###### Abstract

In a previous paper (hep-th/0509071), it was shown that quantum corrections to the BMN spectrum in an effective Landau-Lifshitz (LL) model match with the results from the one-loop gauge theory, provided one chooses an appropriate regularization. In this paper we continue this study for the conjectured Bethe ansatz for the long range spin chain representing perturbative large Super Yang-Mills in the sector, and the “quantum string” Bethe ansatz for its string dual. The comparison is carried out for corrections to BMN energies up to order in the effective expansion parameter . After determining the “gauge-theory” LL action to order , which is accomplished indirectly by fixing the coefficients in the LL action so that the energies of circular strings match with the energies found using the Bethe ansatz, we find perfect agreement. We interpret this as further support for an underlying integrability of the system. We then consider the “string-theory” LL action which is a limit of the classical string action representing fast string motion on an subspace of and compare the resulting corrections to the prediction of the “string” Bethe ansatz. As in the gauge case, we find precise matching. This indicates that the LL Hamiltonian supplemented with a normal ordering prescription and -function regularization reproduces the full superstring result for the corrections, and also signifies that the string Bethe ansatz does describe the quantum BMN string spectrum to order . We also comment on using the quantum LL approach to determine the non-analytic contributions in that are behind the strong to weak coupling interpolation between the string and gauge results.

HUTP-05/A0046

corrections to BMN energies from

the quantum long range Landau-Lifshitz model

J.A. Minahan^{1}^{1}1On leave from Department of Theoretical Physics, Uppsala University, A. Tirziu^{2}^{2}2
and A.A.
Tseytlin^{3}^{3}3Also at
Lebedev Institute, Moscow.

Jefferson Laboratory, Harvard University

Cambridge, MA 02138 USA

Department of Physics, The Ohio State University,

Columbus, OH 43210, USA

Blackett Laboratory, Imperial College, London SW7 2BW, U.K.

## 1 Introduction

Quantum corrections to semiclassical solutions of strings
propagating on play an important part in the
investigation of AdS/CFT duality [2, 3, 4]. In
particular, the
so-called three loop discrepancy
between gauge and string predictions was first found when
computing the leading correction^{1}^{1}1For string
computations, acts
as an inverse string tension or as
, which for semiclassical strings with large total -charge , can formally be traded with . However, it turns out that at higher orders
of perturbation theory there are additional genuine
quantum corrections[5] which are reflected in the presence of terms
non-analytic in . to the two-impurity BMN state [6].
The discrepancy was later found [7] to be present also for the semiclassical
spinning string solutions [8].

The conclusion of [6] (see also [9]) was the result of a complicated calculation, and used the contributions from the full set of world sheet fields, both bosonic and fermionic. Likewise, the quantum superstring corrections were computed for the circular string solution of [8], again employing the full set of the bosonic and fermionic world-sheet fields [10, 11, 12].

In the gauge theory, once one has found the dilatation operator,
the fermionic excitations are not needed to compute corrections in the
or the sectors, which are closed sectors containing no
fermion fields.^{2}^{2}2In deriving the expression for the dilatation
operator one of course uses
the full set of bosonic and fermionic fields of the SYM theory
(for example, already at one loop,
fermions contribute to the scalar self-energy diagrams). At the one-loop level,
that is to linear order in , where is the effective
coupling , the corrections can be determined from the
corresponding Bethe ansätze for these sectors [13, 14]. At
higher loops, one can use the proposed long range Bethe ansätze
in [7, 15, 16].

Since fermions seem to play no role in the Bethe ansatz, we should also be able to ignore them when computing corrections from an effective action. In [17] and our previous paper [18] this was shown to be the case when computing corrections for the one-loop sector. The action used was the Landau-Lifshitz (LL) action, which is the effective action for the ferromagnetic Heisenberg spin chain in the continuum limit, with higher derivative counterterms to account for lattice effects. On the string side, the counterpart of this action is the fast string limit around an subspace of [19, 20, 21]. However, with only bosonic sector modes being quantized, i.e. without the rest of the superstring modes, including fermions, there are infinities that need to be regularized. This can be accomplished with a combination of normal ordering and -function regularization.

A natural extension of [18] is to carry out the computations for higher orders in . In terms of the spin chain, this corresponds to going beyond nearest neighbor interactions, with order contributions coming from interactions between spins separated by up to sites. In [7], Serban and Staudacher (SS) first proposed an all-loop Bethe ansatz that was based on the Inozemtsev spin chain [22] and correctly reproduced the two and three loop predictions in [23]. However, this Bethe ansatz violated the BMN scaling at the 4 loop level, so a different ansatz was proposed by Beisert, Dippel and Staudacher (BDS) [15] that produces identical results as the SS ansatz to order , but also preserves BMN scaling to all loops in the thermodynamic limit.

In order to compare results between a long range Bethe ansatz calculation and an effective action calculation, we need to find the relevant extension of the LL action. The effective LL action to order was derived, both from the spin chain Hamiltonian and the fast string limit, in [20]. To go beyond order on the gauge side, the “string” LL action that follows from the fast string limit [20] can no longer be used, since the results on the gauge and string sides are known to disagree.

In this paper, we are able to find the “gauge” LL action to order indirectly, by using the results from the SS/BDS Bethe ansatz for operators that are dual to circular strings [24]. We construct the LL action by including all possible six derivative terms and varying their coefficients so that the energies agree with the Bethe ansatz predictions. With an effective action now available, we can then directly compute the corrections to BMN states with impurities up to order by quantizing the LL action (assuming normal ordering of the Hamiltonian and using a -function regularization to remove further infinities). Remarkably, comparing the results to the ones found directly from the gauge theory Bethe ansatz, we find perfect agreement.

One can also do the same on the string side, although in some sense the logic is in the reverse direction. Here one starts with the string effective action on and takes the fast string limit, reducing the action to a “string” LL effective action. Results from the ‘string LL action can then be compared with results from the string “quantum” Bethe ansatz of Arutyunov, Frolov and Staudacher (AFS) [25], which itself was originally derived by “discretising” the equations in [26] for general classical string motion on . Again, for the corrections for -impurity BMN states we find, even more remarkably, perfect agreement up to order.

That these results match attests to the underlying integrability of these systems. Any system, integrable or otherwise, should be describable by an effective action. However, the presence of a Bethe ansatz means that all scattering amplitudes can be reduced to products of two body scattering, the hallmark of integrability. Our results seem to indicate that the effective actions we use are consistent with integrability, and that this integrability will be present at the quantum level (or at least the first two orders), even for the string theory.

On the gauge side, the LL action should be interpreted strictly through its series expansion in , since the ’t Hooft coupling is the natural perturbative expansion parameter. However, for the string theory, is the natural semiclassical parameter and so for the string LL action we are formally allowed to expand in while keeping fixed. Hence, when determining the corrections, on the gauge side, one should first expand in and then compute the quantum corrections, while, on the string side, one should first compute the quantum corrections and then expand in . Because of the divergences that arise, the two procedures do not commute and may lead to different results. In particular, for the string theory, this will lead to non-analytic terms in [5, 27] and such non-analytic terms should be included only within the “string” interpretation of the LL computation.

We should stress that the presence of such non-analytic terms in the near-BMN spectrum is non-trivial: on general grounds one expects the energy to have the following expansion

(1.1) |

and while and are known to have a regular expansion in integer powers of , the results of [5] suggest that should contain non-analytic terms with half-integer powers of starting with . Below we will look for such non-analytic terms in the BMN spectrum using the quantum string LL approach, with mixed results. Indeed, we do find half integer powers of in the corrections computing from string LL Hamiltonian, but these come with logarithmic divergences that needed to be regularized. Presumably, for the full superstring calculation the coefficients of these non-analytic terms will be finite, but we are unable to unambiguously find these finite contributions using the string LL action.

This paper is organized as follows: Section 2 describes the structure of the string and gauge LL actions, with the latter determined to order by comparing to results from the Bethe ansatz. Section 3 is a review of the quantization procedure developed in [18] relevant for BMN calculations, now applied to an LL action of more general structure. Sections 4 and 5 contain computations respectively for the and corrections. Section 6 discusses non-analytic corrections and section 7 contains some concluding remarks. Appendix A describes how to fix the structure of the gauge LL action to order. Appendix B presents computations of the energy of impurity BMN states to order from both the gauge and string Bethe ansätze. Appendix C discusses the structure of non-analytic terms in the 1-loop energy of a circular string solution.

## 2 Classical LL action to order

Let us start by describing the structure of the LL action viewed as an effective action for low-energy excitations on either the string or gauge theory side (for a review see [28, 21, 18]). On the gauge side it is understood in a perturbative expansion in , and represents the quantum effective action for the low-energy spin wave modes of the spin chain Hamiltonian equivalent to the perturbative planar dilatation operator in the sector[13]. On the string side it is a “fast-string” expansion of the classical string action in a gauge [20, 21] where the density of the momentum of the “fast” collective coordinate is constant.

It is known [20] that to “2-loop” or order the LL actions obtained from the string theory and gauge theory are the same. At “3-loop” or order, however, they are different. The coefficients in the string LL action were obtained in [20] while for the gauge-theory LL action one can fix them by comparing the energy of particular classical solutions with the one obtained from the spin chain Bethe ansatz. We shall discuss this in Appendix A.

As a result, one may write the LL action in the sector as (we use the gauge where ; )

(2.1) |

where the Lagrangian is

(2.2) | |||||

Here , i.e. is a monopole potential on . Also , where . Here is a unit vector, and we have included all terms which are quadratic in . This exact quadratic part follows from the string action [20] and also from the coherent-state expectation value of the spin-spin part of the dilatatiohttp://www.mozilla.org/products/firefox/start/n operator on the gauge theory side [29]. It reproduces the BMN dispersion relation for small (“magnon”) fluctuations near the BPS vacuum .

The values of the “3-loop” coefficients in the string and gauge theory expressions for (2.2) are:

(2.3) |

(2.4) |

The string coefficients were found in [20]. The gauge
coefficients and are fixed by comparing to the Bethe
ansatz results for the circular solution (see Appendix A), while
the coefficients and can be fixed by matching the
resulting correction to the BMN energy to the
corresponding gauge Bethe ansatz result [15, 25] (see
sect. 4).
^{3}^{3}3In principle, to fix the values of the three coefficients
we could use, instead of the quantum BMN corrections,
the classical folded
string solution, comparing its LL energy to the Bethe ansatz
result of [7]. We shall see in sect.
5 that with these coefficients the corrections
also match, which provides
a strong consistency check.

The difference between the string and gauge values of the coefficients and implies the difference between the LL Lagrangians or the Hamiltonians

(2.5) |

This is a manifestation of the “3-loop disagreement” [6, 7]. Following [5], it can explained by promoting the coefficients and to functions of such that for large they approach the string theory values, while for small they approach the gauge theory values. Subleading terms in the string (strong-coupling) expansion of and ,

(2.6) |

should come from the part of the string quantum corrections which are non-analytic in [5]. The quantum string effective action will then have the structure (2.2) with and as coefficients. We shall return to the discussion of this below.

As in [18], let us now rewrite the LL Lagrangian (2.2) in terms of two independent fields. Solving the constraint as we get the following invariant expression for the Lagrangian in terms of and (; )

(2.7) |

(2.8) | |||||

where we use dot and prime for world-sheet time and space derivatives. The function has a regular expansion near , and so (2.8) may be interpreted as a phase-space Lagrangian with, say, being a coordinate and being related to its momentum.

To simplify the quantization of the LL Lagrangian near a particular solution it is useful to put it into the standard canonical form [18] by doing the field redefinition

(2.9) |

to obtain

(2.10) |

Having the Lagrangian in the standard form , the quantization is straightforward: we promote to operators, impose the canonical commutation relation (cf. (2.1))

(2.11) |

and then decide how to order the “coordinate” and “momentum” operators in .

##
3 Quantization near BPS vacuum:
corrections

to BMN spectrum from LL Hamiltonian

As in [18] our aim will be to use the LL action to compute quantum and corrections to the BMN spectrum of fluctuations near the BPS vacuum solution

(3.1) |

representing the massless geodesic in .
The
corrections can be found
from the Bethe ansatz on the spin chain [13, 30] or from a
direct superstring quantization [9, 6]. As explained in
[18], the derivation from the LL action turns out to be
much simpler
than the string-theory derivation.
Here we shall extend the method of [18] to
and orders.^{4}^{4}4Unfortunately, the exact
(all order in )
form of the and terms in the LL action is not known,
preventing us from computing the and corrections to all
orders in .
The corrections to the BMN spectrum have not yet been obtained from a full superstring computation, and our LL approach
provides a useful short-cut, highlighting several important
issues that will also appear in the exact superstring
approach.

Expanding near this vacuum corresponds to expansion near in (2.8) or in (2.10). Observing that the factor in front of the LL action (2.1) plays the role of the inverse Planck constant, it is natural to rescale as

(3.2) |

so that powers of will play the role of coupling constants for the fluctuations in the non-linear LL Hamiltonian. Expanding the Hamiltonian in (2.8), (2.10) to sixth order in the fluctuation fields we get

(3.3) |

(3.4) |

(3.5) | |||||

(3.6) | |||||

Let us first consider the quadratic approximation. The linearized equations of motion for the fluctuations are

(3.7) |

and their solution may be written as

(3.8) | |||||

(3.9) |

for real and . Upon quantization (3.7) becomes the equations of motion for the operators

(3.10) |

provided we use the canonical commutation relations in (2.11)

(3.11) |

Then the coefficients in (3.8),(3.9) satisfy

(3.12) |

so that and can be interpreted as annihilation and creation operators, with the vacuum state defined by , for all integer . A general oscillator state is

(3.13) |

The integrated Hamiltonian then becomes

(3.14) |

where we have used the normal ordering to ensure that the vacuum energy is zero, since the BMN vacuum is a BPS state in both gauge theory and string theory.

One also needs to impose the extra constraint that the total -momentum is zero [18]. For physical oscillator states we get

(3.15) |

Below we shall consider the “-impurity” states as oscillator states with :

(3.16) |

where for simplicity we shall assume that all are different (generalization to states with several equal is straightforward, at least for corrections). Then the zero-momentum condition (3.15) gives

(3.17) |

and the leading term in the energy of an -impurity state takes the familiar form [31, 2]

(3.18) |

It is useful to make a comment on the choice of parameters. In the LL approach we use as a natural total angular momentum, corresponding to a “fast” collective coordinate. Here is a characteristic of a particular state, while it is that enters into the background-independent form of the LL action (2.1). This is in line with gauge/spin chain intuition, where the use of total or spin chain length as the state-independent parameter is natural. At the same time, on the string side, when expanding near a BPS state, i.e. a massless geodesic with spin , one builds up from quantum excitations, and here it is natural to use and as the basic parameters of, respectively, the vacuum and the state. Thus, compared to generic states in the sector that carry spins with , here we have [18]

(3.19) |

The corresponding gauge-theory states are Tr, and plays the role of the length of the spin chain and is the number of magnons. On the string side, the LL approach is adapted to semiclassical solutions for which is of order rather than to near-BMN states which are small fluctuations near the vacuum and for which . In describing BMN states in the LL approach one has an “unnatural” choice of parameters: , not the usual BMN effective coupling . In the LL description the BMN energy starts with to which we add terms of order , i.e. . while the equivalent string theory expression is

## 4 corrections to the BMN spectrum

Let us now generalize the computation of the corrections to the energy (3.18) in [18] to order. To compute the correction to the energy of an -impurity state one needs to include the quartic term in the Hamiltonian (3.5) integrated over , i.e. , and use the standard quantum mechanical perturbation theory. Written in terms of the creation and annihilation operators, is found to be

(4.1) | |||||

where

(4.2) |

In the expression for the interacting Hamiltonian we have dropped the time dependent phases () since they can be removed by a unitary transformation with the quadratic Hamiltonian . Here and in what follows the summations over , etc., are from to .

As discussed in [18], to obtain the results consistent with both the gauge-theory spin chain and the string-theory expressions one should use a normal ordering prescription for . Doing so we get

(4.3) |

(4.4) | |||||

Then the leading correction to the energy (3.18) of an -impurity state is given by

(4.5) |

Expanding in gives for the correction to the energy

(4.6) |

Plugging in the string-theory and gauge-theory coefficients in (2.3),(2.4) we conclude that this expression is in precise agreement with the full string theory computation in [6, 33] and with the result found using the gauge and string Bethe ansätze in [25], expanded up to order. This agreement confirms, in particular, the values of the coefficients and in the gauge-theory LL action given in (2.4) (see also Appendix A).

Let us recall again that in comparing with the near-BMN results of [25, 33], one should note that as defined there is in the sector notation. To compare with our results, one may define and ; the expressions of [25, 33] should have replaced with and then re-expressed in terms of the parameters which are natural in the present LL approach.

The difference between the order string and gauge theory corrections to the BMN energy is because of the difference of the values of the coefficient : . The energy difference is thus [5]

(4.7) |

We shall return to the discussion of this difference in sect. 6.

In the string case the last double-sum term in (4) is
while in the gauge case
it is
As discussed in Appendix B, the gauge-theory expression in
(4) has a simple spin-chain generalization to all orders in
implied by the gauge Bethe ansatz [15, 25]^{5}^{5}5This is
equivalent to eq. (3.8) in [25] after observing that there
is here and there is here. As
always, we assume the condition (3.17).

(4.8) |

## 5 corrections to BMN spectrum

To find corrections we follow the method in [18] where order terms were computed. We need to combine the second order perturbation theory correction for the quartic Hamiltonian (3.5) with the first order perturbation theory correction for the sixth order Hamiltonian in (3.6). The regularization issues were discussed in detail in [18]: to match string/gauge results we should use the normal-ordered form of the Hamiltonians and apply -function regularization for intermediate-state sums. We shall also need to add a local higher-derivative “counterterm” which (on gauge side) is a lattice correction to the continuum limit of the LL action (see [18] and below).

### 5.1 Second-order perturbation (“exchange”) contribution

Starting with the quartic Hamiltonian (4.1) we need to compute

(5.1) |

where is any possible intermediate state, and . Since in (4.3) contains only terms of the form , the only non-trivial intermediate states can be the -particle states of the form . Then in order for the matrix element to be non-zero, there should be a and such that and , with all other , . In order for to be distinct from , we require that . With these conditions, we then find that if

(5.2) | |||||

where and are not equal to one of the other ’s. The energy difference in (5.1) is

(5.3) |

If , and so has two impurities with the same momenta, then the matrix element is

(5.4) | |||||