Spinor Exchange in Adsd+1

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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 exchange 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 supersymmetric 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 functions [5,24–32]. On the CFT side, the forms of two- and three-point functions are determined 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 problem. 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 . Therefore, the Witten propagators

Σ d 1 , Σ¯ d 1 of the spinor ﬁelds can be seen [8,9] to be m+ 2 + 2 m+ 2 + 2

µ µ Σ∆(z,~x)=U(z x )K∆(z,~x) , − P− (2.9) Σ¯ (z,~x)= K (z,~x)U(zµ xµ), ∆ P+ ∆ − where are the ‘chiral’ projection operators given by = 1 (1 Γ0). P± P± 2 ± For the interacting case i.e.λ= 0, it is convenient to introduce the bulk-to-bulk 6 ‘regularized’ propagator deﬁned by

1 d+1 (D m) S(z, w)=S(z, w) ←D− m = δ (z w), (2.10) 6 − −6 − √g − with the boundary condition

(0) d d 0 ψ (z)= d w~ − S(z, w) w Γ ψ(w~). (2.11) | 0= Z

The explicit form of the bulk-to-bulk propagator S(z, w) is discussed in Appendix A, although we do not need it in this paper, except when verifying that

(0) d d 0 ψ¯ (z)= d w~ − ψ¯(w~)Γ S(w, z) w . (2.12) − | 0= Z

In the limit 0, S(z, w) turns out to be → 1 1 0 −2 2 S(z, w)= D+Γ +m z0 Gd+m 1(z, w) + G d +m+ 1 (z, w) + w0 , (2.13) − 6 2 −2 P− 2 2 P h i where G d +m 1 (z, w) is the bulk-to-bulk scalar propagator given [35,36] by 2 ∓ 2 +1 1 1 Γ( 42 )Γ( 42 ) +1 1 G (z, w)= d+1 F 4,4 ,ν+1; (2.14) 4 (u +1) Γ(ν +1) 2 2 (u+1)2 4π 2 4 4 with = = d + m 1 and ν = d.Thevariableuis the chordal distance; ∓ 2 2 2 (z4 w)2 4 ∓ 4− u = − . The scalar propagator satisﬁes that 2z0w0

2 1 ( ∆+m )G∆(z, w)= δ(z, w). (2.15) − √g

The function F (α, β, γ; z) is a hypergeometric function[37]. The derivation of the propagator (2.13) is given in Appendix A. Using the propagator (2.13), the solution of the equations of motion (2.3) is given by a set of recursion relations [9]

(0) d+1 ψ(z)=ψ (z) λ d w√gwS(z, w)φ(w)ψ(w), − Z (0) d+1 ψ¯(z)=ψ¯ (z) λ d w√gwψ¯(w)φ(w)S(w, z), − Z (2.16) d d+1 φ(z)= d ~xK(z,~x)φ(~x) λ d w√gwG(z, w)ψ¯(w)ψ(w) − Z Z (0) d+1 ¯ = φ (z) λ d w√gwG(z, w)ψ(w)ψ(w), − Z where K(z,~x) is the ‘regularized’ Witten propagator [6,5] of the scalar field. G(z, w)is the ‘regularized’ bulk-to-bulk propagator [5] of it. In this paper, we will not use the explicit form of them. Solving the equation (2.16) recursively and substituting the solution into the action (2.2), we find the bulk action of the spinors vanishing and obtain S = SB + SF , where SB gives the two-point function of the scalar field in the boundary CFT and

d SF = G d ~x√hψ¯(~x)ψ(~x) Z∂M d ¯(0) (0) d+1 ¯(0) (0) (0) = G d ~x√hψ (~x)ψ (~x)+2λG d z g(z) ψ (z)φ (z)ψ (z) ∂M M Z Z p 2 d+1 d+1 ¯(0) (0) ¯(0) (0) (2.17) 2λ G d z g(z)d w g(w) ψ (z)ψ (z)G(z, w)ψ (w)ψ (w) − M Z p p 2 d+1 d+1 (0) (0) (0) (0) 2λ G d z g(z)d w g(w) ψ¯ (z)φ (z)S(z, w)φ (w)ψ (w) − ZM + O(λ3), p p

d 3 with √h = − . Note that the terms included in O(λ ) have more than four external legs. Here we can see in (2.17) that the first term gives the two-point function of the spinors in the boundary CFT [9,8] and that the second term gives the three-point function with two spinors and a scalar [10]. As in the case for scalar fields [6,5], we should take the limit 0 → 5 after the calculation when we evaluate the two-point function [9]. But, since it seems that there is nothing wrong with the exchange of the order for the other multi-point functions, we will take the limit 0 at the beginning of the calculation. The third and fourth term → contribute to four-point functions in the boundary CFT. The third term Sψψψ¯ ψ¯ has four legs of the spinor fields and can be easily related to a scalar exchange amplitude with four legs of scalars

Sψψψ¯ ψ¯

2 d+1 d+1 ¯(0) (0) ¯(0) (0) 2λ G d z g(z)d w g(w) ψ (z)ψ (z)G(z, w)ψ (w)ψ (w) 0 → − M Z p p (2.18) 2 d d d d + + = 2λ G d ~x d ~x d ~x d ~x ψ¯ (~x ) ~x ψ−(~x ) ψ¯ (~x ) ~x ψ−(~x ) − 1 2 3 4 0 1 6 12 0 2 0 3 6 34 0 4 Z d+1 d+1 d+1 d+1 I (~x ,m+ ;~x ,m+ ;~x ,m+ ;~x ,m+ ) × ∆ 1 2 2 2 3 2 4 2 with ~x = ~x ~x and ~x = ~x ~x , where the scalar exchange amplitude 12 1 − 2 34 3 − 4 I∆(~x1, ∆1; ~x2, ∆2; ~x3, ∆3; ~x4, ∆4)isgivenby

dd+1z g(z)dd+1w g(w) Z (2.19) pK (z,~x )Kp (z,~x )G (z, w)K (z,~x )K (z,~x ) × ∆1 1 ∆2 2 ∆ ∆3 3 ∆4 4 which has been calculated by Liu [29] and by D’Hoker and Freedman [30] using general d+1 values for the conformal dimensions ∆a, but here with ∆a = m + for a =1, ,4. 2 ··· The fourth term with two spinor- and two scalar-legs

Sψφψφ¯

2 d+1 d+1 (0) ¯0 0 (0) 2λ G d z g(z)d w g(w) φ (z)ψ (z)S(z, w)ψ (w)φ (w) →0 − → Z 2 d d pd d pd+1 d+1 =2λ G d ~x1d ~x2d ~x3d ~x4 d z g(z)d w g(w) φ0(~x2)K∆2 (z,~x2) (2.20) Z Z + p p ¯ ¯ d 1 d 1 ψ0 (~x1)Σm+ + (z,~x1)S(z, w)Σm+ + (w,~x3)ψ0−(~x3)K∆4 (w,~x4)φ0(~x4) × 2 2 2 2 2 d d d d + = 2λ G d ~x d ~x d ~x d ~x φ (~x )ψ¯ (~x )A(~x ,~x ,~x ,~x )ψ−(~x )φ (~x ) − 1 2 3 4 0 2 0 1 1 2 3 4 0 3 0 4 Z remains to be calculated. The evaluation of this term is the main purpose of this paper, and we will show that this term can also be related to the scalar exchange amplitude I∆ in the next section.

6 3. Spinor Exchange Amplitude In this section, we will evaluate the spinor exchange diagram

A(~x1,~x2,~x3,~x4)

= dd+1z g(z)dd+1w g(w) K (z,~x ) − ∆2 2 Z Σ¯p (z,~x )S(z,p w)Σ (w,~x )K (w,~x ) (3.1) × ∆1 1 ∆3 3 ∆4 4 = dd+1z g(z)dd+1w g(w) K (z,~x )K (z,~x ) − ∆1 1 ∆2 2 Z p p +U(z x1)S(z, w)U(w x3) K∆ (w,~x3)K∆ (w,~x4) ×P − − P− 3 4 with the general value of the weights ∆i (i =1, ,4). The translational invariance ··· on the boundary implies that A(~x1,~x2,~x3,~x4)=A(~x13,~x23, 0,~x43). Under the inversion zµ zˆµ = zµ/z2 and ~x ~xˆ = ~x/ ~x 2, the integration measure is invariant and → → | | K (ˆz,~xˆ)= ~x2∆K (z,~x), ∆ | | ∆ ¯ ˆ ~x z Σ∆(ˆz,~x)= 6 K∆(z,~x)U(z x) 6 , −~x 2∆+2 P− − z (3.2) | |− | | w ∆ 1 − 2 Σ∆(ˆw, 0) = c∆ 6 w0 , w P− | | ~x √ 2 Γ(∆) i03 ˆ with z = z ,wherec∆ = d . Substituting ~xi3 = ~x 2 = ~x0i3 (i =1,2,4) into | | π 2 Γ(∆ d ) i03 − 2 | | A(~x1,~x2,~x3,~x4) and using the above equations (3.2), we can see that

~x13 A(~x1,~x2,~x3,~x4)=c∆3 6 B(~x130 ,~x230 ,~x430 )(3.3) ~x 2∆1 ~x 2∆2 ~x 2∆4 | 13| | 23| | 43| where

d+1 d+1 B(~x1,~x2,~x4)= d z g(z)d w g(w) K∆ (z,~x2)K∆ (z,~x1) P− 2 1 Z (3.4) p pz w ∆ 1 3− 2 U(z x1) 6 S(ˆz, wˆ) 6 w0 K∆4 (w,~x4) . × − z w P− | | | | As explained in Appendix A, S(ˆz,wˆ) turns out to be

1 z 1 2 1 w 0 2 S(ˆz,wˆ)= 6 D+Γ m G∆ (z, w) + + G∆+ (z, w) w0 6 , (3.5) z 6 − z − P P− w | | 0 | | where ∆ = d + m 1 . Putting (3.5) into (3.4), performing partial integration, and then ± 2 ± 2 using the formulas

d +1 0 K∆(z,~x)U(z x)D←−z = ∆ Γ K∆(z,~x)U(z x), − 6 − − 2 − (3.6) ∂ d z Γµ K (z,~x)=∆Γ0K (z,~x) 2 ∆ (z x)K (z,~x), 0 ∂zµ ∆ ∆ − − 2 6 −6 ∆+1 7 we obtain

d 1 B(~x ,~x ,~x )=(2∆ d)J(~x ,~x ,~x )+ m+∆ + I(~x ,~x ,~x )(3.7) 1 2 4 2− 1 2 4 12 − 2 2 1 2 4 with ∆ =∆ ∆ ,where 12 1− 2

d+1 d+1 I(~x1,~x2,~x4)= d z g(z)d w g(w) K∆2 (z,~x2) Z p p ∆3 K∆1 (z,~x1)G∆+ (z, w)w0 K∆4 (w,~x4), × (3.8) d+1 d+1 J(~x1,~x2,~x4)= d z g(z)d w g(w) U(z x1)U(z x2) P− − − P− Z pK (z,~x )pK (z,~x )G (z, w)w∆3 K (w,~x ). × ∆2+1 2 ∆1 1 ∆+ 0 ∆4 4 From the following equations

2 i i i (z x2) Γ (z x ) U(z x1)U(z x2) = − x12 − , P− − − P− z0 −6 z0 P− (3.9) Γi(zi xi) 1 − K∆+1(z,~x)= d ∂xK∆(z,~x), z0 2(∆ ) 6 − 2 we ﬁnd that 1 J(~x ,~x ,~x )= [2∆ ~x ∂ ] I(~x ,~x ,~x ). (3.10) 1 2 4 2∆ d 2−6 12 6 2 1 2 4 2 − Thus, we can obtain

B(~x ,~x ,~x )=[ ~x ∂ +(Σ +∆ d)] I(~x ,~x ,~x )(3.11) 1 2 4 −6 12 6 2 12 + − 1 2 4 with Σ12 =∆1+∆2. As seen in [29] and [30], it is obvious that I(~x1,~x2,~x4) is essentially a scalar exchange amplitude. In this paper, we will follow the method which has been given by Liu [29] to give our amplitude B(~x1,~x2,~x4) by using the Mellin-Barnes representation of the hypergeometric functions. From the result of the paper [29], I(~x1,~x2,~x4)canbe read to be

I(~x1,~x2,~x4)

Σ12 +i C 2 ∞ ds Σ12 Σ34 ∆+ ˜ = Γ( s + )Γ( s + )Γ( s + ) I(s) (3.12) Σ 2∆3 Σ12 2c∆3 ~x24 − i 2πi − 2 − 2 − 2 | | Z 2 − ∞ s+ Σ12 ∆34 ∆12 ξ η− 2 F (s ,s+ , 2s;1 ), × − 2 2 − η

8 where Σ d Σ12+∆+ d Σ34+∆+ d 1 Γ( − )Γ( − )Γ( − ) C = 2 2 2 , 3 d d 4 d 4π 2 Γ(∆ +1 ) Γ(∆i ) + − 2 i=1 − 2 Γ( ∆12 + s)Γ( ∆12 + s)Γ( ∆34 + s)Γ( ∆34 + s) I˜(s)= 2 2 Q 2 2 − Σ+∆ d − (3.13) Γ(2s)Γ( +− s) 2 − Σ12 Σ34 ∆+ d Σ 3F2 , , s;∆+ +1 , s;1 , × 2 2 2 − −2 2 − ! e e e with Σij =Σij +∆+ d and Σ=Σ+∆+ d. Additionally, Σij =∆i+∆j,∆ij =∆i ∆j, − 2 − 2 − 4 ~x14 ~x24 and Σ = i=1 ∆i.Hereη=|~x |2 and ξ = |~x |2 . The function 3F2(a, b, c; e, f; z)isa e e | 12| | 12| generalizedP hypergeometric function [37]. Therefore, putting (3.12) into (3.11), we can immediately calculate B(~x1,~x2,~x4). Finally, from this B(~x1,~x2,~x4) and (3.3), we obtain

~x ~x ~x ~x A(~x ,~x ,~x ,~x )=Λ(~x ,~x ,~x ,~x ) 6 13 A (η, ξ)+ 6 12 6 24 6 43 A (η, ξ) , (3.14) 1 2 3 4 1 2 3 4 ~x 1 ~x ~x ~x 2 | 13| | 12| | 24| | 43| where C Λ(~x1,~x2,~x3,~x4)= , ~x Σ12 ~x ∆12 1 ~x 2Σ23 Σ ~x ∆43 ~x Σ34 | 12| | 13| − | 23| − | 24| | 34| Σ12 +i 2 ∞ ds Σ12 Σ34 ∆+ ˜ A1(η, ξ)= Γ( s + )Γ( s + )Γ( s + ) I(s) Σ12 i 2πi − 2 − 2 − 2 Z 2 − ∞ s ∆+ d ∆34 ∆12 ξ η− (s + − )F (s ,s+ , 2s;1 ), (3.15) × 2 − 2 2 − η Σ12 +i 2 ∞ ds Σ12 Σ34 ∆+ ˜ A2(η, ξ)= Γ( s + )Γ( s + )Γ( s + ) I(s) Σ12 i 2πi − 2 − 2 − 2 Z 2 − ∞ ∆12 ∆34 s 1 (s 2 )(s 2 ) ∆34 ∆12 ξ η− − 2 − − F (s ,s+ , 2s +1;1 ). × 2s − 2 2 −η

2 2 2 2 ~x13 ~x24 ~x14 ~x23 Note that η = |~x |2|~x |2 and that ξ = |~x |2|~x |2 . This is one of our main results in | 12| | 34| | 12| | 34| this paper. As an easy check, we can see that the amplitude A(~x1,~x2,~x3,~x4) consistently ˆ ~x transforms under the inversion ~x ~x = ~x 2 , thanks to the structure of the gamma → | | matrices ~x and ~x ~x ~x . 6 13 6 12 6 24 6 43

4. Discussion

In this paper, we have shown that the spinor exchange amplitude can be reduced to a scalar exchange amplitude and its derivative. By making use of this fact, we have calculated

9 the spinor exchange amplitude explicitly. From our final result (3.14) and (3.15), we can see that three series of poles emerge from the three gamma functions, which is similar to the case of scalar fields [29]. as is similar to the case of scalar fields [29]. Therefore, we can easily see that the appearance of the logarithmic terms in the short distance expansion, for example when ~x 0, has the same cause as in the scalar exchange diagram. Such 12 → terms appear because we have double poles in the integration over s in (3.15) when the position of the pole from one of the gamma functions coincides with that from another. Apart from the issue of the logarithmic terms, we would like to discuss the operator product expansion in the CFT, according to the AdS/CFT correspondence. Let us suppose that we are now considering a Yukawa model with three spinor fields ψ1, ψ3, ψ and two scalar fields φ2, φ4, which is the same situation as in section 3. The masses of the spinor

ﬁelds ψ1, ψ3,andψcorrespond to the weights ∆1,∆3,and∆+ respectively, and the scalar

fields φ2, φ4 to the conformal dimensions ∆2,∆4. The fields interact only through the ¯ ¯ following Yukawa interactions: λφ2ψ1ψ and λφ4ψψ3. In the rest of this section, we will show that the operator product expansions which can be determined from the two- and three-point functions are consistent with at least the first two terms of the short distance expansion of the spinor exchange diagram which we calculated in the previous section,

∆+ when we take account of only the contribution from the poles s = 2 in the exchange amplitude. For this purpose, let us recall the GKP/W relation (1.1), in our case

¯+ exp ψ0 χ +¯χψ0− + φ0 =exp IYukawa(φcl) . (4.1) * ∂(AdS) O !+ − Z CFT As suggested in section 2, the two-point correlation function of the spinors is given [8,9] by Γi(xi yi) χ∆(~x)¯χ∆(~y) =2c∆ − . (4.2) h i ~x ~y2∆ P− | − | d+1 Note that the weight ∆ was ∆ = m + 2 in section 2. But, here, we do not assume any particular value for the conformal dimension, as we have mentioned above. Similarly, we can ﬁnd all the non-vanishing three-point functions [10]:

2C∆1∆2∆+ ~x13 χ∆1 (~x1) ∆2 (~x2)¯χ∆+(~x3) = λ 6 , O ~x Σ12 ∆+ ~x ∆+ ∆12 ~x ∆++∆12 | 12| − | 23| − | 31| (4.3)

2C∆+∆4∆3 ~x13 χ∆ (~x1) ∆ (~x2)¯χ∆ (~x3) = λ 6 , + O 4 3 ~x ∆+ ∆34 ~x Σ34 ∆+ ~x ∆++∆34 | 12| − | 23| − | 31|

10 where the ‘structure constant’ C∆1∆2∆3 is ∆ +∆ +∆ d Σ ∆ Σ ∆ Σ ∆ Γ 1 2 3− Γ 12− 3 Γ 23− 1 Γ 31− 2 C = 2 2 2 2 . (4.4) ∆1∆2∆3 2πd Γ ∆ d Γ ∆ d Γ ∆ d 1 − 2 2 − 2 3 − 2

Note that C∆1∆2∆3 is the structure constant of three-point functions of scalar fields with conformal dimensions ∆1,∆2,and∆3 [5,6]. From this data (4.2), (4.3), we can determine the operator product expansions of the spinor field and the scalar field in the CFT, which turn out to be C ∆3 D ∆3 ∆1∆2 ∆1∆2 ~ χ∆ (~x1) ∆ (~x2) χ∆ (~x2)+ ~x12 ∂2χ∆ (~x2) 1 O 2 ∼ ~x Σ12 ∆3 3 ~x Σ12 ∆3 · 3 | 12| − | 12| − (4.5) ∆3 S∆1∆2 + ~x12 ∂2χ∆3 (~x2)+ , ~x Σ12 ∆3 6 6 ··· | 12| − C¯∆+ D¯ ∆+ ∆4∆3 ~ ∆4∆3 ∆ (~x2)¯χ∆ (~x3) χ¯∆ (~x2) + ~x23 ∂2χ¯∆ (~x2) O 4 3 ∼ + ~x Σ43 ∆+ · + ~x Σ43 ∆+ | 23| − | 23| − (4.6) S¯∆+ ∆4∆3 +¯χ∆+(~x2)←∂−2 ~x23 + , 6 6 ~x Σ43 ∆+ ··· | 23| − where the coefficients are C C ∆3 = C¯∆3 = λ ∆1∆2∆3 , ∆1∆2 ∆2∆1 c∆3 Σ ∆ D ∆3 = D¯ ∆3 = λ 13 2 C ∆3 , ∆1∆2 ∆2∆1 − ∆1∆2 (4.7) − 2∆3 Σ ∆ ∆3 ¯∆3 23 1 ∆3 S∆ ∆ = S = λ − C∆ ∆ . 1 2 − ∆2∆1 − 4∆ (∆ d ) 1 2 3 3 − 2 It is convenient for later use to introduce coefficient A defined by A = λ2 C12∆C34∆ . ∆ ∆ c∆ Now we are ready to consider the four-point function χ χ¯ . By using the h ∆1 O∆2 ∆3 O∆4 i operator product expansions (4.5), (4.6), we can obtain the short distance expansion of the four-point function in the limit ~x ,~x 0, for terms lower than the order O(~x2 )or 12 34 → 12 2 O(~x34); χ (~x ) (~x )¯χ (~x ) (~x ) h ∆1 1 O∆2 2 ∆3 3 O∆4 4 i A∆+ ~x14 ~x13 (∆+ +∆34)(~x43 ~x24) 6 ∼ ~x Σ12 ∆+ ~x 2∆+ ~x Σ34 ∆+ 6 − · ~x 2 | 12| − | 24| | 34| − | 24| ~x (∆ +∆ )(∆ +∆ ) (~x ~x ) (∆ +∆ )(~x ~x ) 6 23 + 12 + 34 12 · 43 − + 12 12 · 24 ~x 2 − 2∆ ~x 2 (4.8) | 24| + | 24| (~x ~x )(~x ~x ) (∆ +∆ )(∆ +∆ ) ~x ~x ~x 2(∆ +1) 12 24 43 24 ~x + + 21 + 43 12 24 43 + · 4 · 24 d 6 6 26 − ~x24 6 4∆ (∆ ) ~x24 | | + + − 2 | | + . ··· 11 This result is in complete agreement with the first two terms of the Taylor expansion of the 2 spinor exchange amplitude 2λ A(~x1,~x2,~x3,~x4) with respect to ~x12, ~x34, if we only take the contribution from the pole s = ∆+ into account and assume that ∆+ Σ12 + Z, ∆34 + Z: 2 2 6∈ 2 2 the condition that we do not have any logarithmic terms. On the other hand, under this

Σ12 Σ34 condition, we have another contribution from the poles s = 2 , 2 and the subsequent poles.Inthecasethat Σ12 Σ34 Z, we do not have the logarithmic terms from those 2 − 2 6∈ Σ12 poles, either. Then we can see that the contribution from the pole s = 2 does not agree with the short distance expansion of the four-point function which can be obtained by combining two- and three-point functions in the same way as before, even though we do not assume any particular value for cΣ12 and C∆1∆2Σ12 in the two- and three-point functions, respectively. This discrepancy has been similarly observed in scalar exchange diagrams [29]. If we do not impose any of the above condition on the weights, which is the case in the type IIB SUGRA on AdS S5, we would have logarithmic terms in the spinor exchange 5 × diagram. These terms have been discussed first in [25] and then in [26–31]. Although these logarithmic terms had been expected to cancel each other in a realistic four-point function, the four-point function of scalar fields was calculated in [31] to show that such terms remain even in a realistic correlation function. Therefore it is likely that they also exist in realistic four-point functions with spinor fields.

Acknowledgements We are grateful to Eric D’Hoker for discussion and for his encouragement. T.K. was supported in part by Grant-in-Aid for Scientiﬁc Research in a Priority Area: “Supersym- metry and Uniﬁed Theory of Elementary Particles”(#707) from the Ministry of Education, Science, Sports and Culture. K.O. was supported in part through a grant from the JSPS Research Fellowship for Young Scientists.

Appendix A. The Bulk-to-Bulk Spinor Propagator

In this appendix, we will give the derivation of the bulk-to-bulk propagator of spinor

ﬁelds on the Euclidean AdSd+1 space (see [36,38] for the Lorentzian counterpart). In addi- tion, we will explain how the propagator transforms under the inversion, as we mentioned in the text.

12 Firstly, we seek the solution of the Dirac equation (2.3) with λ =0,i.e.

a d (D m) ψ = z Γ ∂a Γ m ψ =0, 6 − 0 − 2 0 − (A.1) ¯ ¯ a d¯ ¯ ψ ←D− m = z0∂aψΓ ψΓ0+mψ =0. −6 − −2 In momentum space, the solution is given by

(K) ~ (K) ~ ~ ψ (z0, k)=φ (z0,k)a−(k), ψ(I)(z ,~k)=φ(I)(z ,~k)b (~k), 0 0 − ~ (A.2) ¯(K) ~ ~ ¯(K) ~ ~ /k (K) ~ ψ (z0,k)=¯a−(k)φ (z0,k)=¯a−(k)ik φ (z0, k), ~ − ¯(I) ~ ¯ ~ ¯(I) ~ ¯ ~ /k (I) ~ ψ (z0,k)=b−(k)φ (z0,k)= b−(k)i φ (z0, k), − k − ~ ~ ~ ~ 0 with k = k ,wherea−(k)andb−(k) are functions of k satisfying that Γ (a−,b−)= | | ~ ¯ ~ ¯ 0 ¯ (a−,b−). Similarly fora ¯−(k)andb−(k), (¯a−, b−)Γ = (¯a−, b−). Additionally, − −

d+1 ~k/ (K) ~ 2 φ (z0, k)=z0 Km+ 1 (kz0) i Km 1 (kz0) , " 2 − k − 2 # (A.3) d+1 ~k/ (I) ~ 2 φ (z0, k)=z0 Im+ 1 (kz0)+i Im 1(kz0) , " 2 k −2 # where Kν (z)andIν(z) are modiﬁed Bessel functions [34]. For z , the modiﬁed | |→∞ Bessel functions asymptotically behave [34] as

z π z e Kν (z) e− ,Iν(z) , (A.4) ∼ √2z ∼√2πz on the other hand, for z 0, ∼ ν ν 1 ν z 1 Kν (z) 2 − Γ(ν)z− ,Iν(z) . (A.5) ∼ ∼2 Γ(ν +1) The propagator S(z, w) should satisfy that 1 (D m) S(z, w)=S(z, w)( D m)= δd+1(z w) (A.6) 6 − −6 − √g − with the regularity condition and the boundary condition

lim S(z, w) = lim S(z, w)=0, z w 0→∞ 0→∞ (A.7) lim S(z, w) = lim S(z, w)=0. z 0 w 0 0→ 0→ 13 Since the modiﬁed Bessel functions satisfy the relation

1 Km+ 1 (z)Im 1 (z)+Im+1(z)Km 1(z)= , (A.8) 2 − 2 2 −2 z we can verify that

S(z, w) d~ d k i~k (~z w~) (K) ~ ¯(I) ~ = ke− · − θ(z0 w0)φ (z0, k) φ (w0, k) (2π)d − P− − Z (I) ~ ¯(K) ~ θ(w0 z0)φ (z0, k) φ (w0, k) (A.9) − − P− −

1 1 1 − 2 0 2 = z0 D + Γ + m G d +m 1 (z, w) + G d +m+ 1 (z, w) + w0 − 6 2 2 − 2 P− 2 2 P h i surely satisﬁes (A.6) and (A.7), if we use the expression of the bulk-to-bulk propagator [35,36] of scalar ﬁelds

d~ d k d i~k (~z w~) G (z, w)= (z w ) 2 e− · − θ(z w )Kν(kz )Iν(kw ) ∆ (2π)d 0 0 0 − 0 0 0 Z (A.10)

+ θ(w z )Kν(kw )Iν(kz ) 0 − 0 0 0 d with ∆ = 2 + ν, of which another expression can be seen in (2.14) of the text. It will be useful later to diﬀerentiate the scalar propagator with respect to u:

d ∆C∆ ∆ ∆+1 1 G (z, w)=G0 (z, w)= F ( +1, ,ν+1; ) (A.11) du ∆ ∆ −(1 + u)∆+1 2 2 (1 + u)2

∆ ∆+1 Γ( 2 )Γ( 2 ) with C∆ = d+1 . By using some of the ﬁfteen relations of Gauss on hypergeometric 4π 2 Γ(ν+1) functions, we show that

(1 + u)G∆0 (z, w) (∆ d)G∆(z, w)=G∆0 1(z, w), − − − (A.12) (1 + u)G∆0 (z, w)+∆G∆(z, w)=G∆+10 (z, w).

See [39] for the ﬁrst formula and [40] for the second. From (A.11) and (A.12), an alternative expression of the spinor propagator S(z, w) can be seen to be

1 1 S(z, w)= ( )2 (z w)G (z, w)+(z w)G (z, w) , (A.13) + ∆0 + ∆0 + − z0w0 6 P− −P 6 − 6 P −P− 6 h i 14 where ∆ = d + m 1 . ± 2 ± 2 Now we proceed to consider the propagator S(z, w) under the inversion transformation µ µ zµ z zˆ = z2 .SinceG∆0 (z, w) is invariant under the inversion (3.2) because of the → ± invariance of the chordal distance u, it is easy to verify from the expression (A.13) that

z 1 1 w 0 −2 2 S(ˆz, wˆ)= 6 D+Γ m z0 G∆+(z, w) + G∆ (z, w) + w0 6 . (A.14) z 6 − P− − P w | | | | Note that the middle factor in the right-hand side of (A.14) is ( 1) times the propagator − (A.13) with mass m instead of m. − Finally, we will give the explicit form of the regularized bulk-to-bulk spinor propagator

S(z, w) defined by (2.10) and (2.11) in the text, which satisfies the same differential equation as S(z, w) but with the boundary condition (2.11). By making use of the solution (A.3), we obtain

S(z, w) d d ~k ~ Im+ 1 (k) (A.15) ik (~z w~) 2 (K) ~ ¯(K) ~ = S(z, w)+ d ke− · − φ (z0, k) φ (w0, k), (2π) Km+ 1 (k) P− − Z 2 which can be veriﬁed to satisfy the boundary condition (2.11).

Appendix B. The Boundary-to-Bulk Spinor Propagator

In this appendix, we will review the boundary-to-bulk propagator of spinor ﬁelds which has been derived in [5,8]. As in the scalar case [3], one can obtain the boundary-to- bulk spinor propagator by performing the inversion to the solution of the Dirac equation, which turns out to depend only on z0. As pointed out in [8], upon the inversion of the solution, we need to perform the local Lorentz transformation to preserve the gauge-ﬁxing condition of the vielbein. Instead of this procedure, we will give an alternative derivation of the boundary-to-bulk propagator. The inversion of the Dirac operator is given by

1 1 D(ˆz)= U(z)− D(z)U(z)+ Γ , (B.1) 6 − 6 2 0 where U(z) is deﬁned in (2.7). This relation follows from

1 DµU(z)= ΓµU(z)Γ0, 2 (B.2) 1 U(z)− ΓµU(z)= Jµν (z)Γν , − 15 zµzν where Jµν (z)=δµν 2 2 is the conformal Jacobian deﬁned by − z ∂zˆ 1 µ = J (z). (B.3) 2 µν ∂zν z

From (B.1), the Dirac operator in the coordinate z canbewrittenas

1 1 D(z) m= U(z) D(ˆz)+m Γ U(z)− . (B.4) 6 − − 6 −2 0 h i Using this relation, we can easily see that

d mΓ + 1 2 − 0 2 Σ=U(z)ˆz0 (B.5) satisﬁes the Dirac equation (D(z) m)Σ = 0. Since we impose on the boundary-to-bulk 6 − propagator of spinor ﬁeld Σ the boundary condition that Σ = 0 on the boundary z0 =0, it should be projected on Γ = 1. 0 − The boundary-to-bulk propagator can also be obtained as the limit of the bulk-to-bulk propagator with one point approaching the boundary. From (A.13), we can see that

1 ∆ S(z; , ~x) +−2Σ (z,~x)(B.6) →− ∆+ in the limit 0. Note that a similar property for the scalar propagator is used in → [41] to relate the boundary behavior of the bulk ﬁeld and the expectation value of the corresponding operator in the boundary theory.

References

[1] J. Maldacena, “The Large N Limit of Superconformal Field Theories and Supergrav- ity”, Adv. Theor. Math. Phys. 2, 231-252 (1998), hep-th/9711200. [2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge Theory Correlators from Non-Critical String Theory”, Phys. Lett. B428, 105-114 (1998), hep-th/9802109. [3] E. Witten, “Anti de Sitter Space and Holography”, Adv. Theor. Math. Phys. 2, 253- 291 (1998), hep-th/9802150. [4] I. Ya. Aref’eva and I. V. Volovich, “On Large N Conformal Theories, Field Theories in Anti-De Sitter Space and Singletons”, hep-th/9803028; “On the Breaking of Con- formal Symmetry in the AdS/CFT Correspondence”, Phys. Lett. B433, 49-55 (1998), hep-th/9804182. [5] W. M¨uck and K. S. Viswanathan, “Conformal Field Theory Correlators from Classical

Scalar Field Theory on AdSd+1”, Phys. Rev. D58, 041901 (1998), hep-th/9804035.

16 [6] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Correlation functions

in the CFTd/AdSd+1 correspondence”, Nucl. Phys. B546, 96-118 (1999), hep- th/9804058. [7] G. Chalmers, H. Nastase, K. Schalm and R. Siebelink, “R-Current Correlators in = 4 Super Yang-Mills Theory from Anti-de Sitter Supergravity”, Nucl. Phys. N B540, 247-270 (1999), hep-th/9805105. [8] M. Henningson and K. Sfetsos, “Spinors and the AdS/CFT correspondence”, Phys. Lett. B431, 63-68 (1998), hep-th/9803251. [9] W. M¨uck and K. S. Viswanathan, “Conformal Field Theory Correlators from Classical Field Theory on Anti-de Sitter Space II. Vector and Spinor Fields”, Phys. Rev. D58, 106006 (1998), hep-th/9805145. [10] A. M. Ghezelbash, K. Kaviani, P. Parvizi, and A. H. Fatollahi, “Interacting Spinors- Scalars and AdS/CFT Correspondence”, Phys. Lett. B435, 921 (1998), hep-th/9805162. [11] C. E. Arutyunov and S. A. Frolov, “On the Origin of Supergravity Boundary Terms in the AdS/CFT Correspondence”, Nucl. Phys. B544, 576-589 (1999), hep-th/9806216. [12] H. Liu and A. A. Tseytlin, “D = 4 Super Yang-Mills, D = 5 Gauged Supergravity, and D = 4 Conformal Supergravity”, Nucl. Phys. B533, 88-108 (1998), hep-th/9804083. [13] W. M¨uck and K. S. Viswanathan, “The Graviton in the AdS-CFT correspondence: Solution via the Dirichlet boundary value problem”, hep-th/9810151. [14] G. E. Arutyunov and S. A. Frolov, “Three-Point Function of the Stress Energy Tensor in the AdS/CFT Correspondence”, hep-th/9901121. [15] S. Corley, “The Massless Gravitino and the AdS/CFT Correspondence”, Phys. Rev. D59, 086003 (1999), hep-th/9808184. [16] A. Volovich, “Rarita-Schwinger Field in the AdS/CFT Correspondence”, JHEP 9809, 022 (1998), hep-th/9809009. [17] A. S. Koshelev and O. A. Rytchkov, “Note on the Massive Rarita-Schwinger Field in the AdS/CFT Correspondence”, Phys. Lett. B450, 368-376 (1999), hep-th/9812238.

[18] G. E. Arutyunov and S. A. Frolov, “Antisymmetric Tensor Field on AdS5”, Phys. Lett. B441, 173-177 (1998), hep-th/9807046. [19] W. S. l’Yi, “Generating Functionals of Correlation Functions of P-form Currents in AdS/CFT Correspondence”, hep-th/9809132. [20] W. S. l’Yi, “Correlators of Currents Corresponding to the Massive P-form Currents in AdS/CFT Correspondence”, Phys. Lett. B448, 218-226 (1999), hep-th/9811097. [21] A. Polishchuk, “Massive Symmetric Tensor Field on AdS”, hep-th/9905048. [22] E. D’Hoker, D. Z. Freedman, W. Skiba, “Field Theory Tests for Correlators in the AdS/CFT Correspondence”, Phys. Rev. D59, 045008 (1999), hep-th/9807098. [23] S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, “Three-Point Functions of Chiral Operators in D =4, = 4 SYM at Large N”, Adv. Theor. Math. Phys. 2, 697-718 N 17 (1998), hep-th/9806074. [24] H. Liu and A. A. Tseytlin, “On Four-Point Functions in the CFT/AdS Correspon- dence”, Phys. Rev. D59, 086002 (1999), hep-th/9807097. [25] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Comments on 4-point Functions in the CFT/AdS Correspondence”, Phys. Rev. D59, 086002 (1999), hep- th/9808006. [26] J. H. Brodie and M. Gutperle, “String Correction to Four Point Functions in the AdS/CFT Correspondence”, Phys. Lett. B445, 296-306 (1999), hep-th/9809067.

[27] E. D’Hoker and D. Z. Freedman, “Gauge Boson Exchange in AdSd+1”, Nucl. Phys. B544, 612-632 (1999), hep-th/9809179.

[28] G. Chalmers and K. Schalm, “The Large Nc Limit of Four-Point Functions in =4 N SuperYang-Mills Theory from Anti-de Sitter Supergravity”, hep-th/9810051;“Holo- graphic Normal Ordering and Multi-Particle States in the AdS/CFT Correspon- dence”, hep-th/9901144. [29] H. Liu, “Scattering in Anti-de Sitter Space and Operator Product Expansion”, hep- th/9811152.

[30] E. D’Hoker and D. Z. Freedman, “General Scalar Exchange in AdSd+1”, hep- th/9811257. [31] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Graviton

and Gauge Boson Propagators in AdSd+1”, hep-th/9902042; “Graviton Exchange and Complete 4-Point Functions in the AdS/CFT Correspondence”, hep-th/9903196. [32] E. D’Hoker, D. Z. Freedman and L. Rastelli, “AdS/CFT 4-Point Functions: How to Succeed at z Integrals without Really Trying”, hep-th/9905049. [33] M. Henneaux, “Boundary Terms in the AdS/CFT Correspondence for Spinor Fields”, hep-th/9902137. [34] I. S. Gradshteyn and I. M. Ryzhik; A. Jeffrey ed., “Table of Integrals, Series, and Products (5th ed.)”, (Academic Press, 1994). [35] C. Fronsdal, “Elementary Particles in a Curved Space II”, Phys. Rev. D10, 589 (1974); C. P. Burgess and C. A. Lutken, “Propagators and Effective Potentials in Anti-de Sitter Space”, Phys. Lett. 153B, 137 (1985); T. Inami and H. Ooguri, “One Loop Effective Potential in Anti-de Sitter Space”, Prog. Theor. Phys. 73, 1051 (1985). [36] C. J. Burges, D. Z. Freedman, S. Davis, and G. W. Gibbons, “Supersymmetry in Anti-de Sitter Space”, Ann. Phys. 167, 285 (1986). [37] A. Erdélyi, “Higher Transcendental Functions, Vol. 1”, Bateman Manuscript Project, (McGraw-Hill Book Company, Inc., 1953). [38] B. Allen and C. A. Lutken, “Spinor Two-Point Functions in Maximally Symmetric

18 Spaces”, Comm. Math. Phys. 106, 201-210 (1986) . [39] reference [37], page 103, (35). [40] reference [37], page 102, (21) and (20). [41] I. R. Klebanov and E. Witten, “AdS/CFT Correspondence and Symmetry Breaking”, hep-th/9905104.