Representation of fermions in the Pati-Salam model

In this paper, a representation of fermions in the Pati-Salam model is suggested. The semi-leptonic and beyond standard model flavor changing neutral currents of the Lagrangian in this representation of fermions are discussed. A pair of possible Cabibbo-Kobayashi-Maskawa and Pontecorvo-Maki-Nakagawa-Sakata matrices are defined. An effective Lagrangian for this model is given.


INTRODUCTION
The Pati-Salam model [1] is a grand unified theory (GUT) [1][2][3][4][5][6] and has the gauge group structure of SU (4) L × SU (4) R × SU (4 ), where SU (4) L × SU (4) R is the chiral flavor gauge group, and SU (4 ) is the color gauge group.The gauge group structure of the Pati-Salam model is beneficial in several aspects • The minimal simplicity group SU (5) GUT [3] encounters the issue of proton decay, and the modifications used to address the proton decay problem in SU (5) GUT always encounter issues of naturalness.
• If we use a semi-simplicity group as the GUT gauge group instead of a simplicity group, the standard model (SM) particles phenomena could be unified with the Pati-Salam gauge group SU (4) L × SU (4) R × SU (4 ), where SU (4) L × SU (4) R is the chiral flavor gauge group, and SU (4 ) is the color gauge group.While the gauge group SU (2) L × SU (2) R × SU (4 ) model could be used to reproduce the neutral current (NC) and charge current (CC) weak interaction phenomena, the six flavor fermions and flavor mixing phenomena are difficult to reproduce.
• "Lepton number as the fourth color" [1] is a clean and straightforward assumption when visualizing the fermions from a unified viewpoint.
• The fundamental representations of SU (4) are 4, 6 and 4. In a GUT, the fermions always fill in the fundamental representation of a gauge group.We know that fermions have six flavors and four colors, and each fermion has corresponding antifermion.Thus, fermions (antifermions) can be filled in the Pati-Salam gauge group fundamental representation 4 × 6 ( 4 × 6).
• Dirac matrices are 4 × 4 matrices.If we do not add (or reduce) the degrees of freedom by hand, the fermions should fill in the 4 × 4 matrix.
The original fermions representation in the Pati-Salam model [1] includes only two families of quarks and leptons.In this paper, however, we suggest a representation of fermions in the Pati-Salam model comprising all three families of quark and lepton states as the eigenstates of Lagrangians.We discuss the fermion-antifermionboson vertexes new physics of semi-leptonic processes transpoted by X bosons and beyond standard model flavor changing neutral currents (FCNCs) processes transported by neutral bosons Y , based on the novel representation of fermions.We also present a possible construction of the CKM and PMNS matrices based on this representation of fermions.Finally, we illustrate an effective total Lagrangian density for this model.

REPRESENTATION OF FERMIONS
The well-established Pati-Salam model [1] has the following gauge group where SU (4) L and SU (4) R are the chiral flavor gauge groups, SU (4 ) is the color group.
Fermions have six flavors of quarks and leptons.If we gauge the flavor symmetry according to SU (6) group, the fermions should fill in a 4 × 6 matrix.The SU (6) flavor symmetry will engage with nine gauge bosons at least that transport flavor gauge interactions.To date, the experimental data only showed us three flavor gauge interaction bosons, which are W + , W − and Z.The problem relates to how to reduce the nine flavor gauge bosons naturally to three, reveal the Standard Model interaction vertexes and reproduce flavor mixing phenomena.Furthermore, it will be hard to reproduce the Gell-Mann-Nishijima formula and flavor mixing phenomena, and the SU (6) × SU (4 ) gauge group is not minimal for GUT.
This SU (4) L × SU (4) R flavor gauge group symmetry restricts the representation matrix of fermions to 4 × 4 matrix.For this 4 × 4 fermion matrix, it needs to be established whether the flavor degrees of freedom will take on the shape of a column or row?A minimal coupling Lagrangian is constructed as follow for the color and flavor interaction to answer this question where f, g ∈ R are coupling constants.V µ and W µ are 4× 4 Hermitian matrices and can be decomposed as follows where T a (a = 1, 2, • • • , 15) are generators of SU (4) and an example can be found in Appendix A, V a µ and W a µ are gauge bosons.The first term in Lagrangian (2) is a kinematic term.The flavor interaction can be chiral decomposed but the color interaction cannot.We observe that the second term in Lagrangian ( 2) is difficult to decompose due to chiral symmetry, but the third term can be decomposed (the proof is presented in Appendix ) as follow where the chiral fermions are defined Accordingly, the second term in Lagrangian (2) describes the SU (4 ) color gauge interaction, and the third term in Lagrangian (2) describes the SU (4) L × SU (4) R chiral flavor gauge interaction.We then derive that the column of the 4 × 4 fermion matrix corresponding to color and the row corresponding to flavor.Such as "lepton number as the fourth color", it was then easy to fill four colors of fermions, i.e., R, G, B and L, into the four rows of fermion matrix.The next approach was to derive how to fill the six flavor fermions into the four columns of the fermion matrix?Reminding the six flavor fermions were divided into three families, and each family included two flavor fermions.The action in the path integral formulation of quantum field theory is a phase each phase term should with 0-dimension and 0-charge, and the fermion matrix should result in the model being anomaly free.Consider that the fermions in quantum field theory are the operator valued field, and the quantum states are the eigenstates of operator valued field.
In quantum mechanics, one operator can correspond to several eigenstates.Then, we suggest a representation of fermions where

GAUGE BOSONS IN THE MINIMAL COULPING LAGRANGIAN
The possibility of chiral decomposition infers that W a µ are gauge bosons transporting flavor gauge interactions and V a µ transporting color gauge interactions.We will discuss the decomposition of the Lagrangian of the flavor and color interactions in detail using the minimal coulping model (2).
The gauge boson bears the exchange of quantum numbers charge.For two different fermion-antifermion-boson vertexes, when the exchange of the charge is the same, the quantum numbers of two gauge bosons in two fermionantifermion-boson vertexes are the same, except the possibility of masses difference (thanks the comments from anonymous referees point out that even through the quantum number of the particles are the same, the masses of the particles might not the same).The Z boson is a charge free gauge boson and transports weak NC in the SM, Z boson should on the diagonal of matrix W µ , i.e., then the Lagrangian can be decomposed as follows (see Fig. 1) where According to the fermion matrix and Lagrangian charge free assumption, it is easy to find that W ± µ in this model are Furthermore, W ± transports the CC in the weak interaction.Then, the Lagrangian, i.e., W a µ T a + {L → R} can be decomposed as follows (see Fig. 2) The electric charge of W + and W − are 1 and -1, respectively.
There are new physics chiral flavor processes described by the Lagrangian For example, the predicted beyond SM FCNCs [50-52] The W µ matrix is The corresponding electric charge matrix of W µ is FIG. 3. The fermion-antifermion-boson vertexes of Y are derived by Lagragian (12), where all three external legs of the vertexes are momentum in.
Color SU (4 ) processes We selected V 15 µ as the the photon.Then the vertexes of photon from Lagrangian (2) are written as Except the neutrinos, the electric charge number preseding each flavor fermion Lagrangian term is correct.
As an example, the 2 3 preceding the Lagragian term ūC γ µ V 15 µ u C is the electric charge number of quark u.The experiments show that the neutrino is charge free, such that the neutrino should satisfy the formulas Under the restriction (18), the Lagrangian of neutrinos and photon interaction vertexes degenerates into Then the fermion-antifermion-boson vertexes about photon γ on this minimal coupling model are show in Fig. 4.

The gauge bosons
µ are gluons and transport color SU (3 ) strong interaction and reveal QCD.
There are exotic semi-leptonic processes [40] trans- ported by X ±C µ particles and the related Lagrangian is where The related fermion-antifermion-boson vertexes about X bosons are show in Fig. 5.The Lagrangian charge free restriction derives that the charge of X −C and X +C particles are − 1 3 and 1 3 , respectively.The V µ matrix is where G CC µ (C, C = R, G, B = 1, 2, 3) are gluons and V 15  µ is photon.Then, the electric charge matrix of V µ is 6. Amplitude of quark pair slips to lepton pair is zero because of electric charge conservation.The qC , qC and l, l are particular quarks and leptons.The vertexes in the diagram are described by Lagrangian (2), especially Lagrangian (20).
Three examples of the nonzero semi-leptonic Feynman diagrams in the tree level amplitudes are shown in Fig. 9, where the Fig. 9a and b are the t-channel and u-channel of In addition, the Fig. 9c is the s-channel of the quark lepton interaction The masses of X ±C bosons must have been very large because the s, t and u-channels were still not observed.The amplitudes in Fig. 6 are zero at least on the oneloop level in the model described by Lagrangian (2) because Note that all external fermions in Fig. 6 are not antiparticles.The electric charge is not conserved in the process shown in Fig. 6 such that the total amplitude M total = 0. Which means electric charge conservation avoids quark pair slips to lepton pair in minimal coupling model (2).

FLAVOR MIXING
The left-handed flavor eigenstates of d, s, b quark states can be defined as follows: where |d LC , |s LC and |b LC are flavor eigenstates of d, s and b quarks with left-handed chirality and C color, respectively.The kinematic term of fermions in the Lagrangian ( 2) is The kinematic term of fermions can be decomposed as follows: The left-handed mass eigenstates of the d, s and b quarks are The CKM matrix is The right-handed d, s and b quark states can be defined after L → R.
Similarly, the left-handed flavor eigenstates of neutrinos are The left-handed mass eigenstates of neutrinos are The PMNS matrix is The right-handed eigenstates of neutrinos can be defined similarly after L → R.

EFFECTIVE TOTAL LAGRANGIAN AND GAUGE INVARIANCE
An effective total Lagrangian for color, flavor and Higgs interactions is where φ is the Higgs field; V (φ) is Higgs potential; f, g, ξ ∈ R are coupling constants and the gauge field strength tensors are The second line of Lagrangian ( 43) represents Yang-Mills theory terms, and the third line is magnetic moment terms.Lagrangian ( 43) is invariant under local gauge transformations of color space and flavor space rotation Ũ and U , respectively, where such that

GAUGE ANOMALY
The Lagrangian (43) is flat space-time version of Yang-Mills theory (Pati-Salam type) in curved space-time and Einstein-Cartan gravity [53,54].The curved version theory has deep motivation from point of views of logic and geometry, which derived from square root metric and self-parallel transportation principle and quantized by sheaf quantization and path integral quantization.The anomaly in quantum field theory always means a symmetry is preserved in classial theory but violated in quantum version.The golbal symmetry anomaly might be accessed by quantum field theory, but the locally gauge symmetry anomaly (gauge anomaly) is belived to be a consistence condition for a gauge theory.We have to check the anomaly free condition for Pati-Salam model with this representation of fermions.
In 4-dimenional space-time, the quantum gauge anomaly free condition can be checked by triangle Fey- mann diagram in Fig. 7.The amplitude of Fig. 7 propotional to then for SU (4) L × SU (4) R chiral Yang-Mills theory, the current conservative equation has the formulation such that the indices bc satisfy commutation and anticommutation relations The analyse about SU (4 ) color gauge Yang-Mills theory is similar.Note that a fermions loop cannot interact with flavor and color gauge bosons in one triangle anomaly Feymann diagram at the same time.So the SU (4) L × SU (4) R × SU (4 ) Pati-Salam model is anomaly free.

MONOPOLE AND THE TOPOLOGY OF SPACE-TIME
As an example, we choose SU (4) L × SU (4) R flavor gauge bosons to analyse the problem of monopole.We can combine the SU (4) L × SU (4) R minimal coupling, Yang-Mills and tolopogical terms of flavor gauge bosons as follow which related with monopole where are eletro-magnetic dual gauge strength tensor of F µν .The Euler-Lagrangian equation of W µ for the Lagrangian ( 54) is We decompose the equation (56) as follow where J ν e is electro current and J ν m is monopole current.The fundamental thing in quantum field theory is action where ω is volume form and M is the base manifold of space-time.Note that the monopole related topological term in ( 54) is the second Chern class, the action S about the topological term only relies on the topological structure of the manifold M and propotional with the second Chern number We can easily calulate the second Chern number C 2 with M equals topologies S 4 and S 1 × S 3 Which means, for base manifold M with topology S 4 , the monopole about flavor gauge bosons W µ , there are monopoles currents; for base manifold M with topology S 1 ×S 3 , the monopole currents are depressed.The analyse in this section could apply equally to SU (4 ) color gauge bosons V µ , and SU (5) GUT also.

CONCLUSIONS AND DISCUSSION
Based on the gauge group SU (4) L × SU (4) R × SU (4 ) of Pati-Salam model, a representation of fermions is suggested in this paper.The boson-fermion-antifermion vertexes bring by the SU (4) L × SU (4) R chiral flavor and the SU (4 ) color gauge group were discussed.The electric charge of each particle was consistently defined, and a pair of possible CKM and PMNS matrix formulations were illustrated.An effective total Lagrangian of the model was given.
The experimental data restricts the masses of particles X ±C , Y 1 , Y 1 * , Y 2 and Y 2 * were superheavy.How the masses be generated for these particles requires further discussions.
We thank prof.Hong-Fei Zhang, Zhao Li, Cai-Dian Lv, Ming Zhong, Yong-Chang Huang and Chao-Guang Huang for valuable discussions.If without prof.Zhao Li kind helps, this work is impossible.
Appendix A: Generators of SU (4) group Generators of the SU (4) group are as follows Appendix B: The possibility of the chiral symmetry breaking of flavor gauge interaction γ 5 = iγ 0 γ 1 γ 2 γ 3 such that The gamma matrices satisfy The cross section of Fig. 9c is given as follows:

2 , 3
are color indices; u, c and t are operator valued fields of three flavor quarks; e, µ and τ are operator valued fields of the electron, mu and tau.Furthermore, ν and d C are operator valued fields of neutrinos and d family quarks.Additionally, the |ν e , |ν µ , |ν τ neutrino states and |d C , |s C , |t C quark states are eigenstates related to flavor interaction Lagrangian terms containing ν and d C , respectively.

FIG. 1 .
FIG. 1.In this suggested representation of fermions of Pati-Salam model, the Lagrangian (10) gives us the fermionantifermion-boson vertexes of weak Z boson interaction with left handed fermions.For right handed fermions, the L symbol should be alternated by R.

FIG. 2 .
FIG. 2. The fermion-antifermion-boson vertexes of W boson derived by Lagrangian(11), where all three external legs of vertexes in this figure are momentum in.
not yet being observed and the mass generating mechanism of gauge bosons Y 1 , Y 2 , Y 1 * and Y 2 * is interesting.The electric charges of gauge bosons Y 1 , Y 2 , Y 1 * and Y 2 * are 0. The fermion-antifermion-boson vertexes about Y are show in Fig. 3. Two examples of beyond SM tree level FCNCs are in Fig. 8.