Meson spectrum of SU(2) QCD1+1 with quarks in Large representations

We consider SU(2) quantum chromodynamics in 1 + 1 dimensions with a single quark in the spin J representation of the gauge group and study the theory in the large J limit where the gauge coupling g2 → 0 and J → ∞ with λ = g2J2 fixed. We work with a Dirac spinor field for arbitrary J, and with a Majorana spinor for integer J since the integer spin representations of SU(2) are real, and analyze the two cases separately. The theory is reformulated in terms of global colour non-singlet fermion bilocal operators which satisfy a W∞ × U(2J + 1) algebra. In the large J limit, the dynamics of the bilocal fields is captured by fluctuations along a particular coadjoint orbit of the W∞ algebra. We show that the global colour-singlet sector of the bilocal field fluctuations satisfy the same integral equation for meson wavefunctions that appears in the ’t Hooft model. For Majorana spinors in the integer spin J representation, the Majorana condition projects out half of the meson spectrum, as a result of which the linear spacing of the asymptotic meson spectrum for Majorana fermions is double that of Dirac fermions. The Majorana condition also projects out the zero mass bound state that is present for the Dirac quark at zero quark mass. We also consider the formulation of the model in terms of local charge densities and compute the quark spectral function in the large J limit: we see evidence for the absence of a pole in the quark propagator.

Quantum chromodynamics in 1 + 1 dimensions is expected to exhibit confinement due to the Coulomb potential being linear in two spacetime dimensions.The SU(N ) theory with quarks in the fundamental representation was solved by 't Hooft in the planar large N limit [1,2], and was further analyzed extensively by many authors [3][4][5].Indeed, at leading order in large N , it was first shown in [1] that (1) there is no pole in the fermion-antifermion two-point function, indicating that there is no asymptotic coloured state, and (2) the gauge invariant quark-antiquark wavefunction satisfies an integral eigenvalue equation whose solutions describe an infinite, discrete spectrum of bound states.
The meson spectrum was derived in an entirely different way in [6] by noting that the Hamiltonian of the model in lightcone gauge can be written in terms of the bilocal field i ψ i (x) ψi (y) which is nothing but the manifestly gauge-invariant Wilson line in the lightcone gauge.It was shown in [6] that the bilocal field (i.e., the gauge invariant Wilson line in lightcone gauge) satisfies a W ∞ algebra.In the large N limit, the gauge-invariant Wilson line behaves semiclassically due to factorization, and the classical phase space of the bilocal field is an appropriate coadjoint orbit of the W ∞ algebra [6][7][8][9].The equations of motion for fluctuations along this coadjoint orbit then directly gave the integral equation of 't Hooft for the gauge invariant quark-antiquark wavefunction.
In this paper, we consider SU(2) gauge theory in 1 + 1 dimensions with a single quark in the spin J representation of the gauge group with the spin J being very large. 1 The quark field will be a Majorana spinor since integer J representations of SU (2) are real, whereas we consider Dirac fermions for arbitrary J.We perform lightcone quantization a la 't Hooft [1] by choosing the lightcone direction x + as time and work in the lightcone gauge A − = 0.The advantage of lightcone quantization is that the sole dynamical degree of freedom is the chiral fermion ψ − (x).We solve for the meson spectrum of the model in a 't Hooft-like large J limit where J → ∞ and the gauge coupling g → 0 with λ = g 2 J 2 kept fixed. 2  In Section 2, we describe the large J limit of the theory.In taking the large J limit, we treat the spin J representation semiclassically and replace the discrete representation index m which runs over −J, −J + 1, . . ., J − 1, J, by a continuum angle θ ∈ [0, π] with cos θ = m/J.In this continuum description, the fermion components ψ m (x) coalesce into a field ψ(θ, x) along the θ direction, and we work with this continuum field in our analysis.The four-fermi term in the Hamiltonian takes a nice suggestive form where a potential also appears in θ space, in addition to the Coulomb potential in spacetime.Schematically, it takes the form where, the factor |x − − y − | is the Coulomb potential along the 'spatial' direction x − , and the integrals over θ, θ ′ are the continuum versions of the sums over the representation indices on the fermions.Our subsequent analysis in Section 3 is based on the W ∞ algebra and its coadjoint orbits that appear in [6], but with one big conceptual difference.For the SU(N ) theory with fundamental quarks considered in [6], the effective four-fermi interaction due to the Coulomb force between the quarks could be written as a product of two bilocal fields which were in fact gauge invariant Wilson lines evaluated in lightcone gauge.In our model, the four-fermi term cannot be written in terms of such gauge invariant Wilson lines even in lightcone gauge due to the potential in representation space (see equation (1.1)).However, the four-fermi term can be written in terms of the non gauge-singlet bilocal field ψ i − (x)ψ * j− (y) and hence we work with these variables. 3he non gauge-singlet bilocal fields satisfy a W ∞ × U (2J + 1) algebra and lie on a particular coadjoint orbit of W ∞ × U (2J + 1).The equation of motion for fluctuations along the coadjoint orbit gives an integral equation, but now involving the non gaugesinglet bilocal field.Tracing over the gauge indices of the non gauge-singlet bilocal field, we then get an integral equation for the gauge invariant bilocal field (gauge invariant because it is the gauge invariant Wilson line in lightcone gauge) which is the same integral equation that was derived by 't Hooft [1].In [14][15][16], the meson spectrum of the large N 't Hooft model was interpreted as the spectrum of a string in two spacetime dimensions with massive particles at the ends by demonstrating that the on-shell equation for the string states is the same as the 't Hooft equation.Since we also get the same integral equation, such a string interpretation also holds in our case.This is surprising since we do not have an obvious genus expansion for Feynman diagrams like in the large N limit.
In Section 4, we treat Majorana quarks in integer spin J representations of SU (2).To obtain the meson spectrum in this case, we apply the Majorana condition on the bilocal fields discussed above.This projects out half the solutions of the integral equation and gives a spectrum that is asymptotically linearly spaced, but with the spacing being twice of that of the Dirac fermion case.In the case where the quark bare mass is zero, the same condition projects out the zero mass bound state in the spectrum of the 't Hooft integral equation.We confirm that there is no meson bound state with zero mass by solving the equation numerically.
In Section 5, we work in an axial gauge A 1 = 0 and formulate the theory in terms of collective fields, one of which is the charge density ρ j (x) = ψ † j (x)ψ j (x), or its large J version ρ(θ, x) = ψ † (θ, x)ψ(θ, x).The other collective field σ(θ, x) appears as a fictitious U(1) gauge field minimally coupled to the fermions.The effective action is rewritten in terms of a scalar ϕ with ρ = ∂ 1 ϕ, and the curvature η = ∂ 1 σ of the fictitious gauge field σ.
In the large J limit, we focus on a translation-invariant saddle point of the effective action at which ϕ and η are constant.The fermions appear quadratically and can be integrated out and the resulting log-determinant is the Euler-Heisenberg effective action for the fictitious field strength η.We are not able to solve the saddle point equations.However, assuming that there is a reasonable solution for η and ϕ at the large J saddle point, we are able to compute the spectral function for the gauge invariant fermion two-point function and demonstrate that there is no pole in the two-point function.
The paper also contains several appendices which supplement the main text with details of the various calculations.
2 SU(2) gauge theory with spin J matter

The theory
As described in the introduction, we consider SU(2) gauge theory in 1 + 1 dimensions with a matter quark in the spin J representation of SU(2). 4 It is a well-known fact that irreducible representations of SU(2) are real / pseudoreal when J is integer / half-integer respectively.Thus, when the quark is in an integer spin representation of SU(2), the minimal field content is that of a Majorana spinor.A Dirac spinor in an integer spin representation is equivalent to two flavours of Majorana spinors (see Appendix A for our spinor conventions).For half-integer spin representations, the single quark must be a Dirac spinor. 5In this section, we work with Dirac fermions in arbitrary spin representations and treat the case of integer spin Majorana fermions in Section 4.
The action of our SU(2) gauge theory with a single Dirac fermion ψ in a half-integer spin J representation of SU(2) is where ψ = ψ † γ 0 .The covariant derivative is defined as where g is the coupling constant and the generators T a in (2.2) are hermitian and are in the spin J representation.The field strength F µν = F a µν T a in (2.1) is written with T a being matrices in the fundamental representation of SU (2); in this case, we have tr(T a T b ) = 1 2 δ ab .The action can be written explicitly in components as where the vector indices on the field strength and the covariant derivatives are lightcone indices defined via x ± = x 0 ± x 1 (for more details on our notation and conventions, see Appendix A).We choose the direction x + as time.The component A + is then nondynamical and the Gauss' law is the field equation for A + : where the electric field E a is defined as E a := −F +−,a = 4F a +− .We choose the gauge representative A − = 0 for the orbits generated by the Gauss' law.In this gauge, the electric field is given by The component ψ + is also non-dynamical since its action does not involve a ∂ + derivative.Its equation of motion is a constraint: When m = 0, the ψ + fermions are no longer related to ψ − , and decouple in this gauge.We take the Hamiltonian to be the Noether charge for time translations: (in deriving the above form, we have used the Gauss' law (2.4) and the ψ + field equation (2.6) in the A a − = 0 gauge.)We use the appropriate Fourier transforms in Appendix A to rewrite the Hamiltonian as The colour-singlet condition: The Gauss' law in A − = 0 gauge can be integrated over (2.9) A state |Ψ⟩ is a colour-singlet when the E a (∞) = E a (−∞) on the state.From the integrated Gauss' law (2.9), this is the same as requiring where the current ψ † T a ψ is normal-ordered.

The large J limit
Next, we would like to study the large J limit of the above model.Let us focus on the current ψ † − T a ψ − (x) that appears in the four-fermi term in the Hamiltonian (2.8).Expanding the fermion ψ − in the |J, m⟩ basis as ψ − = m ψ m − (x)|J, m⟩, the components of the current are where we have used the standard expressions for the matrix elements of the generators T ± and T 3 in the |J, m⟩ basis of the spin J representation of SU(2) (see (B.2) in Appendix B for a recap).We begin with the following manipulation.Multiply and divide the coefficients in (2.11) by the square root of the Casimir C 2 (J) = J(J + 1) of the spin J representation, and write the expressions in terms of the following quantities: , m J(J + 1) . (2.12) In the large J limit, the expression J(J + 1) is approximated by J at leading order.Thus, at leading order in large J, the above expressions are approximated by Since the range of m is m = −J, −J + 1, . . ., J − 1, J, the range of u m := m/J is from −1 to 1 in steps of 1/J.In the large J limit, this range becomes a continuum from −1 to 1 and it is appropriate to define angles where C 2 (J) = J(J + 1) ∼ J in the large J limit.

2.3
The continuum field ψ − (θ, x) The values u m = m/J lie in the interval [−1, 1] and become closer and closer as J becomes large.Thus, they serve to discretize the interval [−1, 1] in steps of 1/J.Consequently, the fermion components ψ m − (x) also coalesce into a function ψ− (u, x) on the u-interval [−1, 1].For purposes which will be clear later, we define the function ψ− (u, x) with an additional prefactor √ J: (2.16) As J goes to ∞, the fields ψ− ( m J , x) indeed cluster together on the u interval u ∈ [−1, 1].This enables us to study the variation of ψ− (u, x) w.r.t.small changes in u which are changes of order 1/J in the large J limit.
The prefactor of √ J in (2.16) can be motivated as follows.The fermion components ψ m − are discrete for finite J and the neighbouring components need not be close in any sense since we integrate over all possible configurations of ψ m − in the path integral.In the large J limit, we have defined the fields ψ− ( m+1 J , x) and ψ− ( m J , x) such that (2.17) For large J, two things happen to the left-hand side: 1.The two fields are situated infinitesimally close to each other in the u direction, and 2. The difference in the two fields is suppressed by the large J factor.
Thus, in the large J limit, we assume that the difference ψ m+1 − and ψ m − is parametrically small, which in turn implies the existence and continuity of the field ψ− (u, x).Later, this assumption will turn out to be justified since there will be a smooth solution for ψ(u, x) at the large J saddle point.The continuity in the variable u will also allow us to conceive derivatives of ψ− (u, x) in the u direction.
Another justification for the factor of √ J in the definition of ψ− (u, x) is as follows.Upon canonical quantization, the fermions ψ m − (x) satisfy the equal-time anticommutation relations In terms of ψ− (u, x) we have (2.19) In the large J limit, the Kronecker delta δ m n goes over to a Dirac delta function in the u variable in the continuum limit J → ∞ as so that the continuum limit of (2.19) becomes Note that without the prefactor √ J in the definition (2.16), the anticommutation relations (2.21) would have a 1/J on the right-hand side; the right hand side would then go to zero in the large J limit.The explicit prefactor √ J in the definition of the continuum field ψ− (u, x) ensures that the quantum nature of the fermions persists in the large J limit.
We finally make a transformation from the variable u to the angle θ related as u = cos θ and define ψ − (θ, x) := ψ− (u, x) where we have omitted the ˜in the definition of ψ − (θ, x) on the left-hand side to avoid cluttering of notation.However, we must emphasize that the scaling m → m/J persists in the definition of ψ − (θ, x) which in turn allows us to study derivatives w.r.t.θ as alluded to earlier in this subsection.The equal time anticommutation relations (2.21) become (2.23) Thus, in the continuum limit, the fermion ψ m − (x) will be replaced by The sum over m is also similarly replaced by an integral over u, or an integral over θ: (2.25)

A derivative along the θ direction
Let us look at the fermion bilinear ψ † − T − ψ − that appears in the four-fermi term in the Hamiltonian: Since the term relates the neighbouring components ψ m and ψ m+1 , we expect such a coupling to involve a derivative in the continuum variable u = m/J, or equivalently θ = cos −1 u.We write In terms of ψ− (u, x), we have In the large J limit, the above simply becomes the derivative of ψ− (u, x) along the u direction: (2.28) That the derivative exists rests on the assumption that the difference between ψ m+1 − and ψ m − is suppressed as 1/ √ J; see the comments after (2.17).In terms of the θ coordinate, we have (2.24), Thus, in the large J limit, so that (2.31)

The Hamiltonian in the large J limit
Recall the continuum expressions for the coefficients (2.14), and those for the fermion fields (2.24) and (2.30), and for the sum over the representation index (2.25) Using the formulas (2.32), (2.33) and (2.34) above, the fermion current can be written in the continuum J → ∞ limit as Observe that we can drop the ∂ θ terms in the large J limit since they are suppressed by 1/J.In principle, we can compute subleading terms in an 1/J expansion by retaining these terms.
At leading order in 1/J, Hamiltonian (2.8) can be written in a simple form where there is a potential in the θ direction: (2.36) Similarly, the colour-singlet conditions (2.10) on states |Ψ⟩ become 3 Mesons and W ∞ coadjoint orbits

A 't Hooft-like limit
We reproduce the Hamiltonian of our theory (2.8): Let us focus on the mass term where the trace tr is simply the sum over the representation indices as indicated.As is familiar from studies of the large N limit of quantum field theories, the trace is expected to scale as 2J + 1 in the large J limit since it has 2J + 1 terms.Thus, the first observation in the large J limit is that 1.Every trace over representation indices is O(J) in the large J limit.
It is easy to see that the Hamiltonian can be written as a single trace over the representation indices.To see this, and for subsequent calculations, we introduce the bilocal field or, in more detail, The Hamiltonian (2.36) can now be written as a single trace over the gauge representation indices (which are parametrized by the angle θ in the continuum limit): where M (x, y; θ, θ ′ ) = cos(θ − θ ′ )M (x, y; θ, θ ′ ), and the bilocal fields are all at equal time x + = y + .Since the Hamiltonian is a single trace over the representation indices, we have our second observation: 2. The Hamiltonian is O(J) in the large J limit provided the combination g 2 J 2 is kept fixed in the limit.
Again, in analogy with the large N limit of gauge theories, We define the 't Hooft-like coupling λ = g 2 J 2 , (3.6) and work in the limit Since the Hamiltonian is O(J) in the 't Hooft-like large J limit, and it is expressed completely in terms of the bosonic bilocal variables M , we can treat M semiclassically in this limit.The advantage of this formulation is that one may be able to directly extract the meson, i.e., gauge invariant quark-antiquark, spectrum and wavefunctions from a saddle-point analysis of the bilocal variables.
The above idea was successfully applied to the large N limit of SU(N ) QCD in 1 + 1 dimensions coupled to quarks in the fundamental representation [6], where it was used to derive the integral equation for the quark-antiquark wavefunction first obtained by 't Hooft [1].In this paper, we follow the steps of [6] closely in spirit but keeping one crucial difference in mind: we work with bilocal operators M (x − , y − ; θ, θ ′ ) that are non gaugesinglets whereas the bilocal operators in [6] were gauge-invariant.At the end, we perform the additional step of extracting the gauge invariant information from our results.
We also note an identity satisfied by the equal-time bilocal operators which will be useful for our semiclassical analysis of the model in the 't Hooft-like large J limit: ) where Q is the conserved global U(1) charge operator supported on a constant time slice x + = const.: The charge Q is nothing but the baryon number which measures the number of gauge invariant states of the form : Note that such a charge operator does not exist when the fermions ψ i transform in the Majorana representation for integer J. Indeed, using the 7 This can be seen by the following manipulation.The left hand side is Majorana condition ψ * m = (−1) J+m ψ −m (see Appendix B), we see that the charge operator is simply zero since the terms corresponding to −m and m in the summation cancel each other and the m = 0 term vanishes after normal ordering since : (ψ 0 ) 2 := 0.

The W ∞ algebra and coadjoint orbits
Next, we show that the quantum operators M (x − , y − ; θ, θ ′ ) satisfy a Lie algebra which is the W 2J+1 ∞ algebra.Recall that the continuum fermions ψ − (θ, x) satisfy the equal time anti-commutation relations (2.23) which we reproduce here for convenience: Using (3.10), it is easy to see that the equal-time bilocal variables satisfy The above algebra of the equal-time bilocal variables [6][7][8]).We next obtain a kinetic term for the bilocal fields M by appealing to the theory of quantization of coadjoint orbits of the W ∞ algebra.The orbit method has found many applications in physical problems, such as understanding the non-relativistic fermion description of the c = 1 matrix model [7][8][9]21], the 't Hooft model of two dimensional QCD with fundamental quarks [6], gravity duals for the SYK model [22], a non-linear bosonization of fermi surfaces [23], and so on.
Following previous treatments in [6][7][8][9]21], we describe the coadjoint orbit of the W 2J+1 ∞ algebra which is relevant for us.We only give a summary here and refer the reader to Appendix C for more details.The coadjoint orbit is described by certain constraints on the bilocal field M (x − , y − ; θ, θ ′ ).To describe these constraints, it is convenient to introduce a matrix notation for the bilocal fields.Since we have one field for each (x − , y − , θ, θ ′ ), we can write down a matrix M such that In terms of M , the constraints that describe the coadjoint orbit are where Tr is the trace over both the representation indices and spatial labels x, y, . . ., and Q is the baryon number charge defined in (3.9). 8The coadjoint orbit we choose corresponds choosing Q = 0.The above constraints can be shown to be invariant under the action of the W 2J+1 ∞ algebra on the M .

3.3
The action for M and the large J saddle point equations Next, we describe an action for the classical field M based on the discussion above.First, the Hamiltonian for M is nothing but the Hamiltonian which we derived from the fermion field theory (3.5) in terms of the bilocal variables: where the trace Tr is in the (x, θ) space, and we have defined S and M that have the matrix elements The 'kinetic' term for the M can be borrowed from the theory of quantization of coadjoint orbits where it is derived from the symplectic structure on the coadjoint orbit.It takes the form i where Σ is the two-dimensional surface with coordinates s and x + , with s a semi-infinite coordinate, −∞ < s ≤ 0, which is customary in the orbit method.The field M is a function of both s and x + with boundary conditions Thus, the total action for the field M is where the Tr is on the x − and θ labels of the field M (3.12).Since the action (3.18) involves a single trace over the θ labels, it is O(J) in the large J limit.Thus, the large J dynamics is described by the saddle points of (3.18).To obtain the saddle point equations, we perform an infinitesimal variation of M along the coadjoint orbit: that is, Under a variation of the above kind, the variation of the action (3.18) is (3.21) Ignoring the partial derivative in time ∂ + , we get the equation of motion Define the Fourier components of M (x − , y − ; θ, θ ′ ) as Here onward, we drop the − in the superscripts of coordinates and subscripts of momenta to avoid clutter.All one dimensional integrals in the equations of motion will refer to positions and momenta in the spatial directions x − , k − .The equation of motion becomes where M (p, r; θ, θ ′ ) = cos(θ − θ ′ )M (p, r; θ, θ ′ ), and we have employed matrix notation for the θ, θ ′ labels but explicitly display the spacetime momentum labels on M .Recall the constraints (3.13) for a coadjoint orbit of the W 2J+1 ∞ algebra.The coadjoint orbit we choose is the one corresponding to Q = 0, i.e., zero U(1) charge.Then, any representative M of the coadjoint orbit has to satisfy M 2 = M and Tr(1 − M ) = 0.The following choice satisfies this condition: See [6] for some motivation for this choice.Fluctuations along the orbit about the classical solution M 0 are parametrized as where the 1/ √ J is present to indicate that we consider fluctuations that are subleading in the large J limit.The fluctuating field W is expected to be the quark-antiquark wavefunction, though one has to take an additional trace over the θ, θ ′ labels to arrive at the gauge invariant wavefunction.We will perform this step at the very end of this analysis in Section 3.4.
In terms of spatial momenta, we have We obtain an action for the fluctuations W by substituting the above ansatz for M and retaining terms with an explicit 1/J in them.Due to the trace Tr in the action (3.18) which is of order J, the action for W will be of order 1.Following [6], we split the momentum components of W into four different fields.Let q and r be positive momenta.Then, let The action for W is computed in detail in Appendix D. The final form of the action for the W modes is where W (s, t; θ, θ ′ ) = W (s, t; θ, θ ′ ) cos(θ − θ ′ ) and tr is the trace over the θ labels.Varying the action w.r.t.W pn (s, t), we get the following equation of motion for W np (s, t): (3.30) Displaying the θ labels explicitly, we get

The 't Hooft equation for the meson wavefunction
Displaying the x + dependence explicitly for the W np (t, s), we define the Fourier transform Also, let us define the centre-of-mass momentum r − and fractional momenta x We write the Fourier mode ϕ(t, s; r + ) = ϕ(r − x, r − (1 − x); r + ) as ϕ(x) for short: Let us also define a new variable of integration: The equation of motion (3.30) can be written in terms of these quantities as Define Φ(x) to be the trace of ϕ(x, θ, θ ′ ) over the θ labels: The above trace is gauge invariant since it originates from the gauge-invariant Wilson line in A − = 0 gauge m ψ m (x)ψ * m (y).Taking the trace over θ,θ ′ in (3.36) we get This is the integral equation for the meson wavefunction obtained by 't Hooft for SU(N ) gauge theory coupled to a fundamental quark in the large N limit.The wavefunctions satisfy the boundary conditions where the exponent β satisfies β < 1 and The above conditions can be obtained by demanding that the right hand side of (3.38) is regular as x → 0 and x → 1 respectively (see Appendix E).
4 The meson spectrum for integer spin J In this section, we analyze the model for integer spin J where the minimal representation of the quarks is the 2J + 1 dimensional real (or Majorana) representation of SU(2).We have described the representation theory of SU(2) in Appendix B and just state the relevant facts here.Irreducible representations of SU(2) with integer spin J are real, and hence fermions in such representations can be taken to be Majorana.The Majorana condition on a Dirac fermion Ψ in an integer spin J representation is where Ψ m are the components in the |J, m⟩ basis.To obtain the results for Majorana fermions, we consider the results for a Dirac fermion in the integer spin J representation and study the consequences of the reality condition (4.1).

Reality condition on the meson wavefunction
Recall from the Dirac fermion analysis in Section 3.4 that the gauge-variant wavefunction ϕ(x, θ, θ ′ ) is simply the 1/J fluctuation of the bilocal field written in momentum space Ψ − (θ, p − )Ψ * − (θ ′ , q − ), with x = p − /(p − + q − ).Let us apply the reality conditions (4.1) on its discrete version Ψ m Since we are interested in the gauge-invariant version Φ(x) = dθ sin θ ϕ(x, θ, θ), we only need the reality properties on the above bilocals with m = n: Recall from (3.33) that x is the fraction of the total momentum p − + q − carried by the first fermion in the bilocal.Thus, when the two momenta are interchanged p − ↔ q − in (4.3), we have x ↔ 1 − x.Thus, the reality condition (4.3) on the bilocal translates to the following on ϕ(x; θ, θ): The reality condition (4.5) on Φ(x) thus dictates that we must retain only those solutions of (4.6) which satisfy (4.5).Let us study the consequences of projecting out the other solutions.The 't Hooft equation (4.6) for the Dirac fermion has the asymptotic solutions indexed by a positive integer k Of the above, only solutions with k even satisfy the reality condition (4.5) and hence only those solutions must be retained.Writing k = 2ℓ, the asymptotic meson spectrum for a Majorana fermion is The asymptotic meson spectrum for a Majorana fermion has double the spacing compared to that of a Dirac fermion.

Zero quark mass
For the case m = 0, 't Hooft [1] has demonstrated that there is a solution of the integral equation (4.6) which has zero eigenvalue M 2 = 0 with eigenfunction Φ 0 (x) = 1.Clearly, this is incompatible with the reality condition (4.5) and is projected out.However, there could be zero mass bound states of the integral equation if we choose boundary conditions compatible with (4.5).Indeed, from the boundary analysis for the m = 0 case (see eqs. (3.39), (3.40) and Appendix E), it is clear that Φ(x) has to approach constants c 0 , c 1 as x → 0 or as x → 1 respectively.To find if such a zero mass bound state exists, we appeal to numerical analysis.Since the boundary conditions (3.39) tells us that for m = 0, the eigenfunction need not vanish at the boundaries, we choose the ansatz Φ(x) = The eigenvalues and eigenvectors of the matrix B are then calculated numerically for increasingly large values of integer K to ensure convergence.We find that the lowest eigenvalue is non-zero and is given by M 2 0 ≈ 1.8722 λ. 9 Therefore, even for m = 0, the spectrum of the integral equation (4.6) satisfying the condition (4.5) is gapped.

Local collective variables
In this section, we analyze the dynamics of the model in terms of bosonic collective variables ρ(θ, x) = ψ * (θ, x)ψ m (θ, x) and σ(θ, x) which are local in the coordinates, as opposed to the bilocal variables M (x, y) ij of the previous sections.The collective field σ acts as an effective electric field for the fermion -we assume the electric field to be constant at the large J saddle point.The fermion determinant is then the standard Euler-Heisenberg lagrangian for a fermion in a constant electric field.We then analyze the large J saddle point equations and compute the spectral function for the gauge-invariant fermion two-point function and observe that there is no pole in the propagator.The lack of a pole in the gauge invariant two-point function signifies that a single fermion is not observed in asymptotic states, thus reinforcing the picture of quark confinement that emerged in the discrete meson spectrum computed in the previous sections.

The effective action in terms of collective variables
In this section, we will work with the usual notion of time as x 0 .The action can be written explicitly in terms of the gauge field as where J µ,a = g ψγ µ T a ψ is the SU(2) Noether current.Integrating out the non-dynamical field A a 0 in the A 1 = 0 gauge, we get an effective action only in terms of the fermions: where we have used the Fourier transform of 1/∂ 2 1 given in (A.5) of Appendix A. Note that the four-fermi term is simply the pairwise Coulomb interaction between charge densities J 0,a (x)|x 1 − y 1 |J 0,a (y).In the large J limit, using the simplifications of Section 2.2, the above action can be written as where λ is the 't Hooft coupling λ = g 2 J 2 , and the notation (ψ † ψ)(θ, x) stands for ψ † (θ, x)ψ(θ, x) with the spinor indices contracted between ψ † and ψ.Next, we rewrite the four-fermi term as a quadratic term in an auxiliary local bosonic variable ρ by inserting 1 in the path integral in terms of auxiliary fields σ and ρ: (5.4) 10 The effective action involving the fermions and auxiliary fields is To handle the 1/∂ 2 1 in the last term above, we further introduce a scalar field ϕ(θ, x) such that ρ(θ, x) = ∂ 1 ϕ(θ, x) .

A large J saddle point
Since the effective action is a single trace over the θ labels, the action is O(J) in the large J limit.Hence, we can use saddle-point approximation to compute the path integral.As usual, we look for translation-invariant saddle points.This corresponds to a constant value for the scalar field ϕ along the spacetime directions (ϕ can still be a non-trivial function of θ).The field σ is not a scalar since it couples to ψ † ψ which is the zeroth component of a vector current.But it may be interpreted as the zeroth component of a fictitious U(1) gauge field A µ , and indeed it appears in that role in the fermion kinetic term in (5.7).The spatial component of the gauge field A 1 is zero, and this can be thought of as a gauge choice for the fictitious gauge field.Since the field strength F µν = −iε µν ∂ 1 σ is a pseudoscalar in 1 + 1 dimensions, we can take it to be constant at our translation-invariant saddle point.Thus, we assume that η = ∂ 1 σ is a constant (along spacetime, but not necessarily along θ).
On this saddle point, the problem becomes that of fermions in a constant (imaginary) electric field background in 1 + 1 dimensions.The effective action for this is well-known: it is the Euler-Heisenberg effective action [24,25], which was re-derived by Schwinger using the proper time method [26] (also see [27,28] for a pedagogical exposition).The result of integrating out the fermions in a constant η background is where we have given the expression for the fermion determinant in d dimensions for the purposes of regularization.We discuss the regularization in detail in Appendix F and only give the regularized final expression here.Partially integrating the ∂ 1 derivative in the σ∂ 1 ϕ term in (5.7), the effective action for η and ϕ becomes (up to some constant terms) The saddle point equations are then (5.10) We have not been able to solve the above equations explicitly.For the next two subsections, we assume that we have a solution for (5.10).

The spectral function
Recall from (5.8) that integrating out the fermions gave the log determinant (5.12) The left-hand side is nothing but the trace (over spacetime, Dirac indices and gauge indices) of the fermion two-point function at coincident points x = y.Since we expect our saddle-point to be unitary and translation invariant, we can write down a Källén-Lehmann representation for the left-hand side so that (5.13) 12 By recasting the left-hand side of the above equation ( 5.13) into a proper-time integral, we can hope to extract ϱ(µ) which contains spectral information of the theory.Introducing the Schwinger proper-time integral for the denominator of the integrand, we get (5.16) In the limit η → 0, we get ϱ(µ) = 2mδ(µ − m 2 ) , (5.17) which is the correct spectral density for a free fermion up to a factor of 2 (see Footnote 12).This factor of 2 is from the integral over θ which is in fact a sum over the number of components 2J + 1 of the fermion in units of 1/J in the large J limit (we refer the reader to the discussion in Section 2.2).We can obtain some partial information about ϱ(µ) by 12 We write this in analogy with the case of a single free Dirac fermion of mass m in d dimensions: for the free fermion.
looking at the late time behaviour t → ∞ of the right hand side.When t → ∞, the coth tη factor goes to 1 so that the right hand side of (5.16) becomes where the constant C is C = π 0 dθ sin θ η(θ).The ϱ(µ) that corresponds to the above Clearly, this does not signify a simple pole in the propagator since it is not a δ-function.
In fact, the full spectral density corresponding to (5.16) can be computed by an inverse Laplace transform.We present the calculation in Appendix G and give the result here: where the derivative on the δ-function is w.r.t.µ.The k = 0 term indeed corresponds to the late time answer (5.19) (after choosing Θ(µ − m 2 )| µ=m 2 = 1 2 ).

Majorana fermions
For Majorana fermions, the same calculation as for the Dirac fermions goes through, but with the density variables ρ(θ, x) satisfying a constraint that arises from the Majorana condition.Recall that, for a Majorana fermion Ψ(θ, x), the density variable is ρ(θ, x) = Ψ † (θ, x)Ψ(θ, x).Using the same manipulations as in (4.3) in Section 4, we see that ρ(θ, x) must satisfy ρ(π − θ, x) = −ρ(θ, x) . (5.21) In particular, this implies the boundary condition ρ(θ = π/2, x) = 0.This corresponds to the fact that the component Ψ 0 corresponding to the SU(2) representation index n = 0 of the fermion is real or imaginary (depending on J being even or odd) so that Ψ † 0 Ψ 0 is identically zero.Without loss of generality, we can assume that the field σ(θ, x) also satisfies the same condition (5.21).The rest of the analysis is the same as for the Dirac fermion in an integer spin J representation.

Discussion
In this work, we have analyzed SU(2) gauge theory coupled to a single massive quark in the spin J representation of the gauge group.In a 't Hooft-like large J limit where we keep λ = g 2 J 2 fixed as J is taken large, we have obtained an integral equation for the gauge invariant meson wavefunctions.This integral equation is the same for the two cases of Dirac fermions for arbitrary J and Majorana fermions for integer J.However, in the case of Majorana fermions for integer J, the solutions Φ(x) of the integral equation have to be odd under x → 1 − x as a consequence of the Majorana reality condition.This retains only half of the spectrum of the integral equation.For the case of zero quark mass m = 0, this projection removes the zero-mass bound state solution of the integral equation for meson wavefunctions, resulting in a gap in the meson spectrum.
To the extent that we can compare with results at finite J (in a systematic expansion in 1/J by retaining subleading terms in the effective action), the above results match with well-known facts about the the SU(2) theory with quarks in the adjoint representation: (1) the model with massive quarks has a completely discrete spectrum due to the linear confining potential [11,[29][30][31][32][33][34][35][36][37], and (2) the model with massless quarks does not have a zero-mass bound state.However, it has been argued in [38,39] that the SU(2) theory with massless spin J quarks for any J > 2 is gapless: there is a massless state in the spectrum of the Hamiltonian which is created by the operator T − T ′ where T and T ′ are the stress tensors of the SO(2J + 1) 1 and the SU(2) I(J) WZW models respectively, where I(J) = J(J + 1)(2J + 1)/3 being the index of the spin J representation.This is not in contradiction with our results in the large J limit since our method diagonalizes the Hamiltonian only in the two fermion sector (in other words, the integral equation is for the meson wavefunctions only) whereas the massless state involves contributions from four-fermion states as well since the stress tensor T ′ for the SU(2) I(J) model involves four fermions (as is clear from the Sugawara form of the stress tensor).It is plausible that such massless states may indeed be observed if one is able to extend the methods of this paper to include states with higher numbers of fermions.
The SU(2) theory coupled to an adjoint quark is known to have a screening vs. confinement transition at zero quark mass.Does this transition persist for larger integer J? For J = 1, the adjoint quark is certainly not capable of screening external charges in the fundamental representation.However, it was shown in [33] that there exist certain topological excitations at m = 0 which transform in the fundamental representation of SU(2) and hence are capable of screening fundamental external charges.The same was argued using mixed 't Hooft anomalies between the 0-form Z 2 chiral symmetry and the 1-form Z 2 centre symmetry [36], and non-invertible topological line operators for the SU(N ) adjoint theory [37].It would be interesting to see if such an analysis can be performed for spin J massless quarks in SU(2) 1+1 QCD.
Our method of calculation can be generalized to SU(N ) QCD with quarks in representations having large representation weights (J 1 , . . ., J N −1 ).Recall that the discrete representation index m = −J, . . ., J for the spin J representation was the eigenvalue of the Cartan generator T 3 , and it was replaced by a continuum angle θ ∈ [0, π].This suggests that, in the large weight limit J i → ∞ of SU(N ) representations, we should associate one continuum angle for each Cartan generator of SU(N ), giving a total of N − 1 angles.It would also be interesting to explore the double limit where J i → ∞ and N → ∞ with some ratios of J i and N fixed in the limit.

B Representations of SU(2) and Majorana fermions
Recall that an irreducible representation of SU( 2) is labelled by a positive number J which is either an integer or a half-integer which we call the 'spin'.This terminology should cause no confusion since there is no other notion of spin in 1 + 1 dimensions, e.g., from the representations of the Lorentz group.The spin J representation is simply described in the so-called J-m basis which consists of the basis kets |J, m⟩ , m = −J, −J + 1, . . ., J − 1, J . (B.1) We write the generators of the Lie algebra of SU(2) as T + , T − and T a and they satisfy the Lie algebra [T + , T − ] = 2T 3 , [T 3 , T ± ] = ±T ± .Their matrix elements in the above basis are There is a concrete realization of the above basis in terms of degree 2J polynomials in one complex variable ζ 3) The generators act on the polynomials as A general element of the representation is then a general degree 2J polynomial: In the polynomial basis, there is a convenient definition of the complex conjugate representation which allows us to concretely understand the reality properties of the various representations of SU(2).The complex conjugation consists of three steps: For instance, the above steps applied to ψ(ζ) in (B.5) gives Note that the end result is again a polynomial of degree 2J and the transformation is Let us see what happens if we apply the above operation twice: The final expression ψ * * (ζ) is simply Observe that when J is half-integer (resp.integer), the 'complex conjugation' operation squares to −1 (resp.+1).Thus, when J is an integer, we have a legitimate complex conjugation operation which squares to +1.We can use this to impose a reality condition on the representation: For components, the reality condition can be obtained by comparing the same powers of ζ in (B.5) and (B.7): Since the complex conjugation operation squares to +1, it has two eigenspaces corresponding to the eigenvalues ±1.In the reality conditions above, we have projected down to the +1 eigenspace.We could also project to the −1 eigenspace which corresponds to These two conditions (B.11) and (B.12) split the complex representation m ψ (m) ζ J+m (where J is integer), into a real and an imaginary part which transform separately as irreducible representations of SU(2).
Suppose we have a Dirac spinor in 1 + 1 dimensions which transforms in the spin J representation of SU(2) with J integer.Then, the Majorana conjugation operation and the above complex conjugation operation (B.11) for integer J can be simultaneously applied.Thus, a Dirac spinor ψ transforming in the spin J representation of SU(2) with J integer splits into two Majorana spinors Ψ 1 and Ψ 2 transforming in the spin J (integer) representation.The Majorana condition in our choice of γ-matrices for an SU(2) inert Dirac spinor Ψ is (see Appendix A) where Ψ ± are the spinor components of Ψ.For a Dirac spinor in a spin J representation with J integer, we combine the Majorana condition with the reality condition (B.11 We have dropped the parentheses around the indices m in Ψ m ± since there is no danger of confusing them with exponents as in ζ J+m . We make one final comment about half-integer J representations.Since the 'complex conjugation' operation (B.7) squares to −1 in this case (see (B.9)), imposing a reality condition like (B.11) kills the representation.However, if we have two instances of a half-integer spin representation, then a reality condition mixing the two can be imposed.Suppose the representations are Ψ Am with A = 1, 2. Then, the new reality condition uses the totally antisymmetric tensor on two indices ε AB : It is easy to see that the above operation squares to identity rather than minus identity: where we have used ε AB ε BC = −δ A C .The new reality condition can be adopted for a pair of Dirac spinors as In this paper, we work with a single fermion in the spin J representation of the SU(2) gauge group.When J is an integer, we can take the fermion to be either a Majorana spinor or a Dirac spinor.When J is a half-integer, we must take the fermion to be a complex Dirac spinor since the new reality condition (B.15)only applies to two fermions (or, in general, any even number of fermions).

∞
In this section, we give a brief overview of the coadjoint orbit method of Kirillov [40,41] as applied to the W 2J+1 ∞ = W ∞ ⊗ U(2J + 1) algebra [6][7][8][9]21].We set N = 2J + 1 for convenience.The algebra W N ∞ can be described by generators g(α, where x and p are the position and momentum operators on the single particle Hilbert space and E ij are the elementary matrices which generate the Lie algebra of U(N ).Note that the g(α, β) are elements of the Heisenberg-Weyl group and satisfy the group multiplication The generators (C.1) satisfy the commutation relations which define the W N ∞ algebra: An element Θ of the W N ∞ algebra can be expanded in the above basis as For our current situation, we take the dual space Γ of the W N ∞ algebra to be a copy of the underlying vector space of the W N ∞ algebra itself.Consequently, an element ϕ of Γ can also be expanded in the above basis of generators as ϕ = dαdβ ϕ(α, β) ij g(α, β) ij . (C.5) The pairing between the Lie algebra and its dual space is then The adjoint action of the W N ∞ algebra is given by the commutator: 14 The coadjoint action is then inferred from the inner product (C.6) as According to the theory of quantization of coadjoint orbits [41][42][43], representations of the W N ∞ algebra can be obtained from appropriate orbits in the dual space Γ of the coadjoint action of W N ∞ .One interprets a coadjoint orbit as the classical phase space of a dynamical system and supplies a symplectic form on the phase space that arises from the Lie bracket of the Lie algebra.One then quantizes the phase space by any method that is admissible in the context.In the current paper, we employ path integral quantization of coadjoint orbits [44,45].The application of the path integral method to the current context was worked out in detail in [6][7][8][9]21].
A typical coadjoint orbit of a Lie algebra is specified by the values of the Casimir operators of the Lie algebra which are invariant under the coadjoint action.In the present problem, the information about all the Casimirs can be fixed in terms of two constraints involving the coadjoint field ϕ: where Q is the conserved charge corresponding to the U(1) ⊂ U(N ) symmetry.If the entire U(N ) group arises as a gauge invariance of a model, then we must set Q = 0 to focus on gauge invariant configurations.If the gauge invariance is SU(N ), then Q is the baryon number: different choices of Q corresponds to sectors with different baryon number.In the current paper, we look at the sector with no baryons i.e.Q = 0.The path integral quantization of the above coadjoint orbit proceeds as follows.The symplectic form Ω at the point ϕ on the coadjoint orbit is described by its contraction with two tangent vectors δ 1 ϕ, δ 2 ϕ at the point ϕ: where ℓ 1 and ℓ 2 are the W N ∞ algebra elements whose action on the coadjoint orbit generate the tangent vectors δ 1 ϕ and δ 2 ϕ respectively.The action can then be written as follows.Let us consider a two dimensional region spanned by the time variable t (which is x + in the current situation) and an auxiliary variable s and let us take the field ϕ to be a function of both t and s with the boundary condition Then we can take the two tangent vectors at ϕ to be ∂ t ϕ and ∂ s ϕ.The corresponding W N ∞ elements ℓ t and ℓ s are defined as It is easy to check that the following satisfy the above equations by repeatedly using the defining equation ϕ 2 = ϕ of the coadjoint orbit and its derivatives w.r.t.t and s: The action for the coadjoint orbit is then where we have included a Hamiltonian term as well; the last equality is again achieved by plugging in (C.13) and using ϕ 2 = ϕ repeatedly. 15 We now make contact with the problem considered in this paper.Consider the bilocal operator M (x, y) ij = ψ i (x)ψ * j (y) defined in the main text (3.4) (for the sake of brevity, we replace the continuum U(2J + 1) labels θ,θ ′ by their discrete counterparts i,j, and suppress the dependence on time x + , and the minus signs on x − and y − ).Write this as the matrix element of an operator M between two states |x⟩, |y⟩ in the single particle Hilbert space, M (x, y) = ⟨x|M |y⟩.Then, M can be thought of as an element of the W N ∞ algebra (recall N = 2J + 1) and can be decomposed in the g(α, β) ij basis as The above relations (C.15),(C.16)follow from the formula Recall the commutation relations of the M (x, y) ij from the main text (3.11): It is then straightforward to check that the W (α, β) ij satisfy the defining commutation relations of the W N ∞ algebra: Recall from the main text that the bilocal operators satisfy the constraint (3.8): 15 Compare this with the standard way in which the action C pdq arises in classical mechanics from the integral of the symplectic form over a two dimensional region Σ in phase space with C = ∂Σ being the boundary of Σ: where Q is the conserved global U(1) charge operator (baryon number operator) (3.9): We can take the expectation value of the above equations in states of the fermion Hilbert space corresponding to zero baryon number Q = 0.These states form an orbit under the action of the group GW N ∞ corresponding the W N ∞ algebra, which consists of the elements for all sufficiently well-behaved functions ϕ(α, β), (C.22) where |0⟩ is the fermionic vacuum.The above are the generalized coherent states according to Perelomov [20] and they are elements of the coset space GW N ∞ /H where H is the subgroup that preserves |0⟩ [7].But coadjoint orbits are also precisely of the form GW N ∞ /H, and thus, the states |ϕ⟩ are in one-to-one correspondence with points on a coadjoint orbit.Clearly, the expectation value M ϕ = ⟨ϕ|M |ϕ⟩ of the fermi bilocal operator M in these coherent states |ϕ⟩ satisfy the constraints where we have used the factorization property of coherent state expectation values in the classical limit ⟨ϕ|M 2 |ϕ⟩ = (M ϕ ) 2 + O(ℏ) which becomes exact in the limit ℏ → 0. These equations precisely coincide with the coadjoint orbit of W N ∞ considered previously in (C.9) and consequently, the classical phase space of the fermionic field theory is the coadjoint orbit (C.9) of the W N ∞ algebra.Thus, the above procedure results in a bosonization of the fermionic theory in terms of the bosonic bilocal field M ϕ .Further, by virtue of the M ϕ field being an element of a W N ∞ coadjoint orbit, there is also a ready-made action, viz., (C.14), that governs its dynamics.This is the starting point of Section 3.3.In the main text, we do not display the subscript ϕ on M ϕ for the sake of brevity.

D Derivation of the action for the fluctuations W
We reproduce the action (3.18) for the classical field M here for convenience: where the tr is over the θ and x − space labels.The fluctuations about the saddle point solution M 0 are described by the momentum space expression (3.26): Writing M = M 0 + δM with δM containing the W -dependent terms in (3.27), the kinetic term in (3.18) becomes We drop the ∂ + term since the fluctuation is assumed to die away at temporal infinity.Then, we get where an integral with a subscript like p without an explicitly displayed measure is shorthand for ∞ −∞ dp 2π .Finally, relabelling q → −q, we get tr (W (p, −q)∂ + W (−q, p)) .(D.5) Using the notation, the kinetic term can be written as The Hamiltonian part of the action in momentum space is where the traces are in the θ space and M (q−k, p−k; θ, θ ′ ) = M (q−k, p−k; θ, θ ′ ) cos(θ−θ ′ ).The mass term gets the leading contribution from the 1 J W W term in (3.27) because the 1 √ J W term evaluates to zero.Thus, The interaction term has three contributions at order 1 J , schematically of the form where X(θ, θ ′ ) corresponding to any bilocal field The first two terms are equal and combine to give M 0 (q − k, p − k; θ, θ ′ ) cos(θ − θ ′ )W (p, r; θ ′ , φ)W (r, q; φ, θ) , (D.11) Plugging in M 0 (q − k, p − k; θ, θ ′ ) = 2πΘ(q − k)δ(q − p) δ(θ−θ ′ ) sin θ , we get The above expression is divergent from the t ∼ 0 (ultraviolet) region of the integrand and has to be regularized.Following Dittrich [47], we use the following formula from Gradshteyn and Ryzhik [48, 3.551, no. 3]: We can Taylor-expand the remaining factors in ϵ to get the behaviour near the pole: where, in going to the second equality, we have used where γ is a positive real number, perhaps infinitesimally close to zero.Recall that η is a function of θ.We can close the contour by a large semicircle in the left-half plane Re(t) < γ, provided the integrand is suppressed as the radius of the semicircle goes to infinity.In the left-half plane, we have The t coth tη factor is coth γη + Rη(cos φ + i sin φ) , (G.3) which is bounded as R → ∞.The remaining factors e t(µ−m 2 ) t become e γ(µ−m 2 ) e R cos φ(µ−m 2 ) e iR sin φ(µ−m 2 ) Rη(cos φ + i sin φ) . (G.4) The exponential factor e R cos φ(µ−m 2 ) decays as R → ∞ provided µ > m 2 since cos φ < 0. Thus, for µ > m 2 , the contour in the integral (G.(G.12)

Contents 1 B1
Introduction and summary 2 SU(2) gauge theory with spin J matter 2.1 The theory 2.2 The large J limit 2.3 The continuum field ψ − (θ, x) 2.4 A derivative along the θ direction 2.5 The Hamiltonian in the large J limit 3 Mesons and W ∞ coadjoint orbits 3.1 A 't Hooft-like limit 3.2 The W ∞ algebra and coadjoint orbits 3.3 The action for M and the large J saddle point equations 3.4 The 't Hooft equation for the meson wavefunction 4 The meson spectrum for integer spin J 4.1 Reality condition on the meson wavefunction 4.2 Zero quark mass 5 Local collective variables 5.1 The effective action in terms of collective variables 5.2 A large J saddle point 5Representations of SU(2) and Majorana fermions C Coadjoint orbits of W 2J+1 ∞ D Derivation of the action for the fluctuations W E Boundary condition analysis for the 't Hooft integral equation F Dimensional regularization of the Euler-Heisenberg action G Spectral density for m ̸ = 0 Introduction and summary

bB kn b n , ( 4
n cos (2n − 1)πx .(4.9)This forms a complete basis for functions satisfying the reality condition (4.5) on the interval [0, 1].Plugging this ansatz in the 't Hooft equation (4.6) and exploiting the orthogonality of cosines, we get a matrix equation for the coefficients b n M 2 b k = n x) 2 cos (2n − 1)πy .(4.11)