Reviving the interference: framework and proof-of-principle for the anomalous gluon self-interaction in the SMEFT
CCP3-20-58
Reviving the interference: framework and proof-of-principle for the anomalous gluonself-interaction in the SMEFT
C´eline Degrande ∗ and Matteo Maltoni † Centre for Cosmology, Particle Physics and Phenomenology (CP3),Universit´e catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
Interferences are not positive-definite and therefore they can change sign over the phase space. Ifthe contributions of the regions where the interference is positive and negative nearly cancel eachother, interference effects are hard to measure. In this paper, we propose a method to quantify theability of an observable to separate an interference positive and negative contributions and thereforeto revive the interference effects in measurements. We apply this method to the anomalous gluonoperator in the SMEFT for which the interference suppression is well-known. We show that we canget contraints on its coefficient, using the interference only, similar to those obtained by includingthe square of the new physics amplitude.
Introduction
The Standard Model Effective FieldTheory (SMEFT) explores the deviations in SM cou-plings due to interactions among Standard Model (SM)particles and new states, too heavy to be produced atthe LHC or any other considered experiment. Nonethe-less, those new states affect the interactions betweenthe SM particles and accurate measurements of theirstrengths should, thus, reveal or constrain the presenceof new physics. In this framework, heavy new degreesof freedoom are integrated out and the new physics isparametrised by higher-dimensional operators [1][2], L SMEF T = L SM + X i C i Λ O i + O (Λ − ) , (1)where Λ is the new physics scale. As a results, observ-ables such as differential cross-sections display the sameexpansion, dσdX = dσ SM dX + X i C i Λ dσdX + O (Λ − ) (2)where X is a generic name for a measurable variable.While constraints should ideally come from the secondterm, i.e. the term linear in the coefficients, they of-ten come in practice from the term quadratic in C i orfrom terms of even higher power of C i . This phenomenonmainly originates from the fact that the linear term is aninterference between the SM amplitudes and the ampli-tudes linear in C i , and this interference has been shownto be suppressed [3] for 2 → Framework
In this work we concentrate on thedimension-6 operator O G = g s f abc G a,µν G b,νρ G c,ρµ , (3)with G µν the gluon field strength. While this operatoris expected to contribute to multijets and top-pair pro-duction, its interference vanishes for dijet and is stronglysuppressed for the other processes. Constraints on thisoperator affect the sensitivity over other operators in-volved, for example, in top quark production [5]. High-multiplicity jet measurements strongly constrain this op-erator but mainly from the O (Λ − ) or even higher orderterms [6, 7].The stricter bound on this operator comes from the O (Λ − ) in dijet measurements [8] and reads C G Λ < (0 .
031 TeV) − (4)at 95% confidence level (CL).We use the SMEFT@NLO [9] Universal FeynRules Out-put (UFO) [10], written from a FeynRules model [11]containing the dimension-six operators, to generate theLO partonic events needed for our study. All the opera-tors coefficients are set to zero but the O G one, which istaken equal to 1 with Λ = 5 TeV. Madgraph@NLO [12]is then used to generate events for the SM, the square ofthe 1 / Λ amplitudes and their interference. Throughoutthis paper, we truncate the amplitude at O (cid:0) / Λ (cid:1) andtherefore O (cid:0) / Λ (cid:1) terms always come from the squareof the 1 / Λ amplitudes. Namely, multiple insertions ofthe dimension-six operators are not allowed. We use theNNPDF2.3 parton distribution function (PDF) set [13]and the results are given for LHC at 13 TeV at the par-tonic level. We leave the study of the effect of NLO a r X i v : . [ h e p - ph ] D ec p T >
50 GeV p T >
200 GeV p T > σ [pb] w > σ [pb] w > σ [pb] w > t ¯ t t ¯ tj · −
62% 1.13 · −
60% 1.37 · − jjj ·
52% 5.90 · −
52% 4.91 · − jjjj -2.89 ·
45% -2.50 · −
44% -4.12 · − O (Λ − ) cross-sections and percentages of positive-weighted events for processes with a non-null interference be-tween the SM and the O G operator and a large cross-section.These results are calculated for jets separated by ∆ R > p T corrections, parton shower and detector effects for futurestudies.The cancellation over the phase space is efficient if the in-tegrals of the interference in the phase space part whereits matrix element is positive and negative are almostequal in absolute value. Those two integrals are obtainedfrom the sum of the weights of events generated accord-ing to the interference, keeping respectively only positiveor negative weighted events. In table I, we use the per-centage of positive unweighted events to quantify the effi-ciency of this cancelation for top and jet processes. Sincethe strongest cancellation occurs for three-jets and thisprocess has the large cross-section necessary for accuratedifferential measurement, in the remaining of this letter,we will restrict ourself to this process and leave the otherfor future analyses. The integral of the absolute valuedinterference differential cross-section, σ | int | ≡ Z d Φ (cid:12)(cid:12)(cid:12)(cid:12) dσ int d Φ (cid:12)(cid:12)(cid:12)(cid:12) (5)is computed from the sum of the absolute values of theweights and is an upper bound of the total measurableeffect of the interference over the whole phase space Φ.This quantity is given in table II together with the SM,the interference and the O (cid:0) / Λ (cid:1) total cross-sections.The comparison of those four quantities shows the strongsuppression of the interference total cross-section, andhow it is lifted by σ | int | . Unfortunately, σ | int | is not ameasurable quantity as it requires to measure not onlythe momenta of the jets, but also their flavours and helic-ities, as well as those of the incoming partons. Therefore,we define the measurable absolute value cross-section, σ | meas | ≡ Z d Φ meas (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X { um } dσd Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6)where { um } is the set of unmeasurable quantities of theevents. For other processes, the sum can be replaced,at least partially, by integrals over continuous unmea-surable quantities, such as the longitudinal momenta ofa neutrino. This is the difference between the positive and negative contributions of the interference to the to-tal cross-section using all the information experimentallyavailable (and assuming perfect measurements of the jetsmomenta). As a result, this is an upper bound for anyasymmetry build on one or a few kinematic variables aim-ing at restoring the interference, and therefore can beused to assess the efficiency of such asymmetry. σ | meas | is estimated by σ | meas | = lim N →∞ N X i =1 w i ∗ sign X um M E ( ~p i , um ) ! (7)where ME is the part of the squared amplitude due tothe interference and w i and ~p i label the weight and themomenta of the jets of the event i . Therefore, this can beseen as a matrix element method [14–19] at the partoniclevel to revive the interference. The values of σ | meas | for the three-jet final state and different cuts are givenin table II. The cancellation among positive and negativeweighted events decreases with the p T cut while the ratio σ | meas | /σ | int | remains roughly constant. Differential distributions
We tested the ability toseparate positive and negative weight for various differ-ential and double differential cross-sections. Tested dis-tributions include the transverse momenta p T and thepseudorapidities η of the jets, their angular distances∆ R , their invariant masses, the normalised triple productamong the three-momenta of the jets, and some event-shape variables, including the transverse thrust, the jetbroadening [20] and the transverse sphericity [21]. Sev-eral variables such as the p T of the first jet, p T [ j ], thetransverse trust and the angular distance between thetwo lowest p T jets , ∆ R [ j j ] achieve an efficiency ofabout 40% compared to σ | meas | . For comparison, theefficiency of the total cross-section is about 2%. Thebest efficiency, however, is obtained for the transversesphericity and is about 80%. Moreover, this efficiencybarely varies with the global lower cut on each of thethree jets p T . The transverse sphericity Sph T is definedby using the eigenvalues λ ≥ λ of the transverse mo-mentum tensor: M xy = N jets X i =1 p x,i p x,i p y,i p y,i p x,i p y,i ! , Sph T = 2 λ λ + λ . (8)Therefore, sign flip occurs between the events that aremore two-jets like ( Sph T ∼
0) and those that are threewell separated and balanced jets (
Sph T ∼ p T cut, as strong hierarchy between the jets be-comes then unlikely. The separations of the negative andpositive contributions for some of those variables are il-lustrated in figure 1, where the full distributions as wellas those of the positive and negative weighted events aredrawn separately. Contrarily to inefficient variables, the SM O (1 / Λ ) O (1 / Λ ) p T,min [GeV] σ [pb] σ [pb] wgt > σ | meas | [pb] σ | int | [pb] σ [pb]50 9.70 · · · ·
200 8.96 · · − · · · − · − · − · − · − · − · − · − · − TABLE II. Cross-sections for three-jet production, for different values of the p T -cut, ∆ R > .
4, Λ = 5 TeV and renormalisationscales fixed respectively at 150, 250, 500, 1000 and 2000 GeV, with up to one O G insertion. The percentages of the totalamount of positive-weighted events, the percentages of the positive and negative measurable matrix elements (mme) and σ | int | are shown for the interference distribution of the positive and negative weighted eventsare different, resulting in a non-zero and changing signdistribution for the full interference. FIG. 1. Differential distributions for pp → j at the LHCwith p T >
200 GeV for the jets. The red (blue) line repre-sents the differential cross-section contribution by the positive(negative) weighted events. Their difference, the green line, isthe differential cross-section distribution for the interference;the dashed portion is the opposite of the negative differentialdistribution. The black line reproduces the SM cross-sectiondistribution, divided by 100. The last bins contain the over-flow
Using the transverse sphericity to split the positive andnegative contributions, we now estimate the limits thatcould be obtained on C G Λ , either for the interference onlyor including the O (1 / Λ ) contribution, too. The boundsare obtained, for each double distribution, from the fol-lowing χ -squared χ = X i (cid:18) x expi − x thi σ i (cid:19) = X i C G Λ x / Λ i σ i ! (9)where x expi and x thi = x SMi + C G Λ x / Λ i are respectivelythe measured and predicted content of each bin. Sincethe experimental results for the distributions we are in-terested in have not been published yet, we assume thatthe experimental data will follow the SM distributionsfor the considered quantities (resulting in the last step ofEq. (9)) and that the uncertainty, σ i , for the i th bin is10% of its SM content. This estimate of the uncertaintyseems consistent with available experimental results [22].We choose our binning such that each bin would containenough events, assuming the SM only to ensure that thestatistical errors are below 10%, for a luminosity of 100fb − . The best results are displayed in table III.Finally, to assess the validity of the SMEFT with ourapproach, we display in figure 2 how the limits on Λ variesif a cut on the center-of-mass energy is applied, assum-ing C G = 1. In principle, the EFT is valid if √ s < Λ,which is only satisfied for C G slightly bigger that 1 withthe low p T cuts. The situation improve for the strongerconstraints derived with higher cuts. In both cases, theconstraints barely change when the events with √ s (cid:38) O (Λ − ) contribution too, as it is expected because oftheir different dependency on Λ. The bounds obtainedby using the S T variable, defined in [6], are also shownfor comparison. As expected, our distribution shows anice improvement for the bounds at O (Λ − ). Conclusions
We used the sign of the measurablematrix element as a tool to revive the interference andto quantify the efficiency of differential distributions toseparate negatively and positively contributing regions of p T,min [GeV] Distribution
Sph T cut Bins Upper bound on C G Lower bound on C G p T [ j ] vs Sph T · − (1.1 · − ) -2.5 · − (-1.2 · − )200 S T vs Sph T · − (2.3 · − ) -7.5 · − (-2.4 · − )500 M [ j j ] vs Sph T · − (5.3 · − ) -5.5 · − (-3.5 · − )1000 M [ j j ] vs Sph T · − (1.9 · − ) -2.6 · − (-1.8 · − )TABLE III. Best bounds on the C G coefficient for different cuts on the p T , for Λ =1 TeV and 68% CL. The number of binsis reported, for each distribution; the cut on the sphericity is the value, between 0 and 1, in which we separated the two binsused for this variable. In the bounds columns, the first numbers are obtained through the O (Λ − ) contribution only, the onesinto brackets take into account the O (Λ − ) data, tooFIG. 2. Upper bounds on Λ (for C G = 1) as functions of theupper cut over the center-of-mass energy √ s , inferred fromthe best distribution for each p T -cut. The red line shows thebounds from the O (Λ − ) term only, which are symmetricalwith respect to 0, while the blue line take into account the O (Λ − ) one, too. The orange and purple lines reproduce thebounds, obtained through the S T variable, considered in [6],at O (Λ − ) and O (Λ − ). The axis on top of the plots quantifiesthe percentage of events, in the interference sample, that getlost form the cut on √ s the phase space. We used it to find efficient distributions to look for the interference effect of anomalous gluon in-teractions, as predicted by the SMEFT, and to put onthe corresponding operators, for the first time, contraintswhich are dominated by the leading ( O (Λ − )) interfer-ence and not by the O (Λ − ) term, coming from the newphysics amplitude squared. Therefore, the observablewould also be sensitive to the sign of the coefficient. Inaddition, the proposed measurement can be easily rein-terpreted in other BSM scenarios if SMEFT assumptionsturn out not to be valid, as they are purely kinematicdistributions. While the method has been tested on thisparticular case, it is fully generic and can be applied forany interference suppression due to sign flips over thephase space. Acknowledgements
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