Limits on anomalous top quark gauge couplings from Tevatron and LHC data

We review and update current limits on possible anomalous couplings of the top quark to gauge bosons. We consider data from top quark decay (as encoded in the $W$-boson helicity fractions) and single-top production (in the $t$-, $s$- and $Wt$-channels). We find improved limits with respect to previous results (in most cases of almost one order of magnitude) and extend the analysis to include four-quark operators. We find that new electroweak physics is constrained to live above an energy scale between 430 GeV and 3.2 TeV (depending on the form of its contribution); strongly interacting new physics is bounded by scales higher than 1.3 or 1.5 TeV (again depending on its contribution).

We review and update current limits on possible anomalous couplings of the top quark to gauge bosons. We consider data from top quark decay (as encoded in the W -boson helicity fractions) and single-top production (in the t-, s-and W t-channels). We find improved limits with respect to previous results (in most cases of almost one order of magnitude) and extend the analysis to include four-quark operators. We find that new electroweak physics is constrained to live above an energy scale between 430 GeV and 3.2 TeV (depending on the form of its contribution); strongly interacting new physics is bounded by scales higher than 1.3 or 1.5 TeV (again depending on its contribution).

I. MOTIVATIONS AND NOTATION
Precision studies of the interaction vertices of the top quark to the gauge bosons provide an important tool in testing the standard model (SM) and searching for new physics contributions. Currently available measurements from the Tevatron and the LHC allow to set limits on possible deviations in the values of the top quark couplings from their SM values. A model independent framework to study these anomalous couplings is effective field theory, where the modifications are encoded into the coefficients of a set of higher dimensional operators that parametrize the effects of new physics at low energy. The language of effective field theory is very helpful in identifying the source of top quark anomalous couplings and guiding us toward the identification of possible new physics contributions.
The top quark has both strong and electroweak (EW) interactions. While all interactions come together in collider physics, it is possible to separate in most processes the EW from the strong part so that the anomalous vertices can be discussed separately. In this paper we concentrate on the study of the W tb vertex; the strong interaction Gtt vertex was recently studied in [1]-the results of which are here reported. Possible deviations in the interaction between the top quark and the neutral bosons Z and γ-they enter the associated productions ttZ and ttγ-are left out because still poorly measured.
A. Effective W tb vertex Following the literature [2], we write the effective lagrangian that describes, after EW symmetry breaking, the most general W tb vertex as where g is the SU (2) L gauge coupling, P L,R the chiral projectors (1 ± γ 5 )/2 and σ µν = i[γ µ , γ ν ]/2. The coefficients V L , V R , g L and g R are, in general, complex dimensionless constants. In this work we will restrict ourselves to the CP-conserving case and these couplings are taken to be real. In the SM the W tb vertex in eq. (1) reduces at the tree level to the Dirac vertex with V L = 1 (after mass diagonalization, V L = V tb 1) and we take it to be positive.
Corrections to V L , as well as the others non-zero anomalous couplings V R , g L , g R can be generated by new physics. If the physics beyond the SM lies at an energy scale Λ that is larger than the EW scale υ, then we can parametrize its effects via higher dimensional operators respecting the SM symmetries in a series suppressed by inverse powers of the scale Λ. The leading contributions arise at dimension six and can be written as an expansion of local operators where c k are dimensionless coefficients.

arXiv:1406.5393v1 [hep-ph] 20 Jun 2014
The complete list of independent dimension six SM effective operators is reported in [3] of which we follow the notation. The subset of operators that contributes to the anomalous couplings in eq. (1) consists of the following operatorsÔ 2), υ = 246 GeV, we can express the anomalous couplings in terms of the effective field theory coefficients as follows There is another operator that enters in the processes we consider, it is the four fermion operator where q T = (u d) and, as before, q T = (t b). It does not give a direct contribution to the anomalous couplings but its interplay with the other operators modifies the limits. For this reason it must be included and its effect parametrized by a new coefficient which can be further identified by taking c where the coefficient 2π represents the strength of a strongly interacting new-physics sector-the integration of which gives rise to the effective operator. The effect of these operators on the top-quark EW anomalous couplings can be best studied in two processes: top decay (by means of the W polarizations) and single-top production.

B. Effective Gtt vertex
The effective vertex for the Gtt interactions can be written as where g s is the strong SU (3) C coupling constant, G µ = T A G A µ is the gluon field, T A are the SU (3) C group generators, q µ is the momentum carried by the gluon, t denotes the top quark field. The interaction in eq. (8) is the most general after assuming that the vector-like nature of the gluon-top quark vertex is preserved by the underlying dynamics giving rise to the composite state.
The leading contributions come from the following two higher dimensional operators: BC are the SU (3) C structure constants. In eq. (9) the operatorÔ 1 andÔ 2 are, respectively, of dimension six and five. We limit ourselves to the CP -conserving case and the dimensionless coefficients C 1 and C 2 are taken to be real. The operatorÔ 1 gives the leading q 2 dependence to F 1 whileÔ 2 gives the q 2 -independent term of F 2 : Operators of higher dimensions can in general contribute-they gives further terms in the expansion of the form factors-but their effect is very much suppressed. The form of the coefficients in front of the operators in eq. (9) is conventional and dictated, in our case, by the analogy with the electromagnetic form factors. In addition, the operator O 2 is written for convenience with an extra factor υ/m t because it can be thought as coming, after EW symmetry breaking, from a dimension six SU (3) C × SU (2) L × U (1) Y gauge invariant operator that includes the Higgs boson field.
These vertices were analyzed (most recently in [1] where references to all related works can be found) by means of the tt production cross section and spin correlations. The following limits are found: These limits give a direct bound on the magnetic moment µ and the RMS radius r 2 , traditionally used to characterize the size of extended objects. They correspond to We take the coefficients C 1 and C 2 to be zero when studying EW anomalous couplings-they only enter one channel of the single-top production-and assume that the strong interaction of the top quark follows the SM in all production processes.

II. METHODS
In order to study the effect of anomalous couplings on top decay rates and single top production cross sections at the LHC and Tevatron, we first use FeynRules [4] to implement our model, which has been defined to be the SM with the addition of the effective operators in eq. (3) and eq. (5). The dependence on these operators is encoded in the coefficients c i = V L , V R , g L , g R , C 4f .
FeynRules provides the Universal FeynRules Output (UFO) with the Feynman rules of the model. The UFO is then used by MadGraph 5 [5] (MG5) to compute the branching ratios in the decay rates and production cross sections, which we denote by F MG5 L,0 and σ MG5 , respectively. MG5 computes the square of the amplitude for each single channel. The partonic level result thus obtained can be compared with the partonic experimental cross section that is extracted by the experimental collaborations.
We compute, using MG5, the top-quark decay width and the single-top production cross section varying the values of the anomalous couplings c i in a range that goes from −2 to 2, except for V L that is varied only for positive values. These different values of branching ratios F MG5 L,0 (c i ) and cross sections σ MG5 (c i ) are used to obtain limits on the coefficients by comparing the MG5 computation with the measured cross sections and helicity fractions at the LHC at the center-of-mass (CM) energies √ s = 7 and 8 TeV, as discussed in the next section. By proceeding in the same way, we have also computed the rate and production cross section at the Tevatron and compared it with the measured cross section at the CM energy √ s = 1.98 TeV. Analytical expressions for the amplitudes that we study numerically can be found in [6].

A. Statistical analysis
To obtain 95% confidence level (CL) limits on the coefficients c i from the production cross sections, we use the quantity ∆σ exp , defined to be the difference between the central value of the measured cross sectionσ exp and that of the SM theoretical valueσ th : The uncertainty on ∆σ exp is given by summing in quadrature the respective uncertainties: FIG. 1: Tree-level diagrams for, from left to right, s-, t-and W t-channel.
Using the cross sections σ MG5 (c i ) calculated with MG5 we compute the value of the cross section coming from new physics as: The quantity ∆σ MG5 (c i ) represents the contribution to the cross section coming from the interference between SM leading order and new physics diagrams as well as terms quadratic in the anomalous couplings.
Values of c i for which ∆σ MG5 (c i ) is more than two standard deviations from ∆σ exp , namely: are considered excluded at 95% CL. To obtain limits from the branching ratios in top decay, as measured by the W -boson helicity fractions, a similar procedure is used. The experiments measure the three fractions F 0 , F L and F R (see below for the definition). Being extracted from the same data set and from the same observable, these fractions are not independent. F R is constrained to be equal to 1 − F 0 − F L and is therefore not considered in the limit extraction. In addition, there is a correlation factor ρ different from 0 between the measured values of where: and the constrain L(c i )dc i = 1 is imposed. Then, the quantity L 95% is defined as: (20) and values of c i for which L(c i ) < L 95% are considered excluded at 95% CL.

III. RESULTS
Top-quark decay and single-top production are the two processes where we can directly probe the top coupling to the weak bosons.
The W bt vertex in top-quark decay is best studied by means of the helicity fractions F 0 and F L defined, respectively, as the ratios between the rates of polarized decay of the top quark into zero and left-handed W bosons and the total decay width.
the helicity fraction into right-handed W bosons is vanishingly small. These branching ratios receive corrections from the operatorsÔ tW andÔ bW . The SM values at the next-to-next-to-leading order (NNLO) order in QCD are computed in [7,8].
We study the effect of the operators in eq. (3) and eq. (5) by varying the coefficients of two of them at the time and we derive the related limits following the statistical analysis described in the previous section. The results are depicted in Figs. 2, 3 and 4.
The determination of the limits on the coefficients g R and g L is dominated by the top-quark decay but the single-top production cross section is useful in eliminating larger values (the small area on top of Fig. 2). Accordingly, only the lower region is allowed and − 0.109 ≤ g L ≤ 0.076 and − 0.142 ≤ g R ≤ 0.024 .
The interplay among the various channels of single-top production is crucial in delimiting the allowed region for the parameters C 4f and V L in Fig 3. Unfortunately it is not sufficient in completely delimiting the allowed range of the coefficients to a single region. The W t-channel is not sensitive to V R and C 4f . Accordingly, the final bound is weaken when both coefficients are allowed to vary simultaneously and we find 0.584 ≤ V L ≤ 1.145 and − 0.037 ≤ C 4f ≤ 0.120 .
A future improvement in the W t-channel measurement at the LHC could be instrumental in delimiting the range to a single region. Finally, it is the interplay between top decay and the t-channel of single-top production that provides the limit on the coefficients V R and V L (see Fig. 4). We find 0.891 ≤ V L ≤ 1.081 and − 0.121 ≤ V R ≤ 0.173 . Notice that the coefficient V R is very much constrained by flavor physics [18] from which it should be When this limit is included, Fig. 4 should be read only along the line V R 0 with a slightly improved bound Our results are summarized in Table I in the case in which the various anomalous couplings are turned on one at the time. The limits found can be interpreted in terms of the energy scale of the effective operators in eq. (3) (see second column of Table I). By inspection, we see that all EW limits are around 10 −1 which translates into a characteristic scale Λ 700 GeV (actually from 430 GeV to 1 TeV, depending on the contribution) except for the four-quark interaction which has Λ 3.2 TeV, if we follow the definition in eq. (6). Above these limits, there is still room for new physics. These results can be compared with the strong sector interactions of the top quark where the corresponding energy scale is Λ 1.3 and 1.5 TeV for, respectively, the operatorsÔ 1 andÔ 2 in eq. (9) [1].

IV. DISCUSSION
The previous status of limits on anomalous top quark gauge couplings was the following. Constraints on the W tb vertex from Tevatron and early LHC data were reported in [19]. The bounds were obtained, as in this paper, by combining the experimental measurements of W -boson helicity fractions and single-top production and by varying two operators at the time.
More recently, the measurement of the W -boson helicity fractions has been substantially improved by ATLAS and CMS in [20], [21] and [22] and limits on the coefficients g R and g L were extracted by the experimental collaborations based on these new measurements. Concerning the single top production, many detailed results have been released by both Tevatron and LHC experiments [13][14][15][16][17], as reported in the previous section.
We confirm and improve the above limits by means of the most recent data set at 8 TeV by CMS [9] and the combined use of the helicity fractions and single-top production cross sections. The improvement is substantial, in most cases of almost one order of magnitude. As expected, the simultaneous presence of more than one anomalous coupling weaken the limits. The result for the four-fermion operator (5), and its impact on the determination of the coefficient V L is new.