A Model of Four Generation Fermions and Cold Dark Matter and Matter-Antimatter Asymmetry
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, P. R. China
Abstract: I suggest a practical particle model as an extension to the standard model. The model has a TeV scale symmetry and it contains the fourth generation fermions with the TeV scale masses, in which including a cold dark matter neutrino. The model can completely account for the fermion flavor puzzles, the cold dark matter, and the matter-antimatter asymmetry through the leptogenesis. In particular, it is quite feasible and promising to test the model in future experiments.
Keywords: new model beyond SM; fermion flavor; dark matter; leptogenesis
PACS: 12.60.-i; 12.15.Ff; 14.60.Pq; 95.35.+d
The precise tests for the electroweak scale physics have established plenty of knowledge about the elementary particles [1, 2]. The standard model (SM) has been evidenced to be indeed a very successful theory at the electroweak energy scale. Nevertheless, there are a lot of the issues in the flavor physics and universe observations for which the SM is not able to account . During the past more decades a series of experiment results of physics and neutrino physics have told us a great deal of the detailed information of flavor physics . What are paid more attention are some facts in the following. Firstly, the fermion mass spectrum emerges a very large hierarchy. The quark and charged lepton masses range from one MeV to one hundred GeV or so , while the neutrino masses are only at the Sub-eV level . Secondly, the quark flavor mixing pattern is distinctly different from the lepton one. The former are only three small mixing angles , whereas the latter has bi-large mixing angles and a bit small but non-zero . In addition, the violation in the quark sector has been verified to be non-zero but very small , however, it is in suspense whether the violation in the lepton sector vanishes or not . Thirdly, that the light neutrino nature is Majorana or Dirac particle has to be further identified by the experiments such as . On the other hand, the astrophysics observations and researches lead to some impressive puzzles in the universe, in particular, the genesis of the matter-antimatter asymmetry and the original nature of cold dark matter . Finally, a very important and unsolved problem is whether flavor physics are truly connected with the baryon asymmetry and/or cold dark matter in the universe or not? All the problems have a great significance for both particle physics and the early universe , so they always attract great attention in the experiment and theory fields.
The researches for the above-mentioned problems have motivated many new theories beyond the SM. The various theoretical approaches and models have been proposed to solve the intractable issues . For instance, the Froggatt-Nielsen mechanism with the family symmetry can account for mass hierarchy , the discrete family group can lead to the tri-bimaximal mixing structure of the lepton mixing matrix , the non-Abelian continuous group is introduced to explain the neutrino mixing , the model with the family group can accommodate the experimental data of the quarks and leptons by the fewer parameters , some grand unification models based on the symmetry group can also give some reasonable interpretations for fermion masses and flavor mixings . In addition, some suggestions of the baryon asymmetry and cold dark matter are very constructive . Although these theories are successful in explaining some specific aspects of the above problems, it seems very difficult for them to solve the whole problems all together. Indeed it is especially hard for a model construction to keep the principle of the simplicity, economy and the less number of parameters, otherwise the theory will be excessive complexity and incredible. On all accounts, a better theory beyond the SM has to be confronted with the integration of particle physics and the early universe, in other words, it should simultaneously account for flavor physics, the baryon asymmetry and cold dark matter. Of course, it is still a large challenge for theoretical particle physicists to find a desirable theory to uncover these mysteries of the universe .
In this work, I construct a practical model to integrate three parts of the fermion flavor, cold dark matter and leptogenesis. The model has the local gauge symmetries . The subgroups only appear above the TeV scale, and then are spontaneously broken to the supercharge subgroup below that scale. Besides the SM particles, some new non-SM particles are introduced into the model. They are the fourth generation quarks and leptons, two vector gauge fields related to , and three scalar fields, which are respectively one neutral singlet, one charged singlet and one symmetric triplet under . The breaking is accomplished by the neutral singlet scalar developing a non-vanishing vacuum expectation value (VEV) at the TeV scale. This breaking leads that one of the two new gauge fields obtains a TeV scale masse by the Higgs mechanism, in the meantime, the fourth generation neutrino is given rise to a TeV scale mass. The fourth generation neutrino has some unique natures in the model, which ensure that it is a cold dark matter particle. All of the SM particle masses are generated after the electroweak breaking. The triplet scalar field takes part in the symmetry breakings but it develops only a tiny VEV by virtue of its very heavy mass. This is a real source of the tiny masses of the light neutrinos. The leptogenesis is implemented by the charged singlet scalar decaying into a SM charged lepton and a cold dark matter neutrino. The decay process simultaneously satisfies the three conditions of the violation, violation and being out-of-equilibrium. This mechanism can naturally generate the asymmetry, subsequently it is converted into the baryon asymmetry through the electroweak sphaleron process . In the model, the flavor physics is intimately associated with the cold dark matter and the matter-antimatter asymmetry in the universe. The model can completely accommodate and fit all the current experimental data of the fermion masses and mixings, the cold dark matter and the baryon asymmetry, furthermore, it also predicts some interesting results. Finally, the model is feasible and promising to be tested in future experiments. I give some methods of searching the non-SM particles of the model in the experiments at the LHC .
The remainder of this paper is organized as follows. In Section II I outline the model. In Sec. III, the model symmetry breakings, the particle masses and mixings are introduced. In Sec. IV, the leptogenesis and cold dark matter are discussed. In Sec. V, I give the detailed numerical results. Sec. VI is devoted to conclusions.
The local gauge symmetries of the model are the direct product groups of . The subgroup symmetry of only appears above the TeV scale, and it will be broken to the supercharge subgroup below that scale. The model particle contents and their gauge quantum numbers are in detail listed in the following,
All kinds of the notations are self-explanatory as usual. and are two new vector gauge fields related to the gauge subgroups and , respectively. The last two numbers in the brackets are exactly the and charges. The fermion subscript indicates the first three generation fermions, in addition, the model newly includes the fourth generation quarks and leptons. It should also be noted that the fourth generation left-handed fermions are all singlets under . The right-handed neutrinos are absent in (1). The reason for this is that the neutral fermions are considered as Majorana particles, so the corresponding are not independent fields but rather determined by the relation , in which is a charge conjugation matrix. Besides the SM doublet Higgs field, two complex singlet scalar and , and a symmetric triplet scalar are introduced in the model. After the gauge symmetry breakings, and have respectively a zero electric charge and a negative electric charge. These scalar field representations are such as
In addition, will also be referred. In brief, all the non-SM particles in (1) compose the new particle spectrum, and they play key roles in the new physics beyond the SM. In general, the new particle masses are the TeV scale or above it, so they should appear in the TeV scale circumstances, for example, in the early universe.
Under the model gauge symmetries, the invariant Lagrangian of the model is composed of the three parts in the following. Firstly, the gauge kinetic energy terms are
where stands for all kinds of the fermions in (1). The covariant derivative is defined as
where and are Gell-Mann and Pauli matrices, and are respectively the charge operators of and , and are four gauge coupling constants. After the gauge symmetry breakings, some gauge fields will generate their masses and mixings by the Higgs mechanism.
Secondly, the model scalar potential is given by
where all kinds of the coupling parameters are self-explanatory. In general, the size of the coupling coefficients are . For , and are original masses of the particles and , respectively. Their values are TeV and TeV in the model. For , both the singlet and the doublet will develop non-zero VEVs. is responsible for the breaking , in succession, completes the electroweak breaking . The triplet also takes part in the two breakings because it is involved in the couplings of the last term in (5). However, it is about to be seen that has only a very tiny VEV due to the heavy , therefore only plays a secondary role in these breakings but it plays a key role in the generation of the tiny neutrino masses. Lastly, the singlet does not participate in any breakings because of its VEV vanishing. In comparison with the SM Higgs sector , the phenomena of the model scalar sector are variety and more interesting.
Thirdly, the model Yukawa couplings are written as
These Yukawa couplings have apparently uniform and regular frameworks. are exactly the masses of the fourth generation quarks and charged lepton, which are directly allowed by the model symmetries, however, are about few TeV based on the model theoretical consistency. The coupling coefficients, , are chosen some real numbers since their complex phases can be absorbed by the redefined fermion fields. Since every coupling coefficients scale the order of magnitude of itself term, they are arranged such a hierarchy as in the model. The coupling matrices, , characterize the flavor mixings among the four generation fermions. At present we are lack of an underlying understand for the fermion flavor origin, however, it is believed that some flavor family symmetry is embedded in an underlying theory at a certain high-energy scale, but it is broken at the low-energy scale. The coupling matrices should imply some information of the flavor symmetry and its breaking. Therefore I suggest that the flavor structures of the coupling matrices have such a style as
The flavor structures are both simple and reasonable, in particular, there are only few flavor parameters. The size of the flavor parameters are normally . The majority of their complex phases can be removed by the redefined fermion fields. The remaining complex phases will become the sources of the violations in the quark and lepton sectors. In (6), the couplings have evidently a full flavor symmetry among the first three generation fermions but they are relatively smaller. The couplings only keep such a discrete symmetry as between the second and third generation fermions. The couplings between the first three and fourth generation fermions, , break the flavor symmetry , but they are relatively bigger. Lastly, the couplings involving the charged scalar , , are the smallest ones. After the gauge symmetries are broken spontaneously, all kinds of the fermion masses, , , , will be generated by the corresponding couplings and the VEVs of , respectively. Finally, I in particular point out that the matrices can be diagonalized by the unitary matrix as follows
The mixing angles of are . It evidently distinguishes from the tri-bimaximal mixing matrix . For , the first and second eigenvalues are approximately the same size, and the third one is the biggest. This property of (8) plays a key role in the neutrino mass and mixing in the model.
In summary, the above features of the particle contents and Lagrangian are very important not only for the particle masses and mixings, but also guarantee the cold dark matter and leptogenesis in the model. In the following sections of the paper, it is about to be seen that has unique natures and plays a special role in the model. It is actually a cold dark matter particle. The leptogenesis is really implemented by the decay . In a word, the above contents form the theoretical framework of the model.
III. Symmetry Breakings and Particle Masses and Mixings
The gauge symmetry breakings of the model go through two stages. The first step of the breakings is , namely the breaking, in succession, the second step is , i.e. the electroweak breaking. The former is achieved by the real part component of developing a non-vanishing VEV at the TeV scale, while the latter is accomplished by the neutral component of developing a non-vanishing VEV at the electroweak scale. In addition, the neutral component of also develops a very tiny but non-zero VEV owing of the last term couplings in (5). The scalar vacuum structures and the conditions of the vacuum stabilization are easy derived from the scalar potential (5). The details are as follows
The stable conditions include , , and . In addition, should be sufficient small so that is one order of magnitude larger than , in this way, this ensures that the breaking precedes the electroweak breaking. in (9) is a effective mass of the particle when the breakings are completed (see (10)). Provided that TeV, TeV, GeV, and , thus this naturally leads to eV, consequently, gives the tiny neutrino masses. Thus it can be seen that the tiny nature of the neutrino masses essentially originates in the very heavy in the model. In this sense, this is a new form of the seesaw mechanism . Finally, the field has a vanishing VEV, so it does not actually participate in the breakings. In short, all the conditions are not difficult to be satisfied so long as the parameters are chosen as some suitable values.
After the model gauge symmetry breakings are over, the following massive scalar bosons, , , , , appear in the scalar sector. Their masses and mixings are such as
where is the mixing angle between and . Provided that , then , thus the two neutral boson masses are approximately and . In a similar way, there is also a very weak mixing between and , or between and . However, these very weak mixings in the scalar sector can all be ignored throughout. and are respectively the effective masses of the particles and . Provided that , then . In view of , obviously, there is . At present, has been measured by the LHC , it’s value is GeV. However, the model predicts that and are about TeV, it are quite feasible to find the two bosons at the LHC, but the particles are too heavy to be detected.
In the gauge sector, the gauge symmetry breakings give rise to masses and mixings for some of the vector gauge bosons through the Higgs mechanism. The breaking procedure of is such as
where are respectively the gauge coupling constant, gauge field, supercharge operator of , is an obtained mass neutral gauge field, and is a new charge operator related to . There are two gauge parameters and in (11), however, is not a free parameter but determined by the electroweak relation , only the mixing angle is a free parameter. In addition, the last equation in (11) implies , so should be few TeV or so. It should also be pointed out that the mixing between and , which is the SM weak neutral gauge boson, is very small. Their mixing angle is given by
Because of and , this mixing can indeed be ignored.
Below the electroweak scale, the Yukawa sector becomes clear and simple since the fourth generation heavy quarks and charged lepton have decoupled. After they are integrated out from (6), the effective Yukawa coupling matrices of the three generation quarks and charged lepton are given by
Each term physical meaning is very explicit. According to the standard procedures, in (6) the symmetry breakings give rise to all kinds of the fermion masses as follows
Obviously, there is the mass hierarchical relation . In addition, the hierarchical coefficients, , will lead to the hierarchical masses of the three generation quarks and charged lepton. On the other hand, there is not such a term as in , so is distinguished from . This is a primary source that the lepton flavor mixing is greatly different from the quark one. In short, the interesting features of the fermion mass matrices dominate the fermion masses and flavor mixings.
Finally, the fermion mass eigenvalues and flavor mixing matrices are solved by diagonalizing the above mass matrices. The quark and charged lepton mass matrices are hermitian, while the light neutrino mass matrix is symmetry. Therefore, the mass matrix diagonalizations are accomplished as such
In the light of the characteristic structures of , the mass eigenvalues of the quarks and charged lepton are certainly some hierarchy, and the flavor mixing matrices are all closed to an unit matrix. In contrast, an exact solution of the diagonalization can be given by use of (8) as
Obviously, is completely different from , moreover, the two mass-squared differences can explain the neutrino data very well. The above results are convenient for the following numerical analysis. The flavor mixing matrix in the quark sector and one in the lepton sector are respectively defined by [27, 28]
The two unitary matrices and are parameterized by the standard form in particle data group . are two Majorana phases in the lepton mixing. All kinds of the mixing angles and -violating phases can be calculated numerically. Finally, all of the results can be compared with the current and future experimental data.
IV. Cold Dark Matter and Leptogenesis
The model can naturally and elegantly account for the cold dark matter and leptogenesis after the model symmetry breakings are completed. The fourth generation Majorana neutrino own some unique properties. It has been seen from the model lagrangian that has only the three types of the couplings, , , and . Provided that the mass order , the only decay channel of is via an off-shell boson because there are some weak mixings between the fourth generation quarks and the first three generation ones. If the coupling coefficients are in the last line of (6), then the decay width is estimated as GeV. In other words, the lifetime is actually two orders of magnitude longer than the now age of universe, therefore it becomes a relatively stable particle in the universe. On the other hand, a pair of can annihilate into other particle pair. The annihilate processes are mediated by either the gauge boson or the scalar boson . Because the mass is derived from the breaking, should be around one TeV. Consequently, the Majorana neutrino is genuinely a weak interactive massive particle (WIMP), of course, it also belongs to one of the fewer species of particles which can survive from the early universe to the now epoch. Therefore is a good candidate of the cold dark matter .
The annihilate channels of have two ways. The principal annihilate process is that a pair of annihilate into all kinds of the SM fermion pairs by the intermediate gauge boson , as shown in the figure (1).
The other annihilate process is that two neutrinos annihilate into two Higgs bosons by the TeV scale boson mediating. Because is smaller, the cross section of the latter case is normally far smaller than one of the former except for some Breit-Wigner resonance points, so I can ignore the latter and only consider the former. The annihilate cross section of the figure (1) is calculated as follows
where is the squared center-of-mass energy, is the velocity of in the center-of-mass frame. The sum for count all kinds of the SM fermions who are permitted by kinematics. In fact, all of the relatively lighter in (18) can been approximated to zero except for the relatively heavier . On the basis of WIMP, the relic abundance of in the current universe is determined by the annihilation cross section as such
where is the relative velocity of the two annihilate particles. In addition, the heat average (19) can be calculated by , in which is the freeze temperature of . A rough estimate is as follows. Because of , a weak cross section is naturally obtained as , eventually, it leads to , which is closed to the observation value.
The baryon asymmetry through the leptogenesis can be implemented in the model. On the basis of the relevant couplings in (5) and (6), the main decay channel of the charged scalar boson is (not including a heavy ), in addition, can be ignored because its decay branching ratio is smaller. The figure (2) draws the tree and loop diagrams of , of course, its conjugate process has the corresponding diagrams.
However, this decay process has simultaneously three items of the notable characteristics. Firstly, the decay evidently violates one unit of the quantum number, namely . Secondly, because there is a -violating source in the leptonic Yukawa sector, a asymmetry of the decay is surely generated by the interference between the tree diagram and the loop one. The asymmetry is defined and calculated as
where provided with . Thirdly, the decay is really an out-of-equilibrium process. Provided that the coupling coefficient as before, then the decay rate is far smaller than the Hubble expansion rate of the universe, namely
where has been approximated to zero, GeV, and is the effective number of relativistic degrees of freedom. In brief, the decay process can indeed satisfy Sakharov’s three conditions . Therefore, this mechanism can naturally generate an asymmetry of at the TeV scale. It can be seen from (20) that the asymmetry linearly depends on the three quantities , in particular, the major contribution comes from , namely the process by the inner mediating. In addition, has no relation with though the decay rate depends on . In the model, there are , thus TeV will give .
The above asymmetry occurs at the TeV scale. At that time the heavy and have been the non-relativistic particles, but the produced leptons are truly the relativistic states, moreover, their energy are normally more than GeV. Therefore the sphaleron electroweak transition can smoothly put into effect . Consequently, the asymmetry will eventually be converted into an asymmetry of the baryon number through the sphaleron process. According to the standard discussions, the baryon asymmetry is determined by
where is a ratio of the entropy density to the photon number density, is a coefficient of the sphaleron conversion, stands for the asymmetry, which is related to by the expression in the parentheses. is a dilution factor, it can actually be approximated to on account of the very weak decay rate. At the TeV scale, only the SM particles are the relativistic statuses, whereas the non-SM particles are the non-relativistic statuses, so exactly equal to the effective number of degrees of freedom of the SM particles, namely . In short, the baryon asymmetry can be calculated by the relations of (20) and (22). The model can achieve the leptogenesis at the TeV scale.
V. Numerical Results
In the section I present the model numerical results. The model involves a number of the new parameters besides the SM ones. In the light of the III and IV section discussions, the parameters involved in the numerical calculations are collected together in the following. The gauge sector has the gauge coupling and the mixing angle . The scalar sector includes the three VEVs, , the two scalar boson masses , and the scalar coupling . The Yukawa sector has , all kinds of the coupling coefficients, and the flavor parameters, see (6) and (7). Among which, the three parameters, , in fact belong to the SM parameters. Their values have been determined in the electroweak scale physics and by the recent LHC experiments , namely . In addition, the three parameters, , also belong to the fundamental parameters in the model. Based on an overall consideration, the above six parameters are fixed throughout to the following values,