De-Sheng Li. Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold

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Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold

De-Sheng Li1,2,3* Review Article 1College of Physics, Mechanical and Electrical Engineering, Jishou University, China Volume 5 Issue 2 2Interdisciplinary Center for Quantum Information, National University of Defense Technology, Received Date: July 07, 2021 China Published Date: August 11, 2021 3 Institute of High Energy Physics, Chinese Academy of Sciences, China DOI: 10.23880/psbj-16000186

*Corresponding author: De-Sheng Li, College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, P. R. China, Email: [email protected]

Abstract

There is a UU(4') × (4) -bundle on four-dimensional square root Lorentz manifold. Then a Pati-Salam model in curved space-time (Lagrangian) and a gravity theory (Lagrangian) are constructed on square root Lorentz manifold based on self-parallel transportation principle. An explicit formulation of Sheaf quantization on this square root Lorentz manifold is shown. Sheaf quantization is based on superposition principle and construct a linear Sheaf space in curved space-time. The

in Standard Model (SM) of particle physics and Einstein gravity are found in square root metric and the connections of bundle. transition amplitude in path integral quantization is given which is consistent with Sheaf quantization. All particles and fields

±c Thegravity interactions theory is betweenEinstein-Cartan particles/fields kind with are torsion. described There by are Lagrangian new particles, explicitly. right There handed are neutrinos, few new physics dark photon, in this Fiona,model. TheX

01212 and YYYYY,,,,**.

Keywords: Lorentz manifold; Pati-Salam model; Curved space-time; Gravity theory; Yang-Mills theory; Sheaf quantization

Introduction of extra dimensions. Four-dimensional pseudo-Riemann geometry with signature (−+++,,,) , Lorentz manifold, is the geometry Mills theory takes gauge invariance as its basic principle and background of the general relativity, space-time is described to beLater, the theoretical the Yang-Mills framework theory of electromagnetic, [1] was confirmed. weak Yang- and strong interaction in Standard Model (SM) of particle physics the curve of space-time. In general relativity, the geodesic [2-7]. The Yang-Mills theory is the theoretical framework by the metric, and the gravitational field is described as equation describes the trajectories of free particles, and the of SM and has a good correspondence with the complex Einstein equation determines how matter curves space-time. At the last life time of Einstein, he attempted to establish a new geometry unifying electromagnetic interaction and theorystructure also, group except G fiber that bundle the G theorystructure [8]. Generalgroup of relativity general gravity. This idea was developed by Weil into the early idea can actually be rewritten in the framework of fiber bundle of gauge invariance and by the Kaluza and Klein into the idea the tangent and cotangent bundles. Another way to build relativity is real, and the corresponding fiber bundles are

Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold Phys Sci & Biophys J 2 Physical Science & Biophysics Journal

extraunified dimension field theory were is made.introducing Is it possible extra dimensions to fuse the tangent to give (cotangent)all fields their bundle geometric of general positions. relativity And lotswith of the attempts complex in orthonormal frames describe µgravitational∂ field (Figure 1). θθaa( xx) = ( ) µ . structure group G-bundle of Yang-Mills theory? ∂x spinors through making square root of the Klein-Gordon equation,Inspired we byresearched the Dirac’s the way four-dimensional of finding his equation square rootand Lorentz manifold, which similar with the papers in Clifford algebra or Clifford bundle [9-52], spin-gauge theory in Riemann-Cartan space-time [53,54], sedenion [55] and Einstein-Cartan theory [56-58] etc. Four-dimensional square root Lorentz manifold has extra UU(4') × (4) principal bundle than Lorentz manifold. Two Lagrangians based on four-dimension square root Lorentz manifold are constructed which describe a U(4') × UU(4) × ( 4) Pati- LR Figure 1: Picture visualization of geometry background of Salam model in curved space-time and a gravity theory, (a) general relativity (b) Yang-Mills theory and (c) square respectively. In the Pati-Salam model [59], the ' is SU (4 ) root metric. (a) Geometry background of general relativity color group with “lepton number as the fourth color”, and the is pseudo-Riemann geometry, which is a smooth curved SU44× SU ( )LR( ) manifold. (b) Geometry background of Yang-Mills theory Einstein kind with torsion. We realize an explicit formulation of Sheaf quantizationis chiral [60-76] flavor scheme group. which The gravity consistent theory with is gauge group of Yang-Mills theory. (c) Square root metric path integral quantization. The particles spectrum on this geometryis complex in G-bundlefour-dimension with flat has base everything manifold, of Gpseudo- is the model is analyzed. Riemann geometry and extra UU(4') × (4) -bundle.

Geometry and Lagrangian

The notations are introduced here. abcd,,, represent The definition of gamma matrices is γγa b+= γγ b a2. η ab I (4) frame indices and abcd,,, are equal to 0,1,2,3 . µ,,v ρσ , 44× represent coordinates indices and µ,,v ρσ , are equal to 0,1,2,3 . α represent group indices with α equals to The Hermiticity conditions for gamma matrices are 0,1,....,15. i, jklm , ,, are equal to 1,2,3,4. C is quarks color a b†† b a ab and equals to RGB, ,( 1, 2, 3) . k is Sheaf space index. Repeated γγ+= γ γ 2,II44× (5) indices are summed by default. where I ab is diag (1,1,1,1) The pseudo-Riemann manifold is described by a metric 0†a l= iγγik( x) kj. (We xe) define j⊗ e j θ a ( x), (6) g( x) = -, g( x) dxµν dx (1) µν  a 0† l= iγγik( x) kj( xe) j⊗ e i θ a ( x), (7) where the metric is symmetric

where ei are the orthogonal bases expanding four-dimension 4 4 gµν ( x) = gxνµ ( ), (2) complex space  . The orthogonal bases of  satisfy

µ  ††⊗= =δ (8) and{x|, x=( x) = ( tx)} is a four-dimensional topological tr( ej e i) e i e j ij. space. Here we discuss the four-dimensional pseudo- One simple choice of e is Riemann manifold with signature (−+++,,,) , Lorentz i manifold. And it can be described by orthonormal frame θθ = ii12= (9) (vierbein) formalism as ee12( ,0,0,0) , e( 0, e ,0,0) ,

− θ θ gx1 = −ηθab x θ x, e= 0,0, eei 3 ,0 ,= 0,0,0, ei 4 . (10) ( ) ab( ) ( ) (3) 34( ) ( ) where After using γaa†= γγγ 00 η ab =diag (1, −−− 1, 1, 1) . , we find that

1 UU(44' )× ( ) -bundle are Hermitian -1 éù . (11) g( x)= trëûêú l( x) l( x) 4 †* The gauge fieldsVxVxWxWxµ (on) the= µµ( ),.i j ( ) = µji ( ) Then lx( ) and lx ( ) are the square root of metrics in some a sense. The representation freedom of γ ij( x) can be shown from restriction (4), (5) and (8). The equation as follow can as beThe derived uniqueness from (16) of definition of gauge fields is originated

0''aa†0 a ∇=γγa0 γγaa00 − γγ γik( xx) γ kj ( ) = ψik( x) γ kl γ lm ψ mj ( x) = ψi ( x) γψj ( x), µµ( ) iVx( ) Vx µ( ) , (18)

'' γa( xx) γ0( ) = ψ †0( x) γγ aa ψ( x) = ψ( x) γ† ψ( x), ik kj ik kl lm mj i j where we write where ψ 00 i Vx ( ) = γγ V. index i equals to 1,2,3,4. And µ are the Dirac fermions field with flavor related V (x) and W (x) can be decomposed by the ψ( x) = ψ†0( x) γψ,4( xU)∈ ( ) µ µij ii generators of the U(4) group is 44× The gauge field = α α follow Vxµ ( ) Vµ ( x) , (19) matrix. So, the squarea root† metrics are defined as lx( ) = iψi( x) γψ j( xe) j⊗ e ia θ ( x), (12) αα Wµµij( xWx) = ( )ij, (20)  a †† l( x) = iψi( x) γψj( xe) j⊗ e ia θ( x). (13) where α is equals to 0,1,2,....15. There are Hermitian gauge The square root Lorentz manifold is described by square root α†† αα α bosons fieldsVµ( xVxWxWx) = µµ( ),.( ) = µ( ) (12) and (13) satisfy (11) and metric (12), (13). Direct calculations show that the definition The  α are the generators of U (4) and an expilict one ††=−=− lxlxlxlx( ) ( ),.( ) ( ) can be seen in appendix. An equation is constructed which satisfying the UU(44′)× ( ) gauge invariant, locally Lorentz invariant and generally covariant principles

The coefficients of the affine' × connections on coordinates, UU(44) ( ) tr∇=lx( ) 0. (21) followscoefficients of spin connections on orthonormal frame [77] and gauge fields on the -bundle are defined as ρ This equation originated from generalized self-parallel ∇∂ =Γ( x) ∂, (14) µµvv p transportation principle. Eliminating index x, the explicit ∇=Γθθ( x) b ( xx) ( ), (15) formula of equation (21) is µµa ab

0a 00aa ∇=µµ(γγ) iVx( ) γγ − γγ Vx µ( ) , (16)  ∇=e iW( x) e†. (17) µµi ij j

A relation between coefficients of the affine connections We define a Lagrangian orthonormal frame can be easily found a µi b aµ on coordinates and coefficients of spin connections on  =ψγi(iV ∂+ µ ψ i µ ψ i − ψ j W µji) θ a + ψγψ i i Γ bµ θ a . (22) bvρρ ρ 2 Γaµµ( xx)θ b( ) =∂+Γ θθ a( x) av( x) µ( x). The last term in Lagrangian (22) is Yukawa coupling term ψ φψ If the covariant derivative is compatible with metric, ii

∇=gx( ) 0, and the scalar (Higgs) field is gamma matrix valued and originated from gravitationali field. then φγ=ba Γ θ µ. (23) 2 bµ a Γabµµ = −Γ ba .

Then, the Lagrangian (22) describes UU(44′)× ( ) Yang-Mills where the transformation matrices satisfy theory in curved space-time (Figure 1). The Lagrangian (22) has relation with (21) † * bc c UU= I,,. Uji U jk =δδ ik ΛΛ a b = a † Then the transformation elements tr∇=−lx( ) . (24) UU ∈ (4,′) Hermitian UU∈ 4 , If equation (21) being satisfied, the Lagrangian (22) is ( ij ) ( ) † = . (25) a Λ∈b O(1, 3) , consistent with generalized self-parallel transportation principleSo, the unitary(21). The principle equations of of quantum motion for field the theory Lagrangian (25) and U (4' ) is color group, U (4) O(1, 3) is (22) are locally Lorentz group. Then we complete the proof of the Lagrangian (22) is UU(44′)× ( is) gauge flavor invariant, group, locally Lorentz invariant and generally covariant. a µi b aµ γ(iV∂+µ ψ i µ ψ i − ψ j W µji) θ a + γψi Γ bµ θ a =0, (26) 2 and this equation’s conjugate transpose. We point out that (21) is density matrix version of (26). The effective equation The gauge field strength tensors and curvature tensor are

defined as followsHµν =∂−∂−+ µ V νν V µµ iV V ν iV ν V µ, the massless Dirac equation in curved space-time of this modelof motion is of (26) has signature (1, −1, −1, −1). For example, Fµν ij=∂ µ W νν ij −∂ W µij − iW µik Wν kj + iW ν ik W µkj ,

a a a ca ca aµ R bµν=∂Γ µ b νν −∂Γ bµ +Γ b ν Γ cµ −Γ bµ Γ cν. iγθaµi∂= ψ 0. (27) The square of equation (27) is massless Klein-Gordon equation in curved space-time If the covariant derivative is compatible with metric, the

ab µ ν η θθabµ∂∂ν ψ i =0. (28) curvature tensor satisfiesRRabµνν= − baµ . The signature of equation (28) is (1, −−−1, 1, 1) and consistent with special relativity. Then, a Lagrangian (22) which U(4’)×U(4)-bundle match describes the UU(44′)× ( ) Yang-Mills theory in curved the Hermiticity conditions The gauge fields strengths on the †* Hµν= HF µ νν, µ ij = F µ ν ji . space-time is all other fields (except Higgs field) dynamical background which satisfies the characteristic of the U(4) gravitational field in our real world. generators Lagrangian (22) is UU(44′)× ( ) gauge invariant, locally The gauge field strength can be decomposed by the Lorentz invariant and generally covariant. So, Lagrangian HH= α α, F= Fαα . (22) is demanded invariant under the transformations µνν µ µ νν ij µ ij

After the torsion being definedaa (29) Tνρ =2, Γ [νρ] (33)

we have the Ricci identity and Bianchi identity [78] on this Then, the transformation rules have to be derived as follows geometry structure as follows

'†   † ∂=H H V − VH , (34) Vµµ= UV U −∂( µ U) U , (30) [][][]µνρ µµ ν ρ νρ

'* * ∂=[]µFνρ ij F[[µν ik Wρ ]kj − WFµ ik νρ ]kj , (35) Vµ= U kiµlj WU +∂ U ki( µkj U ), (31)

aa 'b cd b c b TT σρ []µ ν = R[ρνµ ] +∇[]ρ Taµ ν , (36) Γaµ =Λ a Γ cµ Λ d −Λ a ∂ µ Λ c , (32)

∇=aa (37) † ab b†† a [ρR bµν ] RTbµσρ[] ν . −−ψiH ab ( γγ γ γ) ψi , (43)

There is Yang-Mills Lagrangian for gauge bosons in this where we use the notations as follow, model ν ρ νρ (38) ∂µdx ⊗ dx ∂=σδδ µσ,

µ ν νµ dx⊗ dx ∂∂ρσ =δδ ρσ, where ζ ∈ is constant. µν R FFabij= abijθθ a b , = θθµν For the gravity, the Einstein-Hilbert action in Lorentz HHab µν a b . manifold be showed as follow UU(4')× (4 ) gauge invariant, locally Lorentz SR= ∫ω , (39) invariant, generally covariant Lagrangian So, we define a

† abij† a b b†† a volume form g=−−Rψψ ii iF( ψj( γ γ γ γ) ψi (44) where is the Ricci scalar curvature in Lorentz manifold, ω is † ab b†† a 012 3 +−iHψi ab ( γγ γ γ) ψi . ω =−gv dx ∧∧ dx dx ∧ dx and The Lagrangian (44) is Hermitian

† gg= . (45) gvµ= det( gν ( x)).

The ψ †ψ in Lagrangian (44) gives us the Einstein-Hilbert In this geometry framework, the equations can be derived R ii as follows action. The equation (43) and the Einstein tensor can be derived from the Lagrangian (44). The gravity theory in this model is Einstein-Cartan kind with torsion. −1 ∇∇l = (ψγψa FF−* ψγψ aa + ψ H γψ []µ ν2 i kµkjνν µkik j iµν j Sheaf Quantization and Path Integral a i ba† a −+ψγiH µνν ψ j ψγψ i j Ree bµ), j ⊗ i θ (40) Quantization 2 − The entities lx( ) and lx ( ) are two sections of the two  1 a† *† aa † ∇∇[]µ νl = (ψγψi kµkj FFνν− µkik ψγψ j + ψ iµ Hν γψ j bundles, respectively, where these two bundles are dual to 2 each other. Further, the Sheaf valued entities lxˆ( ) and lx ( ) i −+ψγa†H ψ ψγba †† ψRee), ⊗ θ a (41) i µνν j2 i j bµ j i are superposition of sections lx( ) and lx ( ) (Figure 2). where we write

00 ˆ  l( x) = ηκκ( x) κκ,, x xl( x), (46) HHµµνν= γγ. ∑ κ

l x= η x κκ,, x xl x , (47) 2 µ ν ( ) ∑ κκ( ) ( ) We define ∇=∇∇[]µ ν dx ∧ dx , κ the equation of a gravity theory is constructed

2  tr∇= l( xlx) ( ) 0. (42)

This equation (42) is obviously UU(4')× (4 ) gauge invariant, locally Lorentz invariant and generally covariant. The explicit formula of equation (42) is Figure 2: † abij† a b b†† a a section of the bundle. Right: A Sheaf is a collection of the Rψψii= iF( ψj( γ γ− γ γ) ψi sections. lx ˆThe( ) isfiber Sheaf bundle valued. structure is shown. Left: l(x) is

where g and g are Lagrangian multipliers with where ηκ ( x)∈[0 , 1 ] are probability of corresponding section gg,. ∈   lxκ ( ) and lxκ ( ), κ is Sheaf space index and evaluated in an For pure state ρ ( x) =ΨΨ( xx) ( ) , (55) abelian group. The density matrix corresponds to lxˆ( ) and lx is ( ) Ψ=( xx) ∑ακκ ( ) , (56) κ ρ( x) = ∑ ηκ ( xx) κκ, ,. x (48) κ where the Ψ ( x) We have orthogonal bases in Sheaf space. The orthogonal bases in Sheaf space satisfy probability complete formulas integral formula is the quantum state of quantum field theory. The transition amplitude can be defined through path ˆ  κ κ′′= δ− ′′ δκ− κ (49) α= κα′ iωκ ( tx, )  (57) ,|x , x( xx) ( ). κκ(tx,) ∫ D( t, xe) ( t0 ,, x) t′∈( t0 ,t ) where ω is volume form. trρη( xx) =∑ κ ( ) = 1. (50) ê In mathematics, a Sheaf is a collection of sections, the index Particles Spectrum κ of each section correspond to an abelian group element. α and α (α is equals to here) are gauge In physics, the Sheaf spaces Sh( x) and Sh ( x) are expanded Vµ Wµ 0,1 , · · · ,1 5 by all possible sections lx( ) and lx ( ) of the two bundles, α always preserves the possibility of chiral symmetry respectively. The Sh( x) and Sh ( x) are linear spaces, which Wµ bosons fields. The interactions related with the gauge bosons means, for example, any two entities lxˆ ( ) and lxˆ ( ) in breaking in the Lagrangian (54) such that the gauge group 1 2 canfields be decomposed to ′ ×× where ′ UU(44) ( )LR U( 4) , U (4 ) Sh x , there is an entity lxˆ in Sh x equals to the mixing is color group and × ( ) ( ) ( ) UU(44)LR( ) 0 0 0 0 of the two entities Vµ is dark photon and Wµ is Fiona particle. Vµ and Wµ are related with ′ and gauge isgroup chiral in flavor ′ group.and The lˆˆˆ( x) =ηη( xl) ( x) + ( xl) ( x), U (1 ) U (1) U (4 ) U (4) 11 2 2 , respectively. The left-over gauge group is a Pati-Salam gauge lˆˆ x, l x∈ Sh x ; (51) group ′ ×× (Figure 3) [59,79]. The 12( ) ( ) ( ) SU(4) SU( 4)LR SU (4) SU (4′) gauge group can be decomposed as follow ⇒∈lˆ( x) Sh( x), ′ = ′ ⊕ ′ ++ (58) where the probability of each section U(4) SU( 31) U( ) UXX+− U .

ηη12( xx),( )∈[ 0,1] and

ηη12( xx) +=( ) 1.

The Sheaf spaces Sh( x) and Sh ( x) are dual to each other. We call it Sheaf quantization which switching study objects from single section to all possible sections of the bundle. The equation (50) derives to equation of motion for density matrix

dtrρρ ( x) = tr dx ( ) = 0, (52) ( ) ( ) Figure 3: Weight diagram of SU (4) adjoint representation and corresponding gauge bosons. The decomposition of where d is exterior differential derivative. The equations adjoint representation is ⊗ * . (a)  SU (4) 15=8 1+3+3 of motion for entities lxˆ( ) and lx ( ) in Sheaf quantization α The wight diagram of Vµ which contained gauge bosons method are ±C α are gluons, photon and X . (b) The wight diagram of Wµ 01212 ± related gauge bosons are YYYYY,,,,** andWZ, . tr∇=lxˆˆ( ) 0, tr  lxlx ( ) ( ) = 0. (53)   The SU (3′) is the gauge group of quantum chramodynamics α The corresponding total Lagrangian density is (QCD) and the corresponding gauge bosons Vµ (α is equals to 1 , 2 · · · , 8 here) are gluons. The U (1′) in equation (58) is ˆ =++ gg , (54) electro-magnetic interaction gauge group and corresponding ∑ κκκgY,, 15 ±C κ gauge boson Vµ X particles transport

is photon γ. The De-Sheng Li. Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold. Copyright© De-Sheng Li. Phys Sci & Biophys J 2021, 5(2): 000186. 7 Physical Science & Biophysics Journal semi-leptonic processes and as the fourth color” [59]. The fundamental representation 6 corresponds to 6 ±+CC89 + C X= Vµµ ± iV . (59) diagram coordinates in Chevalley basis of representation 6 (see in Table 1) have flavors good of correspondencequarks and leptons. with The the weight quark The electric charges of each X +C and X −C are 1/3 and can be representation 64⊗ similarly. The weight diagram of SU (4)LR, decomposed as (Figure 4). fermionsquantum is number shown [80].in Figure The antifermions4. Both left-handed be filled and into right- the −1/3, respectively. The chiral gauge group handed fermions for all quarks and leptons are existed.

= ⊕ +++− (60) Especially, the existence of right-handed neutrinos is SU(4)LR, SU( 31)Y U( )Z UWW U , predicted. This is compatible with experimental results [81] α and the well know Seesaw mechanism [82-86]. The 2I z and related gauge bosons Vµ (with α equals to1, 2, · · · ,1 5 ) contain weak bosons W ± and Z being used in Table I are not no reasons because we have the Gell-Mann-Nishijima formula [80].

± =±=±9 10 11 12 Wµµ W iW µ W µ iW µ (61) P ++++SC BT 13 14 = + (64) = ± QIz . Wµµ iW , 2

15 ZWµµ= . (62) An explicit fermions representation [79] in this model might be 222u c td' R R RR ' 222uG c G td GG (65) ψ i = , '  222uB c B td BB  e µ τν′

where uct,, and d′ eµ, ,τ and ν ' are

fermions electric charges are of quarks (65) are fields, electron, mu, tau and neutrinos fields. The corresponding 222 1 Figure 4: Weight diagram of SU (4) fundamental − 333 3 representation 46⊗ . Representation 4 = 3 +1, 3 is 3  222 1 kinds of color, red, green and blue, 1 is the lepton number. − Q = 333 3. (66) Representation 6 gives us 6  222 1 6 − has 2 kinds of quarks or leptons. flavors of quarks and leptons. 333 3 flavors have 3 generations, I, II and III, each generation  111 0 01212 The left over gauge bosons are YYYYY,,,,** with 0 1212 The quarks states like dsb,, and neutrinos states electric charge. The gauge bosons YYYY,,,** transport νννeu,,τ are eigen states of the Lagrangian [79]. non-SM flavor changing neutral currents (FCNCs) and 2Iz S+C B+T H1 H2 H3 (63) u 1 0 0 1 -1 1 d -1 0 0 -1 1 -1

±C 1212 c 0 1 0 1 0 -1 The X and YYYY,,,** must be super heavy from the restrictions of experimental data. s 0 -1 0 -1 0 1 t 0 0 1 0 1 0 b 0 0 -1 0 -1 0 ψ i transfer as the UU(44′)× ( ) fundamental representation according to (29). So, fermions Table 1: Corresponding relations between quarks quantum The fermionicSU fields(4) fundamental representation 46⊗ number and weight diagram coordinates in Chevalley basis naturally. The fundamental representation 4 corresponds to H1, H2 and H3 of SU (4) representation 6 are shown. 3are colors filled and into 1 thelepton and leads us reobtain “Lepton number

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