View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server UT-848 KEK-TH-624 hep-th/9905130 Spinor Exchange in AdSd+1 Teruhiko Kawano Department of Physics, University of Tokyo Hongo, Tokyo 113-0033, Japan [email protected] and Kazumi Okuyama High Energy Accelerator Research Organization (KEK) Tsukuba, Ibaraki 305-0801, Japan [email protected] We explicitly calculate a Witten diagram with general spinor field ex- change on (d + 1)-dimensional Euclidean Anti-de Sitter space, which is nec- essary to evaluate four-point correlation functions with spinor fields when we make use of the AdS/CFT correspondence, especially in supersymmet- ric cases. We also show that the amplitude can be reduced to a scalar exchange amplitude. We discuss the operator product expansion of the dual conformal field theory by interpreting the short distance expansion of the amplitude according to the AdS/CFT correspondence. PACS: 11.25.Hf, 11.25.Sq, 11.15.-q keywords: AdS/CFT correspondence, Spinor field, 4-point function 1. Introduction Recently it has been conjectured [1] that string/M theory on (d + 1)-dimensional Anti-de Sitter space AdSd+1 times an Einstein manifold is dual to the large N limit of d-dimensional conformal field theory CFTd on the boundary of AdSd+1. According to this conjecture, the dual of type IIB string theory on AdS S5 is four-dimensional = 5 × N 4 SU(N) supersymmetric Yang-Mills theory. In [2,3], a more precise dictionary of the AdS/CFT correspondence was proposed to be that the partition function of the boundary CFT with source terms is equal to the partition function of the string theory on AdSd+1 2 with fixed boundary values of fields. Since the ’t Hooft coupling gYMN is proportional to the fourth power of the ratio of the radius of AdSd+1 to the string scale, the string partition function at the leading order in the strong coupling expansion can be estimated by classical type IIB supergravity. In a schematic form, the correspondence can be written [3] as exp φφ0 =exp ISUGRA(φcl) (1.1) * @(AdS) O !+ − Z CFT where φcl denotes the solution of the equations of motion of the fields satisfying the Dirich- let boundary condition in the supergravity and the boundary value of φcl is identified with the source φ0 up to a conformal factor. This “GKP/W relation” (1.1) has been checked for two- and three-point functions [2–23] and precise agreement between the two sides of (1.1) was shown. It is interesting and important to check the duality mapping (1.1) for four-point func- tions [5,24–32]. On the CFT side, the forms of two- and three-point functions are de- termined by conformal symmetry, but four-point functions are only determined up to a function of the cross ratios, so they contain more information about the dynamics of the theory. Although correlation functions of the four-dimensional strongly coupled CFT have not been understood very well, the GKP/W relation may give us some clues to the prob- lem. In [24,25], the four-point function of operators φ, C corresponding to dilaton field O O φ and axion field C was considered. It was found in [25], and in subsequent papers [27, 29–31], that logarithmic terms of the cross ratios appear in various diagrams contributing to scalar four-point functions. In particular, such terms were shown [31] to exist in the four-point function of φ, C after summing all the contributing diagrams. In [29], it has O O been argued that the logarithmic terms are due to the mixture of a single-trace operator and double-trace operators in the operator product expansion which correspond to the 1 exchanged scalar field and the two external scalar fields respectively, and that the double- trace operators in the CFT correspond to two-particle bound states in the supergravity on AdS. Since the new states have not fully been understood, it is important to study whether two-particle bound states contribute to the other correlation functions. In this paper, as a model, we will consider the Yukawa theory on AdSd+1:afieldtheory on AdSd+1 of spinors and scalars with Yukawa interaction. In particular, we will calculate a diagram with spinor exchange in the model. This is a substantial step forward from the calculation of bosonic field exchange diagrams [29–31]. From the GKP/W relation, this diagram on AdS may contribute to four-point correlation functions of two spinor and two scalar operators in CFT. Therefore, our diagram is needed to obtain four-point functions in CFTs with spinor fields, for example, in supersymmetric models. In this paper, we will show that the spinor exchange amplitude can be reduced to the scalar exchange amplitude and its derivative. This allows us to confirm the structure of the spinor indices of the four-point function and explicitly determine the function of cross ratios, which cannot be determined solely from conformal invariance. This paper is organized as follows. In section 2, using the Yukawa theory as a model, we will illustrate how the spinor exchange diagram appears as the Witten diagram in the GKP/W relation. In section 3, we will show the calculation of the spinor exchange diagram by relating it to the scalar exchange diagram and present our result explicitly. Section 4 is devoted to discussion about the operator product expansion of the spinor exchange amplitude. In Appendices A and B, the derivations of the bulk-to-bulk propagator and the Witten propagator of spinor fields are explained, respectively. We have also included some useful formulae regarding the propagators, which are indicated in the text. 2. Yukawa model on the AdS Space In this section, we will consider spinor exchange diagrams of four-point correlation functions in a Yukawa model on (d + 1)-dimensional Euclidean AdS space. In this paper, 2 1 µ µ we will use the metric in the Poincar´e representation; ds = 2 dz dz . The corresponding z0 a a 1 a vielbein eµ is eµ = δµ. The Dirac operator D for spinor fields ψ(z)is z0 6 µ a 1 bc Dψ = e Γ ∂µ + ω Ωbc ψ 6 a 2 µ (2.1) a d = z Γ ∂a Γ ψ, 0 − 2 0 2 where ωbc is the spin connection given by ωab = 1 (δaδb δbδa). The gamma matrices Γa µ µ z0 0 µ 0 µ a b ab ab 1 a− b satisfy Γ , Γ =2δ ,andΩbc is defined by Ω = [Γ , Γ ]. { } 4 In the Yukawa model, we have a spinor field ψ and a scalar field φ with Yukawa interaction. The action S is given by d+1 1 µ 2 2 d d x√g ψ¯ (D m) ψ + µφ φ + M φ + λφψψ¯ + G d ~x√hψψ,¯ (2.2) 6 − 2 ∇ ∇ ZM Z@M where ∂M denotes a regularized boundary of AdS, i.e. z0 = . hij is the induced metric on the surface ∂M. After the completion of our calculation, we will take the regularization parameter to zero. This prescription has been discussed for spinor fields in [9]. In [8], it was noted that the surface term must be added to the bulk Lagrangian, and the normalization of the term was determined in [11] and [33] to be G =1. Now we will consider the solution of the equations of motion a d (D m) ψ = z Γ ∂a Γ m ψ = λφψ, 6 − 0 − 2 0 − − ¯ ¯ a d ¯ ¯ ¯ (2.3) ψ ←D− m = z0∂aψΓ ψΓ0 + mψ = λφψ, − 6 − − 2 − ∆ M2 φ = λψψ,¯ − from the action (2.2), where ∆ = 1 ∂ ggµν ∂ , imposed by the Dirichlet boundary √g µ√ ν condition on z0 = ψ(z)=ψ(~z), ψ¯(z)=ψ¯(~z),φ(z)=φ(~z). (2.4) Recalling the discussion in [9], we can only choose the boundary condition on either of the left- or right-handed ‘chiral’ components of these fields, as we will see below soon. For the (0) (0) free case λ = 0, the solutions ψ = ψ , ψ¯ = ψ¯ can be immediately obtained [9] by using the Modified Bessel function Kν (z) [34] d d+1 i i d k ~ z0 2 Km+ 1 (kz0) Γ k Km 1 (kz0) (0) ik ~z 2 − 2 ~ ψ (z)= d e− · i ψ−(k), (2π) " Km+ 1 (k) − k Km+ 1 (k) # Z 2 2 (2.5) d d+1 i i d k ~ z0 2 Km+ 1 (kz0) Γ k Km 1 (kz0) ¯(0) ik ~z ¯+ ~ 2 − 2 ψ (z)= d e− · ψ (k) + i , (2π) " Km+ 1 (k) k Km+ 1 (k) # Z 2 2 0 where ψ− is the ‘chiral’ component of ψ which is defined such that Γ ψ− = ψ−. Similarly − + 0 + for ψ¯, ψ¯ Γ = ψ¯ . In the limit 0, we have → d µ µ ψ(z)= d ~xU(z x )Km+d+1 (z,~x)ψ0−(~x), − − 2 2 Z (2.6) ¯ d ¯+ µ µ ψ(z)= d ~xψ0 (~x)Km+d+1(z,~x)U(z x ), 2 2 − Z 3 where U(z x) was introduced to be − Γµ(zµ xµ) U(z x)= − . (2.7) − √z0 K d 1 (z,~x) is the Witten propagator [3,6] of a scalar field with mass m m+ 2 + 2 Γ(∆) z ∆ K (z,~x)= 0 (2.8) ∆ d d 2 π 2 Γ(∆ ) z0 + ~z w~ − 2 | − | d 1 + with ∆ = m + + . The boundary spinors ψ−(~x)andψ¯ (~x) in the limit 0were 2 2 0 0 → m d ¯+ m d ¯+ defined such that ψ0− = − 2 ψ− and ψ0 = − 2 ψ .