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Predictions for PP-wave string amplitudes from perturbative SYM

Umut Gursoy

TL;DR

Using perturbative SYM in the BMN limit, the paper shows that the anomalous dimensions of all two-impurity, multi-trace BMN operators at order ${g_2^2\lambda'}$ are determined by the corresponding single-trace dimensions due to suppression of connected diagrams. The analysis reveals a degeneracy between single- and multi-trace operators at this order, requiring degenerate perturbation theory, and yields the result that $\Delta_i = \left(\frac{J_1}{J}\right)^2 \Delta_1$ for two-impurity operators in the BMN limit. These gauge-theory insights translate into explicit predictions for pp-wave light-cone string field theory: the matrix elements of the P$^-$ Hamiltonian in the 2-2 and 1-3 sectors are fixed by non-contractible gauge-theory diagrams and basis-transformations to the string basis. The work thus provides a concrete, testable bridge between perturbative SYM and pp-wave SFT, with broader implications for operator mixing, degeneracy handling, and higher-impurity extensions.

Abstract

The role of general two-impurity multi-trace operators in the BMN correspondence is explored. Surprisingly, the anomalous dimensions of all two-impurity multi-trace BMN operators to order g_2^2λ' are completely determined in terms of single-trace anomalous dimensions. This is due to suppression of connected field theory diagrams in the BMN limit and this fact has important implications for some string theory processes on the PP-wave background. We also make gauge theory predictions for the matrix elements of the light-cone string field theory Hamiltonian in the two string-two string and one string-three string sectors.

Predictions for PP-wave string amplitudes from perturbative SYM

TL;DR

Using perturbative SYM in the BMN limit, the paper shows that the anomalous dimensions of all two-impurity, multi-trace BMN operators at order are determined by the corresponding single-trace dimensions due to suppression of connected diagrams. The analysis reveals a degeneracy between single- and multi-trace operators at this order, requiring degenerate perturbation theory, and yields the result that for two-impurity operators in the BMN limit. These gauge-theory insights translate into explicit predictions for pp-wave light-cone string field theory: the matrix elements of the P Hamiltonian in the 2-2 and 1-3 sectors are fixed by non-contractible gauge-theory diagrams and basis-transformations to the string basis. The work thus provides a concrete, testable bridge between perturbative SYM and pp-wave SFT, with broader implications for operator mixing, degeneracy handling, and higher-impurity extensions.

Abstract

The role of general two-impurity multi-trace operators in the BMN correspondence is explored. Surprisingly, the anomalous dimensions of all two-impurity multi-trace BMN operators to order g_2^2λ' are completely determined in terms of single-trace anomalous dimensions. This is due to suppression of connected field theory diagrams in the BMN limit and this fact has important implications for some string theory processes on the PP-wave background. We also make gauge theory predictions for the matrix elements of the light-cone string field theory Hamiltonian in the two string-two string and one string-three string sectors.

Paper Structure

This paper contains 10 sections, 2 theorems, 158 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Operator mixing at $g_2^2$ level
  3. Dependence on $g_2$ and $J$ of an arbitrary gauge theory correlator
  4. Anomalous dimension of a general multi-trace BMN operator at ${\cal O}(g_2^2)$
  5. Matrix elements of pp-wave Hamiltonian in 2-2 and 1-3 sectors
  6. Discussion and outlook
  7. Disconnectedness of GT correlators
  8. Field theory basis change at finite $g_2$
  9. Sum formulas
  10. Computation of $G^{13}$, $G^{23}$, $\Gamma^{13}$ and $\Gamma^{23}$

Key Result

theorem 1

For scalar impurity BMN operators ${\cal O}(g_{YM}^2)$ interactions will not change the scaling law of (gen) at all.

Figures (12)

  • Figure 1: Left figure shows that connected contribution to $2\rightarrow 3\rightarrow 2$ process is suppressed by $1/J$. Right figure shows similar suppression of mixing of double trace operators with single-traces.
  • Figure 2: A representation of planar contributions to $\langle:\bar{O}^{J_1}_n\bar{O}^{J_2}::O^{J_3}_m O^{J_4}:\rangle$ at ${\cal O}(g_2^2\lambda')$. Dashed lines represent scalar impurities. We do not show $Z$ lines explicitly. Vertices are of order $g_2\sqrt{\lambda'/J}$. a Connected contribution. b, c, d Various disconnected contributions.
  • Figure 3: Connected contribution to double-triple correlator, $\langle\bar{O}_{ny}O_{myz}\rangle$, does give non-zero contribution in $1\rightarrow 1$ process. For example this diagram will show up in the computation of ${\cal O}(g^6)$ scale dimension of single-trace operators.
  • Figure 4: A typical planar contribution to $C^{1,i}$. Circles represent single-trace operators. Dashed lines denote impurity fields. $Z$ lines are not shown explicitly and represented by "$\cdots$".
  • Figure 5: All distinct topologies of planar Feynman diagrams that contribute to $\langle:\bar{O}^{J_1}_n\bar{O}^{J_2}::O^{J_3}_mO^{J_4}O^{J_5}:\rangle$. Nodes represent the operators while solid lines represent a bunch of $Z$ propagators. Line between the nodes 1 and 3 also include two scalar impurities $\phi$ and $\psi$. All other topologies are obtained from these two classes by permutations among 3,4,5 and 1,2 separately. Other planar graphs are obtained from these by moving the nodes within the solid lines without disconnecting the diagram. For example 4 in the first diagram can be moved within the solid line 1-3.
  • ...and 7 more figures

Theorems & Definitions (2)

  • theorem 1
  • theorem 2