Unitarity in KK-graviton production: A case study in warped extra-dimensions
UUnitarity in KK-graviton production
A case study in warped extra-dimensions
A. de Giorgi a 1 and S. Vogl a , b 2a Max-Planck-Institut f¨ur Physik (MPP)F¨ohringer Ring 6, 80805 M¨unchen, Germany b Albert-Ludwigs-Universit¨at Freiburg, Physikalisches InstitutHermann-Herder-Str. 3, 79104 Freiburg, Germany
Abstract : The Kaluza-Klein (KK) decomposition of higher-dimensional gravity gives riseto a tower of KK-gravitons in the effective four-dimensional (4D) theory. Such massive spin-2fields are known to be connected with unitarity issues and easily lead to a breakdown of theeffective theory well below the naive scale of the interaction. However, the breakdown of theeffective 4D theory is expected to be controlled by the parameters of the 5D theory. Workingin a simplified Randall-Sundrum model we study the matrix elements for matter annihilationsinto massive gravitons. We find that truncating the KK-tower leads to an early breakdownof perturbative unitarity. However, by considering the full tower we obtain a set of sum rulesfor the couplings between the different KK-fields that restore unitarity up to the scale of the5D theory. We prove analytically that these are fulfilled in the model under considerationand present numerical tests of their convergence. This work complements earlier studiesthat focused on graviton self-interactions and yields additional sum rules that are required ifmatter fields are incorporated into warped extra-dimensions. [email protected] [email protected] a r X i v : . [ h e p - ph ] D ec Introduction
Extra dimensions have been considered in physics for many different purposes since the sem-inal work of Kaluza [1] and Klein [2]. In particular from the early ’90s onward modelsfeaturing large [3–5] and warped extra-dimensions [6, 7] have received considerable attentiondue to their potential to resolve outstanding questions of the Standard Model such as thehierarchy problem. However, even without a concrete particle physics model in mind, extra-dimensional models are interesting laboratories for the physics of massive spin-2 fields, seefor example Sec. 10 of the review [8]. Most particle phenomenology inspired investigationsof extra-dimensional models are chiefly interested in the role of the spin-2 fields as media-tors between initial and final states consisting of matter fields [9–11]. In contrast, studiesmotivated by a more theoretical interest in massive spin-2 fields have recently investigatedthe physics of KK-graviton scattering [12–15]. However, the production of spin-2 particlesfrom matter has not received as much attention. This work aims to close this gap. On theone hand, we expect our results to be relevant for phenomenological studies, for example forgravitationally interaction dark matter in extra-dimensional theories [16–19]. On the otherhand, such a study is also of interest from a more theoretical perspective since matter fieldsare a necessary ingredient in any realistic theory. In the following, we will focus on a concretemodel of warped extra-dimension with two branes originally put forward by Randall andSundrum [6].Interactions between massive spin-2 fields and matter pose a subtle problem since scatter-ing amplitudes involving them are plagued by unitarity issues, see [8] and references therein.The breakdown of the theory at high energies is already expected at the Lagrangian levelbut studies of the scattering amplitudes of spin-2 fields show a rapid growth in the highenergy limit that indicates the break-down of perturbativity at scales much lower than thefundamental scale of the theory. This issue has received attention in the context of massivegravity [20–23] and the construction of theories that avoid this behavior is still being inves-tigated [12, 24]. In contrast, higher dimensional gravity is expected to be well-behaved up tothe fundamental cut-off of the theory and, therefore, these issues should not arise in the asso-ciated 4D theories. Clearly, an individual KK-graviton cannot avoid the conclusions obtainedfrom considerations of general spin-2 fields and, therefore, the other particles of the 4D theoryhave to be involved in the unitarization process that restores the fundamental scale of theunderlying theory [21]. This is reminiscent of the unitarity problem in massive vector bosonscattering in the Standard Model which is resolved by including the Higgs boson [25]. Thedetails of the cancellation mechanism depend on the geometry of the extra-dimension [21, 26]and are not know in general. However, the unitarization of KK-graviton scattering in warpedextra-dimensions has recently been studied [15], see also [13, 14]. So far, only scattering ofKK-gravitons has been considered. However, a theory that describes a phenomenologicallyviable Universe also contains matter fields. We take a look at this previously neglected direc-1ion and investigate the origin and the resolution of unitarity issues in processes connectingmatter and KK-gravitons. For simplicity, we consider only a toy matter Lagrangian andinclude just a single fundamental scalar on the brane. Building on the approach of [15] weanalyze the matrix elements of scalar annihilations into KK-gravitons and find that unitarityis restored up to the fundamental scale once the full tower of KK-gravitons and the radion isincluded in the computation.The paper is organized as follows. In Sec. 2 we briefly introduce warped extra-dimensionsand comment on the connection between gravity in higher dimensions and the effective theoryin 4D. Next, we analyze the matrix elements scalar annihilations into final states consisting ofKK-gravitons and radions. We pay close attention to the high energy behavior and identifysum rules involving the three-KK-graviton (and KK-graviton-radion) couplings required torestore perturbative unitarity up to the cut-off of the full theory. These sum rules are shownto be fulfilled in the RS-model both analytically and numerically in Sec. 4. Finally, we presentour conclusions in Sec. 5
We analyze a simplified version of the Randall-Sundrum model [6] with a toy matter sectorinstead of the full Standard Model field content. To be concrete, our matter Lagrangianconsists of a single scalar with only gravitational interactions. This setup is sufficient tomake the point we are interested in and we expect that our key observations will carry overto a more realistic construction with minor modifications.
Before starting our discussion it is helpful to introduce some basic notation. We use capitalLatin and lower case Greek letters, e.g. M = 0 , , , , µ = 0 , , ,
3, to indicate 5-dimensional (5D) and 4-dimensional (4D)-indices, respectively. Thus, the coordinate of thefull 5D space-time is denoted x M = ( x µ , y ). The 5D space-time is compactified under an S / Z orbifold symmetry yielding a 5D bulk bounded by two 4-dimensional (4D) braneslocated at y = 0 and y = πr c where y indicates the coordinate of the fifth dimension and r c its size. This compactification symmetry leads to the identification ( x, y ) = ( x, − y ) whichallows to extend the coordinate range to y ∈ [ − πr c , πr c ]. It is often convenient to work withdimensionless quantities instead of dimensional ones which can be achieved by normalizingwith respect to r c , e.g. ϕ = y/r c . Gravity permeates the bulk while matter fields are takento be localized on the branes.The action of the theory is given by S = S bulk + S UV + S IR , (2.1)2ith S bulk = 12 M (cid:90) d x π (cid:90) − π dϕ √ G ( R − B ) ,S UV = (cid:90) d x π (cid:90) − π dϕ √− g UV ( − V UV + L UV ) δ ( ϕ ) ,S IR = (cid:90) d x π (cid:90) − π dϕ √− g IR ( − V IR + L IR ) δ ( ϕ − π ) , (2.2)where G is the determinant of the 5D metric, R the Ricci scalar and M the 5D Planck mass.Λ B denotes the vacuum energy of the bulk while V UV and V IR are the vacuum energy termson the brane. L IR and L UV are the Lagrange densities of fields that are localized to the 4Dbranes while g IR/UV are the 4D metric on the respective branes. For simplicity we will take L UV = 0 and L IR = 12 ∂ µ φ∂ µ φ − m φ (2.3)where φ is a scalar field without any interactions besides gravity. Neglecting the matter part,Einstein’s equation is solved by the metric G MN = (cid:32) w ( x, y ) g µν − v ( x, y ) (cid:33) . (2.4)Choosing the vacuum energy contributions on the branes and in the bulk such that thesolution respects 4D Poincar´e invariance allows to fix w ( x, y ) and v ( x, y ) and leads to aninvariant distance interval ds = e − k | y | η µν d x µ d x ν − d y , (2.5)where η µν = Diag(+1 , − , − , − , ) is the flat metric in 4D and k denotes the warpingparameter defined as k ≡ (cid:113) − Λ B . It should be noted that in order to ensure 4D Poincar´einvariance, the branes’ vacuum energies are constrained to be V UV = − V IR = 6 M k . Byperforming the integral over the 5th dimension in eq. 2.2 we can re-express the theory interms of an effective Lagrangian in 4D. In the usual Randall-Sundrum model this allowsto alleviate the hierarchy problem of the SM since the vacuum expectation value (vev) ofthe Higgs v defined in 5D is related to the one in 4D by the warping factor e − kπr c . Forconvenience, we define the dimensionless parameter µ = kr c to simplify the exponent of thewarp factor. For values of µ ≈
12 the exponential factor allows for a TeV scale vev even if allfundamental mass-dimensional parameters of the 5D theory are O ( M P l ), thus resolving thehierarchy problem. As we will not consider the SM explicitly we do not have a preference fora specific value of the warp factor but we will focus on the limit e − µπ (cid:28) G MN (cid:55)−→ G MN + κ h MN , (2.6)where κ is an expansion parameter defined as κ = 2 /M / . The expansion generates scalar,vector and tensor perturbations, corresponding to h µν , h µ and h , respectively. In theRandal-Sundrum model it is possible to choose the gauge such that the vector componentvanishes even though this does not hold for general higher dimensional models [27]. The tensorperturbation correspond to a spin-2 field, i.e. a 5D-graviton, while the scalar perturbation,the radion, is related to the width of the 5th dimension. We utilize the Einstein frameparameterization [28] which amounts to the following replacement in Eq. 2.4 w ( x, y ) = e − k | y | +ˆ u ) , v ( x, y ) = 1 + 2ˆ u , (2.7)where ˆ u contains the radion field. This ansatz eliminates the mixing between the radion andthe gravitons. We take g µν to be weakly perturbed around a flat background g µν = η µν + κ ˆ h µν , (2.8)where ˆ h µν denotes a symmetric tensor field that includes the graviton. The metric is thengiven by G MN = (cid:32) e − k | y | +ˆ u ) ( η µν + κ ˆ h µν ) 00 − (1 + 2ˆ u ) (cid:33) . (2.9)Denoting the radion field ˆ r , we define ˆ u asˆ u ( x, y ) ≡ κ ˆ r ( x )2 √ e k | y | , (2.10)where the fact that the y dependence of ˆ r can be removed by an appropriate choice ofcoordinates [27] has been employed. By expanding the full Lagrangian of the theory inpowers of κ we obtain, order by order, a theory of interacting 5D graviton and radion fieldsˆ h µν and ˆ r . We expand the bulk Lagrangian to third power in the fields since this includesthe three-graviton interaction Lagrangian that is crucial for our studies. In addition, we needthe first two interactions between the scalar field φ and the gravitons and radions. The keyresults of this expansion are summarized in Appx. A. By integrating out the 5th dimension, this model can be reduced to an effective theory in4D. To achieve this, we employ the Kaluza-Klein (KK) decomposition of the 5D fieldsˆ h µν ( x, y ) = ∞ (cid:88) n =0 √ r c h ( n ) µν ( x ) ψ n ( ϕ ( y )) , ˆ r ( x ) = 1 √ r c ψ r r ( x ) , (2.11)4here ψ n ( ϕ ) absorb the 5D-dependence of the fields and the unhatted h and r fields carry the x dependence. As indicated above, ψ r is independent of ϕ . The decomposition transformsthe single 5D-graviton into a tower of 4D-gravitons. In order to get the canonical massiveFierz-Pauli Lagrangian for the gravitons [29], the 5D-components of the KK-decompositionmust satisfy the following differential equation [30]1 r c ddϕ (cid:20) A ( ϕ ) dψ n dϕ (cid:21) = − m n A ψ n , (2.12)where we have introduced the shorthand A ( ϕ ) = e − µ | ϕ | and m n is the mass of the n -thgraviton. This equation is a particular case of the more general Sturm-Liouville equation. Itcan be proved that m n ∈ R with m n < m n +1 and the solutions ψ n ( ϕ ) are orthogonal andnormalized with respect to the scalar product (cid:104) ψ n , ψ m (cid:105) = π (cid:90) − π dϕ A ( ϕ ) ψ n ( ϕ ) ψ m ( ϕ ) = δ n,m . (2.13)Consistency with a phenomenological acceptable 4D gravity requires the graviton with n = 0to correspond to the massless graviton of General Relativity and, hence, m = 0 and ψ =constant. ψ and ψ r are fixed by the normalization to ψ = (cid:114) µ − e − µπ (cid:39) √ µ , ψ r = (cid:114) µe πµ − (cid:39) √ µ e − µπ . (2.14)The functions ψ n> ( ϕ ) and the masses can be determined by solving Equation 2.12 with theboundary conditions ∂ ϕ ψ n | ϕ =0 ,π = 0. In the limit e − µπ (cid:28) ψ n ( ϕ ) (cid:39) e µ | ϕ | N n J (cid:16) γ n e µ ( | ϕ |− π ) (cid:17) and m n (cid:39) kγ n e − µπ , (2.15)where J i denoted the i th Bessel J -function while γ n is the n th zero of J ( x ). The normaliza-tion factors N n are given by N n (cid:39) − e µπ √ µ J ( γ n ) . (2.16)Performing the integration over the 5th dimension on the quadratic pieces of the puregravity Lagrangian yields the kinetic terms of a massless spin-2 field, i.e. the graviton ofgeneral relativity, a massless spin-0 field, i.e. the radion, and an infinite number of spin-2 fields with Fierz-Pauli mass terms, i.e. a tower of massive KK-gravitons. Decomposingthe first order weak field expansion of the matter Lagrangian leads to following interactionbetween matter and gravitons L (1)int = − κT µν ˆ h µν ( x, ϕ = π ) = − κT µν (cid:32) ∞ (cid:88) n =0 √ r c h µνn ( x ) ψ n ( π ) (cid:33) . (2.17)5equiring that the massless graviton matches the expectation from GR allows to fix therelation between the (reduced) Planck mass in 4D, M P l , and the parameters of the 5Dtheory 12 κ √ r c ψ = 1 M P l or, equivalently, M P l = M k (cid:0) − e − µπ (cid:1) , (2.18)which simplifies to M (cid:39) kM P l in the limit e − µπ (cid:28)
1. Due to the different normalization,the strength of the interaction of the other KK-fields h µνn> is controlled by a combined scaleΛ defined by Λ − = M − / ψ n ( π ) / √ r c which leads to Λ (cid:39) M P l e − µπ in the large µ limit. Theradion contribution to the interaction Lagrangian is L (1)int,r = 1 √ rT , (2.19)where T = η µν T µν is the trace of the energy-momentum tensor of the matter field. In the4D reduction of the higher powers of the expanded Lagrangian, interactions between allcombinations of massless graviton, massive KK-modes and the radion with matter appear.The strength of these interactions is given by a generalized scale Λ Nn ,n m ,n r = Λ n m + n r M n P l where n ( n m ) is the number of massless (massive) gravitons, n r the number of radions and N = n + n m + n R .In addition to the interactions between matter and the gravitons or radions we also needthe cubic interactions between these fields which are substantially more involved. Instead ofconsidering a handful of simple interactions on the brane we now need to treat the interactionsbetween all constituents of the KK-tower. The strength of the interaction is controlled by theoverlap of the wave functions or their derivatives in the bulk. As it will turn out to be helpfullater, we first introduce a more general parameterization of the coupling coefficients thanstrictly needed to define the interactions in 4D . The 5D Lagrangian consists of contributionsthat possess either no or two 5D derivatives. The interaction between 3-gravitons does notcontain any derivatives and we can define a coefficient a parameterizing the wave-functionoverlap. In addition, we can also define a different class of coefficients b that include two 5Dderivatives . Labeling the fields from n to n the a ’s and b ’s read a (cid:126)n := π (cid:90) − π d ϕ A ( ϕ ) ψ n ( ϕ ) ψ n ( ϕ ) ψ n ( ϕ ) ,b (cid:126)n := π (cid:90) − π d ϕ A ( ϕ ) ψ (cid:48) n ( ϕ ) ψ (cid:48) n ( z ) ψ n ( ϕ ) , (2.20)where (cid:126)n = ( n , n , n ) indicates the set of fields involved. As can be seen from the definition, a -type coefficients are symmetric under permutation of all indices while b -type coefficientsare only symmetric under permutation of the first two fields. The a - b notation is inspired by [15], however, we prefer to normalize our coefficients differently. The b type integrals do not appear directly in the Lagrangian but they will turn out to the useful later. a nkm √ r c M / ∝ b nkr r / c M / ∝ c nrr √ r c M / Figure 1: Cubic interactions between gravitons and radions.In addition, there are processes involving radion fields. In the two-graviton-radion vertex, asecond kind of b -coefficients appear since the relevant Lagrangian includes two 5D derivatives.One finds b n ,n ,r := π (cid:90) − π d ϕ A ( ϕ ) ψ (cid:48) n ( ϕ ) ψ (cid:48) n ( z ) ψ r (2.21)which is also symmetric in the first two indices. The two-radion graviton vertex is simpler.We denote the coefficient c nrr and find that it is given by c nrr := π (cid:90) − π d ϕ A ( ϕ ) − ψ n ( ϕ ) ψ r ψ r (2.22)Integrating out the 5D and considering all the powers of M and of r c deriving from theLagrangian expansion and the KK-decomposition, the strength of the cubic interactions inthe 4D theory are given by the vertices of Fig. 1. In the large µ -limit these expressionssimplify and we can split off the µ dependence, which combines with M to set the overallscale of the interactions. This simplifies the integrals over the fifth dimension, which separateinto numerical constants that do not depend on the parameters of the theory any more, andinto powers of ke − µπ (cid:39) m n /γ n . In this case we get a knm √ r c M / −→ χ knr √ µe µπ √ r c M / = χ knm Λ (2.23) b knr r / c M / −→ ˜ χ knr µ / e − µπ r / c M / = ˜ χ knr Λ (cid:0) k e − µπ (cid:1) (2.24) c nrr √ r c M / −→ χ nrr √ µe µπ √ r c M / = χ nrr Λ (2.25)7here the numerical coefficients are given by χ nkj ≡ − J ( γ n ) J ( γ k ) J ( γ j ) (cid:90) d u u J ( γ n u ) J ( γ k u ) J ( γ j u ) , ˜ χ knm ≡ γ n γ k J ( γ n ) J ( γ k ) (cid:90) d u u J ( γ n u ) J ( γ k u ) ,χ nrr ≡ − J ( γ n ) (cid:90) d u u J ( γ n u ) = − J ( γ n ) γ n J ( γ n ) . (2.26)Comparing Eq. 2.23, 2.24 and 2.25, it can be seen that the term with the b nkr couplingdoes not have the same energy-dimension as the other two. This follows directly from thepresence of 5D-derivatives which give rise to this term and that are absent in the other two.Consequently, in the 4D Lagrangian, the a knm and c nrr couplings are multiplied either bymasses of the gravitons or by their momenta while the term proportional to b knr just containscombinations of the flat metric η µν .For completeness we also list the coefficient of the interaction between three radions χ rrr .In this case the integration over the fifth dimension is trivial and we find χ rrr ≡ (cid:90) d u u = 12 . (2.27)The numerical couplings above can be generalized to a higher number of participating inter-acting particles by inserting in the integrals a factor of u for every radion, and a factor of − u J ( γ j u ) J ( γ j ) for every j -graviton. We report the Feynman rules for all relevant interactions inthe large µ limit in Appx. B. In order to assess the validity of the theory we analyze the matrix elements of processes thatinvolve the KK-gravitons. We focus on two classes of interactions: 1) KK-graviton productionin φ annihilations ( φφ → G n G m ) and 2) mixed graviton-radion production ( φφ → G n r ). Thefirst class is most interesting from a theoretical point of view and will allow us to identifynew sum rules for the couplings of three gravitational fields while the second class leads toan independent relation for the radion couplings. φφ → G k G n We are interested in the high energy limit of the φφ → G n G k matrix element. There are fourclasses of diagrams that contribute. First, there is the emission of G via t- and u-channel φ φφ → G k G n annihilations.exchange and a contribution from the φφG n G k contact interaction. In addition, there is aninfinite set of diagrams with all possible graviton-modes in the s-channel and one diagramfrom s-channel radion exchange. A representative set of diagrams is shown in Fig. 2.We work in the center of mass frame and take the initial state particle to travel along the z -axis while the final state particle are emitted back to back in some arbitrary direction. Usingthe helicity formalism for spin-2 fields, the spin-2 polarization tensor can be decomposed asa combination of spin-1 polarization tensors [31]: (cid:15) µν = 1 √ (cid:0) (cid:15) µ ± (cid:15) ν ∓ + 2 (cid:15) µ (cid:15) ν + (cid:15) µ ∓ (cid:15) ν ± (cid:1) ,(cid:15) µν ± = 1 √ (cid:0) (cid:15) µ ± (cid:15) ν + (cid:15) µ (cid:15) ν ± (cid:1) ,(cid:15) µν ± = (cid:15) µ ± (cid:15) ν ± . (3.1)An explicit form for the spin-1 polarization tensor of a massive particle moving in an arbitrarydirection, i.e. a particle with with mass m and momentum (cid:126)p = | (cid:126)p | (sin θ cos φ, sin θ sin φ, cos θ ),is given by (cid:15) µ ± = 1 √ e ∓ iγ (0; ∓ cos θ cos φ + i sin φ, cos θ sin φ − i cos φ, ± sin θ ) ,(cid:15) µ = Em (cid:32)(cid:114) − m E ; sin θ cos φ, sin θ sin φ, cos θ (cid:33) . (3.2)Since we only consider scalar particles in the initial state φ and γ can be chosen to be zero.In the following, we neglect m φ and focus on the high energy limit of the annihilations. Weexpand the amplitudes in powers of the center of mass energy √ s . Since we are dealing withan effective theory and the interaction vertex comes with a suppression scale Λ, it is clearthat we have to find contributions to the matrix elements with M ∝ s . However, lookingat the polarization vectors of the longitudinal modes we observe an additional growth for9 (cid:29) m . Thus we expect to find contributions that grow even faster with s . Truncating thes-channel diagrams after the first graviton this is indeed the case and we find contributionswith M ∝ s . This is a bit of a puzzle since this would imply a breakdown of the theory wellbelow the fundamental scale. However, there is no guarantee that a truncated 4D theory willrespect the properties of the 5D theory. Therefore, we expect that the anomalous growth withhigher powers of s cancels once the full theory, i.e. the untruncated KK-tower, is included inthe analysis .We now present the expanded matrix elements in the large s limit order by order in √ s .This allows us to identify sum rules for the couplings of the theory that ensure the cancellationof contributions that grow faster than s . We will demonstrate explicitly that these sum rulesare fulfilled in the Randall-Sundrum model in section 4. Note that we show only the finalstate helicities that are non zero at a given order. In general, we find that the amplitudesrelated to 0-helicity states are the most relevant since they include two of the longitudinalgraviton modes that get enhanced in the high energy limit. Order s : At this order only the helicity zero final state contributes. We find M (0 ,
0) = − is (cid:0) sin ( θ ) (cid:1) ( (cid:80) m χ nkm − m k m n + O ( s ) (3.3)As can be seen the O ( s ) contribution vanishes only if ∞ (cid:88) m =1 χ nkm = 1 . (3.4)This is indeed the case in the Randall-Sundrum model and we will prove it analytically inSection 4. Order s : Here we find more final state helicities that contribute to the expanded matrixelement, see Tab. 1. As can be seen all final states except the (0 , s contribution to M is fulfilled. The (0 , M (0 ,
0) = − is m k m n (cid:34) ( m k + m n ) (cid:32) θ )(4 (cid:88) m χ nkm −
5) + (4 (cid:88) m χ nkm − (cid:33) +4 (cid:88) m χ nkm (cid:0) m k − m n (cid:1) /m m + (3 cos(2 θ ) + 1) (cid:88) m m m χ nkm +24 cos ( θ ) m φ (cid:32)(cid:88) m χ nkm − (cid:33) + 24 k e − µπ ˜ χ nkr (cid:105) + O ( s / ) . (3.5) Recently, this was shown explicitly for the elastic scattering of gravitons, which exhibits an even worsehigh energy behavior [15]. This is resolved by sum rules relating the 3-point interactions of the KK-gravitonsto the 4-point interactions. In contrast, our amplitudes do not depend on the 4-point interactions and ourrelations only involve the three-KK-graviton coupling. n λ k Amplitude-2 0 − is sin ( θ ) ( (cid:80) m χ nkm − ) √ m k -1 -1 − is sin ( θ ) ( (cid:80) m χ nkm − ) m k m n − is sin ( θ ) ( (cid:80) m χ nkm − ) √ m n − is sin ( θ ) ( (cid:80) m χ nkm − ) √ m n +1 +1 − is sin ( θ ) ( (cid:80) m χ nkm − ) m k m n +2 0 − is sin ( θ ) ( (cid:80) m χ nkm − ) √ m k Table 1: Contribution to the matrix element for the production of the k -th and n -th gravitonswith relative helicities λ n and λ k at O ( s ).In contrast to the previous case, this expression also depends on ˜ χ nkr , i.e. the radion con-tribution is crucial for the cancellation. The different angular dependence allows to identifytwo separate sum-rules at this order. The part proportional to cos(2 θ ) vanishes only if ∞ (cid:88) m =1 χ nkm m m = m n + m k (3.6)while the remaining part requires ∞ (cid:88) m =1 χ nkm m m ( m n − m k ) − ( m n + m k ) + 6 k e − µπ ˜ χ nkr = 0 . (3.7)While these sum rules look more daunting than eq. 3.4, it can be shown that they hold inthe model under consideration. Order s / : Finally, we find contributions to the matrix element that scale as s / thatare summarized in tab. 2. Clearly, four of the eight non-zero entries in the table vanishif the O ( s )-sum rule holds. The remaining contributions (containing a 0-mode), are allproportional to ∝ (cid:32) ( m k + m n ) (cid:32)(cid:88) m χ nkm − (cid:33) + (cid:88) m m m χ nkm (cid:33) . (3.8)After imposing eq. 3.4 this just reduces to eq. 3.6. Therefore, all contributions at this ordervanish if the sum rules derived for the contributions at higher power in s hold. φφ → G n r Now we turn towards the the annihilation of DM-particles into a graviton and a radion. Arepresentative set of diagrams is shown in Figure 3. As in the previous Section, we expand11 n λ k Amplitude-2 -1 is / sin(2 θ ) ( (cid:80) m χ nkm − ) m k -1 -2 is / sin(2 θ ) ( (cid:80) m χ nkm − ) m n -1 0 − is / sin(2 θ ) ( ( m k + m n ) ( (cid:80) m χ nkm − ) + (cid:80) m m m χ nkm ) √ m k m n − is / sin(2 θ ) ( ( m k + m n ) ( (cid:80) m χ nkm − ) + (cid:80) m m m χ nkm ) √ m k m n is / sin(2 θ ) ( ( m k + m n ) ( (cid:80) m χ nkm − ) + (cid:80) m m m χ nkm ) √ m k m n +1 0 is / sin(2 θ ) ( ( m k + m n ) ( (cid:80) m χ nkm − ) + (cid:80) m m m χ nkm ) √ m k m n +1 +2 − is / sin(2 θ ) ( (cid:80) m χ nkm − ) m n +2 +1 − is / sin(2 θ ) ( (cid:80) m χ nkm − ) m k Table 2: Contribution to the amplitudes for the production of the n -th and k -th gravitonswith relative helicities λ n and λ k at O ( s / ). M in √ s and report only contributions that grow faster than s . Since a radion is produced,only one longitudinal graviton can be produced and the leading contribution to the matrixelement grows as O ( s ). We find M = is (cid:16)(cid:80) m k e − µπ ˜ χ nmr m m + χ nrr − (cid:17) m n + O ( s ) , (3.9)which leads to our only sum rule from radion final states: (cid:88) m k e − µπ ˜ χ nmr m m = 1 − χ nrr . (3.10)Figure 3: φφ −→ rG n — Representative diagrams contributing at leading order.12his condition relates the graviton-radion-radion coupling and the graviton-graviton-radioncoupling. After having identified the sum rules that ensure the cancellation of contributions that growfaster than s , we now show that these hold in the Randall-Sundrum model. Before startingin earnest, it is important to note that the sum rules for the χ -couplings can be derived fromrelations that only involve ˜ χ since their sums are related. In fact this even goes beyond thelarge µ limit and one can already show this at the level of the a and b coupling. Integrationby parts of b nkm leads to b nmk ≡ π (cid:90) − π d ϕ A ∂ ϕ ψ n ∂ ϕ ψ m ψ k I.b.P. = − π (cid:90) − π d ϕ ψ n ∂ ϕ (cid:0) ∂ ϕ ψ m A ψ k (cid:1) == − π (cid:90) − π d ϕψ n ψ k ∂ ϕ (cid:0) A ∂ ϕ ψ m (cid:1) − π (cid:90) − π d ϕ A ∂ ϕ ψ k ∂ ϕ ψ m ψ n = ( m m r c ) a nkm − b mkn . (4.1)The large µ limit we are interested in allows to separate the part that depends on µ from therest. This was already achieved with the use of χ, ˜ χ that are related to a, b through a nmk = χ nmk √ µe µπ ,b nmk = ˜ χ nmk µ / e − µπ . (4.2)Thus, the relation in eq. 4.1 reduces to γ k χ nmk = ˜ χ knm + ˜ χ kmn . (4.3)In the following we will first prove the sum rules analytically before investigating their im-plications numerically. The relations we need for the cancellations are summarized in Tab. 3. In the following wewill employ various properties of the Bessel functions, see for example [32,33]. Note that it iseasier to prove the first sum rule if the second one has already been established. Therefore,we will demonstrate that they hold in the order that makes the proves simpler and not inthe order in which they appear in the matrix element expansion.13um rule 1: ∞ (cid:80) m =1 χ nkm = 1Sum rule 2: ∞ (cid:80) m =1 χ nkm γ m = γ n + γ k Sum rule 3: ∞ (cid:80) m =1 χ nkm γ m ( γ n − γ k ) = ( γ n + γ k ) − χ nkr k = n = ⇒ ˜ χ nnr = γ n Sum rule 4: ∞ (cid:80) m =1 ˜ χ nmr γ m = 1 − χ nrr Table 3: Sum rules needed for the cancellations.Our starting point is the Fourier-Bessel expansion, see for example [34]. If a function f ( x )is continuous on [0 ,
1] such that f (1) = 0, the integral (cid:90) d x x / f ( x ) (4.4)exists and is absolutely convergent, and f ( x ) has limited total fluctuation, it can be expandedin series in terms of any Bessel-function J ν : f ( x ) = ∞ (cid:88) k =1 a ν,k J ν ( γ ν,k x ) , (4.5)where γ ν,k is the k − th root of J ν and a ν,k = 2 J ν +1 ( γ ν,k ) (cid:90) d u uf ( u ) J ν ( γ ν,k u ) . (4.6)Since we will work only with the roots of J , we define γ k as the k − th root of J and we willset also ν = 1 such that f ( x ) = 2 ∞ (cid:88) k =1 J ( γ k x ) J ( γ k ) (cid:90) d u uf ( u ) J ( γ k u ) . (4.7)Let us recall that:˜ χ knm ≡ − γ k γ n J ( γ k ) J ( γ n ) J ( γ m ) (cid:90) d u u J ( γ k u ) J ( γ n u ) J ( γ m u ) . (4.8)A convenient choice of f ( x ) that satisfies f (1) = 0 is given by: f ( x ) = γ n J ( γ n ) J ( γ m ) J ( γ n x ) J ( γ m x ) x . (4.9)14sing the fact that J ( γ n ) = − J ( γ n ) and multiplying and dividing by γ k , we have: f ( x ) = ∞ (cid:88) k =1 J ( γ k x ) γ k J ( γ k ) ˜ χ knm . (4.10)Differentiating both sides: f (cid:48) ( x ) = 12 (cid:88) k ˜ χ knm (cid:20) J ( γ k x ) − J ( γ k x ) J ( γ k ) (cid:21) x =1 → − (cid:88) k ˜ χ knm . (4.11)But on the other side: f (cid:48) ( x ) | x =1 = − γ n . (4.12)We then have a very helpful relation that will be useful soon: (cid:88) k ˜ χ knm = γ n . (4.13)Combining this expression and eq. 4.3 directly leads to (cid:88) k χ nmk γ k = ( γ n + γ m ) (4.14)thus proving sum rule 2 . By making further use of eq. 4.3 we can derive other sum rules.Permuting the indices we have two equations:˜ χ nmk = γ m χ nkm − ˜ χ mkn , ˜ χ nkm = γ k χ nkm − ˜ χ mkn . (4.15)Using the symmetry properties of χ (cid:126)n and ˜ χ (cid:126)n :˜ χ nmk + ˜ χ mkn γ m = ˜ χ nkm + ˜ χ mkn γ k , = ⇒ ˜ χ mkn (cid:18) γ m − γ k (cid:19) = ˜ χ nkm γ k − ˜ χ nmk γ m , = ⇒ ∞ (cid:88) n =1 ˜ χ mkn (cid:18) γ m − γ k (cid:19) = (cid:88) n ˜ χ nkm γ k − (cid:88) n ˜ χ nmk γ m Eq. . = 0 . (4.16)This implies that: ∞ (cid:88) k =1 ˜ χ nmk = 0 if n (cid:54) = m . (4.17)At this point we have finally the last relation for the case n (cid:54) = m . Using again eq. 4.3 we find γ k (cid:88) n χ nkm = (cid:88) n ˜ χ mkn + (cid:88) n ˜ χ nkm = 0 + γ k , (4.18)which proves sum rule 1 for n (cid:54) = m . For the case with n = m we need to work a littleharder. The starting point is eq. 4.10 combined with the definition of f ( x ) shown in eq.15.9. Multiplying both sides by x and integrating from 0 to 1 we get rid of the extra Besselfunctions in the sum: (cid:90) d x x f ( x ) = (cid:88) k ˜ χ knm γ k . (4.19)In the special case n = m the integral can be performed analytically (cid:90) d x x J ( γ n x ) J ( γ n x ) = J ( γ n ) γ n . (4.20)Including the coefficients of f ( x ), it follows that (cid:88) k ˜ χ knn γ k = 12 . (4.21)Using again Equation 4.3 and setting n = m we get γ k χ nnk = 2 ˜ χ knn = ⇒ (cid:88) k χ nnk = 2 (cid:88) k ˜ χ knn γ k = 1 , (4.22)which proves sum rule 1 for m = n .Next we tackle the relation involving χ nrr . Let us recall the definition of the radioncoefficients: ˜ χ knr ≡ γ k γ n J ( γ n ) J ( γ m ) (cid:90) d u u J ( γ k u ) J ( γ n u ) ,χ nrr ≡ − J ( γ n ) (cid:90) d u u J ( γ n u ) = − γ n J ( γ n ) J ( γ n ) . (4.23)As before, the way to get the desired result is a good choice of f ( x ); in this case we use: f ( x ) = x J ( γ n x ) J ( γ n x ) γ n , (4.24)from which follows f ( x ) = (cid:88) k J ( γ k x ) γ k J ( γ k ) ˜ χ knr . (4.25)Multiplying both sides by x and integrating from 0 to 1 we find (cid:88) k ˜ χ knr γ k = − γ n J ( γ n ) (cid:90) d x x J ( γ n x ) = − J ( γ n ) − γ n J ( γ n ) γ n J ( γ n ) . (4.26)16sing the properties of the Bessel functions, this last result can be written as: − γ n J ( γ n ) (cid:90) d x x J ( γ n x ) = 1 − χ nrr . (4.27)Then the desired result follows (cid:88) k ˜ χ knr γ k = 1 − χ nrr , (4.28)which proves sum rule 4 . Now we turn to sum rule 3, which will turn out to be the mostcomplicated one. With eq. 4.3 we get γ n χ nmk = ˜ χ nmk + ˜ χ nkm , γ m χ nmk = ˜ χ mnk + ˜ χ mkn = ⇒ ( γ n − γ m ) χ nmk = ˜ χ nkm − ˜ χ kmn . (4.29)Multiplying both sides by ( γ n − γ m ), dividing by γ k and summing over k we get (cid:88) k χ nmk ( γ n − γ m ) γ k = ( γ n − γ m ) (cid:88) k ˜ χ knm − ˜ χ kmn γ k . (4.30)To shorten the notation, let us define the following quantity: f nm ≡ (cid:88) k ˜ χ knm γ k = ⇒ (cid:88) k χ nmk ( γ n − γ m ) γ k = ( γ n − γ m )( f nm − f mn ) . (4.31)We proceed by using the result of eq. 4.19. One finds f nm ≡ (cid:88) k ˜ χ knm γ k = N γ n (cid:90) d x x J ( γ n x ) J ( γ m x ) , (4.32)where we have defined a normalization factor N ≡ J ( γ n ) J ( γ m ) (4.33)to get more compact expressions. We can express this integral in terms of ˜ χ nmr , which is thequantity we are interested in. To see this connection we need the following two relations. One: γ m f nm + γ n f mn = 3 ˜ χ nmr For this we exploit the following property of the Besselfunctions: J ( αx ) = 1 α (cid:18) x J ( αx ) − ∂ x ( J ( αx )) (cid:19) , ∀ α ∈ C . (4.34)17hen, it follows that: f nm = γ n N (cid:90) d x x J ( γ n x ) J ( γ m x ) Eq. . = γ n N (cid:90) d x x J ( γ n x ) 1 γ m (cid:18) x J ( γ m x ) − ∂ x ( J ( γ m x )) (cid:19) = 12 γ m ˜ χ nmr − γ n γ m N (cid:90) d x x J ( γ n x ) ∂ x ( J ( γ m x )) I.b.P. = 12 γ m ˜ χ nmr + γ n γ m N (cid:90) d x ∂ x ( x J ( γ n x )) J ( γ m x ) = I.b.P. + Eq. . = 3 ˜ χ nmr γ m − γ m f mn , (4.35)which proves the statement above. Two: f nm + f mn = 1 This is simpler and it relies on the following relation: x J ( αx ) = ∂ x (cid:18) α x J ( αx ) (cid:19) , ∀ α ∈ C . (4.36)It follows that: f nm = γ n N (cid:90) d x x J ( γ n x ) J ( γ m x ) Eq. . = N (cid:90) d x x ∂ x (cid:0) x J ( γ n x ) (cid:1) J ( γ m x ) = I.b.P. = 1 − N (cid:90) d x x J ( γ n x ) ∂ x (cid:0) x J ( γ m x ) (cid:1) Eq. . = 1 − f mn (4.37)and hence f nm + f mn = 1 . (4.38)Using the previous two relations it follows that: f nm − f mn Eq. . = − f nm . (4.39)Combining relation one from above and Equation 4.38: f nm = γ n − χ nmr γ n − γ m . (4.40)18hen the final result follows: (cid:88) k χ nmk ( γ n − γ m ) γ k = ( γ n − γ m ) (cid:88) k ˜ χ knm − ˜ χ kmn γ k = ( γ n − γ m )( f nm − f mn ) == ( γ n + γ m ) − χ nmr , (4.41)which proves sum rule 3 . Thus we have demonstrated that the sum rules required to cancelthe contributions that grow faster than s are fulfilled in the Randall-Sundrum model in thelarge µ limit.A few comments about the limitations of our analysis are in order. First, we only consideredthe large µ limit. In principle, we expect the cancellation to work even without this restriction.However, a possible generalization to this case comes with a number of complications. Thezero-mode of the graviton, which does not contribute in the limit we considered, can no longerbe neglected. This can be accommodated with minimal changes if we switch the discussionfrom χ and ˜ χ to a and b . However, we cannot employ the limiting form of the wave-functionsand masses. Therefore, relations that directly rely on the properties of the wave-functionsand their Fourier-Bessel expansion need to be generalized. Second, it is also worth pointingout that the radion is massless in our analysis. A massless scalar that with a coupling strongerthan gravity would spoil General Relativity. Therefore, this is not acceptable if the model issupposed to incorporate the world we live in. The problem can be solved by a radion mass.This can be achieved by a mechanism that stabilizes the radius of the extra-dimension; aconcrete example is the Goldberger-Wise mechanism [35]. It relies on a new bulk scalar thatmixes with the radion and, as a consequence, the single massless radion gets replaces bya KK-tower of massive scalar fields [28]. An analysis of the impact of the Golberger-Wisemechanism on the unitarization of the matrix elements goes beyond the scope of this work.However, we expect that the basic conclusions will remain the same. To test this assumptionwe have added a radion mass m r by hand and find that the sum rules remain unaffected for m r (cid:28) √ s . We have shown that the sum rules are fulfilled analytically but the question of how fast theyconverge remains. In particular if one is interested in a quantitative study outside of the highenergy limit it is often not feasible to impose the sum rules directly on the matrix elementand one might want to work with a truncated KK-tower. Therefore, it is of great interest toinvestigate how many gravitons need to be summed to get a meaningful numerical result.Let us start by looking at the coefficients χ nkm , ˜ χ nkr and χ nrr . We show their valuesfor a few representative combinations of gravitons in Fig. 4. Looking at the χ n ,n ,m (leftpanel) we observe that they are relatively small with the largest in the ballpark of 0 . − . m = | n ± n | and falls of quite fast whenthis is not fulfilled. This is an important observation that allows us to optimize the number19 a) (b) (c) Figure 4: Numerical values of the relevant coefficients for a representative set of gravitons.of gravitons we need to include in order to get a good approximation of the results. Thesituation is somewhat similar in the case of ˜ χ nkr . Here, the couplings are largest when n = k and fall off quite fast away from this. We observe a growth of ˜ χ nnr with n which is expectedsince we found analytically that ˜ χ nnr ∝ γ n . However, this growth has no implication forsum rule 4 since only the combination ˜ χ nmr /γ m enters. The third type of coefficients is lessinteresting than the others. As can be seen in the third panel, χ nrr decreases rather fast as n increases and has no notable features.Now we turn to study the sum rules in the case of a truncated KK-tower. In Fig. 5 and Fig. 6we show the difference of the sum rule from zero as a function of the number of gravitons inthe sum for some representative benchmarks. For light gravitons the convergence is relativelyfast and considering the first ten massive KK-modes leads to a reduction of the numericalcoefficient by a factor ≤ − for both the first and the second sum rule. However, if weconsider heavier gravitons the situation is a bit different. Since the χ n n m coefficients peakat around m = | n ± n | adding light graviton does not improve the cancellation significantlyuntil N (cid:38) n + n . This can be partially ameliorated if we do not add the gravitons bymass but by largest χ factor. This trick works best for sum rules 1 and 2 and speeds upthe cancellation for moderate values of N while the large N behavior is essentially identical.However, even in this case the precision to which the sum rules are fulfilled lags behind theone achieved for the light spin-2 fields. In the case of sum rule 3, it does not help muchand can even inhibit the cancellation or certain choices of N as can be seen in Fig. 6. Fortwo identical gravitons in the final state sum rule 3 reduces to an analytical expression for χ nnr . As it does not depend on the third graviton any more the sum is trivial and the levelof precision and any deviation from zero is just related to the numerical precision of theevaluation of χ nnr . Therefore, we do not show it in the figure.The level of precision to which the high energy growth has to cancel in a numerical study will20igure 5: Illustration of the first (left) and second (right) sum rule for a truncated KK-tower as a function of the number of included gravitons N. For each sum rule we show threerepresentative choices of the graviton final state as indicated by the inset in the panels. Weshow two prescriptions for the order in which gravitons are added. The solid lines showa standard truncation where all gravitons up to the N th are added while the dashed linecorresponds to in improved prescription where the gravitons are added by the size of χ knm .depend on the computation and the desired precision. However, it appears that just addingone or two gravitons is not enough for most applications. Even for the pair production of thelightest graviton the first five modes need to be considered if a suppression of the unphysicalcontributions by a factor of ≈ − is desired. Extra-dimensional theories have received considerable interest in recent years. They provideattractive models for physics beyond the Standard Model that can help address unsolvedproblems of high energy physics such as the hierarchy problem, dark matter, or the structureof the SM Yukawa couplings. Higher dimensional theories are interesting for other reasons aswell. They allow for intriguing modifications of gravity and give rise to massive spin-2 fields.Despite many years of study, massive gravity is still a topic of current research and far frombeing completely understood. Extra-dimensions provide test cases for more general theoriesand can help out in our understanding of massive spin-2 fields.We choose a particular realization of warped extra-dimensions for our study. To be con-crete, we focused on a simplified version of the well-known Randall-Sundrum model with atoy matter sector consisting of a scalar with gravitational interactions. The structure of the21igure 6: Illustration of the third (left) and forth (right) sum rule for a truncated KK-toweras a function of the number of included gravitons N. For the third sum rule we show twoways of adding the gravitons as in Fig. 5.matrix elements for KK-graviton production from pairs of scalars in the initial state are veryintricate and we find that individual contributions grow as fast as s in the high energy limit.Taken at face value, this points towards a breakdown of perturbative unitarity well belowthe fundamental scale of high dimensional gravity. However, a closer study reveals that thetheory enforces correlations between the graviton (or radion) self-interactions which conspireto cancel terms that grow faster than s . These correlations can be interpreted as sum rulesfor the coefficients of the interactions between gravitational fields. We proved analyticallythat they hold in the large µ limit of the Randall-Sundrum model. Thus the validity ofthe theory is restored once the full KK-tower is included in the calculations. Similar resultshave been obtained in studies of KK-graviton scattering both in warped extra-dimensionsand other geometries [15, 26]. These works found sum rules that relate the coefficients of thegravitation three-point interaction to the four point vertices. We find a separate set of sumrules that connect the graviton and radion interactions to the coupling with matter fieldsthus complementing earlier work. This is interesting from a purely theoretical perspectivesince it adds a second set of conditions for a realistic theory. In addition, our results are alsorelevant to phenomenological studies. For example, we expect effects in the production ofgravitationally interacting dark matter that we intend to study in a forthcoming publication.22 cknowledgments In our calculations we used xAct [36] and xPert [37] for the expansion of the Lagrangian,
FeynRules [38] to derive the Feynman rules and
FeynCalc [39–41] for symbolic manipulationof the matrix elements.
Appendices
A Lagrangian Expansion
We present the results of the weak-field expansion of the metric ds = A ( z ) (cid:16) e − u ( η µν + κ ˆ h µν )d x µ d x ν − (1 + 2ˆ u ) d z (cid:17) , (A.1)where the coordinate z is related to y throughd z = A ( y ) − d y , ∂∂z = A ( y ) ∂∂y (A.2)and where we have defined κ ≡ M / and ˆ u asˆ u ( x, y ) ≡ κ ˆ r ( x )2 √ e k | y | . (A.3)We also adopt the notation ∂ µ ≡ ,µ and ∂ z ≡ (cid:48) . In this coordinate, eq. 2.12 reads ddz (cid:20) A ( z ) dψ n dz (cid:21) = A (3 A (cid:48) ψ (cid:48) n + Aψ (cid:48)(cid:48) n ) = − m n A ψ n . (A.4)For completeness, we first report the volume elements of the bulk √ G = A (cid:34) κ (cid:18)
12 ˆ h − ˆ r √ A (cid:19) + κ (cid:32) − ˆ r ˆ h √ A −
14 ˆ h λµ ˆ h µλ + 18 ˆ h (cid:33) ++ κ (cid:32) ˆ r ˆ h λµ ˆ h µλ √ A − ˆ h ˆ r √ A + 16 ˆ h λµ ˆ h νλ ˆ h νµ −
18 ˆ h ˆ h µν ˆ h νµ + 148 ˆ h + ˆ r √ A (cid:33)(cid:35) + O ( κ ) , (A.5)where ˆ h ≡ η µν ˆ h µν , and of the branes: (cid:112) − g UV/IR = A κ
12 ˆ h − (cid:113) ˆ rA + κ (cid:32) − ˆ r ˆ h √ A −
14 ˆ h λµ ˆ h µλ + 18 ˆ h + ˆ r A (cid:33) ++ κ ˆ r ˆ h λµ ˆ h µλ √ A − ˆ r ˆ h √ A + ˆ r ˆ h A + 16 ˆ h λµ ˆ h νλ ˆ h νµ −
18 ˆ h ˆ h µν ˆ h νµ + 148 ˆ h − (cid:113) ˆ r A ++ O ( κ ) . (A.6)23n the following we use a rescaled expansion parameter ˜ κ ≡ κ . We are going to present theexpansion of L IR to O (˜ κ ) and of the RS-action up to O (˜ κ ). A.1 L IR -Expansion The IR-Lagrangian is given by L IR = √− g IR (cid:20) g µν IR ∂ µ φ∂ ν φ − m φ φ (cid:21) (A.7)We can expand the brane metric g IR and obtain the expansion L IR = L (0)IR + ˜ κ L (1)IR + ˜ κ L (2)IR + O (˜ κ ) (A.8) First order:
The Lagrangian at first order is given by L (1)IR = L (1)IR (ˆ h ) + L (1)IR (ˆ r ) , (A.9)with L (1)IR (ˆ h ) = 12 (cid:104) − m φ ˆ hφ + φ ,µ (ˆ hφ ,µ − h µν φ ,ν ) (cid:105) , L (1)IR (ˆ r ) = 1 √ r (2 m φ φ − φ ,µ φ ,µ ) . (A.10) Second order:
At second order the expanded IR-Lagrangian is given by L (2)IR = L (2)IR (ˆ h ) + L (2)IR (ˆ h ˆ r ) + L (2)IR (ˆ r ) , (A.11)with L (2)IR (ˆ h ) = 14 (cid:104) m φ φ (2ˆ h µν ˆ h νµ − ˆ h ) + φ ,µ ( − h νλ ˆ h λν φ ,µ + ˆ h φ ,µ + 8ˆ h µλ ˆ h λν φ ,ν − h µν ˆ hφ ,ν ) (cid:105) , L (2)IR (ˆ h ˆ r ) = 1 √ r (2 m φ ˆ hφ + φ ,µ ( − ˆ hφ ,µ + 2ˆ h µν φ ,ν )) , L (2)IR (ˆ r ) = 16 ˆ r ( − m φ φ + φ ,µ φ ,µ ) . (A.12) A.2 Bulk-Expansion
We expand the bulk Lagrangian as
L ⊂ L + ˜ κ L + O (˜ κ ) , (A.13)where L already includes the prefactors of the action and the determinant of the metric, i.e. L = √ GM R , such that S RS = (cid:90) d x z IR (cid:90) − z IR d z (cid:16) L − √ GM Λ B (cid:17) . (A.14)24o make the discussion more clear and ordered, the Lagrangian will be separated in two partsdepending on whether they contain 5D-derivatives or not; these will be denoted L A and L B ,respectively. Also some other terms are generated in the expansion; these are responsible forthe vacuum energies cancellations and we do not report them here. We separate each piecefurther depending on how many ˆ h and ˆ r fields they include. Second order: L = L (ˆ h ) + L (ˆ h ˆ r ) + L (ˆ r ). • L (ˆ h ): L A (ˆ h ) = A (cid:104) ˆ h (cid:3) ˆ h − ˆ h µν (cid:3) ˆ h µν − h ˆ h µν ,µν + 2ˆ h µν ˆ h ρν,ρµ (cid:105) , L B (ˆ h ) = − A (cid:104) ˆ h (cid:16) h (cid:48) A (cid:48) + A ˆ h (cid:48)(cid:48) (cid:17) − ˆ h µν (cid:16) h (cid:48) µν A (cid:48) + A ˆ h (cid:48)(cid:48) µν (cid:17)(cid:105) . (A.15) • L (ˆ h ˆ r ): L A (ˆ h ˆ r ) = 0 , L B (ˆ h ˆ r ) = 0 . (A.16) • L (ˆ r ): L A (ˆ r ) = 12 A ˆ r ,λ ˆ r ,λ , L B (ˆ r ) = 0 . (A.17) Third order: L = L (ˆ h ) + L (ˆ h ˆ r ) + L (ˆ h ˆ r ) + L (ˆ r ). • L (ˆ h ): L A (ˆ h ) = A (cid:20)
12 ˆ h µν ˆ h ρσ ˆ h ρσ,µν −
12 ˆ h ˆ h µν ˆ h ,µν − h µν ˆ h ,µ ˆ h ρν ,ρ − ˆ h µν ˆ h νρ ˆ h ,µσσρ + ˆ h µν ˆ h ρσ,µ ˆ h νρ,σ −
14 ˆ h ˆ h µν (cid:3) ˆ h µν + 34 ˆ h µν ˆ h µν,ρ ˆ h ,ρ + 12 ˆ h µν ˆ h νρ (cid:3) ˆ h µρ −
12 ˆ h µν ˆ h ρσ,ρ ˆ h µν,σ + 12 ˆ h ˆ h νρ,µ ˆ h µρ,ν − ˆ h ˆ h µν ,µ ˆ h νρ,ρ + 18 ˆ h (cid:3) ˆ h (cid:21) , L B (ˆ h ) = A (cid:20) −
14 ˆ h (cid:16) h (cid:48) A (cid:48) + A ˆ h (cid:48)(cid:48) (cid:17) + 34 ˆ h µν ˆ h µν (cid:16) h (cid:48) A (cid:48) + A ˆ h (cid:48)(cid:48) (cid:17) + 12 ˆ h µν ˆ h (cid:16) h (cid:48) µν A (cid:48) + A ˆ h (cid:48)(cid:48) µν (cid:17) − ˆ h µλ ˆ h νµ (cid:16) h (cid:48) νλ A (cid:48) + A ˆ h (cid:48)(cid:48) νλ (cid:17)(cid:105) . (A.18) • L (ˆ h ˆ r ): L A (ˆ h ˆ r ) = 0 , L B (ˆ h ˆ r ) = A (cid:114)
32 ˆ rA (cid:104) ˆ h µν (cid:48) ˆ h (cid:48) µν − (ˆ h (cid:48) ) (cid:105) . (A.19)25 L (ˆ h ˆ r ): L A (ˆ h ˆ r ) = 1 A (cid:20) ˆ h µν ˆ r ˆ r ,µν −
16 ˆ h µν ,µν ˆ r −
12 ˆ h ˆ r (cid:3) ˆ r − (cid:3) ˆ h ˆ r (cid:21) , L B (ˆ h ˆ r ) = 0 . (A.20) • L (ˆ r ): L A (ˆ r ) = − (cid:114)
23 ˆ r ˆ r ,µ ˆ r ,µ A , L B (ˆ r ) = 0 . (A.21) B Feynman Rules
List of the Feynman rules that are relevant for the considered processes. At the vertex allmomenta are taken to be directed inwards.
Propagators: = ik − m φ , (B.1)= ik − m r , (B.2)= iP µναβn ( k ) k − m n , (B.3)where P nµναβ ( p ) ≡ (cid:88) s (cid:15) sµν ( p ) (cid:15) sαβ ( p ) ∗ = 12 (cid:18) G µα G νβ + G να G µβ − G µν G αβ (cid:19) , with G µν ≡ η µν − p µ p ν m n . (B.4) B.1 Vertices
For convenience, we define the quantity: C µναβ ≡ η µα η νβ + η µβ η να − η µν η αβ . (B.5)26ith this: A µναβικ ≡ η αι C βκµν + η ακ C βιµν + η αµ C βνικ + η αν C βµικ − η µν C αβικ − η αβ ( η ιν η κµ + η ιµ η κν ) , (B.6) B µναβικ [ k ] ≡ − C ικαβ k µ k ν − C µναβ k ι k κ − C µνικ k α k β + (cid:16) − η βκ η µν k α − η ακ η µν k β + C βκµν k α + C ακµν k β (cid:17) k ι + (cid:16) − η βι η µν k α − η αι η µν k β + C βιµν k α + C αιµν k β (cid:17) k κ + (cid:16) C βµικ k α + C αµικ k β + C αβκµ k ι + C ιµαβ k κ (cid:17) k ν + (cid:16) C βνικ k α + C ανικ k β + C αβκν k ι + C ιναβ k κ (cid:17) k µ − (cid:16) A µναβικ + 2 η µν C αβικ + η αβ η ικ η µν (cid:17) k , (B.7) C µναβικ [ k, p ] ≡ (cid:16) C βκµν p ι + C βιµν p κ + C βνικ p µ + C βµικ p ν − C ικµν p β − p κ η βι η µν − p ι η βκ η µν (cid:17) k α + ( C ακµν p ι + C αιµν p κ + C ανικ p µ + C αµικ p ν − C ικµν p α − p κ η αι η µν − p ι η ακ η µν ) k β + (cid:16) C αβκν p µ + C αβκµ p ν + 2 C µνβκ p α + 2 C ακµν p β − C αβµν p κ − η αβ η κν p µ (cid:17) k ι + (cid:16) C αβιν p µ + C αβιµ p ν + 2 C µνβι p α + 2 C αιµν p β − C αβµν p ι − η αβ η ιν p µ (cid:17) k κ − η αβ (( η κµ p ν + η µν p κ ) k ι + ( η ιµ p ν + η µν p ι ) k κ ) − (cid:16) − C ικβν p α − C ικαν p β − C αβκν p ι − C ιναβ p κ + C ικαβ p ν (cid:17) k µ − (cid:16) − C ικβµ p α − C ικαµ p β − C αβκµ p ι − C ιµαβ p κ + C ικαβ p µ (cid:17) k ν − ( k · p ) (cid:16) A µναβικ + 7 η µν C αβικ + 3 η µν η αβ η ικ + η αβ ( η κµ η ιν + η ιµ η κν ) (cid:17) . (B.8)27 .1.1 Vertices involving only RS particles = i χ nmk (cid:104) A µναβικ m n + A αβµνικ m m + A ικµναβ m k (B.9)+4 (cid:16) B µναβικ [ k ] + B αβµνικ [ k ] + B ικµναβ [ k ] (cid:17) +2 (cid:16) C µναβικ [ k , k ] + C µνικαβ [ k , k ] + C αβικµν [ k , k ] (cid:17)(cid:105) . = i (cid:113) e − πµ µ ˜ χ nkr (cid:0) − η αβ η µν + η αµ η βν + η αν η βµ (cid:1) Λ r c . (B.10)= iχ nrr (cid:0) k + k ) η µν + ( k · k ) η µν + k µ ( k ν − k ν ) (B.11)+ k µ ( k ν − k ν )) . = − i (cid:113) (cid:0) k + k + k (cid:1) χ rrr Λ . (B.12)28 .1.2 Vertices involving φ = i (cid:16) C µναβ p α p β − η µν m φ (cid:17) Λ . (B.13)= i (cid:113) (cid:16) p · p + 2 m φ (cid:17) Λ . (B.14)= − i Λ (cid:16) − m φ C µναβ + p α (cid:16) − η µν p β + p µ η βν + p ν η βµ (cid:17) (B.15)+ p β ( − η µν p α + p µ η αν + p ν η αµ ) + p α p µ η βν + p β p µ η αν − p µ p ν η αβ + p α p ν η βµ + p β p ν η αµ − p µ p ν η αβ − ( p · p ) η αν η βµ − ( p · p ) η αµ η βν + ( p · p ) η αβ η µν (cid:17) . = − i (cid:113) (cid:16) − η µν (cid:16) p · p + 2 m φ (cid:17) + p µ p ν + p µ p ν (cid:17) Λ . (B.16)29 − i (cid:16) p · p + 4 m φ (cid:17) . (B.17) References [1] T. Kaluza, “Zum Unit¨atsproblem der Physik,”
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