NUPHB114619114619S0550-3213(19)30099-910.1016/j.nuclphysb.2019.114619The AuthorQuantum Field Theory and Statistical SystemsAnalyticity properties of scattering amplitude in theories with compactified space dimensionsJnanadevaMaharanaabc⁎maharana@iopb.res.inaNISER, Bhubaneswar, IndiaNISERBhubaneswarIndiabInstitute of Physics, Bhubaneswar - 751005, IndiaInstitute of PhysicsBhubaneswar751005IndiacMax-Planck Institute for Gravitational Physics, Albert Einstein Institute, Golm, GermanyMax-Planck Institute for Gravitational PhysicsAlbert Einstein InstituteGolmGermany⁎Correspondence to: Institute of Physics, Bhubaneswar - 751005, India.Institute of PhysicsBhubaneswar751005IndiaEditor: Hubert SaleurAbstractWe consider a massive scalar, neutral, field theory in a five dimensional flat spacetime. Subsequently, one spatial dimension is compactified on a circle, S1, of radius R. The resulting theory is defined in the manifold, R3,1⊗S1, consists of a states of lowest mass, m0, and a tower of massive Kaluza-Klein states. The analyticity property of the elastic scattering amplitude is investigated in the frame works of Lehmann-Symanzik-Zimmermann formulation of this field theory. In the context of nonrelativistic potential scattering, for R3⊗S1 spatial geometry, it was shown that the forward scattering amplitude does not satisfy analyticity for a class of potentials which might have important consequences if same attribute holds in relativistic quantum field theories. We address this issue with R3,1⊗S1 geometry. We show that the forward scattering amplitude of the theory satisfying LSZ axioms does not suffer from lack of analyticity. The importance of the unitarity constraint is exhibited in displaying the properties of the absorptive part of the forward amplitude.1IntroductionThe analyticity properties of scattering amplitude is a cardinal attribute which has been derived in the frameworks of general relativistic quantum field theories. The scattering amplitude, F(s,t), is an analytic function of the center of mass energy squared, s, for fixed t which is the momentum transfers squared. The dispersion relations in s have been proved for t lying within the Lehmann-Martin ellipse. This is derived from the axiomatic approach of Lehmann-Symanzik-Zimmermann (LSZ) [1] and in the general frameworks of axiomatic formulation of field theories [2–14]. The underlying structure of such theories are locality, microcausality, Lorentz invariance to mention a few. It is generally accepted that if the dispersion relations are violated one might question the validity of the axioms of general field theories since these are the pillars on which the structure of such field theories rest. Moreover, there are host of rigorous results, derived in the form of theorems, which have been tested against experimental data. One of the most important result is the celebrated Froissart-Martin bound [15,16] that the total cross section is bounded from above: σt≤4πt0(logs)2 where t0 is determined from the first principles for a give scattering process. The high energy data for hadronic total cross sections respect this bound over a wide range of energies accessible by accelerators.It is well known that the scattering amplitude in nonrelativistic potential scattering enjoys certain analyticity properties in energy k for a large class of potentials. This topic has been studied quite thoroughly in the past [17–19]. One intriguing point is, in contrast to relativistic quantum field theories (QFT), that the analyticity of the amplitudes in potential scattering is not so intimately related to a principle like microcausality as is the case with QFT. Moreover, the principle of microcausality has its root in the existence of the limiting velocity in the special theory of relativity. In other words, no signal can travel faster than light. Moreover, there is an intimate relationship between microcausality, as postulated in quantum field theory, and the analyticity properties of scattering amplitudes. On the other hand, when we consider nonrelativistic quantum mechanical scattering in potential models the theory is invariant under Galilean transformations. If we encounter a situation, in potential scattering, where the amplitude fails to satisfy analyticity in the momentum, k, it is not so much a matter of serious concern as would be case for a relativistic QFT, especially one which satisfies LSZ or Wightman axioms.It is now generally accepted that theories which exist in higher spacetime dimensions, Dˆ>4, might play important roles in our attempt to unify fundamental interactions. We mention, in this context, supersymmetric theories, supergravity theories and the string theories which have been investigated intensively over past several decades. It is generally postulated in the context of such higher dimensional theories that some of the spatial dimensions be compactified so that one eventually constructs an effective four dimensional theory of fundamental interactions to describe and understand physics in the domain of present accessible energies. It is proposed, in certain scenarios, that signatures of the extra spatial dimensions might be observed in current high energy colliders [20,21]. Consequently, there has been a lot of phenomenological studies to investigate and build models for possible experimental observations of the decompactified dimensions at the present high energy accelerators such as LHC. Indeed, the scale of the extra compact dimensions is extracted from the LHC experiments and it puts the compactification scale to be more than 600 GeV.Khuri [22], motivated by the large compactification scenario, had set out to study analyticity properties of the scattering amplitude in potential scattering where a spatial dimension is compactified on a circle; the so called S1 compactification. He discovered that the amplitude does not always satisfy the analyticity properties. On the other hand the analyticity properties of amplitudes were investigated [18,19] with noncompact spatial coordinates (for d=3) (when there was no S1 compactification) the amplitude satisfied the dispersion relations. In fact it was shown by Khuri [22], through counter examples, within the framework of perturbation theory, how the analyticity of the forward scattering amplitude breaks down in the presence of S1 compactification for a class of nonrelativistic potential models under certain circumstances as we shall describe later.The purpose of this investigation is to study the analyticity properties of the scattering amplitude in a field theory with an S1 compactified spatial dimension. It is an analog of the Kaluza-Klein (KK) compactification although gravitational interaction is not incorporated. Moreover, after compactification, we retain the entire tower of KK states. For sake of simplicity and to bring out the essential features, we consider a single, neutral, massive scalar field in D=5. On compactification to D=4, not only we have a massive scalar field with the same mass as the 5-dimensional theory but also we have tower of massive scalars as a consequence of compactification.Recently, we have studied analyticity properties and the high energy behavior of the four point function for a massive, neutral scalar field in higher dimensions D>4 [23]. It was shown, in the LSZ formalism, that the scattering amplitude has desire attributes in the following sense: (i) We proved the generalization of the Jost-Lehmann-Dyson theorem for the retarded function [24,25] for the D>4 case [26]. (ii) Subsequently, we showed the existence of the Lehmann-Martin ellipse for such a theory. (iii) Thus a dispersion relation can be written in s for fixed t when the momentum transfer squared lies inside Lehmann-Martin ellipse [27,28]. (iv) The analog of Martin's theorem can be derived in the sense that the scattering amplitude is analytic the product domain Ds⊗Dt where Ds is the cut s-plane and Dt is a domain in the t-plane such that the scattering amplitude is analytic inside a disk, |t|<R˜, R˜ is radius of the disk and it is independent of s. Thus the partial wave expansion converges inside this bigger domain. (v) We also derived the analog of Jin-Martin [29]upper bound on the scattering amplitude which states that the fixed t dispersion relation in s does not require more than two subtractions. (vi) Therefore, a generalized Froissart-Martin bound was be proved.Our principal goal, in this investigation, is to examine the four point function derived from the compactified theory in D=4, which originated from D=5 field theory, through the S1 compactification, possesses desired analyticity properties as was derived for the higher dimensional (uncompactified) theory. We shall specifically, investigate analyticity properties of the amplitude for scattering of KK states in the forward direction in view of Khuri's result for scattering of such states in potential scattering with a spatial S1 compactification. We argue in sequel and prove that the forward scattering amplitude satisfies desired analyticity property in s and hence we write down the dispersion relations.We work within the LSZ formalism for the effective four dimensional theory obtained from the D=5 theory after S1 compactification. The main conclusion is that, under certain assumptions, the elastic forward scattering amplitude for the scattering of Kaluza-Klein states satisfies dispersion relation when we consider a massive field theory in the manifold R3,1⊗S1.The paper is organized as follows. The next section (Section 2) is devoted to a brief review of the contents of Khuri's paper [22]. The third section contains the essentials of LSZ axiomatic approach to field theory defined over a five dimensional spacetime. We define the retarded products and elucidate their properties. We introduce kinematical variables which are used in this paper. We also introduce notations and conventions followed in this article. The next section (Section 4), deals with the definitions, conventions and kinematics. Then we consider scattering of the zero modes (the fields which carry lowest mass) of the 4-dimensional theory. This problem is similar to scattering of equal mass scalars in a four dimensional theory. However, there are some subtleties since we have to account for set of complete intermediate states (including entire KK tower) in certain spectral representations. In section 4.1.2 deals with elastic scattering of the lowest mass state with an excited KK state. This is the case of unequal mass scattering where one state carries KK charge (KK momentum) and the other state has no KK momentum. The fifth section is devoted to elastic scattering of two states carrying KK charge. Therefore, some caution is to be exercised in deriving analyticity. Our conclusion is that the forward scattering amplitude possesses nice analyticity properties. Thus if we accept the axioms of LSZ and adopt the standard procedures to investigate analyticity properties of the forward scattering amplitudes we arrive at the same result as is known for field theories defined in flat Minkowski spacetime and we show that the forward scattering amplitude satisfies the dispersion relation. The sixth section summarizes our results and contains discussion.2Non-relativistic potential scattering for R3⊗S1 geometryKhuri [22] envisaged scattering of a particle in a space with R3⊗S1 topology. We provide a brief account of his work and incorporate his important conclusions. We refer to the original paper to the interested reader. The notations of [22] will be followed. The compactified coordinate is a circle of radius R and it is assumed that the radius is small i.e. 1R>>1 where dimensionless units were used. We mention here that the five dimensional theory is defined in a flat Minkowski space. The only mass scale available to us is the mass of the particle; therefore, 1R is much larger than this scale. The potential, V(r,Φ), is such that is periodic in the angular coordinate, Φ, of S1; r∈R3 and r=|r|. The potential V(r,Φ) belongs to a broad class such that for large r these class of potentials fall off like e−μr/r as r→∞. Moreover, V(r,Φ)=V(r,Φ+2π). The scattering amplitude depends on three variables – the momentum of the particle, k, the scattering angle θ, and an integer n which appears due to the periodicity of the Φ-coordinate. Thus forward scattering amplitude is denoted by Tnn(K), where K2=k2+n2R2. The starting point is the Schrödinger equation(1)[∇2+1R2∂2∂Φ2+K2−V(r,Φ)]Ψ(r,Φ)=0 The free plane wave solutions are(2)Ψ0(x,Φ)=1(2π)2eik.xeinΦ and n∈Z. The total energy is defined to be(3)K2=k2+n2R2 The free Green's function (in the presence of a compact coordinate) assumes the following form(4)G0(K;x,Φ:x′,Φ′)=−1(2π)4∑n=−∞n=+∞∫d3peip.(x−x′)ein.(Φ−Φ′)[p2+n2R2−K2−iϵ] The free Green's function satisfies the free Schrödinger equation(5)[∇2+1R2∂2∂Φ2+K2]G0(K;x,Φ:x′,Φ′)=δ3(x−x′)δ(Φ−Φ) The d3p integration can be performed in the expression (4) leading to(6)G0(K;x−x′;Φ−Φ′)=−1(8π2)∑n=−∞n=+∞eiK2−(n2/R2)|x−x′||x−x′|ein(Φ−Φ′) Khuri introduced the prescription that K2−n2/R2 is defined in such a way that when n2/R2>K2(7)iK2−n2/R2→−n2/R2−K2,n2>K2R2 Note that the series expansion for G0(K;x−x′;Φ−Φ′) as expressed in (6) is strongly damped for large enough |n|. A careful analysis, as was carried out in ref. [22] shows that the Green's function is well defined and bounded, except for |x−x′|→0 similar to the properties of Green's functions in potential scattering for a fixed K2. Khuri [22] expressed the scattering integral equation for the potential V(r,Φ) as(8)Ψk,n(x,Φ)=eik.xeinΦ+∫02πdΦ′∫d3x′G0(K;|x−x′|;|Φ−Φ′|)V(x′,Φ′)Ψk,n(x′,Φ′) The expression for the scattering amplitude is extracted from the large |x| limit when one looks at the asymptotic behavior of the wave function,(9)Ψk,n→ek.xeinΦ+∑m=−[KR]+[KR]T(k′,m:k,n)eikmn′|x||x|eimΦ where [KR] is the largest integer less than KR and(10)kmn′=k2+n2R2−m2R2 He also identifies a conservation rule: K2=k2+(n2/R2)=k′2+(m2/R2). Thus it is argued that the scattered wave has only (2[KR]+1) components and those states with (m2/(R2)>k2+(n2/R2) are exponentially damped for large |x| and consequently these do not appear in the scattered wave (see eq. (7)). Now the scattering amplitude is extracted from equations (8) and (9) to be(11)T(k′,n′;k,n)=−18π2∫d3x′∫02πdΦ′e−k′.x′e−in′Φ′V(x′,Φ′)Ψk,n(x′,Φ′) The condition, k′2+n′2/R2=k2+n2/R2 is to be satisfied. Thus the scattering amplitude describes the process where incoming wave |k,n> is scattered to final state |k′,n′>.Remark: Reader should pay attention to the expression for the discussion of scattering processes in relativistic QFT in sequel and note the similarities and differences in subsequent sections.Formally, the amplitude assumes the following form for the full Green's function(12)T(k′,n′;k,n)−TB=−18π2∫....∫d3xd3x′dΦdΦ′e−i(k′.x′+n′+Φ′V(x′,Φ′)G(K;x′,x;Φ′,Φ)V(x,Φ)ei(k.x+nΦ) Here TB is the Born term.(13)TB=−18π2∫d3x∫02πei(k−k).xV(x,Φ)ei(n−n′)Φ Full Green's function satisfied an equation with the full Hamiltonian(14)[∇2+1R2∂2∂Φ2+K2−V(x,Φ)]G(K;x,x′,Φ,ϕ′)=δ3(x−x′)δ(Φ−Φ′) This is the starting point of computing scattering amplitude perturbatively in potential scattering [17]. Khuri [22] proceeds to study the analyticity properties of the amplitude and it is a parallel development similar to investigations done in the past. In the context of a theory with compact space dimension he analyzed the amplitude Tnn(K) to the second order in the Born approximation for n=1.Khuri [22] explicitly computed the second born term T(2) for the forward amplitude, for the choice n=1. He has discovered that the analyticity of the forward amplitude breaks down with a counter example; where Tnn(k) does not satisfy dispersion relations for a class of Yukawa-type potentials of the form(15)V(r,Φ)=u0(r)+2∑m=1Num(r)cos(mΦ) where um(r)=λme−μrr. Khuri noted an important feature of his studies that in the case when scattering theory was applied perturbatively in R3 space the resulting amplitude satisfied analyticity properties for similar Yukawa-type potentials. Thus there has been concerns11Andre Martin brought the work of Khuri [22] to my attention and persuaded me to undertake this investigation. when non-analyticity was discovered in the non-relativistic quantum mechanics in the space R3⊗S1 by Khuri.We shall describe the framework of our investigation in the next section. We remark in passing that the analyticity of scattering amplitude in nonrelativistic scattering is not such a profound property as in the relativistic QFT although the analyticity in non-relativistic potential scattering has been investigated quite thoroughly in the past [17]. However, it is to be noted that in absence a limiting velocity (in relativistic case velocity of light, c, profoundly influences the study of the analyticity of amplitudes) the microcausality is not enforced in nonrelativistic processes. As we shall show (and has been emphasized in many classic books) there is indeed a deep connection between microcausality and analyticity. When a spatial dimension is compactified on S1, the coordinate on the circle is periodic; we can understand microcausality as follows. The compact coordinate y is periodic. Therefore, we can define spacelike separation between two points. We keep this aspect in mind and we shall undertake a systematic study of the analyticity of scattering amplitude in sequel.3Field theory in five dimensional spacetimeLet us consider a neutral, scalar field theory with mass m0 in flat five dimensional Minkowski space R4,1. It is assumed that the particle is stable and there are no bound states. The notation is that the spacetime coordinates are, xˆ, and all operators are denoted with hat when they are defined in the five dimensional space where the spatial coordinates are noncompact. The LSZ axioms are [1]:A1. The states of the system are represented in a Hilbert space, Hˆ. All the physical observables are self-adjoint operators in the Hilbert space, Hˆ.A2. The theory is invariant under inhomogeneous Lorentz transformations.A3. The energy-momentum of the states are defined. It follows from the requirements of Lorentz and translation invariance that we can construct a representation of the orthochronous Lorentz group. The representation corresponds to unitary operators, Uˆ(aˆ,Λˆ), and the theory is invariant under these transformations. Thus there are hermitian operators corresponding to spacetime translations, denoted as Pˆμˆ, with μˆ=0,1,2,3,4 which have following properties:(16)[Pˆμˆ,Pˆνˆ]=0 If Fˆ(xˆ) is any Heisenberg operator then its commutator with Pˆμˆ is(17)[Pˆμˆ,Fˆ(xˆ)]=i∂ˆμˆFˆ(xˆ) It is assumed that the operator does not explicitly depend on spacetime coordinates. If we choose a representation where the translation operators, Pˆμˆ, are diagonal and the basis vectors |pˆ,αˆ> span the Hilbert space, Hˆ,(18)Pˆμˆ|pˆ,αˆ>=pˆμˆ|pˆ,αˆ> then we are in a position to make more precise statements:• Existence of the vacuum: there is a unique invariant vacuum state |0> which has the property(19)Uˆ(aˆ,Λˆ)|0>=|0> The vacuum is unique and is Poincaré invariant.• The eigenvalue of Pˆμˆ, pˆμˆ, is light-like, with pˆ0>0. We are concerned only with massive stated in this discussion. If we implement infinitesimal Poincaré transformation on the vacuum state then(20)Pˆμˆ|0>=0,andMˆμˆνˆ|0>=0 from above postulates; note that Mˆμˆνˆ are the generators of Lorentz transformations.A4. The locality of theory implies that a (bosonic) local operator at spacetime point xˆμˆ commutes with another (bosonic) local operator at xˆ′μˆ when their separation is spacelike i.e. if (xˆ−xˆ′)2<0. Our Minkowski metric convention is as follows: the inner product of two 5-vectors is given by xˆ.yˆ=xˆ0yˆ0−xˆ1yˆ1−...−xˆ4yˆ4. Since we are dealing with a neutral scalar field, for the field operator ϕˆ(xˆ): ϕˆ(xˆ)†=ϕˆ(xˆ) i.e. ϕˆ(xˆ) is hermitian. By definition it transforms as a scalar under inhomogeneous Lorentz transformations(21)Uˆ(aˆ,Λˆ)ϕˆ(xˆ)Uˆ(aˆ,Λˆ)−1=ϕˆ(Λˆxˆ+aˆ) The micro causality, for two local field operators, is stated to be(22)[ϕˆ(xˆ),ϕˆ(xˆ′)]=0,for(xˆ−xˆ′)2<0 It is well known that, in the LSZ formalism, we are concerned with vacuum expectation values of time ordered products of operators as well as with the retarded product of fields. The requirements of the above listed axioms lead to certain relationship, for example, between vacuum expectation values of R-products of operators. These are termed as linear relations and the importance of the above listed axioms is manifested through these relations. In contrast, unitarity imposes nonlinear constraints on amplitude. For example, if we expand an amplitude in partial waves, unitarity demands certain positivity conditions to be satisfied by the partial wave amplitudes.We summarize below some of the important aspects of LSZ formalism as we utilize them through out the present investigation.(i) The asymptotic condition: According to LSZ the field theory accounts for the asymptotic observables. These correspond to particles of definite mass, charge and spin etc. ϕˆin(xˆ) represents the free field and a Fock space is generated by the field operator. The physical observable can be expressed in terms of these fields.(ii) ϕˆ(xˆ) is the interacting field. LSZ technique incorporates a prescription to relate the interacting field, ϕˆ(xˆ), with ϕˆin(xˆ); consequently, the asymptotic fields are defined with a suitable limiting procedure. Thus we introduce the notion of the adiabatic switching off of the interaction. A cutoff adiabatic function is postulated such that this function controls the interactions. It is 1 at finite interval of time and it has a smooth limit of passing to zero as |t|→∞. It is argued that when adiabatic switching is removed we can define the physical observables.(iii) The fields ϕˆin(xˆ) and ϕˆ(xˆ) are related as follows:(23)xˆ0→−∞ϕˆ(xˆ)→Zˆ1/2ϕˆin(xˆ) By the first postulate, ϕˆin(xˆ) creates free particle states. However, in general ϕˆ(xˆ) will create multi particle states besides the single particle one since it is the interacting field. Moreover, <1|ϕˆin(xˆ)|0> and <1|ϕˆ(xˆ)|0> carry same functional dependence in xˆ. If the factor of Zˆ were not the scaling relation between the two fields (23), then canonical commutation relation for each of the two fields (i.e. ϕˆin(xˆ) and ϕˆ(xˆ)) will be the same. Thus in the absence of Zˆ the two theories will be identical. Moreover, the postulate of asymptotic condition states that in the remote future(24)xˆ0→∞ϕˆ(xˆ)→Zˆ1/2ϕˆout(xˆ). We may as well construct a Fock space utilizing ϕˆ(xˆ)out as we could with ϕˆ(xˆ)in. Furthermore, the vacuum is unique for ϕˆin, ϕˆout and ϕˆ(xˆ). The normalizable single particle states are the same i.e. ϕˆin|0>=ϕˆout|0>. We do not display Zˆ from now on. If at all any need arises, Zˆ can be introduced in the relevant expressions.We define creation and annihilation operators for ϕˆin, ϕˆout. We recall that ϕˆ(xˆ) is not a free field. Whereas the fields ϕˆin,out(xˆ) satisfy the free field equations [□5+m02]ϕˆin,out(xˆ)=0, the interacting field satisfies an equation of motion which is endowed with a source current: [□5+m20]ϕˆ(xˆ)]=jˆ(xˆ). We may use the plane wave basis for simplicity in certain computations; however, in a more formal approach, it is desirable to use wave packets.The relevant vacuum expectation values of the products of operators in LSZ formalism are either the time ordered products (the T-products) or the retarded products (the R-products). We shall mostly use the R-products and we use them extensively throughout this investigation. It is defined as(25)Rϕˆ(xˆ)ϕˆ1(xˆ1)...ϕˆn(xˆn)=(−1)n∑Pθ(xˆ0−xˆ10)θ(xˆ10−xˆ20)...θ(xˆn−10−xˆn0)[[...[ϕˆ(xˆ),ϕˆi1(xˆi1)],ϕˆi2(xˆi2)]..],ϕˆin(xˆin)] note that Rϕˆ(xˆ)=ϕˆ(xˆ) and P stands for all the permutations i1,....in of 1,2...n. The R-product is hermitial for hermitial fields ϕˆi(xˆi) and the product is symmetric under exchange of any fields ϕˆ1(xˆ1)...ϕˆn(xˆn). Notice that the field ϕˆ(xˆ) is kept where it is located in its position. We list below some of the important properties of the R-product for future use [5]:(i) Rϕˆ(xˆ)ϕˆ1(xˆ1)...ϕˆn(xˆn)≠0 only if xˆ0>max{xˆ10,...ˆxn0}.(ii) Another important property of the R-product is that(26)Rϕˆ(xˆ)ϕˆ1(xˆ1)...ϕˆn(xˆn)=0 whenever the time component xˆ0, appearing in the argument of ϕˆ(xˆ) whose position is held fix, is less than time component of any of the four vectors (xˆ1,...xˆn) appearing in the arguments of ϕˆ(xˆ1)...ϕˆ(xˆn).(iii) We recall that(27)ϕˆ(xˆi)→ϕˆ(Λˆxˆi)=Uˆ(Λˆ,0)ϕˆ(xˆi)Uˆ(Λˆ,0)−1 Under Lorentz transformation Uˆ(Λˆ,0). Therefore,(28)Rϕˆ(Λˆxˆ)ϕˆ(Λˆxˆi)...ϕˆn(Λˆxˆn)=Uˆ(Λˆ,0)Rϕ(x)ϕ1(x1)...ϕn(xn)U(Λ,0)−1 And(29)ϕˆi(xˆi)→ϕˆi(xˆi+aˆ)=eiaˆ.Pˆϕˆi(xˆi)e−iaˆ.Pˆ under spacetime translations. Consequently,(30)Rϕˆ(xˆ+aˆ)ϕˆ(xˆi+aˆ)...ϕˆn(xˆn+aˆ)=eiaˆ.PˆRϕˆ(xˆ)ϕˆ1(xˆ1)...ϕˆn(xˆn)e−iaˆ.Pˆ Therefore, the vacuum expectation value of the R-product dependents only on difference between pair of coordinates: in other words it depends on the following set of coordinate differences: ξˆ1=xˆ1−xˆ,ξˆ2=xˆ2−xˆ1...ξˆn=xˆn−1−xˆn as a consequence of translational invariance.(iv) The retarded property of R-function and the asymptotic conditions lead to the following relations.(31)[Rϕˆ(xˆ)ϕˆ1(xˆ1)...ϕˆn(xˆn),ϕˆlin(yˆl)]=i∫d5yˆl′Δ(yˆl−yˆl′)(□5yˆ′+mˆl2)×Rϕˆ(xˆ)ϕˆ1(xˆ1)...ϕˆn(xˆn)ϕˆl(yˆl′) Note: here mˆl stands for the mass of a field in five dimensions. We may define ‘in’ and ‘out’ states in terms of the creation operators associated with ‘in’ and ‘out’ fields as follows(32)|kˆ1,kˆ2,....kˆnin>=aˆin†(kˆ1)aˆin†(kˆ2)...aˆin†(kˆn)|0>(33)|kˆ1,kˆ2,....kˆnout>=aˆout†(kˆ1)aˆout†(kˆ2)...aˆout†(kˆn)|0> We can construct a complete set of states either starting from ‘in’ field operators or the ‘out’ field operators and each complete set will span the Hilbert space, Hˆ. Therefore, a unitary operator will relate the two sets of states in this Hilbert space. This is a heuristic way of introducing the concept of the S-matrix. We shall define S-matrix elements through LSZ reduction technique in subsequent section.We shall not distinguish between notations like ϕˆout,in or ϕˆout,in and therefore, there might be use of the sloppy notation in this regard.We record the following important remark en passant. The generic matrix element <αˆ|ϕˆ(xˆ1)ϕˆ(xˆ2)...|βˆ> is not an ordinary function but a distribution. Thus it is to be always understood as smeared with a Schwarz type test function f∈S. The test function is infinitely differentiable and it goes to zero along with all its derivatives faster than any power of its argument. We shall formally derive expressions for scattering amplitudes and the absorptive parts by employing the LSZ technique. It is to be understood that these are generalized functions and such matrix elements are properly defined with smeared out test functions.We obtain below the expression for the Källen-Lehmann representation for the five dimensional theory. It will help us to transparently expose, as we shall recall in the next section, the consequences of S1 compactification. Let us consider the vacuum expectation value (VEV) of the commutator of two fields in the D=5 theory: <0|[ϕˆ(xˆ),ϕˆ(yˆ)]|0>. We introduce a complete set of states between product of the fields after opening up the commutator. Thus we arrive at the following expression by adopting the standard arguments,(34)<0|[ϕˆ(xˆ),ϕˆ(yˆ)]|0>=∑αˆ(<0|ϕˆ(0)αˆ>e−ipˆαˆ.(xˆ−yˆ)<αˆ|ϕˆ(0)|0>−(xˆ↔yˆ)) Let us define(35)ρˆ(qˆ)=(2π)4∑αˆδ5(qˆ−pˆαˆ)|<0|ϕˆ(0)|αˆ>|2 Note that ρˆ(qˆ) is positive, and ρˆ=0 when qˆ is not in the light cone. It is also Lorentz invariant. Thus we write(36)ρˆ(qˆ)=σˆ(qˆ2)θ(qˆ0),σˆ(qˆ2)=0,ifqˆ2<0 This is a positive measure. We may separate the expression for the VEV of the commutator (34) into two parts: the single particle state contribution and the rest. Moreover, we use the asymptotic state condition to arrive at(37)<0|[ϕˆ(xˆ),ϕˆ(yˆ)]|0>=iZˆΔˆ(xˆ,yˆ;m0)+i∫mˆ12∞dmˆ′2Δˆ(xˆ,yˆ;mˆ′) where Δˆ(xˆ,yˆ;m0) is the VEV of the free field commutator, m0 is the mass of the scalar. mˆ12>Mˆ2, the multiple particle threshold.We are in a position to study several attributes of scattering amplitudes in the five dimensional theory such as proving existence of the Lehmann-Martin ellipse, give a proof of fixed t dispersion relation to mention a few. However, these properties have been derived in a general setting recently [23] for D-dimensional theories. The purpose of incorporating the expression for the VEV of the commutator of two fields in the 5-dimensional theory is to provide a prelude to the modification of similar expressions when we compactify the theory on S1 as we shall see in the next section.4Compactification of scalar field theory: R4,1→R3,1⊗S1We consider S1 compactification of a spatial coordinate of the five dimensional theory. Let us decompose the five dimensional spacetime coordinates, xˆμˆ, as follows:(38)xˆμˆ=(xμ,y) where xμ are the four dimensional Minkowski space coordinates; y is the spatial coordinate defined on S1 such that y+2πR=y, R being the radius of compactification. We shall capture the essential features of the S1 compactification when a neutral scalar field (in D=5) of mass m0 is described in the geometry R3,1⊗S1. Let us consider a free field version of the theory such as the in and out field, ϕˆin,out(xˆ). The equation of motion is [□5+m02]ϕˆin,out(xˆ)=0. We expand the field(39)ϕˆin,out(xˆ)=ϕˆin,out(x,y)=ϕ0in,out(x)+∑n=−∞,n≠0+n=∞ϕin,outn(x)einyR Note that ϕ0in,out(x) has no y-dependence and it is the so called zero mode. The terms in rest of the series satisfy periodicity in y. We can decompose the five dimensional □5 as sum of a four dimensional □ and a ∂∂y2 term. The equation of motion is(40)[□−∂∂y2+mn2]ϕin,outn(x,y)=0 where ϕnin,out(x,y)=ϕnin,outeinyR and n=0 term has no y-dependence being ϕ0(x). Here mn2=mo2+n2R2. Thus we have tower of massive states. The momentum associated in the y-direction is qn=n/R and is quantized in the units of 1/R and it is an additive conserved quantum number. We term it as Kaluza-Klein (KK) charge although there is no gravitational interaction in the five dimensional theory; we still call it KK reduction. For the interacting field ϕˆ(xˆ), we can adopt a similar mode expansion.(41)ϕˆ(xˆ)=ϕˆ(x,y)=ϕ0(x)+∑n=−∞,n≠0n=+∞ϕn(x)einyR The equation of motion for the interacting fields is endowed with a source term. Thus source current would be expanded as is the expansion (41). Each field ϕn(x) will have a current, Jn(x) associated with it and source current will be expanded as(42)jˆ(x,y)=j0(x)+∑n=−∞,n≠0n=+∞Jn(x)einy/R Note that the set of currents, {Jn(x)}, are the source currents associated with the tower of interacting fields {ϕn(x)}. These fields carry the discrete KK charge, n. Therefore, Jn(x) also carries the same KK charge. We should keep this aspect in mind when we consider matrix element of such currents between stated. In future, we might not explicitly display the charge of the current; however, it becomes quite obvious in the context.The zero mode, ϕ0in,out, create their Fock spaces. Similarly, each of the fields ϕnin,out(x) create their Fock spaces as well. For example a state with spatial momentum, p, energy, p0 and discrete momentum qn (in y-direction) is created by(43)A†(p,qn)|0>=|p,qn>,p0>0 We may recall that in the five dimensional theory, we started with, there was only one neutral scalar field in the spectrum. As a consequence of the S1 compactification, the resulting spectrum consists of a massive neutral scalar of mass m0 and a tower of ‘charged’ massive field. Moreover, each level in this tower is characterized by a mass and a ‘charge’, (mn,qn), respectively; the zero mode has qn=0. Let us consider the Hilbert space of the compactified theory, keeping in mind the above remarks. The Decomposition of the Hilbert space Hˆ: The Hilbert space associated with the five dimensional theory is Hˆ. It is now decomposed as a direct sum of Hilbert spaces where each one is characterized by its qn quantum number compactified theory(44)Hˆ=∑⊕Hn Thus H0 is the Hilbert space which is spanned by states built from the creation operators {a†(k)} which acting on the vacuum create the complete set of states that span H0. A single particle state is a†(k)|0>=|k> and multiparticle states are created using the procedure out lines in (32) and (33). If we consider a field ϕn(x,y) with a charge qn, we can create a Fock space through the set of creation operators {A†(p,qn)}. Moreover, two state vectors with different ‘charges’ are orthogonal: for example(45)<p,qn′|p′,qn′>=δ3(p−p′)δn,n′ RemarkWe have stated earlier and repeat here that we assume that there are no bound states in the theory and all particles are stable. There exists a possibility that a particle with charge 2n and mass m2n2=m02+4n2R2 could be a bound state of two particles of charge n and masses mn each under certain circumstances. We have excluded such possibilities from the present investigation.Now we can adopt the LSZ formalism for the four dimensional spacetime with an extra compact dimension. If we keep in mind the steps introduced above, it is possible to envisage field operators ϕnin(x) and ϕnout(x) for each of the fields for a given n. Therefore, each Hilbert space, Hn will be spanned by the state vectors created by operators a†(k), for n=0 and A†(p,qn), for n≠0. Moreover, we are in a position to define corresponding set of interacting field {ϕn(x)} which will interpolate into ‘in’ and ‘out’ fields in the asymptotic limits.RemarkNote that in (39) sum over {n} runs over positive and negative integers. If there is a parity symmetry y→−y under which the field is invariant we can reduce the sum to positive n only. However, since qn is an additive discrete quantum number, a state with qn>0 could be designated as a particle and the corresponding state qn<0 can be interpreted as its antiparticle. Thus a two particle state |p,qn>|p,−qn>,qn>0andp0>0 is a particle antiparticle state, qn=0; in other words the sum of the total charges of the two states is zero. Thus it has the quantum number of the vacuum. For example, it could be two particle state of ϕ0 satisfying energy momentum conservation, especially if they appear as intermediate states. We shall keep this fact in mind for future references.Let us momentarily return to the Källen-Lehmann representation (34) in the present context and utilize the expansion (41) in the expression for the VEV of the commutator of two fields defined in D=5: <0|[ϕˆ(xˆ),ϕˆ(xˆ′)]|0>(46)<0|[ϕˆ(x,y),ϕˆ(x′,y′)]|0>=<0|[ϕ0(x)+∑−∞+∞ϕn(x,y),ϕ0(x′)+∑−∞+∞ϕl(x′,y′)]|0> The VEV of a commutator of two fields given by the spectral representation (34) will be decomposed into sum of several commutators whose VEV will appear:(47)<0|[ϕ0(x),ϕ0(x′)]|0>,<0|[ϕn(x),ϕ−n(x′)]|0>,... Since the vacuum carries zero KK charge, qvac=0=q0, the commutator of two fields (with n≠0) should give rise to zero-charge and only ϕn and ϕ−n commutators will appear. Moreover, commutator of fields with different qn vanish since the operators act on states of different Hilbert spaces. Thus we already note the consequences of compactification. When we wish to evaluate the VEV and insert complete set of intermediate states in the product of two operators after opening up the commutators, we note that all states of the entire KK tower can appear as intermediate states as long as they respect all conservation laws. This will be an important feature in all our computations in what follows.4.1Conventions, definitions and kinematicsWe have stated earlier that our goal is to study the analyticity property of the four point amplitude in the forward direction. So far we have laid down the requisite procedures for compactification and we have outlined the structure of the Hilbert space in the compactified theory. We defined in and out fields in each of the sectors. Thus we can apply the LSZ reduction technique to derive expressions for the scattering amplitudes keeping in mind the energy momentum conservation rules and conservation of the KK charge.We adopt the following notations: the field associate with the zero mode (earlier denoted as ϕ0) is denoted as ϕ. If we consider scattering of four such particles for the process a+b→c+d, all being ϕ fields we shall denote it as ϕa+ϕb→ϕc+ϕd. The four momenta of ϕ particles will be denoted as k, in the earlier example it will be like ka+kb→kc+kd. The creation and annihilation operators are: a†(k) and a(k) respectively. Any field which belongs to the KK tower is denoted by χn, n being the KK charge it carries and we always denote four momentum of a KK particle as pμ. For conveniences, we use the notation qn=n from now on which amounts to adopting the convention that R=1. For any field belonging to KK-tower, creation and annihilation operators are denoted respectively as A†(p,qn) and A(p,qn). Sometimes, we might not explicitly exhibit presence of the KK charge in a reaction; however, it will be stated in a context whenever required. There will be three types of scattering processes.(i) qn=0 sector: the reaction involves only four ϕ fields.(ii) Scattering of a ϕ field with a χ field such as ϕ+χ→ϕ+χ. Since KK charge is conserved, by our assumption, the initial and final state particles are described by the χ fields with the same charge.(iii) The scattering of four χ fields. They could be of two types: (a) Elastic scattering where χn+χm→χn+χm. Here the initial particles carry KK charges n and m and final particles also carry same charges. (b) Inelastic scattering like χn+χm→χn′+χm′. The total KK charge conservation implies n+m=n′+m′.Of course in all reactions, total energy momentum conservation is to be guaranteed.Let us consider a generic 4-body reaction(48)a˜+b˜→c˜+d˜ The particles (a˜,b˜,c˜,d˜) (the corresponding fields being ϕ˜a,ϕ˜b,ϕ˜c,ϕ˜d) respectively carrying momenta p˜a,p˜b,p˜c,p˜d; these particles may correspond to the KK zero modes (with KK momentum q=0) or particles might carry nonzero KK charge. We shall consider only elastic scatterings. The Lorentz invariant Mandelstam variables are(49)s=(p˜a+p˜b)2=(p˜c+p˜d)2,t=(p˜a−p˜d)2=(p˜b−p˜c)2,u=(p˜a−p˜c)2=(p˜b−p˜d)2 and ∑p˜a2+p˜b2+p˜c2+p˜d2=ma2+mb2+mc2+md2. We shall maintain independent identities of the four particles which will facilitate the computation of the four point function utilizing the LSZ reduction technique. We list below some relevant (kinematic) variables which we need for our future discussions:(50)Ma2,Mb2,Mc2,Md2 These correspond to lowest mass two or more particle states which carry the same quantum number as that of particle a, b, c and d respectively. We also define six more variables to be(51)(Mab,Mcd),(Mac,Mbd),(Mad,Mbc) The variable Mab carries the same quantum number as (aandb) and it corresponds to two or more particle states. Similar definition holds for the other five variables introduced above. We define two types of thresholds: (i) the physical threshold, sphys, and sthr. In absence of anomalous thresholds (and equal mass scattering) sthr=sphys. Similarly, we may define uphys and uthr which will be useful when we discuss dispersion relations. We assume from now on that sthr=sphys and uthr=uphys. We shall outline how a four point function is obtained in LSZ approach. Normally one starts with |p˜d,p˜cout> and |p˜b,p˜ain> and considers the matrix element <p˜d,p˜cout|p˜b,p˜ain>. Then we subtract out the matrix element <p˜d,p˜cin|p˜b,p˜ain> to define the S-matrix element.(52)<p˜d,p˜dout|p˜b,p˜ain>=δ3(p˜d−p˜b)δ3(p˜c−p˜a)−i(2π)3∫d4x∫d4x′e−i(p˜a.x−p˜cx′)KxKx′<p˜dout|R(x′,x)|p˜bin> where Kx and Kx′ are the four dimensional Klein-Gordon operators and(53)R(x,x′)=−iθ(x0−xo′)[ϕ˜a(x),ϕ˜c(x′)] We have reduced fields associated with a and c in (52). In the next step we may reduce all the four fields and in such a reduction we shall get VEV of the R-product of four fields which will be operated upon by four K-G operators. However, the latter form of LSZ reduction (when all fields are reduced) is not very useful when we want to investigate the analyticity property of the amplitude in the present context. In particular our intent is to write the forward dispersion relation. Thus we abandon the idea of reducing all the four fields.RemarkNote that on the right hand side of the equation (52) the operators act on Rϕ˜a(x)ϕ˜(x′)c and there is a θ-function in the definition of the R-product. Consequently, the action of KxKx′ on ϕ˜a(x)ϕ˜c(x′) will produce a term Rj˜a(x)j˜c(x′). In addition the operation of the two K-G operators will give rise to δ-functions and derivatives of δ-functions and some equal time commutators i.e. there will terms whose coefficients are δ(x0−x0′). When we consider Fourier transforms of the derivatives of these δ-function derivative terms they will be transformed to momentum variables. However, the amplitude is a function of Lorentz invariant quantities. Thus one will get only finite polynomials of such variables, as has been argued by Symanzik [30]. His arguments is that in a local quantum field theory only finite number of derivatives of δ-functions can appear. Moreover, in addition, there are some equal time commutators and many of them vanish when we invoke locality arguments. Therefore, we shall use the relation(54)KxKx′Rϕ˜(x),ϕ˜c(x′)=Rj˜a(x)j˜c(x′) keeping in mind that there are derivatives of δ-functions and some equal time commutation relations which might be present. Moreover, since the derivative terms give rise to polynomials in Lorentz invariant variables, the analyticity properties of the amplitude are not affected due to the presence of such terms. This will be understood whenever we write an equation like (54).4.1.1Scattering of n=0 scalars statesWe study the analyticity properties of the scattering amplitude of the KK zero modes of mass m0. Thus two neutral massive scalars elastically scatter. The amplitude is(55)<kd,kcout|kb,kain>−<kd,kcin|kb,kain>=2πδ3(ka+kb−kc−kd)×F(ka,kb,kc,kd) The notation is as follows: although we are considering scattering of identical, equal mass, neutral particles; it is convenient the label them. ka and kb are the four momenta of incoming particles (a and b respectively), kc and kd are the momenta of outgoing particles. All external particles are on mass shell. F(ka,kb,kc,kd) is the scattering amplitude depending on Lorentz invariant variables defined in (49). We apply the LSZ reduction technique to derive the expression for the four point amplitude(56)<kd,kcout|kb,kain>=4ka0kb0δ3(kd−kb)δ3(ka−kc)−i(2π)3∫d4x∫d4x′e−i(ka.x−kc.x′)×KxKx′<kdout|R(x′;x)|kbin> where(57)R(x′;x)=−iθ(x0−x0′)[ϕa(x),ϕc(x′)] We have reduced particles a and c in the above equation. Kx is the Klein-Gordon (KG) operator. This equation is similar to eq. (52) except that we are now considering the zero modes of KK states. We shall resort to the form of (56) and ignore that form of the amplitude where all four fields are reduced. The following essential remarks are in order in the sequel:(i) The KG operators in (56) act on R(x′;x) in the following ways. When the first KG operator acts on the θ-function it will give rise to a δ-function and also it will produce a derivative of the δ-function. The KG operator, with x-derivative acting on ϕa(x) lead to the source current since the interacting field satisfies the equation of motion (□x+m02)ϕa(x)=ja(x).(ii) Thus the operation of the two KG operators leaves us with R-product of two source currents. Moreover, there are derivatives of δ-functions and equal time commutation relations keeping in mind the preceding remarks. There can be only finite number of such δ-function derivatives in a local field theory as has been argued by Symanzik [30]. After Fourier transforms, these derivatives lead to polynomials in momenta. However, the amplitude is Lorentz invariant and therefore, it depends on Lorentz invariant variables such as s,t and u. The presence of such polynomials do not affect the analyticity properties of the amplitude. Thus we shall not explicitly mention their presence unless it is necessary. When we write(58)(□x+m02)(□x′+m02)(R(x′;x))=Rja(x)jc(x′) it is understood that we are omitting presence of extra terms alluded to above. In a strict sense the equality is not valid. If we consider an arbitrary matrix element between Rja(x)jb(x′) it can be brought to a form Rja(x/2)jb(−x/2) using the translation operations in those matrix elements. We may introduce three generalized functions which are vacuum expectation values of operators [5,6] and their relevance will be transparent soon(59)FR(q)=∫∞+∞d4zeiq.zθ(z0)<Qf|[ja(z/2),j(−z/2)]|Qi>(60)FA(q)=−∫∞+∞d4zeiq.zθ(−z0)<Qf|[ja(z/2),jc(−z/2)]|Qi> and(61)FC(q)=∫−∞+∞d4zeiq.z<Qf|[ja(z/2),jb(−z/2)]|Qi> Qi and Qf are states which carry four momenta and these momenta are held fixed and we treat them as parameters as we shall note in ensuing discussions. It is evident from above equations, (59), (60) and (61), that(62)FC(q)=FR(q)−FA(q) Notice that FC is expressed as commutator of two currents. Then let us open up the commutator and thus we get ja(z/2)jb(−z/2)−jb(−z/2)ja(z/2) between the two states. Let us introduce two complete set of physical states: ∑n|Pnα˜n><Pnα˜n|=1 and ∑n′|P¯n′β˜n′><P¯n′β˜n′|=1. Here {α˜n,β˜n′} stand for quantum numbers that are permitted for the intermediate states. However, the situation slightly different in this scenario in contrast to the well known prescriptions in the study of the analyticity domains of the amplitudes when we consider a theory with only a single neutral scalar field. The intermediate states contribute from the entire Hilbert space which is direct sum of the disjoint Hilbert space each one of which is designated by a KK charge. Thus the intermediate physical states are such that their KK charge is zero in the scattering process of zero charge KK particles. An intermediate state could be a two particle or multiparticle state with total zero KK charge. Now we can express (61) as(63)∫d4zeiq.z[∑n(∫d4Pn<Qf|ja(z2)|Pnα˜n><Pnα˜n|jc(−z2)|Qi>)−∑n′(∫d4P¯n′<Qf|jc(−z2)|P¯n′β˜n′><P¯n′β˜n′|ja(z2)|Qi>)] If we use spacetime translation judiciously on each of the matrix elements in (63) the z-dependence in the arguments of currents disappear i.e. they become ja(0) and jc(0); moreover an energy momentum conserving δ-function appears after we integrate over z. As a consequence Pn=(Qi+Qf)2−q and P¯n′=(Qi+Qf)2+q. Therefore,(64)FC(q)=∑n(<Qf|ja(0)|Pn=(Qi+Qf)2−q,α˜n>×<α˜n,Pn=(Qi+Qf)2−q|jc(0)|Qi>)−∑n′(<Qf|jc(0)|P¯n′=(Qi+Qf)2+q,β˜n′>×<β˜n′,P¯n′=(Qi+Qf)2+q|ja(0)|Qi>) A few explanatory remarks are in order: (i) In the sum over intermediate states the lowest mass two particle state will be 4m02. All higher KK charged intermediate states have higher thresholds since they should appear with zero-sum charges. Although we have not specified |Qi> and |Qf> to have zero KK charge, for the problem at hand these two states (when we consider scattering amplitude) will be particle states with zero KK charge. (ii) The second point to note is that entire tower of KK states will not contribute as intermediate states in the above expressions. In fact, for the scattering amplitude one term will be identified as absorptive part of the s-channel amplitude whereas the other term is the u-channel absorptive amplitude. Thus the contributions from KK towers will have finite number of terms. We shall provide a convincing argument when we consider the elastic scattering of states with nonzero KK charges (see section 5.1).Let us consider the contributions of the multiparticle states from zero-KK-charge sector (states belong to Hqn=0 space) to the above expressions. We define(65)2As(q)=∑n′(<Qf|j(0)a|P¯n′=(Qi+Qf)2+q,β˜n′>×<β˜n′,P˜¯n=(Qi+Qf)2+q|jc(0)|Qi>) and(66)2Au=∑n(<Qf|jc(0)|Pn=(Qi+Qf)2−q,α˜n>×<α˜n,Pn=(Qi+Qf)2−q|jl(0)|Qi>) Note that the Fourier transform of FC(q), FC¯(z), vanishes outside the light cone as a consequence of causality argument. We note that with above definition of Au and As(67)FC(q)=12(Au(q)−As(q)) Moreover, FC(q) will vanish as function of q wherever, both As(q) and Au(q) vanish simultaneously. We also remind that the intermediate states are physical states and their four momenta lies in the forward light cone, V+. Consequently, we arrive at(68)(Qi+Qf2+q)2≥0,(Qi+Qf2)0+q0≥0 and(69)(Qi+Qf2−q)2≥0,(Qi+Qf2)0−q0≥0 These two equations, (68) and (69), imply that there ought to be minimum mass parameters in each of the cases satisfying the conditions, (i) (Qi+Qf2+q)2≥M+2 and (ii) (Qi+Qf2−q)2≥M−2. The matrix elements for As(q) and Au(q) will not vanish and if the two conditions stated above, pertinent to each of them, are fulfilled.The content of the above brief discussion is well known. The essential points are that starting from these results it is possible to derive the analyticity properties of the four point amplitude. (i) The Jost-Lehmann-Dyson theorem allows us to find the location of singularities in the retarded function. Consequently, the existence of small and large Lehmann ellipses is proved.(ii) The fixed t dispersion relation can be proved when t lies within Lehmann ellipse.(iii) A host of known results can be derived for the field theory defined on R3,1⊗S1.(iv) For example, if we choose Qi=kb and Qf=kd then we shall derive expressions for the scattering amplitude. Moreover, we shall be able to obtain expressions for As and Au and relate them to absorptive parts. It can be shown, in this case, that there is a region in the q variable where As(q)=Au(q) for real q and q lies in an unphysical kinematical region. This is the coincidence region. The crossing symmetry of the amplitude is proved using the theory of several complex variables. Our intent is not to address the issues related to crossing symmetry in this article.The summary of this subsection, i.e. 4.1.1, is that for the zero mode field of a compactified field theory (R3,1⊗S1) the scattering amplitude satisfies the analytic properties of known massive, neutral, scalar field theory as expected. This conclusion was also reached by Khuri [22] in his model of potential scattering.4.1.2Elastic scattering of scalar n=0 state and n≠0 stateThis subsection is devoted to study the elastic scattering of n=0 particle with n≠0 particle. We shall utilize the formalism developed in the Subsection 4.1 and take into account the necessary modifications required for this case.First thing to note is that the initial states are the n=0 KK zero mode particle and the n≠0 KK particle which are denoted by ϕa and χb respectively with initial four momenta, ka and pb. Similarly, the outgoing particles are denoted by ϕc and χd with final momenta, kc and pd. The masses of ϕ and χ respectively are m02 and mn2=m02+n2R2 and we are focusing only on the elastic process. The source currents associated with the interacting fields ϕ(x) and χ(x) are denoted by j(x) and J(x) respectively. Note that the initial and the final c.m. momenta, k, are the same for this reaction.The amplitude is defined as(70)<pd,kcout|pb,kain>−<pd,kcin|pb,kain>=2πδ3(ka+pb−kc−pd)F(ka,pb,kc,pd)(71)<pd,kcout|pb,kain>=4ka0kb0δ3(pd−pb)δ3(ka−kc)−i(2π)3∫d4x∫d4x′e−i(ka.x−kc.x′)×KxKx′<kdout|R(x′;x)|kbin> Here we have chosen to reduce the states ϕa and ϕc and the two states with n≠0 are not reduced. Thus R(x′;x) has the same definition as in the previous subsection, 4.1.1, (see (57)). Therefore, we may write the matrix element as before and introduce the three analogous functions, FR(q),FA(q) and FC(q) where the commutator of the source current [ja(z/2),jc(−z/2)] is sandwiched between two fixed arbitrary states |Qi> and |Qf>. When we discuss the properties of the scattering amplitude we should keep in mind that the unreduced initial and final states have n≠0.RemarkTherefore, it is convenient to assign states |Qi> and |Qf> nonzero n quantum number but same value of n since KK charge is conserved and ϕ states carry n=0 KK charge. In view of above remarks let us examine the structure of FC(q) matrix element. As before, we open up the commutator [ja(x),jc(x′)] introduce a complete set of states between the products of the two current. Let us write down the relevant equations for the problem at hand(72)∫d4zeiq.z[∑n(∫d4Pn<Qf|ja(z2)|Pnα˜n>×<Pnα˜n|jc(−z2)|Qi>)−∑n′(∫d4P¯n′<Qf|jc(−z2)|P¯n′β˜n′><P¯n′β˜n′|ja(z2)|Qi>)] If we use spacetime translation on each of the matrix elements in (63) the z-dependence in the arguments of currents disappear i.e. they become ja(0) and jc(0); moreover an energy momentum conserving δ-function appears. As a consequence Pn=(Qi+Qf)2−q and P¯n′=(Qi+Qf)2+q. Therefore,(73)FC(q)=∑n(<Qf|ja(0)|Pn=(Qi+Qf)2−q,α˜n>×<α˜n,Pn=(Qi+Qf)2−q|jc(0)|Qi>)−∑n′(<Qf|jc(0)|P¯n′=(Qi+Qf)2+q,β˜n′>×<β˜n′,P¯n′=(Qi+Qf)2+q|ja(0)|Qi>)Remark(i) We have introduced a complete set of physical states as we had in (63). We recall that we have assigned nonzero (and the same) KK charges to the two states in the matrix element. Moreover, KK charge is an additive conserved quantum number. (ii) Therefore, the admissible physical intermediate states appearing in equations (72) and (73) must carry n-units of KK charge. (iii) There are several possibilities: (a) A zero-mode state (n=0) with a state of single KK state carrying n-unit of KK charge. The attributes of these intermediate states can be understood from (51). (b) Several combinations of KK tower states with many n=0 states. We would like to draw attention to the fact that such contributions arise due to the presence of KK tower of states. However, there will be only finite number of such intermediate states in the sum since these multi particle states are physical and they conserve energy momentum. This argument is intuitively sound. We mention in passing that our main purpose is to investigate the analyticity property of the amplitude for elastic scattering of states carrying nonzero KK charges. The problem at hand (in this subsection) is that we have one particle with KK charge n=0 and other one has n≠0.We may correspondingly define Au(q) and As(q) in analogy with equations (65) and (66). We intend to argue, in this case, FC(q) vanishes when each of the two terms in (73) vanish for certain values of q. In order that FC(q) is nonzero, one of the matrix elements Au(q) or As(q) should be nonzero. On this occasion, we also have constraints(74)(Qi+Qf2+q)2≥0,(Qi+Qf2)0+q0≥0 and(75)(Qi+Qf2−q)2≥0,(Qi+Qf2)0−q0≥0 as was derived in the previous case. We may follows the arguments presented in 4.1.1 that the analyticity properties of this amplitude can be studied even when of the particles carries a nonzero KK charge. There is not much complication in the elastic scattering of two unequal mass particles as is the case here. Therefore, the conclusions drawn at the end of subsection 4.1.1 hold good. Moreover, we are allowed to write a fixed t dispersion relation and do not foresee any difficulty. Thus the forward dispersion relation can be proved in the case analyzed in this subsection(76)(n=0)+(n≠0)→(n=0)+(n≠0) scattering was not addressed by Khuri. The conclusion of this subsection is that the scattering amplitude, considered in this subsection, will satisfy forward dispersion relation. In fact the analyticity hold for nonforward direction as long as t lies inside the corresponding Lehmann-Martin ellipse.5Elastic scatting of states with nonzero Kaluza-Klein chargesThe elastic scattering of two particles carrying nonzero Kaluza-Klein charges are studied here. We repeat some of the assumptions alluded to in the beginning. These are neutral, massive, scalar particles. They are termed as neutral in the sense that they do not carry electric charge (more generally they do not carry any charge associated due to a local gauge symmetry). Of course, the states considered in this section, carry the KK charges. These particles are stable and there are no bound states in the theory. If we consider two incoming particles with KK charges m and n their masses are respectively m02+m2R2 and m02+n2R2. We assume, without any loss of generality, that each of the particles carries same KK charge, n>0; consequently, all four participating particles are of equal mass. It will not affect any of the conclusions if we considered elastic scattering of two KK particles with different charges as will be evident later. In fact, the prescription laid down so far is adequate to handle elastic scattering of particles with unequal KK charges and hence unequal masses. Indeed, Khuri [22]has derived the result for the case (n)+(n)→(n)+(n) for forward scattering. Therefore, our goal is to study the analyticity property of forward amplitude. The initial incoming pair of particles are denoted by χa and χb carry four momenta pa and pb respectively. The outgoing particle are χc and χd and carry four momenta pc and pd. Our first step is to define the scattering amplitude for this reaction and proceed systematically as before(77)<pd,pcout|pb,pain>=4pa0pb0δ3(pd−pb)δ3(pa−pc)−i(2π)3∫d4x∫d4x′e−i(pa.x−pc.x′)×K˜xK˜x′<pdout|R¯(x′;x)|pbin> where(78)R¯(x′;x)=−iθ(x0−x0′)[χa(x),χc(x′)] and K˜x=(□+mn2). We let the two KG operators act on R¯(x;x′) in the VEV and resulting equation is(79)<pd,pcout|pb,pain>=<pd,pcin|pb,pain>−1(2π)3∫d4x∫d4x′e−i(pa.x−pc.x′)×<pd|θ(x0′−x0)[Jc(x′),Ja(x)]|pb> Here Ja(x) and Jc(x′) are the source currents associated with the fields χa(x) and χb(x′) respectively. We arrive at (79) from (77) with the understanding that the R.H.S. of (79) contains additional terms as discussed earlier; however, these terms do not affect the study of the analyticity properties of the amplitude. We have mentioned in the previous section that we shall explore the consequences unitarity in this section and the purpose will be transparent presently. Let us define the T-matrix as follows:(80)S=1−iT The unitarity of the S-matrix, SS†=S†S=1 yields(81)(T†−T)=iT†T In the present context, we consider the matrix element for the reaction a+b→c+d. Note that on L.H.S of (81) it is taken between T†−T. We introduce a complete set of physical states between T†T. For the elastic case with all particles of KK charge, n, the unitarity relation is(82)<pd,pcin|T†−T|pb,pain>=i∑N<pd,pcin|T†|N><N|T|pb,pcin> The complete of states stands for |N>=|Pnα˜n>. The unitarity relation reads,(83)T⁎(pa,pb;pc,pd)−T(pd,pc;pb,pa)=2πi∑Nδ(pd+pc−pn)T⁎(n;pc,pd)T(n;pb,pa) We arrive at an expression like the second term of the R.H.S of (77) after reducing two fields. If we reduce a single field as the first step (as is worked out in text books) there will be a single KG operator acting on the field and eventually we obtain matrix element of only a single current. The R.H.S. of (83) has matrix element like (for example) pa+pb→pn. Thus we can express it as22We adopt the arguments and procedures of Gasiorowicz in these derivations. [31](84)δ(pn−pa−pb)T(n:pb,pa)=(2π)3/2<nout|Ja(0)|pb>δ(pn−pa−pb) After carrying out the computations we arrive at(85)T(pd,pc;pb,pa)−T⁎(pd,pc;pb,pa)=∑N[δ(pd+pc−pn)×T(pd,pc;n)T⁎(n;pb,pa)−δ(pa−pc−pn)×T(pd,−pc;n)T⁎(pd,−pc;n)] Let consider the scattering amplitude for the reaction under considerations.(86)F(s,t)=i∫d4xei(pa+pc).x2θ(x0)<pd|[Ja(x/2),Jc(−x′/2)]|pb> We evaluate the imaginary part of this amplitude, F(s,t)(87)ImF(s,t)=12i(F−F⁎)=12∫d4xei(pa+pc).x2<pd|[Ja(x/2),Jc(−x/2)]|pb> Note that F⁎ is invariant under interchange pb→pd and also pd→pb; moreover, θ(x0)+θ(−x0)=1. We open up the commutator of the two currents in (87). Then introduce a complete set of physical states ∑N|N><N|=1. Next we implement translation operations in each of the (expanded) matrix elements to express arguments of each current as Ja(0) and Jc(0) and finally integrate over d4x to get the δ-functions. As a consequence (87) assumes the form(88)F(pd,pc;pb,pa)−F⁎(pb,pa;pc,pd)=2πi∑N[δ(pd+pc−pn)F(pd,pc;n)F⁎(pa,pb;n)−δ(pa−pc−pn)×F(pd,−pa;n)F⁎(pb,−pc;n)] This is the generalized unitarity relation where all external particles are on the mass shell. Notice that the first term on the R.H.S. of the above equation is identical in form to the R.H.S. of (85); the unitarity relation for T-matrix. The first term in (88) has the following interpretation: the presence of the δ-function and total energy momentum conservation implies pd+pc=pn=pa+pb. We identify it as the s-channel process pa+pb→pc+pd.Let us examine the second term of (88). Recall that the unitarity holds for the S-matrix when all external particles are on shell (as is true for the T-matrix). The presence of the δ-function in the expression ensures that the intermediate physical states will contribute for(89)pb+(−pc)=pn=pd+(−pa) The masses of the intermediate states must satisfy(90)Mn2=pn2=(pb−pc)2 It becomes physically transparent if we choose the Lorentz frame where particle ′b′ is at rest i.e. pb=(mb,0); thus(91)Mn2=2mb(mb−pc0),pc0>0 since mb=mc and pc0=mc2+pc2=mb2+pc2; Mn2<0 in this case. We recall that all particles carry KK charge n and hence the mass is mb2=mn2=m02+n2R2. The intermediate state must carry that quantum number. In conclusion, the second term of (88) does not contribute to the s-channel reaction. There is an important implication of the generalized unitarity equation: Let us look at the crossed channel reaction(92)pb+(−pc)→pd+(−pa);−pa0>0,and−pc0>0 Here pb and pc are incoming (hence the negative sign for pc) and pd and pa are outgoing. The second matrix element in (88) contributes to the above process in the configurations of the four momenta of these particle; whereas the first term in that equation does not if we follow the arguments for the s-channel process.RemarkWe notice the glimpses of crossing symmetry here. As we have argued earlier in subsection 4.1.1. Indeed, the starting point will be to define FC(q) and look for the coincidence region. Notice that q is related to physical momenta of external particles when |Qi> and |Qf> are identified with the momenta of the ‘unreduced’ fields. Indeed, we could proceed to prove crossing symmetry for the scattering process; however, it is not our present goal.Important observations are in order:(i) We could ask whether entire Kaluza-Klein tower of states would appear as intermediate states in the unitarity equation.33I would like to thank Luis Alvarez Gaume for raising this issue. It is obvious from the unitarity equation (88) that for the s-channel process, due to the presence of the energy momentum conserving δ-function, pn2=Mn2=(pa+pb)2; consequently, not all states of the infinite KK tower will contribute to the reaction in this, (s), channel. Therefore the sum would terminate after finite number of terms, even for very large s as long as it is finite. Same argument also holds for the crossed channel process.(ii) We could proceed to prove crossing symmetry from this point. However, our goal, in this investigation, is to specifically examine whether the forward scattering amplitude satisfies in the present case. Thus we focus on the forward amplitude. It is convenient to adopt the prescriptions of Symanzik [30] to this end.5.1Dispersion relation for forward scattering amplitudeWe intend to study the analyticity property of the elastic scattering amplitude in the forward direction in this subsection. This is of paramount interest to us since it was shown by Khuri, in the case of potential scattering in a quantum mechanical model with R3⊗S1 geometry, that the elastic forward scattering amplitude T(k,k;n,n) does not have analyticity property as was the case with potentials with noncompact coordinates. We utilize the formalism developed in Section 4 and in this section. The configuration for the forward scattering amplitude is pc=pa and pd=pb. Moreover, all the particles are scalars and of equal mass, mn2=m02+n2R2. Recall that the amplitude is a function of the Lorentz invariant variables, s and t=0. The forward amplitude is(93)F(pb,pa;pb,pa)=∫d4xeipa.x(□x+mn2)2<pb|Rχa(x)χa(0)|pb> leading to(94)F(pb,pa;pb,pa)=∫d4xeipa.x<pb|RJa(x)Ja(0)|pb>We remind that in the nonforward case the reader that we had an exponential term exp(i(pa+pc).x/2) and there were two KG operators KxKx′ acting on two currents RJc(x)Ja(0) while we considered nonforward amplitude.We shall adopt the procedure of Symanzik who proved the dispersion relations for forward scattering amplitude of pion-nucleon scattering. Indeed, this is a simpler case of equal mass scattering. Alternatively, one could adopt mathematically more rigorous formulation of Bogoliubov; however, we have resorted to the approach of [30]. We go to a frame where particle ′b′ is at rest, pb=0. Introduce the Lorentz invariant variable, ω=pa.pbmn which is the incident energy of ′a′ in this frame. The amplitude, (93) takes the form(95)F(pb,pa:pb,pa)=i∫0∞∫d3xeipa0x0−i(pa0)2−mn2eˆ.xf˜(x,x0) where eˆ is the unit vector along direction of the three momentum pa. We can read off expression for f˜ from (93); at this stage it is sufficient to note that f˜(x,x0) vanishes unless x0>|x| due to microcausality arguments (see more discussions below). We may carry out the angular integration and the resulting expression is(96)F(pb,pa;pb,pa)=∫0∞F(ω,r)dr where(97)F(ω,r)=4πirsinω2−mn2reiωrω2−mn2×∫r∞dteiω(r−t)<pb|[Ja(x,Ja(0)]|pb> Notice that F(ω,r) is analytic function of ω in the upper half ω-plane, i.e. for complex ω, Imω≥0. The following features of the R.H.S. of (97) are noteworthy.(i) It appears that there might be a branch point at ω=±mn. However, note that there is really no branch point at ω=±mn since, as we note, sinω2−mn2eiωrrrω2−mn2 is an even function of rω2−mn2. (ii) When ω<mn we might apprehend the behavior of sinω2−mn2r; however, the presence of eiωr dispels any such doubt. We have used, loosely, the equality (□x+mn2)(Rχa(x)χa(0))=RJa(x)Ja(0). However, as alluded to earlier the equality is up to the presence of finite number of derivatives of δ-functions. Thus the amplitude will be, in the case of forward scattering, might have additional terms on R.H.S. which are polynomials in s, in this Lorentz frame ω (see more discussions on this point later).In what follows, we shall incorporate the essential arguments of Symanzik and give some modified steps of his derivation rather than repeat the entire technical details he provided for the π−N scattering. Let us assume that large ω behavior of the forward amplitude requires no subtractions and therefore, for large ω, the integral over ω vanishes in the dispersion integral expressed as(98)F(ω,r)=1π∫∞+∞ImF(ω′,r)ω′−ω−iϵdω′ We conclude from the expression for F(ω,r), (97), that ImF(r,ω)=−ImF(−ω,r); consequently, the amplitude is an even function of energy, ω. We may rewrite (98) as(99)F(ω,r)=1π∫0+∞ImF(ω′,r)[1ω′−ω−iϵ+1ω′−ω+iϵ] Note, from the definition of F(ω,r), (98), that the integration over r is to be completed if we want to derive dispersion relation for the amplitude F(pb,pa;pb,pa). Thus there arises the issue of interchange of integrations over r and ω. Eventually, we are to compute the forward absorptive amplitude, (Im F), as given in (97). We arrive at the following expression from (94)(100)ImF(pb,pa;pb,pa)=12∫d4xepa.x<pb|[Ja(x),Jb(0)]|> The angular integration for eipa.x can be carried out in the Lorentz frame of our choice leading to an expression analogous to (97). Thus(101)ImF(pb,pa;pb,pa)=12∫dr4πrsinω2−mn2rω2−mn2×∫−∞+∞eiωt<pb|[Ja(x),Ja(0)]|pb>dt We recall that in deriving the generalized unitarity relation for the nonforward amplitude, we had opened up the commutator of the two currents and we inserted complete set of states in the products of currents in the each term. Notice that in the present context, the initial and the final states are identical; this is the route to derive the optical theorem. However, our goal is different here. As has been our practice earlier, we use translation operation to get rid of the x-dependence in the argument of one of the current. Consequently, a factor of eipn.x or e−ipn.x would appear where pn is the momentum of the intermediate state (we are still in the same Lorentz frame). Therefore, as before, the angular integration can be carried out resulting the factors depending on r only(102)ImF(ω,r)=2πrsinω2−mn2rω2−mn2∑nsin|pn|r|pn|r×|<pb|Ja(0)|n>|2[δ(ω+pn0−mn)−δ(ω+mn−pn0)] Here pn is the four momentum of the intermediate state (recall we used completeness relation ∑N|N><N|=1 where N stood for intermediates permitted by energy momentum conservation and collection of all discrete quantum numbers such that KK charge is conserved). The same logic applies here. Notice that the source current Ja carries KK charge of n units. Thus the state Ja(0)|n> has to be such that its KK charge is also n unit since <pb| carries n unit of KK charge. If we carry out d4x integration in (100) then we shall get an expression similar to (87), now in the forward direction.(103)ImF(pb,pa;pb,pa)=12(2π)4∑n|<pb|Ja(0)|pb>|2×[δ4(pb+pa−pn)−δ4(pb−pa+pn)] It is worthwhile to draw the similarities of the present investigation with the work of Symanzik [30] who proved dispersion relation for the forward scattering amplitude of pion-nucleon scattering. We shall discuss the situation where there is departure of the present work from that of [30]. The isospin quantum numbers of pion and nucleon were not accounted for and both the particles were taken to be spinless. Consequently, the complications due to the nucleon spin were not encountered in this situation. Thus the problem at hand was scattering of two unequal mass spinless particles. However, the nucleon was assigned an additive conserved quantum number. Symanzik reduced the two nucleans of initial and final states when he implemented LSZ formalism. In the process of computation of the amplitude, while introducing complete set of states between product of currents in the matrix elements all the conservation laws were accounted for. The dispersion integral was obtained keeping in mind the issues alluded to above. At that juncture, he argued that, at high energy, the amplitude will have, at most, polynomial growth since the works of Jin and Martin and the general results of Epstein, Glaser and Martin were yet to proved.In order to contrast our work with Symanzik's, we note the important feature that all the particles carry additive discrete quantum charge, n, and these are equal mass bosons. Consequently, once we reduce ‘a’ and ‘c’, see (103), (before considering the case of forward scattering) the two unreduced states (‘b’ and ‘d’) each carry n-units of charge. Thus the intermediate states sandwiching between one current with either |b> or <d| must respect the desired conservation laws. We have not proved the Jin-Martin bound in this article. Nonetheless it will suffice if the amplitude has at most a polynomial growth in s for large s. We can write a subtracted dispersion relation in such a case. Notice that in the case of pion-nucleon scattering, there is a stable nucleon carrying the baryon number and the one-particle intermediate state is the nucleon. For the case at hand the intermediate state is to carry two units of n-charge. If the particles of mass m02+4n2/R2 appears as a pole in the amplitude, we can account for this pole as the presence of nucleon pole was taken care of in the pion-nucleon process.Notice that, keeping the above remarks in mind, the rest of the computation could be developed in parallel to the case of the pion-nucleon scattering. The interchange of r and ω integrations in defining the ImF in the dispersion integral can be justified if we adopt arguments of Symanzik and Gasiorowicz [30,31]. On the other hand, Bogoliubov introduced a prescription to obtain the dispersion relation for the scattering amplitude. It is now obvious that if we follow either of the procedures Symanzik [30] or Bogoliubov [14] the forward scattering amplitude for elastic the process(104)pa(n)+pb(n)→pa(n)+pb(n) satisfies dispersion relation in ω and hence in s. In view of the above discussions, the forward scattering amplitude might admit a pole with KK charge of 2n units. Therefore, besides this pole term, if we resort to s and =4mn2−s, since t=0, the dispersion relation can be written down in the familiar form(105)F(pb,pa;pb,pa)=1π∫sthr∞ds′ImF(s′)s′−s+1π∫uthr∞du′ImF(u′)u′−u Therefore, it is demonstrated under the stated assumptions that the imaginary part of the amplitude is tends to zero as s→∞ (i.e. sufficiently convergent) the forward amplitude satisfies dispersion relation. If the amplitude is polynomially bounded as s tends to large values, we can introduce finite number of subtractions as the analyticity property will continue to hold. In case we invoke Jin-Martin [29] upper bound for the amplitude then the forward amplitude will need no more than two subtractions (see discussion for more details).The important conclusion of our investigation is that the forward scattering amplitude for elastic scattering of KK states with n>0, in a scalar field theory defined in a compactified spacetime, i.e. R3,1⊗S1; satisfies dispersion relation in s in the forward direction. This conclusion is similar to the case of a massive, neutral, scalar field theory defines in a flat four dimensional spacetime, R3,1.6Summary and discussionsWe summarize our results in this section and discuss their implications. Our principal goal of this article is to study the analyticity property of the forward scattering amplitude for a five dimensional scalar field theory which is compactified to R3,1⊗S1. The interest in this problem arose from a work of Khuri in potential scattering in a spatial geometry of R3⊗S1. He adopted the Green's function technique and employed perturbation theory. The main conclusion was that for a class of Yukawa-type potentials, the forward scattering amplitude failed to satisfy the dispersion relation in the second order for the case when the discrete quantum number, n, associated with the periodic coordinate of S1, is nonzero. Indeed, there were some concerns if dispersion relation is invalidated in relativistic quantum field theories.We have worked in the axiomatic frameworks of LSZ (Lehmann-Symanzik-Zimmermann). We considered a neutral, scalar, scalar field of mass m0 in D=5 and compactified the spacetime to R3,1⊗S1. The resulting theory is Lorentz invariant in four dimensions. It has a massive, neutral scalar field of mass m0 (mass of the zero mode) in addition to a tower of KK states. We presented a brief outline of Khuri's result in Section 2. We developed the systematic prescription to study the field theory in the D=4 spacetime with a compact dimension. The sections 4, 4.1.1, 4.1.2 were devoted to setting up the frame work. The first case was to consider elastic scattering of n=0 particles. This problem has been studied long ago in the frameworks of axiomatic field theory. The only departure, from the standard case, is the presence of KK towers. We argued qualitatively that the entire tower of KK states do not appear as intermediate states when we derive spectral representations for absorptive parts. It is to be noted that, for this case, we can derive the analyticity properties rigorously for the forward amplitude as well as for the nonforward amplitude. Moreover, we can prove the existence Lehmann ellipses and thus write down fixed t dispersion relation as long as |t| lies inside the Lehmann-Martin ellipse. In fact, Khuri in his analysis concluded that the amplitude for n=0 sector satisfied analyticity properties in his potential model. Next we considered elastic scattering of the n=0 state with a KK state with nonzero charge. This is a case of unequal mass scattering. We adopted the same prescription as we developed for elastic scattering of n=0 states. In the brief subsection 4.1.2 we outlined the argument that this amplitude would satisfy desired and expected analyticity property in the forward direction. Indeed, fixed t dispersion relations can be written down as well. This case has not been studied by Khuri. In the study of the two cases analyzed in Section 4.1.1 and 4.1.2, the unitarity of the S-matrix was not explicitly invoked in the computations. It was sufficient to require Lorentz invariance and microcausality to arrive at the conclusions.We investigated analyticity property for the forward elastic scattering amplitude of states with nonzero KK charges. In principle, one can study elastic scattering of two states carrying KK charges (l,n). The final particles will also carry the same charge since we focus on elastic process. However, we considered a simpler scenario, without any loss of generality, when initial (two) particles carry the same KK charge, n>0. If these charges were different, say (l,n), then it will be elastic scattering of two unequal mass particles of respectively masses, ml and mn. First we derived an expression for the nonforward scattering amplitude starting from the LSZ reduction particles with equal charges. Our goal is to derive dispersion relation. Therefore, we extracted the imaginary part of the amplitude and took the opportunity to invoke generalized unitarity relation. Thus we derived an expression for product of currents and we inserted a complete set of states between the two currents in the expression for the matrix element. The complete set of physical states included contributions of the entire KK tower in the matrix element as long as it respects energy momentum conservation and discrete charge conservation. To recollect, the S-matrix is obtained when the four external particles are on the mass shell and all the intermediate states are physical states. One important conclusion is that energy momentum conservation (and mass shell condition) ensures that entire KK tower does not contribute to the sum of intermediate states. As long as s is finite, we may choose it to be very large, there is a cut off on the contribution of KK towers. The second important point to note that we see a glimpse of crossing symmetry (see remarks after (88)) in the following way: The first term in that equation contributes to the s-channel process and the second term does not from energy momentum conservation consideration. The second term, to be interpreted as crossed channel reaction, contributed to that reaction and the first term does not. These conclusions hold good for physical processes. That the two amplitudes coincide in the unphysical kinematical region as is well known. We have neither proved crossing symmetry here nor we had any intentions. Finally, we show that the forward scattering amplitude satisfies dispersion relations. We invoked arguments of Symanzik in proving the forward dispersion relation. Our conclusion is that in a relativistic quantum field theory defined on a spacetime geometry, R3,1⊗S1, the forward scattering amplitude for elastic scattering of KK states (with nonzero charge) the dispersion relation is satisfied.Now we proceed to discuss some aspects of this investigation. We have assumed existence of stable particles in the entire spectrum of the theory defined on R3,1⊗S1 geometry. Our arguments is based on the conservation of KK discrete charge qn=nR; it is the momentum along the compactified direction. We have also assumed the absence of bound states. The charge, qn is like a global charge and is assumed to be conserved. This conservation law does not originate from a local gauge invariance. At the moment we have no strong argument about absence of bound states. If we had considered a five dimensional theory with gravity and had utilized KK technique to reduce it to a theory in flat Minkowski space with geometry R3,1⊗S1, there will a U(1) gauge field coupled the resulting KK scalars in the spectrum. The gauge field appears in the standard dimensional reduction of a five dimensional theory (with gravity) coupled to a massive neutral scalar field. The spectrum of the flat four dimensional theory is a (zero mode) scalar, the tower of KK states and a U(1) gauge field couple to all KK states. In this scenario, the charge could be associate with a U(1) gauge charge. However, we cannot use LSZ procedure in the five dimensional curved space. Nonetheless in the compactified theory, defined in a flat space with an S1 coordinate the LSZ formalism can be incorporated. In this scenario, with one U(1) gauge field, may be we could have bound states with very low binding energy (like BPS states); although BPS states appear in spontaneously broken symmetry theory (like BPS monopoles). We mention in passing that the present investigation of analyticity of amplitude is carried out for S1 compactification. S1 compactification does not play any special role. We can compactify a flat space Dˆ dimensional field theory to D=4 theory on a Dˆ−4 dimensional torus [32,33]. The line of arguments followed in this work can be suitably generalized to study the analyticity properties of corresponding theory. At this stage we offer no further remark; leave the details for future investigations.Khuri [22] was motivated by the large extra dimension scenario to undertake the problem. He had raised the question what will be the consequences of his conclusions (in the potential scattering model) if indeed the dispersion relation is not valid at LHC energies. However, the field theory we have considered, the forward dispersion relation holds good. Nevertheless it is important to ask if there is large radius compactified theory which is accessible at LHC energy what are consequences for field theories?AcknowledgementsIt is my great pleasure to thank André Martin who inspired me to undertake this investigation. I am grateful to him for numerous valuable discussions. I wish to record my thanks to Luis Alvarez Gaume, D.Z. Freedman, Adam Schwimmer and Stefan Theisen for very illuminating discussions and e-mail correspondances. The gracious hospitality of Hermann Nicolai and the Max-Planck Institüt für Gravitational Physik and the Albert Einstein Institüt is gratefully acknowledged. 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