Effect of Fermionic Operators on the Gauge Legacy of the LHC Run I

We revisit the extraction of the triple electroweak gauge boson couplings from the Large Hadron Collider Run I data on the $W^+W^-$ and $W^\pm Z$ productions when the analysis also contains additional operators that modify the couplings of the gauge bosons to light quarks and the gauge boson self-energies. We work in the framework of effective Lagrangians where we consider dimension-six operators and perform a global fit to consistently take into account the bounds on these additional operators originating from the electroweak precision data. We show that the constraints on the Wilson coefficients $f_B/\Lambda^2$ and $f_W/\Lambda^2$ are modified when we include the additional operators while the limits on $f_{WWW}/\Lambda^2$ remain unchanged.


II. DIMENSION-SIX OPERATORS
In this work we are interested in deviations from the Standard Model relevant to gauge boson pair production at the LHC. We parametrize those in terms of higher dimension operators as L eff = L SM + n>4,j f n,j Λ n−4 O n,j , (2.1) where the O n,j operators are linearly invariant under SM gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y . Here we assume C and P conservation. The first operators that impact the LHC physics are of n = 6, i.e. dimension-six. The most general dimension-six operator basis respecting the SM gauge symmetry, as well as baryon and lepton number conservation, contains 59 independent operators, up to flavor and hermitian conjugation [22,23]. Since the S-matrix elements are unchanged by the use of the equations of motion (EOM), there is a freedom in the choice of basis [24][25][26][27].
Here we work in that of Hagiwara, Ishihara, Szalapski, and Zeppenfeld [28,29] for the pure bosonic operators.

A. Bosonic Operators
There are nine C-and P -conserving dimension-six operators in our basis involving only bosons that take part at tree level in two-to-two scattering of gauge and Higgs bosons after we employ the EOM to eliminate redundant operators [30]. Of those, five contribute to electroweak gauge boson pair production at LHC after finite renormalizaton effects are accounted for. In particular there is just one operator that contains exclusively gauge bosons In addition there are four dimension-six operators that include Higgs and electroweak gauge fields

(2.3)
Here Φ stands for the Higgs doublet and σ a are the Pauli matrices. We have also adopted the notation B µν ≡ i(g /2)B µν and W µν ≡ i(g/2)σ a W a µν , with g and g being the SU (2) L and U (1) Y gauge couplings respectively. The first three operators in Eqs. (2.2)-(2.3) modify directly the TGC, thus, affecting the ff → V V scattering, where V stands for the electroweak gauge bosons. On the other hand, the operator O BW (O Φ,1 ) is associated to the S (T ) oblique parameter and is constrained by the EWPD. However, these operators also modify the TGC once the Lagrangian is canonically normalized.

B. Operators with fermions
Our operator basis contains 40 independent fermionic operators, barring flavor indices, that conserve C, P and baryon number and that do not involve gluon fields. Of these operators there are 28 of them that do not take part in our analyses since they either modify Higgs couplings to fermions or are contact interactions. Moreover, we also did not considered 6 that give rise to dipole interactions for the gauge bosons. Therefore, the operators that contribute to the processes here analyzed are We have also used the notation of L for the lepton doublet, Q for the quark doublet and f R for the SU (2) L singlet fermions, where i, j are family indices.
In order to avoid the existence of blind directions [31,32] in the analyses of the EWPD we used the freedom associated to the use of EOM to remove from our basis the following combination of operators [30] (2.5) Furthermore, to prevent the generation of too large flavor violation, in what follows we assume no generation mixing in the above operators. For the same reason we will work under the assumption that the coefficient of the potential source of additional flavour violation, O Φud,ij , is suppressed and can be neglected 1 . Also for simplicity we consider the operators to be generation independent. In this case operators O Φe can be bounded by the EWPD, in particular from Z-pole and W -pole observables [33]. In this work we focus on fermionic operators most relevant for gauge boson pair production at LHC which are those leading to modification of the quark couplings to gauge bosons and we will not consider O (1) Φe in our TGC analysis but it is kept in the EWPD studies. Furthermore, for completeness, we also include the effect of the dimension-six four-fermion operator contributing with a finite renormalization to the Fermi constant O LLLL = (Lγ µ L)(Lγ µ L) . (2.6) Altogether the effective Lagrangian considered in this work reads: C. Lorentz and U (1)em invariant Parametrization After accounting for finite renormalization effects, the part of Lagrangian (2.7) relevant for our analyses can be cast in a Lorentz and U (1) em invariant form as: where P L,R are the chirality projectors. The TGC effective couplings are where c W (s W ) stands for the cosine (sine) of the weak mixing angle and c 2θW is the cosine of twice this angle. Notice that there are only three independent TGC due to the linear realization of the SU (2) L ⊗ U (1) Y symmetry in the dimension-six operators. The effective couplings of the fermions can be written as The first contributions to these anomalous couplings originates from finite renormalizations due to O BW and O Φ,1 with the oblique parameters given by The fermionic dimension-six operators in Eq. (2.7) give rise to additional contributions Φe . (2.13) The anomalous TGC and gauge bosons interactions to quarks modify the high energy behavior of the scattering of quark pairs into two electroweak gauge bosons since the anomalous interactions can spoil the cancellations built in the SM. For the W + W − and W ± Z channels the leading scattering amplitudes in the helicity basis are where s stands for the center-of-mass energy and θ is the polar angle in the center-of-mass frame.
We notice that the leptonic operator O Φe does not contribute to the gauge boson production amplitudes at LHC. It only contributes to the decay rate of the Z boson in the ZW channels and in the narrow width approximation its effect is subdominant. We will not consider it in the TGC analysis but it is kept in the EWPD analysis.

III. ANALYSES FRAMEWORK
In order to constrain the parameters in the effective Lagrangian in Eq. (2.7) we study the W + W − and W ± Z productions in the leptonic channel since these are the measurements with the highest sensitivity for charged triple gauge boson vertices. In doing so we consider the same kinematic distributions employed by the experiments for their anomalous gauge boson coupling analyses what allows us to validate our results against the bounds obtained by the experiments in each of the final states. More specifically, the channels that we analyze and their kinematical distributions are For each experiment and channel, we extract from the experimental publications the observed event rates in each bin, N a i,d , as well as the background expectations N a i,bck , and the SM W + W − (W ± Z) predictions, N a i,sm . The procedure to obtain the relevant kinematical distributions predicted by Eq. (2.7) is as follows. First we simulate the W + W − and W ± Z productions using MadGraph5 [34] with the UFO files for our effective Lagrangian generated with FeynRules [35,36]. We employ PYTHIA6.4 [37] to perform the parton shower, while the fast detector simulation is carried out with Delphes [38]. In order to account for higher order corrections and additional detector effects we simulate SM W + W − and W ± Z productions in the fiducial region requiring the same cuts and isolation criteria adopted by the corresponding ATLAS and CMS studies, and normalize our results bin by bin to the experimental collaboration predictions for the kinematical distributions under consideration. Then we apply these correction factors to our simulated W V distributions in the presence of the anomalous couplings. This procedure yields our predicted number of signal events in each bin i for the "a" channel, N a,nosys i,sig . The statistical confrontation of these predictions with the LHC Run I data is made by means of a binned loglikelihood function based on the contents of the different bins in the relevant kinematical distribution of each channel.
Depending on the number of data events in the bin we use a Poissonian or a Gaussian probability distribution for its statistical error. In constructing the log-likelihood function we simulate the effect of the systematic and theoretical uncertainties by introducing two sets of pulls: two globally affecting the predictions of the event rates in all bins in fully correlated form -which parametrize, among others, the luminosity uncertainty, and theoretical errors on the total cross-section for the process and its backgrounds-and N a b independent pulls, one per-bin, to account for the bin-uncorrelated errors arising from the theoretical errors affecting the distributions, experimental energy resolutions and, in general, any energy and/or momentum dependence of the uncertainties. With this, the number of predicted events in bin i for channel a is N a i = (1 + ξ a sig )(1 + ξ a i,unc )N a,nosys i,sig The errors of these pulls are introduced as Gaussian bias in the log-likelihood functions and are extracted from the information given by the experiments. For completness they are reported in the table above.
In order to validate our simulation we obtain first the 95% CL allowed regions for the TGC for each channel and experiment under the same assumptions the collaboration used. For example, we present in Figure 1 our two-dimensional allowed regions using the ATLAS W + W − data and assuming that the only non-vanishing Wilson coefficients are f W W W , f W and f B , two different from zero at a time, as in the ATLAS analysis. As seen in the figure, our results for the 95% CL allowed region (blue region) agrees well with the one obtained by ATLAS, whose border is represented by the black curve.
When including the effect of the additional operators we must also account for their contribution to the EWPD. Our construction of the χ 2 function for the EWPD follows the analysis in Ref. [33] to which we refer the reader for details. In brief in our EWPD analysis we fit 15 observables of which 12 are Z observables [19]: complemented by three W observables M W , Γ W and Br(W → ν) that are, respectively, its average mass taken from [39], its width from LEP2/Tevatron [20], and the leptonic W branching ratio for which the average in Ref. [39] is considered. The correlations among these inputs are presented in Ref. [19] and we take them into consideration in the analyses. The SM predictions and their uncertainties due to variations of the SM parameters were extracted from [40].  Altogether we construct a combined χ 2 function φ,e , f LLLL ) , (3.1) from which we derive the allowed ranges for each coefficient or pair of coefficients after marginalization over all the others. It is worth commenting that this marginalization of the profiled binned log-likelihood is computationally very expensive due the high dimensionality of the parameters space. Achieving an acceptable accuracy in the determination of the statistical confidence bounds for one-and two-dimensional distributions requires typically hundreds of millions of Monte Carlo evaluations for each one of the points used to obtain the 1D and 2D the allowed regions.
Finally, for comparison, we also consider the constraints from LEP2 global analysis of TGC [3]. In order to do so we follow the procedure in Ref. [8] and construct a simplified gaussian χ 2 LEP2 using the central values, σ and correlation matrix for the couplings ∆g 1 Z , ∆κ γ and λ and their correlation coefficients from the final combined LEP2 analysis in Ref. [3] (reproduced in Table I for completness) which was performed in terms of these effective TGC coefficients under the relations implied by dimension-six effective operator formalism for TGC. We notice, however, that in extracting those bounds on the effective TGC couplings, the LEP collaborations did not include the effect of fermion operators. For that reason the combination of those LEP2 bounds with our LHC Run I and EWPD is only shown for the purpose of illustration.

IV. BOUNDS ON TRIPLE GAUGE BOSON INTERACTIONS
We start by showing the results of our analysis in terms of the allowed ranges for the Wilson coefficients of the three "canonical" TGC operators, f W W W /Λ 2 , f W /Λ 2 and f B /Λ 2 . We depict first in Fig. 2 the 95% CL (2 dof) allowed regions in the planes f B /Λ 2 ⊗ f W /Λ 2 , f W W W /Λ 2 ⊗ f B /Λ 2 and f W W W /Λ 2 ⊗ f W /Λ 2 for the W + W − and W ± Z channels and for ATLAS and CMS, as well as the combination of these results. In order to assess the impact of additional operators in the TGC extraction at LHC we performed first the "standard" analysis fitting just these three coefficients, and setting the coefficient of all other operators to zero. The corresponding allowed regions are shown in the left panels after marginalizing over the third coefficient which is not displayed. Conversely the results of the global analysis of the LHC Run I data together with EWPD performed in terms of 11 non-zero Wilson coefficients (see Eq. (2.7)) are shown on the right panels. These regions are obtained after marginalization over the 9 undisplayed coefficients.
One salient feature of Figure 2 is that the W ± Z bounds on the Wilson coefficient f B /Λ 2 are much looser than the ones on f W /Λ 2 and f W W W /Λ 2 , as expected, because O B does not contribute to the leading term of the growth of the scattering amplitudes; see Eqs. (2.14)- (2.20).
For better comparison of the results obtained with and without including the additional operators we overlay in Fig. 3 the 1σ and 95% CL allowed regions obtained combining all channels and experiments for the two scenarios. As we can see from this figure the addition of more parameters leads to the expansion of the allowed regions, as expected. Moreover, the region of f B /Λ 2 suffers the largest shift towards positive values of this parameter while there is a small shift in the f W /Λ 2 direction and there is no appreciable displacement along the f W W W /Λ 2 axis. FIG. 2: Allowed 95% CL regions in the planes fW W W /Λ 2 ⊗ fB/Λ 2 (top row), fB/Λ 2 ⊗ fW /Λ 2 (middle row) and fW W W /Λ 2 ⊗ fW /Λ 2 (lower row) for the different channels as labeled in the figure. In the left panels only fW W W , fW and fB were considered non-zero in the fit, while the right panels display the result from the 11 parameter fit. In each case we marginalize over the undisplayed non-zero variables.
The corresponding dependence of the ∆χ 2 for the two analysis with each of the three coefficients is given in Fig. 4 and from those we read the 95%CL one-dimensional allowed ranges for each coefficient given in Table II. As seen above, the ∆χ 2 distribution for f B /Λ 2 (f W /Λ 2 ) broadens and shifts to positive (negative) values when we compare the results considering only the LHC Run I data and three canonical parameters (solid black line) with the one containing additional operators also constrained by the EWPD (solid blue line). Quantitatively the effect is slightly larger for f B /Λ 2 whose allowed range widens by about 30% versus 20% for f W /Λ 2 .
The effect of each of the six additional operators on the extracted range of the three "canonical" TGC operator coefficients is illustrated in Figure 5 where we depict the two-dimensional correlations between the three TGC coefficients and the additional ones. In each panel of this figure we exhibit the 1σ and 95% CL level (2 dof) allowed regions after marginalizing over the remaining parameters. As we can see, f B /Λ 2 has a significant correlation only with f Φd /Λ 2 ). This is expected as these are the operator coefficients contributing the growth of the scattering amplitudes into longitudinally polarized gauge bosons (Eqs. 2.14-2.20). In particular the correlion with f The impact of the LHC diboson production data on the determination of the parameters directly constrained by the EWPD is illustrated in Fig. 6 that depicts the ∆χ 2 distribution as a function of these parameters where the magenta (blue) line stands for the result obtained using the EWPD (and the LHC Run I diboson production data). The top left and middle panels of this figure show that the addition of the LHC data does not alter the constraints on f BW /Λ 2 and f Φ1 /Λ 2 parameters. This is easy to understand since these parameters do not modify the high energy behavior of qq → V V amplitudes; see Eqs. (2.14)- (2.15). This is expected from O Φ,1 as it only contributes to the amplitudes via finite renormalization effects of the SM parameters. The operator O BW , on the other hand, modifies the TGC directly also, however, its effects on the Z wave-function renormalization cancel the growth with the center-of-mass energy due to the anomalous TGC. From the top right, bottom left and middle panels we can see that the impact of the Run I data on f (1) ΦQ /Λ 2 and f (1) Φu /Λ 2 is marginal. f (1) Φd /Λ 2 is the only parameter whose ∆χ 2 distribution gets significantly affected. The EWPD analysis favours non-vanishing value for f (1) Φd /Λ 2 at 2σ, a result driven by the 2.7σ discrepancy between the observed A 0,b FB and the SM. On the contrary no significant discrepancy is observed between the observed LHC Run I diboson data and the SM. Hence there is a shift towards zero of f Φd /Λ 2 when including the LHC Run I data in the analysis. This slight tension results also into the reduction of the globally allowed range.
We finish this section by comparing our results with the bounds derived from LEP2 diboson data. To do so we plot in Figure 7 the two-dimensional 95% CL allowed regions for the three combination of the canonical TGC parameters for the analysis with and without additional operators together with the LEP2 results. As shown in Ref. [8], the limits emanating from the canonical LHC Run I diboson data (black solid line) are substantially more stringent than those Φu /Λ 2 , and f (1) Φd /Λ 2 after the marginalization over the undisplayed parameters. The magenta line stands for the results using only the EWPD while the blue one is obtained considering the EWPD and LHC Run I diboson production.
imposed by LEP2 (black dashed line). As seen in this figure, enlarging the number of operators in the LHC analyses, together with the EWPD, does not alter this conclusion despite the growth of the allowed regions (solid blue line). For illustration we also show in the figure the allowed regions obtained by naively combining the general LHC Run I + EWPD analysis with the LEP2 information (see discussion at the end of Sec. III). As seen, including LEP2 data in the approximation used leads to a reduction of the allowed regions in the f B /Λ 2 direction, as well as to a shift of it towards negative values (see also the dashed blue line in Fig. 4).

V. SUMMARY
In this work we have quantified the impact of possible anomalous gauge couplings to quarks on the TGC determination performed using the LHC Run I diboson data. In order to carry out a statistically consistent analysis we have included in addition the EWPD to constrain the couplings between quarks and gauge boson as well as the modifications of the gauge boson self energies. We have worked in the framework of effective Lagrangians so our study has been performed including the 11 dimension-six operators given in Eq. (2.7).
As a summary of our findings we present in Table II the 95% CL globally allowed ranges for the Wilson coefficients of the nine operators that contribute to the LHC Run I data considered. The comparison of the the first and third columns of this table shows that the addition of the new operators modifies the TGC bounds on f B /Λ 2 and f W /Λ 2 coming from the LHC Run I diboson. Quantitatively the effect is slightly larger for f B /Λ 2 whose allowed range widens by about 30% versus 20% for f W /Λ 2 . The limits on f W W W /Λ 2 , on the contrary, result almost unaffected. Despite these changes, the constraints on these parameters are still dominated by the LHC Run I data at large, and are still substantially stronger than those obtained from LEP2 data.
We have also learned from our analyses that the LHC Run I diboson data is not precise enough to yield substantial information on the gauge couplings to quarks in addition to what is already known from EWPD; contrast the second and third columns of Table II. The only apparent exception is f (1) Φd /Λ 2 which in the considered family universal scenario is driven to be non-zero in the EWPD analysis by the discrepancy between the measured A 0,b FB at LEP/SLC and the SM while LHC Run I data shows no evidence of any deviation with respect to the SM. Nevertheless, these results allow us to foresee that diboson production at the LHC will play an important role in the analyses of anomalous couplings of gauge bosons to quarks as the LHC increases the integrated luminosity. Hence global analysis of LHC and EWPD are becoming a must for consistent determination of the Wilson coefficients of the full set of dimension-6 operators.