Confining Strings, Infinite Statistics and Integrability

We study confining strings in massive adjoint two-dimensional chromodynamics. Off-shell, as a consequence of zigzag formation, the resulting worldsheet theory provides a non-trivial dynamical realization of infinite quon statistics. Taking the high energy limit we identify a remarkably simple and novel integrable relativistic $N$-body system. Its symmetry algebra contains an additional"shadow"Poincar\'e subalgebra. This model describes the $N$-particle subsector of a $T\bar T$-deformed massless fermion.


Introduction.
SU (N c ) quantum chromodynamics (QCD) turns into a free string theory in the planar (N c → ∞) limit [1]. For its maximally supersymmetric cousin the corresponding string theory has been identified as type IIB critical superstrings on AdS 5 × S 5 [2]. Moreover, the corresponding worldsheet theory is integrable and has been solved, resulting in the exact spectrum of planar N = 4 Yang-Mills (YM) (see [3] for an overview).
It has proven excruciatingly difficult to reproduce this success for planar non-supersymmetric gluodynamics. The corresponding string theory has not even been identified yet, despite a rich 45 year-long history of study. One may suspect at first that the question is not sufficiently well-posed. However, a sharp version of the problem can be formulated as follows [4]. Consider a background of YM theory with a single infinitely long confining string (flux tube). In the strict planar limit the string excitations decouple from bulk degrees of freedom and give rise to a microscopic two-dimensional model. The challenge is to build this worldsheet theory.
With this sharp formulation at hand, one immediately understands the source of the difficulties. Namely, in the absence of additional massless degrees of freedom on the worldsheet, the flux tube theory in D = 4-dimensional space-time has irreducible particle production [5,6], associated with the Polchinski-Strominger term [7]. Lattice YM simulations [8][9][10][11][12][13] (see [14,15] for reviews) do in fact exclude the presence of additional massless excitations on the worldsheet.
Note that at D = 3 integrability does not require additional degrees of freedom [4]. However, a closer look at the lattice data shows that the worldsheet theory is not integrable in D = 3 YM either [16][17][18].
Hence, unlike for the N = 4 case, in the planar limit of non-supersymmetric YM we are left with an interacting non-integrable two-dimensional model. To make life harder (and more interesting), at high energies this model exhibits characteristically gravitational behavior instead of that of a conventional quantum field theory [19,20].
This leaves two directions for further progress. First, as gluodynamics does not have any relevant deformations, the worldsheet theory is likely isolated. This turns it into a natural target for the modern S-matrix bootstrap [21,22], following the success of the conformal bootstrap in describing another isolated theory-the 3D Ising model [23]. A first promising step in this direction has been made very recently [24].
The other possibility is to identify a nearby integrable model, and to obtain a description of the worldsheet dynamics by performing a systematic perturbative expansion around this integrable theory.
One may object that there is no a priori reason for a controlled integrable approximation to exist. However, the analysis of lattice data provides a number of tantalizing hints supporting this program. This motivated a proposal for an integrable approximation-the Axionic String Ansatz (ASA)-both at D = 3 and D = 4 [4,17,25,26].
The successes of ASA have a clear physical origin [20,27]. At low energies the worldsheet degrees of freedom are translational Goldstone bosons of the nonlinearly realized Poincaré symmetry [5,[28][29][30][31][32]. Their low energy dynamics is well-approximated by a classically integrable Nambu-Goto action. On the other hand, at high energies worldsheet degrees of freedom correspond to partons of perturbative QCD (gluons in the YM case). Asymptotic freedom then implies that hard particle production is suppressed also at high energies.
This reasoning suggests that the violation of worldsheet integrability may be a transient phenomenon dominated by intermediate energies, E ∼ Λ QCD . A combination of low and high energy expansions may then allow one to describe the worldsheet dynamics at all scales.
As argued in [20], a natural playground to test these ideas is provided by adjoint D = 2 QCD. As the worldsheet theory lives in two dimensions for any D, general lessons from the study of the worldsheet should be universally applicable.
The goal of the present paper is to report a solution to the very first step in this program-the identification of an integrable approximation at D = 2.
Hamiltonian Formalism and Infinite Statistics. Massive adjoint QCD 2 is defined by the following action where ψ is a Majorana fermion in the adjoint representation of the SU (N c ) gauge group. We will mostly consider the heavy mass regime arXiv:1907.07799v1 [hep-th] 17 Jul 2019 when one expects a straightforward perturbative treatment to apply. The spectrum of this model was studied extensively in early 90's [33][34][35] (see [36] for a recent update). A study of the worldsheet dynamics has been initiated in [20], we will adopt the same procedure here. Following [37,38], we put the theory on a finite interval, σ ∈ (−L, L) with an infinitely heavy fundamental quark-antiquark pair Q,Q placed at the endpoints. Eventually, we take the infinite volume limit L → ∞. Then the worldsheet theory describes dynamics of states created by gauge invariant single-trace operators of the form Here the integration path C starts at σ = −L and ends at σ = L, but does not have to be straight. Turning points are allowed at the locations σ i of the adjoint quarks insertions because the Polyakov zigzag symmetry [39] is broken in the presence of the adjoint matter. Physically, one may think that the worldsheet theory describes the interior of a heavy quark QQ-meson.
After fixing the spatial gauge, A σ = 0, one can integrate out the remaining non-dynamical component A τ of the gauge field. The resulting Hamiltonian acts in the extended Hilbert space where H f is the free fermion Fock space, and V (V ) are (anti)fundamental representations of the color group. These additional factors represent color degrees of freedom of the endpoint quarks Q(Q). As a consequence of confinement, the physical Hilbert space H ph is the subspace of H ex annihilated by all color charges, Here ρ a is the color density of adjoint fermions, and T a (T a ) are (anti)fundamental generators representing color charges of Q(Q).
To describe the physical states it is convenient to adopt operator notations following from the identification where L(V ) is the space of linear operators acting on V . Then a general state in H ex takes the form with |ψ i F ∈ H f and M i ∈ L(V ). The physical worldsheet Hilbert space H w is generated by color singlets of the form Here T a 's are fundamental SU (N c ) generators considered as elements of L(V ). These are not to be confused with the quantum operators T a 's in (5). Note that H ph contains additional multitrace color singlet states such as the mesonic state ψ a ψ a |0 F ⊗ 1. Multitrace states decouple from the worldsheet in the planar limit.
In the momentum representation the physical worldsheet states can be written as linear combinations of T ai (9) where b a † k are fermionic creation operators. For many purposes it is convenient to also use the coordinate representation. The corresponding basis is In the heavy mass regime (2) eigenstates of the worldsheet Hamiltonian are well characterized by their parton number. In particular, one finds the worldsheet vacuum |0 = |0 F ⊗ 1 and one-particle excitations |k = √ 2 Nc b a † k ⊗ T a , dubbed "free quarks" in [38]. As emphasized in [20], the worldsheet multiparticle states (9), (10) do not describe conventional identical particles. For instance, the configuration space of twoparticle states (10) is the whole plane instead of the halfplane, as |σ 1 , σ 2 = |σ 2 , σ 1 . Equivalently, in the planar limit the exchange term is missing in multiparticle inner products, This is the inner product for a system of N distinguishable particles, indicating that the worldsheet theory provides a non-trivial dynamical realization of infinite quon statistics (see, e.g., [40]). It also serves as an interesting counterexample to the common lore [41] that the conventional Fock space is the only possible arena for Lorentz invariant quantum dynamics.
Up to now our treatment of the worldsheet theory was mostly kinematical. To study dynamics and to implement the program outlined in the introduction we need to evaluate how the worldsheet Hamiltonian acts on the physical space H w and to learn how to develop perturbation theory in this unconventional Hilbert space. We leave this task for a separate publication [42] and will discuss here only the very first step-identification of an unperturbed Hamiltonian H.
Let us first inspect two-particle matrix elements of the full worldsheet Hamiltonian H w . In the c.o.m. frame one finds [20] Here ω k = √ k 2 + m 2 and where U(k, k 1 ) is a smooth function of momenta equal to unity for k, k 1 m, or k, k 1 m and also in the forward limit k = k 1 . P stands for the principal value.
Even though V is multiplied by the 't Hooft coupling, it cannot be entirely treated as a perturbation due to the presence of forward singularities in (12). The physical meaning of these singularities is transparent in position space. Setting U(k, k 1 ) = 1, dropping the last nonsingular term in (12) and transforming (12) into position space we arrive at We see that the conventional statistics gets restored onshell-the growth of the potenital (13) at σ = +∞ reduces the number of scattering states by a factor of two, restoring the agreement with state counting for conventional identical particles. It is straightforward to check that the story repeats for multiparticle states. Namely, all terms in the full Hamiltonian which grow at spatial infinity are diagonal in the particle number and combine into the following potential in the N -particle sector where σ i,i+1 = σ i − σ i+1 . As a result, in the asymptotic regions τ → ±∞ one only finds configurations with σ 1 ≤ σ 2 · · · ≤ σ N and the conventional statistics gets restored on-shell for any number of colliding particles. Zigzags responsible for the emergence of the off-shell infinite statistics cost energy and do not survive on-shell. Integrable Worldsheet Mechanics. These results suggest the following choice of the unperturbed Hamiltonian in the worldsheet perturbation theory, where the subscript N indicates that (15) is a restriction of the unperturbed Hamitlonian H to the N -particle sector. In addition to the conventional free piece, H N,m also incorporates the leading forward singularities, which determine the asymptotic growth of the potential. A similar leading order Hamiltonian was also obtained in the Abelian case (at N = 2) based on counting [37]. To see whether (15) is a good choice it is natural to check at least the following two conditions: (i) Is (15) Poincaré invariant? (ii) Is (15) integrable?
We restrict to the classical analysis of (15). Even though the procedure which lead us to (15) was not manifestly Lorentz covariant, the final result is. Indeed, piecewise the potential (14) is either free or describes a constant electric field acting on some of the particles. Both options correspond to Lorentz invariant dynamics in two dimensions. More precisely, a single particle moving in a constant electric field is invariant under the centrally extended Poincaré group [43]. However, the central charge vanishes for (15) after contributions from all particles are added up. The corresponding boost generator is The Poisson brackets between H, J and the total momentum P satisfy the ISO(1, 1) Poincaré algebra One might expect then that the system (15) is also integrable, i.e. no momentum transfer occurs in multiparticle collisions. Indeed, as a consequence of crossing symmetry, momentum transfer implies particle production. The latter is absent for H. In accord with this argument, in the relativistic two-dimensional 1 N -body systems of [47,48] integrability automatically follows from Poincaré invariance. However, a straightforward Mathematica simulation of (15) shows that the system is not integrable. Most likely this is related to difficulties with preserving the Poincaré algebra (17) at the quantum level in very similar models [49,50]. We leave it to [42] to study whether this may be resolved by including subleading forward singularities in H. Instead, here we observe that the high energy limit of (15) given by does give rise to a Poincaré invariant integrable N -body model. From now on we set the 't Hooft coupling to unity, g 2 N c = 4π. Note that this limit does not contradict (2), as p 2 m 2 g 2 N c . On the other hand, (18) does not actually rely on (2)-this Hamiltonian describes the high energy worldsheet dynamics independently of whether (2) holds or not.
Integrability of (18) is the main observation of this paper. It confirms that (18) does provide a good starting point for the high energy expansion on the worldsheet. The fastest way to check integrability is to solve the equations of motion, Here s i = sign p i , s 0,1 = s N,N +1 = −1 and s j,j+1 = sign σ j,j+1 for 1 ≤ j < N . A general solution of (19) is a piecewise linear function of time. Using Mathematica 2 Time evolution for a sample 5-particle scattering (left) and of a 3-particle configuration in the periodic case (right).
it does not take long to convince oneself that the initial and final asymptotic sets of momenta are always the same. As an illustration, in Fig. 1 we plot a non-trivial five-particle collision and a self-repeating three-particle solution of the integrable periodic version of the model obtained by setting s 0,1 = s N,N +1 = sign (σ N − σ 1 − 2π). We will present a detailed study of the resulting integrable structure in [42] and restrict here to just a few remarks. Given the piecewise linear time dependence of the solutions, it is natural to look for conserved topological invariants T (S) and also for stepwise linear conserved charges of the form where S = (s 0,1 , s 1 , . . . , s n , s n,n+1 ) is the set of all signs. For instance, using (19) it is straightforward to check that stays constant. To see the physical meaning of T 2 let us evaluate it in the asymptotic regions, where all s i−1,i = −1. One finds, where N L(R) are the numbers of left(right)-movers in the asymptotic regions. It is natural to think of S as a sequence of classical spin variables. Interestingly, T 2 turns then into the Ising Hamiltonian. Topological invariants also provide a convenient starting point to construct non-topological charges by looking for F T 's satisfying {F T , H} = T. To see how this works, note that using (19) it is straightforward to construct F T2 in the non-periodic case, F T2 itself is only conserved in the sector with N L = N R . However, it is possible to build a new conserved quantity in all sectors by acting on F T2 with the boost generator. Namely, let us definẽ Then the Poincaré algebra (17)  It was suggested in [20] that in a putative gravitational description of the worldsheet the zigzags should correspond to black holes and the exotic off-shell statistics to black hole complementarity. In this context it is encouraging to find the "shadow" Poincaré subalgebra (H,P , J). It will be interesting to check whether the shadow charges (H,P ) may be identified with the Hamiltonian and momentum seen by infalling observers. Similar techniques also allow one to obtain additional integrals in involution at N > 2. For instance, for N = 3 the following translationally invariant combination is conserved in the N R = 2, N L = 1 sector This establishes Liouville integrability at N = 3. Interestingly, I 12 is not a higher order polynomial in momenta, as typically found using the Lax pair technique in similar integrable models (such as the Toda chain [51]). We expect that also at N > 3 the ansatz (20) leads to N − 2 additional independent charges in involution.
Future Directions and Relation to TT . The presented results open numerous avenues for future research.
In particular, what about the quantum integrability of (18)?
The following argument indicates that it should hold. The classical two-particle phase shift following from (18) is δ = s, which coincides with the exact phase shift of a known quantum integrable model-the TT deformation of a free massless fermion [19,[52][53][54][55]. This indicates that there exists a quantization of (18) resulting in the same phase shift, and that (18) describes the Nparticle subsector of the TT deformed fermion, similarly to how the Ruijsenaars-Schneider model describes the N -particle subsector of the sine-Gordon theory [47]. The appearance of the shadow Poincaré subalgebra in (22) mirrors the presence of dynamical and worldsheet "clocks and rods" in the gravitational formulation of the TT de-formation [56][57][58][59]. This relation also suggests that the infinite quon Hilbert space H w may provide a natural arena for defining off-shell observables in TT deformed theories. It will be interesting to connect this to the recent construction of [60].
As far as the D = 2 QCD physics goes, it was noticed in the numerical studies of the spectrum that meson mass eigenstates have definite parton numbers with a very high accuracy even for small quark masses [33,34]. Also it was observed that the equation for the spectrum becomes exactly solvable in the high energy limit [35]. Both these observations should be related to the integrability found here and it will be useful to make the connection precise.
In addition, it will be very interesting to see which of the presented results can be generalized to D = 3, 4 and to connect this approach to other signs of approximate integrability in QCD, such as [61,62].