Chromomagnetic and chromoelectric dipole moments of the top quark in the 4GTHDM

The chromo magnetic dipole moment (CMDM) and chromo electric dipole moment (CEDM) of the top quark are calculated at the one-loop level in the framework of the two-Higgs doublet model with four fermion generations (4GTHDM), which is still consistent with experimental data and apart from new scalar bosons ($H^0$, $A^0$, and $H^\pm$) and quarks ($b'$ and $t'$) predicts new sources of $CP$ violation via the extended $4\times 4$ CKM matrix. Analytical expressions for the CMDM and CEDM of a quark are presented both in terms of Feynman parameter integrals and Passarino-Veltman scalar functions, with the main contributions arising from loops carrying the scalar bosons accompanied by the third- and fourth-generation quarks. The current bounds on the parameter space of the 4GTHDM are discussed and a region still consistent with the LHC data on the 125 GeV Higgs boson is identified. It is found that the top quark CMDM, which is induced by all the scalar bosons, can reach values of the order of $10^{-4}$-$10^{-3}$, with the dominant contributions arising from the fourth generation quarks, though may also be large cancellations for some parameter values, thereby giving a negligible CMDM. As for the top quark CEDM, it only receives contributions from the charged scalar boson and can reach values of the order of $10^{-19}$-$10^{-18}$ ecm for relatively light $m_{H^\pm}$ and large $m_{b'}$, with the dominant contribution arising from the $b'$ quark. The latter would be the most interesting prediction of this model as can be larger than the value predicted by the usual THDMs by one or two orders of magnitude.


I. INTRODUCTION
Since its discovery in 1995 by the CDF and D0 experiments at Fermilab's Tevatron [1,2], the top quark has played a special role in the study of the phenomenology of the standard model (SM), which stems from the fact that its mass is of the order of the electroweak symmetry breaking scale. Even more, the top quark is unique as it does not hadronize unlike all other quarks, due to its tiny timelife τ t = 5 × 10 −25 s, but it also can decay semi-weakly and has a Yukawa coupling of the order of the unity. At the CERN Large Hadron Collider (LHC), it is pair produced mainly via the processes qq → tt and gg → tt. At a center-of-mass energy of √ s = 14 TeV, about 90% of the top quark production arises from gluon fusion and the remainder from qq annihilation [3]. The LHC is thus a top quark factory, which opens up a plethora of opportunities to test its properties: mass, couplings to other SM particles, CP properties, rare decays, etc. A top quark factory also provides a laboratory to search for new physics effects. Along these lines, the study of the new contributions to the chromo magnetic dipole moment (CMDM) and chromo electric dipole moment (CEDM) of the top quark is a topic worth studying as they could be at the reach of experimental measurement in the near future.
In the context of the SM, there are many unsolved problems. Among them, one of the most interesting is the baryon asymmetry of the universe. According to Sakharov's criteria [4], CP violation is a necessary requirement for this phenomena. In the SM, the complex phase of the CKM matrix [5,6] gives rise to CP violation, though still not enough to explain the baryon asymmetry, which means that new sources of CP violation beyond the SM are required. It is therefore necessary to search for evidences of any CP -violating effects. We are thus interested in looking for evidences of such effects in the ttg vertex, whose anomalous contributions can be written via the following dimension-five effective Lagrangian where G a µ is the gluon field and G a µν its strength tensor, with T a being the color generators. The anomalous couplings a t and d t are known as the CMDM and CEDM, respectively, though alternative definitions for the latter are also used in the literature [7]. The existence of a CEDM implies time-reversal violation, which is equivalent to CP violation because of the CP T theorem, so any evidence of a CEDM of the top quark would indicate a CP -violating effect. In the SM the top quark CMDM is induced at the one-loop level [8], whereas the CEDM appears up to the three-loop level [9] and is negligibly small, therefore a sizeable CEDM would hint new sources of CP violation. The most recent experimental bounds are −0.016 < a t < 0.008 and |d t | < 1.6 × 10 −18 ecm, respectively [10]. It is worth contrasting these values with the electromagnetic properties of the top quark, namely, the anomalous magnetic dipole moment (MDM) and the electric dipole moment (EDM), which have also been calculated in the literature in the framework of the SM and several of its extensions. In the SM, there are three types of contributions to the top quark MDM, namely, QED, EW, and QCD contributions, with the SM prediction being 3.5 × 10 −2 [11]. On the other hand, the top quark EDM has not been calculated yet but several authors estimate that its magnitude is about 10 −30 ecm [12].
In this work, we study the one-loop contributions to the CMDM and CEDM of the top quark in the THDM with a fourth family of fermions (4GTHDM), which was proposed by Bar-Shalom et al. in 2011 [45]. A fourth SM-like fermion family was introduced in the past in the so-called sequential standard model (SM4) [46], which is the most simple extension of the SM with additional up-type and down-type quarks denoted by t and b , respectively. The introduction of a new quark family requires a 4 × 4 CKM matrix, which can be parametrized by six real parameters and three complex phases. The latter imply new sources of CP violation as those required to solve the baryon asymmetry puzzle. Although the SM does not exclude a new fermion family, a fourth generation of SM-like fermions has been ruled out by the measurement of the invisible decay width of the Z gauge boson, which is consistent with three flavors of light neutrinos [47], though extra neutrinos with mass m ν > m Z /2 are still allowed. However, the SM4 is not consistent with the LHC data on Higgs boson production [48][49][50][51] as an extra family of quarks with SM-like couplings would increase the Higgs production [52] at a level not consistent with that experimentally observed [53]. On the other hand, the 4GTHDM can still be consistent with the 125 GeV Higgs boson discovered in 2012 [54]: the theoretical prediction for Higgs boson production at the LHC remains unchanged . This was shown by the authors of the 4GTHDM in Ref. [54,55], where they perform a fit to the parameters of the lightest scalar boson h 0 with the LHC data on the 125 GeV Higgs boson to constraint the masses of the quarks of the fourth family and other parameters of the model. In the experimental side, the current bounds on the fourth-generation quark masses obtained by the ATLAS collaboration are m t > 656 GeV [56] and m b > 480 GeV [57], whereas the CMS collaboration obtain m t ,b > 685 GeV [58]. It turns out that these constraints, obtained within the framework of the SM4, could be relaxed by the addition of an extra Higgs doublet [59].
The contributions to the MDM of a fermion were calculated in the 4GTHDM framework prior the Higgs boson discovery [60], with a post-discovery update presented in [55]. Furthermore, several decay modes of the top quark have been studied within this model [60][61][62], and the inclusion of a fourth generation of chiral fermions was studied in [63]. We present below an analysis of the contributions of the new heavy scalar bosons of the 4GTHDM to the CMDM and CEDM of the top quark, along with the implications of the presence of the quarks of a fourth family.
The rest of this work is organized as follows. In Sec. II we present a brief outline of the framework of the 4GTHDM, with particular emphasis on the Yukawa Lagrangian, from which the couplings of the new scalar bosons with the SM and fourth-generation fermions are extracted. Section III is devoted to the analytical results for the CMDM and CEDM of the top quark in terms of Feynman parameter integrals and Passarino-Veltman scalar functions. In Sec, IV we discuss the most up-to-date constraints on the parameter space of the model, and perform a numerical analysis of the behavior of the CMDM and CEDM of the top quark for the still allowed parameter values. The concluding remarks and outline are presented in Sec. V.

II. THE TWO-HIGGS DOUBLET WITH A FOURTH GENERATION OF FERMIONS
We now present a short overview of the 4GTHDM and refer the interested reader to the original References [45,54]. The Higgs sector of THDMs is composed of two complex scalar doublets Φ 1 and Φ 2 with vacuum expectation values (VEVs) υ 1 and υ 2 , respectively. We use the usual definitions υ ≡ υ 2 1 + υ 2 2 and tan β ≡ υ 2 /υ 1 . The study of this class of models is well motivated as they are simple but offer a great variety of new physics effects, such as new sources of CP violation, new neutral and charged scalar bosons, three-level scalar flavor-changing neutral currents, etc. In addition, the MSSM scalar sector and axion models require two Higgs doublets, which have also been used to conjecture that the top quark mass is so heavy due to a disparity between the two VEVs, namely, υ 1 υ 2 .
Contrary to the usual THDMs that only adds one scalar doublet to the SM, in the 4GTHDM a fourth fermion family is introduced, which can still be in accordance with the LHC data on the 125 GeV Higgs boson [54] and can lead to very interesting new physics effects such as new sources of CP violation. In this model, the Yukawa Lagrangian of the quark sector can be written as follows where q R (q = u, d) is a right-handed quark singlet, Q L is a left-handed SU (2) quark doublet, F and G are general complex 4 × 4 Yukawa matrices in flavor space, I is the 4 × 4 identity matrix and I αqβq q are diagonal matrices defined as I αqβq q = diag(0, 0, α q , β q ). The Higgs doublets can be written as is a variation of type-II THDM, therefore the Yukawa Lagrangian (2) has a Z 2 symmetry, with the fields transforming as shown in Table I, where we introduce the notation Φ 1 = Φ and Φ 2 = Φ h , with VEVs υ 1 = υ and υ 2 = υ h .
The fermions of the fourth family can get their masses via the following three scenarios [45]: In this work we only consider the case (i), which is still compatible with the LHC data on the 125 GeV Higgs boson [54,55]. In this scenario the agreement with experimental data requires that tan β 1.
As for the Higgs potential, it has the most general form. The physical fields H ± , h 0 , H 0 , A 0 are obtained after the diagonalization of the neutral and charged Higgs mass matrices: where G + and G 0 are the charged and neutral Goldstone bosons, α is the mixing angle in the CP -even neutral Higgs sector, with the usual shorthand notation c a ≡ cos a and s a ≡ sin a. It is customary to assume that h 0 is lighter than H 0 . In this model, flavor changing neutral currents (FCNCs) arise at the tree level in the scalar sector. After introducing the mass eigenstates, the Yukawa interactions can be written as [45] where φ = h 0 , H 0 , A 0 and H ± . For the neutral scalar bosons, the subscripts i and j run over up or down quarks, whereas for the charged scalar boson H + i (j) runs over up (down) quarks. The coupling constants f φ , S φ ij , and P φ ij depend on the model parameters and are shown in Table II. In general S φ ij and P φ ij are given in terms of the complex entries of the 4 × 4 CKM matrix elements U ij and the mixing matrix elements Σ u,d ij . In the scenario (i) described above, the matrices Σ u,d are given as [55] where D R and U R are the unitary rotation matrices that diagonalize the quark mass matrix. Note that Σ d and Σ u depend on the elements of the fourth row of D R and U R , respectively. Since D R,4i and U R,4i are the mixings between the quarks of the fourth-generation and the first three generations, Σ d ij and Σ u ij (i, j = 1, 2, 3) are expected to be very small. This fact becomes evident in the parametrization introduced in [64] in terms of one complex parameter where 0 is the 2 × 2 zero matrix. A similar expression for Σ u is given in terms of the complex parameter t = | sin θ tt |e iδt .
TABLE II: f φ constants along with the scalar S φ ij and pseudoscalar P φ ij couplings of the physical scalar bosons of the 4GTHDM. The subscript i and j run over up (down) quarks for neutral scalar bosons, whereas i (j) runs over up (down) quarks for the charged scalar boson. Here Iq is the weak isospin (I d = − 1 2 , Iu = 1 2 ), whereas Σ u,d ij are elements of the new complex mixing matrix Σ u,d , and Uij are elements of the 4 × 4 CKM matrix. In addition, When sin (β − α) = 1, the alignment limit, the h 0 couplings to the SM particles are identical to those of the SM Higgs boson. So, it is natural to introduce the following parameter which parametrizes the deviation of the h 0 couplings from the SM Higgs boson couplings. The latter are recovered when χ = 0. In terms of this parameter we have

III. CHROMO DIPOLE MOMENTS OF THE TOP QUARK IN THE 4GTHDM
The most relevant contributions to the CMDM and CEDM of the top quark arise from the heaviest quarks, thus we only consider the contributions from the quarks of the third and fourth families. From the general Lagrangian (5), one can deduce that the one-loop level scalar boson contributions to the CMDM and CEDM of the top quark arise through the generic Feynman diagram of Fig. 1, where Q = t, t for the neutral scalar bosons, whereas Q = b, b for the charged scalar boson. After writing out the corresponding invariant amplitude for thettg vertex, we have used both the Feynman parameter technique and the Passarino-Veltman reduction scheme to solve the loop integrals, which turn out to be free of ultraviolet divergences. We thus write the contribution of the Feynman diagram of 1 to the top quark CMDM and CEDM as follows: where for convenience we introduce the dimensionless parameters Our result is consistent as the CEDM requires a complex phase to be non-vanishing, whereas a non-zero CMDM does not require such a phase. The F (x, y) and G(x, y) functions are given in terms of Feynman parameter integrals as follows and whereas we obtain the respective expressions in terms of Passarino-Veltman scalar functions with the help of the FeynCalc routines [65] as follows with , and B 0 (a, b, c) being two-point scalar functions Passarino-Veltman written as usually. These alternative expressions are useful to cross-check the numerical results.
To obtain the total contribution of the 4GTHDM to a t , we must sum over all the scalar bosons, along with the third-and fourth-generation quarks, whereas d t only receives contribution from the charged scalar boson as discussed below. It is worth noting that since χ 1, the contribution to the top quark CMDM from the loop with the lightest neutral Higgs boson h 0 and the top quark does not deviate considerably from that of the SM Higgs boson h 0 SM , which follows straightforwardly from Eq. (9) after substituting φ → h 0 SM , r Q = 1, f φ = S φ tt = 1, and P φ tt = 0: which agrees with results previously reported in the literature [8,66] and is also in accordance with the corresponding contribution to the top quark anomalous MDM. By using m t = 173 GeV and m h 0 SM = 125 GeV, we can obtain the following numerical value As a h 0 SM q is proportional to the quark mass, the CMDMs of light quarks are more suppressed, thus the top quark offers the best opportunity to study this property.

IV. NUMERICAL ANALYSIS AND DISCUSSION
We now analyze the parameter space of the 4GTHDM and the most up-to-date constraints.

A. Constraints and parameter space of the 4GTHDM
According to the results given in Eqs. (9) and (10) along with Table II, we need the following parameters for our calculation: t β , χ, the masses of the heavy scalar bosons and the fourth-generation quarks, the 4 × 4 CKM matrix elements U ij (i = t, t and j = b, b ), and the mixing matrix elements Σ u ij and Σ d ij (i, j = 3, 4). We have already noted that to be consistent with the LHC data on the 125 GeV scalar boson, the scenario (i) of the 4GTHDM requires t β 1 and χ of the order of 10 −1 o less. The remaining free parameters are worth a special discussion.

Masses of the heavy scalar boson and the fourth-generation quarks
The existence of new scalar bosons has been explored by the ATLAS [67][68][69] and CMS [70][71][72][73][74] Collaborations: a heavy scalar boson H 0 has been searched for in the γγ [72], ZZ [69] and h 0 h 0 [70,74] channels, whereas the pseudoscalar boson A 0 has been looked for in the γγ [72] and Zh 0 [68,70,71,74] channels. On the other hand, the presence of a charged scalar H ± has been explored via the decays t → H ± b [67,73] and H ± → τ + ν τ [73]. In all these analyses, the heavy scalar boson masses were scanned from 150 GeV to 1000 GeV. In our analysis we will consider the interval 400 GeV-1000 GeV for the heavy scalar boson masses.
As far as the masses of fourth-generation quarks are concerned, when t → W b is assumed to be the dominant decay mode, the lower bounds m t > 450 GeV and m t > 557 GeV can be obtained from the CMS analysis of the semileptonic (pp → tt → νqqb) and dileptonic (pp → tt → + − ννbb) channels [59]. On other hand, if t → ht is taken as the dominant decay mode, the lower bound is m t 350 GeV [59]. Along these lines, Ref. [54] analyze the LHC data on the 125 Higgs boson in the context of the 4GTHDM. Those authors consider the 400 GeV< m t ,b < 600 GeV range and find a good agreement with experimental data. However, the splitting between the masses of the fourth-generation quarks ∆m 4 = m t − m b is restricted to be below 200 GeV [45]. In our analysis we use the range 400 GeV < m t < 1000 GeV and fix ∆m 4 = 100 GeV, which is within the allowed region [55].

Matrix elements
The diagonal and non-diagonal elements of the 4 × 4 U and Σ q matrices are involved in our analysis, consequently a more detailed discussion is required. We first write the corresponding matrix elements in exponential form and discuss the implications of unitarity and hermicity on the moduli and phases. A 4 × 4 unitary matrix can be parameterized by six mixing angles and three CP -violating complex phases [75], but we only need the U ij (i = t, t and j = b, b ) elements for our analysis. For the diagonal elements, ρ ii = 0 due to unitarity and we can assume |U ii | 1. Furthermore, we can take |U t b | |U tb |, ρ t b 0 and ρ tb = 0 without losing generality [75]. Thus, |U tb | and ρ tb will be the only free parameters involved in the CMDM and CEDM. From the experimental data on Z, K, and B decays as well as B-meson mixing, the upper bound |U tb | < 0.12 was extracted [64]. We will then use |U tb | 10 −1 and ρ tb ∈ [−π, π].
On the other hand, Σ q is hermitian [see Eq. (6)], so its diagonal elements must be real (η q ii = 0), whereas its nondiagonal elements must obey |Σ u ij | = |Σ u ji | and η u ij = −η u ji . This leaves |Σ u 33 |, |Σ u 34 |,|Σ u 44 | and η u 34 as free parameters, along with an identical number of free parameters associated with the Σ d matrix. As explained above, these matrices parametrize the mixing between the fourth-generation quarks and those of the first three generations. Instead of the parametrization of Eq. (17), we will use the parametrization of Eq. (7) in terms of the complex parameters t and b , which means that η u 43 ≡ δ t and η d 43 ≡ δ b . However, there are no direct experimental bounds on these parameters, though the authors of [64] . In our calculation we will consider two illustrative scenarios: whereas the complex phases δ t,b will be taken in the interval [−π,π]. We summarize in Table III the values of the free parameters of the 4GTHDM that we will use below, unless stated otherwise, to present a numerical analysis of the behavior of the CMDM and CEDM of the top quark.  (7), with an analogue parametrization for Σ d , and two scenarios for the values of t,b and δ t,b .

B. Top quark CMDM and CEDM in the 4GTHDM
For the evaluation of the CMDM and CEDM of the top quark we used the Mathematica routines for the numerical integration of Eqs. (11) and (12). A cross-check was done by evaluating the respective expressions in terms of Passarino-Veltman scalar functions [Eqs. (13) and (14)] via the LoopTools routines [76,77].

Top quark CMDM
In the 4GTHDM there are new contributions to the top quark CMDM arising from all the scalar bosons, but in our analysis we only consider the new physics contributions, so we remove the pure SM Higgs boson contribution given in Eq. (16). The total contribution of the 4GTHDM is thus given as a 4GTHDM where the new physics contribution δa 4GTHDM t is given as follows where a 3rd t and a 4th t are the contributions of the loops with internal quarks of the third and fourth generations, respectively, which can be written as and with being the new physics correction to a h 0 t arising from the loop with h 0 and t quark exchange. Notice that in this model the H −b t coupling depends on m b and m t , so the third-generation quark contribution also depends on the masses of the fourth-generation quarks.
We start our analysis by assessing the impact of the presence of the new heavy quarks on a t as they are the new ingredient of the 4GTHDM as compared to the usual THDMs. We first assume that all the heavy scalar bosons have a degenerate mass m φ and show in Fig. 2 the behavior of the partial contributions of the light and heavy scalar bosons to a t as functions of m φ for the parameter values of Table III (19) and (20)]. We first examine the partial contributions from the third-generation quarks (left plots), which in scenario II (small mixing) are about the same as the contribution of the usual type-II THDM with three quark families. We observe from the left plots of Fig. 2 that the heavy neutral bosons give contributions that are about the same size than the contribution of the lightest Higgs boson h 0 , whereas the charged Higgs boson contribution is slightly smaller. However, all these contributions are of opposite signs and tend to cancel out. This is evident in scenario II, where a 3rd We now we fix the scalar boson masses to their lowest allowed values and analyze the behavior of δa 4GTHMD t as a function of m t , whereas the remaining parameters are given the same values of Fig. 2. The corresponding results are shown in Fig. 3. Note that a 3rd t (left plots) does depend on m t and m b through the coupling constant H −b t appearing in the H ± contribution, which however is much smaller than those of the neutral scalar bosons, with H 0 giving the dominant contribution. Again, there is a cancellation effect between partial contributions, so a 3rd t is smaller than the H 0 contribution alone, though in scenario II it is slightly larger than in scenario I. As for a 4th t (right plots), the lightest CP -even scalar h 0 gives a negligible contribution, below the 10 −5 level and it is not shown in the plots. In scenario I the H ± contribution increases with m t , whereas the H 0 and A 0 contributions are almost independent of it. On the other hand, in scenario II the H 0 and A 0 contributions cancel each other out, so a 4th t is indistinguishable from the H ± contribution. We conclude that in scenario I a 4th t gives the dominant contribution to δa 4GTHDM t , whereas in scenario II the largest contribution arises from a 3rd t . While δa 4GTHDM t can reach values of the order of 10 −3 for a heavy m t in scenario I, it is independent of m t in scenario II and can reach slightly smaller values than in scenario I.
Finally we consider the scenario I of the mixing matrices Σ u,d along with the parameter values given in Table III, with m t = 400 GeV, and show in the top plots of Fig. 4 Table III. We show separately the partial contributions to a 3rd t (left plots) and a 4th t (right plots) as well as the total contribution for each generation (dash-dotted lines). We set m t = 400 GeV and for the remaining parameters we use the values shown in Table III. The total new physics contribution δa 4GTHDM t is denoted by the solid lines in the right plots. Also, the h 0 contribution arising from the t quark is negligible and is not shown in the right plots. can reach values of the order of 10 −3 , which is about the same size of the pure SM Higgs boson contribution. We also examined the dependence of a 4GTHDM t on other free parameters of the model, but an enhancement above the 10 −2 level was not found.

Top quark CEDM
We now turn to the analysis of the top quark CEDM in the 4GTHDM. As discussed above, new sources of CP violation can arise in this model via the new phase of the extended CKM matrix but also through the mixing matrices Σ u,d . The analysis simplifies considerably since due to the hermicity of the mixing matrix Σ u,d , the contributions from the neutral scalar bosons to d t vanishes and there is only contributions from the charged scalar boson. The contribution from the 4GTHDM to the top quark CEDM can thus be written as In Fig. 5 we show the CEDM of the top quark in the 4GTHDM as a function of m H ± (m b ) for fixed m b (m H ± ) and two values of the complex phase ρ tb entering into the 4 × 4 CKM mixing matrix. We consider the scenarios for the mixing matrices Σ u,d discussed above and for the remaining parameters we use the values shown in Table III. We first note that the dominant contribution to d t is that of the b quark, whereas the contribution of the b quark is much smaller. Thus the total contribution from this model to the top quark CEDM basically coincides with that of the b quark, which can reach values of the order of 10 −19 − 10 −18 ecm. We also observe that d t decreases as m H ± increases, but it increases as m b increases, so the largest values of d t would be reached for very heavy m b . However, in general d t shows little variation when both m H ± and m b are varied in the interval of 400 GeV to 1000 GeV.  Table III. We show separately the partial contributions to a 3rd t (left plots) and a 4th t (right plots). We set m H 0 = 500 GeV, mA = 300 GeV, m H ± = 600 GeV, and for the remaining parameters we use the values shown in Table III. The total new physics contribution δa 4GTHDM t is denoted by the solid lines in the right plots. Also, the h 0 contribution of the t quark is negligible and it is not shown in the right plots.
We now analyze the dependence of d t on the complex phases. We found that there is little dependence on the phase δ b appearing in Σ d , so we refrain from presenting a detailed analysis along this line and focus instead on the dependence on ρ tb and δ t , the complex phase of Σ u . Since these phases can interfere, we introduce the phase δ = ρ tb + δ t and plot in the top row of Fig. 6 the contour lines of d t in the planes δ vs m H ± and δ vs m t in scenario I for the mixing matrices Σ u,d and the parameter values of Table III. We note that d t vanishes when δ = 0 and δ = π, whereas its largest values, of the order of 10 −18 ecm, are reached for δ = π/2 and either light m H ± or heavy m b . On the other hand, in the lower plots of Fig. 6 we show the contour lines of d t in the planes δ vs t and δ vs |U tb | for fixed m H ± and m t . The remaining parameters were fixed to the values of Table III in scenario I. Again, d t is larger for δ = π/2, but its largest values are reached for large t and |U tb |, though there is more sensitivity to a change in the latter. We conclude that d 4GTHDM t can reach values not much larger than about 10 −18 ecm for δ = π/2, m H ± close to its lower bound, and large values of m b . However, there is little variation with respect to these parameters and also with respect to other parameters such as d and ∆m 4 .
Finally, it is worth comparing the results for the CMDM and CEDM of the top quark in the 4GTHDM with the predictions of other popular extension models. In Table IV we show the corresponding predictions, if available, of the top quark CMDM and CEDM in the usual THDMs, multiple Higgs-doublet models (MHDMs), 331 models, technicolor, extra dimensions, little Higgs models, SUSY theories, unparticles, and models with vector-like multiplets. We conclude that the 4GTHDM can give contributions to the CMDM of similar order of magnitude than these extension models, though the contribution to the CEDM can be larger than that predicted by the usual THDMs by one or two orders of magnitude, which is due to the presence of the new b quark.

V. CONCLUSIONS AND OUTLOOK
We have presented a calculation of the one-loop contributions to the chromo magnetic and chromo electric dipole moments of the top quark within the two-Higgs doublet model with four fermion families, which predicts new sources of CP violation arising through the complex phases of two mixing matrices and the extended 4 × 4 CKM matrix. Unlike the standard model with a sequential fourth generation of fermions, which is already excluded by the LHC data  Table III within the scenario I for the mixing matrices Σ u,d described there.  [12] on the SM Higgs boson, there is a region of the parameter space of the 4GTHDM still allowed. The new contributions to the CMDM of the top quark arise from loops carrying the new neutral scalar bosons H 0 and A 0 accompanied by the t quark and the fourth-generation t quark, together with loops carrying the charged scalar boson H ± along with the  Table III. b quark and the fourth-generation b quark. There are also new contributions from the lightest scalar boson h 0 , which is identified with the SM Higgs boson, via loops carrying the t and t quarks, though the latter is negligibly small, whereas the former is due to a tiny correction to the htt coupling. On the other hand, the CEDM of the top quark only receives the contribution from loops with the charged scalar boson along with the b and b quarks. We present analytical expressions for all these contributions in terms of Feynman parameter integrals and Passarino-Veltman scalar functions. We focus our numerical analysis of the behavior of the CMDM and CEDM of the top quark in the region of the parameter space of the 4GTHDM that is still consistent with the LHC data on the 125 GeV Higgs boson. In such a region the top quark CMDM can reach values of the order of 10 −4 − 10 −3 , with the dominant contribution arising from the loops with the fourth generation quarks, though it is also possible that all the contributions interfere destructively for some parameter values, thereby giving a negligible CMDM. As for the top quark CEDM, it does not suffer from possible cancellations as the dominant contribution arise from the loop with the charged scalar boson and the b quark, whereas the loop with the b quark gives a negligible contribution. It is found that the top quark CEDM can reach values of the order of 10 −18 ecm for relatively light m H ± and large m b . This value can be larger than the value predicted by the usual THDM by two orders of magnitude and it is mainly due to the presence of the new b quark, though it can be more suppressed for a tiny complex phase.  Table III within scenario I for the mixing matrices Σ u,d described there.