Hadamard function and the vacuum currents in braneworlds with compact dimensions: Two-branes geometry
aa r X i v : . [ h e p - t h ] D ec Hadamard function and the vacuum currents in braneworldswith compact dimensions: Two-branes geometry
S. Bellucci ∗ , A. A. Saharian † , V. Vardanyan ‡ INFN, Laboratori Nazionali di Frascati,Via Enrico Fermi 40,00044 Frascati, Italy Department of Physics, Yerevan State University,1 Alex Manoogian Street, 0025 Yerevan, Armenia
August 2, 2018
Abstract
We evaluate the Hadamard function and the vacuum expectation value (VEV) of the cur-rent density for a charged scalar field in the region between two co-dimension one branes on thebackground of locally AdS spacetime with an arbitrary number of toroidally compactified spatialdimensions. Along compact dimensions periodicity conditions are considered with general valuesof the phases and on the branes Robin boundary conditions are imposed for the field operator. Inaddition, we assume the presence of a constant gauge field. The latter gives rise to Aharonov-Bohmtype effect on the vacuum currents. There exists a range in the space of the Robin coefficients forseparate branes where the vacuum state becomes unstable. Compared to the case of the standardAdS bulk, in models with compact dimensions the stability condition imposed on the parameters isless restrictive. The current density has nonzero components along compact dimensions only. Thesecomponents are decomposed into the brane-free and brane-induced contributions. Different repre-sentations are provided for the latter well suited for the investigation of the near-brane, near-AdSboundary and near-AdS horizon asymptotics. The component along a given compact dimension isa periodic function of the gauge field flux, enclosed by that dimension, with the period of the fluxquantum. An important feature, that distinguishes the current density from the expectation valuesof the field squared and energy-momentum tensor, is its finiteness on the branes. In particular, forDirichlet boundary condition the current density vanishes on the branes. We show that, dependingon the constants in the boundary conditions, the presence of the branes may either increase ordecrease the current density compared with that for the brane-free geometry. Applications aregiven to the Randall–Sundrum 2-brane model with extra compact dimensions. In particular, weestimate the effects of the hidden brane on the current density on the visible brane.
PACS numbers: 04.62.+v, 04.50.-h, 11.10.Kk, 11.25.-w
In quantum field theory the vacuum is defined as the state of a quantum field with zero number ofquanta. The field operator does not commute with the operator of the number of quanta and, hence, ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
1n the vacuum state the field has no definite value. The corresponding quantum fluctuations are knownas zero-point or vacuum fluctuations. The properties of these fluctuations and, hence, of the vacuumstate, crucially depend on the geometry of the background spacetime (for general reviews see [1]).Not surprisingly, exact results for the physical characteristics of the vacuum can be found for highlysymmetric backgrounds only. Continuing our previous research [2, 3], in this paper we investigatethe changes in the properties of the vacuum state for a charged scalar field induced by three types ofsources: by the curved geometry, by nontrivial topology and by boundaries.As a background geometry we will consider locally anti-de Sitter (AdS) spacetime. AdS spacetimeis the maximally symmetric solution of the vacuum Einstein equations with a negative cosmologicalconstant and because of its high symmetry numerous physical problems are exactly solvable in thisgeometry. In particular, quantum field theory in AdS background has long been an active field ofresearch. There are a number of reasons for that. Much of the early interest in the seventies wasmotivated by principal questions of the quantization procedure on curved backgrounds. Among thenew features, having no analogues in quantum field theory on the Minkowski bulk, are the lack ofglobal hyperbolicity and the presence of both regular and irregular modes. In addition, the naturallength scale of the AdS geometry provides a convenient infrared regulator in interacting quantum fieldtheories without reducing the number of symmetries [4]. The natural appearance of AdS spacetimeas a ground state in supergravity and Kaluza-Klein theories and also as the near horizon geometryof the extremal black holes and domain walls has triggered a further increase of interest to quantumfield theories on AdS bulk. This motivated the developement of a parallel line of research, i.e. that ofsupersymmetric field theory models in AdS background spacetime, see e.g. [5].The AdS geometry plays the crucial role in two recent developments in high-energy physics suchas the AdS/CFT correspondence and the braneworld scenario. The AdS/CFT correspondence (see,for instance, [6]) relates string theories or supergravity in the AdS bulk with a conformal field theorylocalized on its boundary. This duality has many interesting consequences and provides a powerfultool for the investigation of gauge field theories in the strong coupling regime. Among the recentdevelopments of the AdS/CFT correspondence is the application to strong-coupling problems in con-densed matter physics. The braneworld scenario (for reviews see [7]) offers a new perspective for thesolution of the hierarchy problem between the Planck and electroweak mass scales. The main ideato resolve the large hierarchy is that the small coupling of four-dimensional gravity is generated bythe large physical volume of extra dimensions. Braneworlds naturally appear in the string/M-theorycontext and present intriguing possibilities to solve or to address from a different point of view variousproblems in particle physics and cosmology.The global geometry considered in the present paper will be different from the standard AdSone. Namely, we will assume that a part of spatial dimensions, described in Poincar´e coordinates, arecompactified to a torus. Note that the extra compact dimensions are an inherent feature of braneworldmodels arising from string and M-theories. The nontrivial topology of the background space can haveimportant physical implications in quantum field theory. The periodicity conditions imposed on fieldsalong compact dimensions modify the spectrum for zero-point fluctuations and, related to this, thevacuum expectation values (VEVs) of physical observables are changed. A well-known effect of thiskind, demonstrating the relation between quantum phenomena and global properties of spacetime,is the topological Casimir effect [8]. The Casimir energy of bulk fields induces a nontrivial potentialfor the compactification radius, providing a stabilization mechanism for moduli fields and effectivegauge couplings. The Casimir effect has also been considered as an origin for the dark energy inKaluza-Klein-type and braneworld models [9].For charged fields an important characteristic of the vacuum state is the expectation value of thecurrent density. In addition to describing the local physical structure of the quantum field, the currentacts as the source in the Maxwell equations and plays an important role in modeling a self-consistentdynamics involving the electromagnetic field. The VEV of the current density for a charged scalar fieldin the background of locally AdS spacetime with an arbitrary number of toroidally compactified spatial2imensions has been considered in [2] (for a recent review of quantum filed-theoretical effects in toroidaltopology see [10]). Both the zero and finite temperature expectation values of the current density forcharged scalar and fermionic fields in background of the flat spacetime with toroidal dimensions wereinvestigated in [11, 12]. The vacuum current densities for charged scalar and Dirac spinor fields inde Sitter spacetime with compact spatial dimensions are considered in [13]. The effects of nontrivialtopology induced by the compactification of a cosmic string along its axis have been discussed in [14].As the third source for the vacuum polarization, we will consider two co-dimension one branesparallel to the AdS boundary. The effects induced by a single brane were studied in [3]. The influenceof boundaries on the vacuum currents in topologically nontrivial flat spaces are studied in [15, 16]for scalar and fermionic fields. Note that, motivated by the problems of radion stabilization andthe cosmological constant generation, the investigations of the vacuum energy and related forces forbranes on AdS bulk have attracted a great deal of attention (see, for instance, the references in [17]).The Casimir effect in higher-dimensional generalizations of the AdS spacetime with compact internalspaces has been discussed in [18, 19, 20].The organization of the paper is as follows. The next section is devoted to the description ofthe background geometry, the configuration of the branes, the boundary conditions, and the fieldcontent. In section 3, we evaluate the Hadamard function in the region between the branes. Thesingle brane contributions are explicitly separated and an integral representation for the interferencepart is obtained well adapted for the investigation of the VEVs for physical quantities bilinear in thefield operator. In section 4, the expression for the Hadamard function is used for the investigation ofthe vacuum current in the region between the branes. The behavior of the current density in variousasymptotic regions of the parameters is discussed. Numerical examples are presented in the case whenthe Robin coefficients on separate branes are the same. The applications of the results to the Randall-Sundrum 2-brane model with extra compact dimensions are given in section 5. The main results of thepaper are summarized in section 6. Alternative representations for the Hadamard functions, adaptedfor the investigation of the near-brane asymptotic of the vacuum current, are provided in Appendix.
First we will describe the bulk geometry. The corresponding metric tensor is given by the ( D + 1)-dimensional line element ds = g µν dx µ dx ν = e − y/a η ik dx i dx k − dy , (2.1)where i, k = 0 , . . . , D − a is a constant, and η ik = diag(1 , − , . . . , −
1) is the metric tensor for D -dimensional Minkowski spacetime. In addition to the y coordinate, −∞ < y + ∞ , we will use thecoordinate z , defined as z = ae y/a , 0 z < ∞ . In terms of the latter, the line element is written in amanifestly conformally flat form: ds = ( a/z ) ( η ik dx i dx k − dz ) . (2.2)The local geometry given by (2.2) coincides with that for AdS spacetime described in Poincar´e coor-dinates. The hypersurfaces z = 0 and z = ∞ present the AdS boundary and horizon, respectively.The constant a is related to the Ricci scalar by the formula R = − D ( D + 1) /a and the metric tensorcorresponding to (2.2) is a solution of the vacuum Einstein equations with a negative cosmologicalconstant Λ = − D ( D − a − / y -coordinate has the topology R p × T q ,with p and q being integers such that p + q = D −
1, and T q stands for a q -dimensional torus. So, forthe ranges of the coordinates x i in (2.1) one has − ∞ < x i < + ∞ , i = 1 , , . . . , p, x i L i , i = p + 1 , . . . , D − , (2.3)3ith L i being the coordinate length of the i th compact dimension. Note that the proper lengthmeasured by an observer with a fixed z is given by L ( p ) i = ( a/z ) L i = e − y/a L i . The latter decreaseswith increasing y . This feature is seen in figure 1 where we have displayed the spatial geometry in thecase D = 2, embedded into the 3-dimensional Euclidean space. The circles correspond to the compactdimension and the thick circles are the locations of the branes (see below).Figure 1: The spatial section of the geometry at hand for D = 2 embedded into a 3-dimensionalEuclidean space. The thick circles present the locations of the branes.Here we are interested in the VEV of the current density for a charged scalar field ϕ ( x ) in thebackground geometry specified above. In addition, we will assume the presence of an external classicalgauge field A µ . The dynamics of the field is governed by the equation (cid:0) g µν D µ D ν + m + ξR (cid:1) ϕ ( x ) = 0 , (2.4)where ξ is the curvature coupling parameter, D µ = ∇ µ + ieA µ , with ∇ µ being the covariant derivativeoperator, m and e are the mass and the charge of the field quanta. The most important special casescorrespond to minimally and conformally coupled fields with ξ = 0 and ξ = ( D − / (4 D ), respectively.The background topology is nontrivial and for the complete formulation of the problem, in additionto the field equation, the periodicity conditions should be specified along compact dimensions. Herewe consider quasiperiodicity conditions ϕ ( t, x , . . . , x l + L l , . . . , y ) = e iα l ϕ ( t, x , . . . , x l , . . . , y ) , (2.5)with constant phases α l , l = p + 1 , . . . , D − α l = 0) and twisted ( α l = π ) scalars.Now we turn to the description of the boundary geometry. It consists of two co-dimension onebranes, located at y = y and y = y , y < y , on which the field operator obeys the gauge invariantRobin boundary conditions (1 + β j n µj D µ ) ϕ ( x ) = 0 , y = y j , (2.6)with j = 1 ,
2. Here, β and β are constants, n µj is the inward pointing (with respect to the regionunder consideration) normal to the brane at y = y j . In the region between the branes, y y y ,in the coordinates ( x i , y ) one has n µj = δ j δ µD , where δ = 1 and δ = −
1. The locations of the branesin terms of the conformal coordinate z we will denote by z and z , z j = ae y j /a . For the properdistance between the branes one has y − y = a ln( z /z ). Boundary conditions of the type (2.6)appear in a number of physical problems, including the considerations of vacuum effects for a confinedcharged scalar field in external fields [21], gauge field theories, quantum gravity and supergravity[22, 23], and in models where the boundaries separate different gravitational backgrounds [24]. TheRobin boundary conditions naturally arise in braneworld models (see below). A more general class ofboundary conditions in the context of AdS/CFT correspondence, that include tangential derivativesof the field on the boundary, has been discussed in [25, 26]. In the corresponding approach the4oundary conditions are implemented by adding the surface term in the action for a scalar field thatcontains a boundary kinetic term. The latter leads to the modification of the standard Klein-Gordoninner product by a boundary term. It has been shown that the appropriate choice of the surfaceaction makes the modes with non-Dirichlet boundary conditions on the AdS boundary normalizable.However, because of the lack of a manifestly positive inner product, ghosts may appear in the bulktheory. The presence of the brane sufficiently far from the AdS boundary can serve as a mechanismto banish these ghosts [26].In what follows we will consider the gauge field configuration with constant A µ . In this case, bythe gauge transformation ϕ ( x ) = e − ieχ ( x ) ϕ ′ ( x ), A µ = A ′ µ + ∂ µ χ ( x ), with the function χ ( x ) = A µ x µ ,we can pass to a new gauge with the zero vector potential, A ′ µ = 0. However, the vector potentialof the former gauge will not completely disappear from the problem. It will enter in the periodicityconditions for the new field operator: ϕ ′ ( t, x , . . . , x l + L l , . . . , y ) = e i ˜ α l ϕ ′ ( t, x , . . . , x l , . . . , y ) , (2.7)where ˜ α l = α l + eA l L l , l = p + 1 , . . . , D − . (2.8)Hence, the presence of a constant gauge field is equivalent to the shift in the phases of the quasiperiod-icity conditions along compact dimensions. In particular, nontrivial phases are generated for untwistedand twisted scalars. The phase shift is expressed in terms of the magnetic flux Φ l enclosed by the l thcompact dimension: eA l L l = − π Φ l / Φ , with Φ = 2 π/e being the flux quantum (for physical effectsof gauge field fluxes in higher dimensional models with compact dimensions see, for example, [27]). We are considering a free field theory (the only interactions are with the background gravitational andelectromagnetic fields) and all the information on the properties of the quantum vacuum is encoded intwo-point functions. As such we will choose the Hadamard function, defined as the VEV G ( x, x ′ ) = h | ϕ ( x ) ϕ + ( x ′ ) + ϕ + ( x ′ ) ϕ ( x ) | i , where | i corresponds to the vacuum state. In what follows it willbe assumed that the field is prepared in the Poincar´e vacuum state. The latter is realized by themode functions of the field which are obtained by solving the field equation in Poincar´e coordinatescorresponding to (2.1) or (2.2). The VEVs of physical observables, bilinear in the field operator,such as the energy-momentum tensor and current density, are obtained from the Hadamard functionafter some differentiations and limiting transition to the coincidence limit of the arguments (with anappropriate renormalization). In what follows we will present the evaluation procedure in the gaugewith the fields ( ϕ ′ ( x ) , A ′ µ = 0), omitting the prime.By expanding the field operator in terms of a complete set of positive- and negative-energy modefunctions { ϕ ( ± ) σ ( x ) } (upper and lower signs respectively), specified by the set of quantum numbers σ and obeying the quasiperiodicity and boundary conditions of the problem at hand, the Hadamardfunction can be presented as the mode sum G ( x, x ′ ) = X σ X s = ± ϕ ( s ) σ ( x ) ϕ ( s ) ∗ σ ( x ′ ) , (3.1)where P σ includes the summation over the discrete quantum numbers and the integration over thecontinuous ones. The problem under consideration is plane symmetric and the mode function can beexpressed in the factorized form ϕ ( ± ) σ ( x ) = z D/ Z ν ( λz ) e ik r x r ∓ iωt , (3.2)5here k r x r = P D − r =1 k r x r , ω = p λ + k , k = D − X l =1 k l . (3.3)In (3.2), Z ν ( x ) is a cylinder function and ν = p D / − D ( D + 1) ξ + m a . (3.4)For a conformally coupled massless scalar field ν = 1 / ν the vacuum state becomesunstable [29]. In the discussion below we assume the values of the parameters for which ν > −∞ < k l < + ∞ , l = 1 , . . . , p , and the eigenvalues for the components along compact dimensions are obtained from theconditions (2.7): k l = (2 πn l + ˜ α l ) /L l , l = p + 1 , . . . , D − , (3.5)where n l = 0 , ± , ± , . . . . In what follows, we will denote by k q ) the squared momentum in thecompact subspace: k q ) = D − X l = p +1 k l = D − X l = p +1 (2 πn l + ˜ α l ) /L l . (3.6)Assuming that | ˜ α i | π , for the lowest value of this momentum, denoted here by k (0)( q ) , one has k (0)2( q ) = D − X i = p +1 ˜ α i /L i . (3.7)In particular, for an untwisted scalar field and for the zero gauge field one has k (0)( q ) = 0.The branes divide the space into three regions: −∞ < y < y , y < y < y , and y > y . Ingeneral, the curvature radius a can be different in these three sections, as the branes may separatedifferent phases of theory. In the braneworld scenario with two branes based on the orbifolded versionof the model the region between the branes is employed only (see below). The Hadamard functionsin the regions y < y and y > y coincide with the corresponding functions in the geometry of asingle brane located at y = y and y = y , respectively, with the same boundary conditions. Thelatter geometry is considered in [3] and here we will be mainly concentrated on the region between thebranes, y y y . In this region, the function Z ν ( λz ) in (3.2) is a linear combination of the Besseland Neumann functions J ν ( λz ) and Y ν ( λz ). Imposing the boundary condition (2.6) (with D µ = ∂ µ )on the brane y = y we find Z ν ( λz ) = C σ g ν ( λz , λz ) , (3.8)with the function g ν ( u, v ) = J ν ( v ) ¯ Y (1) ν ( u ) − ¯ J (1) ν ( u ) Y ν ( v ) . (3.9)Here and below, for a given function F ( x ), the notations with the bars are defined as¯ F ( j ) ( x ) = B j xF ′ ( x ) + A j F ( x ) , j = 1 , , (3.10)where the coefficients are given by B j = δ j β j /a, A j = 1 + DB j / . (3.11)6ote that in the special cases A j = ± νB j one has¯ F ( j ) ν ( x ) = ± B j xF ν ∓ ( x ) , for F ν = J ν and F ν = Y ν . These special cases correspond to the values β j /a = − δ j D/ ∓ ν , (3.12)for the Robin coefficients and, hence, B j = − / ( D/ ∓ ν ).From the boundary condition on the brane y = y it follows that the eigenvalues of λ are solutionsof the equation ¯ J (1) ν ( λz ) ¯ Y (2) ν ( λz ) − ¯ Y (1) ν ( λz ) ¯ J (2) ν ( λz ) = 0 . (3.13)Firstly we will assume that all the roots of this equation are real. The changes in the evaluationprocedure in the case when purely imaginary eigenvalues are present for λ will be discussed below.We denote by λ = λ n , λ n < λ n +1 , n = 1 , , . . . , the positive roots of (3.13). Note that, for a fixedinterbrane distance y − y and Robin coefficients β j , the product z λ n does not depend on the locationof the branes and on the lengths of compact dimensions. The set of quantum numbers σ specifyingthe mode functions are given by σ = ( n, k p , n q ), where k p = ( k , . . . , k p ) is the momentum in the non-compact subspace and n q = ( n p +1 , . . . , n D − ) determines the momentum in the compact subspace.The normalization coefficient C σ in (3.8) is found from the condition Z d D x p | g | g ϕ ( s ) σ ( x ) ϕ ( s ′ ) ∗ σ ′ ( x ) = δ ss ′ ω δ nn ′ δ ( k p − k ′ p ) δ n q , n ′ q , (3.14)where s, s ′ = + , − and the y -integration goes over the region between the branes, y y y . Bytaking into account that the function g ν ( λz , λz ) is a cylinder function of the order ν with respectto the second argument (containing the integration variable) and using the standard integral for thesquare of the cylinder functions (see, for instance, [30]) we get the following result | C σ | = π λ n T ν ( χ, z λ n )4 ωa D − (2 π ) p V q z , χ = z z , (3.15)where we have introduced the notation T ν ( χ, u ) = u ( ¯ J (1)2 ν ( u )¯ J (2)2 ν ( χu ) (cid:2) ( χ u − ν ) B + A (cid:3) − ( u − ν ) B − A ) − . (3.16)In (3.15), V q = L p +1 · · · L D − is the volume of the compact subspace.Substituting the mode functions into (3.1), the Hadamard function is presented in the form G ( x, x ′ ) = a − D ( zz ′ ) D/ p +1 π p − V q z X n q Z d k p e ik r ∆ x r ∞ X n =1 λ n ω n × T ν ( χ, z λ n ) g ν ( λ n z , λ n z ) g ν ( λ n z , λ n z ′ ) cos( ω n ∆ t ) , (3.17)where ∆ x r = x r − x ′ r , ∆ t = t − t ′ , and ω n = p λ n + k . The eigenvalues λ n are given implicitly, asroots of (3.13), and for that reason this representation is not well adapted for the evaluation of theVEVs. Another drawback is that the terms in the series with large n are highly oscillatory. A moreconvenient representation, free of these disadvantages, is obtained by making use of the generalizedAbel-Plana formula [31, 32] ∞ X n =1 h ( z λ n ) T ν ( χ, z λ n ) = 2 π Z ∞ h ( x ) dx ¯ J (1)2 ν ( x ) + ¯ Y (1)2 ν ( x ) − π Z ∞ dx Ω ν ( x, χx ) [ h ( ix ) + h ( − ix )] , (3.18)7ith the notations Ω ν ( u, v ) = ¯ K (2) ν ( v )¯ K (1) ν ( u ) F ( u, v ) , (3.19)and F ( u, v ) = ¯ K (1) ν ( u ) ¯ I (2) ν ( v ) − ¯ K (2) ν ( v ) ¯ I (1) ν ( u ) . (3.20)Here, I ν ( u ) and K ν ( u ) are the modified Bessel functions and for the functions with the bars we usethe notation defined by (3.10). In the case of the function h ( x ) corresponding to (3.17), the conditionsof the validity for (3.18) are satisfied if z + z ′ + | ∆ t | < z . Note that in the coincidence limit and inthe region between the branes this condition is satisfied for points away from the brane at z = z .Let us denote by G (1)1 ( x, x ′ ) the contribution to the Hadamard function coming from the first termin the right hand-side of (3.18): G (1)1 ( x, x ′ ) = ( zz ′ ) D/ (2 π ) p a D − V q X n q Z d k p e ik r ∆ x r Z ∞ dλ λ × cos(∆ t √ λ + k ) √ λ + k g ν ( λz , λz ) g ν ( λz , λz ′ )¯ J (1)2 ν ( λz ) + ¯ Y (1)2 ν ( λz ) . (3.21)It coincides with the Hadamard function in the region z > z in the geometry of a single brane at z = z and has been investigated in [3]. As a result, the application of (3.18) leads to the representation G ( x, x ′ ) = G (1)1 ( x, x ′ ) − zz ′ ) D/ (2 π ) p +1 a D − V q X n q Z d k p e ik r ∆ x r Z ∞ k du u × Ω ν ( uz , uz ) √ u − k X (1) ν ( uz , uz ) X (1) ν ( uz , uz ′ ) cosh(∆ t p u − k ) , (3.22)where X ( j ) ν ( u, v ) = I ν ( v ) ¯ K ( j ) ν ( u ) − ¯ I ( j ) ν ( u ) K ν ( v ) , j = 1 , . (3.23)For special values (3.12) of the Robin coefficients one has¯ I ( j ) ν ( x ) = B j xI ν ∓ ( x ) , ¯ K ( j ) ν ( x ) = − B j xK ν ∓ ( x ) , (3.24)with B j = − / ( D/ ∓ ν ). The second term in the right-hand side of (3.22) is induced by the presenceof the brane at z = z . Note that, extracting the Hadamard function for the bulk in the absence ofthe branes, G ( x, x ′ ), the function (3.21) is expressed as [3] G (1)1 ( x, x ′ ) = G ( x, x ′ ) − zz ′ ) D/ (2 π ) p +1 a D − V q X n q Z d k p e ik r ∆ x r Z ∞ k du × u cosh(∆ t √ u − k ) √ u − k ¯ I (1) ν ( uz )¯ K (1) ν ( uz ) K ν ( uz ) K ν ( uz ′ ) , (3.25)with the last term being the brane-induced contribution.Another representation for the Hadamard function is obtained by using the identity¯ K (2) ν ( uz )¯ I (2) ν ( uz ) I ν ( uz ) I ν ( uz ′ ) − ¯ I (1) ν ( uz )¯ K (1) ν ( uz ) K ν ( uz ) K ν ( uz ′ )= X j =1 , δ j Ω jν ( uz , uz ) X ( j ) ν ( uz j , uz ) X ( j ) ν ( uz j , uz ′ ) , (3.26)8here Ω ν ( u, v ) = ¯ I (1) ν ( u )¯ I (2) ν ( v ) F ( u, v ) . (3.27)Combining this with the expressions (3.22) and (3.25), one gets G ( x, x ′ ) = G (2)1 ( x, x ′ ) − a − D ( zz ′ ) D/ p − π p +1 V q X n q Z d k p e ik r ∆ x r Z ∞ k du u × Ω ν ( uz , uz ) √ u − k X (2) ν ( uz , uz ) X (2) ν ( uz , uz ′ ) cosh(∆ t p u − k ) . (3.28)In this formula, the function G (2)1 ( x, x ′ ) = G ( x, x ′ ) − a − D ( zz ′ ) D/ p − π p +1 V q X n q Z d k p e ik r ∆ x r Z ∞ k du × u ¯ K (2) ν ( uz )¯ I (2) ν ( uz ) I ν ( uz ) I ν ( uz ′ ) √ u − k cosh( p u − k ∆ t ) (3.29)is the Hadamard function in the geometry of a single brane at y = y when the brane y = y is absent(see also [3]).In the discussion above we have assumed that all the roots λ of the equation (3.13) are real.However, depending on the values of the coefficients in the Robin boundary conditions on the branes,this equation can have purely imaginary roots, λ = iη , η > ϕ ( ± )(im) σ ( x ) = C (im) σ z D/ X (1) ν ( ηz , ηz ) e ik r x r ∓ iω ( η ) t , (3.30)where ω ( η ) = p k − η and the function X (1) ν ( ηz , ηz ) is defined by (3.23). If η > k (0)( q ) , then for themodes with k (0)( q ) k < η the energy is purely imaginary and the vacuum state becomes unstable. Inorder to escape this instability, we will assume that η < k (0)( q ) . (3.31)Note that in the absence of compact dimensions any imaginary root for the eigenvalue equation wouldlead to the vacuum instability. Hence, in models with compact dimensions the constraints given bythe stability condition are less restrictive. The functions (3.30) obey the boundary condition on thebrane at y = y . From the boundary condition on the second brane it follows that η is the root of theequation F ( ηz , ηz ) = 0 , (3.32)with the function F ( u, v ) defined by the expression (3.20). Of course, this equation could directly beobtained from (3.13).By using the integration formula Z ηz ηz du uX (1)2 ν ( ηz , u ) = 12 h ( u + ν ) X (1)2 ν ( ηz , u ) − u ( ∂ u X (1) ν ( ηz , u )) i ηz ηz , (3.33)from the normalization condition (3.14) (with the replacement δ nn ′ → δ ηη ′ ) for the coefficient in (3.30)we find the expression | C (im) σ | = (2 π ) − p a − D η V q ω ( η ) ¯ I (1)2 ν ( ηz ) X j =1 , A j − ( η z j + ν ) B j δ j ¯ I ( j )2 ν ( ηz j ) − . (3.34)9ere we have used the relations X (1) ν ( ηz , ηz ) = − B , [ ∂ x X (1) ν ( ηz , x )] x = ηz = A / ( ηz ) . (3.35)and X (1) ν ( ηz , ηz ) = − B ¯ I (1) ν ( ηz )¯ I (2) ν ( ηz ) , [ ∂ x X (1) ν ( ηz , x )] x = ηz = A ηz ¯ I (1) ν ( ηz )¯ I (2) ν ( ηz ) . (3.36)From (3.32) it follows that ¯ I (1) ν ( ηz ) / ¯ I (2) ν ( ηz ) = ¯ K (1) ν ( ηz ) / ¯ K (2) ν ( ηz ) and, hence, in (3.34) we canreplace the I -functions by the K -functions.By taking into account the equation (3.32), it can be seen that X j =1 , B j ( η z j + ν ) − A j δ j ¯ I ( j )2 ν ( ηz j ) = η∂ u [ F ( uz , uz )] u = η ¯ I (1) ν ( ηz ) ¯ I (2) ν ( ηz ) . (3.37)Now, for the contribution of the modes (3.30) to the Hadamard function, by using the relation (3.37),we find the following expression G (im) ( x, x ′ ) = − zz ′ ) D/ (2 π ) p V q a D − X n q Z d k p e ik r ∆ x r X η ηω ( η ) I (2) ν ( ηz ) I (1) ν ( ηz ) × cos[ ω ( η )∆ t ][ ∂ u F ( uz , uz )] u = η X (1) ν ( ηz , ηz ) X (1) ν ( ηz , ηz ′ ) . (3.38)The contribution to the Hadamard function from the modes with real λ is still given by the expres-sion (3.17). However, in the evaluation procedure by using the Abel-Plana formula differences arisecompared to the case in the absence of purely imaginary roots. In the presence of the imaginary roots λ = ± iη the function used in the derivation of the Abel-Plana summation formula (see [31, 32]) haspoles on the imaginary axis. These poles should be avoided by semicircles of small radius in the righthalf-plane. The integrals over these semicircles lead to the term − i X x = ηz ¯ K (2) ν ( χx )¯ K (1) ν ( x ) h ( ix ) − h ( − ix ) ∂ x F ( x, χx ) , (3.39)which should be added to the right-hand side of (3.18). In addition, in the presence of the imaginaryroots, the second integral in (3.18) is understood in the sense of the principal value. Now, we can seethat, after the application of the generalized Abel-Plana summation formula (with the additional term(3.39)) to the series over n in (3.17), the part of the Hadamard function coming from the term (3.39)is equal to − G (im) ( x, x ′ ). Hence, this part cancels the contribution of the purely imaginary modes inthe Hadamard function. As a result, the expressions (3.22) and (3.28) remain valid in the presence ofthe imaginary modes with η < k (0)( q ) .In addition to the modes discussed above, a mode may be present for which λ = 0 and, hence, ω = k . For this mode the function Z ν in (3.2) is a linear combination of z ν and z − ν . The relativecoefficient in this combination is determined from the boundary condition at z = z and the modefunctions are presented as ϕ ( ± )(s) σ ( x ) = C (s) σ z D/ [( z/z ) ν − b ( z/z ) − ν ] e ik r x r ∓ ikt , (3.40)10ith the notation b j = 1 + ( D/ ν ) δ j β j /a D/ − ν ) δ j β j /a , j = 1 , . (3.41)Here we have assumed that β j /a = − δ j / ( D/ ∓ ν ). From the boundary condition at z = z it followsthat b ( z /z ) ν = b . (3.42)For a given interbrane distance, the equation (3.42) gives the relation between the Robin coefficients.For Dirichlet boundary condition on the branes this equation has no solutions. For Neumann boundarycondition it has a solution in the case ν = D/ z . In the Robin case, if the coefficients β j are the same for both the branes, β = β = β ,the equation (3.42) has no solutions for a/β ν − D/ β there are no modes with λ = 0.Note that for a minimally coupled field ν > D/ C (s) σ in (3.40) is determined from the normalization condition. As a result, thenormalized mode functions for the special mode with λ = 0 are presented in the form ϕ ( ± )(s) σ ( x ) = Ω( z ) ϕ ( ± )(M) σ ( x ) , (3.43)where ϕ ( ± )(M) σ ( x ) = e ik r x r ∓ ikt p π ) p kV q , (3.44)are the mode functions for a massless scalar field in D -dimensional Minkowski spacetime with thespatial topology R p × T q andΩ( z ) = z D/ ( z/z ) ν − b ( z/z ) − ν a ( D − / z (cid:20) b ν − b − ν + 2 − b b χ (cid:18) b ν − b − ν + 2 (cid:19)(cid:21) − / . (3.45)In the cases β j /a = − δ j / ( D/ ∓ ν ) the mode functions for the special mode with λ = 0 have theform (3.43) with the conformal functionΩ ( z ) = 2 (1 ∓ ν ) z D ∓ ν a D − ( z ∓ ν − z ∓ ν ) . (3.46)In the case ν = D/ λ = 0 one has ϕ ( ± )(0) σ ( x ) = C (s) σ e ik r x r ∓ ikt and for the corresponding function Ω( z ) one getsΩ ( z ) = ( D − z D − a D − [1 − ( z /z ) D − ] . (3.47)This case will be considered in the numerical examples below.As a consequence of the relation (3.43), the contribution of the mode with λ = 0 to the Hadamardfunction, G (s) ( x, x ′ ), is expressed in terms of the corresponding function for a massless scalar fieldin D -dimensional Minkowski spacetime with the spatial topology R p × T q . Denoting the latter by G (M) R p × T q ( x, x ′ ), one has G (s) ( x, x ′ ) = Ω( z )Ω( z ′ ) G (M) R p × T q ( x, x ′ ), or by making use of the expression forthe Minkowskian function: G (s) ( x, x ′ ) = 2Ω( z )Ω( z ′ )(2 π ) p +1 / V q X n q e ik l ∆ x l k p − q ) f ( p − / ( k ( q ) rX pl =1 (∆ x l ) − (∆ t ) ) . (3.48)11here k l ∆ x l = P D − l = p +1 k l ∆ x l , and f µ ( x ) = x − µ K µ ( x ) . (3.49)Note that the dependence on the mass of the field in this expression appears through the parameter ν in (3.45).Under the condition (3.42), the contribution (3.48) coming from the special mode with λ = 0should be added to the part (3.17) for the modes with λ = λ n . However, we can show that therepresentations (3.22) and (3.28) are not changed. Indeed, in the presence of the mode with λ = 0 thefunction h ( x ) in the generalized Abel-Plana formula (3.18) corresponding to the series over n in (3.17)has a simple pole at x = 0. Now, rotating the integration contour in the derivation of the Abel-Planaformula we should avoid this pole by arcs of a circle of small radius. The terms coming from theintegrals over these arcs exactly cancel the contribution (3.48) of the special mode. As a result, theformulas (3.22) and (3.28) remain valid in the presence of the mode λ = 0 as well. Having the Hadamard function, one can evaluate the VEVs of various local physical observablesbilinear in the field. The VEV of the energy-momentum tensor for a scalar field in the geometrywith two branes is investigated in [33, 34, 35] for the background with trivial topology and in [19]for models with extra compact subspaces. Our main interest here is the VEV of the current density, h | j µ ( x ) | i ≡ h j µ ( x ) i . For a charged scalar field the corresponding operator is given by the expression j µ ( x ) = ie [ ϕ + ( x ) D µ ϕ ( x ) − ( D µ ϕ + ( x )) ϕ ( x )] . (4.1)The VEV is obtained from the Hadamard function by making use of the formula h j µ ( x ) i = i e lim x ′ → x ( ∂ µ − ∂ ′ µ + 2 ieA µ ) G ( x, x ′ ) , (4.2)First of all, we can see that h j µ i = 0 for µ = 0 , , . . . , p, D . Hence, the VEVs of the charge densityand of the current density components along uncompact dimensions (including the one perpendicularto the branes) vanish.By using the expressions (3.22) and (3.28) for the Hadamard function and integrating over themomentum in the uncompact subspace, for the component of the vacuum current along the l th compactdimension one finds two equivalent representations h j l i = h j l i + h j l i ( j )1 − eC p z D +2 p − a D +1 V q X n q k l Z ∞ k ( q ) dx x × ( x − k q ) ) ( p − / Ω jν ( xz , xz ) X ( j )2 ν ( xz j , xz ) , (4.3)with j = 1 , l = p + 1 , . . . , D −
1. Here we have introduced the notation C p = π − ( p +1) / Γ(( p + 1) / . (4.4)In the formula (4.3), h j l i is the current density in the absence of the branes and (cid:10) j l (cid:11) ( j )1 is the currentdensity induced by the presence of the brane at y = y j when the second brane is absent. Hence, thelast term in the right-hand side can be interpreted as the contribution induced by the second braneat y = y j ′ , j ′ = 1 , j ′ = j . 12he contribution h j l i is investigated in [2] and is given by the expression h j l i = 2 ea − − D L l (2 π ) ( D +1) / X n q n l sin( ˜ α l n l ) cos( X i = l ˜ α i n i ) × q ( D +1) / ν − / (1 + X i n i L i / (2 z )) , (4.5)where q µα ( x ) = ( x − − µ/ e − iπµ Q µα ( x ), with Q µα ( x ) being the associated Legendre function of thesecond kind. In the region z z z , for the single brane-induced parts in (4.3) one has h j l i (1)1 = − eC p z D +2 p − a D +1 V q X n q k l Z ∞ k ( q ) dx x ( x − k q ) ) p − ¯ I (1) ν ( xz )¯ K (1) ν ( xz ) K ν ( xz ) , h j l i (2)1 = − eC p z D +2 p − a D +1 V q X n q k l Z ∞ k ( q ) dx x ( x − k q ) ) p − ¯ K (2) ν ( xz )¯ I (2) ν ( xz ) I ν ( xz ) . (4.6)The properties of these single brane contributions are discussed in [3].All the contributions to the VEV of the current density and, hence, the total current as well,are periodic functions of the phases ˜ α i with the period equal 2 π . In particular, the current is aperiodic function of the magnetic flux enclosed by compact dimensions with the period equal to theflux quantum. The VEV of the component for the current density along the l th compact dimensionis an odd function of the phase ˜ α l corresponding to the same direction and an even function of theremaining phases ˜ α i , i = l . The appearance of nonzero vacuum currents discussed here is a consequenceof the nontrivial spatial topology (though influenced by the local geometry and boundaries). This isan Aharonov-Bohm type effect related to the sensitivity of the wave function phase to the globalgeometry. For ˜ α i = πm i , with m i being an integer, the current density vanishes. The relation (2.8)shows two interrelated reasons for the appearance of the currents: nontrivial phases in the periodicityconditions and the magnetic flux enclosed by compact dimensions.By taking into account the expressions for the single brane-induced parts (4.6), the total currentdensity in the region between the branes can also be presented in the form h j l i = h j l i − eC p z D +2 p − a D +1 V q X n q k l Z ∞ k ( q ) dx x × ( x − k q ) ) p − " ¯ K (1) ν ( xz ) ¯ I (2) ν ( xz )¯ K (2) ν ( xz ) ¯ I (1) ν ( xz ) − − × " I ν ( xz ) X (1) ν ( xz , xz )¯ I (1) ν ( xz ) − K ν ( xz ) X (2) ν ( xz , xz )¯ K (2) ν ( xz ) . (4.7)The second term in the right-hand side is the brane-induced contribution. Alternatively, extractingthe single-brane contributions we can write the following decomposition h j l i = h j l i + X j =1 , h j l i ( j )1 + h j l i int , (4.8)13ith the interference part h j l i int = − eC p z D +2 p − a D +1 V q X n q k l Z ∞ k ( q ) dx x × ( x − k q ) ) p − " ¯ K (1) ν ( xz ) ¯ I (2) ν ( xz )¯ I (1) ν ( xz ) ¯ K (2) ν ( xz ) − − × " I ν ( xz ) X (2) ν ( xz , xz )¯ I (2) ν ( xz ) − K ν ( xz ) X (1) ν ( xz , xz )¯ K (1) ν ( xz ) . (4.9)The integrand in this expression decays exponentially in the upper limit for all points including thoseon the branes..Let us consider some limiting cases of the general formulas. In the limit of the large curvatureradius, a ≫ y j , m − , one has z ≈ a + y , z j ≈ a + y j , and both the order and arguments of the modifiedBessel functions in the integrand of (4.3) are large. By using the corresponding uniform asymptoticexpansions (given, for example, in [36]), to the leading order, the result for the geometry of two parallelRobin plates on the Minkowski bulk with the topology R p +1 × T q (see [16]) is obtained: h j l i (M) = h j l i (M)0 + eC p p V q X n q k l Z ∞ q m + k q ) dx ( x − m − k q ) ) p − × P j =1 , c j ( x ) e x | y − y j | c ( x ) c ( x ) e x ( y − y ) − . (4.10)where c j ( x ) = β j x − β j x + 1 , j = 1 , . (4.11)In (4.10), the current density in the boundary-free Minkowskian geometry with compact dimensionsis given by the formula [14] h j l i (M)0 = 2 eL l m D +1 (2 π ) ( D +1) / X n q n l sin( n l ˜ α l ) cos( X i = l ˜ α i n i ) f D +12 ( m ( X i n i L i ) / ) , (4.12)with f µ ( x ) defined by (3.49).For a conformally coupled massless field one has ν = 1 / h j l i = ( z/a ) D +1 h j l i (M)0 + eC p p V q X n q k l Z ∞ k ( q ) dx × ( x − k q ) ) p − P j =1 , ˜ c j ( xz j ) e x | z − z j | ˜ c ( xz )˜ c ( xz ) e x ( z − z ) − , (4.13)where we have introduced the notation˜ c j ( u ) = u − a/β j − δ j ( D − / u + a/β j + δ j ( D − / . (4.14)Note that the expression in the square brackets of (4.13), with the functions ˜ c j ( u ) replaced by c j ( u )from (4.11), coincides with the corresponding result in the region between two Robin boundaries in14inkowski bulk with spatial topology R p +1 × T q . The difference in the functions ˜ c j ( u ) and c j ( u ) isrelated to that, though the bulk geometry is conformally flat and the field equation is conformallyinvariant, the coefficient in the Robin boundary condition is not conformally invariant.As it has been shown in [16] for the Minkowski bulk and in [3] for the geometry of a single braneon AdS bulk, unlike the VEVs of the field squared and of the energy-momentum tensor, the currentdensity is finite on the branes. For the geometry under consideration, the VEV of the current densityon the brane is obtained from (4.3) with z = z j . By taking into account that X ( j ) ν ( xz j , xz j ) = − B j ,we get h j l i z = z j = h j l i + h j l i ( j )1 ,z = z j − eC p z D +2 j B j p − a D +1 V q X n q k l × Z ∞ k ( q ) dx x ( x − k q ) ) p − Ω jν ( xz , xz ) , (4.15)where the last term is the current density induced by the second brane on the brane at z = z j . ForDirichlet boundary condition the single brane and the second brane induced parts and, hence, alsothe total current, vanish on the branes.In the limit when the left brane tends to the AdS boundary, z →
0, one gets h j l i ≈ h j l i + h j l i (2)1 − − p − ν eC p a D +1 V q A + B νA − B ν z ν z D +2 ν Γ ( ν ) X n q k l × Z ∞ k ( q ) dx x ν +1 ( x − k q ) ) p − X (2)2 ν ( xz , xz )¯ I (2)2 ν ( xz ) . (4.16)To the leading order we obtain the VEV in the geometry of a single brane at z = z . The contributioncoming from the brane at z = z , corresponding to the last term in (4.16), decays as z ν . When theright brane is close to the AdS horizon, z → ∞ , our starting point will be the expression (4.3) with j = 1. In the limit under consideration one hasΩ ν ( xz , xz ) ≈ (2 δ B − πe − xz ¯ K (1)2 ν ( xz ) . (4.17)The dominant contribution to the integral in the right-hand side of (4.3) comes from the region nearthe lower limit of the integration and from the term with n q = 0 with the minimal value of k ( q ) = k (0)( q ) .In the leading order one gets h j l i ≈ h j l i + h j l i (1)1 + (1 − δ B ) e ˜ α l z D +2 p π ( p − / a D +1 L l V q × X (1)2 ν ( k (0)( q ) z , k (0)( q ) z )¯ K (1)2 ν ( k (0)( q ) z ) e k (0)( q ) z ( k (0)( q ) /z ) ( p +1) / , (4.18)with k (0)( q ) z ≫
1. Hence, when the right brane tends to the AdS horizon, its contribution to the currentdensity, for a fixed value of z , is suppressed by the factor exp( − k (0)( q ) z ).Now we turn to the asymptotics in the limiting cases for the lengths of compact dimensions.Firstly, consider the limit when the length of the one of compact dimensions, say L r , r = l , is muchlarger than other length scales, L r ≫ L i , z . In this case the dominant contribution to the series over n r in (4.7) comes from large values of | n r | and the corresponding summation can be replaced by theintegration in accordance with P + ∞ n r = −∞ → ( L r /π ) R ∞ dk r . Then, passing to a new integration variable15 = q x − k q ) and introducing polar coordinates in the plane ( k r , u ), after the integration over theangular part, we can see that, to the leading order, the result is obtained for the current density inthe model where the r th dimension is uncompactified. In the opposite limit of small lengths of the r th dimension, L r ≪ L i , z , under the condition | ˜ α r | < π , in the expression (4.7) the contribution ofthe term with n r = 0 dominates. The behavior of the component of the current density along the l thcompact dimension crucially depends whether the phase ˜ α r is zero or not. In the case ˜ α r = 0, theleading term obtained from the right-hand side coincides with the current density in the D -dimensionalmodel, with the excluded r th compact dimension divided by L ( p ) r = aL r /z . Recall that the latter isthe proper length of the r th dimension measured by an observer with the fixed coordinate z . For thecase ˜ α r = 0, the arguments of the modified Bessel functions in the integrand of (4.7) are large. Bymaking use of the corresponding asymptotic formulas, we can see that the contribution of the singlebrane at z = z j is suppressed by the factor exp( − | ˜ α r || z − z j | /L r ), whereas the interference partdecays as exp[ − | ˜ α r | ( z − z ) /L r ].If the length of the l th compact dimension is much smaller than the remaining lengths, L l ≪ L i ,the main contribution to the series over n q − = ( n p +1 , . . . , n l − , n l +1 , . . . , n D − ) in (4.3) comes fromlarge values of | n i | , i = p + 1 , . . . , D − i = l . In this case the corresponding summation can bereplaced by the integration in accordance with X n q − f ( k ( q − ) → − q π − ( q − / V q L l Γ(( q − / Z ∞ du u q − f ( u ) , (4.19)where k q − = k q ) − k l . As the next step, instead of x we introduce a new integration variable w = q x − u − k l . Passing to the polar coordinates in the plane ( u, w ), after the evaluation of theintegral over the angular variable, we find h j l i ≈ h j l i + h j l i ( j )1 − − D π (1 − D ) / ez D +2 a D +1 L l Γ(( D − / + ∞ X n l = −∞ k l Z ∞| k l | dx x × ( x − k l ) D − Ω jν ( xz , xz ) X ( j )2 ν ( xz j , xz ) . (4.20)A similar transformation is done with the single brane part h j l i ( j )1 and the right-hand side of (4.20)coincides with the corresponding result in the model with a single compact dimension of the length L l . When, in addition to the condition L l ≪ L i , one has L l ≪ z j , in the integration range of (4.20)the arguments of the modified Bessel functions are large and we use the corresponding asymptoticexpressions [36]. For the single brane and interference parts one gets h j l i ( j )1 ≈ − γ j eL l ( z/L l ) D +1 sgn( ˜ α l )(4 π ) ( D − / a D +1 | ˜ α l | ( D − / e − | ˜ α l || z − z j | /L l ( | z − z j | /L l ) ( D − / , h j l i int ≈ γ γ eL l ( z/L l ) D +1 sgn( ˜ α l )(4 π ) ( D − / a D +1 | ˜ α l | ( D − / e − | ˜ α l | ( z − z ) /L l [( z − z ) /L l ] ( D − / , (4.21)with L l ≪ | z − z j | . Here, γ j = 2 δ B j − j = 1 ,
2. As it is seen, both the single brane andinterference contributions decay exponentially and the decay of the interference part is stronger. Inthe same limit, for the boundary-free part one has h j l i ≈ e Γ(( D + 1) / π ( D +1) / a D +1 L l ( z/L l ) D +1 ∞ X n =1 sin( ˜ α l n ) n D , (4.22)and it dominates in the total VEV. Hence, in the limit under consideration the brane-induced partsare mainly concentrated near the branes within the range | z − z j | . L l .16nother representation for the VEV of the current density is obtained from the decomposition (A.2)for the Hadamard function, where the part G l ( x, x ′ ), given by (A.3), is induced by the compactificationof the l th dimension. The first term in the right-hand side does not contribute to the current densityand, after the integrations, in the absence of the modes with λ <
0, for the VEV we find the followingexpression h j l i = ea − − D z D +2 π ) p/ − V q L pl z ∞ X s =1 sin ( s ˜ α l ) s p +1 ∞ X n =1 λ n T ν ( χ, λ n z ) × g ν ( λ n z , λ n z ) X n q − g p/ ( sL l q λ n + k q − ) , (4.23)with the function g µ ( x ) = x µ K µ ( x ) . (4.24)In the model with a single compact dimension x l , the corresponding formula is obtained from (4.23)putting p = D − k ( q − = 0, V q = L l and omitting the summation P n q − . The vanishing of thevacuum currents on the branes in the case of Dirichlet boundary condition is explicitly seen from (4.23)by taking into account that g ν ( λ n z , λ n z j ) = 0 for j = 1 ,
2. Due to the presence of the Macdonaldfunction, the series in the right-hand side are strongly convergent for all values of z . In particular,the representation (4.23) explicitly shows the finiteness of the current density on the branes. For theVEV of the current density on the brane at z = z j from (4.23) one gets h j l i z = z j = − ea − − D B j z D +2 j (2 π ) p/ V q L pl ∞ X s =1 sin ( s ˜ α l ) s p +1 ∞ X n =1 λ n ¯ J ( j )2 ν ( z j λ n ) × X i =1 , ( z i λ n − ν ) B i + A i δ i ¯ J ( i )2 ν ( z i λ n ) − X n q − g p/ ( sL l q λ n + k q − ) , (4.25)for j = 1 , λ λ = iη , η >
0, assuming that η < k (0)( q − , with k (0)( q − being the minimal value for k ( q − , the corresponding contribution to thecurrent density is formally obtained from (4.23) by making the replacements λ n → iη , P n → P η .If | ˜ α i | π , then one has k (0)2( q − = P D − i = p +1 , = l ˜ α i /L i . For the possible special mode with λ = 0 itscontribution to the current density is given by the expression h j l i (s) = 4 e Ω ( z )( z/a ) (2 π ) p/ V q L pl ∞ X n =1 sin ( n ˜ α l ) n p +1 X n q − g p/ ( nL l k ( q − ) . (4.26)The factor ( z/a ) in this expression arises when one passes from the covariant component of thecurrent density to the contravariant one. In models with a single compact dimension from here we get h j l i (s) = 2 e Γ( D/ ( z )( z/a ) π D/ L D − ∞ X n =1 sin ( n ˜ α l ) n D − . (4.27)In particular, for Neumann boundary conditions on both the branes and for ν = D/
2, the factor Ω ( z )does not depend on z and is given by (3.47).The representation (4.23) is well adapted for the investigation of the asymptotic behavior for largevalues of L l compared with the other length scales. In this limit one has L l λ n ≫ g p/ ( x ) is large. The dominant contribution to the VEV comes from the mode with17he lowest λ n and from the term with s = 1. By using the asymptotic for the function g p/ ( x ),it is seen that the current density is suppressed by the factor exp( − L l q λ + k (0)2( q − ). If the modewith λ = 0 is present, the corresponding asymptotic directly follows from (4.26). For k (0)( q − >
0, thecontribution of the term with n = 1 and n q − = 0 dominates and the corresponding current densitydecays as exp( − L l k (0)( q − ). This decay is weaker than for the modes with positive λ . In the case k (0)( q − = 0, the decay of the contribution of the zero mode goes down like a power-law, i.e. 1 /L p +1 l .The limit of small distances between the branes, compared with the AdS curvature radius, corre-sponds to the ratio z /z close to 1. With decreasing z /z the eigenvalues λ n increase and tend toinfinity in the limit z /z →
1. With this feature, from (4.23) it follows that the contribution of themodes with positive λ to the VEV of the current density tends to zero in the limit when the distancebetween the branes tends to 0. This is not necessary to be the case for the contribution of the specialmode with λ = 0. For example, in the case of Neumann boundary condition and for ν = D/
2, thecurrent density from the special mode is given by (4.26) with Ω ( z ) defined by (3.47). It diverges inthe limit z /z → h j l i = h j l i + h j l i (1)1 + 4 ea − D z D (2 π ) p/ V q L pl ∞ X n =1 sin ( n ˜ α l ) n p +1 X n q − Z ∞ k ( q − dx × xw p/ ( nL l q x − k q − )Ω ν ( xz , xz ) X (1)2 ν ( xz , xz ) . (4.28)with the notation w ν ( x ) = x ν J ν ( x ) . (4.29)By taking into account the corresponding formula from [3] for the contribution h j l i (1)1 , this expressionis presented as h j l i = h j l i + 4 ea − D z D (2 π ) p/ V q L pl ∞ X n =1 sin ( n ˜ α l ) n p +1 X n q − Z ∞ k ( q − dx × xw p/ ( nL l q x − k q − ) " ¯ K (1) ν ( xz ) ¯ I (2) ν ( xz )¯ K (2) ν ( xz ) ¯ I (1) ν ( xz ) − − × " X (1) ν ( xz , xz )¯ I (1) ν ( xz ) I ν ( xz ) − X (2) ν ( xz , xz )¯ K (2) ν ( xz ) K ν ( xz ) . (4.30)In the presence of the mode with λ <
0, the representations (4.28) and (4.30) are valid under thecondition η < k (0)( q − .In the figures below all the graphs are plotted for a minimally coupled massless scalar field in D = 4 (except the figure in section 5, where we consider the model with D = 5) in the model witha single compact dimension ( q = 1, p = D −
2) of the length L and with the phase ˜ α l = ˜ α . For thismodel one has ν = 2 and, hence, ν = D/
2. As it has been discussed above, in this case for Neumannboundary condition there is a special mode with λ = 0. The corresponding contribution to the currentdensity is given by (4.27) with Ω ( z ) from (3.47). In the numerical examples with Robin boundaryconditions we assume that the Robin coefficients for the branes are the same: β = β ≡ β . In thiscase there are no modes with imaginary λ for β
0. In order to see the behavior of the imaginarymodes as a function of the Robin coefficient in the range β >
0, in figure 2 we have plotted the rootsof the equation (3.32), multiplied by L , as a function of β/a for fixed values z /L = 0 . z /L = 1.In the range 0 < β/a < . β/a there are two roots. For β/a > . β/a . For a given ˜ α , the vacuum is stable if β β > β c where β c is the root of the equation Lη ( β/a ) = ˜ α . This root is the abscissa of the intersection pointof the left curve in figure 2 with the horizontal line Lη = ˜ α . For example, in the case ˜ α = π/ β c /a ≈ . λ versus the Robin coefficient for fixed values z /L = 0 . z /L = 1,and ˜ α = π/ D − x l = const. The latter is given by the quantity n l h j l i , where n l = a/z is the normal tothe hypersurface. This quantity is the current density measured by an observer with a fixed value forthe coordinate z . Indeed, in order to discuss the physics from the point of view of that observer, it isconvenient to introduce rescaled coordinates x ′ i = ( a/z ) x i . With these coordinates the warp factor inthe metric for the subspace parallel to the branes is equal to one and they are physical coordinates ofthe observer. For the current density in these coordinates one has h j ′ l i = ( a/z ) h j l i which is exactlythe quantity presented in the graphs below. In figure 3 we have displayed the dependence of thecurrent density on the phase ˜ α for fixed values z /L = 0 . z /L = 1, z/L = 0 .
75. The graphs areplotted for Dirichlet, Neumann and Robin (for β/a = − , −
3, numbers near the curves) boundaryconditions. The dashed curve presents the current density in the geometry without branes. Recallthat the current density is an odd periodic function of ˜ α with the period 2 π .As it is seen from figure 3, for β | β | . In order toshow the dependence of the current density on the coefficient in Robin boundary condition, in figure4 the current density is plotted versus β/a for z /L = 0 . z /L = 1, z/L = 0 .
75, and ˜ α = π/
2. Aswe have noted above, for these values of the parameters and in the region 0 < β/a < .
845 there aremodes with imaginary λ = iη for which Lη > ˜ α . This means that in this region the vacuum state isunstable. In figure 4, the instability region is between the ordinate axis and the dotted vertical line,corresponding to β/a = 4 . | β | theresults for Robin boundary condition tend to the one for Neumann condition, whereas for β → − β < β >
0) the modulus of the currentdensity for Robin boundary condition is smaller (larger) than that for the Neumann case.In figure 5, the current density is plotted in the region between the branes as a function of z/L for z /L = 1 (left panel) and z /L = 2 (right panel). For the other parameters we have taken z /L = 0 . α = π/
2. The graphs are presented for Dirichlet, Neumann and for Robin boundary conditions (withthe numbers near the curves being the values of β/a ). For z /L = 2 the dependence of the roots ofthe equation (3.32) for imaginary modes on β/a is qualitatively similar to that depicted in figure 2.The dashed curve in the right panel corresponds to the current density in the brane-free geometry.19igure 3: The vacuum current density in the region between the branes as a function of the phase inthe quasiperiodicity condition along the compact dimension. The graphs are plotted for scalar fieldswith Dirichlet, Neumann and Robin (for β/a = − , −
3, numbers near the curves) boundary conditionsand for fixed values z /L = 0 . z /L = 1, z/L = 0 .
75. The dashed curve corresponds to the currentdensity in the absence of branes.Figure 4: The VEV of the current density as a function of the Robin coefficient for z /L = 0 . z /L = 1, z/L = 0 .
75, and ˜ α = π/
2. The dashed lines present the current densities for Dirichlet andNeumann boundary conditions. 20igure 5: The current density as a function of z/L for two different values of the location of theright brane: z /L = 1 (left panel) and z /L = 2 (right panel). The graphs are plotted for Dirichlet,Neumann and Robin (the numbers near the curves are the corresponding values of the ratio β/a )boundary conditions for fixed z /L = 0 .
5, ˜ α = π/ z/L = 1 as a function of the ratio z /L . Thehorizontal dashed line corresponds to the current density in the absence of the branes. We assumethat the observation point has equal proper distances from the left and right branes. This meansthat z = √ z z and, hence, for the example in figure 6 one has z /L = L/z . Under this conditionthe proper distance between the branes is related to the ratio z /L by y − y = 2 a ln( z /L ). Hence,the figure 6 presents the current density as a function of the proper distance between the branesat the fixed observation point in the middle between the branes. For all the boundary conditions,except the Neumann one, the current density tends to zero in the limit y − y →
0. For Neumannboundary condition the current density tends to infinity. This behavior of the VEV, as a function ofthe interbrane distance, is in accordance with the general analysis given above. For the values of theparameters we have taken one has ν = D/ λ = 0. The contribution diverging in the zero distance limit comes from thismode. The contribution of the modes with λ > α ) is smaller than the boundary-free part and for the Neumann caseit is bigger. In particular, this means that the branes with Dirichlet (Neumann) boundary conditionssuppress (enhance) the vacuum currents. By using the results given above we can obtain the vacuum current densities in generalized Randall–Sundrum braneworld models [37] with extra compact dimensions. In these models the y -direction iscompactified on an orbifold, S /Z , with − b y b and with the fixed points y = 0 and y = b . Thelatter are the locations of the hidden and visible branes, respectively. The corresponding line elementis given by (2.1) with the replacement y → | y | . Because of this, the Ricci scalar contains δ -functionterms located on the branes: R = R AdS + 4 D [ δ ( y ) − δ ( y − b )] /a . In addition, the action for a scalarfield may involve the contributions of the form S b = − Z d D +1 x p | g | [ c δ ( y ) + c δ ( y − b )] ϕ + ( x ) ϕ ( x ) , (5.1)21igure 6: The current density at fixed observation point corresponding to z/L = 1 as a function ofthe ratio z /L . The observation point has equal proper distances from the left and right branes and,hence, z /L = L/z .where the constants c and c are the so-called brane mass terms. Now, the equation for the radial partof the mode functions contains the δ -function terms coming from the Ricci scalar and from the branemass terms. The boundary conditions for these functions are obtained by integrating the equationnear the branes. For fields even under the reflection y → − y (untwisted scalar fields) the boundaryconditions obtained in this way are of the Robin type with the coefficients (see [34, 38, 39]) β j a = − ac j + 4 Dξδ j , j = 1 , . (5.2)For odd fields (twisted scalars) Dirichlet boundary conditions are obtained on both the branes.Now the integration over y in the normalization integral (3.14) goes over the range − b y b . Asa consequence of this an additional factor 1 / / y = 0, y = b .In braneworld models of the Randall–Sundrum type the standard model fields are located on thebrane at y = b (visible brane) and it is of interest to consider the current density induced by a bulkscalar field on this brane. In figure 7, in the model with D = 5, q = 1 (i.e. the Randall–Sundrummodel with a single compact extra dimension), the current density is plotted on the visible brane, z = z , as a function of the location of that brane for the length of the compact dimension L = a .For the phase in the quasiperiodicity condition we have taken ˜ α = π/ β /a (the Robin coefficient for the visible brane). In the numerical evaluations we haveused the representation (4.25) with j = 2 and with an additional factor 1/2. Note that in the caseof Neumann boundary condition on both the branes the contribution of the special mode with λ = 0should be added separately. This contribution is given by (4.27) with the function Ω ( z ) from (3.47)and, again, with an extra factor 1/2.In the original Randall–Sundrum 2-brane model, the hierarchy problem between the gravitationaland electroweak scales is solved for the interbrane distance about 37 times larger than the AdScurvature radius a . In the setup we have considered this corresponds to large values of z comparedwith z . If, in addition, one has z ≫ L i , the effect of the hidden brane on the current density atthe location of the visible brane can be estimated by using the representation (4.15) with j = 2.22igure 7: The current density on the visible brane for a minimally coupled massless scalar field in theRandall–Sundrum model with a single extra compact dimension as a function of the brane location.The graphs are plotted for L = a and ˜ α = π/ β /a ) boundary conditions are imposed.The contribution induced by the hidden brane is given by the last term in the right-hand side. Thecorresponding expression is simplified by taking into account that for z ≫ L i the argument xz ofthe modified Bessel functions is large in the integration range. The dominant contribution comes fromthe region near the lower limit of the integral and, under the assumption | ˜ α l | π , from the term n q = 0, corresponding to the lowest value of the momentum in the compact subspace. By using theasymptotic expressions of the modified Bessel functions for large arguments, to the leading order wefind h j l i RS ,z = z ≈ h j l i + 12 h j l i (2)1 − eπ (1 − p ) / z q +22 ˜ α l p a D +1 V q × ¯ I (1) ν ( k (0)( q ) a )¯ K (1) ν ( k (0)( q ) a ) ( z k (0)( q ) ) p − e − k (0)( q ) z , (5.3)where for the location of the hidden brane we have taken z = ae y /a = a . The contributions h j l i and h j l i (2)1 are given by the formulas from the previous section with z = z . As is seen, the effects ofthe hidden brane on the visible brane are suppressed by the exponential factor exp( − k (0)( q ) z ). We have investigated the combined effects of the background geometry, the nontrivial topology and thebranes on the VEV of the current density for a charged scalar field with a general curvature couplingparameter. In order to have an exactly solvable problem, a highly symmetric locally AdS geometry isconsidered with an arbitrary number of toroidally compactified spatial dimensions. Along the compactdimensions the field obeys quasiperiodicity conditions with arbitrary constant phases and, in addition,we have assumed also the presence of a constant gauge field. By a gauge transformation, the latter isequivalent to the shift of the phases in the periodicity conditions equal to the magnetic flux enclosedby a compact dimension, measured in units of the flux quantum. As the geometry of boundaries wehave considered two branes, parallel to the AdS boundary, on which the field operator obeys Robinboundary conditions, in general, with different coefficients. In the model at hand, all the properties23f the vacuum state can be extracted from the two-point functions and, as the first step, we haveevaluated the Hadamard function.In the region between the branes the eigenvalues of the quantum number corresponding to thecoordinate perpendicular to the branes are roots of the equation (3.13). In addition to an infinitenumber of modes with positive λ , depending on the coefficients in the boundary conditions on thebranes, this equation may have modes with purely imaginary λ = iη and also modes with λ = 0.In order to escape the vacuum instability, we have assumed the condition (3.31) with k (0)( q ) being theminimal value of the momentum in the compact subspace. Note that in the corresponding modelswith trivial topology the presence of any mode with imaginary λ leads to the vacuum instability.The modes with λ = 0 are present under the condition (3.42) on the parameters of the model. Fora given interbrane distance, this condition gives a relation between the Robin coefficients. In theexpression of the Hadamard function, for the summation of the series over the positive roots of theeigenvalue equation (3.13) we have employed the generalized Abel-Plana formula. This allowed usto extract explicitly the single brane contribution and to present the second brane-induced part interms of the integral rapidly convergent for points away from the branes. Then, we have shownthat the representations (3.22) and (3.28), obtained in this way, remain valid in the presence ofthe modes with λ l th compact dimension, are given in Appendix. In these representations the Hadamardfunction is decomposed into the contribution corresponding to the model with an uncompactified l thdimension and the part induced by the compactification of the latter to S . The first term does notcontribute to the component of the current density along the l th dimension. The representationsobtained in this way are well adapted for the investigation of the VEVs on the branes.Given the Hadamard function, the VEV of the current density is evaluated with the help of therelation (4.2). The VEVs of the charge density and of the current components along uncompactifieddimensions vanish. The component of the current density along the l th compact dimension is presentedin two equivalent forms given by (4.3) with j = 1 ,
2. Another representation, in which the brane-induced contribution is presented in the form of a single integral, is provided by (4.7). The currentdensity along the l th compact dimension is an odd periodic function of the phase ˜ α l and an evenperiodic function of the remaining phases ˜ α i , i = l . In both cases the period is equal to 2 π . Inparticular, the current density is a periodic function of the magnetic flux having the period equalto the flux quantum. In order to clarify the behavior of the vacuum current as a function of theparameters of the model, we have considered various limiting cases. First of all, we have shown thatin the limit of the large curvature radius of the background spacetime the corresponding result forthe Robin plates in the Minkowski bulk with partially compactified dimensions is obtained. For aconformally coupled massless field the current density in the AdS bulk is connected to the one inthe Minkowski spacetime by the conformal relation with an appropriate transformation of the Robincoefficients (see (4.13)). In the limit when the brane at z = z tends to the AdS boundary, z → z ν and, to the leading order, theresult for the geometry of a single brane at z = z is obtained. For a fixed location of the left brane,when the right brane tends to the AdS horizon, z → ∞ , its contribution to the vacuum current at agiven observation point decays as exp( − k (0)( q ) z ). If the length of the r th compact dimension is muchsmaller than the remaining length scales and the observation point is not too close to the branes, L r ≪ | z − z j | , for 0 < | ˜ α r | < π the single brane contribution to the current density is suppressedby the factor exp( − | ˜ α r || z − z j | /L r ) for the brane at z = z j , and the interference part decays asexp[ − | ˜ α r | ( z − z ) /L r ]. In this limit, the current density is localized near the branes within theregion | z − z j | . L r . For small values of L r and ˜ α r = 0, the component of the current density alongthe r th dimension vanishes whereas the current density along other directions, to the leading order,coincides with that in the D -dimensional model, with the excluded r th compact dimension, dividedby the proper length of the r th dimension. 24or the investigation of the asymptotic for large values of the length L l , it is more convenient to usethe representation (4.23). In this limit, the contribution to the VEV of the l th component of the currentdensity coming from the modes with positive λ is suppressed by the factor exp( − L l q λ + k (0)2( q − ). Ifthe mode with λ = 0 is present, for large values of L l the corresponding current density decays asexp( − L l k (0)( q − ) in the case k (0)( q − > /L p +1 l for k (0)( q − = 0. The representation (4.23) is alsowell adapted for the investigation of the near-brane asymptotics of the current density. An importantresult seen from this representation is the finiteness of the current density on the branes. This featureis in drastic contrast compared to the cases of the VEVs for the field squared and energy-momentumtensor. It is well known that the latter diverge on the boundaries and in the evaluation of the relatedglobal quantities, like the total vacuum energy, an additional renormalization procedure is required.The current densities on the branes are directly obtained from (4.23) by putting z = z j and aregiven by the expression (4.25) with j = 1 , λ tends to zero in the limit of small interbrane distances.In the numerical examples, discussed in section 4, we have considered a minimally coupled masslessscalar field in the D = 4 model with a single compact dimension and with the same Robin coefficientsfor the left and right branes. In this case, for β λ . In thisrange, for fixed values of the other parameters, the current density is an increasing function of | β | .In particular, it takes the minimum value for Dirichlet boundary condition and the maximum valuefor the Neumann one. In the range β > β c >
0, where β c is the critical value of the Robin coefficientfor the stability of the vacuum (the vacuum is unstable in the range 0 < β < β c ), the situation isopposite: the current density decreases with increasing β .In section 5 we have applied the general result to the 2-brane Randall–Sundrum type model withextra compact dimensions. The corresponding boundary conditions are obtained by the integrationof the field equation near the branes. For untwisted scalar fields the boundary conditions are of theRobin type with the coefficients given by (5.2) with c j being the brane mass terms. For twisted fieldsDirichlet boundary conditions are obtained. The corresponding expressions for the vacuum currentsare obtained from those in section 4 with an additional coefficient 1 / y = 0, y = b . Forthe values of the interbrane distance solving the hierarchy problem between the electroweak and Planckscales, the current density induced by the hidden brane on the visible one is suppressed exponentiallyas a function of the location of the visible brane. Acknowledgments
A. A. S. was supported by the State Committee of Science Ministry of Education and Science RA,within the frame of Grant No. SCS 15T-1C110.
A Other representations for the Hadamard function
In section 3, by using the generalized Abel-Plana formula (3.18), for the Hadamard function we haveprovided the representation (3.22). In the expression for the VEV of the current density obtained fromthis representation (see (4.3)), the convergence of the integral is too slow for points near the branesand it is not convenient for the investigation of the near-brane asymptotic of the current density. Herewe apply to the mode sum for the Hadamard function another type of Abel-Plana formula that allowsone to extract the contribution induced by the compactification. The representation obtained in thisway is well suited for the evaluation of the vacuum currents on the branes.Let us apply to the series over n l in the representation (3.17) the Abel-Plana type summation25ormula [40] 2 πL l ∞ X n l = −∞ g ( k l ) f ( | k l | ) = Z ∞ du [ g ( u ) + g ( − u )] f ( u )+ i Z ∞ du [ f ( iu ) − f ( − iu )] X λ = ± g ( iλu ) e uL l + iλ ˜ α l − , (A.1)where k l is given by (3.5). In the special case g ( u ) = 1 and ˜ α l = 0, this formula is reduced to theAbel-Plana formula in its standard form. The contribution of the first term in the right-hand sideof (A.1) gives the Hadamard function in the geometry of two branes in ( D + 1)-dimensional locallyAdS spacetime with compact dimensions ( x p +1 , . . . , x l − , x l +1 , . . . , x D − ) and with the l th dimensionbeing uncompactified. The latter corresponds to the spatial topology R p +2 × T q − and the respectiveHadamard function will be denoted by G R p +2 × T q − ( x, x ′ ). As a consequence, the Hadamard functionis decomposed as G ( x, x ′ ) = G R p +2 × T q − ( x, x ′ ) + G l ( x, x ′ ) , (A.2)where the last term comes from the second integral in the right-hand side of (A.1) and is induced bythe compactification of the l th dimension. It is presented in the form G l ( x, x ′ ) = a − D L l ( zz ′ ) D/ p π p − V q z ∞ X s =1 X n q − Z d k p e ik r ∆ x r Z ∞ dw cosh( w ∆ t ) × ∞ X n =1 λ n T ν ( χ, λ n ) g ν ( λ n z , λ n z ) g ν ( λ n z , λ n z ′ ) × cosh (cid:16) u ∆ x l + is ˜ α l (cid:17) e − suL l u | u = √ w + λ n + k ( l )2 , (A.3)where n q − = ( n p +1 , . . . , n l − , n l +1 , . . . , n D − ), k ( l )2 = k − k l , and the summation over r in k r ∆ x r is over r = 1 , . . . , D − r = l . Note that the part (A.3) is finite in the coincidence limit of thearguments, including the points on the branes. The physical reason for this feature is related to thatthe toroidal compactification does not change the local geometry and the structure of the divergencesis the same as that for AdS bulk without the compactification. The first term in the right-hand side(A.2) does not contribute to the component of the current density along the l th dimension.Yet another representation is obtained applying to the series over n in (A.3) the summation formula(3.18). This leads to the following decomposition G l ( x, x ′ ) = G (1) l ( x, x ′ ) − L l ( zz ′ ) D/ (2 π ) p +2 a D − V q ∞ X s =1 X n q − Z d k p e ik r ∆ x r × Z ∞ dw cosh( w ∆ t ) Z ∞ dv X j = ± cos (cid:16) v ∆ x l + js ˜ α l (cid:17) e − ijsvL l × Ω ν ( uz , uz ) X (1) ν ( uz , uz ) X (1) ν ( uz , uz ′ ) | u = √ v + w + k ( l )2 . (A.4)Here the term G (1) l ( x, x ′ ) = 4 ( zz ′ ) D/ L l (2 π ) p +1 a D − V q ∞ X n =1 X n q − Z d k p e ik r ∆ x r Z ∞ dλ λ × g ν ( λz , λz ) g ν ( λz , λz ′ )¯ J (1)2 ν ( λz ) + ¯ Y (1)2 ν ( λz ) Z ∞ dw cosh( w ∆ t ) × e − nuL l u cosh( u ∆ x l + in ˜ α l ) | u = √ w + λ + k ( l )2 , (A.5)26omes from the first integral in the right-hand side of (3.18). It is the part of the Hadamard functioninduced by the compactification of the l th dimension in the geometry of a single brane at y = y . References [1] N.D. Birrell and P.C.W. Davies,
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