Logarithmic corrections to higher twist scaling at strong coupling from AdS/CFT

Abstract

We compute 1-loop correction E 1 to the energy of folded string in AdS 5 × S 5 (carrying spin S in AdS 5 and momentum J in S 5 ) using a "long string" approximation in which S ≫ J ≫ 1. According to the AdS/CFT the function E 1 should represent first subleading correction to strong coupling expansion of anomalous dimension of higher twist SL(2) sector operators of the form TrD S Z J . We show that E 1 smoothly interpolates between the ln S regime (previously found in the J → 0 case) and the λ/J 2 ln 3 (S/J) regime (which is the leading correction to the thermodynamic limit on the spin chain side). This supports the universality of the ln S scaling. As in the previous work, we also find "nonanalytic" corrections related to non-trivial 1-loop phase in the corresponding Bethe ansatz S-matrix.

A generalized scaling function for AdS/CFT

“…A very interesting question is how f (g, j) behaves at strong coupling. Indeed we would like to make contact with the already known results from string theory [15,16,3,20,21,23]. A potential trouble is that in the semi-classical computations pioneered by Frolov and Tseytlin [15] the coupling constant g is intricately entangled with the, respectively, AdS 5 and S 5 charges M and L. In [3] the strong coupling limit of the dimension ∆ of the operators (1.1) was predicted from the results of [15,16] on the energy of a folded string soliton (see also the discussion in [20,21,23,24]).…”

The strong coupling limit of the scaling function from the quantum string Bethe ansatz

“…one finds that the semi-classical [18] as well as the one loop energy [19] can be written down in a closed form as a function of z. Furthermore, the formula obtained for the string energy interpolates smoothly between small and large values of z and the large-z expansion looks as (3) just with the replacement log S → log( S J ).…”

“…We start in section 2 by recalling from reference [19] the description of the folded string rotating on AdS 3 × S 1 ⊂ AdS 5 × S 5 in the limit given by eqn. (4).…”

Quantum corrections to energy of short spinning string inAdS5

“…The analytic form of the quantum correction can be found in the limit of large S when the ends of the string reach the boundary of the AdS 5 . Then the solution drastically simplifies (ρ becomes linear in σ) [3,4] and one finds that E 1 = c 1 ln S + ..., c 1 = − 3 ln 2 π . Since rotation of the string balances the contracting effect of its tension, smaller values of the spin correspond to smaller values of the length of the string whose center of mass is at ρ = 0: S essentially measures the length of the string.…”

“…Here the order of limits was different (we first expanded in ǫ for fixed ν) and that could be a possible reason for a disagreement between (B.24) and (B.25). 4 To recover the standard fast string result one would need to start with the short string fluctuation operators in (B.13), where no assumption on S ν was made, use them and (B.17) without expanding in ǫ, compute the determinants needed in (B.11), then expand in large ν with S ν kept fixed, and at the end take S ν to be small. One can then consider the 1-loop correction in the small ν region by taking ǫ to zero while keeping the parameter x ≡ ν ǫ fixed, i.e.…”