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

We compute the top quark right 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 of the order of 2 × 10−4. We also present detailed results of the one loop analytical computation.


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The discovery of the top quark was done at TEVATRON [9,10] where CDF an D0 collaborations obtained a first bound on the tbW anomalous couplings [11][12][13] in agreement with the SM predictions. Nowadays, top quark physics is being extensively studied at LHC [14][15][16][17][18] where the ATLAS and CMS collaborations are looking for any signal of new physics [19][20][21][22] in their data on top decays. The study of the tbW vertex structure has been usually done by considering observables built on the measurement of the different helicity components (± and 0) of the W in the decay of the top quark. In recent works [23][24][25][26][27][28] 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 also transverse helicities of the W coming from the top decay. 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 studied [29] as a window for new physics, in the 2HDM-framework with a CP -violating potential. The enormous amount of collected data by the LHC (and in the future by the HL-LHC) will hopefully determine the complete structure of the tbW vertex, with a precise determination of the properties of top quark anomalous couplings to the W boson and b quark.
As already said in the previous paragraph and as it is also shown in detail in refs. [8,30,31], at LHC, the observables considered in the literature to bound the anomalous couplings of the tbW vertex are based on the measurement of the helicity components of the W boson product of the top decay, for pp → tt → W + bW −b events with one (or both) of the W bosons decaying leptonically. Those observables are the F ± and F 0 fractions of W bosons produced with helicity ± and 0 [32], respectively, angular asymmetries in the W rest frame [23,30,31,33,34], angular asymmetries in the top rest frame [23,30,[35][36][37][38] and spin correlations [23,30,39,40]. These observables are not, in general, very sensitive to the right coupling V R . This is due to the fact that the V R coupling has the same parity and chirality properties as the leading coupling V L , so that the observables receive contributions from both terms on an equal footing. In ref. [8] some of these observables are redefined in such a way as to cancel the leading V L contribution to them. Then, they are directly sensitive to V R and can be an important tool in order to search for new physics contributions to this coupling. Based on 1.04 fb 1 of pp data at s = √ 7 TeV, with a combined analysis of the W helicity fractions F 0 and F L , for single-lepton and dilepton events, and angular asymmetries, A + and A − , for single-lepton channels [41], the direct bound on the W tb right coupling obtained is [3] − 0.20 < V R < 0.23.
On the other hand, the current 95% C.L. bound on V R from the precise measurement of the radiative B-meson decayB → X s γ, is [42] − 0.0007 < V R < 0.0025, (1.2) so that a combined analysis of collider data with these precise measurement of B-meson decay may be useful to bound the V R top coupling. A simple and widely studied extension of the electroweak theory is to consider a second scalar doublet added to the SM [43,44]. However, tree level flavour changing neutral JHEP03(2017)128 currents (FCNC) arise unless new hypothesis are introduced. A recent proposal in order to address this issue is the aligned two Higgs doublet model A2HDM [45], 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 and masses of the new particles so as to cover the meaningful parameter space. 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 [43,44]. Similarly to what happens in the SM, after symmetry breaking, the neutral components of the two doublets get non zero vacuum expectation values 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 JHEP03(2017)128 {S 1 (x), S 2 (x)} will not mix with the CP-odd one (S 3 (x)) so that: where γ is the neutral scalar 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: , u L (x) and l R (x), are three-dimensional vectors in the flavour space, M f (f = d, u, l) 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. For different values of the ξ f parameter (see [45]) 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.

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The neutral Yukawa terms are flavour-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: In the particular case of a CP-conserving potential they can be written as: (2.8) 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 [3]. 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 associated phenomenology. In ref. [46] 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, |ς u ς * l |/m 2 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 newphysics corrections to the observables are those generated by the charged scalar; then |ς u | < 1.91 for m H ± = 500 GeV [46,47].
Bounds on ς d are more difficult to get from phenomenology so an upper bound as big as |ς d | ≤ 50 can be used [47]. 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 [46,47]. More constraints on the ς u -ς d plane can also be set fromB decays and are given in refs. [46,47] Recently, direct searches of light charged scalar Higgs in t → H + b decay in ATLAS and CMS [48] give an upper bound [49] 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:

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where the outgoing W + momentum, mass and polarization vector are q = p − p , m W and µ , respectively. The couplings are all dimensionless; V L and V R parameterize 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 at low energy of 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 [32,50,51].
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 matrix element V L = V tb 1 [3]. The values of the anomalous tensor couplings at one loop have been calculated in ref. [4] for the SM, and in ref. [5] for a general A2HDM.
The SM contribution to V R has been calculated in ref. [8]. 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 central values of the standard masses and couplings given in ref. [3], they are: We would like to call the attention of the reader to the fact that, as can be seen from ref. [8], the EW contribution to the real part of the V R coupling, from most of the EW diagrams considered there, is of the order of 10 −5 but, due to accidental cancellations among them, the final result is two orders of magnitude smaller. We also found that these cancellations are very sensitive to values of the SM parameters used in the calculation in such a way that the final result for Re [V R (EW)] is strongly dependent on small changes (within experimental errors given in ref. [3]) on the SM masses and couplings. Thus, for example, we see that the value for the real part of the coupling moves from +0.091×10 −5 to −0.115×10 −5 , when m t runs over its PDG-allowed range (m t = (173.21±0.87) GeV), showing that it remains (in that sense) undetermined. Only a global bound, |Re [V R (EW)] | 10 −6 , can be set when considering the uncertainties of the present SM data. The value shown in eq. (3.2) is the one we obtain using the central values (given by PDG. [3]) for the SM parameters and it is given just as a reference value for the SM-EW prediction in this work (SM-EW central value). The imaginary part instead, remains stable, of the order of 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.
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) JHEP03(2017)128 Figure 1. One-loop contributions to the V R coupling in the t → bW + vertex.
Type Particles in the loop ABC 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 parameterized 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. 1 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 rightcoupling in the A2HDM -diagrams type (3) to (7) -are identical to the corresponding ones in the SM obtained in ref. [8].

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 V R . 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 [3,52,53]. 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 [54]. 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 [55] 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 [56,57] and are not considered here.
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

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prediction. In all scenarios the value of the heaviest (scalar or pseudoscalar particle) mass, 866 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 (V R ) = V Re R . For the 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 [46,47].
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 central prediction, given by eq. (3.2), for scenarios II and IIi, while it can be only 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 crucially depend on the particular choice of θ d . 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 central 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 central 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 crucially changing 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 central 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 central value prediction, for scenarios II and IIi.
In figure 5 we present the dependence of V R on the coupling parameter ρ u . The plots show that V Re R is three (two) orders of magnitude larger than the SM-EW central value, for scenarios II and IIi (I and Ii). Besides, for large values of the ρ u parameter, V Re R grows JHEP03(2017)128 , as a function of the γ scalar mixing angle, for different θ u values and ρ u,d = 1, θ d = π/4.

Model
ς d ς u Type-I cot β cot β Type-II − tan β cot β Table 3. Values for ς u,d that reproduce the Type-I and Type-II 2HDM.
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 central prediction for large values of ρ u .
Finally, we compute V R for Type-I [58,59] and Type-II [59,60] 2HDM. 2 In table 3 we show the ς u,d values that reproduce the Type-I and Type-II models.
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 central one, for 1 < tan β < 4 and for all mass scenarios; for tan β 4 it rapidly decreases. Note that for Type-I 2HDM the Yukawa JHEP03(2017)128  ) and Q Im V as function of the θ u parameter, with γ = π/4, ρ d = 1 and θ d = π/4. 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 for the parameters of the model, the magnitude of V Re R can be up to 10 −3 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 a few times larger. For Type-II (Type-I) 2HDM, V Re R can grow up to 10 −4 (10 −5 ), 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.
For the HL-LHC with a high top statistic, the sensitivity to the V R coupling will be improved by at least two orders of magnitude. In addition, with the new generation of High Luminosity Super B Factories [62], the current bound on V R from radiative B-meson decay is expected to be improved at least in a few percent. Then the A2HDM prediction for V R is expected to be very close to be observable.

Acknowledgments
This work has been supported, in part, by the Ministerio de Economía y Competitividad A A2HDM contribution to V R Following the notation of ref. [8], we define y H u = cos γ + sin γ ς * u , y h u = − sin γ + cos γ ς * u , y A u = −i ς * u . (A.7) Then, we have the following expressions for the new contributions, listed in table 1: -Type (1) diagrams. (A.9) -Type (2) diagrams.
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