WUE-ITP-2004-035 hep-ph/0411258 November 2004 Quantization and High Energy Unitarity of 5 0 0 5D Orbifold Theories with Brane Kinetic Terms 2 n a J 1 Alexander Mu¨cka,b, Lars Nilsec, Apostolos Pilaftsisc and Reinhold Ru¨ckla 3 3 aInstitut fu¨r Theoretische Physik und Astrophysik, Universit¨at Wu¨rzburg, v 8 Am Hubland, 97074 Wu¨rzburg, Germany 5 2 bSchool of Physics, University of Edinburgh, 1 1 King’s Buildings, Edinburgh EH9 3JZ, United Kingdom 4 0 cSchool of Physics and Astronomy, University of Manchester, / h Manchester M13 9PL, United Kingdom p - p e h : v ABSTRACT i X Five-dimensional field theories compactified on an S1/Z orbifold naturally include local r 2 a brane kinetic terms at the orbifold fixed points at the tree as well as the quantum level. We study the quantization of these theories before the Kaluza–Klein reduction and derive the relevant Ward and Slavnov–Taylor identities that result from the underlying gauge and Becchi–Rouet–Stora symmetries of the theory. With the help of these identities, we obtain a generalization of the equivalence theorem, where the known high-energy equivalence relation between the longitudinal Kaluza–Klein gauge modes and their respective would-be Goldstone bosons is extended to consistently include the energetically suppressed terms in the high-energy scatterings. Demanding perturbative unitarity, we compute upper limits on the number of the Kaluza–Klein modes. We find that these limits weakly depend on the size of the brane kinetic terms. 1 1 Introduction Field theories that realize extra compact dimensions offer new perspectives [1,2] to address problems associated with the breaking of the electroweak gauge symmetry of the Standard Model (SM). One interesting aspect of these theories is that the extra components of the gauge fields may acquire a vacuum expectation value through the so-called Hosotani mechanism [2] and so play the role of the Higgs field [3]. Another novel possibility for electroweak symmetry breaking emerges if the gauge symmetry is broken explicitly by boundary conditions, e.g. by compactifying the theory on an interval [4]. Higher dimensional field theories, however, are not renormalizable, in the sense that only a finite number of counterterms would be needed to fix the ultra-violet (UV) infinities to all loop orders. Therefore, higher dimensional field theories should be regarded as effective theories, and as such, they can play a significant role in understanding the low- energy properties of a more fundamental theory that could, for example, be of stringy origin. Their consistent formulation within the context of perturbation theory should still respect basic field-theoretic properties, such as gauge invariance and perturbative unitarity. The present study focuses on five-dimensional (5D) field theories compactified on an S1/Z orbifold. Orbifolding provides a viable mechanism to introduce chiral fermions after 2 the so-called Kaluza–Klein (KK) reduction of 5D theories to 4 dimensions. However, a consistent description of orbifold field theories requires the inclusion of brane kinetic terms (BKTs) which are localized at the orbifold fixed points [5,6]. These new operators are local and need be included as counterterms at the tree-level to renormalize the UV infinities of local operators of the same form that are generated at the quantum level. Inthispaperweextend ourearlier approachtoquantization[7]to5Dorbifoldtheories that include BKTs. An important aspect of our approach is that the higher dimensional theory is quantized, e.g. within the framework of generalized R gauges, before the KK ξ reduction. Such a gauge-fixing procedure is not trivial, since both the gauge-fixing and ghost sectors should contain terms localized at the orbifold fixed points. After the KK decomposition and integration of the extra dimension, the resulting effective 4D theory coincides with the one that would have been obtained if the KK gauge fields had been quantized mode by mode [8] in the conventional R gauge. ξ One of the advantages of our 5D quantization formalism is that both the usual gauge andthe so-called Becchi–Rouet–Stora (BRS) transformations [9]take on a simple and finite form. In close analogy to the ordinary case of 4D Quantum Chromodynamics (QCD), the invariance of the Lagrangian under these transformations gives rise to the corresponding 2 5D Ward and Slavnov–Taylor (ST) identities [10,11]. With the aid of these identities, we can derive a generalization of the Equivalence Theorem (ET) [12,13] within the context of 5D orbifold theories [14] that takes account of the presence of BKTs and the energetically suppressed terms in high-energy scatterings. Such a generalization is obtained by following a line of argument very analogous to that for the ordinary 4D case [15] and has been assisted by the introduction of a new formalism of functional differentiation that preserves the orbifold constraints on the 5D fields. As any perturbative framework of field theory, higher dimensional theories should also satisfy perturbative unitarity so as to be able to make credible predictions for physical observables, such as cross sections and decay widths. Requiring the validity of perturbative unitarity for the s-wave amplitude, we compute upper limits on the number of the KK modes. The restoration of high-energy unitarity differs in our models from the one taking place in 5D orbifold theories without BKTs [14,16]. In the presence of BKTs, unitarity is achieved by delicate cancellations among the entire infinite tower of the KK modes. We find, however, that the upper limits established from perturbative unitarity have a weak dependence on the actual size of the BKTs. In particular, large values of BKTs do not seem to screen the effect of the bulk terms. This weak dependence of the unitarity lim- its on the size of the BKTs may be understood from first principles as follows. High-energy unitarity strongly depends on the UV structure of the theory, since at high energies dis- tances much smaller than the compactification radius of the theory are probed. Therefore, only terms related to the bulk dynamics appear to be relevant when studying perturbative unitarity constraints. Evidently, such considerations will be of immediate phenomenolog- ical relevance when constraining more realistic higher-dimensional models into which the well-established Standard Model (SM) can be embedded. The structure of the paper is as follows. In Section 2, after briefly reviewing the ordinary QCD case, we study the quantization of 5D orbifold field theories with BKTs before the KK reduction. In addition, we derive the relevant Ward and ST identities which are a consequence of the underlying gauge and BRS symmetries of the theory. With the help of these identities, we derive in Section 3 a generalization of the ET, which we call the 5D generalized equivalence theorem (GET). The 5D GET provides a complete equivalence relation between the longitudinal KK gauge modes and their respective would- beGoldstonebosonsthattakes consistently into account theenergetically suppressed terms in high-energy scatterings. Section 4 is devoted to the computation of upper limits on the number of the KK modes which are obtained by requiring the validity of perturbative unitarity. We also analyze the dependence of these unitarity limits on the actual size of the 3 BKTs. Technical aspects of our study are presented in the appendices. Finally, Section 5 summarizes our conclusions. 2 Ward and Slavnov–Taylor Identities We will first review the basic formalism of quantization and the derivation of the Ward and ST identities for the conventional 4D QCD case. This will help us to set up our conventions and obtain insight into analogous considerations concerning quantization and derivation of the corresponding Ward and ST identities within the context of 5D orbifold field theories with BKTs. 2.1 4D QCD The ordinary 4D QCD Lagrangian quantized in the R gauge is given by ξ 1 = Fa Faµν + + , (2.1) LQCD − 4 µν LGF LFP where Fa = ∂ Aa ∂ Aa + gfabcAbAc is the field-strength tensor of the gluon field Aa. µν µ ν − ν µ µ ν µ Moreover, and denote the parts of the Lagrangian, associated with the gauge- GF FP L L fixing scheme and the Faddeev–Popov ghosts: 1 1 = F[Aa] 2 = (∂µAa)2 , LGF − 2ξ µ − 2ξ µ δF(cid:0)[Aa] (cid:1) = ca µ cb = ca δab∂2 gfabc∂µAc cb . (2.2) LFP δθb − µ (cid:0) (cid:1) In the absence of and , the remaining (classical) part of the QCD Lagrangian GF FP QCD L L L is invariant under the usual gauge transformations δAa = Dabθb = δab∂ gfabcAc θb . (2.3) µ µ µ − µ Hence,thegauge-invariantpartofthetree-le(cid:0)veleffectiveaction,(cid:1)Γ[Aa] = 1 d4x Fa Faµν, µ −4 µν satisfies the relation Γ[Aa] = Γ[Aa +δAa], from which the following master Ward identity µ µ µ R for QCD is easily derived: δΓ δΓ ∂ gfabc Ac = 0 . (2.4) µδAa − δAb µ µ µ There is a similarity of the Ward identity (2.4) with the one derived with the background field method (BFM) in QCD [17], which is a consequence of the gauge-invariance of the 4 effective action with respect to gauge transformations of the background fields. Therefore, the results derived below from the classical part of will be valid within the BFM QCD L framework as well. After functionally differentiating (2.4) with respect to gluon fields and subsequently setting all the fields to zero, we obtain Ward identities that relate tree-level n-point corre- lation functions to each other. With the convention that all momenta flow into the vertex, these Ward identities may be graphically represented as follows: cρ q (2.5) q p ik aµ = gfabd + gfacd µ − dν cρ dρ bν k p bν aµ cρ dσ dσ k r r q ik = gfabe eν + gface eρ µ − p k+p q k+q p r cρ bν bν dσ (2.6) cρ q + gfade eσ k+r p bν We now turn our attention to the complete QCD Lagrangian (2.1) quantized in the covariant R gauges. Althoughthe gauge-fixing andghost terms break the gaugeinvariance ξ of , the complete QCD Lagrangian is still invariant under the so-called BRS QCD QCD L L transformations [9]: sAa = Dabcb , µ µ gfabc sca = cbcc , (2.7) − 2 F[Aa] ∂µAa sca = µ = µ . ξ ξ Note that the above form of BRS transformations is nilpotent after the anti-ghost equation of motion is imposed. The nilpotency of the BRS transformations is an important property thatensures unitarity oftheS-matrixoperatorinthesubspace ofthephysical fields [18,19]. 5 Following the standard path-integral quantization approach, we define the generating functional Z of the connected Green functions through the relation eiZ = DADcDc¯exp i d4x +JaµAa +Daca +caDa +KaµsAa +Masca . LQCD µ µ Z h Z (cid:0) ((cid:1)2i.8) In the above, Jaµ, Da and Da are the sources for the gluons, ghosts and anti-ghosts, respectively. As usual, we have included the additional sources Kaµ and Ma for the BRS variations sAa and sca that are non-linear in the fields. µ Given the invariance of the path-integral measure under BRS transformations, it is not difficult to show that Z Jaµ, Da, Da, Kaµ, Ma = Z Jaµ ω ∂µDa, Da, Da, Kaµ+ωJaµ, Ma ωDa , (2.9) −ξ − (cid:2) (cid:3) (cid:2) (cid:3) where ω is a Grassmann parameter, i.e. ω2 = 0. An immediate consequence of (2.9) is the master ST identity for QCD δZ δZ 1 δZ Jaµ Da + Da∂µ = 0. (2.10) δKaµ − δMa ξ δJaµ As before, a number of ST identities can be derived from (2.10), using functional differen- tiation techniques. An alternative and perhaps more practical approach to deriving ST identities isto use the BRS invariance of the Green function defined by means of the time-ordered operator formalism. To give an example, let us consider the BRS invariance of the Green function associated with the fields c¯a(x)Ab(y)Ac(z), i.e. ν ρ s 0 Tca(x)Ab(y)Ac(z) 0 = 0 . (2.11) h | ν ρ | i Employing the BRS transformations given by (2.7), it is straightforward to obtain the expression 1 ∂µ 0 TAa(x)Ab(y)Ac(z) 0 ∂y 0 Tca(x)cb(y)Ac(z) 0 ∂z 0 Tca(x)Ab(y)cc(z) 0 ξ x h | µ ν ρ | i − νh | ρ | i − ρh | ν | i +gfbde 0 Tca(x)cd(y)Ae(y)Ac(z) 0 + gfcde 0 Tca(x)Ab(y)cd(z)Ae(z) 0 = 0. h | ν ρ | i h | ν ρ | i (2.12) The last two terms on the LHS of (2.12) depend on bilinear fields evaluated at the same space-time point y or z. These terms do not have one-particle poles for the external lines attached to those points, and therefore cancel on-shell. The resulting on-shell ST 6 identity for the involved one-particle irreducible Green functions may then be represented graphically as follows: pµ aµ p r q cρ − qν a .(cid:27)..p......r...........q. cρ − rρ a .(cid:27)..p......r...........q. c = 0 , (2.13) bν b bν with q2 = r2 = 0. It is not difficult to check the validity of the above on-shell ST identity for the tree-level QCD Lagrangian (2.1). For our purpose of studying high-energy unitarity in 2 2 scatterings, more useful → are ST identities that apply to on-shell 4-point functions, e.g. to the product of fields: Aa(x)Ab(y)Ac(z)Ad(w). Again, starting from the identity ν ν ρ σ s 0 T c¯a(x)[∂νAb(y)][∂ρAc(z)][∂σAd(w)] 0 = 0, (2.14) h | ν ρ σ | i we arrive at the ST identity which may be given diagrammatically by µa ρc k q k p q r = 0 . (2.15) µ ν ρ σ p r νb σd As we will see in the next section and Section 3, it is the 5D analog of the above on-shell ST identity which lies at the heart of the proof of the GET for the 5D orbifold field theories. 2.2 5D Orbifold Theories with Brane Kinetic Terms We will now study a 5D Yang–Mills theory, such as 5D QCD, compactified on an S1/Z 2 orbifold. As has been observed in [5,6], orbifold compactification introduces BKTs for the 5Dgluon field at theorbifoldfixed points y = 0 andy = πR. These BKTs naturally emerge as counterterms at the tree level, so as to cancel the UV infinities of the same form that are generated by radiative corrections [3]. Here and in the following, we denote Lorentz indices enumerating the 5 dimensions with capital Roman letters M,N etc., with M = µ,5 (µ = 0,1,2,3), and the fifth compact dimension with y x5 ( πR,πR]. ≡ ∈ − 7 To avoid excessive complication in our analytic results, we consider 5D QCD com- pactified on an S1/Z orbifold with one BKT at y = 0. However, our discussion can very 2 analogously carry over to the general case with two BKTs at y = 0 and y = πR. To be specific, the 5D QCD Lagrangian of interest reads 1 (x,y) = 1 + r δ(y) Fa FaMN + + , (2.16) L5DQCD − 4 c MN L5DGF L5DFP (cid:16) (cid:17) where Fa = ∂ Aa ∂ Aa + g fabcAb Ac is the corresponding field-strength tensor MN M N − N M 5 M N for the 5D QCD gluon Aa , and r is taken to be a positive dimensionful coupling for the M c BKT at y = 0.1 In addition, the terms and describe the 5D gauge-fixing and 5DGF 5DFP L L Faddeev–Popov ghost sectors to be discussed in detail below. In order that the 5D orbifold theory includes ordinary QCD with a massless gluon, it is sufficient for the 5D gluon field Aa to obey the following constraints: M Aa (x,y) = Aa (x,y +2πR), Aa(x,y) = Aa(x, y), Aa(x,y) = Aa(x, y) . M M µ µ − 5 − 5 − (2.17) Observe that Aa (Aa) is even (odd) under the Z reflection, y y. Although the BKT at µ 5 2 → − y = 0 breaks explicitly the higher-dimensional Lorentz invariance of the Lagrangian (2.16) down to 4 dimensions, the classical part of the 5D QCD Lagrangian remains invariant under the 5D gauge transformations δAa = Dab θb = δab∂ g fabcAc θb. (2.18) M M M − 5 M (cid:0) (cid:1) For consistency, the 5D gauge parameter θa(x,y) has to be periodic and even under S1/Z : 2 θa(x,y) = θa(x,y +2πR) and θa(x,y) = θa(x, y). − Given the S1/Z parities of the 5D gluon field Aa (x,y) and the gauge parameter 2 M θa(x,y),thesequantitiescanbeexpandedininfiniteseriesintermsoforthonormalfunctions f (y) and g (y) that are even and odd in y, respectively. More explicitly, we have n n ∞ Aa(x,y) = Aa (x)f (y) , µ (n)µ n n=0 X ∞ Aa(x,y) = Aa (x)g (y) , (2.19) 5 (n)5 n n=1 X ∞ θa(x,y) = θa (x)f (y) . (n) n n=0 X 1If rc is negative and rc 2πR, the theory will contain undesirable negative norm states [20]. | |≤ 8 An important property of the orthonormal functions f (y) and g (y) is that the resulting n n coefficients Aa (x) and Aa (x), the so-called KK modes, are mass eigenstates of the (n)µ (n)5 effective theory, after integrating out the extra dimension. For example, in the limit of a vanishing BKT term, r 0, one recovers the standard Fourier series expansion, where c → the functions f (y) and g (y) are proportional to cos(ny/R) and sin(ny/R), respectively. n n In Appendix B we derive the analytic forms of the orthonormal functions f (y) and n g (y) for the complete 5D orbifold theory with BKTs. To properly deal with the singular n behaviour of the BKTs, e.g. at y = 0, we introduce a regularization method to analytically define f (y) and g (y) in the interval ( ǫ,ǫ), where ǫ is an infinitesimal positive constant n n − which is taken to zero at the end of the calculation. In their definition interval ( πR + − ǫ,πR+ǫ], the ǫ-regularized orthonormal functions are given by cosm y 1 m r sinm y, for πR+ǫ < y ǫ n − 2 n c n − ≤ − N f (y) = n  cosm y, for ǫ < y < ǫ (2.20) n √2δn,0πR × n −   cosm y + 1 m r sinm y, for ǫ y πR+ǫ n 2 n c n ≤ ≤     sinm y + 1 m r cosm y, for πR+ǫ < y ǫ n 2 n c n − ≤ − N g (y) = n  sinm y, for ǫ < y < ǫ (2.21) n √2δn,0πR × n −   sinm y 1 m r cosm y, for ǫ y πR+ǫ , n − 2 n c n ≤ ≤   where  N 2 = 1+r˜ +π2R2r˜2m2 , (2.22) n− c c n with r˜ = r /(2πR) 0. Notice that the orthonormal functions f (y) and g (y) are c c n n ≥ gauge-independent and related to each other through ∂ f = m g , ∂ g = m f , (2.23) 5 n n n 5 n n n − where the derivatives are understood to be piecewise defined within the respective sub- intervals given in (2.20) and (2.21). Finally, the physical masses m of the KK gauge n modes Aa are obtained by solving numerically the transcendental equation [6]: (n)µ m r n c = tan m πR . (2.24) n 2 − (cid:0) (cid:1) In addition to the massless solution m = 0, in the limits r 0 and r , (2.24) has 0 c c → → ∞ the simple analytic solutions m = n/R and m = (2n 1)/2R, respectively. n 1 n 1 ≥ ≥ − Employing Fourier-like convolution techniques developed in Appendix C.1, we can calculate the Feynman rules of the effective 4D KK theory. These are listed in Appendix A. 9 For example, applying the convolution techniques to the 5D gauge transformations (2.18), we find the effective gauge transformations for the KK modes δAa = ∂ θa gfabc ∞ √2−1−δn,0−δm,0−δl,0 θb Ac ∆ , (n)µ µ (n) − (m) (l)µ n,l,m m,l=0 X δAa = m θa gfabc ∞ √2−1−δm,0 θb Ac ∆˜ , (2.25) (n)5 − n (n) − (m) (l)5 n,l,m m=0,l=1 X where g = g /√2πR is a dimensionless coupling constant. In the limit r 0, these 5 c → transformations reduce to the ones stated in [7] without BKTs, with the identifications: ∆ = δ and ∆˜ = δ˜ . The definitions of ∆ and ∆˜ are given in n,l,m n,l,m n,l,m n,l,m n,l,m n,l,m Appendix C.2. A technical drawback of (2.25) is that the number of the effective gauge-field trans- formations becomes infinite. Our task of deriving Ward and ST identities will be greatly facilitated if we introduce a formalism of functional differentiation on an S1/Z orbifold. 2 To this end, we need first to define the δ-function relevant to our S1/Z theory with BKTs. 2 An appropriate construction can be achieved by exploiting the completeness relation ∞ δ(y y ;r ) = f (y)f (y ) + g (y)g (y ) . (2.26) ′ c n n ′ n n ′ − n=0 X (cid:2) (cid:3) In the limit r 0, the δ-function goes over to the known form: c → ∞ 1 ny δ(y;r = 0) δ(y) = cos . (2.27) c ≡ 2δn,0 πR R n=0 (cid:18) (cid:19) X Foranyperiodictestfunctionh(y)definedonS1, itcanbeshownthatthedefining property of the δ-function, πR dy[1 + r δ(y)] h(y)δ(y;r ) = h(0), (2.28) c c Z πR − holdstrue. Infact, theproofof(2.28)isbasedontheorthonormalityofthemasseigenmode wavefunctions f (y) and g (y). n n We are now in a position to introduce a functional differentiation formalism that respects the S1/Z properties of the fields Aa(x,y) and Aa(x,y). To be specific, functional 2 µ 5 differentiation on S1/Z may be defined as follows: 2 δAa(x ,y ) 1 µ 1 1 = gνδab δ(y y ;r ) + δ(y +y ;r ) δ(4)(x x ) , δAb(x ,y ) 2 µ 1 − 2 c 1 2 c 1 − 2 ν 2 2 h i δAa(x ,y ) 1 5 1 1 = δab δ(y y ;r ) δ(y +y ;r ) δ(4)(x x ) . (2.29) δAb(x ,y ) 2 1 − 2 c − 1 2 c 1 − 2 5 2 2 h i 10