An improved correspondence formula for AdS/CFT with multi-trace operators

An improved correspondence formula is proposed for the calculation of correlation functions of a conformal field theory perturbed by multi-trace operators from the analysis of the dynamics of the dual field theory in Anti-de Sitter space. The formula reduces to the usual AdS/CFT correspondence formula in the case of single-trace perturbations. In a recent paper [1] Witten proposed an improved boundary condition to be used in the AdS/CFT correspondence when the boundary conformal field theory is perturbed by multi-trace operators, IQFT[O] = ICFT[O] + ∫ dxW [O] , (1) where O denotes a scalar primary operator, W [O] is an arbitrary function of O, and dx stands for the covariant volume integral measure. In the case of W [O] = 1 2 fO, and O of conformal dimension ∆ = d/2, Witten demonstrated that the proposed boundary condition yields the correct renormalization formula for the coupling f . However, it is not difficult to realize that the usual AdS/CFT correspondence formula [2, 3], exp(−IAdS) = 〈exp(− ∫ dxαO)〉, where IAdS is the regularized and renormalized bulk onshell action, and α is a generating current, does not give the correct result for the oneand two-point functions of the boundary field theory. The purpose of the present letter is to present an improved correspondence formula, which gives correct boundary field theory correlators for multi-trace perturbations. The new method will naturally apply to both regular and irregular boundary conditions of the bulk fields [4], which correspond to field theory operators of dimensions ∆ = d/2+λ and ∆ = d/2−λ, respectively [5]. (λ is positive and in the second case satisfies the unitarity bound λ < 1.) We shall for simplicity consider a single scalar field in AdS bulk space and first explain the general method. Later, we will test the formula for a free bulk field and W = ∫ dx (βO+ 1 2 fO), where β is a finite soucre, and f is a coupling constant. First, to clarify our notation, we shall consider an AdS bulk of dimension d+ 1 with a metric ds = r(dr + dx) , (2) where dx denotes the Euclidean metric in d dimensions. The asymptotic boundary (horizon) is located at r = 0. For small r, a scalar field behaves asymptotically as φ(r, x) = rφ̂(x) + rφ̌(x) + · · · , (3) where λ is related to the mass of the bulk field by λ = √ d2/4 +m2. The two independent series solutions are determined by φ̂ and φ̌, which are called the 1 regular and irregular boundary data [4], respectively, and the ellipses denote all subsequent terms of the two series. Regularity of the bulk solution as well as bulk interactions determine the relation between φ̂ and φ̌ uniquely. The main outcome of the AdS bulk analysis [2, 3] is the regularized and renormalized bulk on-shell action as a functional of the regular boundary data, I[φ̂]. Moreover, the two boundary data φ̂ and φ̌ are canonically conjugate and satisfy (with suitable normalization) [5] φ̌ = − δI[φ̂] δφ̂ and φ̂ = δJ [φ̌] δφ̌ , (4) where J [φ̌] = I − ∫ dx φ̂ δI[φ̂] δφ̂ (5) is the Legendre transform of I. It has been proven in [6] that the formulae (4) and (5) hold for interacting bulk fields to any order in perturbation theory. Armed with these results, we can proceed to construct a generating functional of the boundary field theory. In order to calculate correlation functions in the quantum field theory with an action IQFT, we need to add to IQFT a source term, ∫ dxαO, which can be viewed as a single-trace perturbation of the theory. In the usual AdS/CFT calculation [W = 0 in eqn. (1)], one identifies one of the boundary data φ̂ or φ̌ with the source α. As a result (c.f. eqn. (4)) the other one corresponds to the boundary field theory operator, O. It turns out that this second property is the fundamental one and can be used when multi-trace perturbations are switched on. Let us first assume that O corresponds to φ̌. Then, the perturbation of the conformal field theory, ∫ dx (W [O]+αO), is a functional of φ̌, and we shall add it to J [φ̌] in order to construct a generating functional, S = J [φ̌] + ∫ dx (W [φ̌] + αφ̌) . (6) The relation between the field φ̌ and the source α is determined by demanding δS δφ̌ = δJ δφ̌ + dW dφ̌ + α = 0 . (7) For λ = 0, the formula (3) degenerates to φ(r, x) = r(φ̂ ln r + φ̌) + · · · . 2 By virtue of eqn. (4), this is identical to Witten’s improved boundary condition [1] (up to a sign that depends on the convention for the Green function), and it reduces to the regular boundary condition for W = 0. Furthermore, we can re-write S as S = S − ∫ dx φ̌ δS δφ̌ = I + ∫