LHC Data and Aspects of New Physics

We consider the implications of current LHC data on new physics with strongly interacting sector(s). We parametrize the relevant interaction Lagrangian and study the best fit values in light of current data. These are then considered within a simple framework of bosonic technicolor. We consider first the effective Lagrangian containing only spin-0 composites of the underlying theory, which corresponds to a two Higgs doublet model. With respect to this baseline, the effects of the vector bosons, a staple in strong interacting theories, are illustrated by considering two cases: first, the case where the effects of the vector bosons arise only through their mixing with the electroweak SU(2)_L gauge fields and, second, the case where also a direct interaction term with neutral scalars exists. We find that the case of a W' coupling to the Higgs boson only via the mixing of vector fields produces a negligible improvement in the fit of the present data, while even a small direct coupling of the composite vector fields to the Higgs allows the tested model to fit optimally the experimental results.


I. INTRODUCTION
The recent discovery of a light scalar with properties compatible with those of the Standard Model (SM) Higgs boson, h 0 , imposes new experimental tests on previously viable beyond the Standard Model (BSM) theory frameworks. Fervent activity in this direction has focused mostly on the possibility that a new charged particle could enhance the Higgs decay rate to two photons [1][2][3][4][5][6][7][8][9][10][11][12][13]. This is measured at LHC and Tevatron, with the full dataset, to be slightly enhanced compared to the SM prediction [14][15][16], with the diphoton signal strength equal to 1.65 ± 0.32 at ATLAS and 1.11 ± 0.31 at CMS. 1 While most of the efforts were focused on the Higgs physics associated with a new charged scalar or a vector fermion, both of which naturally arise in supersymmetry [17][18][19][20][21][22][23] or composite Higgs frameworks [24][25][26][27], less extensive research has been conducted recently on the possibility that a heavy charged vector boson be responsible for the observed deviations of the Higgs couplings from the corresponding SM predictions [9,[28][29][30].
In this paper we want to explore the effect of couplings of a heavy W boson to the Higgs on LHC and Tevatron observables. To approach this task we first, in Section II, perform a fit with the parameters of a simple effective Lagrangian featuring rescaled Higgs couplings to SM particles as well as to a heavy charged scalar and a heavy charged vector boson. To have a concrete model to test against the results of the general fit, in Section III we introduce a low energy effective Lagrangian for a simple bosonic technicolor model, the bosonic Next to Minimal Walking Technicolor (bNMWT) [37][38][39][40]. The low energy effective Lagrangian for the scalar sector of bNMWT corresponds to a Type-I Two Higgs Doublet Model (2HDM) [41]. We scan the allowed parameter space of this model for data points viable under direct search constraints and electroweak (EW) precision tests, compare the data points to the measured Higgs physics observables, and find the optimal fit of the model. In Section IV we introduce two composite vector boson triplets in the low energy Lagrangian, while conserving gauge invariance at the microscopic level, which both mix with the SM W ± and directly couple to h 0 , and repeat the goodness of fit analysis above to determine the optimal values of the W and W couplings to h 0 . We illustrate separately the features which originate from the mixing and from the direct coupling. Our essential conclusion is that mixing alone produces a negligible improvement of the fit to the current data. A direct interaction on the other hand makes the bNMWT model a perfectly viable candidate.

II. LHC AND TEVATRON DATA FIT
The experimental results are expressed in terms of the signal strengths, defined aŝ .
where Ω is the efficiency associated with the given final state Ω in an exclusive search, while for inclusive searches one simply has σ tot = σ pp→h(X) , the h production total cross section.
The signal strengths from ATLAS, CMS, and Tevatron are given in Table I  of integrated luminosity at 8 TeV and 5 fb −1 at 7 TeV. The bb quark pair is produced in association with a vector boson, with an efficiency assumed equal to one, while the other searches are inclusive. We also include in our analysis the dijet associated γγ production based on the full dataset [50,51], with signal strengths and efficiencies 2 listed in Table II.
Within the class of models we will study, the part of the Lagrangian relevant for the recent LHC data is of the form where the third and fourth terms involve, respectively, charged (but color singlet) vector and scalar bosons. We fix the mass parameters to the physical mass of the corresponding particle and v w to the EW vacuum expectation value (vev), v w = 246 GeV.
Consequently, the cross sections and branching rates relevant for Higgs physics are related to the corresponding quantities of the SM in a simple way. We definê and then, in terms of the coupling coefficients in Eq. (2), we havê where the gg and h final states are produced through a loop triangle diagram with only quarks as virtual particles.
The calculation of the Higgs decay rate to two photons is more involved. By using the formulas given in [52], we can write where the index i is summed over the SM charged particles as well as S ± and W ± , N i is the number of colors, e i the electric charge in units of the electron charge, and the factors F i are defined by with In the limit of heavy W ± and S ± , one finds Given the experimental lower bounds on m W and m S [53, 54], the error on F W is irrelevant while |F S | gets enhanced by about 10% for m S = 150 GeV: since the experimental error on µ γγ is large and constructive interference of the S ± and W ± is favored by the experiment, we also assume the error involved by the above approximation for F S to be negligible.
We notice also that in the limit of heavy masses for the charged scalar and vector bosons the light Higgs decay to such (virtual) states is highly suppressed by kinematics, and therefore no additional decay channels have to be taken into account besides those of the SM.
To evaluate the theoretical predictions for the measured observables, we need the SM production cross sections for the Higgs boson and the SM branching ratios for its decay.
The production cross sections at the LHC and Tevatron for the final state Ω are given [55] in Table III.
To determine the experimentally favored values of the free parameters a f , a V , a V , a S , we minimize the quantity where the measured values and errors of the observables are given in Tables I,II, while the numerical predictions of the theory are easily determined from Eqs. (3)(4)(5)(6)(7)(8), with the SM input values given in Table III. In defining χ 2 above, we assumed the correlation matrix to be simply the identity matrix. We note that it is not possible to constrain both a S and a V since they both contribute only to the diphoton decay. The optimal values given below then refer to either of the two taken equal to zero: with The probability to get a minimum value of χ 2 larger than the optimal value above is naively expected to be around 50%, and therefore the corresponding value obtained above shows that the simple parametrization of Eq. (2) fits satisfactorily the data.
As a comparison, the SM produces for only the Higgs physics data, which indeed is a rather ideal result. The inclusion of the EW parameters S and T (S = T = 0 for SM) [57][58][59] in the fit improves further the quality of the fit: which shows that the SM is still perfectly viable in light of current collider data.
In the next section we use the Higgs physics constraints derived here to test the viability of a simple bosonic walking technicolor model [40] whose low energy effective Lagrangian belongs to the class specified by the generic Lagrangian of Eq. (2).
In Technicolor (TC) an additional, confining gauge interaction causes techniquarks, charged under TC and the EW interaction, to condense and break spontaneously the EW symmetry [31,32]. This mechanism allows the W and Z bosons, and the composite states of the strongly coupled TC sector to acquire mass, while the SM fermions remain massless.
The TC sector we consider has SU(2) L ×SU(2) R chiral symmetry and is described in terms of the complex composite meson field M T = (φ + , φ 0 ) by an effective Lagrangian To provide mass for ordinary matter fermions, an additional interaction linking them to the TC condensate has to be provided. In bosonic TC models, this link is provided by one (or more) elementary scalar(s) [60][61][62][63][64]. While in this paper we consider nonsupersymmetric theories, bosonic technicolor effective Lagrangians also arise as low energy realizations of supersymmetric technicolor theories [65][66][67]. In the context of bosonic TC, it is therefore the techniquark condensate that breaks EW symmetry, while the scalar plays the role of "spectator". The Higgs Lagrangian is written in terms of the usual complex doublet H as The link between the technicolor and the SM matter fields obtained at the effective Lagrangian level is due to the Yukawa couplings of the Higgs field H. In addition to the usual couplings to the SM matter fields, these include also the couplings to techniquarks, y T CΨL HΨ R . When constructing the effective Lagrangian for the composite sector of the theory, this coupling generates further terms in the effective TC Lagrangian so that the technicolor sector is described by [40] 3 where c i are unknown parameters and f is the vev of M . The model that we consider is therefore specified by the effective Lagrangian where L SM is the usual SM Lagrangian containing the sectors L Higgs and L Yuk . The coefficients c i in Eq. (20) are estimated by naive dimensional analysis [68,69] to be Two of the parameters on the r.h.s. of Eqs. (18,20) are determined by the extremum conditions of the potential. Furthermore, the electroweak scale constrains the vevs of M and H by Finally, the requirement for the potential to be bounded from below imposes The mass eigenstates are obtained by diagonalizing first the kinetic terms and then applying a rotation to diagonalize the mass terms in the scalar and pseudoscalar sectors. First, the Higgs fields M and H are expressed as After this transformation, the fields M 1,2 are written in terms of the charge eigenstates as The rotation angles α and β determine the physical states in the scalar and pseudoscalar sectors so that the Goldstone bosons, G ± and G 0 , provide the respective longitudinal components of the W ± and Z bosons, while h 0 , H 0 , A 0 , and H ± are the neutral scalars, pseudoscalar, and charged scalar mass eigenstates, respectively: The mixing angle β is defined so that tan β = f 2 /f 1 . The masses of the lightest composite states, including neutral scalars, are naturally expected to be of O(Λ T C ) ∼ 1 TeV. For bosonic TC the strong dynamics effect on the Higgs mass can be somewhat tamed by the mixing of the composite and elementary neutral scalar states, since the latter state can have a squared mass term much smaller than the former. This mechanism is analogous to the TeV-scale seesaw recently put forward in [70]. Moreover, it has been shown [71] that the topquark loop contribution can greatly reduce the tree level TC prediction on the Higgs mass.
A further suppression of the light Higgs mass is expected in NMWT because of walking dynamics [38]. From here on we assume that one or a combination of the mechanisms above is at work and use m h 0 = 125 ± 1 GeV as an input to fix the value of one of the free parameters of the low energy effective theory.
To compare the model predictions with the LHC and Tevatron measurements, we need the coefficients of the SM Higgs linear couplings introduced in Eq.
(2) to be expressed in terms of the bNMWT parameters: where s α , c α , t α are shorthands for sin α, cos α, tan α, respectively, with α, β defined by the rotation matrices in Eq. (27) and ρ by We are now ready to test the particle spectrum and its couplings against the latest experimental data. First, we scan the parameter space looking for data points that produce the right SM mass spectrum and satisfy the direct searches for charged particles at LEP [54] and a heavy neutral scalar at LHC [72] as well as the EW precision constraints from the S and T parameters [57][58][59]. More specifically, we impose the constraints The quantity r(S, T ) is the correlation coefficient for the S and T parameters [59]. The constraint on the trigonometric functions is to ensure that the heavy Higgs does not couple to SM fermions more strongly than a SM Higgs with the same mass does; this allows us to use straightforwardly the LHC constraint on m H 0 . The upper bounds on m A 0 , m H 0 are enforced by the cutoff of O(Λ T C ≈ 4πv w ) of the effective Lagrangian. We also require the free parameters to produce the remaining SM mass spectrum and satisfy the bounds in Eq. (24). Then, we scan for such viable points in the domain with m 2 H , m 2 M determined by the extremum conditions ∂V ∂h 0 = 0 , where V is the scalar potential of the effective Lagrangian in Eqs. (18,20,21). The results that we present in the following of this section can be applied directly to the Type-I 2HDM by using the formulas in Appendix A. The disclaimer is that we are testing only a portion of the parameter space available to such a model, and more precisely the range of values typical for underlying strong dynamics.
The distribution of 5000 viable data points of the bNMWT allowed parameter space in the (S, T ) plane is shown in Fig. 1. The 90% Confidence Level (CL) allowed region is shaded in green while the viable data points featuring m 2 H > 0 (m 2 H < 0) are plotted in black (grey). We make this distinction because for positive m 2 H the SM Higgs sector alone would not break EW symmetry, and therefore EW symmetry breaking is generated through bosonic TC interactions. The black dots are, thus, relevant for bNMWT while the grey ones refer more generically to the Type-I 2HDM. It is clear that the viable region in S and T accessible by bNMWT is very limited.
In Fig. 2 we plot the viable data points in the (a S , a f ) and (a S , a V ) planes, respectively, together with the 68% (green), 90% (blue), and 95% (yellow) CL regions obtained in the previous section: these plots represent a slice of the a f , a V , and a S parameter space passing through the optimal point (blue star) given in Eq. (13). There is a perfectly specular viable region, which we do not show here, intersecting another χ 2 global minimum point, obtained by flipping the signs of a V , a f , and a S in Eq. (13). In Fig. 2, left panel, the upper viable region, containing the best fit point, obtains the observed slight enhancement of the Higgs diphoton decay rate exclusively by the charged scalar contribution which interferes constructively with the W boson contribution. This results from a rather large linear coupling of h 0 to S ± . In the lower viable region, the slight enhancement of the Higgs diphoton decay rate is entirely due to the SM fermions which couple to h 0 with the same sign as W ± and, therefore, give constructive interference, while the scalar interferes destructively to balance an otherwise excessive enhancement of the decay rate to two photons. The same comments apply to the viable regions presented in Fig. 2, right panel.
The bNMWT data points are closely clustered around the SM values. This was somewhat expected, because of the small mixing of the two neutral Higgs fields for a heavy H mass.
The choice of a heavy masses for the new states is naively dictated by strong dynamics, which in the scaled up QCD case would predict masses of O(Λ T C ) ≈ TeV.
Finally, in Fig. 3 we show the bNMWT data points in the (a V , a f ) plane that passes through the SM point (a S = a f = a V = 0) to which they approximately belong. In bNMWT the Higgs-vector boson coupling is always reduced compared to its SM value, as shown in We note that Fig. 1 also includes bNMWT points with flipped sign of a f , a V , and a S .
These points belong to the plane passing through the specular optimal point, and therefore produces the following statistical results: These numbers should be compared to the corresponding results in the SM, Eq. (16), which indeed produces a better fit. Even without including S and T in the fit, we obtain the same optimal data point as the one in Eq. (33), with goodness of fit given by: After adding the S and T parameters as well to the fit, we obtain a V = 1.00 , a f = 0.99 , a S = −0.4 , S = 0.01 , T = 0 ; Indeed, the data points featuring m 2 H < 0 produce a better fit than that of the bNMWT with EW symmetry breaking determined by strong dynamics, Eq. (34).
However, the results obtained so far for bNMWT do not take into account the contributions to Higgs physics coming from heavy charged vector bosons, which are a staple of strong dynamics. In the next section we study this subject by introducing some simple interaction terms in the Lagrangian and by working out the corresponding Higgs physics phenomenology. To introduce, in the effective Lagrangian, direct Higgs couplings to an extra massive vector boson which conserves gauge symmetry at the fundamental scale, we use the hidden local symmetry principle [73,74], which has been already applied to NMWT in [39,75]: here we just outline the main steps required to introduce composite vector bosons while conserving gauge invariance in the fundamental theory.
We begin by defining the following covariant derivatives where A µ is the vector boson associated with the G ≡ SU (2) × SU (2) global symmetry of L T C , which we have gauged above, and N L (N R ) is a scalar field in the fundamental of SU (2) L (U (1) Y ) and antifundamental of G. From the equations above, we can define a new vector field and its transformation under the full gauged symmetry by where u L is a unitary transformation operator of SU (2). The definition and transformation law of P R are obtained simply by replacing L with R in Eq. (39). Among the possible dimension four, gauge invariant P µ coupling terms to M , we retain only the following where M is the matrix representation of the EW doublet M . Assigning non-zero vevs for N L and N R , their kinetic terms generate a squared mass term for two vector boson combinations: The resulting massless eigenstates give the ordinary W µ and B µ vector bosons, which instead acquire mass through EW symmetry breaking. In addition, there are two vector boson triplets, one vectorial (V µ ) and the other axial (A µ ). Since their interaction terms, given by Eq. (40) evaluated at the vev defined in Eq. (41), respect custodial symmetry and give the same contribution to the axial-axial and vector-vector EW vector boson polarization functions [76], the total contribution of the vector bosons to the EW precision parameters is identical to the SM one, and S and T are, therefore, zero [39,75]. Moreover the vector boson contributions to FCNC have been tested against the experimental results and shown to be phenomenologically viable [77]. To simplify our analysis we fix so that onlyW µ and the vector resonance, V µ , couple to the neutral Higgs fields.
The charged vector boson mass matrix in the (W , V, A) basis can be written in a compact form as  with and We now study the implications of this setup in light of the LHC and Tevatron data fit we have at our disposal.

A. Mixing of Vector Fields
Let us begin with a rather general and simple case. We require that onlyW , the elementary gauge field, couples to h 0 , and therefore the W coupling to the light Higgs is generated only through terms mixingW with the composite vector fields V and A. The squared mass matrix in the gauge basis (W , V, A) is obtained simply by setting s = 0 in Eqs. (43,44): We define the rotation to the mass eigenbasis in terms of x and , Eqs. (45), by with corresponding eigenvalues The mixing matrix in Eq. (47) shows that only W and W contribute to the gauge field W .
We checked that the Fermi coupling, G F , determined by evaluating the amplitude for the muon decay (µ − → ν µνe e − ), respects the usual relation The vector coupling coefficient a V is suppressed, compared to the result in Eq. (28), because of mixing: with The W coupling to h 0 is zero instead because we set r 2 = −r 1 in Eq. (40). Finally, the fermion and scalar coupling coefficients in Eq. (28) remain unchanged.
The lower limit on the mass of a sequential 4 W from direct searches at ATLAS [53], equal to 2.55 TeV at 95%CL, can be readily applied to the case above by properly rescaling the lower limit: In this limit the effect of mixing on the W and W couplings is negligible. Moreover, because the sum of a V and a V is independent of , so is the Higgs decay rate to two photons.
In the next subsection we study the phenomenologically more appealing scenario in which the composite vector fields feature a direct coupling to neutral Higgs fields.

B. Direct Higgs Coupling to W and W
Next we want to study the effects of a direct Higgs coupling to the composite vector field V , and consider s = 0. In this case, the charged vector boson mass matrix in the (W , V, A) basis is given by Eq. (43). The mass eigenvalues are lengthy cubic roots. These can be expanded in x and , which in TC are both expected to be small: where contributions of O(x n 5−n ) are neglected, with n = 0, . . . , 5. TheW and V coupling terms to the light Higgs can be derived from the mass matrix by taking its derivative with respect to v w and introducing a factor ζ to take into account the rotation of M to the mass eigenbasis: The vector boson coupling coefficients get an enhancement factor because of the s coupling: where at O(x n 5−n ), with n = 0, . . . , 5. We collected together the W and W contributions to the Higgs decay to diphoton by summing up their respective coupling coefficients in Eq. (56).
It is interesting to notice that, at all orders in x: The fermion and scalar coefficients are still determined by Eqs. (28). We obtain the optimal value of s by performing the global fit in the limit of negligible vector mixing ( = 0) and decoupled neutral heavy Higgs: a f = a V = 1 , a S = 0 , a V = 2s 2 1 + 2s 2 , ⇒ s = 0.32 +0. 17 −0.32 .
We use the same set of 5000 viable points scanned over the bNMWT parameter space with no W and W , and re-calculate the coupling coefficients a V and a V at each data point for random values of s and , with 0 ≤ s ≤ 1 and 0 ≤ ≤ 0.1, and m A = 1 TeV. We plot the resulting bNMWT data points together with the experimentally favored regions in the (a V , a V ) and (a V , a f ) planes in Fig. 4, and in the (a V , a f ) plane in Fig 5, respectively, all passing through the optimal data point defined in Eq. (13). The plots are again limited to the positive a V half-plane. For m 2 H < 0 the mixing factor ζ, Eq. (55), can be negative, which makes a V flip sign, compared to a V , because of Eqs. (57). Already by visual inspection, it is clear that the s coupling allows bNMWT to cover a large portion of the 68% CL favored region. We verified that all the three parameters a f , a V , and (to a lesser degree) a V are free (meaning that they are little correlated and with a range of values comparable to the error affecting the optimal values in Eq. (13)), which was expected since we introduced two new The cases above are equally favored by the experiment and both feature a probability greater than the one for the general fit, Eq. (13), which does not include the S and T EW parameters (and therefore has two less d.o.f.) but gives a similar value of χ 2 min . As a last remark, we note that a V is rather unconstrained by the chosen set of observables, and so a broader set of observables related to W physics would be necessary to further test the viability of bNMWT.
In this paper we have considered quantitatively how much the coefficients of Higgs couplings to electroweak gauge bosons (a V ) and fermions (a f ) as well as to possible extra scalars (a S ) can deviate from their corresponding values in the Standard Model (a V = a f = 1, a S = 0) in light of the current LHC and Tevatron data. We then considered a bosonic technicolor model, bNMWT [40], and studied its viability by performing a scan on the parameter space which implements the direct constraints on the mass spectrum as well as the constraints from precision EW data (in terms of the S and T parameters). The scalar sector of the bosonic technicolor model we considered can be more generally viewed as a type-I 2HDM.
The essential consequence of underlying strong dynamics is the existence of new vector resonances in the particle spectrum. We implemented these new states in an effective Lagrangian to study their effects. We considered in detail the implications of the effective Lagrangian on the couplings of the Higgs boson to the physical W and Z bosons as well as to fermions. Then, we studied first the simple case of minimal coupling to the SM fields, which amounts to considering only the mixing of the two new triplets of vector bosons and the SU(2) L gauge fields without direct interaction between the composite vector bosons and SM fields. In this simple scenario we determined the 95%CL upper bound on the amount of mixing allowed by experimental data. We showed that this generic scenario in bNMWT cannot be resolved by the current data.
Finally we illustrated the possible effects of the direct coupling between composite vector bosons and neutral scalar fields within our effective Lagrangian scheme. We showed that the direct coupling allows for an optimal fit by the bNMWT predictions of the current experimental data. A more refined analysis of the model including additional observables can further test the possibility of a strongly coupled sector underlying the electroweak sector of the SM. boson, while h 0 , H 0 , A 0 , and H ± are the neutral scalars, pseudoscalar, and charged scalar mass eigenstates, respectively: The angles α , β differ from α, β only because of the extra rotation in Eq.(A2) that makes only H 2 couple to SM fermions.