Fermion Generations from "Apple-Shaped" Extra Dimensions
aa r X i v : . [ h e p - t h ] A ug Fermion Generations from ”Apple-Shaped” Extra Dimensions
Merab Gogberashvili ∗ Andronikashvili Institute of Physics6 Tamarashvili St., Tbilisi 0177, GeorgiaandJavakhishvili State UniversityFaculty of Exact and Natural Sciences,3 Chavchavadze Avenue, Tbilisi 0128, Georgia
Pavle Midodashvili † Chavchavadze State University32 Chavchavadze Avenue, Tbilisi 0179, GeorgiaandTskhinvali State University2 Besiki St., Gori 1400, Georgia
Douglas Singleton ‡ Physics Dept., CSU Fresno, Fresno, CA 93740-8031, USA (Dated: October 24, 2018)We examine the behavior of fermions in the presence of an internal compact 2-manifold which inone of the spherical angles exhibits a conical character with an obtuse angle. The extra manifoldcan be pictured as an apple-like surface i.e. a sphere with an extra ”wedge” insert. Such a surfacehas conical singularities at north and south poles. It is shown that for this setup one can obtain,in four dimensions, three trapped massless fermion modes which differ from each other by havingdifferent values of angular momentum with respect to the internal 2-manifold. The extra angularmomentum acts as the family label and these three massless modes are interpreted as the threegenerations of fundamental fermions.
PACS numbers: 11.10.Kk, 04.50.+h, 11.25.Mj
I. INTRODUCTION
One of the open questions in the Standard Model of particle physics is the fermion family puzzle - why the firstgeneration of quarks and leptons are replicated in two other families of increasing mass. It is not clear how to explainthe mass hierarchy of the generations and the mixing between the families characterized by the Cabbibo-Kobayashi-Maskawa matrix. Several ideas have been suggested such as a horizontal family symmetry [1].Recently the brane world idea [2] has been used to find new solutions to old problems in particle physics andcosmology. A key requirement for theories with extra dimensions is that the various bulk fields (with the exception ofgravity) be localized on the brane. Brane solutions with different matter localization mechanisms have been widelyinvestigated in the scientific literature [3]. A pure gravitational trapping of zero modes of all bulk fields was given in[4, 5].The main emphasis of the present paper is to explain some properties of fermion families in the framework of a branemodel. For the other attempts using extra dimensions see [6, 7]. We introduce an extra 2-dimensional compact surfaceand investigate the properties of higher dimensional fermions place in this space-time. In 6-dimensional models theinternal compact 2-manifold usually is considered as having rugby(football)-ball shaped geometry with a deficit angle[8, 9]. As shown in this paper one can address the generation puzzle using an internal 2-surface with a profuse angle,or having an ”apple-like” geometry. Using the brane solution of [5] we show that for apple-shaped extra dimensionsthree fermion generations naturally arise from the zero modes of a single ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] model [7]. A mass hierarchy and mixings between the three zero modes are obtained by introducing of a Yukawa-typecoupling to a single 6-dimensional scalar field.
II. SOLUTION OF -DIMENSIONAL EINSTEIN EQUATIONS In this article we consider 6-dimensional space-time with the signature (+ − − − −− ). Einstein’s equations in thisspace have the form R AB − g AB R = 1 M ( g AB Λ + T AB ) , (1)where M and Λ are the 6-dimensional fundamental scale and the cosmological constant. Capital Latin indices runover A, B, ... = 0 , , , , , ansatz ds = φ ( θ ) g µν ( x α ) dx µ dx ν − ε (cid:0) dθ + b sin θdϕ (cid:1) , (2)where ε and b are constants. Here the metric of ordinary 4-space-time, g µν ( x α ), has the signature (+ − −− ) (theGreek indices α, µ, ν... = 0 , , , θ and ϕ (0 ≤ θ ≤ π, ≤ ϕ ≤ π ). We take this 2-surface to be attached to the braneat the point θ = 0. Thus the geodesic distance into the extra dimensions goes from north to south pole of the extra2-spheroid when θ changes from 0 to π .If in (2) the constant b = 1 then the extra 2-surface is exactly a 2-sphere with the radius ε . If b = 1 the extramanifold is a 2-spheroid with either a deficit or profuse angle ϕ , i.e. its conical sections, θ = const , are either missingsome angle, δϕ , or have some extra angle, δϕ . The metric for this 2-manifold will take usual form with b = 1 if weredefine ϕ so it ranges from 0 to 2 πb . One can think of the extra 2-surface as being of sphere with cut out (if b < b >
1) the ”wedge” having an angle δϕ = 2 π ( b − δ -like contribution to the curvaturetensor localized at the points with sin θ = 0. These singularities can be canceled by introduction of 3-branes at thesepositions [8]. Usually in the literature one considers the case b < b > ansatz for the energy-momentum tensor of the bulk matter fields we take in the form T µν = − g µν E ( θ ) , T ij = − g ij P ( θ ) , T iµ = 0 . (3)small Latin indices correspond to the two extra coordinates. The source functions E and P depend only on the extracoordinate θ .For these ans¨atze Einstein’s equations (1) take the following form:3 φ ′′ φ + 3 φ ′ φ + 3 φ ′ φ cot θ − ε M [ E ( θ ) − Λ] , φ ′ φ + 4 φ ′ φ cot θ = ε M [ P ( θ ) − Λ] , (4)4 φ ′′ φ + 6 φ ′ φ = ε M [ P ( θ ) − Λ] , where the prime denotes differentiation d/dθ . For the 4-dimensional space-time we have assumed zero cosmologicalconstant and Einstein’s equations in the form R (4) αβ − g αβ R (4) = 0 , (5)where R (4) αβ and R (4) are 4-dimensional Ricci tensor and scalar curvature.In [5] a non-singular solution of (4) was found for the boundary conditions φ (0) = 1 , φ ′ (0) = 0. The solution wasgiven by φ ( θ ) = 1 + ( a −
1) sin ( θ/ , (6)where a is the integration constant. The source terms for this solution were given by E ( θ ) = Λ (cid:20) a + 1)5 φ ( θ ) − a φ ( θ ) (cid:21) , P ( θ ) = Λ (cid:20) a + 1)5 φ ( θ ) − a φ ( θ ) (cid:21) (7)and with the radius of the extra 2-spheroid given by ε = 10 M / Λ. For simplicity in this paper we take a = 0 so thatbelow we will use the warp factor φ ( θ ) = 1 − sin ( θ/
2) = cos ( θ/ . (8)This warp factor equals one at the brane location ( θ = 0) and decreases to zero in the asymptotic region θ = π , i.e.at the south pole of the extra 2-dimensional spheroid.The expression for the determinant of our ansatz (2), which will be used often in what follows, is given by √− g = p − g (4) ε φ ( θ ) sin θ , (9)where p − g (4) is determinant of 4-dimensional space-time. III. FERMIONS IN SIX DIMENSION
Let us consider spinors in the 6-dimensional space-time (2), where the warp factor φ ( θ ) has the form (8). Theaction integral for the 6-dimensional massless fermions in a curved background is S Ψ = Z d x √− g h i Ψ h B e A Γ e A D B Ψ + h.c. i , (10) D A denote covariant derivatives, Γ e A are the 6-dimensional flat gamma matrices and we have introduced the sechsbein h e AA through the usual definition g AB = h e AA h e BB η e A e B , (11) e A, e B, ... are local Lorentz indices.In six dimensions a spinor Ψ( x A ) = (cid:18) ψξ (cid:19) (12)has eight components and is equivalent to a pair of 4-dimensional Dirac spinors, ψ and ξ .In this paper we use the following representation of the flat (8 ×
8) gamma-matrices (for simplicity we drop thetildes on the local Lorentz indices when no confusion will occur)Γ ν = (cid:18) γ ν − γ ν (cid:19) , Γ θ = (cid:18) −
11 0 (cid:19) , Γ ϕ = (cid:18) ii (cid:19) , (13)where 1 denotes the 4-dimensional unit matrix and γ ν are ordinary (4 ×
4) gamma-matrices. It is easy to check thatthe representation (13) gives the correct space-time signature (+ − − − −− ). The 6-dimensional analog of the γ matrix in the representation (13) has the form Γ = (cid:18) γ γ (cid:19) . (14)From (12) one finds that the 6-dimensional left-handed (right-handed) particles correspond to a pair of 4-dimensionalparticles, ψ and ξ which correspond to the particle (anti-particle) from the 4-dimensional point of view.The 6-dimensional massless Dirac equation, which follows from the action (10), has the form (cid:16) h µ e B Γ e B D µ + h θ e B Γ e B D θ + h ϕ e B Γ e B D ϕ (cid:17) Ψ( x A ) = 0 , (15)with the sechsbein for our background metric (2) given by h B e A = (cid:18) φ δ B e µ , ε δ B e θ , bε sin θ δ B e ϕ (cid:19) . (16)From the definition ω f M e NM = 12 h N f M (cid:16) ∂ M h e NN − ∂ N h e NM (cid:17) − h N e N (cid:16) ∂ M h f MN − ∂ N h f MM (cid:17) − h P f M h Q e N (cid:16) ∂ P h Q e R − ∂ Q h P e R (cid:17) h e RM (17)the non-vanishing components of the spin-connection for the sechsbein (16) can be found ω e θ e ϕϕ = − b cos θ, ω e θ e νν = − φ ′ ε = sin θ ε . (18)The covariant derivatives of the spinor field have the forms D µ Ψ( x A ) = (cid:20) ∂ µ + sin θ ε Γ θ Γ ν (cid:21) Ψ( x A ) ,D θ Ψ( x A ) = ∂ θ Ψ( x A ) , (19) D ϕ Ψ( x A ) = (cid:18) ∂ ϕ − b cos θ θ Γ ϕ (cid:19) Ψ( x A ) . Then Dirac’s equation (15) takes the form [10, 11] (cid:20) φ Γ µ ∂∂x µ + sin θ εφ Γ ν Γ θ Γ ν + 1 ε Γ θ ∂∂θ + 1 bε sin θ Γ ϕ ∂∂ϕ − cot θ ε Γ ϕ Γ θ Γ ϕ (cid:21) Ψ( x A ) == (cid:20) φ Γ µ ∂∂x µ + 1 ε Γ θ (cid:18) ∂∂θ − sin θφ + cot θ (cid:19) + 1 bε sin θ Γ ϕ ∂∂ϕ (cid:21) Ψ( x A ) = 0 . (20)This system of first order partial differential equations can be treated using the following separation of variablesΨ( x A ) = X l e ilϕ √ πφ ( θ ) (cid:18) α l ( θ ) ψ l ( x ν ) β l ( θ ) ξ l ( x ν ) (cid:19) , (21)where ψ l ( x ν ) and ξ l ( x ν ) are 4-dimensional Dirac spinors. Here we note that since dimension of Ψ( x A ) in six dimensionsis m / then dimensions of α l ( θ ) , β l ( θ ) and ψ l ( x ν ) , ξ l ( x ν ) should be m and m / respectively.At the end of the section we note that our case is unlike the model studied in [11], which examined spin-1 / ψ l ( x ν ) and ξ l ( x ν ) in (21) which must have spinorrepresentations. So the wave-function given in (21) is single-valued for 2 π rotations around the brane by the extraangle ϕ . Thus the quantum number l takes integer values – l = 0 , ± , ± , ... – and not half-integer values. IV. FERMION GENERATIONS
We are looking for 4-dimensional fermionic zero modes. To this end we take ψ l ( x ν ) and ξ l ( x ν ) in (21) to obey the4-dimensional, massless Dirac equations γ µ ∂ µ ψ l ( x ν ) = γ µ ∂ µ ξ l ( x ν ) = 0 . (22)There will also be very massive KK modes whose masses will go a integer multiples of the inverse size of the extra2-dimensional space i.e. as 1 /ε . However, we will assume later that 1 /ε ≃ ψ l ( x ν )and ξ l ( x ν ) are indistinguishable from the 4-dimensional point of view and we can write ψ l ( x ν ) = ξ l ( x ν )Inserting (21) and (22) into (20) converts the bulk Dirac equation into (cid:20) Γ θ (cid:18) ∂∂θ + cot θ (cid:19) + ilb sin θ Γ ϕ (cid:21) (cid:18) α l ( θ ) β l ( θ ) (cid:19) = 0 . (23)Using the representation for Γ θ , Γ ϕ gives the following system of equations for α l ( θ ) and β l ( θ ) (cid:18) ∂∂θ + cot θ − lb sin θ (cid:19) α l ( θ ) = 0 , (cid:18) ∂∂θ + cot θ lb sin θ (cid:19) β l ( θ ) = 0 . (24)The solutions of these equations are α l ( θ ) = A l tan l/b ( θ/ √ sin θ , β l ( θ ) = B l tan − l/b ( θ/ √ sin θ , (25)where A l and B l are integration constants with the dimension of mass.The normalizable modes are those for which Z √− g d x ¯ΨΨ = Z p g (4) d x (cid:0) ¯ ψ l ψ l + ¯ ξ l ξ l (cid:1) . (26)In other words we want the integral over the extra coordinates, ϕ and θ , to equal 1. Thus inserting (21), (25) and thedeterminant (9) into (26) the requirement that the integral over ϕ and θ equal 1 gives ε Z π dθ h A ∗ l A l tan l/b ( θ/
2) + B ∗ l B l tan − l/b ( θ/ i = 1 , (27)where the integral over ϕ contributes 2 π .Using the formula Z π dθ tan c ( θ/
2) = π/ cos( cπ ) , − < c < − b < l < b . (29)Recall that the parameter b in (2) is an integration constant of Einstein’s equations and governs the topology of theinternal 2-spheroid. If b = 1 the internal 2-surface is exactly a sphere. For this case, as it clear from (29), there existonly one zero mode with l = 0. If on the other hand 2 < b ≤ l = 0 and l = ±
1. To be concrete we will set b = 4 in the following. Other choices of b from thisinterval will only slightly change the numerical results below. From the normalization condition (27) we now find thefollowing relation for the constants A l and B l πε ( A ∗ l A l + B ∗ l B l ) = cos( lπ/ , (30)where l = 0 , ± ( x A ) = 1 √ π sin θ φ ( θ ) (cid:18) A B (cid:19) ψ ( x ν ) , Ψ ( x A ) = 1 √ π sin θ φ ( θ ) e iϕ (cid:18) tan / ( θ/ A tan − / ( θ/ B (cid:19) ψ ( x ν ) , (31)Ψ − ( x A ) = 1 √ π sin θ φ ( θ ) e − iϕ (cid:18) tan − / ( θ/ A − tan / ( θ/ B − (cid:19) ψ − ( x ν ) , where the constants A l and B l obey the relations (30).These three normalizable modes all appear as massless 4-dimensional fermions on the brane. To explain the observedmass spectrum and mixing between these fermions one needs to couple these particles to a scalar (Higgs) field. V. COUPLING WITH HIGGS FIELD
In the previous section it was shown that by adjusting the integration constant b in our gravitational background(2) it is possible to get three zero-mass modes on the brane. To make this model more realistic we have two problems:a) There is no mixing between the different generations due to the orthogonality of the angular parts of the higherdimensional wave functions. Overlap integrals like R dϕ ¯ ψ l ψ l ′ , which characterize the mixing between the differentstates, vanish since Z π dϕ e − ilϕ e il ′ ϕ = 0 . l = l ′ (32)b) All the fermionic states (31) are massless, whereas the fermions of the real world have masses that increase witheach family.Following [12] we address both of these issues by introducing a coupling between the fermions with the bulk scalarfield Φ p ( x A ) (which has dimensions (mass) ) by adding to the action an interaction term of the form S int = 1 F Z d xdϕdθ √− g Φ p ¯Ψ l Ψ l ′ , (33) F is the coupling constant between the scalar and spinor fields and has the dimensions of mass.For simplicity we take the massless, real scalar field to be of the formΦ p ( x A ) = κ p Φ p ( θ ) e ipϕ , (34)i.e. the scalar field only depends on the bulk coordinates θ, ϕ , not on the brane coordinates x µ . In (34) the angularquantum number p is an integer and κ p are the 4-dimensional constant parts of Φ p ( x A ) having dimensions of mass.The equation of motion of a massless real scalar field in six dimensions has the form:1 √− g D A (cid:2) √− g g AB D B Φ( x A ) (cid:3) = 0 . (35)Using the form of Laplace operator on our 2-spheroid∆ = − ε (cid:18) ∂ ∂θ + cot θ ∂∂θ + 4 φ ′ φ ∂∂θ + 1 b sin θ ∂ ∂ϕ (cid:19) , (36)where φ is given by (8), the equation (35) can be written asΦ ′′ p + (cid:18) cot θ − θ θ (cid:19) Φ ′ p − p b sin θ Φ p = 0 . (37)It is possible to give an exact solution to this equation in quadratures. However, this solution is a complicatedfunction. In order to make understandable estimates of the masses and mixings we will use approximate solutions.Close to the origin ( θ → θ → φ → ′′ p + cot θ Φ ′ p − p b sin θ Φ p = 0 . (38)For p = 0 a solution to this equation isΦ ( θ ) = D { θ/ } , p = 0 (39)where D is an integration constant.For non-zero p one of the solutions of (38) isΦ p ( θ ) = D p cosh n pb ln [cot( θ/ o , p = 0 (40)where D p are integration constants. Note that these solutions (as well as the spinor fields (21) and (25)) are singularat sin θ = 0, however, because of the determinant (9) the various integrals done with these fields are finite.We determine the constants D p by requiring that the scalar field is normalized over the extra coordinates, i.e. using(9) we require 2 πε Z π dθ sin θ φ ( θ ) Φ p ( θ ) = 1 . (41)For the values of a and b used in this paper ( a = 0, b = 4) from (41) we find D = 1 ε q π − π ≈ . ε , D ± = 1 ε q π + √ π ≈ . ε , D ± = 1 ε q π + π ≈ . ε . (42)Substituting (34) and (21) into (33) we find S int = U pl,l ′ Z d x p − g (4) ¯ ψ l ( x µ ) ψ l ′ ( x µ ) , (43)with U pl,l ′ = ε κ p πF Z π dϕe i ( p − l + l ′ ) ϕ Z π dθ sin θ Φ p ( θ ) [ A ∗ l A l ′ α l ( θ ) α l ′ ( θ ) + B ∗ l B l ′ β l ( θ ) β l ′ ( θ )] , (44)where D p are expressed by (42) and A l , B l obey the relations (30). Below we will use the new definition f p = κ p /F for the ratios of the 4-dimensional constant values of Higgs field from (34) and of the coupling constant from (32).The first integral in (44) for the quantities U pll ′ will be non-zero if p − l + l ′ = 0 . (45)When l = l ′ and p = 0 this gives a mass term; when l = l ′ and p = 0 this gives mixings between the l and l ′ modes. VI. MASSES AND MIXINGS
To find mass terms appearing because of coupling of the three fermionic zero modes (31) with the Higgs field (34)for the angular momentum quantum numbers in (44) we should use the values, p = 0 , l = l ′ , or calculate only thecomponents of the matrix (44) with the zero upper index. Using (25) and (39) from (44) we get U , = f D ε π ( A ∗ A + B ∗ B ) = f D ,U , = f D ε π √ π ) A ∗ A + (2 − π ) B ∗ B ] = f D (cid:18) − π √ ε π | A | (cid:19) , (46) U − , − = f D ε π √ (cid:2) (2 − π ) A ∗− A − + (2 + π ) B ∗− B − (cid:3) = f D (cid:18) π − √ ε π | A − | (cid:19) . To obtain the last equality in each term above we have used (30) to eliminate | B ± | in favor of | A ± | .As a concrete example of how the realistic mass hierarchy can arise let us take 1 /ε ≃ D ≃ ≃ /ε ) will be much heavier thanthe three zero mass modes, even after they are given a mass via the Higgs mechanism. Next let us examine threequarks from the ”down” sector, i.e. d , s and b quarks. This is meant as a toy model since we do not have an ”up”sector and we do not have three generations of leptons. Our aim here is just to show that it is possible to generate arealistic fermion mass hierarchy from an extra dimensional model.Making the association that s -quark → l = − b -quark → l = 0 and d -quark → l = +1, we get the followingconditions on U l,l from (46) U − , − = m s ≈ M eV , U , = m b ≈ M eV , U , = m d ≈ M eV , (47)where we have taken average values of the quark masses from [13]. Solving the system (46) and (47) gives f ≈ . × − , | A | ≈ . /ε , | A − | ≈ . /ε . (48)Note these values of | A ± | are consistent with the condition in (30) which requires | A ± | , | B ± | < . ε . The largestmass corresponds to the l = 0 quantum number. This can be understood from the point of view that this state hasa non-zero effective wavefunction near the brane, θ = 0, and thus has a large overlap with the scalar field (39). (Byeffective wavefunction we mean the combination of the wavefunctions from (31) and the square root of the determinantfrom (9). In this way the singular sin θ term cancels out). The d and s quarks, which correspond to the l = +1 , − θ = 0 and thus have a smaller overlap with the scalar field.For mixings between the different families, characterized by different angular momentum l , the selection rule (45)indicates that we must consider components of the matrix (44), which have a nonzero upper index p . There are threeindependent components whose indices are given by U , = U − , = f ε D (1 + √ π A ∗ A + B ∗ B ) = f ε D − (1 + √ π A ∗ A + B ∗ B ) ,U , − = U − − , = f ε D (1 + √ π A ∗ A − + B ∗ B − ) = f ε D − (1 + √ π A ∗− A + B ∗− B ) , (49) U , − = U − − , = f ε D √ π ( A ∗ A − + B ∗ B − ) = f ε D − √ π ( A ∗− A + B ∗− B ) . From [13] one finds that the mixing between the first and second generation is of order 0 . V us ≈ . .
01 (i.e. V cb ≈ . .
001 (i.e. V ub ≈ . l (+1 , − ,
0) wearrive at the following connections for the mixings from (49) | U , − | → V us ≃ . , | U , − | → V cb ≃ . , | U , | → V ub ≃ .
001 (50)In terms of ratios we want to fix A l , B l such that from (49) we get | U , || U , − | ≃ . , | U , − || U , − | ≃ . . (51)To simplify the analysis we assume that all A l , B l are purely real. Then from (30) using (48) we have | B | ≈ . /ε , | B − | ≈ . /ε . (52)Also we take B = kA where k is some real constant, i.e. from (30) B is determined once A is given.Applying all this to the first condition from (51) we find | U , || U , − | = 0 . . k . . k = 0 . . (53)Solving for k gives k = − . k we find from (30) that A = 0 . /ε , B = − . /ε . (54)Inserting all these real values for A l , B l into the second condition from (51) we find that | U , − || U , − | = 1 . f f . (55)It is clear that if we set f /f ∼ . κ and κ ) we reproduce the mixings between the differentgenerations as given by the rough estimate (50). VII. SUMMARY AND CONCLUSIONS
We have given a higher dimensional model to address the fermion generation puzzle. Three zero mass modes arisein an ”apple” geometry given by (2) and (8). Exactly three zero modes are obtained by adjusting the shape of theinternal 2-dimensional space via b giving a profuse angle rather than the more common case of a deficit angle. Weinterpret these three zero mass modes as a toy model for the three generations of fermions. This is a toy model sincewe do not reproduce the full flavor structure of the Standard Model fermions. For example in this paper we took thethree zero mass modes as the down quarks, d, s, b leaving out the up quarks and leptons. The family number in thismodel was the quantum number l associated with angular momentum of fermions with respect to the extra 2-space.To give masses and mixings one had to couple the zero mass modes to a scalar field. Thus in this model the massesand mixings arose from the same mechanism. We demonstrated that one could get a realistic mass spectrum andmixings by taking our zero mass modes to be the family of down quarks. That we are able to reproduce a realisticmasses and mixings is not surprising since there are number of free parameters involved especially in terms of thenormalization constants, κ p , A l , B l for the higher dimensional wavefunctions. The central point of this paper wasnot so much to obtain a realistic masses and mixings (since in any case the model does not contain complete set ofparticles of the Standard Model) but rather to give a higher dimensional model for the fermion generation puzzle.In addition to the zero mass modes there will be massive KK modes whose masses will be of the order 1 /ε . Here,since 1 /ε ≃ Acknowledgment:
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