Integrability of Chern-Simons Theory
Six Loops and Beyond
Dongsu Bak, Hyunsoo Min, Soo-Jong Rey
a) Physics Department, University of Seoul, Seoul 130-743 KOREA
b) School of Physics and Astronomy, Seoul National University, Seoul 151-742 KOREA
c) School of Natural Sciences, Institute for Advanced Study, Princeton NJ 08540 USA
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We study issues concerning perturbative integrability of Chern-Simons theory at planar and weak ‘t Hooft coupling regime. By Feynman diagrammatics, we derive so called maximal-ranged interactions in the quantum dilatation generator, originating from homogeneous and inhomogeneous diagrams. These diagrams require proper regularization of not only ultraviolet but also infrared divergences. We first consider standard operator mixing method. We show that homogeneous diagrams are obtainable by recursive method to all orders. The method, however, is not easily extendable to inhomogeneous diagrams. We thus consider two-point function method and study both operator contents and spectrum of the quantum dilatation generator up to six loop orders. We show that, of two possible classes of operators, only one linear combination actually contributes. Curiously, this is exactly the same combination as in super Yang-Mills theory. We then study spectrum of anomalous dimension up to six loops. We find that the spectrum agrees perfectly with the prediction based on quantum integrability. In evaluating the six loop diagrams, we utilized remarkable integer-relation algorithm (PSLQ) developed by Ferguson, Baily and Arno.
The AdS/CFT correspondence continues revealing remarkable relations between gauge and gravity theories. The most extensively studied so far is the correspondence between the four-dimensional superconformal Yang-Mills theory and the Type IIB superstring theory on AdS. Importantly, both theories admit Lagrangian formulation, which involve two coupling parameters: the rank of the gauge group and the ‘t Hooft coupling constant for the former, and the string coupling and the curvature scale (as measured in string unit) for the latter. As such, perturbatively, one can compute physical observables in both theories in double series of the respective parameters and test the correspondence by comparing a given observable extracted from each sides. In the planar limit, and , remarkable agreement was discovered between the two sides for a variety of observables. The agreement is largely attributed to the integrability structure-.
Recently, a Type IIA counterpart was discovered and added to the list of the AdS/CFT correspondence. The ABJ(M) theory is (2+1)-dimensional superconformal Chern-Simons theory and was proposed as the holographic dual to the Type IIA superstring theory on AdS[18, 19]. A question of interest is whether the two sides in this Type IIA counterpart also have an integrability structure. Recently, positive indications toward the quantum integrability were accumulated-. At strong coupling side, Lax pair construction of the integrability was shown at leading order. At weak coupling side, there were more indications. At two loops, spin chain Hamiltonian was computed explicitly for the subsector and was shown integrable[21, 22, 25]. See also Ref. . At four loops, spectrum of the spin chain Hamiltonian was shown to agree with the prediction of the integrability[35, 36].
In this paper, we further continue our previous investigations [22, 25, 35] concerning integrability of the Chern-Simons theory in the weak ’t Hooft coupling regime by computing the dilatation operator to six loops. Given that the integrability is in place up to four loops, why bother six loops? We contend that there are two important aspects that arise beginning at six loops and beyond: operator contents and recursive structure of the long-range spin chain. With these two issues on focus, we shall test the integrability of the Chern-Simons theory at six loop order. As in , we shall focus on magnon excitation in the subsector, compute operator structure and spectrum of the spin chain Hamiltonian up to six loops and check them against the prediction based on the integrability and the centrally extended superalgebra of excitation symmetry.
The off-shell superalgebra of the excitation symmetry is spanned by the two rotation generators , , the supersymmetry generator and the superconformal generator . The off-shell configuration is characterized by central charges . Their (anti)commutators are given by 
The central charges is related to the energy by , while introduced at off-shell are related to momentum of the magnon. Acting on a magnon transforming in the fundamental representation, closure of the superalgebra leads to the relation among the central charges
The central charges are in turn related to an excitation momentum . More generally, a bound-state of elementary magnons transforming in higher-dimensional representations can be studied. The off-shell analysis was sufficient to determine the dispersion relation. It is
where is a function of the ‘t Hooft coupling parameter .
Restricting to large limit and sector of the ) superconformal symmetry group, the quantum dilatation operator was computed explicitly at two loops from which an integrable alternating spin chain Hamiltonian and Bethe ansatz equations were identified [21, 22, 25]. Aspects of the integrability were explored further beyond two loops by focusing on diagrams generating maximal-ranged terms. These are the diagrams in which interaction vertices range over lattice sites of the spin chain Hamiltonian maximally. In , we computed four-loop contribution to these terms and found that the spectrum fits with the prediction based on the integrability and the excitation symmetry.
At each order in perturbation theory, depending on the range the ‘spin’ flavors at different sites are permuted, maximal-ranged terms in the dilatation operator are further classifiable into maximal-shuffling and next-to-maximal-shuffling terms. In SYM theory, it was found by Gross, Mikhailov and Roiban  that maximal-shuffling terms are computable recursively. Inspection of relevant Feynman diagrams indicates that all diagrams contributing to maximal-shuffling terms are related by a recursion relation and hence resummable to an exact all-order result. On the other hand, diagrams contributing to nonmaximal-shuffling terms are combinatorially too complicated to be resummable. One might anticipate that a similar argument exists for the ABJ(M) theory since conformal interactions are tightly constrained by large amount of supersymmetry. Indeed, we shall find that the maximal-shuffling terms in the dilatation operator originates from the same class of skeleton diagrams which we call homogeneous diagrams. We were able to perform all-order resummation and show that they match with the structure of dilatation operator predicted by integrability. The nonmaximal-shuffling terms receive contribution from another class of skeleton diagrams which we call inhomogeneous diagrams. As these diagrams are not recursively resummable and afflicted with potential infrared divergences, we need to resort to an alternative approach for direct evaluation. In the second half of work, we thus adopt two-point function method and compute six-loop contribution to maximal-ranged interactions from both homogeneous and inhomogeneous diagrams. This method amounts in dual Type IIA string theory to deriving time evolution Hamiltonian of a single non-interacting string.
Key results of our work point to the followings. The dilatation operator at six loops are extractable free of infrared divergences from two-point correlation functions of gauge invariant operators, which was already utilized in our earlier study at four loops . Moreover, the maximal-ranged interactions are consistent with the integrability and that, curiously, operator contents of the long-range spin chain Hamiltonian is identical to those of the super Yang-Mills theory, viz. Inozemtsev spin chain [38, 8].
This paper is organized as follows. We begin with description of the expected spectrum based on the integrability and prediction for the maximal shuffling coefficients to all orders. In section 3, we recapitulate all-loop computation of maximal shuffling terms in super Yang-Mills theory, obtained first by Gross, Mikhailov and Roiban in . We then extend the method to Chern-Simons theory and find two classes of diagrams contributing to the maximal shuffling. The first class of diagrams, homogeneous diagrams, is computable by a straightforward extension of the method in  and yields a result exactly parallel to the super Yang-Mills theory. The second class, inhomogeneous diagrams, is not computable by the method of  or variants of it: these diagrams are afflicted with infrared divergences. We conclude that all-order derivation for the maximal shuffling part of the dilatation operator is not straightforward in Chern-Simons theory. We thus resort to computing operator contents and spectrum of their anomalous dimensions order by order in perturbation theory. In section 4, we study 6-loop contribution to the anomalous dimension directly defined by two point correlation functions of operators. In section 5, we identify the matrix structures of the homogeneous and the inhomogeneous maximal-ranged interactions at 6-loops. In section 6, we compute the maximal-ranged part of the 6-loop Hamiltonian and confirm the prediction based on the integrability. We also show that only one particular choice arises among two possible maximal shuffling operators. This operator is the same as the one arising in the super Yang-Mills theory. In section 7, we extend our results to parity non-conserving ABJ theory, whose gauge group is U(U with . The last section is devoted to the concluding remarks. In the appendices, we relegate several technical details. Appendix A illustrate a comparative calculation of inhomogeneous term in operator-mixing method. Appendix B presents several lattice operator identities. Appendix C discusses derivation of skeleton 4-loop diagrams. Appendix D explains implementation of numerical integration and the remarkable integer relation PSLQ algorithm.
2 Hamiltonian and Spectrum From Integrability
We begin with consequence of the quantum integrability and the off-shell superalgebra to the spin chain Hamiltonian of a sub-sector of our interest. Consider single-trace bosonic operators in the ABJM theory of the type
Here, is the total number of the sites. We shall take the infinite volume limit and view a particular ordering of the operator as a lattice spin chain wave function . Gauge invariant operators place at odd sites the elementary scalar fields transforming as under the R-symmetry and at even sites the conjugate fields. Under the R-symmetry, these fields transform as and , respectively. We denote as where () transform under the and the SU(2) subgroups of . We then consider a subset of the operators (2.1), where we put only / fields at the odd/even sites, respectively. Explicitly, they are the following restricted set of operators
Since the only possible interaction between and fields is the contraction group theoretically, there cannot be any interaction between and representations acting on the above type of states. Therefore, for this restricted set of operators, the odd-site chain and even-site chain are decoupled from each other. Thus, there will be -type magnon and -type magnon propagating independently without any interactions between them. For unrestricted operators, there are interactions between them, but the above choice avoids unnecessary complexity in investigating the integrability. For the reference vacuum, we take the ferromagnetic state . In , we explained how elementary excitations above the reference vacuum are organized by the centrally extended superalgebra. The first acts on the flavors of the magnons formed by exciting odd sites of the spin chain, while the second acts on the magnons at even sites. The bosonic subalgebra of superalgebra corresponds to exciting for the odd sites and for the even sites. In , we carried out careful study of the representations of the centrally extended superalgebra for the asymptotic spin chain where is sent to infinity. The quantum integrability of the dilatation operator implies the factorization of multi-magnon S-matrices into product of two-magnon S-matrices satisfying the Yang-Baxter equations 111The all-loop proposal of the Bethe ansatz and S-matrices of ABJM theory was put forward in Ref. .. For both the ABJM and the ABJ theories, we also showed [22, 25] that dynamics of the magnons on even sites and on odd sites are governed by two separate transfer matrices for arbitrary spectral parameters and that, using the Yang-Baxter equations, they are mutually commuting
Their moments are
of which is proportional to the dilatation operator. They all depend on the spectral parameter – a nonzero value of is an indication that the dilatation operator and all other higher moments are not invariant under lattice parity transformation. We found in [22, 25] that not only for the ABJM theory but surprisingly also for the ABJ theory, which is parity non-conserving. From (2.4), it followed that there are two sets of mutually commuting, infinitely many conserved charges
It was argued  that these mutually commuting conserved charges are responsible for reflectionless property of the S-matrix elements between a magnon on even sites and a magnon on odd sites.
Invariance of the S-matrices under the off-shell superalgebra transformations was crucial to fix the representation as well as the spectrum of elementary magnons. The analysis (recapitulated in the previous section) shows that the dispersion relation of an elementary magnon takes the form (1.3) with and is pseudo-momentum of the magnon and is an interpolating function of the ’t Hooft coupling . In addition, the pseudo-momentum that specifies the central charge and the magnon spectrum can also be a function of the lattice momentum defined by translation in the spin chain. The functional form of the interpolating function and the pseudo-momentum are not determinable by the symmetry alone and require extra inputs of explicit computations either of Chern-Simons theory at weak coupling or of string worldsheet sigma-model at strong coupling. In the previous work , we found that holds up to four-loop order. In this work, we shall assume this as an input and proceed for computation of six-loops and beyond.
Perturbatively, the interpolating function is expandable as
where, for the leading term, we use the result of the two-loop computation in [21, 22]. Recently, in , the next coefficient was computed to be . Thus, in terms of the lattice momentum , the magnon dispersion relation can be expanded as
Note that corresponds to the classical scaling dimension of the elementary scalar fields and is required by one-third of the supersymmetry preserved by the ferromagnetic vacuum state. 222In Chern-Simons theory, the choice of regularization method is known to be a subtle issue. Detailed study in [22, 25] utilized the dimensional reduction and obtained at two loops. This confirms that the dimension reduction is a gauge invariant and supersymmetric regulator at least up to two loop order. Whether the corresponding Ward identities are satisfied at higher loops is an open problem that needs to be checked. All higher loop diagrams involving gauge and ghost fields are afflicted by the problem but diagrams involving matter fields only are not. Our previous  and the present works deal only with Feynman diagrams of the latter type and hence are free from this open problem. The two-loop coefficient was found by explicit computation in [21, 22]. The four-loop coefficient was computed in , while was argued in .
On the other hand the expected spectrum based on the quantum integrability and the off-shell superalgebra representation theory is expandable as
where we use the expansion for in (2.6). Note that the coefficients are fixed completely by the dispersion relation:
It is the coefficient of the term . Therefore, at order of the perturbation theory, the coefficient of the eigenvalue is uniquely fixed as
As we shall explain below graphically, is the eigenvalue of the maximal-shuffling operator on the spin chain lattice. From this argument, we conclude that the coefficients of the maximal-shuffling operators are fixed by the assumption of the integrability and the representation theory of off-shell symmetry superalgebra. In this work, we shall explicitly compute these coefficients at six loops and compare with (2.10). It constitutes a nontrivial check of the quantum integrability of Chern-Simons theory.
To derive the spectrum, following , we use the lattice-momentum eigenstates of elementary magnon. For the A-spin magnon propagating on the odd-site chain, we have
Here, refers that we put at the ’s site while we put for the remaining odd-sites, and similarly for the even-site chain. Hence, for an elementary magnon state, we may consider two kinds of states and . Below we shall focus on the odd-site chain as the odd-site and even-site chains behave independently for the above subset of magnon states we are interested in.
The corresponding integrable Hamiltonian at each order is well known for the above set of states333All order generalization to the full spin chain that takes account of interactions would be extremely interesting.. Since the structure of the even chain is identical to the odd chain, it is sufficient to focus on the odd-site chain only. The zeroth order spin chain Hamiltonian counts the classical scaling dimension of the spins:
and is the permutation operator defined by . We shall take the infinite volume limit, , and do not consider wrapping interactions. The corresponding 4-loop Hamiltonian can be identified as
Here, is an arbitrary coefficient. We thus see that a new feature arises beginning at six loop order. Up to four loops, candidate spin chain operators consistent with the quantum integrability are uniquely fixed. At 6-loops, operators consistent with the integrability are no longer unique: there are two commuting, mutually independent operators
Notice that these operator contents are identifiable with the Inozemtsev spin chain system [38, 8]. Acting on the momentum eigenstate in (2.11), these operators are diagonalized with the same eigenvalue
So, the operator content of the six-loop Hamiltonian is not determinable uniquely by probing a single magnon dispersion relation. On the other hand, the operators and are distinguishable by acting on a state containing two or more magnon excitations. Note that both of them include the maximal-shuffling operators that produce the eigenvalues . From the purely integrability point of view of the long-ranged Heisenberg spin chain, both operators are allowed and fit perfectly. Below, we shall approach this issue by computing relevant Feynman diagrams explicitly. We then determine the coefficient and, from it, the operator contents of the spin chain Hamiltonian.
3 Recursive Method for Maximal-Ranged Interactions
We begin in this section with a recursive method, first put forward by Gross, Mikhailov and Roiban , of extracting maximal-shuffling terms among maximal-ranged interactions. We first redo the computation in super Yang-Mills theory and then repeat the computation in Chern-Simons theory.
Of maximal-ranged interactions, we are particularly interested in the coefficients of the maximal shuffling operators — they will provide a direct test of the integrability. In super Yang-Mills theory, Feynman diagrams contributing to maximal-shuffling term are easily identifiable. As will be briefly reviewed below, at each order in perturbation theory, there is only one type diagram contributing to maximal-shuffling term. We refer to it as homogeneous diagram. For example, at 3-loop order, this maximal-shuffling term is responsible for the operator . There are also non-maximal-shuffling terms generated from maximal-ranged diagrams.
For Chern-Simons theory, despite different interaction structures, we find a strikingly similar pattern repeated. Among the maximal-ranged interactions, maximal-shuffling term arises from homogeneous diagram. It turns out this diagram gives rise to the operator , as in the situation of the super Yang-Mills theory. We shall further confirm the coefficients of the maximal shuffling terms. Among the maximal-ranged interactions, there are also inhomogeneous diagrams that give rise to nonmaximal-shuffled terms. At six-loop order, for instance, these diagrams are responsible for the operator . However, by a direct computation, we shall find the coefficient vanishes identically. This is interesting since, a priori, this operator could be generated given that interactions in the Chern-Simons theory are different from those in the super Yang-Mills theory, this operator could be generated.
3.1 super Yang-Mills theory
We first rederive the all-loop, maximal ranged interactions in super Yang-Mills theory obtained by Gross, Mikhailov and Roiban , emphasizing aspects directly relevant for similar considerations in Chern-Simons theory.
Consider in the super-Yang-Mills theory renormalization of composite operators. Introduce in the theory the dimensional regularization parameter , of ultraviolet divergences. To avoid infrared divergences, a set of external momenta are also injected to the operator. Denote by a collection of regularized Feynman diagrams that contribute to the renormalization of composite operators of the type (2.1). Omitting wave function renormalization part proportional to the identity part, multiplicative renormalizability of the operators asserts that
ought to be finite in the limit where goes to zero. The generator is the quantum dilatation operator generating renormalzation group transformation and its eigenvalue corresponds to the magnon dispersion relation in the spin chain interpretation. In perturbation theory,
From the -independence of , we get the relation
for the operators which do not include the identity part. In planar super Yang-Mills theory, the maximal-ranged interactions are generated only by quartic scalar interactions. Homogeneous diagrams among them are depicted in Fig. 1. To avoid infrared divergences, we inject a finite momentum to the field from the right in Fig. 1. It suffices to keep zero momentum for all other fields. The -loop contribution to the maximal shuffling diagrams can be evaluated recursively from the -loop contribution :
where denotes and denotes the lattice momentum such that generates shift one lattice site to the left or right in the spin chain. From the integrability, we expect the Hamiltonian to be
One can check finiteness of the renormalized diagrams in (3.1) order by order in . For instance, we checked this explicitly to the order using the Mathematica and found that the renormalized diagrams to this order are indeed finite, canceling all singular terms. As was done in , we now show asymptotically that the singular terms of the regularized amplitude are canceled by the expected Hamiltonian (3.5) in the renormalization factor
in the limit goes to zero. To show this, we take while holding finite and sum the all-loop contribution by the Euler-McLaughlin formula:
We have relegated all sub-leading remainder to :
where is the poly-gamma function. The integral can be evaluated by the saddle-point approximation. At the saddle-point
the integral is evaluated as
The pole term is precisely inverse of the renormalization factor (3.6) dictated by the integrability. As it should be, the renormalization factor is independent of the infrared regularizing momentum – the dependence resides in the finite remainder function .
Among the maximal-ranged interactions, there are also inhomogeneous diagrams. At four-loop order, they were studied in . These diagrams are distinguished from homogeneous diagrams that some of the scalar quartic vertices do not connect to the operator at all. They are responsible for generating nonmaximal-shuffling terms in the dilatation operator. One can convince that these diagrams proliferate rapidly at higher orders in perturbation theory and, even worse, do not show any recursive pattern. Therefore, their contribution needs to be computed individually. In section 6, adopting two-point function method, we will find that the inhomogeneous diagrams can be computed without ambiguity.
3.2 Chern-Simons theory
We now extend the recursive method to Chern-Simons theory. One easily see that relevant Feynman diagrams are classifiable again into homogeneous and inhomogeneous diagrams. The homogeneous diagrams are planar irreducible diagrams all of whose interaction vertices are connected to the operator by two internal lines. The inhomogeneous diagrams: planar irreducible diagrams some of whose interaction vertices are connected to the operator by one internal line or not connected to the operator at all.
Here, we first consider the homogeneous diagrams as they are easier to evaluate.
The homogeneous diagrams include all maximal shuffling terms. let us first state the expected scaling of operators via the anomalous dimension.
Here, as explained in detail in [22, 25], we adopt dimension reduction for regularizing ultraviolet divergences in Feynman diagrams. To avoid infrared divergences, we again inject momenta to the composite operator. Multiplicative renormalizability of composite operator asserts that
is independent of the regulator . Recall that, in Chern-Simons theory, ultraviolet divergences arise only at even loops. So, in perturbation theory,
where denotes the -loop regularized diagrams of order . From -independence of , we get the relation
for the operators which do not include the identity part.
As we explained in the previous section, from the integrability, we expect that the Hamiltonian for the maximal shuffling term is given by
Here, and should now be interpreted as a left/right shift operator by one lattice spacing on even- or odd-sites of the alternating spin chain. Therefore, the integrability asserts that the renormalization factor is given by
To check if ultraviolet divergence of the bare diagrams is inverse of (3.15), we now evaluate the homogeneous diagrams. At elementary -loops, from Fig. 2, we see that the diagram can be evaluated using many skeleton propagators defined by 1-bubble diagram:
one skeleton propagator defined by 2-bubble diagram:
and using recursively the skeleton 1-loop integral:
Denote by the coefficient that skeleton -loop contributes to the permutation in the maximal-shuffling term in the dilatation operator. Label -th loop momenta by and inject an external momentum at the last skeleton vertex to regulate the loop integrals. At other vertices, there is no need to inject external momenta. Taking vertex and symmetry factors into account, the skeleton 0-loop (which is actually elementary 2-loop) contribution reads
The skeleton 1-loop (which is actually elementary 4-loop) contribution reads
The skeleton 2-loop (which is actually elementary 6-loop) contribution reads
Recursive pattern is evident. The skeleton -th loop (note that this is actually elementary -loop) contribution reads
Multiplying to , we find the regularized -loop amplitude has the expression
Using Mathematica, we checked up to 10-loop orders that the renormalized diagram in (3.11) with the dictated by the integrability is indeed finite in the limit where goes to zero.
As in the super Yang-Mills theory, we can estimate the asymptotic behavior of the bare diagram in the limit , while holding constant. By the Euler-McLaughlin formula, we have
where denotes a sub-dominant remainder
We evaluate the integral by the saddle-point approximation. At the saddle point:
the integral is given by
We see that the pole term is precisely the inverse of the renormalization factor in (3.15). It is remarkable that this all-loop agreement between the homogeneous diagrams and the integrability is closely parallel to the situation in the super Yang-Mills theory. Because of this, in the next section, we adopt the two-point function method for deriving quantum dilatation operator.
We also need to take account of inhomogeneous diagrams. It is easy to see that this class of diagrams does not include the maximal-shuffling terms. Nevertheless, the interaction range is still maximal, viz. maximal-ranged, at a given order in perturbation theory, as illustrated in Fig. 4. In so far as one just focuses on maximal-shuffling part the spectrum, the homogeneous diagrams are sufficient. If one would like to identify operator contents of the spin chain Hamiltonian, however, it is indispensable and crucial to take account of the inhomogeneous diagrams. For instance, at 6-loop order, the inhomogeneous diagrams are responsible for the coefficient of the operator .
One would like to see if all-loop contribution of the inhomogeneous diagrams is also obtainable from the recursive method, much as for the homogeneous diagrams. Here, as in super Yang-Mills theory, the relevant diagrams proliferate rapidly at each higher order in perturbation theory and do not exhibit recursive pattern in any obvious way.
4 Anomalous dimension matrix
From the dual Type IIA string theory viewpoint, this method amounts to deriving time-evolution Hamiltonian of a single non-interacting string propagating in AdS spacetime. We use the dimensional reduction to regularize ultraviolet divergences. The two-point correlation functions are related to Feynman loop diagrams by
where denote the Euclidean scalar propagator in the position space
Here we consider all Feynman diagrams, connected or not, contributing to a given order of . As the definition of the anomalous dimension matrix takes the logarithm, it suffices to compute connected diagrams only. The anomalous dimension matrix is then extractable as coefficient of in the exponent. Up to 6-loop orders, the spin chain Hamiltonians classified in section 2 are given by
Note the extra factor multiplied. It arises combinatorially from extracting coefficients of from the -loop contribution ,
At -loop order, we have
Summing over all loops,
and we have