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Abstract
In this talk, I present a new framework to understand the longstanding fermion mass hierarchy puzzle. We extend the Standard Model gauge symmetry by an extra local symmetry, broken spontaneously at the electroweak scale. All the SM particles are singlet with respect to this . We also introduce additional flavor symmetries, ’s, with flavon scalars , as well as vectorlike quarks and leptons at the TeV scale. The flavon scalars have VEV in the TeV scale. Only the top quark has the usual dimension four Yukawa coupling. EW symmetry breaking to all other quarks and leptons are propagated through the messenger field, through their interactions involving the heavy vectorlike fermions and , as well as through their interactions involving the vectorlike fermions and . In addition the explaining the hierarchy of the charged fermion masses and mixings, the model has several interesting predictions for Higgs decays, flavor changing neutral current processes in the top and the b quark decays, decays of the new singlet scalars to the new boson, as well as productions of the new vectorlike quarks. These predictions can be tested at the LHC.
:
12.60.Cn, 14.80.Cp6x9
OSUHEP0907
]Fermion Mass Hierarchy and
New Physics at the TeV Scale
address=Department of Physics and
Oklahoma Center for High Energy Physics,
Oklahoma State University,
Stillwater, OK 74078,email=s.nandi@okstate.edu
1 Introduction
Fermion mass hierarchy is a longstanding problem in particle physics. The charged fermion masses vary by five orders of magnitude, while the quark mixing angles vary by almost two orders of magnitude. There are two main approaches to understand this puzzle [1, 2, 3, 4, 5]. This hierarchy is caused by physics at the high scale (GUT scale or the Planck scale: so called FroggattNielsen type mechanism; or this is caused by some new physics at the TeV scale. In this work, I present a new model for this second approach with new physics at the TeV scale [6].
What are the new physics possibilities at the TeV scale? Supersymmetry is highly motivated, and predicts new superpartners and the Higgs boson at the TeV scale. Extra dimensions are somewhat motivated, and predicts new KaluzaKlein excitations at the TeV scale. Extra is somewhat string theory motivated, and predicts new gauge boson at the TeV scale. However these are all theory motivated. The experimental clues so far are that the charged fermion masses are highly hierarchial, quark mixing angles are hierarchial, and FCNC processes are strongly suppressed. The natural question is what sort of new physics can explain these, and can be observed at the LHC. In this talk, I present one such possibility.
In the SM, the Yukawa interactions of the fermions, in the mass basis, are parameterized by
(1) 
with , where . Note that , whereas the Yakawa couplings of all the light quarks and leptons, are . Thus the top quark is directly connected to the EW symmetry breaking sector, and has dimension 4 Yukawa interactions. The lighter quarks are probably not directly connected to the EW symmetry breaking sector, They may be connected via some messenger fields.
We know that the FCNC interactions among the quarks are highly suppressed. This hints at the existence of some sort of flavor symmetry. Furthermore, if we let all the SM fermions except , and H carry nonzero flavor charges, then the dimension 4 Yukawa couplings for the light quarks with H will be prevented. These flavor symmetries need to be global, and have to be spontaneously broken at the TeV scale. What sort of fields in addition to the SM we need to achieve this scenario? We shall see that one possibility is to have vectorlike quarks, and leptons at the TeV scale, and additional flavor symmetries, . Again, since the light quarks and leptons are not directly connected to the EW symmetry breaking scale, we need a messenger field to achieve this, and a SM singlet complex scalar field with an extra local symmetry will serve our purpose. Thus the new physics in our scenario will involve , and .
2 Model and Formalism
We extend the gauge symmetry of the SM model by a local symmetry and global symmetries. All of the SM fermions are neutral with respect to , while all of the SM fermions apart from the third generation quark doublet and righthanded top are charged under the global . We introduce a complex scalar field which has charge 1 under , is neutral under , and is a SM singlet. We also introduce complex scalar fields , the “flavons”, which have charges under , are neutral under , and are SM singlets. The Higgs field is taken as neutral under and . We assume that the charges of the SM fermions are such that only the top quark has an allowed dimension 4 Yukawa interaction.
The field acquires a vev at the EW scale that spontaneously breaks the symmetry. The pseudoscalar component of is eaten to give mass to the gauge boson . The field acts as a messenger to both the flavor symmetry breaking and EW symmetry breaking. The symmetries are broken by the vev of the flavon scalar fields, at the TeV scale. There are additional vectorlike fermions at the TeV scale charged under both and .
In this framework, the Yukawa interactions of the light fermions, after integrating out the heavy vectorlike fermions appear as higher dimensional operators in a hierarchial pattern given by
with similar expressions for the up sector. The observed fermion mass hierarchy and mixings are reproduced in powers of
which we call the ”little” hierarchy. We can absorb the dependence into fielddependent dimensionless complex couplings , where , are generation labels. The values of these couplings we will then take to be of order 1.
In the model we propose, the observed fermions mass hierarchy is generated from the following low energy effective interactions:
(2)  
where all the couplings are assumed to be of order 1.
Note that the above interactions are very similar to those proposed in reference [7, 8], except our interactions involve suppression by powers of , instead of .
2.1 Fit to Fermion Masses and CKM Mixing
The gauge symmetry of our model is the usual SM symmetry, plus an additional symmetry. The SM symmetry is broken spontaneously by the usual Higgs doublet, at the EW scale. We assume that the extra symmetry is also broken spontaneously at the EW scale by a SM singlet complex scalar field, . The pseudoscalar part of the complex scalar field, is absorbed by the to get its mass. Thus after symmetry breaking, the remaining scalar fields are and . Parameterizing the Higgs doublet and singlet in the unitary gauge as
(3) 
with GeV, and defining an additional small parameter
(4) 
we obtain, from Eqs. (24) the following mass matrices for the up and down quark sector:
(5) 
The charged lepton mass matrix is obtained from by replacing the couplings appropriately. Note that these mass matrices are the same as in Ref. [7], and as was shown there, good fits to the quark and charged lepton masses, as well as the CKM mixing angles are obtained by choosing , and all the couplings of order one. To leading order in , the fermion masses are given by
(6)  
while the quark mixing angles are
(7)  
2.2 Yukawa Interactions and FCNC
Our model has flavor changing neutral current interactions in the Yukawa sector. Using Eqs.(14), the Yukawa interaction matrices , , , for the up and down sector, for and fields are obtained to be
(8) 
with the charged lepton Yukawa coupling matrix obtained from by replacing .
(9)  
(10) 
with the charged lepton Yukawa coupling matrix obtained from by replacing .
There are several important features that distinguish our model from the proposal of Refs. [7, 8, 9]. i) Note, from Eqs.(5) and (8), in our model, the Yukawa couplings of to the SM fermions are exactly the same as in the SM. This is because the fermion mass hierarchy in our model is arising from . This is a distinguishing feature of our model from that proposed in [7, 8] where the Yukawa couplings of are flavor dependent, because the hierarchy there arises from . ii) In our model, we have an additional singlet Higgs boson whose coupling to the SM fermions are flavor dependent as given in Eq. (9, 10). Again, this is because the hierarchy in our model arises from . In particular, does not couple to the top quark, and its dominant fermionic coupling is to the bottom quark. This will have interesting phenomenological implications for the Higgs searches at the LHC. iii) We note from Eq. (58) that the mass matrices and the corresponding Yukawa coupling matrices for are proportional as in the SM. Thus there are no flavor changing Yukawa interactions mediated by . However, this is not true for the Yukawa interactions of the singlet Higgs as can be seen from Eqs. (5) and (9, 10). Thus exchange will lead to flavor violation in the neutral Higgs interactions.
2.3 Higgs Sector and Extra
The Higgs potential of our model, consistent with the SM and the extra symmetry, can be written as
(11) 
Note that after absorbing the three components of in and , and the pseudoscalar component of in , we are left with only two scalar Higgs, and . The squared mass matrix in the basis is given by
(12) 
where .
The mass eigenstates and can be written as
(13) 
where is the mixing angle in the Higgs sector.
In the Yukawa interactions discussed above, as well as in the gauge interactions involving the Higgs fields, the fields appearing are and , and these can be expressed in terms of and using Eq. (13).
The mass of the gauge boson is given by
(14) 
Note that the does not couple to any SM particles directly. Its coupling with the neutral scalar Higgs coupling) is also zero. The coupling to the SM particles will be only via dimension six or higher operators. Such couplings are generated by the vectorlike fermions in the model.
3 Phenomenological Implications: Constraints, predictions and new physics signals
Constraint on the mass of s: Experiments at LEP2 have set a lower limit of GeV for the mass of the SM Higgs boson. This is due to the nonobservation of the Higgs signal from the associated production . In our model, since the singlet Higgs can mix with the doublet , there will be a limit for depending on the value of the mixing angle, . For , the bound of applies also for [10]. However, can be lighter if the mixing is small.
Constraint on the mass of the : We have assumed that the extra symmetry in our model is spontaneously broken at the EW scale. But the corresponding gauge coupling, is arbitrary and hence the mass of is not determined in our model. However, very accurately measured properties at LEP1 put a constraint on the mixing to be or smaller [11, 12]. In our model, the does not couple to any SM particle directly. mixing can take place at the one loop level with the new vectorlike fermions in the loop. The mixing angle is
(15) 
where M is the mass of the vectorlike fermions with masses in the TeV scale. Even with , we get or less. Thus there is no significant bound for the mass of this from the LEP1 [13].
Higgs signals: As can be seen from Eq. (8), the couplings of the doublet Higgs to the SM fermions are identical to that in the SM, whereas the couplings of the singlet Higgs are flavor dependent. In particular, the singlet Higgs does not couple to the top quark, whereas its coupling to involve the flavor dependent factors respectively. This is, of course, in the limit of zero mixing between and . Including the mixing, these factors will be modified by the appropriate mixing factors. Thus our model will be distinguished from the SM by the fact that the Higgs couplings, in general, will be fermion flavor dependent.
Higgs decays: The couplings of the Higgs bosons and to the fermions and the gauge bosons can be obtained from Eqns. (8) and (9, 10). Because of the flavor dependency of the couplings of (and hence for both and via mixing) to the fermions, the branching ratios (BR) for to various final states are altered substantially from those in the SM. These branching ratios (BR) for to the various final states are shown in Figs. 1, 2 for the values of the mixing angle, and [14].
For , these BR’s are the same as for the SM. Note that for , the and the BR’s are enhanced substantially compared to the SM. This is due to drastic reduction for the mode due the almost cancelation in the corresponding coupling In particular, for , the effect is quite dramatic. For a light Higgs ( around GeV), the usually dominant mode is highly suppressed and the mode is enhanced by a factor of almost compared to the SM. This is to be contrasted with the proposal of Refs. [7, 8] in which the mode is reduced by about a factor of . Thus the Higgs signal in this mode for a Higgs mass of GeV gets a big enhancement making its potential discovery via this mode much more favorable at the LHC. Such a signal may be observable at the Tevatron for a Higgs mass GeV as the luminosity accumulates, but would require about 10 of data [15].
Another interesting effect is the Higgs signal via the for the light higgs. In the SM, this mode becomes important for the Tevatron search staring at GeV, where the BR to is approximate equal to that of . Currently Tevatron Run2 experiments have excluded SM Higgs mass in the range of to GeV (where the BR to is around percent) for this mode. In our model, for for example, note that this cross over between the mode and the mode takes place sooner than GeV. Thus the Tevatron will be more sensitive to lower mass range than in the SM, and will be able to exclude mass ranges much smaller than GeV.
Top quark physics: In the SM, mode is severely suppressed with a BR [16]. In our model, as can be seen from Eqs.(8) and (9, 10), although is zero at tree level, we have a large coupling for . This gives rise to significant BR for the mode for a Higgs mass of up to about GeV. If the mixing between the and is substantial, both decay modes, and will have BR . With a very large cross section , pb at the LHC, this can be a major discovery mode for higgs bosons at the LHC. Observation of signals for two different Higgs masses will also show clear evidence for new physics beyond the SM.
physics: Our model has a boson in the EW scale from the spontaneous breaking of the extra symmetry. As discussed before, since the mixing is very small or less, its mass is not constrained by the very accurately measured Z properties at LEP. Its mass can be as low as few GeV from the existing constraints. This does not couple to the SM particles with dimension 4 operators. It does couple to at tree level via the interaction. Thus it can be produced via the decay of (or if there is a substantial mixing between and ). This gives an interesting signal for the Higgs decays, , if allowed kinematically.
: In our model this decay gets a contribution from an FCNC interaction mediated by exchange. The amplitude for this decay is . Taking , , and with the couplings , we obtain the branching ratio, . Current experimental limit for this BR is [11], and thus this decay could be observed soon at the Tevatron as the luminosity accumulates.
Vectorlike fermions, productions and decays: Our model requires vectorlike quarks and leptons, both doublets, and singlets and , with masses at the TeV scale. These will be pair produced at high energy hadron colliders via strong interaction. For example, for a 1 TeV vectorlike quark, the production cross section at the LHC is fb [17]. We need several such vectorlike quarks for our model. So the total production cross section will be few hundred fb. These will decay to the light quarks of the same electric charge and Higgs bosons ( or ): . Thus the signal will be two high jets together with the final states arising from the Higgs decay. For a heavy Higgs, in the golden mode (, this will give rise to two high jets plus four bosons. In the case of a light , the final state signal will be two high jets plus 8 charged leptons in the final state (with each lepton pair having the invariant mass of the ).
4 A concrete model
We have constructed a concrete model giving rise to the phenomenological Lagrangian. In addition to the SM, the model has a local symmetry, and the () global symmetries. The global symmetries are slightly broken explicitly in the Higgs potential so that there are no unwanted Goldstone bosons. In addition to the SM fermions, the model has a complex scalar, , the flavon fields ’s, and several vectorlike quarks, both weak doublets and weak singlets. The details of their charge assignments under these symmetries, and how one obtains the interaction Lagrangian can be found in [6] .
5 Conclusions
We have presented a proposal in which only the top quark obtains its mass from the Yukawa interaction with the SM Higgs boson via dimension four operators. All the other quarks receive their masses from operators of dimension six or higher involving a complex scalar Higgs whose vev is at the EW scale. The successive hierarchy of light quark masses is generated via the expansion parameter , where . All the couplings of the higher dimensional operators are of order one. We are able to generate the appropriate hierarchy of fermion masses with this small parameter . Since is at the EW scale, the physics of the new scale, is at the TeV. Because of the new degree of freedom at the EW scale, we have an EW singlet neutral scalar , which gives rise to interesting new physics signals which can be tested at the LHC and at the Tevatron. There are new scenarios for the Higgs decays and the top quark physics. The model has a light which can be produced via the Higgs decays at the LHC, and can give rise invisible Higgs decays, displaced vertices for the decays, or multilepton final states arise from the decays, depending on the mass and lifetime of the . We have presented a model in which an effective interaction given in Eq. (1) can be realized. This requires the existence of vectorlike quarks and leptons, both EW doublets and singlets, at the TeV scale. These can be probed at the LHC. Their decays give rise to final states with 4 ’s or 4 ’s and interesting new physics signals at the LHC.
6 Acknowledgement
The work presented in this talk was done in collaboration with J. D. Lykken and Z. Murdock. This research was supported in part by Grant Numbers DOEFG0204ER41306 and DOEFG02ER46140.
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