and Laplace [8], p ( λ | x ) = p ( x | λ ) · p ( λ ) p ( x ) , (1)where p ( x | λ ) is the probability of the data, or likeli-hood, and p ( λ ) is the prior probability of the param-eters λ . In Bayesian literature, the normalisation con-stant in the denominator, p ( x ) = (cid:90) d λ p ( x | λ ) · p ( λ ) , (2)is often referred to as the evidence.In the following, we distinguish two types of mod-els: i) those for which the parameters can be directlymeasured from the data, and ii) those for which this isnot the case. Models of type ii) typically predict valuesof physical quantities, or observables, y , which dependon the parameters of the model, λ , so that y = y ( λ ).For example, the predicted cross sections in scatteringprocesses depend on the couplings and masses of theparticles described by physics models. These couplingsand masses are often not predicted by the models them-selves, in which case they are free parameters. Cou-plings, e.g., can often only be estimated indirectly fromthe measurements of cross sections and other observ-ables. This case will be discussed further in Section 3.On the other hand, models of type i) are often used forthe plain combination of measurements in which thephysical quantities themselves, e.g. cross sections, an-gular distributions or branching ratios, are interpretedas model parameters, i.e., y = λ . We will discuss thiscase in the following.2.1 Combination of measurementsFollowing the notation of Ref. [9], we assume to have N observables, y i ( i = 1 , . . . , N ), which are estimated based on n measurements, x i ( i = 1 , . . . , n ). Each quan-tity y i is measured n i ≥ n = (cid:80) Ni =1 n i ≥ N . We adopt the common assumption that the like-lihood terms in Eqn. (1), p ( x | y ), have a multivariateGaussian shape, and that the uncertainties of the mea-surements of x i can be correlated. The elements of thesymmetric and positive-semidefinite covariance matrixare M ij = cov[ x i , x j ] . (3)Assuming M different sources of uncertainty, the co-variance matrix can be decomposed into contributionsfrom each source,cov[ x i , x j ] = M (cid:88) k =1 cov ( k ) [ x i , x j ] . (4)The likelihood can then be expressed as − p ( x | y ) = n (cid:88) i =1 n (cid:88) j =1 [ x − U y ] i M − ij [ x − U y ] j , (5)where the elements U ij of the n × N -matrix U are unityif x i is a measurement of the observable y j , and zerootherwise.The best linear unbiased estimator (BLUE) [9,10]can be found by minimising the expression in Eqn. (5),while an estimator in the Bayesian approach is con-structed by inserting Eqn. (5) into the RHS of Eqn. (1),and by specifying prior probabilities for the parameters.The prior probabilities can include physical constraints,e.g. the requirements that cross-sections can only takepositive values or that branching ratios lie between zeroand one. Prior knowledge can also come from auxiliarymeasurements or theoretical considerations. It is worthnoting that combined estimates of physical quantitiesbased on individual posterior probabilities need to becleaned from the corresponding prior probabilities, i.e.the prior information about a parameter should onlybe included once in the overall combination. It shouldalso be noted that prior probabilities, and in particularphysical constraints, can lead to a strong non-Gaussianshape of the resulting posterior probability distribution,even if the input measurements are assumed to be de-scribed by Gaussian probability densities.Typical estimators are the set of parameters whichmaximise the posterior probability, the mean values ofthe posterior probability distribution, or the set of pa-rameters which maximise the marginal probabilities, p ( y i | x ) = (cid:90) (cid:89) j (cid:54) = i d y j p ( y | x ) . (6)For uniform prior probabilities, and in the absence offurther constraints, the global mode of the posteriorcorresponds to the BLUE solution [10]. The uncertainty on y i can be defined as the central interval containing68% probability, the set of smallest intervals contain-ing 68% probability or simply the standard deviationof the marginalised posterior. These three measures areequal for Gaussian distributions. Similarly, simultane-ous estimates of the uncertainties on y i and y j can beobtained by the two-dimensional contours of the small-est intervals containing, e.g., 39% or 68% probability. Upper and lower limits on y i are typically set by calcu-lating the 90% or 95% quantiles of the correspondingmarginal posterior distribution.2.2 Uncertainties of the correlationAlthough it is often straightforward to obtain estimatesof the quantities y and of their uncertainties, it is nottrivial to quantify the correlation induced by the dif-ferent sources of uncertainties. If, e.g., sources of sys-tematic uncertainty have an impact on several of thosemeasurements, the correlation is often assumed to beextreme ( ρ = ± x i and x j , are obtained from common setsof simulated data and the linear correlation coefficient ρ ij = cov[ x i , x j ] /σ i σ j between the estimates is calcu-lated. Here, σ i and σ j are the standard deviations of x i and x j , respectively.If no reliable estimate of the correlation is possible,one can associate correlation coefficients with nuisanceparameters ν and choose suitable prior probabilities forthese parameters. Necessary requirements for the pri-ors are that the correlation coefficients are constrainedto be in the interval [ − , +1], and that the covariancematrix remains positive-semidefinite for all possible val-ues of ν . They should, however, parameterise the priorknowledge about the correlation, e.g. by restricting thecorrelation coefficients to be positive or by favouringmild correlations. Depending on the problem, it can bedifficult to formulate such priors analytically. In par-ticular, it is advisable to not allow values resulting incorrelation coefficients of ρ = ± M , and thus the likelihood, arethen functions of ν , so that p ( x | y , ν ). The prior prob-ability can be factorised, p ( y , ν ) = p ( y ) · p ( ν ), if y and ν are assumed to be independent, which is typically the The classical one-sigma contour contains 39% probabilitywhile the 68% contour is typically shown in the field of par-ticle physics [11]. case. In order to obtain a function depending only on y alone, all nuisance parameters are integrated out, p ( y | x ) = (cid:90) d ν p ( y , ν | x ) , (7)and the estimates of y are obtained as before from theresulting marginal posterior probabilities.2.3 Propagation of uncertaintyIn cases where the probability density for a quantity f ( y ) is needed, the uncertainty on y needs to be prop-agated to f . For Gaussian posterior probabilities, thepropagation of uncertainty is often done using the well-known rules for uncertainty propagation. These implythat the function f can be linearised in y and that theposterior probability for the quantity of interest alsohas a Gaussian shape. Since this is not always the case,and since the posterior of the combination does nothave to be a Gaussian due to the additional prior in-formation, we instead propose to use a numerical eval-uation of the uncertainty: if it is possible to samplefrom the posterior distribution p ( y | x ), one can calcu-late the target quantity f ( y ) for each sampled point.The obtained frequency distribution for f converges tothe posterior probability distribution p ( f | x ) in the largesample limit. As discussed in Section 4, such samplingcan be done on-the-fly when using Markov Chain MonteCarlo (MCMC).2.4 Ranking the impact of individual measurementsOne is often interested in how much a single measure-ment contributes to a combination. In BLUE averages,the contributions are added linearly using weights: thelarger the weight, the more important the measure-ment. Although counter-intuitive at first sight, theseweights can also take negative values, induced by strongnegative correlations, see e.g. the discussion in Refs. [9,10].There are several ways to estimate the impact ofindividual measurements in a combination. For the ap-proach discussed here, we propose to repeat the com-bination while removing one measurement from thecombination at a time. The measurements can thenbe ranked according to the resulting increase in uncer-tainty compared to the uncertainty of the overall com-bination. For combinations with two ( n ) physical quan-tities, the area (volume) of the smallest contour (hyper-sphere) covering, e.g., 68.3% of the posterior probabil-ity, can be used as a rank indicator. The motivation for this choice is to answer the question how the combina-tion would change if a particular measurement had notbeen considered in the combination.Regardless of the choice of rank indicator, theestimators themselves, their uncertainties and thecorrelations between the physical quantities should bemonitored during this procedure. Note that the uncer-tainty can also decrease if particular measurements areremoved, as in the case of outliers.2.5 Ranking the impact of individual sources ofuncertaintyThe situation is similar when one aims to rank thesources of uncertainties in order of their importance. Wepropose to repeat the combination while removing onesource of uncertainty from the combination at a time.The sources of uncertainty can then be ranked accord-ing to the resulting decrease of uncertainty comparedto the uncertainty of the overall combination. The rankindicator can be extended to n -dimensional problems.From a practical point of view, this approach allowsanswering the question which source of uncertainty ismost important to improve in future iterations of themeasurements. Estimating the parameter values λ of a complex physicsmodel M based on (the combination of) measurementsand the subsequent propagation of uncertainties is aninverse problem which is often ill-posed and in mostcases difficult to solve. We propose here to re-formulatethe problem discussed in the previous section in the fol-lowing way: instead of directly identifying the observ-ables with the fit parameters, we instead fit the free pa-rameters of the physics model under study based on therelation between the observables and the parameters.If the model predicts observables y i for each set of pa-rameter values λ , y i = y i ( λ ), these are then comparedto the measurements x i using a multivariate Gaussianmodel. The same formalism as in Section 2.1 can beused to estimate λ for a given data set. The likelihoodof the model is p ( x | λ ) = (cid:90) d y p ( x | y ) · p ( y | λ ) , where p ( y | λ ) = δ ( y − y ( λ )). It is worth noting thatone has to formulate prior probabilities in terms of the model parameters and not in terms of the measure-ments themselves. Physical constraints can be incorpo-rated into the model predictions, ‘external knowledge’can be viewed as an additional measurement. Using the same framework also helps to includemeasurements and physical quantities in the analysiswhich would otherwise be combined separately, e.g.by a working group concerned with cross sections andanother one interested in angular distributions. How-ever, due to common sources of systematic uncertaintiesand overlapping data sets, the posterior probability ofthe two measurements shows a correlation—a fact thatneeds to be considered in the global fit to a physicsmodel.
The EFT fitter builds on the Bayesian Analysis Toolkit(BAT) [15] which is a software package written in C++that allows the implementation of statistical modelsand the inference on their free parameters. Severalnumerical algorithms can be used to perform the com-bination and interpretation steps introduced in this pa-per: marginal distributions can be calculated using, forexample, Markov Chain Monte Carlo, while global opti-misation can be done using the Minuit implementationof ROOT [16].4.1 Definition of a model and observablesThe key component of the EFT fitter is the user’s defi-nition of a model. It is simply characterised by a set offree parameters, e.g. couplings and masses, and by thepredictions of observables as a function of the model’sparameters. Note that the model is not automaticallyderived from a user-defined Lagrangian and it is thusnot constrained to a particular class of models. As aconsequence, however, the predictions from the modelare not required to be consistent, and it is the user’sresponsibility to formulate a meaningful set of predic-tions. Of course, interfaces to more complex softwaretools can be included in the model. It is a virtue of the Bayesian formalism that an update ofknowledge is trivially obtained by defining results from pre-vious measurements or other considerations as prior proba-bilities of a new measurement. The code is available at https://github.com/tudo-physik-e4/EFTfitterRelease and includes thecode for the example discussed in this paper. The BAT code is available at http://mppmu.mpg.de/bat/ . fitter is a set of measurementsincluding a break-down of the uncertainties into sev-eral categories, e.g. statistical uncertainties and differ-ent sources of systematic uncertainties. In addition, thecorrelation between the measurements for each categoryof uncertainty needs to be provided.It is worth noting that the different measurementsneed to be unified in a sense that the sources of uncer-tainties are treated in the same categories throughoutall measurements, e.g. uncertainties related to the re-construction of objects in a collider experiment shouldhave one or more well-defined categories: uncertain-ties on the luminosity, efficiencies or acceptances shouldeach have their own categories, etc.The measurements can be any measurable quantity,most commonly a cross section, a mass or a branchingratio. It can also be an unfolded spectrum, in whichcase each bin is treated as an individual measurementand the full unfolding matrix is needed as an input. Theunfolding matrix provides the acceptances and efficien-cies, which can be treated consistently in the EFT fitter as described in Sec. 4.3.All of these inputs are provided in a configurationfile in xml format. The file contains information aboutthe number of observables, the number of measure-ments of these observables, the number of uncertaintycategories and the number of nuisance parameters tobe used in the fit. The allowed range for each observ-able has to be provided. For each measurement, thename of the observable, its measured value as well asits uncertainty in each uncertainty category have to beprovided as well. Furthermore, measurements can beomitted from the fit by flagging them as inactive. Thecorrelation matrix is provided in the same configurationfile. The correlation coefficients between measurementscan be treated as nuisance parameters in the fit. Foreach nuisance parameter, the measurements that thecorrelation coefficient refers to as well as its prior prob-ability have to be specified.Apart from the measurements, the user needs tospecify prior probability densities for each model pa-rameter. These can be chosen freely, e.g. uniform,Gaussian or exponentially decreasing functions. Physi-cal boundaries for parameters and observables are ad-dressed by the definition of the prior probabilities andthe predictions for observables, respectively. A secondconfiguration file in xml format is used to specify theprior probability densities for the parameters, theirminimum and maximum values, as well as their SMpredictions. Sometimes it is necessary to include external inputin a combination, e.g. measurements from low-energyobservables, b -physics or cosmology. These can eitherbe treated as additional measurements or as priors onthe parameters, depending on the type of informationprovided. In both cases, it is usually difficult to esti-mate the correlation between the different inputs andthe user should carefully consider whether the choice ofcorrelation made has a strong effect on the interpreta-tion of the data.4.3 Treatment of acceptance and efficiency correctionsA problem often encountered when measurements areinterpreted in terms of BSM contributions, is that theacceptances and efficiencies may be different for dif-ferent BSM processes, while measurements are mostlyperformed assuming the acceptances and efficiencies ofSM processes. An example is the measurement of thecross section for the pair production of a particle, whichin BSM scenarios might also be produced from an addi-tional resonance decay. The acceptance times efficiencyfor the BSM process may be different than for theSM process if the measurement requires a minimummomentum for the final-state particles. These require-ments may be more frequently fulfilled in the BSM sce-nario if the resonance has a very high mass. It is alsoclear that the acceptance and efficiency may depend onthe parameters of the BSM theory, such as the mass ofthe resonance in this example.The EFT fitter addresses this problem by separat-ing observables, measurements and parameters, so thatthe acceptance and efficiency can be corrected for whencomparing the prediction for the observables with themeasurements. For the SM process, this acceptance andefficiency correction is equal to unity, but for BSM mod-els, the correction may differ from one and it is, in gen-eral, a function of the parameters of the theory.4.4 OutputA brief summary of the fit results is provided in a textfile, while four figure files are provided for a more de-tailed analysis of the fit results. In one figure file, allone-dimensional and two-dimensional marginalised dis-tributions of the fitted parameters are saved. In twofigure files, the estimated correlation matrix of the pa-rameters and a comparison of the prior and posteriorprobability density functions for the different param-eters are shown, respectively. A last figure file showsthe relations between the parameters of the model andthe observables as defined by the underlying model. An additional text file provides the post-fit ranking of themeasurements determined as described in Sections 2.4and 2.5.4.5 Structure of the implementationThe code is structured such that a new folder needsto be created for a specific analysis. Folders are pro-vided for the example discussed in this paper (Sec. 5)as well as a blank example a user can start from. Eachsuch folder contains a subfolder for the input configura-tion files and an empty folder for the result files. It alsocontains a rather generic run file which holds the mainfunction for the run executable. The model specific de-tails, such as the relation between the parameters andthe observables, the acceptance and efficiency correc-tion etc. are implemented in a class inheriting from BCMVCPhysicsModel . In this class, it is necessary toprovide in particular a concrete implementation of thevirtual method
CalculateObservable(...) . The codeis then compiled using a Makefile, which is provided. AREADME file contains further details for users to getstarted.
As an example for applications of the EFT fitter , wediscuss the constraints on anomalous top-quark cou-plings from two sets of observables: the measurementof the polarisation of W -bosons produced in top-quarkdecays and the measurements of the t -channel top-and antitop-quark cross sections. Similar cases havebeen discussed in Refs. [17,18,19,20,21,22,23,24,25].The concrete model we use is that of Ref. [17], wherethe Lagrangian describing the W tb -vertex does notonly include a purely left-handed coupling with relativestrength V L , but also a right-handed vector couplingwith strength V R as well as left- and right-handed ten-sor couplings g L and g R . The generalised Lagrangianthen takes the form L = − g √ bγ µ ( V L P L + V R P R ) tW − µ − (8) g √ b iσ µν q ν M W ( g L P L + g R P R ) tW − µ + h.c. , (9)where g is the weak coupling constant, P L and P R areleft- and right-handed projection operators and M W isthe mass of the W -boson. In the SM, the left-handedcoupling strength is given by V L = | V tb | ≈
1, while thethree other coupling strengths are V R = g L = g R = 0. The physical model defined by the Lagrangian inEqn. (8) has four free parameters: V L , V R , g L , g R . Forsimplicity, we assume these parameters to be real.5.1 Observables and predictionsThe observables described in the following are calcu-lated based on Refs. [17,18,26]. The masses of the topquark, the W -boson and the bottom quark are assumedto be 172 . . . W -bosons produced in top-quark decays can beleft-handed, right-handed and longitudinally polarised.The fraction of events with either of these polarisationsare f L , f R and f , and often referred to as helicity frac-tions . At NNLO accuracy in the strong coupling, thesefractions are predicted to be f L = 0 . ± . f R =0 . ± . f = 0 . ± .
005 in the SM [27].Assuming the Lagrangian defined in Eqn. (8), thesefractions are functions of the four coupling strengths.As an example, Fig. 1 shows the predicted fraction ofleft-handed W -bosons as a function of any of the cou-plings in the range [ − , +1], while keeping the otherthree couplings fixed to their SM values. While thevariation of V R and g L results in variations of f L fromabout 15% to a maximum of roughly 30% at V R = 0and g L = 0, g R has a stronger impact. The value of f L ranges from 0% to about 70% with a minimum atapproximately g R = 0 .
5. As expected, there is no de-pendence of the helicity fractions on V L . Since the sumof the three fractions is unity, only two of the three, f L and f , are considered as well as their correlation.Similarly, the cross sections for single top- andantitop-quark production in the t -channel at a centre-of-mass-energy of √ s = 7 TeV are predicted in the SMto be σ t = 41 . +1 . − . pb and σ ¯ t = 22 . +0 . − . pb at NNLOaccuracy in the strong coupling [28]. As an example,the single top-quark cross section as a function of thefour coupling strengths is illustrated in Fig. 2, againfixing the other couplings to their SM values for eachof the curves. All four couplings change the predictedcross section resulting in values of σ t between 0 pb and140 pb, where a minimum can be found at couplingvalues of about 0.5.2 Measurements and assumptionsThe measurements of the W -boson polarisation and the t -channel cross sections considered in this example aretaken from Refs. [29] and [30], respectively. The un-certainties on these measurements are assumed to bederived from multivariate Gaussian distributions. The Fig. 1
Fraction of left-handed W -bosons from top-quark de-cays as a function of anomalous couplings. The asterisk indi-cates the SM values. Fig. 2
Cross section for t -channel single top-quark produc-tion as a function of anomalous couplings. The asterisk indi-cates the SM values. measured values are f L = 0 . ± . f = 0 . ± . σ t = 46 ± σ ¯ t = 23 ± . The correlation between the measurements of f L and f is quoted in the reference as ρ ( f L , f ) = − . ρ ( σ t , σ ¯ t ) = +0 . Table 1
Total uncertainties and the correlation matrix of thehelicity fraction and t -channel cross section measurements.Uncertainty f L f σ t σ ¯ t f L f σ t σ ¯ t The efficiency times acceptance of the measured t -channel cross section depends on the anomalous cou-plings assumed because they have an impact on thekinematic distributions of the final-state particles. Sinceits evaluation would require further studies includingsimulations of the detector setup and a repetition of theanalysis procedure, they are not considered in this ex-ample. In general, these corrections should be providedby the experimental collaborations as such an evalua-tion often requires access to unpublished material. Asdescribed in Section 4.3, the EFT fitter code is preparedto include such correlations.5.3 Interpretation of measurementsWe assume no prior knowledge on the values of the fourcoupling strengths, i.e. we choose the prior probabilitydensity for each coupling to be uniform. The values of g L and g R are limited to a range [ − , V L and V R are constrained to be within [ − . , . V L = 1 and V R = 0. Taking all four measurements and their corre-lations into account, Figure 3 shows the contours of thesmallest areas containing 68.3% and 95.5% probabilityin the two-dimensional plane of g R vs. g L . In compari-son, the dark and coloured lines indicate these contoursif only the measurements of the W -helicity or of the Fig. 3
Contours of the smallest areas containing 68.3%,95.5% and 99.7% posterior probability in the ( g L , g R )-plane.The filled areas consider all four measurements, while theopen ones take into account only two measurements. Alsoindicated is the SM prediction. t -channel cross sections are considered. While the mea-surement of the W -helicity alone constrains two sepa-rate regions in ( g L , g R )-space, one centred around theSM prediction of (0 ,
0) and another, smaller one around(0 , . t -channel cross sectionshas less constraining power, but excludes the secondregion. Using all four measurements thus reduces theavailable parameter space for anomalous couplings byexcluding the second region and, if only marginally, byreducing the area of the first region.The relative importance of each of the four mea-surements is illustrated in Tab. 2, which shows a rank-ing based on the relative increase of the area withinthe 68.3% contour if individual measurements are ig-nored. This is compared to a ranking based on theuncertainties of the one-dimensional marginal distribu-tions. As expected from Fig. 3, the measurements ofthe W -helicity have a larger impact on the constraintsthan the t -channel measurements. It is worth noting,however, that the ranking only addresses the size ofthe uncertainties, but not the topology of the contours,i.e. the appearance of a second, disconnected allowedregion. The small negative values associated with themeasurement of σ t can be explained by the fact thatthe measured value is furthest away from the SM pre-diction in comparison to the other three measurements.As an example, we test the impact of the correlationbetween σ t and σ ¯ t on the estimate of g R . Fig. 4 showsthe one-dimensional marginal posterior probability for g R as a function of the linear correlation coefficient inthe range [ − . , . ρ in the range [ − . , . Table 2
Relative increase of the area contained in the 68.3%posterior probability contour when removing one measure-ment at a time. Also indicated is the rank.Measurement Relative increase [%] (rank)( g L , g R ) g L g R f f L σ ¯ t σ t - 3.2 (4.) - 5.4 (4.) - 2.9 (4.) Fig. 4
Contours of the smallest areas containing 68.3%,95.5% and 99.7% posterior probability for g R as a function ofthe linear correlation coefficient ρ between the measurementsof σ t and σ ¯ t . to a factor of two for large, negative values and theybecome slightly larger for large, positive values. Also,the median is shifted towards smaller values of g R inboth extreme cases.Instead of fixing the correlation coefficient betweenthe two cross section measurements, one can also as-sign an uncertainty to that correlation. Assuming aGaussian prior on the corresponding nuisance parame-ter with a mean value of 0.5 and a standard deviationof 0.1, the uncertainty on g L and g R does not changesignificantly. This is expected from Fig. 4 because thecorrelation does not have a significant impact in thiscase.For the second interpretation, we assume all fourcouplings to be free parameters of the fit. As an ex-ample, Figs. 5, Fig. 6 and Fig. 7 show the marginalposterior distributions in the ( V L , V R )-plane, in the( V L , g R )-plane, and in the ( g L , g R )-plane, respectively.All three distributions have highly non-Gaussian shapesand some show disconnected regions. In each of the pro-jections, the local mode is consistent with the predic-tions of the SM and the absence of anomalous couplings. Fig. 5
Marginal posterior distributions in the ( V L , V R )-planewhen all four parameters are left free in the fit and all fourmeasurements are considered. Fig. 6
Marginal posterior distributions in the ( V L , g R )-planewhen all four parameters are left free in the fit and all fourmeasurements are considered. The global mode is found at V L = 1 . V R = − . g L = 0 .
03, and g R = − .
01. Since the smallestfour-dimensional hypervolume containing 68.3% poste-rior probability is strongly non-Gaussian in shape andfeatures several disconnected subsets, one-dimensionalmeasures, such as the standard deviation or smallestintervals, are rendered useless.
Fig. 7
Marginal posterior distributions in the ( g L , g R )-planewhen all four parameters are left free in the fit and all fourmeasurements are considered. We have presented a tool for interpreting measurementsin the context of effective field theories. This EFT fitter allows implementing a user-defined model, either di-rectly or via interfaces to other software tools, includingpredictions of observables based on the free parametersof the model. Measurements of these observables arethen combined and used to constrain the free param-eters. A variety of features of the EFT fitter helps toquantify and visualise the results. An example in thefield of top-quark physics was shown, for which anoma-lous couplings of the
W tb -vertex were constrained basedon measurements of the W -boson helicity fractions andthe single-top t -channel cross sections. Acknowledgements
The authors would like to thankFabian Bach, Kathrin Becker, Dominic Hirschb¨uhl and Miko-laj Misiak for their help and for the fruitful discussions. Inparticular, the authors would like to thank Fabian Bach forproviding the code for the single-top cross sections. N.C. ac-knowledges the support of FCT-Portugal through the con-tract IF/00050/2013/CP1172/CT0002.
References
1. F. Beaujean, C. Bobeth, D. van Dyk, Eur. Phys. J.
C74 ,2897 (2014). [Erratum: Eur. Phys. J.
C74 , 3179 (2014)]2. P. Bechtle, K. Desch, P. Wienemann, Comput. Phys.Commun. , 47 (2006)3. H. Flacher, et al., Eur. Phys. J.
C60 , 543 (2009). [Erra-tum: Eur. Phys. J.
C71 , 1718 (2011)]4. M. Czakon, A. Mitov, JHEP , 080 (2013)05. M. Czakon, A. Mitov, Comput. Phys. Commun. ,2930 (2014)6. M. Czakon, P. Fiedler, A. Mitov, Phys. Rev. Lett. ,252004 (2013)7. P. Langacker, M. Luo, Phys. Rev. D44 , 817 (1991)8. R.T. Bayes, Phil. Trans. Roy. Soc. Lond. , 370 (1764)9. A. Valassi, Nucl. Instrum. Meth. A500 , 391 (2003)10. L. Lyons, D. Gibaut, P. Clifford, Nucl. Instrum. Meth.
A270 , 110 (1988)11. S. Brandt,
Data Analysis - Statistical and ComputationalMethods for Scientists and Engineers (Springer, 1999)12. T. Leonard, J. Hsu, Ann. Stat. , 1669 (1992)13. J. Barnard, R. McCulloch, X.L. Meng, Stat. Sinica ,1281 (2000)14. A. Huang, M. Wand, Bayesian Anal. , 439 (2013)15. A. Caldwell, D. Kollar, K. Kr¨oninger, Comput. Phys.Commun. , 2197 (2009)16. R. Brun, F. Rademakers, Nucl. Instrum. Meth. A389 , 81(1997)17. J.A. Aguilar-Saavedra, et al., Eur. Phys. J.
C50 , 519(2007)18. J.A. Aguilar-Saavedra, Nucl. Phys.
B804 , 160 (2008)19. J.A. Aguilar-Saavedra, Nucl. Phys.
B812 , 181 (2009)20. C. Zhang, S. Willenbrock, Phys. Rev.
D83 , 034006 (2011)21. A. Buckley, et al., Phys. Rev.
D92 , 091501 (2015)22. A. Buckley, et al., JHEP , 015 (2016)23. C. Bernardo, et al., Phys. Rev. D90 , 113007 (2014)24. M. Fabbrichesi, M. Pinamonti, A. Tonero, Eur. Phys. J.
C74 , 3193 (2014)25. J.L. Birman, et al., (2016)26. F. Bach, T. Ohl, Phys. Rev.
D90 , 074022 (2014)27. A. Czarnecki, J.G. Korner, J.H. Piclum, Phys. Rev.
D81 ,111503 (2010)28. N. Kidonakis, Phys. Rev.