The right top coupling in the aligned two-Higgs-doublet model

We compute the right top quark coupling in the aligned two-Higgs-doublet model. In the Standard Model the real part of this coupling is dominated by QCD-gluon-exchange diagram, but the imaginary part, instead, is purely electroweak at one loop. Within this model we show that values for the imaginary part of the coupling up to one order of magnitude larger than the electroweak prediction can be obtained. For the real part of the electroweak contribution we find that it can be up to three orders of magnitude larger than the standard model one. We also present detailed results of the one loop analytical computation.


Introduction
In 2015 the LHC center-of-mass energy has reached 13 TeV. By the end of 2016 the LHC will be close to a peak luminosity of 1.5 × 10 34 cm −1 s −1 , with an integrated luminosity of 38f b −1 . After 2020, several components of the accelerator will reach the radiation damage or reliability limit so that, by 2024 the LHC will have to be upgraded to the High-Luminosity LHC (HL-LHC), which is expected to accumulate over the next 10 years an impressive integrated luminosity of 3000 f b −1 at energies close to [13][14] TeV [1,2]. The CMS and Atlas experiments have already collected millions of top quark pairs and single top events but in this scenario of very high luminosity, they will detect billions of them in the future. Besides, next generation of colliders, such as CLIC, will eventually be built and it is expected that the top quarks physics will enter in an era of high precision. The top quark is the only quark that decays weakly before hadronization and, up to now, only one decay mode, t → bW + , is known. It was detected for the first time at TEVATRON [3,4] where many of its physical properties were first measured and also some limits on the anomalous tbW couplings were set [5][6][7].
Top quark physics is considered as one of the gateways to new physics [8][9][10] and the study of its decay properties at the LHC [11][12][13][14] is being extensively investigated by the ATLAS and CMS collaborations [15,16]. The determination of other couplings of the top quark, such as the chromoelectric and chromomagnetic of the ttg (top-top-gluon) vertex has been recently suggested [17] as a window for new physics, in the two-Higgsdoublet model (2HDM) framework with a CP -violating potential. The study of the different helicity components of the W in the top decay has been also proposed to investigate the tbW Lorentz vertex structure [18]. In recent works [19][20][21][22][23][24] it has been shown that a precise determination of the Lorentz form factors of the tbW vertex can be done with a suitable choice of observables built from longitudinal and transverse helicities of the W coming from the top decay.
The enormous amount of collected data by the LHC (and in the future by the HL-LHC) will determine the complete structure of the tW b vertex, with a precise determination of the properties of top quark couplings to the W boson and to the b quark.
The most general parametrization of the on-shell vertex needs four couplings. In the Standard Model (SM) the left coupling V L is not zero and takes a value close to one [25]. The other three are zero at tree level: the chiral V R coupling, and the left g L and right g R anomalous tensorial couplings. This is not the case in extended models where, in addition, some of these couplings can also be sensitive to new CPviolation mechanisms. The measurement of the two tensorial couplings g L,R at the LHC was investigated in ref. [26]. The values of g L,R within the SM, the 2HDM and other extended models where recently calculated in refs. [27][28][29] and they will not be considered in this paper. The right top coupling V R was computed in the SM at leading order in ref. [30].
The LHC observables considered in the literature are not, in general, very sensitive to the right coupling V R . This is due to the fact that in the lagrangian the V R coupling has the same parity and chirality properties than the leading coupling V L , so that the observables receive contributions from both terms. Some of these observables are the angular asymmetries in the W rest frame [18,19,31,32], angular asymmetries in the top rest frame [19,[32][33][34] and spin correlations [19,32,35,36]. In ref. [30] some of these observables were redefined in order to be directly proportional to the coupling we are interested in, V R , in such a way as to cancel the leading V L contribution to them. Then, these observables are directly sensitive to V R and can be an important tool in order to search for new physics contributions to this coupling.
A simple and widely studied extension of the electroweak theory is to consider a second scalar doublet added to the SM. However, tree level flavour changing neutral currents (FCNC) arise unless new hypothesis are introduced. A solution to this issue is the aligned two Higgs doublet model A2HDM [37], where the two Yukawa matrices coupled to the same type of right-handed fermion are aligned in flavour space. Then, no FCNCs appear at tree level. Besides, most of the popular versions of the 2HDM are reproduced with particular choices of the A2HDM parameters. In this paper we present a detailed calculation of the new contributions to the V R top right coupling in the general framework of the A2HDM.
This work is organized as follows. In the next section we briefly review the A2HDM, introducing the notation used in the paper and presenting the current limits that constraint the parameters of the model. In section 3 we define the vertex parametrization and show the details of the computation of the different contributions to the right vector coupling V R within the A2HDM. In section 4 we investigate the sensitivity of the V R coupling to the scalar mixing angle and alignments parameters, for a CP-conserving scalar potential. We show the results obtained for values of the parameters of the model and masses of the new particles so as to cover the meaningful parameter space of the model. The results for 2HDM Type-I and II are also shown. We present our conclusions in section 5.

The aligned two-Higgs-doublet model
The 2HDM extends the SM by adding a second scalar doublet φ 2 (x) with the same hypercharge Y = 1/2 [38,39]. Similarly to what happens in the SM, after symmetry breaking, the neutral components of the two doublets get non zero vacuum expectation The so called Higgs basis (Φ 1 (x), Φ 2 (x) is obtained through a rotation of the φ 1 (x), φ 2 (x) states given by the angle β (defined as tan β = v 2 v 1 ), in such a way that only one of the doublets (Φ 1 (x)) gets a non-zero expectation value v = v 2 1 + v 2 2 . In this basis, the three components of the doublets can be written as where G 0 (x) and G ± (x) correspond to the three would-be Goldstone bosons of the SM, H ± (x) are two new charged scalar fields and {S i (x)} i=1,2,3 are three neutral scalars with no defined mass. To get the three mass eigenstates as a linear combination of the later three scalars one has to perform an orthogonal transformation R so that the new three mass eigenstates, can be written as The particular form of the potential will define the matrix R and the structure of the scalar mass matrix and mass eigenstates. If the potential is CP-conserving, the CP-even states {S 1 (x), S 2 (x)} will not mix with the CP-odd one (S 3 (x)) so that: where γ is the neutral scalars mixing angle. The most general Yukawa Lagrangian, with standard fermionic content will have different couplings to Φ 1 (x) and Φ 2 (x) doublets. It means that when one diagonalizes the fermionic mass matrices -in the Higgs basis-this transformation will no diagonalize the fermion-scalar Yukawa matrices. The Yukawa lagrangian can then be written as are the non-diagonal 3×3 fermion mass matrices, and Y f are the fermion-scalar Yukawa couplings that are, in general, also non-diagonal. The rotation to the fermionic mass eigenstates (d(x), u(x), l(x), ν(x)) which diagonalizes the mass matrices M f will, in general, not diagonalize simultaneously the Yukawa matrices Y f , so that they will introduce FCNC at tree level. Among the different approaches to avoid this unwanted effect we choose the one that, before diagonalization, makes both Yukawa matrices -M f and Y f , for each type of right handed fermions-proportional to each other (alignment in the flavour space). Then, they can be simultaneously diagonalized and the diagonal Yukawa matrices satisfy the relations: with ξ f being an arbitrary complex number and M f (f ≡ u, d, l) diagonal mass matrices. This is the so called A2HDM. It has the advantage that for different values of the ξ f parameter (see [37]) it reproduces the 2HDM with discrete Z 2 symmetries, Type-I, II, X, Y and inert model. Obviously if the ς f are taken to be arbitrary complex numbers the Lagrangian incorporate new sources of CP-violation. The Yukawa lagrangian can be then written as: where V is the Cabibbo-Kobayashi-Maskawa matrix and P R,L ≡ 1 2 (1 ± γ 5 ) are the chirality projectors.
The neutral Yukawa terms are flavor-diagonal and the couplings y ϕ i f (ϕ i = h, H, A) are proportional to the corresponding elements of the neutral scalar mixing matrix R: that, in the particular case of a CP-conserving potential can be written as: Then, the CP-conserving A2HDM contains 10 real parameters: the three complex alignment constants ς u,d,l , the three scalar masses m A,H,H ± , and the scalar mixing angle γ. We will assume that the light CP-even Higgs h is the SM-like Higgs with a mass of 125.09 GeV [25]. The other parameters have not yet been measured and they can be constrainted by indirect phenomenological and theoretical arguments.
The presence of a charged Higgs is a signature of the model that allows some constraints coming from the phenomenolgy associated. In ref. [40] combined bounds on ς u,d,l and m H ± are obtained from: a) tau decays, |ς l |/m H ± ≤ 0.40 GeV −1 , and b) a global fit to the tree leptonic and semi-leptonic decays of pseudoscalar mesons, Bounds can be improved by looking at loop-induced processes, Z → bb, B 0 -B 0 and K 0 -K 0 mixing, andB → X s γ, assuming that the dominant new-physics corrections to the observables are those generated by the charged scalar; then |ς u | < 1.91 for m H ± = 500 GeV [40,41].
Bounds on ς d are more difficult to get from phenomenology so an upper bound as big as |ς d | ≤ 50 can be used [41]. Studies of the radiative decaysB → X s,d γ, show that the combination |ς * u ς d | is strongly correlated with the mass of the scalar charged boson m H ± , thus one find that |ς * u ς d | ≤ 25 for m H ± ∈ (100, 500) GeV [40,41]. More constraints on the ς u -ς d plane can also be set fromB decays and are given in ref. [40,41] Recently, direct searches of light charged scalar Higgs in t → H + b decay in ATLAS and CMS [42] give an upper bound [43] on the combination |ς * u ς d | that excludes part of the allowed regions constrained byB decays.
All these limits put constraints on the parameter space of the model. In this paper we only consider the ones that are related to the top physics.

V R top coupling in the A2HDM
The most general Lorentz structure of the amplitude M tbW , for on-shell particles, in the t(p) → b(p ′ )W + (q) decay is: where the outgoing W + momentum, mass and polarization vector are q = p − p ′ , m W and ǫ µ , respectively. The couplings are all dimensionless; V L and V R parametrize the left and right vector couplings while g L and g R are the so called left and right anomalous tensor couplings, respectively.
In an effective Lagrangian approach these couplings arise as contributions of low energy non-renormalizable lagrangian terms, originated in a high energy theory. This approach assumes that the new physics spectrum is well above the electroweak (EW) energy scale [44][45][46].
The couplings V R , g R and g L are zero at tree level within the SM, and V L is given by the Kobayashi-Maskawa matix element V L = V tb ≃ 1 [47]. The values of the anomalous tensor couplings at one loop have been calculated in ref. [27] for the SM, and in ref. [28] for a general A2HDM.
The SM contribution to V R has been calculated in ref. [30]. There, the QCD one loop gluon exchange and the one loop contribution from the EW sector of the SM have been explicitly calculated. For the values of the standard masses and couplings given in [47], they are: Note that, as can be seen from ref. [30], the EW contribution to the real part of the V R coupling from most of the EW diagrams is of the order of 10 −5 but, due to accidental cancellations among them, the final result is two orders of magnitude smaller. In fact this real part, within the precision of our calculation and considering the uncertainties of the data used, is compatible with zero, at 10 −7 precision. 1 The imaginary part, instead, remains of order 10 −5 , and it is purely EW.
In the 2HDM, the couplings structure of the tbW remains unchanged at tree level. However, at one loop, in addition to the usual particle contents of the SM, the three new neutral scalars h, H and A, and the new charged scalars H ± of the 2HDM may circulate in the internal lines of the loop and new contributions to the V R coupling arise. The structure of the one loop diagrams contributing to the V R top right-coupling is given in figure 1. Figure 1: One-loop contributions to the V R coupling in the t → bW + vertex. 1 Notice that the result quoted here differs (even in sign) from the one of ref. [30]. As explained, this is so for two reasons: 1) the set of PDG values used here for the SM parameters is different and, 2) the accidental cancellation among diagrams makes the result very sensitive to these values, and consequently, the final result is not well determined and strongly depends on small changes on the SM masses and couplings within the experimental errors given in [47].
We denote each diagram by the label ABC according with the particles running in the loop. In table 1 we shown the 17 new diagrams to be considered, ordered by the position (A, B, or C) of the neutral scalars ϕ i , where ϕ i stands for one of the neutrals h, H and A in the diagram types from (1) to (3), while for diagrams types (4) to (7), ϕ i runs only for the neutral scalar bosons h and H. It is important to notice that diagrams type (5) and (7) always have an imaginary part while, depending on the mass of the new scalar charged Higgs (H ± ), diagrams type (2) may or may not develop it. Chirality imposes that all the contributions are proportional to the bottom mass and can be written as:

Type Particles in the loop
where r b = m b /m t and I ABC is the Feynman integral corresponding to the given diagram. In appendix A we give the analytical expressions of all these integrals, for the diagrams shown in table 1.
The V R coupling depends on the scalar mixing angle γ and on the alignment parameters ς u and ς d . The mass dependence is parametrized by the dimensionless variable r X = m X /m t , where m X is the mass of the particle X circulating in the loop. For the neutral scalar masses above the TeV scale, the Feynman integrals give negligible values when compared to the SM contributions. However, the V R coupling is very sensitive to the new particles masses when they take lower values.
As in the SM, some of the diagrams are ultraviolet divergent, but we know that the total result must be finite. In appendix A it can be seen that the sum of diagrams (3), (6) and (7), to the SM diagrams G 0 tb, tG 0 G − and bG + G 0 , respectively, cancel all the ultraviolet divergences and the total result is finite. This fact has been also used as a test of our analytical calculation 2 .
We recover the SM expressions from the A2HDM just by taking the ς u,d → 0 limit and setting γ = −π/2, in such a way that the neutral scalar h has the same couplings as the SM Higgs boson. In that limit we explicitly checked that the contributions to the top right-coupling in the A2HDM -diagrams type (3) to (7)-are identical to the corresponding ones in the SM obtained in ref. [30].

Results
In this section we present the one loop corrections to the top right-coupling V R in the A2HDM. As already stated, these corrections depend on the alignment parameters ς u,d , the scalar mixing angle γ and on the masses of the new particles: 2 neutrals scalars h and H, one axial A, and two charged scalars H ± . We write the alignment parameters as: and we investigate separately the effects of modulus and phases on the top V R rightcoupling. In addition to the masses of the new particles we have five free parameters: ρ u , ρ d , θ u , θ d and the mixing angle γ.
We chose different sets of values for the masses of the new neutral and charged scalar particles; the scenarios we consider are shown in table 2. The new scalar masses are taken to be of the order of 10 2 GeV [48,49]. In the framework of 2HDM and under certain assumptions on its dominant decays, the charged scalar mass, m H + , is excluded to be below 85 GeV by LEP data [50]. Then, it can take values below the top quark mass, so that the decay t → bH + is kinematically possible and therefore, type (2) diagrams may develop an absorptive part. These scenarios are called (i) in our paper and we fix for them the mass of the charged scalar, m H + , to be 150 GeV. For the other cases, where m H + > m t , we take m H + = 320 GeV, as shown in table 2. In addition, for a CP conserving scalar potential [51] we have to impose that m h ≤ m H . We define four different mass scenarios: two with three light neutral scalars (I and Ii) and two with h as the only light scalar (II and IIi). The other possible two, with the CP-odd scalar A being the lightest one, are disfavored by present LHC data [52,53] and are not considered here.

Scalar mass scenarios (in GeV)
Type  The set of scenarios given in table 2 allows us to investigate the whole meaningful parameter space and to determine the regions where V R strongly differs from the SM-EW prediction. In all scenarios the value of the heaviest (scalar or pseudoscalar particle) mass, 866.05 GeV, is fixed by setting r heaviest = (m heaviest )/m t = 5.
For our numerical analysis we define Q Im V as the ratio of the imaginary part of the V R coupling in the A2HDM to the SM-EW: Regarding the analysis of the V R real part, due to the uncertainty already commented in the SM-EW, we present the results for the A2HDM in terms of Re For the four different mass scenarios defined in table 2, we study the V R dependence on the four alignment parameters ρ d,u , θ u,d , and on the scalar mixing angle γ. We show the results for conservative values of the modulus, i.e. for ρ u,d ∼ 1. Larger values of these modulus will certainly produce large deviations from the SM predictions but these values are disfavoured with present data [40,41]. In figure 2 we show the dependence of V Re R on the γ mixing angle, for different values of the θ u parameter, with ρ u,d = 1 and fixing θ d = π/4. V Re R in the A2HDM can be three orders of magnitude bigger than the SM-EW prediction for scenarios II and IIi, while it can be one order of magnitude larger for scenarios I and Ii. The behaviour with the γ parameter always exhibits the usual oscillating dependence. We checked that these results do not depend crucially on the particular θ d value chosen. Similar values -with a slight shift of the central values of the V R coupling-are found when fixing θ u = π/4 and varying θ d . In figure 3 we show the behaviour of Q Im V for the same set of parameters as given in figure 2. For ρ u,d = 1, θ d = π/4, and θ u given in the plots, it can be up to three times larger than the SM-EW value, as can be seen in the third plot of figure 3 (scenarios II and IIi). For scenarios I and Ii, the deviation from the SM-EW value is much smaller. The figures show the expected dependence of the observable with γ as a combination of sin γ and cos γ. As in the real part of the coupling, the plots for Im(V R ) present similar behaviour as the one shown in figure 3, with a small shift of their central values, when interchanging θ u ↔ θ d .
The V R coupling is more sensitive to the values of ρ u than to those of ρ d . The last one may move over a wide range of values (1 < ρ d < 10) without changing crucially the results. In the following we fix the values of the ρ d = 1 and θ d = π/4 as a representative choice of these parameters and we study the dependence of V R with the rest of the parameters of the model.  In figure 4 we show V R (real and imaginary parts) as functions of the θ u angle, for the scalar mixing angle γ = π/4. As seen there, the real part can be three (two) orders of magnitude bigger that the SM-EW one for scenarios II and IIi (I and Ii), while the imaginary part can take values up to three times larger than the SM-EW prediction, for scenarios II and IIi.
In figure 5 we present the dependence of V R with the coupling parameter ρ u . The plots show that V Re R is three (two) orders of magnitude larger than the SM-EW value, for scenarios II and IIi (I and Ii). Besides, for large values of the ρ u parameter, V Re R grows with ρ u independently of the values of the other parameters of the model, such as γ and θ u . A similar behaviour is found for the imaginary part of V R , that can be a factor seven larger than the SM-EW one for large values of ρ u .
Finally, we compute V R for Type-I [54,55] and Type-II [55,56] 2HDM 3 . In table 3 we show the ς u,d values that reproduce the Type-I and Type-II models. These models have a discrete Z 2 symmetry in order to avoid tree level FCNC.

Model
ς d ς u Type-I cot β cot β Type-II − tan β cot β  For Type-I and Type-II 2HDM, we present the results as a function of tan β, for the different mass scenarios considered. We work on the alignment limit, γ = −π/2, where the neutral scalar h has SM-like couplings to the photon and to the weak bosons. The results for the real part of V R are shown in figure 6. For Type-I model, V R takes values one order of magnitude larger than the SM-EW one, for 1 < tan β < 4 and for all mass scenarios; for tan β >> 4 it approaches the SM-EW value. Note that for Type-I 2HDM the Yukawa couplings go to zero in the large tan β limit. For Type-II model the value of V R grows with tan β, reaching values close to 10 −4 (10 −5 ) for tan β ≃ 50 in the mass scenarios I and Ii (II and IIi). We also find that, within these models, V Im R is very close to the SM-EW value and almost constant for the considered mass scenarios.

Conclusions
We computed the one-loop contribution to V R in the A2HDM. In the SM, accidental cancellation among the one loop EW parts results in values for V Re R two orders of magnitude smaller than expected from each diagram. This cancellation does not take place in the A2HDM. Then, depending on the values of the parameters of the model, the magnitude of V Re R can be three orders of magnitude larger than the SM-EW prediction (i.e. close to 10 −4 ) and close to the leading QCD contribution. V Im R can be one order of magnitude larger than the SM prediction for ρ u,d > 4 but, for ρ u,d ∼ 1, its magnitude is only a few times larger. For Type-II (Type-I) 2HDM, V Re R can grow up to two (one) orders of magnitude with respect the SM-EW value, for tan β ≈ 10 and depending of the mass scenarios considered, while the imaginary part remains basically of the same order as in the SM-EW. As it is shown in our previous work [30], new observables for the LHC and next generation colliders can provide a direct measurement of the right top coupling V R . Appendix A A2HDM contribution to V R Following the notation of ref. [30], we define

Acknowledgments
Then, we have the following expressions for the new contributions, listed in with -Type (2) diagrams. with -Type (3) diagrams.
I bG + G 0 + I bG + H + I bG + h = 1 16πs 2 w r 2 The contribution from type (4), {tϕ i W }, and type (5), {bW ϕ i }, diagrams (ϕ i = h, H) is zero as in the SM: Notice that in the limit ς u,d → 0, and fixing γ = −π/2 to identifying h with the standard Higgs, we recover the SM result [30].