Three-Point Functions of Quarter BPS Operators in N=4 SYM

Abstract

In a recent paper hep-th/0109064, quarter-BPS chiral primaries were constructed in the fully interacting four dimensional N=4 Super-Yang-Mills theory with gauge group SU(N). These operators are annihilated by four supercharges, and at order g^2 have protected scaling dimension and normalization. Here, we compute three-point functions involving these quarter-BPS operators along with half-BPS operators. The combinatorics of the problem is rather involved, and we consider the following special cases: (1) correlators < half half BPS > of two half-BPS primaries with an arbitrary chiral primary; (2) certain classes of < half quarter quarter > and < quarter quarter quarter > three-point functions; (3) three-point functions involving the Delta < 8 operators found in hep-th/0109064; (4) < half quarter quarter> correlators with the special quarter-BPS operator made of single and double trace operators only. The analysis in cases (1)-(3) is valid for general N, while (4) is a large N approximation. Order g^2 corrections to all three-point functions considered in this paper are found to vanish. In the AdS/CFT correspondence, quarter-BPS chiral primaries are dual to threshold bound states of elementary supergravity excitations. We present a supergravity discussion of two- and three-point correlators involving these bound states.

Figures (7)

• Figure 1: Structures contributing to two-point functions of scalars at order $g^2$ through four-scalar blocks and the propagator. Thick lines correspond to exchanges of the gauge boson, and of the auxiliary fields $F_i$ and $D$ (in the $\cal N$=1 formulation; after integrating out $F_i$ and $D$, the $z z \bar{z} \bar{z}$ vertex). The scalar propagator remains diagonal in both color and flavor indices at order $g^2$.
• Figure 2: Building blocks for $g^2$ corrections to three-point functions. The three-points are $x$ (with two legs attached) and $y$ and $w$ (with a single leg each).
• Figure 3: $F$-term contributions to $\langle \hbox{$\cal O$}_{\rm BPS}(x) \hbox{$\cal O$}'_{\rm BPS}(y) \hbox{$\cal O$}_{\tiny \hbox{$1\over2$}}(w) \rangle_{g^2}$ in the case when the correlator can be "partitioned into two flavors:" (a) proportional to $\tilde{B}(x,y)$; (b) proportional to $\tilde{C}(x;y,w)$; (c) proportional to $\tilde{C}(y;x,w)$.
• Figure 4: Order $g^2$ corrections to correlators of the form (\ref{['disjoint-flavors']}): (a) and (b) includes a gauge boson exchange; (c) and (d) $F$-terms. Self energy contributions (not shown) also include a gauge boson exchange.
• Figure 5: Order $g^2$ corrections to extremal correlators: (a) and (b) within a single peddle; (c) and (d) between the two peddles. Self energy contributions (not shown) and diagrams (a) and (c) are gauge dependent, while (b) and (d) diagrams arise from the $F$-terms.