Fermionic zero modes in the vortex field in arbitrary dimensions and index of Dirac operator with Majorana-like interaction
aa r X i v : . [ h e p - t h ] N ov Fermionic zero modes in the vortex field in arbitrary dimensionsand index of Dirac operator with Majorana-like interaction
G. P. Bednik
Institute for Nuclear Research of RAS,60th October Anniversary Prospect 7a, Moscow 117312, Russia andPhysics Department, Moscow State University,Vorobjevy Gory, 119991, Moscow, Russia
Abstract
In this work we consider fermionic zero modes in the external scalar and electromagnetic fieldforming the vortex on a sphere. We find the correspondence between the equations for the fermionsin different dimensions, find their explicit expressions through the vortex fields in case of masslessfermions, asymptotics near the poles in case of massive fermions and check the number of thesolutions by proving index theorem for the fermions on a sphere. As a part of deriving the index,we write a detailed calculation of the Green function of the Heat equation. . INTRODUCTION Recently, considerable interest has been attracted to localized fermionic states in topo-logically non-trivial external fields. Its revival is stimulated by developments in two verydifferent areas of physics, high-energy particle theory and condensed-matter physics (theoryand experiment).The Standard Model of particle physics, though extremely successful in description ofmost of the experimental results, cannot explain the apparent hierarchy between two funda-mental scales, the electroweak symmetry-breaking scale v ∼
200 GeV and the gravitational(Planck) scale M Pl ∼ GeV. It has been suggested [1] and discussed in thousands ofresearch papers (for reviews, see e.g. Ref. [2, 3]) that the gravitational scale may be muchlower if there exist additional space dimensions beyond the three observed ones. Theirnon-observation may be related either to compactification of extra dimensions down to avery small size [4, 5] or to localization of visible particles on the observed four-dimensionalMinkowski subspace [6, 7]. The latter may naturally happen due to interaction of a particlewith the external field with non-trivial topological properties (e.g. [8, 9]).Quite recently, it has been realized that a similar situation may occur in a laboratory,where non-trivial topology of the effective bosonic external field results in the localizationof fermionic states at particular points, lines or surfaces in certain materials. The motion offermions is governed in these cases by the very same equations as in the particle-physics mod-els and in many cases, the localized states have been observed in a laboratory. Well-knownexamples include the fractional Hall charges [10], topological insulators [11], topological su-perconductors [12] and graphene [13]. The topological nature of the localized states makesthem in principle suitable for the purposes of quantum computing [14]. Reviews of sometopics in this rapidly developing field may be found in Ref. [15].In this paper, we will study localized fermionic states in the external field of abelianAbrikosov–Nielsen–Olesen vortex [16, 17].The purpose of the present work is to explore various possible fermion-vortex interactionsfor various number of space dimensions. In (3+1) dimensions, it has been shown that forthe vortex winding number n , there exist n localized zero modes of fermions with eitherMajorana-like [9] or Dirac-like [18] equations. These zero modes describe (1+1)-dimensionalmassless fermions which move freely in the core of the vortex line dubbed “superconduct-2ng string”; the existence of these modes is related to the vortex topology by the indextheorem [19]. This feature has been used for extra-dimensional model building [1, 6] in(5+1) dimensions, where the vortex fields depend on two extra spatial coordinates whilethe core corresponds to our three-dimensional space. Two (related to each other) featuresof the localized modes are important for construction of realistic particle models: chiralityand masslessness. The former provides for correct quantum numbers of the standard-modelfermions while the latter opens up a possibility to generate particle masses at a correct scaleby means of an additional interaction with the Higgs scalar field treated perturbatively.Moreover, if three fermionic generations are associated with three zero modes of an n = 3vortex, then an elegant mechanism to explain the mass hierarchy among the families ofcharged fermions may operate [20, 21]. It has been shown [20–22] that within this approach,not only the hierarchical pattern of masses and mixing may be reproduced, but their partic-ular experimental values are obtained with a surprisingly small number of free parameters.Finally, the very same model may automatically explain [23] the observed hierarchical pat-tern of neutrino masses and leptonic mixings whch is very different (e.g. Ref. [24]) from theone observed among charged particles.Though various compactification schemes for the extra dimensions may be considered,a practical working example [21] for the quoted phenomenological results was the M × S manifold, where M is the usual (3+1)-dimensional Minkowski space and S is the extra-dimensional sphere. Though the vortex on the sphere has a somewhat different topologyfrom the flat space (for instance, nontrivial topological configurations may exist for purelygauge fields), it has been shown [21] that the basic prperties of the flat-space solution remainintact. However, to the best of our knowledge, the index theorem for the spherical case hasnot been proven.Returning to condensed-matter examples, we point out that the vortex-on-the-sphereequations are relevant for description of fermionic states on a fullerene molecule, where theeffective vortex-like external field is induced by disclinations in the hexagon vertices [25].Possible applications of the fullerene-localized fermions are yet to be explored.Motivated by these important applications, we consider here the fermions with Majorana-like interaction with the vortex field. The fermions are assumed to live on a M × S manifold,the vortex field depending on the two S coordinates. We will relate Majorana and Diraccases for arbitrary D to each other (and ultimately to the cases of lower dimensions), study3he properties of the fermionic zero modes and derive the index theorem.The rest of the paper is organized as follows. In Sec.II we write the vortex-fermion in-teraction, study the fermion equations of motion and perform the reduction of the cases ofdifferent dimensionality to each other. We find the structure of the fermionic zero modes bya direct analysis of these equations made in B. In Sec.III we derive the corresponding indicesand prove particular index theorems which of course are in agreement with the explicit resultsof Sec.II. We summarize our conclusions, emphasise properties of two phenomenologicallymost interesting cases ( M × S – extra-dimensional extension of the particle-physics Stan-dard Model ) and discuss possible applications of other cases in the section IV. Notationsand technical details are collected in the Appendices. II. STRUCTURE OF ZERO MODES
In this paper we consider fermions on multi-dimensional manifold S × M , where M isMinkovski space. If the dimensionality of the manifold is even, we designate it by D . Ifthe dimensionality is odd, it is convenient to designate it by D + 1. The sphere contains anAbelian gauge and a scalar field which form Abrikosov-Nielsen-Olesen vortex. Its structureis described in detail in ref.[21]. We briefly recall it here. The Lagrangian for these fields is: L = p G S ( − F ab F ab + ( D a φ ) + D a φ − λ φ − v ) ) , (1)where D a φ = ∂ a φ − ieA a φ and F ab = ∂ a A b − ∂ b A a (all kinds of subscripts are described inappendix A). To describe the solution we have to introduce two patches (this discourse wasproposed in [26] ). The first of them covers the sphere with the South Pole excluded. Inthis patch one may use the following Ansatz for the solution: φ = f ( θ ) e iϕ ,A ϕ = A ( θ ) ,A θ = 0 . (2)The other patch covers the sphere with the North Pole excluded. In the overlapping regionits field values are connected to the values on the first patch by the gauge transformationwith the gauge function equal to ( − ϕ ) , A −→ A − e , φ −→ φe − iϕ . (3)4t can be shown that f ( θ ) and A ( θ ) have the following asymptotics: θ → f ∼ C F θ, A ∼ C A θ ,θ → π : f ∼ C πF , A ∼ e , (4)where C F , C A , C πF are constants.Now let us consider fermions interacting with the vortex. To construct the Lagrangianwe introduce velbain e Aα which is: e αθ = 1 R δ αθ ,e αϕ = 1 R sin θ δ αϕ ,e αµ = δ αµ (5)Then we define the spin connection R αβB = e αA ∇ B e A,β . It can be shown that R = − R = cos θ and all other components are equal to zero. The Lagrangian for the fermions is definedas L = √− G (cid:16) i ¯ N Γ α ( D ) e Aα D A N + ( g k N + Γ D ) C ( D ) N ∗ + h.c. ) (cid:17) , (6)where D A = ∂ A + R βγA β ( D ) Γ γ ( D ) − ie ( k + κ ˜Γ D ( D ) ) A A , (7) g can be chosen a real coupling constant without loss of generality (if g had a complexphase, we could include it into N ), k and κ are integer or half-integer charges. So in casewe consider the Lagrangian is: L = √− G (cid:16) i ¯ N Γ µ ( D ) ∂ µ N + i ¯ N Γ D ) R ( ∂ θ + cot θ N + i ¯ N Γ D ) R sin θ ( ∂ ϕ − ieA ϕ ( k + κ ˜Γ D ( D ) )) N + g k N + Γ D ) C ( D ) N ∗ + h.c. ) (cid:1) (8)Let us note that since N is a column of anticommuting variables, the term responsiblefor the interaction with the scalar field in (8) is non-zero only if Γ D ) C ( D ) is antisymmetric.If we take into account (A13) we see that in even number of dimensions a few cases arepossible: 5 D/2 mod 4 = 0, no term with majorana mass • D/2 mod 4 = 1, (a) C ( D ) = C (1)( D ) • D/2 mod 4 = 2, (b) C = C (2)( D ) , (c) C = C (1)( D ) • D/2 mod 4 = 3, (d) C = C (2)( D ) Also if we consider a gauge transformation of the term g Φ k N + Γ D ) C ( D ) N ∗ , we can seethat the term containing κ is allowed by the gauge symmetry only if the anticommutator { Γ D ( D ) , Γ D ) C ( D ) } = 0, that is in cases (a),(d).If the number of dimensions is odd and equal to D + 1, to obtain Lorentz-invariantcouplings we cannot assume C = C (2)( D ) or write the term containing κ . So we are left withthe following cases: • D/2 mod 4 = 0 or 3, No term with Majorana mass • D/2 mod 4 = 1, C ( D ) = C (1)( D ) • D/2 mod 4 = 2, C ( D ) = C (1)( D ) Thereby the equations for odd number of dimensions can be considered as a special case ofthe equations for even number of dimensions except for D = 2.The Lagrangian results in the following equations of motion: i Γ µ ( D ) ∂ µ N + i Γ D ) R ( ∂ θ + cot θ N + i Γ D ) R sin θ ( ∂ ϕ − ieA ϕ ( k + κ ˜Γ D ( D ) )) N + g Φ k C ( D ) N ∗ = 0 (9) N can always be decomposed into a set of plane waves propagating in M . Separating positiveand negative frequency, we get the following expression: N ( x A ) = Z d k ( N + ( k µ , x a ) e ik µ x µ + N − ( k µ , x a ) e − ik µ x µ ) . (10)After substituting this decomposition into eq.(9), multiplying it by Γ and separating theterms with different exponents we have the following equations:( − k − Γ D ) Γ i ( D ) k i ) N + + D kin N + + g Φ k Γ D ) C ( D ) N ∗− = 0 , ( k + Γ D ) Γ i ( D ) k i ) N − + D kin N − + g Φ k Γ D ) C ( D ) N ∗ + = 0 , (11)6here D kin ≡ i Γ D ) Γ D ) R ( ∂ θ + cot θ i Γ D ) Γ D ) R sin θ ( ∂ ϕ − ieA ϕ ( k + κ ˜Γ D ( D ) )) . (12)We introduce a bispinor ˜ N = N + N ∗− (13)and rewrite the equations of motion as k ˜ N = ˜ A i k i ˜ N + D ˜ N , (14)where ˜ A i = Γ D ) Γ i ( D )
00 (Γ D ) Γ i ( D ) ) ∗ and D = D kin g Φ k Γ D ) C ( D ) − g (Φ ∗ ) k (Γ D ) C ( D ) ) ∗ − D ∗ kin . (15)One can note the following anticommutation property of ˜ A i : { ˜ A i ˜ A j } = 2 δ ij , { A i D } = 0 . (16)By making use of the last relation, we conclude that if ˜ N is an eigenvector of D with aneigenvalue m : D ˜ N = m ˜ N , then ˜ N A = k i A i | k | ˜ N , where | k | = k i k i , is also an eigenvector of D but with opposite eigenvalue, D ˜ N A = − m ˜ N A . Also one can see that if ˜ N satisfies Eq. (14), ˜ N A also satisfy this equation. These relationscan be written in the following form: k ˜ N ˜ N A = m | k || k | − m ˜ N ˜ N A . (17)7his system has non-trivial solutions when ( k ) = ( k i ) + m . In this paper we willexplore the zero modes, that are eigenvectors with m = 0.So let us consider the equation D ˜ N = 0 which can also be written as D kin N + + N − g Φ k Γ D ) C ( D ) (cid:18) N + + N − (cid:19) ∗ = 0 ,D kin N + − N − i + g Φ k Γ D ) C ( D ) (cid:18) N + − N − i (cid:19) ∗ = 0 . (18)We see that these two equations have similar structure. Further we will show that theyhave a finite number of fundamental solutions N ( l ) . If we find them, we can write a generalsolution as: N + + N − X l α l N ( l ) ,N + − N − i = X l β l N ( l ) , (19)or N + = ( α l + iβ l ) N ( l ) , (20) N − = ( α l − iβ l ) N ( l ) . (21)Here α l and β l are arbitrary anticommutative coefficients. They are real because the equa-tions are linear only in terms of multiplying by real numbers. Also let us note that since α l , β l are anticommutative N ( l ) are commutative. From the last formulae we conclude that N + have the same structure as N ∗− .Now let us consider the fundamental solutions. We will reduce the higher-dimensionalequations to the case of two dimensions. The asymptotics of the solutions are discussed inthe appendix B. The equation for the fundamental solutions is: i Γ D ) ( ∂ θ + cot θ N ( l ) + i Γ D ) sin θ ( ∂ ϕ − ieA ϕ ( k + κ ˜Γ D ( D ) )) N ( l ) + gR Φ k C ( D ) ( N ( l ) ) ∗ = 0 (22)We note that inside the vortex N ( l ) obeys the effective Dirac equation in M , i Γ µ ( D ) ∂ µ N = 0 . A 2 D/ -component spinor N ( l ) can be represented as a column consisting of two (2 D/ − )-component spinors N , N : N ( l ) = N N . (23)8et us substitute this expression for N ( l ) into the eq. (22). The result is different for eachcase so let us consider them separately.In case (b) Eqs.(22) decouple into two equivalent systems: i Γ D − ( ∂ θ + cot θ N , + i Γ D − sin θ ( ∂ ϕ − ieA ϕ k ) N , + gR Φ k C (1)( D − N ∗ , = 0 . (24)In case (c), the equations decouple into the systems i Γ D − ( ∂ θ + cot θ N + i Γ D − sin θ ( ∂ ϕ − ieA ϕ k ) N + igR Φ k C (1)( D − N ∗ = 0 ,i Γ D − ( ∂ θ + cot θ N + i Γ D − sin θ ( ∂ ϕ − ieA ϕ k ) N − igR Φ k C (1)( D − N ∗ = 0 . (25)By the phase transformation N → e iπ/ N , N → e − iπ/ N (26)we reduce the equations to the same form as in case (b).We can see that Eqs. (24) have the same form as the equations of motion (22) for thefermions in ( D −
1) or ( D − D -dimensional fermions in case (a). After the substitution (23) theequations become: i Γ D − ( ∂ θ + cot θ N + i Γ D − sin θ ( ∂ ϕ − ieA ϕ ( k + κ )) N − igR Φ k C (1)( D − N ∗ = 0 ,i Γ D − ( ∂ θ + cot θ N + i Γ D − sin θ ( ∂ ϕ − ieA ϕ ( k − κ )) N + igR Φ k C (1)( D − N ∗ = 0 . (27)Let us multiply the second equation by C (1)( D − and complex conjugate it. Then we introducea new variable ˜ N = iC (1)( D − N ∗ (28)and obtain the following equations: i Γ D − ( ∂ θ + cot θ N + i Γ D − sin θ ( ∂ ϕ − ieA ϕ ( k + κ )) N − gR Φ k ˜ N = 0 ,i Γ D − ( ∂ θ + cot θ N + i Γ D − sin θ ( ∂ ϕ + ieA ϕ ( k − κ )) ˜ N − gR (Φ ∗ ) k N = 0 . (29)We now introduce a new spinor Ψ defined asΨ = N ˜ N (30)9t describes zero modes of Dirac-like interacting fermions with the Lagrangian L = √− G ¯ΨΓ µ ( D ) ∂ µ Ψ + ¯Ψ i Γ D ) R ( ∂ θ + cot θ i ¯Ψ Γ D ) R sin θ ( ∂ ϕ − ieA ϕ ( κ + k ˜Γ D ( D ) ))Ψ − g Φ k ¯Ψ (1 − ˜Γ D ( D ) )2 Ψ − (Φ ∗ ) k ¯Ψ (1 + ˜Γ D ( D ) )2 Ψ ! (31)and the equations for zero modes i Γ D ) ( ∂ θ + cot θ i Γ D ) sin θ ( ∂ ϕ − ieA ϕ ( κ + k ˜Γ D ( D ) ))Ψ − gR Φ k (1 − ˜Γ D ( D ) )2 Ψ + gR (Φ ∗ ) k (1 + ˜Γ D ( D ) )2 Ψ = 0 . (32)In case (d) we again make the change (28). Then we repeat the steps made above, makethe same change (30) and obtain the equations: i Γ D ) ( ∂ θ + cot θ i Γ D ) sin θ ( ∂ ϕ − ieA ϕ ( κ + k ˜Γ D ( D ) ))Ψ − gR Φ k (1 − ˜Γ D ( D ) )2 Ψ − gR (Φ ∗ ) k (1 + ˜Γ D ( D ) )2 Ψ = 0 . (33)Now let us reduce the system (33) to the case D = 4. If we represent Ψ as a set of N two component spinors Ψ . . . Ψ N where N is equal to 2 D − :Ψ = (cid:16) Ψ . . . Ψ N (cid:17) T and use (A4), then the system (33) decouples into the following set of equations: i Γ ( ∂ θ + cot θ n + i Γ sin θ ( ∂ ϕ − ieA ϕ ( k + κ ))Ψ n − gR Φ k Ψ N − n +1 = 0 ,i Γ ( ∂ θ + cot θ N − n +1 + i Γ sin θ ( ∂ ϕ − ieA ϕ ( κ − k ))Ψ N − n +1 − gR (Φ ∗ ) k Ψ n = 0 , (34) n = 1 ...N/ . One can see that the spinor Ψ n Ψ N − n +1 satisfies Eq.(33) for D = 4. 10n the same way we can make a reduction from four to two dimensions. We write Ψ n andΨ N +1 − n in the component form:Ψ n = ξ n ξ n , Ψ N +1 − n = η N +1 − n η N +1 − n (35)and after using (A3) rewrite Eqs. (34) as i Γ ( ∂ θ + cot θ ξ n η N +1 − n + i sin θ Γ ( ∂ ϕ − ie ( κ + k Γ ) A ) ξ n η N +1 − n − gR − Γ k + 1 + Γ ∗ ) k ! ξ n η N +1 − n = 0 , (36) i Γ ( ∂ θ + cot θ η N +1 − n ξ n + i sin θ Γ ( ∂ ϕ − ie ( κ − k Γ ) A ) η N +1 − n ξ n − gR k + 1 − Γ ∗ ) k ! η N +1 − n ξ n = 0 . (37)One can see that the first of these equations is equivalent to (33) in two-dimensional case,and the second one becomes equivalent after the change k → − k, Φ → Φ ∗ . In the appendixB the component form of these equations is given.Thus we showed that the D dimensional spinor satisfying the Dirac equation for zeromodes (33) can be expressed in terms of a 2-component spinor satisfying the Dirac equation.The structure of fundamental solution is the following:Ψ = ψ ψ , (38)where ψ = n ...0 , ψ = N +1 − n ...0 , Ψ n = ξ n ξ n , Ψ N +1 − n = η N +1 − n η N +1 − n , n = 1 , ..., N/ . (39)11or g = 0 the only non-zero components are ξ n , η N +1 − n . If g = 0, non-zero components arelinearly independent but their structure is more complicated and it is given in the table I.Explicit expressions and asymptotics of the solutions are found in the appendix B. Thespinor N ( l ) has the following structure: • case (a), N ( l ) = ψ − iC (2)( D − ψ ∗ ; (40) • case (b), N ( l ) = N ( l )( a ) , N ( l ) = N ( l )( a ) ; (41) • case (c), N ( l ) = e − iπ N ( l )( a ) , N ( l ) = e iπ N ( l )( a ) ; (42) • case (d), N ( l ) = ψ C (2)( D − ψ ∗ . (43)Finally let us note that though fermions on two-dimensional sphere without time coor-dinate obey to the Dirac equation (22), they are defined by Lagrangian different from (8)because in the case of two dimensions there are no boost transformations: L = p − G S (cid:16) iN + Γ α (2) e aα D a N + g k N + C (2) N ∗ + h.c. ) (cid:17) , (44)where D A = ∂ A + R βγA β (2) Γ γ (2) − ie ( k + κ Γ ) A A , and e aα is defined in the same way as (5). One can show that this Lagrangian produce thesame equations of motion as (22). 12 umber of solutions eigenvalue of component κ < − k κ = − k − k < κ < k κ = k κ > k K ˜ K ξ k + κ k k + κ ξ − κ − k -1 -1 η k − κ k k − κ η κ − k -1 1 a K , ˜ K are defined in the next section TABLE I. The number of non-zero components in case of different κ III. INDEX OF THE DIRAC OPERATOR
In this section we will calculate the index of the operator D defined in section II byEq.(15). Indices of Dirac operators without Majorana-like interaction have been consideredpreviously (see e.g. [27], [28]). The case of the Majorana-like interaction is more involvedbecause the Dirac operator mixes N and N ∗ . For the two-dimensional flat space this problemhas been solved in [19]. We consider the case of a sphere.Let us introduce operators of chirality which distinguish left-handed and right-handedmodes. In the case g = 0 (no Majorana-like interaction) they are: K = i Γ D ) Γ D ) = diag (cid:16) σ . . . σ (cid:17) , (45)˜ K = ˜Γ D ( D ) K . (46)These operators anticommute with the operator D kin . In case g = 0 we consider spinors(13) having doubled number of components and want to generalize these operators to savetheir anticommutation with D . To achieve this we introduce an analog of K defined as K = K K . (47)The analog of ˜ K is ˜ K = ˜ K
00 ˜ K in cases (b), (c) (48)˜ K = ˜ K − ˜ K in cases (a),(d) (49)13ow let us consider an arbitrary Dirak Hamiltonian D (it does not have to have a formconsidered in this paper) and a chirality operator K anticommmuting with D . The factthat { D, K } = 0 implies that D transforms left-handed spinors (eigenvectors of K witheigenvalue equal to 1 ) into right-handed ones (eigenvectors of K with eigenvalue equal to-1 ) and vice versa because (1 + K ) D = D (1 − K ) , (50)(1 − K ) D = D (1 + K ) . (51)Let us introduce D U = D (1 + K )2 , (52) D L = D (1 − K )2 . (53)It is easy to check that if D and K are Hermitian, D U and D L are mutually conjugated: D + U = D L , (54) D + L = D U . (55)Therefore, if we have a Dirac Hamiltonian in the space of left-handed fermions, its Hermitconjugated operator is the Dirac Hamiltonian in the space of right-handed fermions.Now let us define the index as index = dim ( ker D L D U ) − dim ( ker D U D L ) = tr e − t ( D U D L ) − tr e − t ( D L D U ) . This expression can be transformed in the following way: index = tr (cid:16) e − tD − K − e − tD K (cid:17) = tr (cid:18) e − tD − K − e − tD K (cid:19) = − tr (e − tD K ) . (56)Here we made use of a general property of any chirality operator K = 1.To find e − tD we consider an equation ddt G ( x, y, t ) = − D G, G → δ ( x − y ) at t → G is its Green function. According to our notations,in case g = 0 G is a 2 D/ *2 D/ matrix and in case g = 0 it is 2 D/ *2 D/ matrix.14o continue the calculation of G , we introduce a scalar product in the space where D acts.In sec. II we defined this space as a set of spinors having a form (13). Now we consider D in the linear span of fundamental solutions of (13) with commutative coefficients. Wehave to do this because if the coefficients were anticommutative, the scalar square of anyelement would be equal to zero. Certainly the change of coefficients from anticommutativeto commutative does not affect the equations since they are linear, and hence does not affectthe index. A basis vector in this space is: N ( l ) ( N ( l ) ) ∗ . (58)In the case g = 0 the basis element is N ( l ) . The scalar product is defined in a usual way, h χ | Ψ i = Z dV χ + Ψ . (59)In case g = 0 we define the scalar product as h χ | Ψ i = Z dV (cid:16) χ + Ψ + (cid:17) χ Ψ . (60)Here dV is an element of volume in S .The terms in the expression for the index can be transformed in the following way: tr e − t ˜ D = X l h n | e − t ˜ D K | n i = Z dV x dV y X l ( N ( l ) ) i ( x ) + G ij ( x, y, t ) K jk ( N ( l ) ) k ( y ) = Z dV x dV y G ij ( x, y ) K jk δ ik δ ( x − y ) = Z dV x G ij ( x, x ) K ji . Here | n i is a basis vector which coordinate representation is ( N ( l ) ) i ( x ), l numerates thebasis vectors and i numerates the spinor components of each vector. Also we assumed herethat X l ( N ( l ) ) k ( y ) + ( N ( l ) ) i ( x ) = δ ( x − y ) δ ik (61)15or g = 0 it can be reached in a straightforward way. Let us consider the case g = 0. If weuse the representation (13) and note that x = { θ x , ϕ x } the basis spinors are Ψ nmp ( x )Ψ ∗ nmp ( x ) = e p P mn (cos θ x ) e imϕ x e p P mn (cos θ x ) e − imϕ x (62)Here e p are basis spinors whose p-th component is equal to 1 and the others are equal to 0; P mn are Legendre polynomials (In this consideration our basis does not have to be a solutionof the Dirac equation).In this basis, X nmp (Ψ + nmp ) k ( y )Ψ inmp ( x ) = δ ( θ x − θ y ) δ ( ϕ x − ϕ y ) δ ( ϕ x + ϕ y ) δ ( ϕ x + ϕ y ) δ ( ϕ x − ϕ y ) ki (63)and tr e − t ˜ D = Z dV x D/ X i =1 G i,i ( x, x ) + D/ X i =1 ( G i, D/ + i ( x, − x ) + G D/ + i,i ( x, − x )) K. (64)Later (see eq. (76)) we see that D is block-diagonal and therefore (see (C12) ) G is block-diagonal. Consequently, we still can use the expression (61). Finally, index = Z dV x tr ( G ( x, x, t ) K ) (65)Now let us find the index in the case g = 0. The expression for D kin is: D kin = − R ∂ θ + ∂ ϕ sin θ + cot θ∂ θ − ie ( k + κ ˜Γ D ( D ) ) A ϕ sin θ ∂ ϕ + Γ D ) Γ D ) cos θ sin θ ∂ ϕ + C ! , (66)where C = −
12 sin θ + cot θ − e A ϕ sin θ (cid:16) k − κ + 2 kκ ˜Γ D ( D ) (cid:17) +Γ D ) Γ D ) (cid:18) − ie cos θ sin θ ( k + κ ˜Γ D ( D ) ) A ϕ + ie sin θ ( k + κ ˜Γ D ( D ) ) ∂ θ A ϕ (cid:19) . (67)In derivation of this expression, we took into account that A ϕ depends only on θ and doesnot depend on ϕ . From formula (C12) we can find the expression for the Green function of16he operator ( − D ). In case x = y it is equal to G ( x, x, t ) = 14 πR (cid:18) t + 1sin θ ( e ( k + κ + 2 kκ ˜Γ D ( D ) ) A ϕ + ie sin θ Γ D ) Γ D ) ( k + κ ˜Γ D ( D ) ) A ϕ cos θ ) + C (cid:19) =14 πR cot θ −
12 sin θ + ie Γ D ) Γ D ) sin θ ( k + κ ˜Γ D ( D ) ) ∂ θ A ϕ ! . (68)Since we know G , we can now easily find the index by making use of Eq. (65). The indexassosiated with the chirality K is equal to index = Z dθdϕR sin θ tr ( G ( x, x, t ) K ) = 2 D/ Z S dθdϕ π ek∂ θ A ϕ . (69)Since A ϕ is regular in all points except the South Pole, we can rewrite this expression as index = 2 D/ ek π Z L dx a A a . (70)The last integral is taken over a small curve L surrounding the South Pole . We can calculateit using the properties (2,4). So finally the index assosiated with K is equal to2 D/ − k. (71)The index assosiated with ˜ K is calculated in the same way and it is equal to2 D/ − κ. (72)Let us note that by making use of the equations for the vortex, we can transfrom Eq.(69) to express the index through the scalar field. The Lagrangian (1) implies the followingfield equations: ∂ a ( p G S F cd ( G S ) ac ( G S ) bd ) = ( G S ) ab ( ie Φ + ∂ a Φ − ie Φ ∂ a Φ + + 2 e A a | Φ | ) , (73)or A e = ( G S ) be ∂ a ( √ G S F cd ( G S ) ac ( G S ) bd )2 e | Φ | − i e | Φ | (Φ + ∂ e Φ − Φ ∂ e Φ + ) . (74)Substituting this expression for A into Eq. (70), we obtain index = − D/ ik π Z dx a Φ + ∂ a Φ − Φ ∂ a Φ + | Φ | (75)17he index assosiated with ˜ K is obtained from this expression by the change of k to κ .Now let us consider the theory with g = 0. Like the previous case, we find D : D = D kin − g | Φ | k Γ D ) C ( D ) (Γ D ) ) ∗ C ∗ ( D )
00 ( D ∗ kin ) − g | Φ | k (Γ D ) ) ∗ C ∗ ( D ) Γ D ) C ( D ) (76)In this expression, non-diagonal elements are equal to zero due to Eq. (A12). In additionone can notice that Γ C (Γ ) ∗ C ∗ is proportional to a unit matrix. We take into account that tr K = tr ˜ K = 0 and conclude that this term does not contribute to the index. Therebywe have the following expression for the index assosiated with K : index = Z dV x tr ( G ( x, x, t ) K + G ( x, x, t ) ∗ K ) = 2 D/ k. (77)The index assosiated with the chirality ˜ K is calculated in the same way. In cases (b), (c) itis equal to Z dV x tr ( G ( x, x, t ) ˜ K + G ( x, x, t ) ∗ ˜ K ) = 2 D/ κ (78)and in cases (a), (d) it is equal to Z dV x tr ( G ( x, x, t ) ˜ K − G ( x, x, t ) ∗ ˜ K ) = 0 . (79)The result in Eq.(79) can be understood in the following way. By looking at the definitionof ˜ K (49), we see that if this operator acts to a fundamental solution having a form of (58),the obtained vector does not have a form of (58). For this reason we conclude that in thesecases the zero modes are not chiral.Let us emphasize again that we obtained the index in the case g = 0 twice as much asin the case g = 0 only because we calculated them in different spaces. In the first case thespace was formed by N ( l ) and in the second case the space was formed by columns (58). Ifone calculates the index in the case g = 0 using the basis (58) the obtained answer is thesame as in case g = 0. Actually this calculation can be done just by taking g = 0 in theoperator D . IV. CONCLUSIONS
We have considered the Dirac equation with Majorana-like interaction on the vortex ona sphere. We have found the structure of its solutions which is given by (39). Also we have18alculated two indices of this Dirac operator, which correspond to two its chiralities. Inthe case g = 0 they are given by (78, 79). In the case g = 0 they are given by (71, 72).In principle in high-dimensional space one can consider some other chiralities and calculateother indices assosiated with them, but we leave this idea for the further job.Now we consider an example of the case d with D = 6. In section II the structure of thezero modes is written. By looking at the table I one can check explicitly the index for thecase g = 0. In the case g = 0 one can check the index assosiated with K by looking to thestructure of the solutions described by (38 -43) and the list of non-zero components. Alsoone can check that the zero modes are not the eigenvectors of ˜ K .The last fact is important from the phenomenological point of view. One can consider amodel where the observable world is M and extra dimensions form S (see ref. [21], [23]).In this model ˜ K is four-dimensional chirality. So if one assume that the zero modes of N areobservable particles, they are non-chiral from the four-dimensional point of view, but chiralfrom the extra-dimensional point of view. This is the difference from the particels formedby Dirac-like interaction, which are chiral from the four-dimensional point of view (see [21]). V. ACKNOWLEDGEMENTS
The author would like to thank his supervisors S.Troitsky , M.Libanov and E.Nugaevfor helpful discussions. This work was supported in part by the Dynasty Foundation. Thiswork was supported in part by the grants of the President of the Russian Federation NS-5525.2010.2, MK-1632.2011.2.
Appendix A: Notations
The manifold considered in this paper is S × M , and M is the Minkovski space. Allcoordinates of this manifold are numerated by capital Latin letters: A , B , . . . .They runfrom 0 to D − x = θ and x = ϕ . θ is a polar angle and it is countedfrom the North Pole; ϕ is an axial angle. These coordinates are numerated by small Latin19etters: a , b , . . . . So the metric tensor of the sphere is( G S ) ab = R θ (A1)Its determinant is designated by G S : G S = det( G S ) ab . The coordinates of M are numerated by Greek letters: µ , ν , . . . . The zero coordinate x is time. Spatial coordinates are numerated by Latin letters: i , j , . . . .The metric tensor forthe whole manifold is G AB = diag( 1 − ( G S ) ab − . . . − D = 3 gamma-matrices are defined as:Γ = σ , Γ = − iσ , Γ = iσ . (A3)The charge conjugation matrix is C = Γ . Here σ , σ , σ are Pauli matrices. In this paperwe also consider Γ = i Γ Γ as chirality operator.Let us construct gamma-matrices in higher-dimensional spaces. If the number of dimen-sions is even and equal to D , we can define gamma-matrices Γ A ( D ) by using ( D − γ A ( D − in the following way:Γ A ( D ) = A ( D − Γ A ( D − , A = 0 , . . . , D −
2; (A4)Γ D − D ) = −
11 0 . (A5)If we want to consider the case of odd number of dimensions which is equal to D + 1 wealso introduce Γ D ( D ) = i − . (A6)Also we introduce a projection operator ˜Γ D ( D ) = diag (cid:16) , − (cid:17) . One can check that˜Γ D ( D ) = − i Γ D ( D ) = ( − i ) D +1 Γ D ) ∗ . . . ∗ Γ D − D ) . (A7)20ne can notice that our gamma-matrices are imaginary if A = 2 , , . . . , D (any evennumber more than 0) and real otherwise. In the D-dimensional space we can define chargeconjugation matrices as C (1)( D ) = Γ D ) Γ D ) ... Γ D ( D ) , (A8) C (2)( D ) = Γ D ) Γ D ) ... Γ D − D ) . (A9)In the D + 1-dimensional space only C (1)( D ) is suitable.Defined in this way, C -matrices have the following properties in D ≥ A ( D ) C (2)( D ) = ( − D − C (2)( D ) (Γ A ( D ) ) ∗ , (A10)Γ A ( D ) C (1)( D ) = ( − D C (1)( D ) (Γ A ) ∗ . (A11)Matrix C ( D ) written without upper index denotes any of these charge conjugation matri-ces.By making use of the last formulae, one can show thatΓ D ) Γ A ( D ) Γ D ) C ( D ) + Γ D ) C ( D ) (Γ D ) ) ∗ (Γ A ( D ) ) ∗ = 0 . (A12)This relation is useful in derivation of many of the expressions mentioned in this paper.One can show that (Γ D ) C (1)( D ) ) T = ( − D ( D +1) Γ D ) C (1)( D ) , (Γ D ) C (2)( D ) ) T = ( − D ( D − Γ D ) C (2)( D ) , (A13)( C (2)( D ) ) − = ( − D ( D − C (2)( D ) , ( C (1)( D ) ) − = ( − D ( D +1) C (1)( D ) , (A14)( C (1)( D ) ) ∗ C (1)( D ) = ( − D ( D +3) , (A15)( C (2)( D ) ) ∗ C (2)( D ) = ( − ( D +1)( D − . (A16)Also in this paper we use the properties: { Γ D ( D ) , Γ D ) C ( D ) } = 0 if D/2 mod 4 = 1 or 3 (cid:2) Γ D ( D ) , Γ D ) C ( D ) (cid:3) = 0 if D/2 mod 4 = 2 (A17) (cid:2) Γ D ( D ) , C ( D ) (cid:3) = 0 if D/2 mod 4 = 1 or 3 { Γ D ( D ) , C ( D ) } = 0 if D/2 mod 4 = 2 (A18)The structure of C -matrices in the cases used in this paper is the following:21 (a) D/2 mod 4 = 1, C (1)( D ) = i C (1)( D − − C (1)( D − , (A19) • (b) D/2 mod 4 = 2, C D ) = C (1)( D − C (1)( D − , (A20) • (c) D/2 mod 4 = 2, C (1)( D ) = i − C (1)( D − C (1)( D − , (A21) • (d) D/2 mod 4 = 3, C (2)( D ) = C (1)( D − C (1)( D − . (A22) Appendix B: The solution of the equations for zero modes
Here we explore the fundamental solutions of the equations (36 - 37). In the case g = 0we find them explicitly and in the case g = 0 we find their asymptotics. We assume k > ∂ θ + cot θ ξ n − i sin θ ( ∂ ϕ − ie ( κ + k ) A ϕ ) ξ n − igR Φ k η N +1 − n = 0 , ( ∂ θ + cot θ ξ n + i sin θ ( ∂ ϕ − ie ( κ + k ) A ϕ ) ξ n + igR Φ k η N +1 − n = 0 , ( ∂ θ + cot θ η N +1 − n ) + i sin θ ( ∂ ϕ − ie ( κ − k ) A ϕ )( η N +1 − n ) + igR (Φ ∗ ) k ξ n = 0 , ( ∂ θ + cot θ η N +1 − n ) − i sin θ ( ∂ ϕ − ie ( κ − k ) A ϕ )( η N +1 − n ) − igR (Φ ∗ ) k ξ n = 0; (B1)22o get rid of the dependence on ϕ we use the following ansatz: ξ n = u L exp (cid:18) − i ( l + 12 ) ϕ + Z θ dθ eA ϕ sin θ ( κ + k ) (cid:19) ,ξ n = u U exp (cid:18) i ( l + 12 ) ϕ + Z θ dθ eA ϕ sin θ ( − κ − k ) (cid:19) ,η N +1 − n = v L exp (cid:18) − i (2 k + l + 12 ) ϕ + Z θ dθ eA ϕ sin θ ( − κ + k ) (cid:19) ,η N +1 − n = v U exp (cid:18) − i (2 k − l −
12 ) ϕ + Z θ dθ eA ϕ sin θ ( κ − k ) (cid:19) . (B2)Here u L , u U , v L , v U are new variables depending only on θ , and l is an integer number whichfurther will numerate the solutions. Let us note that in this ansatz, we assumed that thespinor components obey antiperiodical boundary conditions. They are chosen because if wemove on a closed curve from ϕ = 0 to ϕ = 2 π we rotate our velbain (5) by 2 π . This resultsin the change of the sign of a fermion. In terms of the new variables the equations read ∂ θ u L − l + 1 / θ u L + cot2 u L − gRf k e − R eκAϕdθ sin θ iv L = 0 ,∂ θ u U − l + 1 / θ u U + cot2 u U + gRf k e R eκAϕdθ sin θ iv U = 0 ,∂ θ v L + 2 k + l + 1 / θ v L + cot2 v L + gRf k e R eκAϕdθ sin θ iu L = 0 ,∂ θ v U − k − l − / θ v U + cot2 v U − gRf k e − R eκAϕdθ sin θ iu U = 0 . (B3)Then we make another change: u L = y L (sin θ/ l (cos θ/ l +1 ,v L = z L (sin θ/ − k − l − (cos θ/ − k − l ,u U = y u (sin θ/ l (cos θ/ l +1 ,v U = z u (sin θ/ k − l − (cos θ/ k − l (B4)to we obtain the following equations: ∂ θ y L − igRf k e − R eκAϕdθ sin θ (tan θ/ − k − l − z L = 0 , (B5) ∂ θ z L + igRf k e R eκAϕdθ sin θ (tan θ/ k +2 l +1 y L = 0 ,∂ θ y U + igRf k e R eκAϕdθ sin θ (tan θ/ k − l − z U = 0 ,∂ θ z U − igRf k e − R eκAϕdθ sin θ (tan θ/ − k +2 l +1 y U = 0 . (B6)23ow let us consider separately the case g = 0. Here we conclude that all variables in theequations (B6) are constants: y L = y L = const ,y u = y u = const ,z L = z L = const ,z U = z U = const . (B7)Substituting them into the expressions (B2 , B4) we find the solutions for the initial variables: ξ n = y L (sin θ/ l (cos θ/ l +1 exp (cid:18) − i ( l + 12 ) ϕ + Z θ dθ eA ϕ sin θ ( κ + k ) (cid:19) ,ξ n = y U (sin θ/ l (cos θ/ l +1 exp (cid:18) i ( l + 12 ) ϕ + Z θ dθ eA ϕ sin θ ( − κ − k ) (cid:19) ,η N +1 − n = z L (sin θ/ − k − l − (cos θ/ − k − l exp (cid:18) − i (2 k + l + 12 ) ϕ + Z θ dθ eA ϕ sin θ ( − κ + k ) (cid:19) ,η N +1 − n = z u (sin θ/ k − l − (cos θ/ k − l exp (cid:18) − i (2 k − l −
12 ) ϕ + Z θ dθ eA ϕ sin θ ( κ − k ) (cid:19) . (B8)These expressions may have singularities at θ → θ → π . To investigate them we writethe asymptotics of Eqs. (B8) which are the following ( ξ = π − θ ): • θ → ξ n ∼ y L ( θ/ l exp (cid:18) − i ( l + 12 ) ϕ (cid:19) ,ξ n ∼ y U ( θ/ l exp (cid:18) i ( l + 12 ) ϕ (cid:19) ,η N +1 − n ∼ z L ( θ/ − k − l − exp (cid:18) − i (2 k + l + 12 ) ϕ (cid:19) ,η N +1 − n ∼ z u ( θ/ k − l − exp (cid:18) − i (2 k − l −
12 ) ϕ (cid:19) . • θ → π : ξ n ∼ y L ζ − k − κ − l − exp (cid:18) − i ( l + 12 ) ϕ (cid:19) ,ξ n ∼ y U ζ κ + k − l − exp (cid:18) i ( l + 12 ) ϕ (cid:19) ,η N +1 − n ∼ z L ζ κ + k + l exp (cid:18) − i (2 k + l + 12 ) ϕ (cid:19) ,η N +1 − n ∼ z u ζ − κ − k + l exp (cid:18) − i (2 k − l −
12 ) ϕ (cid:19) . (B9)24 = π − θ These expressions are regular if θ and ζ have non-negative powers. From this conditionwe find the constraints for l : ξ n : 0 ≤ l ≤ − k − κ − ,ξ n : 0 ≤ l ≤ k + κ − ,η N +1 − n : − κ − k ≤ l ≤ − k − ,η N +1 − n : κ + k ≤ l ≤ k − . Finally we can write the structure of the solutions depending on k and κ (recall that k > κ + k ≤ − : ξ n has | κ + k | modes and η N +1 − n has k − κ modes. κ + k = 0 : Only η N +1 − n has non-zero modes. Their number is 2 k .1 ≤ κ + k ≤ k − : ξ n has κ + k modes and η N +1 − n has k − κ modes. κ + k = 2 k : ξ n has 2 k non-zero modes. κ + k ≥ k + 1 : ξ n has κ + k modes and η N +1 − n has κ − k modes.Now let us consider the case g = 0. A general analytical solution to the equations (B6) isunknown. In this paper we find the asymptotics of the solutions in cases θ → θ → π .To do it we derive the second-order equations containing only y L , y U from Eqs. (B6): ∂ θ y L + (cid:18) k + 2 l sin θ − k ∂ θ ff + 2 eκA sin θ (cid:19) ∂ θ y L − ( gR ) f k y L = 0 ,∂ θ y U + (cid:18) − k + 2 l sin θ − k ∂ θ ff − eκA sin θ (cid:19) ∂ θ y U − ( gR ) f k y U = 0 . (B10)Each of these equations has two linearly independent solutions. After substituting theasymptotics of the vortex fields, eq. (4), we can find the behavior of the unknown functions: • θ → y L ∼ l c (cid:18) g R C k F k + 2 l + 2)(2 k + 1) θ k +2 (cid:19) + 2 l d gθ − l ,y U ∼ l a (cid:18) g R C k F l + 2)(2 k + 1) θ k +2 (cid:19) + 2 l b gθ k − l . θ → π : y L ∼ − l − c π (cid:18) − ( gR ) C kπF ζ k + κ + l ) (cid:19) + 2 − l − d π gζ l +2 k +2 κ +2 ,y U ∼ − l − a π (cid:18) gR ) C kπF ζ k + κ − l ) (cid:19) + 2 − l − b π gζ l − k − κ +2 . (B11)Here a , b , c , d , a π , b π , c π , d π are arbitrary constants and factors 2 l , 2 − l − and g are intro-duced here for convenience.If we substitute the last expressions into Eqs. (B6, B2 , B4), we find the behavior of ourinitial functions: • θ → ξ n ∼ (cid:0) c θ l + d gθ − l (cid:1) e − i ( l +1 / ϕ ,ξ n ∼ (cid:0) a θ l + b gθ k − l (cid:1) e i ( l +1 / ϕ ,η N +1 − n ∼ (cid:18) − ic gRC k F θ k + l +1 k + 2 l + 2 + id (2 l ) RC k F θ − k − l − (cid:19) e − i (2 k + l +1 / ϕ ,η N +1 − n ∼ (cid:18) igRC k F a θ k + l +1 l + 2 + ib (4 k − l ) RC k F θ k − l − (cid:19) e − i (2 k − l − / ϕ . • θ → π : ξ n ∼ (cid:0) c π ζ − k − κ − l − + d π gζ l + k + κ +1 (cid:1) e − i ( l +1 / ϕ ,ξ n ∼ (cid:0) a π ζ k + κ − l − + b π gζ l − k − κ +1 (cid:1) e i ( l +1 / ϕ ,η N +1 − n ∼ (cid:18) − igRC kπF c π ζ − k − κ − l l + 2 k + 2 κ + id π (2 l + 2 k + 2 κ + 2) ζ l + k + κ RC kπF (cid:19) e − i (2 k + l +1 / ϕ ,η N +1 − n ∼ (cid:18) − igRC kπF a π − l + 2 k + 2 κ ζ k + κ − l + − ib π RC kπF (2 l − k − κ + 2) ζ l − k − κ (cid:19) e − i (2 k − l − / ϕ . (B12)To select regular solutions we use the so-called different signs theorem (see also [21],[20]). It states that if y L,U and ∂ θ y L,U have the same sign at one interior point θ of theinterval (0 , π ), then they have the same sign at all points of the interval ( θ , π ). Let usprove it. If a function and its derivative have the same sign which is positive(negative) thefunction cannot change it before its derivative does it. The derivative can change its signin a point of a local maximum(minimum). From Eqs.(B10) we see that if ∂ θ y L,U = 0 and ∂ θ f /f , A ϕ do not have any singularities (this is true for the vortex fields), then y U,L and26 θ y U,L have the same sign. Thus we got the contradictory statement that in the point of alocal maximum(minimun) the second derivative is positive(negative).Now let us consider the solutions ξ n and η N +1 − n . One can see that their linearly inde-pendent components have opposite powers of ζ in case θ → π so only one of them shouldbe non-zero if we want to get a regular expression. So there are two possible cases: − k − κ − l − ≥ c π = 0 , d π = 0 ,k + κ + l ≥ c π = 0 , d π = 0 . (B13)Since the asymptotics have the power-like form, we conclude that in both cases if ζ increases, y L ( ζ ) also increases. If we want ξ n to be regular, only one of the coefficients c , d should benon-zero. If we suppose that d = 0, then from the expression for η N +1 − n we conclude that l < − k − y L ( θ ) increases. This contradicts to the different signs theorem.If we suppose that c = 0, from the expression for ξ n we conclude that l ≥ y L ( θ )also increases if θ increases. This possibility also contradicts to the different signs theorem.The case of ξ n and η N +1 − n is treated in the same way. From the asymptotics of θ → π we see that there are two cases: k + κ − l − ≥ a π = 0 , b π = 0 ,l − k − κ ≥ a π = 0 , b π = 0 . (B14)In these cases (as opposed to the previous ones) we concede that a , b = 0 together.From the condition that ξ n , η N +1 − n cannot have negative powers of θ, ξ , we conclude that0 ≤ l ≤ k −
1. Also, if we force these coefficients to have different signs, we can satisfythe different signs theorem. Let us consider the case when only one of the coefficients a , b is non-zero. For a = 0 we have l > y U ( θ ) does not satisfy the theorem; and for b = 0 we have l < k − y U ( θ ) again does not satisfy the different signs theorem.Thus we have shown that ξ n and η N +1 − n have regular solutions but ξ n and η N +1 − n donot have them.One can note that the solutions in the cases g = 0 and g = 0 have different structure.Nevertheless, if we take the expressions (B12) and find their limit in case g →
0, we obtainthe expressions having the same form as Eqs.(B9).27 ppendix C: The Green function of the Heat equation
Let us consider a general Heat equation ddt G ( x, y, t ) = ( a αβ ∂ α ∂ β + b α ∂ α + c ) G ( x, y, t ) (C1)supplemented by the condition G ( x, y, t ) → δ ( x − y ) at t → . (C2)Here ∂ α = ∂∂x α and a , b , c depend on x α . We can solve this equation in the case x ≈ y and t → G ( x, y, t ) = e S ( x,y ) /t X k = − A k ( x, y ) t k . (C3)By substituting it into Eq.(C1) and equating the terms having equal powers of t , we getthe following equations: − S = a αβ ∂ α S∂ β S, (C4) A − = − a αβ ∂ α ∂ β SA − − a αβ ∂ α S∂ β A − − b α ∂ α SA − , (C5)0 = a αβ ∂ α ∂ β SA + 2 a αβ ∂ α S∂ β A + b α ∂ α SA +( a αβ ∂ α ∂ β + b α ∂ α + c ) A − . (C6)Now we take into account that x ≈ y and assume a, b, c ( x ) ≈ a, b, c ( y ).The equation (C4)has the solution S = − a αβ ∆ x α ∆ x β , (C7)where a αβ = ( a αβ ) − , ∆ x α ≡ x α − y α . Using the expression for S we transform the Eq. (C5) to∆ x α ∂ α A − + b α a γα ∆ x γ A − = 0 . (C8)It’s solution is A − = e − bαaγα ∆ xγ (C9)28y substituting S and A − into Eq. (C6), we get0 = − A − ∆ x β ∂ β A − b α a γα ∆ x γ A + ( − b α b β a αβ c ) A − , (C10)Its general solution is A = − b α b β a αβ c + const qP α (∆ x α ) A − (C11)To avoid a singularity in the last expression we take const = 0.Finally after substituting (C7), (C9), (C11) into (C3) we have G = 14 π (cid:18) t − b α a αβ b β c (cid:19) e − t a αβ ∆ x α ∆ x β − bαaγα ∆ xk (C12)The constant π is introduced here to satisfy the condition (C2). [1] N. Arkani-Hamed, S. Dimopoulos, G. R. Dvali and J. March-Russell, Phys. Rev. D (2002)024032 [arXiv:hep-ph/9811448].[2] V. A. Rubakov, Phys. Usp. (2001) 871 [Usp. Fiz. Nauk (2001) 913][arXiv:hep-ph/0104152].[3] V. A. Rubakov, Phys. Usp. (2003) 211 [Usp. Fiz. Nauk (2003) 219].[4] Kaluza T Sitzungsber. Preuss. Akad. Wiss. Berlin , Math.-Phys. Kl. (1) 966 (1921)[5] Klein O
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