Abstract
AbstractWe consider the rational six-vertex model on anL×Llattice with domain wall boundary conditions and restrictNparallel-line rapidities,N≤L/2, to satisfy length-LXXX$ \mathrm{spin}-\frac{1}{2} $chain Bethe equations. We show that the partition function is an (L− 2N)- parameter extension of Slavnov’s scalar product of a Bethe eigenstate and a generic state, withNmagnons each, on a length-LXXX$ \mathrm{spin}-\frac{1}{2} $chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is Casoratian, and consequently that tree-level$ \mathcal{N}=4 $SYM structure constants that are known to be determinants, remain determinants at 1-loop level.
Highlights
Scalar products of N -magnon states on a length-L spin chain, play a central role in studies of correlation functions in integrable spin chains [3, 4]
In 7, we show that SYM4 tree-level structure constants that are determinants [8], remain determinants when 1-loop corrections are included along the lines of [10, 11]
In [24, 25], I Kostov and F Smirnov independently suggested that the scalar product of a Bethe eigenstate and a generic state can be obtained from Izergin’s (L × L) domain wall partition function, either by sending (L − 2N ) rapidities, N L/2, to infinity, and thereby decoupling them so that one ends up with a partial domain wall partition function, setting N of the remaining rapidities to satisfy appropriate Bethe equations [24], or by reinterpreting Korepin’s domain wall configuration as the scalar product of a Bethe eigenstate that is built on the lowest-weight pseudo-vacuum and a generic state, by requiring an appropriate subset of rapidities to satisfy appropriate Bethe equations [25]
Summary
Scalar products of N -magnon states on a length-L spin chain, play a central role in studies of correlation functions in integrable spin chains [3, 4]. They have appeared in studies of 3-point functions in 4-dimensional N = 4 super Yang-Mills theory, SYM4 [5,6,7,8]. We show that the result is an extended version of Slavnov’s scalar product that depends on (L − 2N ) extra parameters Taking these extra parameters to infinity, they decouple from the partition function, and we obtain a third expression for Slavnov’s scalar product as an L×L determinant. N Gromov and P Vieira [10, 11] to show that SYM4 structure constants that are known to be determinants at tree-level [8], remain determinants at 1-loop level
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