Spin Chains in Superconformal Theories
From the Quiver to Superconformal QCD
In this paper we find preliminary evidence that superconformal QCD, the SYM theory with fundamental hypermultiplets, might be integrable in the large Veneziano limit. We evaluate the one-loop dilation operator in the scalar sector of the superconformal quiver with gauge group, for . Both gauge couplings and are exactly marginal. This theory interpolates between the orbifold of SYM, which corresponds to , and superconformal QCD, which is obtained for . The planar one-loop dilation operator takes the form of a nearest-neighbor spin-chain Hamiltonian. For superconformal QCD the spin chain is of novel form: besides the color-adjoint fields , which occupy individual sites of the chain, there are “dimers” of flavor-contracted fundamental fields, which occupy two neighboring sites. We solve the two-body scattering problem of magnon excitations and study the spectrum of bound states, for general . The dimeric excitations of superconformal QCD are seen to arise smoothly for as the limit of bound wavefunctions of the interpolating theory. Finally we check the Yang-Baxter equation for the two-magnon S-matrix. It holds as expected at the orbifold point . While violated for general , it holds again in the limit , hinting at one-loop integrability of planar superconformal QCD.
- 1 Introduction
- 2 Lagrangian and Symmetries
- 3 One-loop Dilation Operator in the Scalar Sector
- 4 Protected Spectrum
- 5 Two-body scattering
- 6 Yang-Baxter Equation
- 7 Discussion
- A Simplified computation of the one-loop dilation operator
- B The Hamiltonian for SCQCD in the Dimer Picture
The gauge/gravity duality has given crucial insights into the dynamics of four-dimensional gauge theories. The long-standing hope is to find a precise string theory description of realistic field theories such as QCD. At present however we lack a systematic procedure to find the string dual of a given gauge theory, and all well-understood dual pairs fall into the “universality class” of the original example, the duality between super Yang-Mills and IIB on . These dualities are motivated by taking the decoupling limit of brane configurations in critical string theory. Field theories in this class share a few common features, for instance: all fields are in bifundamental representations of the gauge group; the and conformal anomaly coefficients are equal at large ; there is an exactly marginal coupling such that for large the dual worldsheet sigma-model is weakly coupled and the gravity approximation is valid.
To break outside the universality class, an important case study is superconformal QCD, namely the super Yang-Mills theory with gauge group and fundamental hyper multiplets. There is a large number of fundamental flavors, and at large . Nevertheless the theory shares with SYM the crucial simplifying feature of an exactly marginal gauge coupling. In a recent paper  we made some progress towards the AdS dual of SCQCD. We attacked the problem from two fronts: from the bottom-up, we performed a systematic analysis of the protected spectrum using superconformal representation theory; from the top-down, we considered the decoupling limit of known brane constructions in string theory. We concluded that the string dual is a sub-critical string background with seven geometric dimensions, containing both and and an factor. In this paper we take the next step of the bottom-up (=field theory) analysis, by evaluating the one-loop dilation operator in the scalar sector of the theory.
Perturbative calculations of anomalous dimensions have given important clues into the nature of SYM. They gave the first hint for integrability of the planar theory: the one-loop dilation operator in the scalar sector is the Hamiltonian of the integrable spin chain  – a result later generalized to the full theory and to higher loops, using the formalism of the asymptotic Bethe ansatz (see e.g. [3, 4, 5, 6, 7] for a very incomplete list of references.) Remarkably, the asymptotic S-matrix of magnon excitations in the field theory spin chain can be exactly matched with the analogous S-matrix for the dual string sigma-model. Thus perturbative calculations open a window into the structure of the dual string theory.444The calculation of the circular Wilson loop by localization techniques  is another interesting probe of the dual theory. It is natural to attempt the same strategy for SCQCD. The theory admits a large expansion in the Veneziano sense : the number of colors and the number of fundamental flavors are both sent to infinity keeping fixed their ratio in our case) and the combination . We focus on the flavor-singlet sector of the theory, which is a consistent truncation since flavor singlets close under operator product expansion. Let us denote a generic color-adjoint field by , with , and a generic color-fundamental and flavor-fundamental field by , where ; we are suppressing all other quantum numbers. In the Veneziano limit, single-trace “glueball” operators, of the schematic form , are not closed under the action of the dilation operator – this is a major difference with respect to the the standard ’t Hooft limit of large with fixed . Rather, glueball operators mix at order one (in the large counting) with flavor-singlet meson operators of the form . The simplest example is the mixing of with the singlet meson , which occurs at one-loop in planar perturbation theory (order ). The basic “elementary” operators are thus what we call generalized single-trace operators, of the schematic form
where is a color trace. We have introduced a flavor-contracted combination of a fundamental and an antifundamental field, , which for the purpose of the large expansion plays the role of just another color-adjoint field. The usual large factorization theorems apply: correlators of generalized multi-traces factorize into correlators of generalized single-traces. In particular, acting with the dilation operator on a generalized single-trace operator yields (at leading order in ) another generalized single-trace operator, so we may consistently diagonalize the dilation operator in the space of generalized single-traces. The dilation operator acting on generalized single-traces can then be interpreted, in the usual fashion, as the Hamiltonian of a closed spin chain. Just as in the ’t Hooft limit, planarity of the perturbative diagrams translates into locality of the spin chain: at one-loop the spin chain has only nearest neighbor interactions, at two two-loops there are next-to-nearest neighbors interactions, and each higher loop spreads the range interaction one site further.
More insight is gained by viewing SCQCD as part of an “interpolating” superconformal field theory (SCFT) that has a product gauge group , with , and correspondingly two exactly marginal couplings and . For one recovers SCQCD plus a decoupled free vector multiplet, while for one finds the familiar orbifold of SYM. We have evaluated the one-loop dilation operator for the whole interpolating theory, in the sector of operators made out of scalar fields. The magnon excitations of the spin chain and their bound states undergo an interesting evolution as a function of . For (that is, for SCQCD itself), the basic asymptotic excitations of the spin chain are linear combinations of the the adjoint impurity and of “dimer” impurities (we refer to them as dimers since they occupy two sites of the chain). From the point of view of the interpolating theory with , these dimeric asymptotic states of SCQCD are bound states of two elementary magnons; the bound-state wavefunction localizes in the limit , giving an impurity that occupies two sites.
Armed with the one-loop Hamiltonian in the scalar sector, we can easily determine the complete spectrum of one-loop protected composite operators made of scalar fields. It is instructive to follow the evolution of the protected eigenstates as a function of , from the orbifold point to SCQCD. Some of these results were quoted with no derivation in our previous paper , where they served as input to the analysis of the full protected spectrum, carried out with the help of the superconformal index .
An important question is whether the one-loop spin chain of SCQCD is integrable. The spin chain for the orbifold of SYM (which by definition has ) is known to be integrable [12, 13]. We find that as we move away from the orbifold point integrability is broken, indeed for general the Yang-Baxter equation for the two-magnon S-matrix does not hold. Remarkably however the Yang-Baxter equation is satisfied again in the SCQCD limit . Ordinarily a check of the Yang-Baxter equation is strong evidence in favor of integrability. In our case things are more subtle: the elementary excitations freeze in the limit (their dispersion relation becomes constant), while some (but not all) of their dimeric bound states retain non-trivial dynamics. Nevertheless, for infinitesimal the elementary s are propagating excitations, and the Yang-Baxter equation fails only infinitesimally, so it seems plausible that one can define consistent Bethe equations by taking small as a regulator, to be removed at the end of the calculation.
In section 2 we review the Lagrangian and symmetries of SCQCD and of the interpolating superconformal field theory. In section 3.1 we evaluate the one-loop dilation operator of SCQCD (in the scalar sector), and write it as a spin-chain Hamiltonian. In section 3.2 we find the spectrum of magnon excitations of this spin chain. These calculations are repeated in sections 3.3 and 3.4 for the the interpolating SCFT. A simplified derivation of the Hamiltonians is presented in appendix A, while appendix B contains an equivalent way to write the Hamiltonian for SCQCD in terms of composite (dimeric) impurities. In section 4 we study the spectrum of protected operators of the interpolating theory, and follow its evolution in the limit . In section 5 we solve the two-magnon scattering problem and analyze the spectrum of bound states in the spin chain of the interpolating SCFT, with particular attention to the limit. In section 5 we check the Yang-Baxter equation for the two-body S-matrix of the interpolating theory, finding that it holds for and . We conclude in section 6 with a brief discussion of integrability and of future directions of research.
2 Lagrangian and Symmetries
Our main interest is SYM theory with gauge group and fundamental hypermultiplets. We refer to this theory as superconformal QCD (SCQCD). Its global symmetry group is , where is the R-symmetry subgroup of the superconformal group. We use indices for , for the flavor group and for the color group .
Table 1 summarizes the field content and quantum numbers of the model: The Poincaré supercharges , and the conformal supercharges , are doublets with charges under . The vector multiplet consists of a gauge field , two Weyl spinors , , which form a doublet under , and one complex scalar , all in the adjoint representation of . Each hypermultiplet consists of an doublet of complex scalars and of two Weyl spinors and , singlets. It is convenient to define the flavor contracted mesonic operators
which may be decomposed into into the singlet and triplet combinations
The operators decompose into the adjoint plus the singlet representations of the color group ; the singlet piece is however subleading in the large limit.
|Adj + 1|
|Adj + 1|
The Lagrangian is
where stands for the Lagrangian of the vector multiplet and the for the Lagrangian of hypermultiplet. Explicitly555In our conventions, . We raise and lower indices with the antisymmetric symbols and , which obey .
where the potential for the squarks is
Using the flavor contracted mesonic operator (2), can be written more compactly as
2.2 orbifold of and interpolating family of SCFTs
SCQCD can be viewed as a limit of a family of superconformal theories; in the opposite limit the family reduces to a orbifold of SYM. In this subsection we first describe the orbifold theory and then its connection to SCQCD.
As familiar, the field content of SYM comprises the gauge field , four Weyl fermions and six real scalars , where are indices of the R-symmetry group. Under , the fermions are in the representation, while the scalars are in (antisymmetric self-dual) and obey the reality condition666The indicates hermitian conjugation of the matrix in color space. We choose hermitian generators for the color group.
We may parametrize in terms of six real scalars , ,
Next, we pick an subgroup of ,
We use indices for (corresponding to ) and indices for (corresponding to ). To make more manifest their transformation properties, the scalars are rewritten as the singlet (with charge under ) and as the bifundamental (neutral under ),
Note the reality condition . Geometrically, is the group of rotations and the group of rotations. Diagonal transformations () preserve the trace, , and thus correspond to rotations.
We are now ready to discuss the orbifold projection. In R-symmetry space, the orbifold group is chosen to be with elements . This is the well-known quiver theory  obtained by placing D3 branes at the singularity , with and and invariant. Supersymmetry is broken to , since the supercharges with indices are projected out. The symmetry is broken to , or more precisely to since only objects with integer spin survive. The factors are the R-symmetry of the unbroken superconformal group, while is an extra global symmetry under which the unbroken supercharges are neutral.
In color space, we start with gauge group , and declare the non-trivial element of the orbifold to be
All in all the action on the fields is
The components that survive the projection are
The gauge group is broken to , where the factor is the relative777Had we started with group, we would also have an extra diagonal , which would completely decouple since no fields are charged under it. generated by (equ.(12)): it must be removed by hand, since its beta function is non-vanishing. The process of removing the relative modifies the scalar potential by double-trace terms, which arise from the fact that the auxiliary fields (in superspace) are now missing the component. Equivalently we can evaluate the beta function for the double-trace couplings, and tune them to their fixed point .
Supersymmetry organizes the component fields into the vector multiplets of each factor of the gauge group, and , and into two bifundamental hypermultiplets, and . Table 2 summarizes the field content and quantum numbers of the orbifold theory.
The two gauge-couplings and can be independently varied while preserving superconformal invariance, thus defining a two-parameter family of SCFTs. Some care is needed in adjusting the Yukawa and scalar potential terms so that supersymmetry is preserved. We find
where the mesonic operators are defined as888Note that .
and the double-trace terms in the potential are
The symmetry is present for all values of the couplings (and so is the R-symmetry, of course). At the orbifold point there is an extra symmetry (the quantum symmetry of the orbifold) acting as
Setting , the second vector multiplet becomes free and completely decouples from the rest of theory, which happens to coincide with SCQCD (indeed the field content is the same and susy does the rest). The symmetry can now be interpreted as a global flavor symmetry. In fact there is a symmetry enhancement : one sees in (17, 18) that for the index and the index can be combined into a single flavor index .
In the rest of the paper, unless otherwise stated, we will work in the large limit, keeping fixed the ‘t Hooft couplings
We will refer to the theory with arbitrary and as the “interpolating SCFT”, thinking of keeping fixed as we vary from (orbifold theory) to ( SCQCD extra free vector multiplets).
3 One-loop Dilation Operator in the Scalar Sector
At large , the dilation operator of SCQCD can be diagonalized in the sector of generalized single-trace operators, of the form (1), indeed the mixing with generalized multi-traces is subleading. Motivated by the success of the analogous calculation in SYM , we have evaluated the one-loop dilation operator on generalized single-trace operators made out of scalar fields. An example of such an operator is
Since the color or flavor indices of consecutive elementary fields are contracted, we can assign each field to a definite “lattice site”999Up to cyclic re-ordering of course, under which the trace is invariant. and think of a generalized single-trace operator as a state in a periodic spin chain. In the scalar sector, the state space at each lattice site is six-dimensional, spanned by . However the index structure of the fields imposes restrictions on the total space : not all states in the tensor product are allowed. Indeed a at site must always be followed by a at site , and viceversa a must always be preceded by a . Equivalently, as in appendix B, we may use instead the color-adjoint objects , , and (recall the definitions (3)), where the ’s are viewed as “dimers” occupying two sites of the chain.
As usual, we may interpret the perturbative dilation operator as the Hamiltonian of the spin chain. It is convenient to factor out the overall coupling from the definition of the Hamiltonian ,
where is the one-loop anomalous dimension matrix. By a simple extension of the usual arguments, the Veneziano double-line notation (see figure 6 for an example) makes it clear that for large (with fixed) the dominant contribution comes from planar diagrams. Planarity implies that the one-loop Hamiltonian is of nearest-neighbor type, (with ), where . The two-loop correction is next-to-nearest-neighbor and so on. In section 3.1 we present our results for the one-loop Hamiltonian of the spin chain for SCQCD. We then derive (section 3.2) the one-particle “magnon” excitations of the infinite chain above the BPS vacuum . The one-particle eigenstates are interesting admixtures of the adjoint impurity and of the “dimeric” impurities.
The generalization to the full interpolating SCFT is straightforward and is carried out in sections 3.3 and 3.4. The structure of this more general spin chain is in a sense more conventional, and it is somewhat reminiscent of the spin chain [16, 17, 18, 19] for the ABJM  and ABJ  theories.101010An important difference is that our spin chain has an exact parity symmetry, whereas the spin chain of the ABJ theory is expected to violate parity at sufficiently high perturbative order (though somewhat surprisingly the ABJ planar theory appears to be parity invariant to low perturbative order [19, 22, 23].)
There are two types of color indices, for the two gauge groups and , with adjoint fields and carrying two indices of the same type, and bifundamental fields and carrying two indices of opposite type. Of course one must contract neighboring indices of the same type. Now a and a need not be adjacent since they can be separated for fields. The infinite chain admits two BPS vacua, the state with all s and the state with all s. The magnons are momentum eigenstates containing a single or impurity, separating one BPS vacuum on the left from the other vacuum on the right. We will see in section 5 how the “dimeric” impurities of the SCQCD chain arise in the limit from the localization of the bound state wavefunctions of the interpolating chain.
3.1 Hamiltonian for super QCD
We have determined the one-loop dilation operator in the scalar sector by explicit evaluation of the divergent part of all the relevant Feynman diagrams, which can be classified as self energy diagrams, gluon interaction diagrams and quartic vertex diagrams and are schematically shown in figure 1. The calculation is straightforward and its details will not be reproduced here. In appendix A we present a shortcut derivation that bypasses the explicit evaluation of the self-energy and gluon exchange diagrams, whose contribution can be fixed by requiring the vanishing of the anomalous dimension of certain protected operators.
As we are at it, we may as well consider the case of arbitrary , though we are ultimately interested in the conformal case . In the non-conformal case, it is more useful to normalize the fields so that the Lagrangian has an overall factor of in front . This different normalization affects the anomalous dimension of composite operators for , which acquire an extra contribution due to the beta function, but it is of course immaterial for . It is in this normalization that the chiral operator has vanishing anomalous dimension for all .
We find111111The spin chain with this nearest-neighbor Hamiltonian reproduces the one-loop anomalous dimension of all operators with , where is the number of sites. The case is special: the double-trace terms in the scalar potential, which give subleading contributions (at large ) for , become important for and must be added separately. This special case plays a role in the protection of , see section 4.
The indices label the charges of and , in other terms we have defined , , and . The parameter is the gauge parameter that appears in the gluon propagator as . Although the form of nearest-neighbor Hamiltonian depends on gauge choice , it is easy to check that dependence drops when acts on a closed chain. In the following we will set .121212This choice corresponds to setting to zero the self-energy of and . Then our Hamiltonian can also be used as is to calculate the anomalous dimension of operators with open flavor indices, of the schematic form . For there are extra contributions form the self-energy of the and at the edge of the chain.
We may rewrite more concisely (we have set ) as
The symbols and for identity, permutation and trace operators respectively. Their position in the matrix specifies the space in which they act. For example, the operator that appears in the matrix element of