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Advances in High Energy Physics
Volume 2013 (2013), Article ID 672972, 28 pages
Two Higgs Doublets, a 4th Generation and a 125 GeV Higgs: A Review
1Physics Department, Technion-Institute of Technology, 32000 Haifa, Israel
2Theoretische Elementarteilchenphysik, Naturwissenschaftlich Technische Fakultät, Universität Siegen, 57068 Siegen, Germany
3Indian Institute of Technology, North Guwahati, Guwahati 781039, Assam, India
4Theory Group, Brookhaven National Laboratory, Upton, NY 11973, USA
Received 24 June 2012; Accepted 20 November 2012
Academic Editor: George Wei-Shu Hou
Copyright © 2013 Shaouly Bar-Shalom et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We review the possible role that multi-Higgs models may play in our understanding of the dynamics of a heavy 4th sequential generation of fermions. We describe the underlying ingredients of such models, focusing on two Higgs doublets, and discuss how they may effectively accommodate the low-energy phenomenology of such new heavy fermionic degrees of freedom. We also discuss the constraints on these models from precision electroweak data as well as from flavor physics and the implications for collider searches of the Higgs particles and of the 4th generation fermions, bearing in mind the recent observation of a light Higgs with a mass of ~125 GeV.
1. Introduction: The “Need” of a Multi-Higgs Setup for the 4th Generation
The minimal and perhaps the simplest framework for incorporating 4th generation fermions can be constructed by adding to the standard model (SM) a 4th sequential generation of fermion (quarks and leptons) doublets (for reviews see [1–3]). This framework, which is widely known as the SM4, can already address some of the leading theoretical challenges in particle physics:(i)the hierarchy problem [4–11], (ii)the origin of matter/antimatter asymmetry in the universe [12–16], (iii)flavor physics and CKM anomalies [17–30].
Unfortunately, the current bounds on the masses of the 4th generation quarks within the SM4 are rather high, reaching up to ~600 GeV [31–34], that is, around the unitarity bounds on quark masses [35–37]. The implications of such a “superheavy” 4th generation spectrum are far reaching. In fact, the SM4 as such is also strongly disfavored from searches at the LHC [38–41] and Tevatron  of the single Higgs particle of this model, essentially excluding the SM4 Higgs with masses up to 600 GeV [43, 44] and, thus, making it incompatible with the recent observation/evidence of a light Higgs with a mass of ~125 GeV [45, 46] (for a recent comprehensive analysis of the SM4 status in light of the latest Higgs results and electroweak precision data (EWPD), we refer the reader to ). These rather stringent limits on the SM4 raise several questions at the fundamental level: (1)Are superheavy fermionic degrees of freedom a surprise or is that expected once new physics (NP), beyond the SM4 (BSM4), is assumed to enter at the TeV scale? (2)Are such heavy fermions linked to strong dynamics and/or to compositeness at the nearby TeV scale? (3)What sub-TeV degrees of freedom should we expect if indeed such heavy fermions are found? And what is the proper framework/effective theory required to describe the corresponding low energy dynamics? (4)How do such heavy fermions affect Higgs physics? (5)Can one construct a natural framework for 4th generation heavy fermions with a mass in the range 400–600 GeV that is consistent with EWPD and that is not excluded by the recent direct measurements from present high-energy colliders? (6)What type of indirect hints for BSM4 dynamics can we expect in low energy flavor physics?
In this paper we will try to address these questions by considering a class of BSM4 low energy effective theories which are based on multi-Higgs models.
Let us start by studying the hints for BSM4 and strong dynamics from the evolution of the 4th generation Yukawa coupling , under some simplifying assumptions. In particular, one can write the RGE of assuming SM4 dynamics and neglecting the gauge and the top-Yukawa couplings and taking all 4th generation Yukawa couplings equal  This yields a Landau Pole (defined by ) at , giving TeV for GeV. Therefore, within the SM4, the 4th generation Yukawa couplings are expected to “run into” a Landau Pole at the near by TeV scale.
In fact, there are additional strong indications from the Higgs sector that a heavy 4th generation of fermions is tied with new strong dynamics at the near by TeV scale and that the SM4 is not the adequate framework to describe the new TeV-scale physics
The Higgs Mass Correction Due to Such Heavy Fermions Is Pushed to the Cutoff Scale. To see that, one can calculate the self-energy 1-loop correction to the Higgs mass with the exchange of a heavy fermion and set the cutoff to , obtaining indicating that a heavy 4th family fermion with a mass around 400 GeV cannot coexist with the recently observed single light Higgs, since in the absence of fine tuning, the Higgs mass should be pushed up to the cutoff scale where the NP enters (in which case the definition of the Higgs particle becomes meaningless).
The SM4 Higgs Quartic Coupling and a Heavy Higgs. One can again study the RGE for , assuming SM4 dynamics and neglecting the gauge and the top-Yukawa couplings and taking all 4th generation Yukawa couplings equal. One then obtains  giving a Landau Pole (i.e., ) at TeV for GeV and, thus, indicating that a light Higgs is not consistent with the SM4 if the NP scale is at the few-TeV range. Indeed, solving the full RGE for the SM4 one finds that when the cutoff of the theory is set to the TeV scale, that is, to the proper cutoff of the SM4 when GeV . The implications of a heavy Higgs in this mass range was considered, for example, in [49–52], claiming that the heavy SM4 Higgs case can relax the currently reported exclusion on the SM4. However, the heavy SM4 Higgs scenario is now in contradiction with the recent measurements of the two experiments at the LHC, which observe a light Higgs boson with a mass of ~125 GeV [38–41]. On the other hand, as will be shown in this paper (and was also demonstrated before in  for the case of the popular 2HDM of type II with a 4th generation of doublets), a multi-Higgs setup for the 4th generation theory can relax the constraint .
Thus, under the assumption that heavy 4th generation quarks exist, if one assumes a light Higgs with a mass around 125 GeV and seriously takes into account the fact that low energy 4th generation theories possess a new threshold/cutoff (or a fixed point; see, e.g., [53, 54]) at the TeV scale, then one is forced to consider extensions of the naive SM4 with more than one Higgs doublet which, in turn, leads to the possibility that the Higgs particles (or some of the Higgs particles) may be composites primarily of the 4th generation fermions (see, e.g., [55–60]), with condensates , (and possibly also , ). These condensates then induce EWSB and generate a dynamical mass for the condensing fermions. This viewpoint in fact dates back to an “old” idea suggested more than two decades ago ; that a heavy top quark may be used to form a condensate which could trigger dynamical EWSB. Although, this top-condensate mechanism led to the prediction of a too large , this idea ignited further thoughts and studies towards the possibility that 4th generation fermions may play an important role in dynamical EWSB [4–9]. In particular, due to the presence of such heavy fermionic degrees of freedom, some form of strong dynamics and/or compositeness may occur at the near by TeV scale.
In this article, we will review the above viewpoint which was also adopted in : that theories which contain such heavy fermionic states are inevitably cutoff at the near by TeV scale and are, therefore, more naturally embedded at low energies in multi-Higgs models, which are the proper low-energy effective frameworks for describing the sub-TeV dynamics of 4th generation fermions. As mentioned above, in this picture, the Higgs particles are viewed as the composite scalars that emerge as manifestations of the different possible bound states of the fundamental heavy fermions. This approach was considered already 20 years ago by Luty  and more recently in , where an attempt to put 4th degeneration heavy fermions into an effective multi-(composite) Higgs doublets model was made, using a Nambu-Jona-Lasinio (NJL) type approach.
The phenomenology of multi-Higgs models with a 4th family of fermions was studied to some extent recently in [48, 63–69] and within a SUSY framework in [14, 16, 70–72]. In this article, we will further study the phenomenology of 2HDM frameworks with a 4th family of fermions, focusing on a new class of 2HDM’s “for the 4th generation” (named hereafter 4G2HDM) that can effectively address the low-energy phenomenology of a TeV-scale dynamical EWSB scenario, which is possibly triggered by the condensates of the 4th generation fermions.
We will first describe a few viable manifestations of a 2HDM framework with a 4th generation of fermions, focusing on the 4G2HDM framework of . We will then discuss the constraints on such 4th generation 2HDM models from PEWD as well as from flavor physics. We will end by studying the expected implication of such 2HDM frameworks on direct searches for the 4th generation fermions and for the Higgs particle(s), assuming the existence of a light Higgs with a mass of 125 GeV.
2. 2HDM’s and 4th Generation Fermions
Assuming a common generic 2HDM potential, the phenomenology of 2HDM’s is generically encoded in the texture of the Yukawa interaction Lagrangian. The simplest variant of a 2HDM with 4th generations of fermions can be constructed based on the so-called type II 2HDM (which we denote hereafter by 2HDMII), in which one of the Higgs doublets couples only to up-type fermions and the other to down-type ones. This setup ensures the absence of tree-level flavor changing neutral currents (FCNC) and is, therefore, widely favored when confronted with low energy flavor data. The Yukawa terms of the 2HDMII, extended to include the extra 4th generation quark doublet, are (and similarly in the leptonic sector) where are left-(right) handed fermion fields, is the left-handed quark doublet, and , are general Yukawa matrices in flavor space. Also, are the Higgs doublets:
Motivated by the idea that the low energy scalar degrees of freedom may be the composites of the heavy 4th generation fermions, it is possible to construct a new class of 2HDM’s that effectively parameterize 4th generation condensation by giving a special status to the 4th family fermions. This was done in , where (in the spirit of the Das and Kao 2HDM that was based on the SM’s three families of fermions ) one of the Higgs fields (—call it the “heavier” field) was assumed to couple only to heavy fermionic states, while the second Higgs field (—the “lighter” field) is responsible for the mass generation of all other (lighter) fermions. The possible viable variants of this approach can be parameterized as  (and similarly in the leptonic sector) where are the two Higgs doublets, is the identity matrix, and () are diagonal matrices defined by .
The Yukawa interaction Lagrangian of (6) can lead to several interesting textures that can be realized in terms of a -symmetry under which the fields transform as follows: which allows us to construct several models that have a non-trivial Yukawa structure and that are potentially associated with the following compositeness scenario(i)Type I 4G2HDM: denoted hereafter by 4G2HDMI and defined by , in which case gives masses only to and , while generates masses for all other quarks (including the top quark). For this model, which seems to be the natural choice for the leptonic sector, we expect (ii)Type II 4G2HDM: denoted hereafter by 4G2HDMII and defined by , in which case the heavy condensate couples to the heavy quarks states of both the 3rd and 4th generations and quarks, whereas couples to the light quarks of the 1st and 2nd generations. For this model one expects . (iii)Type III 4G2HDM: denoted hereafter by 4G2HDMIII and defined by , in which case , , and , so that only quarks with masses at the EW-scale are coupled to the heavy doublet . Here also one expects . The Yukawa interactions for these models are given by  where is the CKM matrix, or for down or up quarks with weak isospin and , respectively, and . Also, the 4G2HDM type, that is, the 4G2HDMI, 4G2HDMII, and 4G2HDMIII, as well as FCNC effects are all encoded in and , which are new mixing matrices in the down- (up-) quark sectors, obtained after diagonalizing the quarks mass matrices: depending on , which are the rotation (unitary) matrices of the right-handed down and up quarks, respectively, and on whether and/or are “turned on.” This is in contrast to “standard” frameworks such as the SM4 and the 2HDM’s of types I and II, where the right-handed mixing matrices and are nonphysical being “rotated away” in the diagonalization procedure of the quark masses. Indeed, in the 4G2HDM’s described above some elements of and can, in principle, be measured in Higgs-fermion systems, as we will later show.
In particular, inspired by the working assumption of the 4G2HDM’s and by the observed flavor pattern in the up-and down-quark sectors, it was shown in  that the new mixing matrices and are expected to have the following form:
and similarly for by replacing and . The new parameters , are free parameters that effectively control the mixing between the 4th generation and the 2nd and 3rd generation quarks and , respectively. Thus, a natural choice which will be adopted here in some instances is , and .
3. Constraints on 2HDM’s with a 4th Generation of Fermions
3.1. Constraints from Electroweak Precision Data: Oblique Parameters
The sensitivity of EWPD to 4th generation fermions within the minimal SM4 framework was extensively analyzed in the past decade [74–80]. Here we are interested instead in the constraints that EWPD impose on 2HDM’s with a 4th generation family. As usual, the effects of the NP can be divided into the effects of the heavy NP which does and which does not couple directly to the ordinary SM fermions. For the former, the leading effect comes from the decay , which is mainly sensitive to the and couplings through one-loop exchanges of and shown in Figure 1, and which was analyzed in detail in .
On the other hand, the effects which do not involve direct couplings to the ordinary fermions can be analyzed by the quantum oblique corrections to the gauge-bosons 2-point functions, which can be parameterized in terms of the oblique parameters , , and [81, 82]. For the oblique parameters the effects of a 2HDM with a 4th generation are common to any variant of a 2HDM framework (including the 2HDMII, and the 4G2HDMI, 4G2HDMII and 4G2HDMIII described in the preivous section), since the Yukawa interactions of any 2HDM do not contribute at 1 loop to the gauge-bosons self-energies.
In particular, apart from the pure 1-loop Higgs exchanges, one also has to include the new contributions from and exchanges which shift the parameter () and which involve the new SM4-like diagonal coupling as well as the and off-diagonal vertices (see, e.g., ): with and .
The complete set of corrections to the and parameters within a 2HDM with a 4th generation of fermions was considered in [61, 75, 83]. Following the recent analysis in , we show in Figure 2 the results of “blindly” (randomly) scanning the relevant parameter space with 100000 models, where we set the light Higgs mass to be GeV and vary the rest of the relevant parameters within the ranges: , , , , , , , and and , and the CP-even neutral Higgs mixing angle in the range . In particular, we plot in Figure 2 the allowed points in parameter space projected onto the 68%, 95%, and 99% allowed contours in the - plane, and the 95% CL allowed range in the and the planes, corresponding to the 95% CL contour in the - plane.
We find that compatibility with PEWD mostly requires with a small number of points in parameter space having . We also find that the 2HDM frameworks allow 4th generation quarks and leptons mass splittings extended to and , and “solutions” where both the quarks and the leptons of the 4th generation doublets are degenerate. For the cases of a small (or no) 4th generation fermion mass splitting, the amount of isospin breaking required to compensate for the effect of the extra fermions and Higgs particles on and is provided by a mass splitting among the Higgs particles; see .
3.2. Constraints from Electroweak Precision Data:
The effects of the NP in are best studied via the well-measured quantity : which is a rather clean test of the SM, since being a ratio between two hadronic rates, most of the electroweak, oblique, and QCD corrections cancel between numerator and denumerator.
Following , the effects of NP in can be parameterized in terms of the corrections and to the decays and , respectively: where and are the corresponding 1-loop quantities calculated in the SM and are the NP corrections defined in terms of the couplings as where with , and , so that are the SM (1-loop) quantities and are the NP 1-loop corrections.
The corrections to from the 4th generation quarks in the 4G2HDMI, 4G2HDMII, and 4G2HDMIII are of three types (see ), where in all cases one finds that , so that one can safely neglect the effects from .
3.2.1. SM4-Like Corrections
These are the corrections to due to the 1-loop exchanges (denoted here as ), which are given by [18, 79, 84, 85] where is the mixing angle between the 3rd and 4th generation quarks, that is, defining , and the 2nd term is the decrease from the SM’s correction to the W-boson vacuum polarization, which in the 4th generation case is .
The SM4-like effect on is plotted in Figure 3, from which we can see that puts rather stringent constraints on the plane which is the SM4 subspace of the parameter space of any 2HDM containing a 4th generation of fermions. For example, for GeV the mixing angle is restricted to .
The corrections from the 1-loop exchanges are plotted in Figure 1. In the 4G2HDM of types II and III, these charged Higgs exchange diagrams are found to have negligible effects on and are, therefore, not constrained by this quantity. On the other hand, is rather sensitive to the charged Higgs 1-loop exchanges within the 4G2HDMI. This can be seen in Figure 4, where is plotted (for the 4G2HDMI case) as a function of the charged Higgs and masses, fixing and focusing on the values , , and GeV.
In Figure 5 we further plot the allowed ranges in the plane in the 4G2HDMI, subject to the constraint (at ), for in the range 1–15, fixing GeV, GeV, , (which also enters the vertex) and for three representative values of the mixing parameter: , , and . We see, that, as expected, when is lowered, the constraints on the charged Higgs mass are weakened. In particular, while there are no constraints from on the charged Higgs and masses if , for higher values of a more restricted region of the charged Higgs mass is imposed which again depends on . We see for example, that for , is compatible with values ranging from 200 GeV up to the TeV scale, while for , the charged Higgs mass is restricted to be within the range GeV.
For the case of the 2HDMII (i.e., extended with a 4th family of fermions), which is also plotted in Figure 4, we find that there is essentially no constraint in the plane for GeV.
3.2.3. The Flavor Changing Interactions
The 1-loop corrections to which involve the flavor changing (FC) interactions emanate from the nondiagonal 34 and 43 elements in , with , or . These corrections are found to be much smaller than 1-loop exchanges, so that they can be safely neglected, in particular for .
3.3. Constraints from Flavor in Physics
Flavor physics plays an important role in discriminating between the various NP models. In this regard, FCNC decays can provide key information about the SM and its various extensions.
The inclusive radiative decay is indeed known to be a very sensitive probe of NP. The underlying process is induced by the FC decay of the -quark into a strange quark and a photon. The Br() has already carved out large regions of the parameter space of most of the NP models [89–102]. On the other hand, model independent analysis in the effective field theory approach without  and with  the assumption of minimal flavor violation also show the strong constraining power of the decay . Once more precise data from super-B factories are available, this decay will undoubtedly be more efficient in selecting the viable regions of the parameter space in the various classes of NP models.
The calculation of the decay rate of the transition is most conveniently performed after integrating out the heavy degrees of freedom. The resulting effective theory contains various FC dimension-five and -six local interactions and the inclusive decay rate is given by where the Wilson coefficients, , of the effective operators (see below) are perturbatively calculable at the relevant renormalization scale and the Renormalization Group Equations (RGE) can be used to evaluate at the scale . At present, all the relevant Wilson coefficients are known at the Next-to-Next-to-Leading Order (NNLO) [105–116] and is determined by the matrix elements of the operators [107, 108]: which consists of perturbative and nonperturbative corrections. The perturbative corrections are well under control and are fully known at NLO QCD . However, quantitative estimates of all the non-perturbative effects are not available, although they are believed to be ≈5% .
The SM prediction is, thus, consistent with the experiment (both having a 7% error) and is therefore useful for constraining many extensions of the SM.
In the SM4, there are no new operators other than the ones present in the SM. However, there are extra contributions to the Wilson coefficients corresponding to the operators and from -loops [17–20]. In a 2HDM framework with a 4th generation family, the new ingredient with respect to the SM4 is the presence of the charged Higgs 1-loop exchanges which contribute to the Wilson coefficients of the effective theory. In particular, at the parton level within a 2HDM, proceeds via the penguin diagrams depicted in Figure 6. As was shown in , in the 4G2HDMI, 4G2HDMII, and 4G2HDMIII frameworks, the leading effects enter in and from the 1-loop exchanges of , , and .
An important role for constraining NP in the b-quark system is also played by () mixing, the phenomenon of which is described by the dispersive part of the mixing amplitude. The current theory precision is limited by lattice results; the SM prediction still allows NP contributions to of order 20% .
Within a 2HDM setup, the leading contribution to () mixing comes from the box diagrams shown in Figure 7, where the boson is replaced by the charged Higgs , and the fermions are replaced by . Thus, the net contribution to the mass difference is given by  where and , , ( or ), and . Here, , , and are the contributions from the box diagrams with the combination of the gauge bosons , , and in the internal lines ( stands for the charged Higgs), respectively. The detailed expression for the various Inami-Lim functions is given in .
For the B-physics parameters, we use the inputs given in Table 1, and for the 4th generation quark masses, we take GeV and GeV.
3.3.3. Constraints from Physics: Results
For the “standard” 2HDMII with four generations we find that the constraints from and have a simple pattern in the plane. In particular, with GeV we find that GeV for , while GeV for .
For the 4G2HDM’s of types I, II, and III, the combined constraints on their parameter space from both and are summarized below. In Figures 8 and 9 we show a sample of the results obtained in , where the allowed ranges are shown in the and the planes, respectively. In these plots we use —corresponding to the “” scenario with a negligible 4th-3rd generation mixing, that is, with correspondingly. We see, for example, that in the 4G2HDMI, the “” scenario typically imposes with typically larger than about 0.4 when GeV. In the 4G2HDMII and the 4G2HDMIII one observes a similar correlation between and ; however, larger values are allowed for and a charged Higgs mass is typically heavier than 400 GeV.
For the case of a Cabibbo size mixing between the 4th and 3rd generation quarks, we set and show in Figures 10 and 11 the allowed parameter space in the and planes, in the 4G2HDM’s of types I, II, and III, with GeV, GeV, and . In the 4G2HDMII and the 4G2HDMIII we see a similar behavior as in the no-mixing case (i.e., as in the case ), while in the 4G2HDMI we see that “turning on” allows for a slightly larger , that is, up to for . Also, similar to the no-mixing case, larger values of are allowed in the 4G2HDMII and 4G2HDMIII. Furthermore, GeV and are allowed in the 4G2HDMI.
3.4. Combined Constraints and Points of Interest
In Table 2 we give a sample list of interesting points (models) in parameter space of the 4G2HDMI that “survive” all constraints from EWPD and flavor physics in the 4G2HDMI, for GeV, , and . The list includes (see also caption of Table 2) models with a 4th generation mass splitting (between the up and down partners of both the 4th family quarks and leptons) larger than 150 GeV; models where both the 4th generation quarks and leptons are nearly degenerate; models with a light to intermediate neutral Higgs spectrum, that is, GeV and or in the range 150–300 GeV; models with a large inverted mass hierarchy in the quark doublet, that is, GeV; models with a light charged Higgs with a mass smaller than 400 GeV and models with a Cabibbo size as well as an size mixing angle.
4. Other Useful Effects in Flavor Physics
We discuss below some important low energy observables, which are potentially sensitive to the 4th generation dynamics within the multi-Higgs framework, and have shown some degree of discrepancy between their measured values and the SM predictions.
4.1. Muon and Lepton Flavor Violation
The muon anomalous magnetic moment (AMM), , is well known to play an important role in the search for NP. In the SM, the total contributions to the AMM, , can be divided into three parts: the QED, the electroweak (EW), and the hadronic contributions. While the QED [120–125] and EW [126–129] contributions are well understood, the main theoretical uncertainty lies with the hadronic part which is difficult to control [130, 131].
Since the first precision measurement of , there has been a discrepancy between its experimental value and the SM prediction. This discrepancy has been slowly growing due to recent impressive theoretical and experimental progress. Comparing theory and experiment, the deviation amounts to  which corresponds to effect. In order to confirm this result, the uncertainties have to be further reduced.
It is interesting to interpret the difference as a contribution from loop exchanges of new particles. A number of groups have studied the contribution to in various extensions of the SM to constrain their parameters space (for reviews see [133, 134]). In most extensions of the SM, new charged or neutral states can contribute to the AMM at the one-loop (lowest) level. In , we have shown that the ~3 excess in (with respect to the SM prediction) can be accounted for by one-loop exchanges of the heavy 4th generation neutrino () in the 4G2HDMI setup when applied to the leptonic sector (i.e., where the “heavy” Higgs doublet couples only to the 4th generation lepton doublet and the “light” Higgs doublet couples to leptons of the lighter 1st–3rd generations; see ).
The effective vertex of a photon with a charged fermion can in general be written as where, to lowest order, and . While remains unity at all orders due to charge conservation, quantum corrections yield . Thus, since , it follows that .
In the 4G2HDMI [61, 135] the one-loop contribution to the muon anomaly can be subdivided as where contains the charged and neutral Higgs contributions coming from the one-loop diagrams in Figure 12, where the diagrams with and in the loop dominate. The SM4-like contribution, , comes from the one-loop diagram with in the loop and is given by  where is the 24 element of the CKM-like PMNS leptonic matrix, . For values of in the range , one finds , so that for (as expected) the simple SM4 cannot accommodate the observed discrepancy in . The detail expression for has been given in . It is interesting to note that the dominant contribution to , or for that matter to , comes from the charged Higgs loops and the contribution from diagrams with the neutral Higgs exchanges is subleading . In addition, was found to be sensitive only to the product , where and are the new mixing matrices (i.e., in the 4G2HDMI) in the charged (neutral) leptonic sectors. That is, similar to the quark sector (see (10)), these matrices are obtained after diagonalizing the lepton mass matrices where and are the rotation (unitary) matrices of the right-handed charged and neutral leptons, respectively.1
In Figure 13 we plot as a function of the product (assuming its real) for several values of and and fixing . Depending on the mass , we find that is typically required to accommodate the measured value of .
The constraints on the 4G2HDMI parameters and in particular on the quantities and which control the AMM were studied in , by analyzing the lepton flavor violating (LFV) decays and . These decays are absent in the SM and are useful for constraining NP models that can potentially contribute to the muon anomaly.
The amplitude for the transition can be defined as where is the photon polarization. The decay width is then given by
Here also, the new 4G2HDMI contribution to the amplitude, , can be divided as where is the SM4-like -exchange contribution which is much smaller than the charged and neutral Higgs amplitudes, and (calculated from the diagrams in Figure 12). As in the AMM case, the dominant contribution to LFV decays was found to be from the charged Higgs exchange diagrams . In addition, the decays and are sensitive to and through the products and , respectively, so that, in principle, one can avoid constraints on the quantities and if , , , and are sufficiently small.
In , we have shown that it is possible to address both the and the muon anomaly within the 4G2HDMI framework, if and , which is indeed expected if we consider the observed hierarchical pattern of the quark’s CKM matrix as a guide. However, in order to account also for the measured upper limit on (see (31)), one requires that and . Therefore, the typical benchmark texture for the 4th generation elements of the matrices that can account for the observed muon anomaly and still be consistent with the current constraints from the LFV decays and is Where, for example, for GeV.
The above texture implies a hierarchical pattern which is different from what one would expect from the observed hierarchical pattern of the quark’s CKM matrix. Nonetheless, without a fundamental theory of flavor, our insight for flavor should be data driven also in the leptonic sector. Besides, the above texture is sensitive to the current precision in the measurement of the muon which can change for example, if more accurate calculations end up showing that part of the hadronic contributions cannot be ignored.
4.2. Insight from Physics
Among the various rare decays, the purely leptonic decays are highly sensitive to indirect effects of NP, since the quark level decays are based on the FCNC transitions which are severely (loop) suppressed in the SM. In particular, the decay has received special attention in the past decade, since its branching fraction, , can be significantly enhanced by loop exchanges of new particles predicted by various NP scenarios. For example, imposes restrictive constraints on the SUSY parameter space (see, e.g., [138–140]), where in some scenarios better limits than those obtained from direct searches have been claimed. However, the excluded SUSY parameter space depends strongly on the choice of since the rate typically varies as .
In the LHC era the current limit on has been improved. The two different experiments LHCb and CMS, using and data sample, respectively, yield [141, 142] whereas the SM prediction for this decay is 
In fact, LHCb has the sensitivity to measure the down to ~, which is about smaller than the SM prediction.
In general, the matrix element for the decay can be written as  where is the four-momentum of the initial meson and ’s are functions of Lorentz invariant quantities. Squaring the matrix and summing over the lepton spins, we obtain the branching fraction In the SM, the dominant effect in arises from the diagrams shown in Figure 14, which contribute only to in (38).
As in other NP models, in the 4G2HDMI there will be contributions to , , and coming from the charged Higgs exchange penguin and box diagrams (replacing in Figure 14). In , constraints on the 4G2HDMI parameter spaces were estimated, using the recent data on . This was done in the context of the muon , in the sense that only those interactions (in the leptonic vertex) which are associated with have been considered. In particular, considering only the vertex, the only diagrams that contribute to are the Higgs exchange box diagrams in Figure 14, where one or two -bosons are replaced by and are being replaced by both and . It was then found that the contribution from the new box diagrams in the 4G2HDMI that involve the heavy 4th generation neutrino is consistent with the current experimental bound on for values of and that reproduce the observed muon see Figure 15.
It is also interesting to note that the Br, in both the SM4 and the 4G2HDMI, can differ from the SM value by at-most a factor of in either direction (for a detail discussions see ).
Other purely leptonic and semileptonic decays of the meson, such as decays, can also provide useful tests of the SM and its extensions. Of particular interest are the purely leptonic and the semileptonic decays. The SM contribution to the branching ratios of these decays arises at the tree-level from the charged weak interactions. An important NP contribution to these decays is the tree level exchange of a charged Higgs in multi-Higgs models, so that these decays offer interesting probes of the Higgs sector and, particularly, of its Yukawa interactions.
The SM expression for the decay rate of is given by where is the decay constant and is the life time. The SM prediction for is, therefore, sensitive to the decay constant and to the CKM element and is thus limited by the uncertainty in the determination of these quantities. Using the available constraints on and the inclusive determination of : MeV and , the SM prediction for the decay rate is
Furthermore, the SM prediction on , obtained directly from a fit to various other observables (i.e., without using and the lattice results for ) is 
Both results show some degree of discrepancy with the current world average on which is 
We want to indicate here how the 4G2HDM can address this if the discrepancy is confirmed.
From the theoretical point of view, several models of NP predict large deviations from the SM for processes involving third generation fermions. For instance, in a “standard” 2HDM where the two Higgs doublets are coupled separately to up- and down-type quarks (i.e., the 2HDMII setup described in Section 2), the amplitude receives an additional tree-level contribution from the heavy charged-Higgs exchange, leading to so that for large , the r.h.s. of (44) can be significantly different from “1.” However, in this particular case (of the 2HDMII), the charged-Higgs contribution reduces the SM value for the branching ratio, thus further worsening the situation with respect to the experimentally measured value.
In the 4G2HDMI, the effective tree-level interactions that will contribute to can be written as where the second term represents the tree-level charged-Higgs exchange and the first term results from the diagram with boson exchange. Also, , , and are factors coming from the and vertices, respectively, given by A simple calculation, using (45) and (46), yields
Thus, taking, for example, , only a moderate enhancement to is possible at large . If, on the other hand, , then the can be significantly enhanced compared to the SM prediction. Of course, the experimental deviations at the moment are only a few sigmas, but, if they get confirmed, then we have indicated here how we may be able to address them.
Semileptonic decays such as are more complicated to handle than the pure leptonic ones, since the theoretical predictions for these decays to exclusive final states require knowledge of the form factors involved. There are, however, several other observables (besides the branching fraction), such as the decay distributions and the polarization, which can be useful in this cases for probing NP.
As in the case of , the semileptonic decay is also known to be a sensitive mode to the tree-level charged-Higgs exchange. Furthermore, the precise measurement of at the B-factories and the theoretical developments of heavy-quark effective theory (HQET) has improved our understanding of exclusive semileptonic decays [74, 145].
In particular, the ratios reduce considerably the main theoretical uncertainties and, hence, turn out to be a more useful observable . The updated SM predictions of these rates, averaged over electron and muons, are given by [147, 148] so that at this level of precision the experimental uncertainties are expected to dominate.
The measured values, therefore, exceed the SM predictions for and by 2.0 and 2.7, respectively, so it is argued that the possibility of both the measured values agreeing with the SM is excluded at the 3.4 level. In addition, the combined analysis of and rules out the 2HDMII charged Higgs boson with 99.8% confidence level for any value of when combined with ; see [147, 148]. Once again, it is not clear to us how serious to take the indications of the deviations in (49). Nonetheless, we briefly indicate here how this discrepancy (if experimentally confirmed) can be addressed in the 4G2HDMI, for which the effective tree-level interactions that contribute to are given in (45) with the -quark replaced by the -quark. Thus, similar to the case of , we expect a moderate enhancement to both and in the 4G2HDMI if and a larger effect for larger values of .
5. New Aspects of the Phenomenology of the 4G2HDMI
In the 4G2HDMI (i.e., the 4G2HDM with see (6)), one obtains (see (11)) which leads to new interesting patterns (in flavor space) in both the neutral and charged Higgs sectors. For example, the Yukawa interactions of (9) () give rise to potentially enhanced tree-level and FC transitions and absence of “dangerous” tree-level FCNC transitions between the 4th and the 1st and 2nd generations quarks as well as among the 1st-2nd and 3rd generation quarks. In particular, the FC interactions in this case are (taking ) and similarly for the vertices by changing (and an extra minus sign in the coupling).
If , then the above couplings can become sizable, to the level that it might dominate the decay pattern of the (see below). In fact, large FC effects are also expected in transitions since, even for a very small , the FC and Yukawa couplings can become sizable if, for example, , for which case they are . Therefore, such new FCNC and transitions can have drastic phenomenological consequences for high-energy collider searches of the 4th generation fermions, as we be discussed below.
Furthermore, the flavor diagonal interactions of the Higgs species with the up quarks of the 1st, 2nd, and 3rd generations are proportional to in this model, thus being a factor of larger than the corresponding “conventional” 2HDMII (i.e., the type II 2HDM) couplings (which are ). For example, this gives rise to an enhanced flavor diagonal interactions, while suppressing the one, when .
Another important new feature of this model occurs in the charged Higgs couplings involving the 3rd and 4th generation quarks, which are completely altered by the presence of the and matrices and can thus lead to interesting new effects in both leptonic (see, previous section) and quark sectors. For example, taking , and the and Yukawa couplings are given in the 4G2HDMI by
Recalling that in the “standard” 2HDMII (which would underies a supersymmetric four-generation model) the would be , we find that in the 4G2HDMI the coupling is potentially enhanced by a factor of so that if, for example, and , there is a factor of enhancement to the interaction.
These new aspects of phenomenology in the Yukawa interactions sector can have far reaching implications for collider searches of the heavy 4th generation quarks and leptons, as will be discussed in more detail in the next sections. To see that, one can study the new decay patterns of and that follow from the above new Yukawa terms. In particular, in Figure 16 we plot the branching ratios of the leading decay channels (assuming ): ( stands for either on-shell or off-shell depending on ), as a function of the mass. We use GeV, GeV, , , and and . We see that the can easily reach (even for a rather large for which becomes sizable), in particular when ; see for example, points 8–11 in Table 2 for which .
In Figure 17 we plot the branching ratios of the leading decay channels , as a function of for GeV, GeV, , GeV, , and and . We see that in the case the dominance of (if kinematically allowed) should be much more pronounced due to the expected smallness of the mixing parameter, , which controls the FC decay ; see, for example, points 12 and 13 in Table 2 for which . On the other hand, if