Scattering of Giant Holes
We study scalar excitations of high spin operators in super Yang-Mills theory, which are dual to solitons propagating on a long folded string in . In the spin chain description of the gauge theory, these are associated to holes in the magnon distribution in the sector. We compute the all-loop hole S-matrix from the asymptotic Bethe ansatz, and expand in leading orders at weak and strong coupling. The worldsheet S-matrix of solitonic excitations on the GKP string is calculated using semiclassical quantization. We find an exact agreement between the gauge theory and string theory results.
University of Cambridge, United Kingdom\notoc
The discovery of integrability on both sides of the AdS/CFT correspondence has opened the possibility for precise tests of the conjecture. On the gauge theory side, the full spectrum of planar super Yang-Mills theory has been determined by the all-loop asymptotic Bethe ansatz Beisert:2005fw () for an integrable spin chain. Of particular interest is the subsector of twist operators composed of light-cone covariant derivatives acting on complex scalar fields ,
where the ellipsis denotes all other operators that mix under renormalization. At large Lorentz spin , the conformal dimension of the ground state in this sector exhibits logarithmic scaling at all loops
The cusp anomalous dimension interpolates smoothly between small and large values of the coupling constant Basso:2007wd ().111For a review of twist operators and the cusp anomalous dimension, see Freyhult:2010kc (). It also controls the infrared divergences of gluon scattering amplitudes, which are dual to light-like polygonal Wilson loops. There has been much recent interest in studying excitations around this ground state Alday:2007mf (); Gaiotto:2010fk (); Giombi:2010bj (). In particular, their spectrum is an important ingredient in the operator product expansion of light-like Wilson loops Alday:2010ku (). The spectrum of all such excitations has been determined by Basso Basso:2010in (), by way of a set of integral equations for the excitation densities which can be solved at all values of the coupling. The scalar excitations on top of the sea of derivatives (1) are known as holes. In the spin chain language, the holes are gaps in the magnon distribution. The vacuum contains two large holes of order which contribute to the logarithmic scaling Belitsky:2006en (). Excitation of the vacuum introduces additional holes of order .
The string state dual to the ground state operator of fixed spin (1) is a long folded string rotating in , known as the GKP string Gubser:2002tv (). The string energy also exhibits logarithmic scaling due to the length of the string in the target space, whose coefficient coincides with at strong coupling. The two spikes at the end of the string approach the boundary of and are identified with the two large holes. More generally hole excitations correspond to classical solitons that propagate on the GKP string. These are identified with solitons of the sinh-Gordon equation after Pohlmeyer reduction of the string equations of motion Jev (). The classical dispersion relation for these excitations was obtained in Dorey:2010iy () where they were named “Giant Holes” in analogy with the Giant Magnons of Hofman:2006xt ().
In this paper we study the scattering of two holes/solitons for all values of the coupling. On the gauge theory side, the scattering phase is obtained by considering the scattering of the second hole with the change in magnon density due to the first hole Faddeev:1977rm (). Following the techniques introduced in Basso:2010in (); Basso:2009gh (), we obtain an integral expression, which we expand at weak and strong coupling. On the string theory side, the worldsheet S-matrix is calculated from the time-delay Jackiw:1975im () that one soliton accumulates as it passes the other. It receives additional contributions due to the non-trivial relation between worldsheet coordinates and global coordinates and also due to the contribution of each excitation to the length of the string. Once these subtleties are carefully taken into account, we find exact agreement with the gauge theory result at strong coupling.
2 Scattering of spin chain holes
First it is important to be precise about we mean by hole excitations. Consider operators of arbitrary twist
In the spin chain description, we identify the covariant derivatives with magnons propagating on a background of scalars corresponding to the ferromagnetic ground state. An important feature of the non-compact spin chain is that there can be an arbitrary number of magnons at each spin site without increasing the length of the chain which is identified with the number of scalars. In addition, the number of holes in the mode number distribution of the magnons is also equal to . The Bethe ansatz equation provides a quantization condition for the magnon rapidities. The large spin limit we will consider represents a highly excited state in which the total length is held fixed. In this limit, it is natural to view the sea of derivatives as the pseudovacuum on which the scalars act as hole excitations. Moreover, Alday and Maldacena have identified the elementary fluctuations around the GKP string with insertion of twist-one operators in the pseudovacuum Alday:2007mf (). With these motivations, we define the holes as vacancies in the magnon distribution, to be identified with inserting a scalar , as opposed to removing a derivative in the trace (3).
2.1 One-loop S-matrix at weak coupling
The one-loop Bethe ansatz equation for magnon excitations in an spin chain of sites is
where the magnon momentum and S-matrix are defined as
Each Bethe root is associated with a mode number corresponding to a branch of the logarithm
The roots all lie on the real axis. When , the mode numbers are symmetrically distributed as .222For arbitrary , the magnon mode number develops a gap in the middle corresponding to small holes. The gap closes in the large spin limit. In the large spin limit, the magnon roots form a cut in the interval and (4) becomes an integral equation for the magnon density Eden:2006rx ()
Introducing an excited hole of rapidity shifts and the magnon mode number , where is the Heaviside step function. The change in the ground state density due to the hole, , can be written as
We will also use when we do not need to specify the hole rapidity. It satisfies the integral equation
Note that we have extended the integration domain to the entire real line. This is justified because the excitations are of order and is suppressed at large rapidity. We solve for by Fourier transform to obtain
which decays as at infinity.
The S-matrix between two holes carrying rapidities is a phase . It is given by integrating the scattering of the second hole with the magnon density change introduced by the first hole Faddeev:1977rm ().
The integral can be evaluated by Fourier transform. We find
where in recovering the phase from its derivative (11) we fixed the constant of integration such that the expression is antisymmetric in and , as is necessary for a unitary S-matrix. Note the appearance of momentum terms and . They are present because by introducing a hole we have created a vacancy in the magnon distribution. In defining the effective Bethe ansatz for holes, it is more appropriate to absorb such terms as part of the phase due to propagation instead of as part of the S-matrix. 333We thank Benjamin Basso for insightful comments on interpreting the S-matrix and its connection to the effective Bethe ansatz. Assuming factorized scattering, we may write down the effective Bethe ansatz for a twist chain
The two large holes of order are non-dynamical because their positions are fixed by the quantum number of the spin chain. We separate their contribution from the other dynamical holes. Using Stirling’s approximation, we find
where is the hole momentum Belitsky:2006en () and we redefine the S-matrix as
The two large holes define the effective length of the chain (=) as seen by the other dynamical holes. There are contribution to the effective length from the momentum terms in (12) that was absorbed into the propagation phase, as well as an additional phase from separating the two large holes. They are suppressed at large . The effective Bethe ansatz (14) is equivalent to the quantization conditions of Belitsky:2006en () (see equation (3.41) of this reference).
By comparing (10) and (15), we note that the excitation density coincides with the hole scattering kernel . This is not surprising because the definition of (8) can be rewritten as an integral equation for the hole density
Hence can be naturally interpreted as the hole scattering kernel. This general fact has also been observed in the study of Destri-de Vega type non-linear integral equations Fioravanti:1996 (); Freyhult:2007pz ().
2.2 All-loop asymptotic Bethe ansatz
where the magnon S-matrix is defined by the deformed rapidity as
and is the BES dressing phase Beisert:2006ez () that gives the correct answer beyond three loops in weak-coupling and at leading order in strong coupling. As in the one-loop case, we obtain an integral equation for the excitation density in the large spin limit
Following Basso:2010in (), we split the density into parity even and odd parts in as . We also split the magnon scattering kernel into , the dressing contributions into , and the inhomogeneous term into . From their integral expressions Basso:2010in (), Basso observed that the main scattering kernels are also even and odd in . Thus the parity even and odd parts decouple and we obtain an integral equation for each
We proceed to solve for the densities and compute the S-matrix between two excited holes. As before, the scattering phase is given by integrating the excitation density against the inhomogeneous term
where the even and odd inhomogeneous terms admit the integral representations Basso:2010in ()
The symmetric BES kernel can be split into parity even and odd parts that admit expansion over Bessel functions as
Define the Fourier-Laplace transformed densities as
for . By using the integral representations for the main scattering kernel, the dressing contributions, and the inhomogeneous terms Basso:2010in (), one can write the integral equations (20) as equations for
where are the generating functions for the conserved charges sourced by the holes. Their dependence on the excitation densities can be seen from the following integral representations Basso:2010in ()
One recognizes (25) as a generalization of the celebrated BES equation Beisert:2006ez (), which gives an all-loop expression for the density that can be expanded at weak and strong limits of the coupling and is amenable to numerical study at intermediate values of the coupling. The expression for the scattering phase (21) can be written in terms of as
2.3 S-matrix at strong coupling
To expand (27) at strong coupling, it is useful to rescale and introduce the auxiliary densities related to by
The auxiliary functions admit expansion as Neumann series of Bessel functions
and similarly for . We expand (25) in series of Bessel functions and separate into parity even and odd parts. We obtain four equations, two of which relate the higher conserved charges to the auxiliary functions, , which can be used to express the other pair solely in terms of the auxiliary functions
with . In the Fourier space, they become singular integral equations for and
The above relations hold for . For , can be determined by analyticity and is subleading at strong coupling Basso:2009gh (). Thus we can restrict the range of integration to the unit interval. Such singular integral equations have a standard solution
where can be determined from the so-called quantization conditions Basso:2009gh () and is negligible.
We may now compute the scattering phase (27) from the transformed densities . Performing the integral is straightforward but tedious. Remarkably, the answer takes a simple form when expressed in terms of the Zhukovsky variables , which correspond to the soliton velocities in the string worldsheet
where are the energy and momentum of the giant holes (46), is the soliton velocity in the center of mass frame (50), and is the Lorentz factor. The phase is antisymmetric in and , as is expected from a unitary S-matrix.
3 Worldsheet scattering of giant holes
In this section we review the construction of solitonic excitations on the GKP string Dorey:2010iy () and study their scattering. Consider classical strings in , in embedding coordinates
We may form two complex coordinates and express in global coordinates as
Integrability of the classical string is manifest by performing a Pohlmeyer reduction on the equation of motion to obtain the sinh-Gordon equation Jev ()
The GKP string is a vacuum solution with sinh-Gordon angle , which yields
As in Dorey:2010iy (), to regulate the IR divergence in the string energy a cut-off on the range of the worldsheet coordinate is introduced. It is natural to consider solitonic excitations of this vacuum. However, a one-soliton solution does not correspond to a physical state of the string due to the zero total momentum constraint . Instead, we construct a two-soliton solution and then identify one of the solitons when it is well-separated from the other as the “one-soliton solution”. The two-soliton solution in the center of mass frame is Jev ()
where with sinh-Gordon angle
We may study a single soliton by considering a limit of the above solution where the second soliton is sent to infinity. The solution is obtained by taking the large limit of (41) and takes the form Dorey:2010iy ()
where . The sinh-Gordon angle reduces to that of a single soliton so it can be interpreted as a one-soliton solution. At large distance the soliton solution can be matched onto the vacuum or GKP solution using the asymptotics,
In order to match the asymptotics of the GKP string solution, we need to restrict the worldsheet coordinate to where is the IR cut-off Dorey:2010iy (). Note also that global AdS time no longer coincides with the worldsheet time and is shifted by a constant as where we define,
The dispersion relation of the giant holes has been computed in Dorey:2010iy () as,
To move to a frame where the two solitons have generic velocities , we introduce the following boost
where is the soliton rapidity. One may easily check that
As , the two solitons are now located at . We may separate them by zooming into the first soliton in the limit and (for the second soliton we would zoom in the limit and ). The sinh-Gordon angle will be that of a single soliton with a time-shift.
We may read off the time delay experienced by the first soliton from (49) as
The scattering phase can be calculated from the semiclassical formula Jackiw:1975im (), where is the soliton energy evaluated at . Although was computed in the center of mass frame, it is intrinsic to the single soliton so is independent of frame. We integrate the time delay to find
where we used the asymptotic condition