Constraining qqtt operators from four-top production: a case for enhanced EFT sensitivity
PPrepared for Chinese Physics C
Constraining qqtt operators from four-top production:a case for enhanced EFT sensitivity *Cen Zhang( 张 岑 ) Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
Abstract:
Recently, experimental collaborations have reported O (10) upper limits on the signal strength of four-top production at the LHC. Surprisingly, we find that the constraining power of four-top production on the qqtt type of operators is already competitive with the measurements of top-pair production, even though the precisionlevel of the latter is more than two orders of magnitude better. This is explained by the enhanced sensitivity of thefour-top cross section to qqtt operators, due to multiple insertion of operators in the squared amplitude, and to thelarge threshold energy of four-top production. We point out that even though the dominant contribution beyondthe standard model comes from the O ( C / Λ ) terms, the effective field theory expansion remains valid for a widerange of underlying theories. Considering the possible improvements of this measurement with higher integratedluminosity, we believe that this process will become even more crucial for probing and testing the standard modeldeviations in the top-quark sector, and will eventually provide valuable information about the top-quark properties,leading to significant improvements in precision top physics. Key words:
Top quark, LHC, effective field theory
PACS:
As a top-quark factory with more than six milliontop-quark pairs produced at Run-I and much more to ex-pect in the future, the LHC is an ideal place to probe thetop-quark properties. In proton-proton collision, mosttop quarks are produced in t ¯ t pairs. Single top produc-tion has the second largest cross section, which is aboutone third of the t ¯ t . More recently, associated productionmodes such as t ¯ t + X and single t + X , where X is agauge boson or the Higgs boson, have also been exten-sively studied. These are the main channels that are nowpushing the top-quark physics into a precision era [1].Attention has also been paid to the four-top produc-tion mode, pp → tt ¯ t ¯ t , which, despite its tiny rate ( ≈ t ¯ t production (832 pb, [4, 5]), isparticularly sensitive to new physics. It has been noticedthat the total rate of this process can be enhanced sig-nificantly in many scenarios beyond the standard model(BSM) [6–17]. This can be due to the direct productionof new resonant states which subsequently decay intotops, or to the contribution from contact four-top oper-ators, which rises as the energy grows. These operatorsare not directly constrained by other processes at thetree level, and therefore the four-top channel may be thefirst place to see their effects.Nevertheless, a comprehensive model-independent study of four-top production in the context of the stan-dard model effective field theory (SMEFT) approach [18–20] has not yet appeared in the literature. This is notsurprising. The SMEFT framework aims to probe theindirect effects from BSM models that are beyond the di-rect reach of the LHC. These effects are expected to showup as relatively small deviations from the SM prediction,and therefore the most powerful approach is to combineall available precision measurements and perform globalanalyses. In the top-quark sector, such analyses are of-ten based on the most precise ones, such as top-pair andsingle-top cross sections and distributions, branching ra-tio measurements, and recently also on associated pro-duction modes such as t ¯ tZ and t ¯ tγ , see e.g. Refs. [21, 22]for a recent global fit. The four-top production, on theother hand, is still far from being precise. The processhas been searched for in a series of experimental reports[23–32], and the best upper limit from Ref. [30] is about4.6 times the SM signal. Naively, one would not expectan O (10) upper bound to provide competitive informa-tion with respect to all the other precise measurements,except for the four-top operators that are not directlyprobed elsewhere.The goal of this work is to demonstrate that this isnot the case. For a very important class of operators,namely the contact four-fermion interactions with twotop quarks and two light quarks, qqtt , we will show thatthe four-top process, with only a O (10) upper bound, is ∗ Supported by the 100-talent project of Chinese Academy of Sciences1) E-mail: [email protected] a r X i v : . [ h e p - ph ] N ov repared for Chinese Physics C as powerful as t ¯ t measurements with a percentage error.This constraining power is due to an enhanced sensitivityof four-top production, which comes from the fact thatits cross section can depend on up to the fourth powerof the operator coefficients, which scales like ( CE / Λ ) ,where E is the energy of the process, and C/ Λ is thecoefficient of an qqtt operator. Given the large energyscale related to this process, and the current limits onthe coefficient C/ Λ , the factor ( CE / Λ ) significantlyenhances the sensitivity of four-top process to the qqtt operators. We will also show that the validity of SMEFTand its perturbativity can be guaranteed by imposing ananalysis cut on the center of mass energy of the processat a few TeV, without reducing the enhancement factortoo much, and thus the resulting constraints apply toBSM theories that live above this energy scale, if certainassumptions are made to justify the omission of opera-tors at dim-8 and higher.For comparison, we also consider the t ¯ t observablesat the LHC, and study the corresponding exclusion limiton the same class of operators. These observables havebeen incorporated in a global fit by the authors ofRefs. [21, 22]. In this work, however, the approach wefollow is quite different, mainly because we are interestedin the enhancement effect of higher powers of CE / Λ .Even in t ¯ t measurements, the squared term from dim-6operators cannot be neglected with the current precision,and therefore instead of the four linear combinations of qqtt operators used in Refs. [21, 22] (defined in Ref. [33]),we will have to include the complete set of 14 qqtt op-erators. A global fit, including the main cross sectionand asymmetry measurements, as well as a differentialcross section measurement, will be performed to derivethe global constraints in these 14 directions. These con-straints will then be compared with those from the four-top production in the same directions. For the latter pro-cess, we will also consider the impact of including the fullset of tttt type four-fermion operators, which might begenerated together with the qqtt operators, when heavymediator particles in the full theory are integrated out.Note that RG-induced constraints are also available onthe qqtt operators [34], but they are typically consideredas indirect constraints.In this work, our numerical approach will be fullybased on the MadGraph5 aMC@NLO framework [3].We use NNPDF3.0 parton distribution functions (PDF)[35]. A UFO model [36] that contains all 14 qqtt oper-ators and 4 tttt operators is generated using the
Feyn-Rules package [37]. All calculations are done at theleading order (LO). For the four-top production, we as-sume that a SM K -factor of about 1.4 [2, 3] at the next-to-leading order (NLO) can be applied also to the op-erator contributions. This might not be a good approx- imation (see Ref. [38] for an example), but is the bestwe can do given that the NLO prediction for all qqtt operators are not yet available. ∗ The corresponding the-oretical error at the NLO is about ∼ qqtt op-erators. If one naively applies the bound on the SM crosssection, the limits on BSM will be more conservative, asin general the effective operators lead to harder energydistributions. This is indeed the case for the four-topoperators, as have been considered in the experimentalanalyses in Refs. [25–28]. Furthermore, applying the up-per bound of the total cross section on the fiducial crosssection below some center-of-mass energy cut to ensurethe SMEFT validity will also make the results conser-vative. Still, even these conservative constraints on qqtt operator coefficients already compete with those from t ¯ t measurements, so they are sufficient for the goal of thiswork. One should keep in mind that further improve-ments from the experimental side are possible.The paper is organized as follows. In Section wepresent the relevant dim-6 operators in this work. InSection we explain the enhanced sensitivity of the four-top process, and discuss the validity range of the EFT.We compare the constraining powers of the four-top and t ¯ t cross sections in Section . Section is devoted toa global fit using t ¯ t measurements, which will be com-pared with the fully marginalized constraints from four-top cross section. In Section we conclude. In this work we are interested in the four-fermion op-erators that involve two top quarks and two light quarks.This is an important class of operators, as they are com-mon in BSM models where new heavy states couple toboth t ¯ t and q ¯ q , or q ¯ t and t ¯ q currents.Assuming an U(2) u,d,q ) flavor symmetry for the firsttwo generations, the full set of qqtt operators at dim-6 ∗ An NLO implementation of the four-fermion top operators based on the
MadGraph5 aMC@NLO framework is in progress [39]. repared for Chinese Physics C can be written as follows O (8 , Qq = (cid:0) ¯ Q L γ µ T a τ i Q L (cid:1) (cid:0) ¯ q L γ µ T a τ i q L (cid:1) (1) O (8 , Qq = (cid:0) ¯ Q L γ µ T a Q L (cid:1) (¯ q L γ µ T a q L ) (2) O (8) td = (¯ t R γ µ T a t R ) (cid:0) ¯ d R γ µ T a d R (cid:1) (3) O (8) tu = (¯ t R γ µ T a t R ) (¯ u R γ µ T a u R ) (4) O (8) tq = (¯ t R γ µ T a t R ) (¯ q L γ µ T a q L ) (5) O (8) Qd = (cid:0) ¯ Q L γ µ T a Q L (cid:1) (cid:0) ¯ d R γ µ T a d R (cid:1) (6) O (8) Qu = (cid:0) ¯ Q L γ µ T a Q L (cid:1) (¯ u R γ µ T a u R ) (7) O (1 , Qq = (cid:0) ¯ Q L γ µ τ i Q L (cid:1) (cid:0) ¯ q L γ µ τ i q L (cid:1) (8) O (1 , Qq = (cid:0) ¯ Q L γ µ Q L (cid:1) (¯ q L γ µ q L ) (9) O (1) td = (¯ t R γ µ t R ) (cid:0) ¯ d R γ µ d R (cid:1) (10) O (1) tu = (¯ t R γ µ t R ) (¯ u R γ µ u R ) (11) O (1) tq = (¯ t R γ µ t R ) (¯ q L γ µ q L ) (12) O (1) Qd = (cid:0) ¯ Q L γ µ Q L (cid:1) (cid:0) ¯ d R γ µ d R (cid:1) (13) O (1) Qu = (cid:0) ¯ Q L γ µ Q L (cid:1) (¯ u R γ µ u R ) (14)where Q L represents the left-handed doublet for the 3rdgeneration, and q L , u R and d R represent the 1st and the2nd generation quarks. The operators are summed overthe first two generations, but we omit the flavor indices.Other four-fermion operators are excluded by the flavorsymmetry.For later convenience we have written the 14 oper-ators in the form of a top-quark vector current (colorsinglet or octet) contracted with a light-quark vector cur-rent. Their contributions to both q ¯ q → t ¯ t and q ¯ q → tt ¯ t ¯ t are independent of each other. One could also count the14 degrees of freedom in a more physical way: • Both the light and the heavy quark currents can beeither left- or right-handed. This counts 4 degreesof freedom. • The light quark can be up/charm or down/strange.This leads to 8 in total. • SU(2) L symmetry requires that u L u L t R t R and d L d L t R t R interactions have the same coefficient.This reduces the number to 7. • With two possible color structures, i.e. singlet andoctet, the total number of degrees of freedom is 14.In the t ¯ t process, the cross section can be written as aquadratic function of 14 operator coefficients: σ = σ SM + (cid:88) i C i Λ σ i + (cid:88) i ≤ j C i C j Λ σ ij (15)If one truncates the function and keeps only the interfer-ence term, then the 8 color-singlet operators, Eqs. (8)-(14), do not give any contribution at the LO. Further-more, without information from the decay of the tops, the LLLL (LLRR) interactions cannot be distinguishedfrom the RRRR (RRLL) operators. Therefore only 4 de-grees of freedom can be observed [33], which significantlysimplifies the analysis. However, the current limits onthe operator coefficients C/ Λ indicate that the dim-6squared terms are not negligible, and so the full set of 14operator need to be included in t ¯ t production. The four-top production mode is similar, and in particular, therethe dominant terms may come from the fourth power ofdim-6 coefficients.Fortunately, as we will see in Section , the SMEFTanalysis with all 14 operators can be simplified by ob-serving that these operators can be divided into threecategories according to the flavor of the light quarks,without any interference effect across:1. u R : O (8) tu , O (1) tu , O (8) Qu , O (1) Qu ; (16)2. d R : O (8) td , O (1) td , O (8) Qd , O (1) Qd ; (17)3. q L : O (8 , Qq , O (8 , Qq , O (1 , Qq , O (1 , Qq , O (8) tq , O (1) tq . (18)Furthermore, the operators in the first two categories canbe easily related to those in the last category by parity.This implies that one analysis with 14 operators can besimplified into two independent ones, each with only 4operators, from the 1st or the 2nd category, and par-ity can be used to derive results for the 3rd category.This is one of the reasons for choosing the operator basisgiven by Eqs. (1)-(14). As we will see, this simplificationis very important for analyzing the four-top productionprocess, as there the cross section is a quartic function of14 operators, with a large number of interference terms.We also consider the operators that consist of fourtop quarks. These four-top operators are important be-cause unlike the qqtt ones, they are bound to be gener-ated as long as there are BSM particles coupled to thetop quark. The four-top production is the first processto directly probe them (see, for example, discussions inRefs. [7, 8, 16, 40]). In this work we will also provideconstraints on these operators. Note that our main goalis to derive constraints on the qqtt operators, however,reliable constraints need to be obtained by marginalizingover other operators that enter the same process. This isthe main reason to study the contribution from four-topoperators, as we want our conclusion to be independentof their sizes.Five such operators exist in the so-called Warsaw ba-sis [41]: O (1)(3333) qq , O (3)(3333) qq , O (3333) uu , O (1)(3333) qu , O (8)(3333) qu . (19)3 repared for Chinese Physics C Among them only four are independent in four-top pro-duction, which we define as O (+) QQ ≡ O (1)(3333) qq + 12 O (3)(3333) qq , (20) O tt ≡ O (3333) uu , (21) O (1) Qt ≡ O (1)(3333) qu , (22) O (8) Qt ≡ O (8)(3333) qu , (23)while the remaining degree of freedom is chosen as O ( − ) QQ ≡ O (1)(3333) qq − O (3)(3333) qq , (24)with no contribution to the process.These four-top operators in general interfere with theother qqtt operators in the four-top production. For acomplete analysis, one will have to consider each cate-gory in Eqs. (16)-(18) together with these four operators.Parity relation still holds, under which C (+) QQ and C tt isexchanged (neglecting contributions initiated by two b quarks).The relation between our four-fermion operator basisand the more standard basis, i.e. the Warsaw basis inRef. [41], is given in Appendix A.Finally, we briefly explain the notation used in thiswork. The coefficients of dim-6 operators are denoted as C/ Λ . One should keep in mind that the C and Λ indi-vidually do not have any physical meaning. Only theircombination is a physical quantity. We define˜ C i ≡ C i (1 TeV) Λ (25)so that constraints on SM deviations can be convenientlyquoted in terms of ˜ C . The values of ˜ C are constrained byexperiments and are model-independent. On the otherhand, we use Λ NP to denote the characteristic scale atwhich the new physics resides. This is not a model-independent quantity, but it is useful for defining therange of validity of the EFT expansion, which requires E < Λ NP , where E is the typical energy transfer in theprocess of interest. To briefly explain the sensitivity of the four-top pro-cess to four-fermion qqtt operators, let us take O (8) tu as anexample. This operator represents a contact interactionbetween a color octet right-handed up-quark current anda color octet right-handed top-quark current. We firstconsider the t ¯ t process. t ¯ t measurements so far imposethe tightest bounds on qqtt operators. The LO crosssection at 8 TeV, rescaled to the next-to-next-to-leadingorder (NNLO) prediction including the resummation of next-to-next-to-leading logarithmic (NNLL) soft gluonterms [4, 5], is numerically given by (in pb):252 . .
94 ˜ C (8) tu + 0 .
411 ˜ C (8) tu . (26)Using the combined ATLAS and CMS measurement on t ¯ t inclusive cross section [42], we find the following bounds − . < ˜ C (8) tu < . C (8) tu come only from theupper bound of the cross section. In particular for thelower limit ˜ C (8) tu = − .
8, the squared term in Eq. (26)already dominates over the interference.As we have mentioned in the introduction, it is thissame effect, i.e. the dominance of terms with higher pow-ers in ˜ C , that enhances the EFT sensitivity of four-topproduction. In particular, at the LO, the 14 qqtt typeoperators can be inserted at most twice in the amplitude.The squared amplitude at LO is thus a quartic functionwith 14 arguments: O = O SM + (cid:88) i C i Λ O i + (cid:88) i ≤ j C i C j Λ O ij + (cid:88) i ≤ j ≤ k C i C j C k Λ O ijk + (cid:88) i ≤ j ≤ k ≤ l C i C j C k C l Λ O ijkl , (28)where O represents any observable. Focusing again on O (8) tu , without worrying about EFT validity for the mo-ment, the LO total cross section is (in fb)6 . .
10 ˜ C (8) tu +0 .
081 ˜ C (8) tu +0 .
016 ˜ C (8) tu +0 . C (8) tu . (29)The CMS search presented in Ref. [30] gives an upperbound on the signal strength of four-top process, µ < . − . < ˜ C (8) tu < . , (30)which are already complementary to the previous con-straints from t ¯ t . Note, however, that when these con-straints are saturated, it is the ˜ C (8) tu term that givesthe dominant contribution. Had we truncated Eq. (29)to, say, the linear term in ˜ C (8) tu , the resulting constraintswould have been more than one order of magnitudeworse. This implies that the four-top process has an en-hanced sensitivity to qqtt operators, due to the contribu-tion from higher power terms in ˜ C , and this is why sucha process with only a O (10) upper bound on its signalstrength can beat the t ¯ t measurement with a precisionat the percentage level. Note that which term domi-nates depends on the size of the ˜ C (8) tu , and is thereforerelated to the current experimental bounds. The quarticterm dominates if | ˜ C (8) tu | > .
1, while the quadratic one4 repared for Chinese Physics C dominates if 1 . < | ˜ C (8) tu | < .
1. As the experimental con-straints continue to improve in the future, the situationmight change. Also note that a similar effect, i.e. thedominance of the quadratic term, has been observed inmultijet production [43].
SMC (cid:142) tu (cid:72) (cid:76) (cid:61) (cid:142) tu (cid:72) (cid:76) (cid:61)(cid:45) s (cid:64) GeV (cid:68)
Fig. 1. Center-of-mass energy distribution of four-top production, normalized, to illustrate the typi-cal energy scale of this process. Results are shownfor the SM case and for ˜ C tu = ± . The above observation however leads to two ques-tions: why the high power terms dominate, and whetherthe SMEFT expansion is still valid. The first question ismostly explained by the large energy scale related to thefour-top process. The threshold of four-top production is4 m t ≈
690 GeV. Most signal events have a typical centerof mass energy of (cid:38) O (1) TeV, depending on the value ofthe operator coefficients, as illustrated in Figure 1. Theseries in Eq. (29) comes from multiple insertion of thefour-fermion effective interaction in the squared ampli-tude, and by power counting each insertion correspondsto a factor of CE / Λ , where E is the characteristic en-ergy of the process. The current constraints on C/ Λ then implies that CE Λ > C/ Λ are sup-posed to dominate. † Note that this is not true for all op-erators. For example, another important operator thatenters both t ¯ t and four-top production channels is thetop-quark chromo-magnetic dipole operator, O tG = y t g s ( ¯ Qσ µν T A t ) ˜ φG Aµν . (32)The contribution of this operator does not scale as CE / Λ because of the Higgs vev. It is also better con-strained by t ¯ t in the gg initiated channel. As a result, the four-top limit on C tG cannot compete with the onefrom the t ¯ t measurement, and so we will not consider itin this work.The second question is more crucial. The fact thathigher power terms in Eq. (29) dominate seems to implythe breakdown of the EFT expansion, as one could askwhether the contributions from dim-8 and higher oper-ators can be safely ignored in an EFT expansion, giventhat they scale the same way in 1 / Λ as the higher-powerterms in Eq. (29). Therefore the validity of the EFT ex-pansion itself needs to be justified. Here to make thingsclear, it is important to distinguish between two kindsof “expansions”. The EFT expansion comes from inte-grating out heavy degrees of freedom at the energy scaleΛ NP (to be distinguished from the non-physical Λ), aprocedure whose legitimacy is related to E/ Λ NP < CE / Λ >
1. However, this second “expansion” is notrelated to EFT validity, and is strictly speaking not evenan expansion: there are no more terms after the fourthpower of CE / Λ (at LO, with on-shell tops and no fur-ther radiations), so there is no need to truncate. Simplyput, when CE / Λ > CE / Λ . The relative theory error due to neglectinghigher order terms is then controlled by E / Λ NP < CE / Λ > → g SM , while a dimension-six operatorcoming from integrating out the heavy mediator can beas large as g ∗ E / Λ NP , where g ∗ is the BSM coupling ofthe mediator to the SM particles. We have C Λ ∼ g ∗ Λ NP . (33)If the coupling g ∗ is much larger than the SM coupling g SM (which is often the case when experimental con-straints are saturated, if Λ NP is kept larger than E ), theBSM contribution dominates the SM contribution when † Eq. (31) with E ≈ √ s tends to overestimate the effective contribution. The reason is that the energy transfer at the effective verticesis often less than √ s . The only configuration where the energy transfer is equal to √ s is the case where the two initial quarks enterthe same effective vertex, which then produces t ∗ ¯ t → t ¯ tt ¯ t , but in this case the squared amplitude can depend on at most two powers of Cs/ Λ . Still, in this process either ( CE Λ ) with E (cid:46) √ s or ( Cs Λ ) represents a large factor. repared for Chinese Physics C ( g ∗ /g SM ) E / Λ NP >
1, and similarly the BSM squaredterm dominates over the interference term between SMand BSM. The EFT expansion is still valid if E/ Λ NP < → g ∗ , and therefore no g ∗ /g SM factorexists between dimension-six and dimension-eight oper-ators. In general the validity of the EFT expansion isnot spoiled by a large g ∗ , or a large C/ Λ , because themaximum power of g ∗ in a given process is fixed. Asa physics case, in reality the LHC sensitivities to thetriple-gauge-boson couplings are completely dominatedby dim-6 squared contributions, while the global EFTanalyses can be performed without including dim-8 op-erators [47, 48].In four-top production the situation is similar. The q ¯ q → tt ¯ t ¯ t amplitude can be enhanced at most by g ∗ E / Λ NP ∼ ( CE / Λ ) , and the cross section by( CE / Λ ) , as shown in for example Figure 2 (a). Uponintegrating out the heavy mediators, the amplitude inthe EFT is described by Figure 2 (b), i.e. with two in-sertions of dimension-six operators. However, the crucialdifference here is that truncating the higher-dimensionaloperators is not guaranteed as in a 2 → NP andone coupling g ∗ , and that the power counting in the EFTis given by [49] L EFT = Λ NP g ∗ L (cid:18) D µ Λ NP , g ∗ H Λ NP , g ∗ f L,R Λ / NP , gF µν Λ NP (cid:19) . (34)Higher dimensional operators can be constructed in dif-ferent ways. One can use the first expansion param- eter in Eq. (34), D µ / Λ NP , to increase the dimensionwithout changing the field content. This is like ex-panding a heavy mediator propagator ( p − M ) − = − M − (1 + p /M + p /M + . . . ), where M ≈ Λ NP , sothe expansion parameter is simply E / Λ NP . In this caseneglecting higher-dimensional operators is justified. Al-ternatively, one can also use g ∗ f L,R / Λ / NP or g ∗ H/ Λ NP toincrease the dimension, and the expansion parameter isenhanced by g ∗ , so higher-dimensional operators have achance to contribute more. This however cannot be donerepetitively, because at some point the operator will con-tain more than six fields and become irrelevant (assum-ing LO amplitude dominates, and neglecting the vev aswe are interested in the high-energy regime). The ques-tion is where to stop this g ∗ enhanced expansion. Notethat the dim-6 contribution is dominated by amplitudeslike Figure 2 (b) which already scale like g ∗ E / Λ NP .The first relevant operator that is enhanced by g ∗ is adim-10 operator, g ∗ f D , whose contribution scales like g ∗ E / Λ NP . This is still subdominant. For illustration wegive an example in Figure 2 (c) and (d), in a model witha heavy mediator with coupling strength g ∗ . Note thatthe two-to-four process can be enhanced at most by g ∗ atthe tree level, and a SMEFT operator that contains sixfermions is at least at dim-10, because odd-dimensionaloperators do not exist in the SM if B and L number vio-lating operators are ignored [50, 51]. On the other hand,dim-8 operators are enhanced at most by 3 powers of g ∗ . Since both the dim-8 and dim-10 contributions areless than g ∗ E / Λ NP , and further enhancement with g ∗ beyond dim-10 is not possible without adding more par-ticles, we conclude that, under the above assumption,truncating the SMEFT at dim-6 is justified. (a) (b) (c) (d) Fig. 2. The q ¯ q → tt ¯ t ¯ t amplitudes that are enhanced by four powers of BSM coupling g ∗ . Blue lines are heavymediators. Double lines represent the top quarks. The square represents a g ∗ coupling, and the blue blob representseffective operators, coming from integrating out the mediators. Diagrams (a), (c) describe the amplitudes in theunderlying theory, while in the EFT they respectively correspond to (b) and (d). Diagrams (a) and (b) correspondto two insertions of dim-6 operators. They scale like g ∗ E / Λ NP . Diagrams (c) and (d) correspond to one insertionof a dim-10 operator. They scale like g ∗ E / Λ NP . repared for Chinese Physics C It is important to keep in mind that this conclusionis a model-dependent one. In practice one could comeup with theories with more than one scales or couplings,where the dim-8/10 contributions might be important.As a general rule, when interpreting results obtainedwith a dim-6 SMEFT in specific models, one alwaysneeds to check the validity of the EFT by estimatingthe impact of higher dimensional contributions.It remains to show that the validity condition, E/ Λ NP <
1, can be taken under control. As proposed inRef. [46], the standard way to deal with this in a hadroncollider is to apply a mass cut M cut on the center of massenergy of the event, or some other observable that char-acterizes the energy scale of the process. Results of theanalysis should be provided as functions of M cut . TheSMEFT approach is then valid if results are interpretedwith BSM models that satisfy Λ NP > M cut , and theoryerrors due to missing higher dimensional terms can beestimated by M cut / Λ NP . As we have mentioned in theintroduction, since the experimental search is not car-ried out with this kind of strategy, we will simply applyvarious M cut of order a few TeV on the center-of-massenergy in our cross section calculation. The resultingfiducial cross sections are required to be less than theupper bound set on the SM total cross section, whichthen gives conservative constraints. EFTM V (cid:61) V (cid:61) V (cid:61) s (cid:64) GeV (cid:68)
Fig. 3. Center-of-mass energy distribution of four-top production, to illustrate the validity of EFT.Results are shown for the EFT, the SM, and for M V = 5, 6, 8 TeV respectively. To illustrate how well the SMEFT could reproducethe full theory prediction below M cut , we have consid-ered an explicit model with a heavy vector mediator par-ticle of mass M V and width M V / (8 π ) that couples to theright-handed quark currents. The coupling correspondsto ˜ C (1) tu = − C tt = − ‡ The invariant mass dis-tributions for M V = 5, 6, 8 TeV and for the EFT case (equivalent to M V → ∞ ) are compared in Figure 3. Notethat the SM contribution is very small compared with theEFT, as the latter is dominated by the dim-6 quartic con-tributions. Dim-8 and higher-dimensional contributionsare however subleading as illustrated by the differencesbetween the EFT curve and the explicit models. Forthe 5 TeV case, the cross section below M cut = 3 TeV isreproduced by the EFT with about 30% error, roughlycorresponds to ( M cut / Λ NP ) as expected. For larger me-diator masses the EFT approximation becomes better,which implies that the EFT validity issue will becomeless severe as the measurements continue to improve inthe future. This example corresponds to | CE / Λ | ≈ qq → tt ¯ t ¯ t amplitude can go beyond the fourthpower of CE / Λ , if loop corrections are important. Inthis case dim-8 operators are also needed for a consistenttheory prediction, as in general two dim-6 operators canmix with a dim-8 one. Given the precision level of theprocess, we simply impose the following perturbativitycondition CE (4 π ) Λ < CM cut (4 π ) Λ < . (35)to make sure the loop corrections due to additional in-sertion of effective operators are not important. Thiswill be checked for typical values of M cut . A related is-sue is that jets in the final state may allow additionalpowers of CE / Λ . As an example, we find that the tt ¯ t ¯ tjj cross section depends on ˜ C (8) tu , with a coefficientof 5 . × − fb. This term is less important than the˜ C (8) tu term if ˜ C (8) tu <
9, consistent with the perturba-tivity condition for M cut ≈ CE / Λ may come from non-top operators,if on-shell top quarks are not strictly required. We willsimply assume that these operators are more likely to beconstrained by other non-top measurements. The cross section of four-top production is a quar-tic function of the 14 qqtt operator coefficients. Sucha function in general has C = 3060 terms. Nu-merically determining this function will then require atleast 3060 independent simulations at different param-eter space points, which is huge amount of work. For-tunately, as we have explained in Section , the proce-dure can be simplified into two steps. The first step isto determine the cross section as functions of operatorsin the first two categories, separately, which requires aminimum of only C + C − ‡ When Λ NP is large, it may seem that these values would require a large coupling strength which is not compatible with our assump-tion on the width. However it is always possible to obtain large values of ˜ C without using a very strong coupling by arranging more thanone particles, or using group factors from a large representation, etc. repared for Chinese Physics C function of operators in the third category, with the helpof parity. Namely, if one imposes C (8) Qu = C (8) Qd , C (1) Qu = C (1) Qd , (36)then the cross section is invariant under the followingtransformation C ( a ) Qu = C ( a ) Qd ⇔ C ( a ) tq , (37) C ( a ) tutd ⇒ C ( a, Qq ± C ( a, Qq , (38) C ( a, ) Qq ⇒ (cid:16) C ( a ) tu ± C ( a ) td (cid:17) , (39)where a = 1 ,
8. Using these relations, the dependence onthe third category operators can be derived from that ofthe first two.When the four-top operators are included, in eachcategory one has to consider together the 4 qqtt opera-tors and the 4 tttt operators. The tttt operators can beinserted only once in the amplitude, if the qqtt opera-tors are not inserted twice in the same amplitude. Thisincreases the total number of independent terms to 705,which is still manageable. The parity relations can stillbe used to derive the dependence on the third categoryoperators, provided that C (+) QQ ⇔ C tt (40)is added to Eqs. (37)-(39).Following the procedures described above, to deter-mine the dependence of the four-top cross section on the14 qqtt and 4 tttt operator coefficients, we have randomlygenerated ∼ O (1000) points in the parameter space, andcomputed the cross section at these points, applying M cut = 2, 3, 4 TeV respectively. These points are uni-formly distributed roughly within the experimentally al-lowed region of the coefficients. Results are then fitted tothe polynomial described above. We have checked thatthe prediction of the fitted function at all these sampledpoints agree with the simulation within 3% error.With this function we are ready to evaluate the con-straining power of the signal process, and compare theconstraints from four-top production with those obtainedfrom t ¯ t measurements. For this purpose we first considersingle measurements on the t ¯ t total cross sections at theLHC, including: • σ = 242 . ± . •
13 TeV CMS, σ = 888 +33 − pb [53].Corresponding theoretical predictions at NNLO+NNLLare taken from [4, 5]. For the four-top production, weconsider the current upper bound with signal strength µ < . M cut = 3 TeV cut on thecenter-of-mass energy. We further consider the projec-tion for an integrated luminosity at 300 fb − , µ < .
87, estimated by Ref. [32], and apply M cut = 2 TeV and 3TeV respectively.In Figure 4 we show the resulting constraints at 95%confidence level, for the operators in the first category(i.e. those that couple to u R ), with two operators turnedon at a time. Results for the other two categories aresimilar and are given in Appendix B. From these plots,our observations are the following: • Current constraints from four-top productionalready provide competitive constraints (blackdashed), which are close to, and in some cases bet-ter than, the constraints from the 13 TeV t ¯ t mea-surement with only a 4% error (green dashed). • The 8 TeV t ¯ t measurement so far gives bet-ter constraints (green shaded), but even thesewill be superseded in the future by an improvedsearch/measurement of four-top production at 300fb − luminosity with a projected µ < .
87 upperbound (black solid), assuming M cut = 3 TeV. • Lowering M cut to 2 TeV will give somewhat looserconstraints (blue solid), but results can be appliedto more underlying models where the BSM scalesare not so heavy. On the other hand, increasingthis cut can further improve the constraints. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) tu (cid:72) (cid:76) C (cid:142) t u (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) Qu (cid:72) (cid:76) C (cid:142) t u (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) cut (cid:61) (cid:45) M cut (cid:61) (cid:45) M cut (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) Qu (cid:72) (cid:76) C (cid:142) Q u (cid:72) (cid:76) Fig. 4. Constraints from four-top cross sectionand individual t ¯ t cross section measurements,on the operator coefficients in the first category( ˜ C (8) tu , ˜ C (1) tu , ˜ C (8) Qu , ˜ C (1) Qu ), assuming two coefficientsto be nonzero at a time. It is important to point out that the t ¯ t measure-ment is limited by systematic errors, and further im-provements with higher luminosity is difficult. On the8 repared for Chinese Physics C other hand, there is still a lot of room for the four-topsearch/measurement to be improved in the future. Ourresults indicate that, in the near future, the four-top pro-cess could even take place and provide more crucial infor-mation on qqtt operators. One should keep in mind thatthis requires a relatively large cut, M cut (cid:38) ∼ ∼ In the previous section we have shown that the fu-ture four-top measurement can have a better sensitivityto four-fermion operators compared with individual t ¯ t cross section measurements. However, the t ¯ t productionis such an important process that has been measured ex-tensively at many different energies (Tevatron, LHC 7,8, 13 TeV) and in many different ways (cross section,asymmetries, distributions, etc.) A global fit of all avail-able measurements thus provides so far the best avail-able limits. In this section we will investigate whetherthe four-top process can add useful information on topof the global top measurement program.The recent global fit performed by the authors ofRefs. [21, 22] has included the four-fermion operators.The fit is based on four linear combinations of their co-efficients, called C , u,d , which are the only independentdegrees of freedom at the dim-6 interference level [33].However, the theoretical set up in this work is differ-ent, in that we expect that the dim-6 squared terms candominate given the current bounds. This has been con-firmed by, for example, a fit for four-fermion operatorsin Ref. [54], where this domination has been interpretedas the SMEFT being invalid. However, as explained inSection , in this work we distinguish the CE / Λ ex-pansion from the true EFT expansion E / Λ NP , and inany case we expect higher powers of CE / Λ to domi-nate in four-top production and to enhance its sensitivityto BSM. The actual EFT validity is then guaranteed bythe assumption Λ NP > M cut ∼ a few TeV, which is morethan enough for most t ¯ t measurements. In light of the above considerations, the fit we willperform is different than the previous ones, in that thedim-6 squared terms as well as the interference betweentwo dim-6 operators will be fully incorporated. We willhave to abandon the C , u,d language, as this simplificationbreaks down at the dim-6 squared level. Furthermore,the color-singlet operators cannot be neglected due tovanishing interference term. The fit will then include14 independent operators. A complete analysis includ-ing every existing measurements is certainly beyond thescope of this work. Given that the goal is to evaluate therelative constraining powers of t ¯ t and four-top produc-tion, we will follow the approach in Ref. [54], where themost relevant measurements on cross sections and asym-metries have been included. The most recent LHC 13TeV cross section measurements will be added as well.Furthermore, unlike Ref. [54], we shall also consider thedifferential m tt distribution measurement to constrainthe possible shape change from four-fermion operators[16]. In Table 1 we list the measurements that will beused in our fit, together with the corresponding SM pre-dictions. We believe these observables represent the mostsensitive ones to qqtt operators that have been measuredso far.For simplicity, we only consider the SM predictionuncertainties as theoretical uncertainties. We add thetheoretical and experimental errors in quadrature, andwhen the errors are not symmetric, we take the largerone for both sides. We further assume all uncertaintiesare not correlated, except for the theory ones that comefrom the same prediction. The fit for the m tt distribu-tion should be considered at most as a “toy fit” given thatcorrelations between different bins are not available fromthe experimental report. We have dropped the last bindue to the normalization constraint. A K -factor rescal-ing will not improve the normalized distribution. Forthis reason, we use LO prediction and consider varioustheory errors. The scale uncertainty is from variationof µ R and µ F by a factor of 2. The PDF uncertaintyis taken from the envelope of three PDFs, including theNNPDF [35], MMHT [55], and CT14 [56] PDF sets withtheir own uncertainty bands. We have checked that thedifferences between LO and NLO predictions are withinthese errors.9 repared for Chinese Physics CTable 1. Measurements used in the global fit, with corresponding theory predictions and uncertainties. SM prediction MeasurementCross section, Tevatron 1.96 TeV, CDF+D0 7.35 +0 . − . pb [4] 7.60 ± +13 . − . pb [4] 241.5 ± +40 − pb [5] 888 +33 − pb [53]Cross section, LHC 13 TeV, ATLAS 832 +40 − pb [5] 818 +36 − pb [58] A FB , Tevatron 1.96 TeV, CDF+D0 0.095 ± ± A C , LHC 8 TeV, ATLAS 0.0111 ± ± A C , LHC 8 TeV, CMS 0.0111 ± ± m tt distribution, LHC 8 TeV, ATLAS MadGraph5 aMC@NLO+PYTHIA6 [64] Ref. [65] A χ is constructed based on the information in Ta-ble 1, to derive the 95% CL limits on operator coeffi-cients. These limits are compared with the projectionof four-top measurement at 300 fb − . Some results areshown in Figure 5, with two operators turned on at atime. Our observations are the following: • In general, with a 3 ∼ m tt measurement (black dashed vs blue and red).In rare cases they are complementary. • In most cases, combining the t ¯ t inclusive measure-ments, i.e. the cross sections and the asymmetries,provides the most constraining limits, as expected.This is illustrated by the three plots in the first rowin Figure 5. Results from m tt differential measure-ments and four-top cross sections provide slightlyweaker bounds. The t ¯ t global fit, including crosssections, asymmetries, and m tt , is indicated by thegreen shaded area. • Exceptions are the directions that are not effec-tively constrained by asymmetry measurements. Inthis case both m tt differential measurements andthe four-top cross section provide better limits.These cases are illustrated in the second row in Fig-ure 5. The diagonal directions roughly correspondto flat directions between the LLLL/RRRR oper-ators and the LLRR/RRLL operators, whose con-tributions to A FB and A C have the opposite sign. The asymmetry measurements thus do not provideuseful information in these directions. When twooperators are turned on, there can be four such di-rections, as the dominant contributions are comingfrom dim-6 squared terms. This can be seen in Fig-ure 5, second row. Clearly, in these cases both the m tt and the four-top measurements help to furtherimprove our reach in SM deviations.Given that the four-top measurement provides almostthe same information as the m tt differential cross section,we do not expect the four-top to give better constraintsthan a global fit on the qqtt operators. It is howevera valuable add to the precision top physics at the LHC,given that the m tt distribution is already one of the mostsensitive observables to four-fermion operators. In par-ticular, in directions that are not sensitive to asymme-try measurements, information from four-top process isuseful. For this reason we expect that marginalized con-straints from a t ¯ t global fit and those from the four-top process are comparable. However, to really confirmthis point in a model-independent way, we need to takeinto account the four-top operators given in Eqs. (20)-(23), to derive the fully marginalized constraints from thefour-top process. Naively, one would expect that furthermarginalizing over the additional tttt operators wouldmake the constraints on qqtt operators weaker. We willshow that, while this is indeed the case, the effect is notlarge enough to qualitatively change our conclusion.10 repared for Chinese Physics C (cid:45) (cid:45) (cid:45) (cid:45) (cid:142) Qd (cid:72) (cid:76) C (cid:142) Q d (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:142) Qd (cid:72) (cid:76) C (cid:142) Q u (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:142) Qd (cid:72) (cid:76) C (cid:142) t q (cid:72) (cid:76) tt inclusivett m tt tt globaltttt M cut (cid:61) cut (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:142) td (cid:72) (cid:76) C (cid:142) Q d (cid:72) (cid:76) (cid:45) (cid:45)
505 C (cid:142) td (cid:72) (cid:76) C (cid:142) Q d (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:142) td (cid:72) (cid:76) C (cid:142) Q d (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:142) tu (cid:72) (cid:76) C (cid:142) Q u (cid:72) (cid:76) Fig. 5. Selected results for t ¯ t global fit, compared with projected constraints from the four-top production athigh luminosity, at 95% CL. Black solid and dashed contours represent constraints from t ¯ t inclusive measurements(i.e. cross sections and asymmetries) and m tt differential measurement respectively. The green shaded area is thecombined result. Constraints from tt ¯ t ¯ t with M cut = 3, 4 TeV are given by the blue and red curves separately. (cid:45) (cid:45) C (cid:142) tt C (cid:142) QQ (cid:72) (cid:43) (cid:76) C (cid:142) Qt (cid:72) (cid:76) C (cid:142) Qt (cid:72) (cid:76) Current M cut (cid:61) (cid:45) M cut (cid:61) (cid:45) M cut (cid:61) (cid:45) M cut (cid:61) Fig. 6. Marginalized constraints on four-top op-erators, using the current bound as well as theprojection for 300 fb − , for several M cut values. For illustration, in Figure 6 we present the constraintson four-top operators. These are marginalized over otherfour-top operators, but not the qqtt operators. The cur-rent constraints are derived using M cut = 3 TeV, whilethe projected ones for 300 fb − are given with M cut = 2,3, 4 TeV, respectively. The constraints are more con-servative than those directly extracted from a tailoredexperimental analysis, e.g. Ref. [28]. This is expectedbecause we assumed SM signal shape and only use thecross section below M cut . Also note that even for themost constraining limits, the dim-6 squared contribu-tion already dominates over the interference. For this reason, including these operators in our analysis shouldnot significantly affect the constraints on the other 14 qqtt operators.In Figure 7 we present the most important results ofthis work: a comparison of fixed (i.e. one operator ata time) and fully marginalized (i.e. all other operatorsfloated) constraints for all qqtt operators, from the four-top measurement and from the t ¯ t measurements. The t ¯ t constraints are from our global fit, including crosssections, asymmetries, and m tt distribution, while thefour-top constraints are from the 300 fb − projection, µ < .
87, with different M cut values applied. Perturba-tivity in the EFT requires Eq. (35) to hold. This leads toan upper bound of | ˜ C | <
39, 18, and 9.9 respectively for M cut = 2, 3, 4 TeV. The latter two are shown in Figure 7by the vertical dotted lines.11 repared for Chinese Physics C (cid:45) (cid:45)
10 0 10 20 C (cid:142) tu (cid:72) (cid:76) C (cid:142) tu (cid:72) (cid:76) C (cid:142) Qu (cid:72) (cid:76) C (cid:142) Qu (cid:72) (cid:76) C (cid:142) td (cid:72) (cid:76) C (cid:142) td (cid:72) (cid:76) C (cid:142) Qd (cid:72) (cid:76) C (cid:142) Qd (cid:72) (cid:76) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) tq (cid:72) (cid:76) C (cid:142) tq (cid:72) (cid:76) ttbar global: marg.fixed four top: M cut (cid:61) cut (cid:61) cut (cid:61) cut (cid:61) perturbativity: M cut (cid:61) cut (cid:61) Fig. 7. Fixed (i.e. one operator at a time) andfully marginalized (i.e. all other operators floated)constraints for all qqtt operators, from four-topmeasurement and from t ¯ t measurements, at 95%CL. The t ¯ t constraints are from our global fit,while the four-top constraints are form the 300fb − projection. Different M cut values are ap-plied. Perturbativity bounds are derived fromEq. (35). We can see that while the M cut = 2 TeV results areworse, the M cut = 3, 4 TeV marginalized constraints arein general as good as those from the t ¯ t global fit. Thisagrees with our previous expectation: the t ¯ t global anal-ysis gives better individual constraints, thanks to theasymmetry measurements, while the four-top productionis very helpful in directions that are not sensitive to thesemeasurements. Marginalizing over additional four-topoperators does not significantly change our results. Infact, for the four-top process the difference between indi-vidual limits and fully marginalized ones is in general notvery large (see the difference between the red solid andthe red dashed lines), which implies that the cross sectionis dominated by the ( CE / Λ ) terms, while the interfer-ence between different operators or with the SM is small. We want to emphasize that the four-top constraints ob-tained in this work are in general conservative, and inpractice better results can be expected from a tailoredexperimental analysis. For example, in the case of four- t R operator O tt , by assuming the spectrum via the EFTmodel, the constraints can be enhanced by a factor of ∼ qqtt op-erators, and in this case the four-top cross section couldeven be more constraining than a t ¯ t global fit. In addi-tion, in the future combining searches/measurements indifferent channels could further improve the reach.The perturbativity in general is not a problem for M cut = 2, 3 TeV, while for 4 TeV some of the marginal-ized constraints start to approach or even go beyond theperturbative limit. Unitarity gives further constraints.Following Ref. [50], we find that the following constraints C (1) i E Λ (cid:46) √ π , (41) C (8) i E Λ (cid:46) √ π , (42)apply to color singlet and octet operators respectively.These values imply that the limits obtained with M cut =2 ∼ ∼ M cut ≈ t ¯ t global fit, the four-top productioncan provide competitive constraints, due to its enhancedsensitivity to four-fermion qqtt operators. Including thisprocess in the global top-fitting program will definitelyimprove our reach in SM deviations in the top sec-tor. Compared with t ¯ t inclusive measurements which aredominated by systematic errors, the high-mass t ¯ t pro-duction and the four-top production have more room toimprove. In the long term, they should become the cru-cial ones that determine our final reach at the LHC.Before concluding, we remind the reader that the costof such an enhanced sensitivity is a relatively large valueof M cut , which implies that the results are applicable onlyto BSM scenarios above this scale. In the long term, how-ever, we believe that in any case new states below thisenergy scale are likely to be excluded by explicit reso-nance searches. One should also keep in mind that whenthese results are interpreted with explicit BSM models,it typically implies that a large BSM coupling is allowed,and so one should always check the sizes of higher di-12 repared for Chinese Physics C mensional operators, to make sure that the truncationof the SMEFT at dim-6 is valid. Precision measurements are not just for precision it-self. The ultimate goal is the higher reach in testing newphysics and the ability to exclude deviations from theSM, and therefore sensitivity to SM deviations is cru-cial. An observable with an enhanced sensitivity to SMdeviations, even poorly measured, may have a chanceto play an important role. We have demonstrated thislast point, using the four-top production process in thetop EFT context. As a benchmark to assess its sensi-tivity, we use the top-pair production measurements forcomparison. We have found that, as far as the dim-6 four-fermion operators are concerned, the current up-per bound on the four-top signal strength at the O (10)level is already as powerful as the t ¯ t cross section mea-surements which have percentage level precision. Fur-thermore, using the projected bounds for 300 fb − at 13TeV, the four-top measurement can even compete witha global fit using t ¯ t measurements, including the m tt dif-ferential distributions. This comparison is remarkableas the four-top cross section was never considered as aprecision measurement like the t ¯ t process.The origin of the enhanced sensitivity of four-topcross section comes from the fact the four-fermion op-erators can be inserted up to four times in the squaredamplitude, each time with a factor CE / Λ enhancingtheir contribution to the cross section. This factor can belarger than one given the current limits on C/ Λ , and thetypical center-of-mass energy of the four top quarks pro-duced. We have shown that the validity of the EFT, orin other words the validity of expansion in higher dimen-sional operators, can be controlled by E < M cut < Λ NP ,without spoiling the enhancement effect for M cut ∼ a fewTeV, and that the EFT perturbativity CM cut / Λ < (4 π ) is also satisfied in general. The four-top measurementcan thus provide useful bounds for underlying BSM mod-els that live at a scale (cid:38) a few TeV, and is therefore avaluable add to the precision top physics at the LHC, inparticular, given that there is still a lot of room for thisprocess to improve in the future. On the other hand, forBSM scenarios below a few TeV, these results may notapply, but in any case one expects that there the explicitresonant searches provide better exclusion.For comparison purpose we have performed a globalfit for the most relevant t ¯ t measurements, including a dif-ferential measurement on m tt , to which the four-fermionoperators are sensitive. Unlike previous studies, our fit isdone including all dim-6 squared terms as well as interfer-ence effects between all 14 dim-6 operators. We have alsoincluded the four-top operators in our analysis, and have demonstrated that marginalizing over these operators donot qualitatively change our conclusion. Compared withour t ¯ t global fit, the four-top process gives comparablelimits on all operator coefficients. One should howeverkeep in mind that these limits are still relatively conser-vative, given that the upper bound on the total cross sec-tion assumes SM signal shape, and is used regardless ofthe value of the M cut . We expect that future experimen-tal analyses following the SMEFT strategy will furtherimprove the sensitivity of this process to SM deviations.Finally, we would like to point out that potentiallyother processes can have a similar enhanced sensitivity,provided that the following conditions are satisfied: 1)there are multiple heavy particles in the final state, sothat the process is naturally related with a large energyscale; 2) multiple insertion of dim-6 operators are al-lowed, and thus potentially leading to more powers of CE / Λ enhancing the EFT contribution; and 3) thecontribution of dim-6 operators goes like E / Λ , i.e. notsuppressed by any mass or Higgs vev factors. Of course,the validity of EFT has to be checked carefully as onestarts to approach the boundary of its applicability. Still,we hope that this study could inspire new ideas aboutusing observables that are not so precisely measured, tofurther push the frontier of precision measurements inthe EFT context. Acknowledgements
CZ thanks Gauthier Durieux and Fabio Maltoni fortheir invaluable advice.
Appendix A: Operator basis
Here we present the relations between the coefficients ofour four-fermion operators and those of the basis operators repared for Chinese Physics Cin the so-called Warsaw basis, in Ref. [41]. qqtt operator coefficients: C (1 , Qq ≡ C (1)( i i ) qq + 3 C (3)( i i ) qq , (A1) C (3 , Qq ≡ C (1)( i i ) qq − C (3)( i i ) qq , (A2) C (1 , Qq ≡ C (1)( ii qq + 16 C (1)( i i ) qq + 12 C (3)( i i ) qq , (A3) C (3 , Qq ≡ C (3)( ii qq + 16 (cid:16) C (1)( i i ) qq − C (3)( i i ) qq (cid:17) , (A4) C (8) tu ≡ C ( i i ) uu , (A5) C (8) td ≡ C (8)(33 ii ) ud , (A6) C (1) tu ≡ C ( ii uu + 13 C ( i i ) uu , (A7) C (1) td ≡ C (1)(33 ii ) ud , (A8) C (8) tq ≡ C (8)( ii qu , (A9) C (8) Qu ≡ C (8)(33 ii ) qu , (A10) C (8) Qd ≡ C (8)(33 ii ) qd , (A11) C (1) tq ≡ C (1)( ii qu , (A12) C (1) Qu ≡ C (1)(33 ii ) qu , (A13) C (1) Qd ≡ C (1)(33 ii ) qd ; (A14) tttt operator coefficients: C (+) QQ ≡ C (1)(3333) qq + C (3)(3333) qq , (A15) C (1) tt ≡ C (3333) uu , (A16) C (1) Qt ≡ C (1)(3333) qu , (A17) C (8) Qt ≡ C (8)(3333) qu , (A18)where on the l.h.s are the coefficients of the operators used inthis work, while on the r.h.s are the coefficients of the Warsawoperators. i = 1 , Appendix B: More results
We present constraints from four-top production and t ¯ t cross section measurements, similar to Figure 4, but for theoperators in the 2nd and the 3rd categories, i.e. in Eqs. (17)and (18). They are displayed in Figure 8 and Figure 9, re-spectively. (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) C (cid:142) td (cid:72) (cid:76) C (cid:142) t d (cid:72) (cid:76) (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) C (cid:142) Qd (cid:72) (cid:76) C (cid:142) t d (cid:72) (cid:76) (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) cut (cid:61) (cid:45) M cut (cid:61) (cid:45) M cut (cid:61) (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) C (cid:142) Qd (cid:72) (cid:76) C (cid:142) Q d (cid:72) (cid:76) Fig. 8. Constraints from four-top cross sectionand individual t ¯ t cross section measurements, onthe operator coefficients in the second category( ˜ C (8) td , ˜ C (1) td , ˜ C (8) Qd , ˜ C (1) Qd ), assuming two coefficientsto be nonzero at a time. repared for Chinese Physics C (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) Q q (cid:72) , (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) Q q (cid:72) , (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) Qq (cid:72) (cid:76) C (cid:142) Q q (cid:72) , (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) tq (cid:72) (cid:76) C (cid:142) Q q (cid:72) , (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) cut (cid:61) (cid:45) M cut (cid:61) (cid:45) M cut (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) C (cid:142) tq (cid:72) (cid:76) C (cid:142) t q (cid:72) (cid:76) Fig. 9. Constraints from four-top cross section and individual t ¯ t cross section measurements, on the operatorcoefficients in the third category ( ˜ C (8 , Qq , ˜ C (1 , Qq , ˜ C (8 , Qq , ˜ C (1 , Qq , ˜ C (8) tq , ˜ C (1) tq ), assuming two coefficients to be nonzeroat a time. References ,no. 10: 100001 (2016)2 G. Bevilacqua and M. Worek, JHEP : 111 (2012)3 J. Alwall et al, JHEP : 079 (2014)4 M. Czakon, P. Fiedler and A. Mitov, Phys. Rev. Lett. :252004 (2013)5 M. Czakon and A. Mitov, Comput. Phys. Commun. : 2930(2014)6 B. Lillie, J. Shu and T. M. P. Tait, JHEP : 087 (2008)7 A. Pomarol and J. Serra, Phys. Rev. D : 074026 (2008)8 K. Kumar, T. M. P. Tait and R. Vega-Morales, JHEP :022 (2009)9 G. Cacciapaglia, R. Chierici, A. Deandrea et al, JHEP :042 (2011)10 M. Perelstein and A. Spray, JHEP : 008 (2011)11 J. A. Aguilar-Saavedra and J. Santiago, Phys. Rev. D : 034021 (2012)12 L. Beck, F. Blekman, D. Dobur et al, Phys. Lett. B : 48(2015)13 P. S. Bhupal Dev and A. Pilaftsis, JHEP : 024 (2014)Erratum: [JHEP : 147 (2015)]14 B. S. Acharya, P. Grajek, G. L. Kane et al, arXiv:0901.3367[hep-ph].15 T. Gregoire, E. Katz and V. Sanz, Phys. Rev. D : 055024(2012)16 C. Degrande, J. M. Gerard, C. Grojean et al, JHEP : 125(2011)17 Q. H. Cao, S. L. Chen and Y. Liu, Phys. Rev. D , no. 5:053004 (2017)18 S. Weinberg, Phys. Rev. Lett. : 1566 (1979)19 C. N. Leung, S. T. Love and S. Rao, Z. Phys. C : 433 (1986)20 W. Buchmuller and D. Wyler, Nucl. Phys. B : 621 (1986)21 A. Buckley, C. Englert, J. Ferrando et al, Phys. Rev. D , no.9: 091501 (2015) repared for Chinese Physics C
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