Exact Symmetries and Threshold States in Two-Dimensional Models for QCD
PPUPT-2623
Exact Symmetries and Threshold Statesin Two-Dimensional Models for QCD
Ross Dempsey, Igor R. Klebanov, , and Silviu S. Pufu Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544
Abstract
Two-dimensional SU( N ) gauge theory coupled to a Majorana fermion in the adjointrepresentation is a nice toy model for higher-dimensional gauge dynamics. It possesses amultitude of “gluinoball” bound states whose spectrum has been studied using numericaldiagonalizations of the light-cone Hamiltonian. We extend this model by coupling it to N f flavors of fundamental Dirac fermions (quarks). The extended model also contains meson-like bound states, both bosonic and fermionic, which in the large- N limit decouple fromthe gluinoballs. We study the large- N meson spectrum using the Discretized Light-ConeQuantization (DLCQ). When all the fermions are massless, we exhibit an exact osp (1 | N ) current. We also present strong numericalevidence that additional threshold states appear in the continuum limit. Finally, we makethe quarks massive while keeping the adjoint fermion massless. In this case too, we observesome exact degeneracies that show that the spectrum of mesons becomes continuous abovea certain threshold. This demonstrates quantitatively that the fundamental string tensionvanishes in the massless adjoint QCD .January 2021 a r X i v : . [ h e p - t h ] J a n ontents N when N f = 0 . . . . . . . . . . . . . . . . . 164.3 Gluinoball degeneracies at large N when N f = 0 . . . . . . . . . . . . . . . . 184.4 Current blocks and degeneracies at finite N for N f > N for N f > N osp (1 |
4) algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.2 osp (1 |
4) representations of P − eigenstates . . . . . . . . . . . . . . . . . . . 42 . . . . . . 48 Introduction and Summary
Soon after the emergence of Quantum Chromodynamics as the SU(3) Yang-Mills theory ofstrong interactions [1–3], ’t Hooft introduced its generalization to gauge group SU( N ) andthe large N limit where g N is held fixed [4]. To demonstrate the power of this approach,he applied it to the theory now known as the ’t Hooft model [5]: the 1+1 dimensional SU( N )gauge theory coupled to massive Dirac fermions in the fundamental representation. Usinglight-cone quantization, he derived an equation for the meson spectrum. This equation isexact in the large N limit. A crucial simplification in this model is that the meson light-conewave functions describe the bound states of only two quanta, a quark and an antiquark.It is interesting to generalize the ’t Hooft model by replacing the fermions in the funda-mental representation by those in two-index representations; such 1 + 1 dimensional modelscontain discrete analogues of θ -vacua [6]. A minimal model of this type, the SU( N ) gaugetheory coupled to one adjoint Majorana fermion of mass m adj [7] has turned out to be aninteresting playground for studying various non-perturbative phenomena in gauge theory.Instead of the meson spectrum, its large N spectrum consists of glueball-like bound statesthat may be viewed as closed strings. Since the adjoint fermion is akin to a gluino, we willrefer to them as gluinoballs. They are bosonic when the number of adjoint quanta is evenand fermionic when it is odd. The light-cone bound state equations are not separable sincethey involve superpositions of states with different numbers of quanta. Nevertheless, somefeatures of the spectra can be studied with good precision using the Discretized Light-ConeQuantization (DLCQ) [11–14], where one of the light-cone coordinates, which we take tobe x − , is formally compactified on a circle of radius L , thus identifying x − ∼ x − + 2 πL .As m adj →
0, all the bound states of the theory remain massive [7, 15, 16], but the vacuaof the model have interesting topological properties [6, 17–23]. They are described by thetopological coset model
SO( N − SU( N ) N which has a vanishing central charge. Thus, for any N , theSU( N ) gauge theory coupled to a massless adjoint fermion serves as a non-trivial exampleof a gapped topological phase.In another sign of interesting physics, it was argued in [20] that as m adj → N ). In recent literature there have been renewed discussionsof the m adj → N discrete θ -vacua [6], the model possesses a large It is also possible to study models with adjoint scalars [7,8] and supersymmetric models containing bothadjoint scalars and fermions [9, 10]. N . Thishas led to new arguments for the screening of the massless adjoint model. In Section 8 wewill provide additional quantitative evidence for the screening behavior of the theory with m adj = 0.As noted in [19], the massless limit of adjoint QCD exhibits an important simplificationbecause the DLCQ Hilbert space breaks up into separate Kac-Moody current algebra blocks[24]. At large N , this separation of the DLCQ spectrum leads to exact degeneracies betweenthe value of P − for certain single-trace states at resolution K and sums of the values of P − forcertain fermionic single-trace states at resolutions m and K − m [25]. As a result, some single-particle states may be interpreted as threshold bound states. It therefore appears that thespectrum of the massless large N model is built of a sequence of basic states, sometimes called“single-particle states,” which were investigated in [19, 25–28] using DLCQ. In Section 4 wewill review the implications of the Kac-Moody structure of the massless model [19] andexamine the exact DLCQ degeneracies more fully. The massless model was also studiednumerically using a conformal truncation approach (for a review, see [29]), which providesanother basis for light-cone wave functions [30]. While the model with a single adjoint Majorana fermion contains rich physics, it isof obvious interest to consider its generalizations. For example, the SU( N ) gauge theorycoupled to two adjoint Majorana fermions, or equivalently one Dirac adjoint, was studiedin [33]. As m adj →
0, this model is not gapped but is rather described by a gauged
SO(2 N − SU( N ) N Wess-Zumino-Witten (WZW) model with central charge N − . Surprisingly, this CFT turnsout to have N = 2 supersymmetry [33–35]. The U(1) R symmetry of this CFT is traced backto the U(1) phase symmetry of the adjoint Dirac fermion.In this paper we study another extension of the ’t Hooft model by coupling the 1 + 1dimensional SU( N ) gauge theory to N f fundamental Dirac fermions (quarks) q α of mass m fund and an adjoint Majorana fermion Ψ of mass m adj : S = (cid:90) d x tr (cid:18) − g F µν F µν + i /D Ψ − m adj (cid:19) + N f (cid:88) α =1 (cid:0) iq α /Dq α − m fund q α q α (cid:1) . (1.1)Then, similarly to the higher-dimensional large N QCD, the spectrum contains both theglueball-like bound states akin to closed strings, and the meson-like bound states akin to Let us also mention that for m = g Nπ the model becomes supersymmetric [15, 16, 31]. This leads tovery interesting effects in non-trivial θ -vacua, where the supersymmetry is broken [32]. The N f meson states transform in the adjoint representation of the U( N f )global symmetry, while the gluinoballs are U( N f ) singlets. We will consider the large N limit where N f is held fixed. The structure of the mesons is much more intricate than inthe original ’t Hooft model [5] because there can be an arbitrary number of adjoint quantaforming a string which connects the quark and anti-quark at its endpoints. When thisnumber is even, the mesons are bosonic, while when it is odd they are fermionic. Far inthe IR, the mesonic sector of the massless theory is described by a gauged
SO( N − NN f ) SU( N ) N + Nf WZW model with central charge N f (3 N +2 N f N +1)2(2 N + N f ) . Therefore, in contrast to the pure adjointmodel where the corresponding central charge vanishes, the IR limit of the theory (1.1) iscomplicated and dynamical.The bound state equations in the mesonic sector involve mixing of states with differentnumbers of quanta. As a result, the problem of determining the meson masses is muchmore complicated than in [5]. When we apply the DLCQ to the mesonic sector, we findnew surprises in the limit where both m adj and m fund are sent to zero. The values of P − for certain massive meson states does not change as the resolution parameter is increasedand is the same as the sum of the values of P − for one or more fermionic gluinoball states.We trace this exact result to the osp (1 |
4) symmetry of the DLCQ system with antiperiodicboundary conditions around the circle in x − direction. This symmetry helps us prove thatin the continuum limit, there is a large amount of degeneracy in the spectrum of the mesonstates at the same value of masses as possessed by certain gluinoball states; the first ofthem occurs at M ≈ . g Nπ . This fascinating structure of the spectrum is similar to thethreshold bound states in the pure gluinoball sector. Some of new degeneracies we observe(see Figures 3 and 4) suggest that the meson states with growing multiplicities may bethought of as threshold bound states of the singlet gluinoball of mass M and states fromthe CFT sector. After we turn on the quark mass m fund , we continue to find that somemeson states are threshold bound states of gluinoballs and lighter mesons; see Section 8.This structure of some meson states in our model (1.1) is also reminiscent of the exotic,possibly molecular, XY Z mesons observed in the real world (for reviews, see [37, 38]). Inparticular, the mass of the X (3872) charmonium state is extremely close to the sum of themasses of D and D ∗ mesons. Clearly, the intricate and surprising structure of the boundstate spectrum in model (1.1) deserves further studies and a deeper understanding. There are also baryons which consist of N quarks and can include some number of adjoint quanta. Thebaryon masses grow as N in the large N limit [36]. We will not discuss them further in this paper. The large N limit of meson masses does not depend on N f , and in some parts of the paper we willrestrict to N f = 1. P − and the spectrum of masses for both gluinoballs and mesons at leading orderin the large N limit when m adj = m fund = 0. Section 4 contains a review of the Kac-Moodyapproach developed in [19] as well as its implications for the model we study. As we will show,many of the degeneracies we observe can be anticipated from this approach, and many areexpected to survive at finite N . In Section 5 we present the numerical gluinoball spectrumobtained using DLCQ and show that introduction of the adjoint mass lifts the degeneracies.In Section 6 we similarly present the numerical meson spectrum. In Section 7 we explainmany of the degeneracies in the P − meson spectrum by constructing the symmetry generatorsthat form the osp (1 |
4) algebra. Lastly, in Section 8 we examine how the spectrum changeswhen we introduce non-zero fundamental mass m fund and show that some of the degeneraciesare not lifted. We provide additional evidence for the screening behavior of the theory withmassless adjoints. It follows the approach in [19, 20] and relies on the quantitative propertiesof meson spectra in theory (1.1) with massive quarks. We end with a discussion of our resultsin Section 9. When working with fermions in 1 + 1 dimensions, we will use the gamma matrices γ = σ , γ = iσ , obeying the Clifford algebra { γ µ , γ ν } = 2 η µν with η µν = diag {− , } . The actionfor our model for an SU( N ) gauge theory with gauge field A µ as well as an adjoint Majoranafermion Ψ and fundamental Dirac fermions q α was given in (1.1), where the SU( N ) gaugecovariant derivative acts as D µ Ψ ij = ∂ µ Ψ ij + i [ A µ , Ψ] ij , D µ q iα = ∂ µ q iα + i ( A µ ) ij q jα , (2.1)where ( A µ ) ij is hermitian and traceless.To study this action in light-cone quantization, we define the light-cone coordinates x ± =( x ± x ) / √ A ± = ( A ± A ) / √
2, while for the fermions we define Ψ ij = 2 − / (cid:32) ψ ij χ ij (cid:33) q iα = 2 − / (cid:32) v iα φ iα (cid:33) . In the gauge A − = 0, the action (1.1) takes the form S = (cid:90) d x (cid:34) tr (cid:18) g ( ∂ − A + ) + 12 iψ∂ + ψ + 12 iχ∂ − χ + A + J + − i √ m adj χψ (cid:19) + iv † αk ∂ + v kα + iφ † αk ∂ − φ k − i √ m fund (cid:16) φ † αk v kα + φ kα v † αk (cid:17) (cid:35) , (2.2)where J + = J +adj + J +fund is the right-moving component of the gauged SU( N ) current, withcontributions from the adjoint and fundamental fermions given by ( J +adj ) ij ≡ ψ ik ψ kj − N δ ij ψ kl ψ lk , ( J +fund ) ij = v iα v † αj − N δ ij v kα v † αk . (2.3)If we treat x + as the time coordinate, then we see that A + , χ , φ and φ † are non-dynamicaland can be eliminated using their equations of motion, which are J + = 1 g ∂ − A + , ∂ − χ = m adj √ ψ , ∂ − φ = m fund √ v , ∂ − φ † = m fund √ v † . (2.4)Eliminating each of the non-dynamical fields, we can write (2.2) as S = (cid:90) d x (cid:20) tr (cid:18) g J + ∂ − J + + i ψ∂ + ψ + im ψ ∂ − ψ (cid:19) + iv † αk ∂ + v kα + im v † αk ∂ − v kα (cid:21) . (2.5)From this expression, it follows that the light-cone momentum operators are P + = (cid:90) dx − (cid:2)
12 tr ( iψ∂ − ψ ) + iv † αk ∂ − v kα (cid:3) ,P − = (cid:90) dx − (cid:20) − tr (cid:18) g J + ∂ − J + + im ψ ∂ − ψ (cid:19) − im v † αk ∂ − v kα (cid:21) . (2.6)The light-cone momenta P + and P − commute and can be simultaneously diagonalized. Tofind the spectrum of the theory, we will be interested in finding the eigenvalues of the masssquared operator M = 2 P + P − . Equivalently, we can also write the adjoint contribution more simply as the normal-ordered expression J + ij, adj =: ψ ik ψ kj :. However, if we subtract the SU( N ) trace, as in (2.3), then ( J +adj ) ij is manifestly an SU( N )adjoint and we do not have to worry about the normal ordering prescription. Indeed, the normal orderingambiguity is a c -number, which represents a mixing of the current with the identity operator. Since theidentity operator and the current transform in different representations of SU( N ), there can be no mixing.
6n addition, both P + and P − commute with the charge conjugation operator C definedby C ψ ij C − = ψ ji , C v kα C − = v † αk . (2.7)The operator C generates a Z symmetry which we sometimes use to classify states. One can analyze this theory in canonical quantization. The canonical anti-commutationrelations are { ψ ij ( x − ) , ψ kl ( y − ) } = δ ( x − − y − ) (cid:18) δ il δ kj − N δ ij δ kl (cid:19) , { v † αi ( x − ) , v jβ ( y − ) } = δ ( x − − y − ) δ ji δ αβ . (2.8)To construct the states, we first expand ψ and v in Fourier modes in the x − direction ψ ij ( x − ) = 1 √ π (cid:90) ∞ dk + (cid:16) b ij ( k + ) e − ik + x − + b † ji ( k + ) e ik + x − (cid:17) ,v iα ( x − ) = 1 √ π (cid:90) ∞ dk + (cid:16) d iα ( k + ) e − ik + x − + c † αi ( k + ) e ik + x − (cid:17) , (2.9)and then one can check that the canonical commutation relations (2.8) imply (cid:110) c † αi ( k ) , c βj ( k (cid:48) ) (cid:111) = δ ij δ αβ δ ( k − k (cid:48) ) , (cid:110) d † iα ( k ) , d jβ ( k (cid:48) ) (cid:111) = δ ij δ αβ δ ( k − k (cid:48) ) , (cid:110) b † ij ( k ) , b kl ( k (cid:48) ) (cid:111) = δ ( k − k (cid:48) ) (cid:18) δ ik δ jl − N δ ij δ kl (cid:19) , (2.10)where we have omitted the superscript + on the momenta. We treat c αi ( k ), d iα ( k ), and b ij ( k )as annihilation operators and c † αi ( k ), d † iα ( k ), and b † ji ( k ) as creation operators, and we assumethat all annihilation operators annihilate the vacuum | (cid:105) . The space of states is obtained byacting with creation operators on this vacuum. The charge conjugation operator acts on thecreation operators according to C b † ij ( k ) C − = b † ji ( k ) , C c † αi ( k ) C − = d † iα ( k ) , C d † iα ( k ) C − = c † αi ( k ) . (2.11)At leading order in large N , the single-trace states decouple, in the sense that the matrixelements of M = 2 P + P − between single-trace and multi-trace states are suppressed in 1 /N .7ur main interest here will be in the single-trace mesonic states. For a fixed P + componentof the momentum, the most general such single-trace mesonic state can be written as |{ g } αβ ; P + (cid:105) = ( P + ) ( n − / N ( n − / (cid:88) n (cid:90) dx · · · dx n δ (cid:32) n (cid:88) i =1 x i − (cid:33) × g n ( x , . . . , x n ) c † α ( k ) b † ( k ) · · · b † ( k n − ) d † β ( k n ) | (cid:105) , (2.12)where x i = k i /P + are the momentum fractions, g n is the n -bit component of the wavefunc-tion, and { g } denotes the set of all g n with n ≥
2. The overall powers of P + and N are suchthat, at leading order in 1 /N , the inner product on the states (2.12) takes the form (cid:104){ g (cid:48) } αβ ; P (cid:48) + |{ g } γδ ; P + (cid:105) = (cid:88) n (cid:90) dx · · · dx n g (cid:48) n ( x , . . . , x n ) ∗ g n ( x , . . . , x n ) × δ ( P (cid:48) + − P + ) δ αγ δ βδ , (2.13)and thus is of order N at large N . The analogous single-trace gluinoball states previouslystudied in [16] are of the form |{ f } ; P + (cid:105) = (cid:88) n ( P + ) ( n − / N n/ (cid:90) dx · · · dx n δ (cid:32) n (cid:88) i =1 x i − (cid:33) × f n ( x , . . . , x n ) tr (cid:2) b † ( k ) · · · b † ( k n ) (cid:3) | (cid:105) , (2.14)where now, due to the cyclic property of the trace, the n -bit component functions f n ( x , . . . , x n )obey f n ( x , . . . , x n , x ) = ( − n − f n ( x , . . . , x n ). At leading order in 1 /N , the inner producton these states takes the form (cid:104){ f (cid:48) } ; P (cid:48) + |{ f } ; P + (cid:105) = δ ( P (cid:48) + − P + ) (cid:88) n n (cid:90) dx · · · dx n f (cid:48) n ( x , . . . , x n ) ∗ f n ( x , . . . , x n ) . (2.15)Note that a different power of N is needed in (2.12) and (2.14) in order to achieve an innerproduct that does not scale with N . The single-trace meson and gluionoball states in (2.12)and (2.14) are orthogonal. 8 .3 Light-cone momentum in canonical quantization Inserting the mode expansions (2.9) into the expression for P + in (2.6) gives P + = (cid:90) ∞ dk k (cid:16) b † ij ( k ) b ij ( k ) + c † αi ( k ) c αi ( k ) + d † iα ( k ) d iα ( k ) (cid:17) . (2.16)The expression for P − is more complicated since it involves mass terms that are quadraticin the creation operators as well as interaction terms arising from the first term in (2.6). Wethus split up P − as P − = P − mass + P − int (2.17)where P − mass = m (cid:90) ∞ dk k b † ij ( k ) b ij ( k ) + m (cid:90) ∞ dk k (cid:16) c † αi ( k ) c αi ( k ) + d † iα ( k ) d iα ( k ) (cid:17) , (2.18)and P − int = − g (cid:90) dx − tr (cid:18) J + ∂ − J + (cid:19) . (2.19)To evaluate this term, note that the current is J + ij ( x − ) = 12 π (cid:90) ∞ dk dk (cid:48) (cid:34) (cid:16) b ik ( k ) e − ikx − + b † ki ( k ) e ikx − (cid:17) (cid:16) b kj ( k (cid:48) ) e − ik (cid:48) x − + b † jk ( k (cid:48) ) e ik (cid:48) x − (cid:17) + (cid:16) d iα ( k ) e − ikx − + c † αi ( k ) e ikx − (cid:17) (cid:16) d † jα ( k (cid:48) ) e ik (cid:48) x − + c αj ( k (cid:48) ) e − ik (cid:48) x − (cid:17) (cid:35) . (2.20)9hen this is expanded out, J + ij has 8 terms, and so tr (cid:16) J + 1 ∂ − J + (cid:17) has 64 terms. A carefulcalculation gives P − int = g π (cid:90) ∞ d(cid:126)k (cid:20) δ ( k + k − k − k ) (cid:18) k − k ) − k + k ) (cid:19) b † ik ( k ) b † kj ( k ) b il ( k ) b lj ( k )+ δ ( k + k + k − k ) (cid:18) k + k ) − k + k ) (cid:19) b † ik ( k ) b † kl ( k ) b † lj ( k ) b ij ( k )+ δ ( k + k + k − k ) (cid:18) k + k ) − k + k ) (cid:19) b † ij ( k ) b ik ( k ) b kl ( k ) b lj ( k )+ δ ( k + k − k − k ) 1( k − k ) c † αi ( k ) d † iβ ( k ) c αj ( k ) d jβ ( k )+ δ ( k − k − k − k ) c † αj ( k ) c αi ( k ) b ik ( k ) b kj ( k ) − c † αj ( k ) b † jk ( k ) b † ki ( k ) c αi ( k )( k − k ) + δ ( k − k − k − k ) b † jk ( k ) b † ki ( k ) d † iα ( k ) d jα ( k ) − d † iα ( k ) b ik ( k ) b kj ( k ) d jα ( k )( k − k ) + δ ( k + k − k − k ) c † αj ( k ) b † jk ( k ) c αi ( k ) b ik ( k ) + b † ki ( k ) d † iα ( k ) b kj ( k ) d jα ( k )( k − k ) + g Nπ (cid:90) ∞ dk (cid:18) b † ij ( k ) b ij ( k ) + c † αi ( k ) c αi ( k ) + d † iα ( k ) d iα ( k ) (cid:19) (cid:90) k dp ( p − k ) , (2.21)where we ignored terms that are suppressed in 1 /N when acting on the single-trace mesonand gluinoball states. More precisely, we ignored terms whose matrix elements between thestates (2.12) and (2.14) vanish as N → ∞ . To treat the problem numerically, we compactify the x − direction into a circle of circumfer-ence 2 πL and impose antiperiodic boundary conditions for the fermions, namely ψ ij ( x − ) = − ψ ij ( x − + 2 πL ) , v i ( x − ) = − v i ( x − + 2 πL ) . (3.1)Then the allowed momenta are k n = n L , with odd n . All the formulas in the previoussection can be easily rewritten in the discretized case by limiting the range of the x integralsto [0 , πL ] and replacing δ ( k − k (cid:48) ) (cid:55)→ Lδ k,k (cid:48) and (cid:82) dk (cid:55)→ L (cid:80) k , where δ k,k (cid:48) is the Kronecker10elta symbol. Thus, in the discrete case the anti-commutation relations (2.10) become (cid:110) b ij ( k ) , b † lk ( k ) (cid:111) = Lδ k ,k (cid:18) δ il δ jk − N δ ij δ kl (cid:19) , (cid:110) c αi ( k ) , c † βj ( k ) (cid:111) = (cid:110) d iα ( k ) , d † jβ ( k ) (cid:111) = Lδ k ,k δ ij δ αβ . (3.2)To simplify the notation and to exhibit the fact that L drops out from the formula for themasses, we define the dimensionless annihilation and creation operators. The annihilationoperators are defined by B ij ( n ) = 1 √ L b ij (cid:16) n L (cid:17) , C αi ( n ) = 1 √ L c αi (cid:16) n L (cid:17) , D iα ( n ) = 1 √ L d iα (cid:16) n L (cid:17) , (3.3)with n > (cid:110) B ij ( n ) , B † lk ( n ) (cid:111) = δ n ,n (cid:18) δ il δ jk − N δ ij δ kl (cid:19) , (cid:110) C αi ( n ) , C † βj ( n ) (cid:111) = (cid:110) D iα ( n ) , D † jβ ( n ) (cid:111) = δ n ,n δ ij δ αβ . (3.4)In terms of the dimensionless oscillators, the mode decompositions in (2.9) become ψ ij ( x ) = 1 √ πL (cid:88) odd n> (cid:16) B ij ( n ) e − in x L + B † ji ( n ) e in x L (cid:17) ,v iα ( x ) = 1 √ πL (cid:88) odd n> (cid:16) D iα ( n ) e − in x L + C † αi ( n ) e in x L (cid:17) , (3.5)the operator P + is P + = 12 L (cid:88) odd n> n (cid:16) B † ij ( n ) B ij ( n ) + C † αi ( n ) C αi ( n ) + D † iα ( n ) D iα ( n ) (cid:17) , (3.6)and the operator P − can be written as P − = g Lπ ( y adj V adj + y fund V fund + T ) , (3.7)11here we have defined y adj = m πg N and y fund = m πg N , with V adj = (cid:88) odd n> n B † ij ( n ) B ij ( n ) , V fund = (cid:88) odd n> n (cid:16) C † αi ( n ) C αi ( n ) + D † iα ( n ) D iα ( n ) (cid:17) , (3.8)and T = N (cid:88) odd n> (cid:16) B † ij ( n ) B ij ( n ) + 2 C † αi ( n ) C αi ( n ) + 2 D † iα ( n ) D iα ( n ) (cid:17) n − (cid:88) m n − m ) +2 (cid:88) odd n i > (cid:40) δ n + n ,n + n (cid:34) (cid:18) n − n ) − n + n ) (cid:19) B † ik ( n ) B † kj ( n ) B il ( n ) B lj ( n )+ 1( n − n ) C † αi ( n ) B † ik ( n ) C αj ( n ) B jk ( n )+ 1( n − n ) B † ki ( n ) D † iα ( n ) B kj ( n ) D jα ( n )+ 1( n − n ) C † αi ( n ) D † iβ ( n ) C αj ( n ) D jβ ( n ) (cid:35) + δ n ,n + n + n (cid:34) (cid:18) n + n ) − n + n ) (cid:19) B † ij ( n ) B ik ( n ) B kl ( n ) B lj ( n )+ 1( n + n ) C † αj ( n ) C αi ( n ) B ik ( n ) B kj ( n ) − n + n ) D † iα ( n ) B ik ( n ) B kj ( n ) D jα ( n ) + h . c . (cid:35)(cid:41) . (3.9)Since P + and P − commute, we can work at fixed P + = K/ (2 L ) for some integer K . Thismeans we consider single-trace meson states of the form1 N ( p − / C † α ( n ) B † ( n ) · · · B † ( n p − ) D † β ( n p ) | (cid:105) , (3.10)with (cid:80) pi =1 n i = K , as well assingle-trace gluinoball states1 N p/ tr (cid:0) B † ( n ) · · · B † ( n p ) (cid:1) | (cid:105) , (3.11)with the same condition on the sum of the n i . At leading order in large N , one can choosean orthonormal basis of such states. Just as in (2.12) and (2.14), the overall powers of N in(3.10)–(3.11) are such that the states have finite norm as N → ∞ , and from the expression12or P − we dropped all terms whose matrix elements are suppressed in 1 /N in the large N limit.The mass squared operator becomes M = 2 P + P − = g Kπ ( y adj V adj + y fund V fund + T ) , (3.12)Notice that the factors of L canceled from this expression. Removing the cutoff correspondsto taking L → ∞ and K → ∞ with P + fixed. In Sections 5, 6, and 8 we will use theexpression (3.12) to find the spectrum of gluinoballs and mesons numerically at leadingorder in large N . When m fund = m adj = 0, the discretized problem of finding the spectrum of M presented inthe previous section can be somewhat simplified upon using the SU( N ) gauge current algebraand its representations [19]. While we will not use this simplification, let us now review thisapproach because it will be useful for interpreting the numerical results of Sections 5 and 6. The simplification mentioned above is based on the fact that the operator P − in (3.7)–(3.9)can be expressed solely in terms of the Fourier modes of the gauge current J + ij . Let us definethe n th Fourier mode J ij ( n ) (with n ∈ Z ) through J + ij ( x − ) = 12 πL (cid:88) even n J ij ( n ) e − in x − L , J ij ( n ) = J adj ,ij ( n ) + J fund ,ij ( n ) (4.1)with the adjoint and fundamental contributions J adj ,ij ( n ) and J fund ,ij ( n ) being J adj ,ij ( n ) ≡ (cid:88) n + n = n (cid:18) B ik ( n ) B kj ( n ) − N δ ij B kl ( n ) B lk ( n ) (cid:19) J fund ,ij ( n ) ≡ (cid:88) n + n = n (cid:18) D iα ( n ) C αj ( n ) − N δ ij D kα ( n ) C αk ( n ) (cid:19) , (4.2)13here for simplicity we denoted B ij ( − n ) = B † ji ( n ), D iα ( − n ) = C † αi ( n ), and C αi ( − n ) = D † iα ( n ) for n >
0. For n > J ij ( n ) | (cid:105) = 0 because each term in J ij ( n ) involvesat least one annihilation operator. We also have that J ij (0) | (cid:105) = 0 because the J ij (0) are theconserved SU( N ) charge operators that annihilate the Fock vacuum. In fact, J ij (0) | ψ (cid:105) = 0for any gauge-invariant state | ψ (cid:105) . When n < J ij ( n ) | (cid:105) (cid:54) = 0 because in this case J ij ( n )contains at least one term with only fermionic creation operators.Using the commutation relations (3.4) and (4.2), we can compute the commutation rela-tion of J with the fermionic oscillators:[ J ij ( n ) , B kl ( m )] = δ kj B il ( n + m ) − δ il B kj ( n + m ) , [ J ij ( n ) , D kα ( m )] = δ kj D iα ( n + m ) − N δ ij D kα ( n + m ) , [ J ij ( n ) , C αk ( m )] = − δ ik C αj ( n + m ) + 1 N δ ij C αk ( n + m ) . (4.3)From (4.3) and the expression (4.2) for J in terms of fermionic oscillators, we can derive theKac-Moody (KM) current algebra[ J ij ( n ) , J kl ( m )] = δ kj J il ( n + m ) − δ il J kj ( n + m ) + k KM n δ n + m, (cid:18) δ il δ kj − N δ ij δ kl (cid:19) , (4.4)with level k KM = N + N f . Note that (4.3) and the first two terms in (4.4) follow from theSU( N ) transformation properties of the operators being commuted with the current. The op-erators J ij ( n ) carry P + momentum equal to − n/ (2 L ), as can be seen from the commutationrelation [ P + , J ij ( n )] = − n L J ij ( n ) . (4.5)Thus, when J ij ( n ) acts on a state with a definite value of K = 2 LP + , it lowers K by n units.For massless fermions, the expression for P − is P − = − g (cid:90) dx − tr (cid:18) J + ∂ − J + (cid:19) . (4.6) If instead of a single adjoint Majorana fermion we had n adj flavors of adjoint Majorana fermions, wewould have obtained (4.4) with k KM = n adj N + N f . P − = g Lπ (cid:88) even n (cid:54) =0 tr [ J ( − n ) J ( n )] n = 2 g Lπ (cid:88) even n> tr [ J ( − n ) J ( n )] n , (4.7)where there are no n = 0 terms in these sums because J (0) annihilates all gauge-invariantstates, and where in the second equality we used the algebra (4.4) to interchange J ( − n )with J ( n ) for n < c -number energy shift. The requirementthat P − annihilate the vacuum | (cid:105) fixes the regularized value of this divergent term to zero.Unlike the expressions (3.7)–(3.9) which hold only to leading order in large N when actingon single-trace states, the expression (4.7) is exact at finite N as well.As we will explain in more detail below, the advantage of the form (4.7) is to make itmanifest that P − only mixes states that belong to the same representation of the currentalgebra (4.4) [19]. Representations of the Kac-Moody current algebra (4.4) can be con-structed in the free theory of ψ ij , v iα , and v † αi , and they each contain both gauge-invariantstates (i.e., annihilated by J ij (0)) and non-gauge-invariant states. Because P − commuteswith J ij (0), it follows that P − maps gauge-invariant states to gauge-invariant states, so afterconstructing a given KM representation we can restrict our attention to its gauge-invariantsubspace.A representation of the KM algebra (also referred to below as a current block or KMblock) starts with a KM primary state | χ (cid:105) I (generally not gauge invariant with all SU( N )indices grouped together into the multi-index I ) that is annihilated by all J ij ( n ) with n > J ij ( n )with n < J i j ( − n ) J i j ( − n ) · · · J i p j p ( − n p ) | χ (cid:105) I (4.8)with n i >
0. For such a state to be gauge-invariant, all SU( N ) indices of the J ’s and of | χ (cid:105) must be fully contracted. Because J ij ( n ) carries P + momentum equal to − n/ L (see(4.5)), the states of a KM representation are graded by the value of K = 2 LP + , with theKM primary having the smallest value of K .When acting with P − on a state of the form (4.8), we can use the algebra (4.4) tocommute the J ( n ) with n > | χ (cid:105) I .Any J ( n (cid:48) ) with n (cid:48) > Acting with J ij (0) on the KM primary or one of its descendants can be reduced to a linear combinationof the KM primary and the descendants because J ij (0) acts as the corresponding SU( N ) generator. J ( n ) with n > P − takes the states (4.8) to linear combinations of states of the same form, and thusthe diagonalization of P − can be done independently for each current block.As explained in [19], this construction not only shows that P − is diagonal in each currentblock, but also that the eigenvalues of P − within the gauge-invariant subspace of a givencurrent block depend only on the level k KM of the current algebra and on the SU( N ) repre-sentation of the KM primary. Indeed, the structure of the gauge-invariant states does dependon the representation of the KM primary, and when commuting the J ( n )’s with n > k KM . It follows that for any two current blocks whose KM primaries transform in the samerepresentation of SU( N ), either belonging to the same theory or to two different theorieswith the same value of k KM , the eigenvalues of P − will be the same. This universality ofthe massive spectrum, first noticed in [19], will be very important in explaining some of thedegeneracies we observe numerically in the spectrum of P − in the following sections.For a current block to contain singlets it is necessary that the KM primary transformin a representation of SU( N ) with N -ality 0, because the N -ality is conserved under tensorproducts and J ( − n ), being an SU( N ) adjoint, has N -ality zero. (All other KM representa-tions where the KM primary does not have N -ality 0 does not contain any gauge-invariantstates.) The representations of N -ality 0 can be obtained by taking tensor products of theadjoint representation. We will denote by n the smallest number of adjoint factors in suchtensor products that are needed to obtain a given representation. See Table 1 for the SU( N )representations obtained for up to n = 3. For each representation, we stated in the lastcolumn whether or not it belongs to the totally anti-symmetric product of n adjoint factors.This fact will be important in the next subsection. N when N f = 0 Before tackling the theory with one Majorana adjoint and N f fundamental fermions, let usfirst review current block construction when N f = 0, following [19]. Labeling the current16 SU( N ) irreps anti-symmetric product?0 R = [00 . . .
0] (identity) yes1 R = [10 . . .
01] (adjoint) yes2 R = [20 . . . R = [010 . . .
02] yes R (cid:48) = [20 . . . R (cid:48)(cid:48) = [01 . . .
10] no3 R = [30 . . . R ⊕ [0010 . . .
03] yes R (cid:48) = [110 . . . R (cid:48)(cid:48) = [30 . . . R (cid:48)(cid:48)(cid:48) = [0010 . . . N ) representations for various valuesof n . The last column states whether the representations are contained in the anti-symmetricproduct of n adjoint representations. All irreps are real except for the complex representa-tions R and R and their conjugates R and R .blocks as above by n , we have the following primaries: n = 0 : | (cid:105) , n = 1 : B † ji (1) | (cid:105) , n = 2 : (cid:18) B † ji (1) B † lk (1) − N δ kj J il ( −
2) + 1
N δ il J kj ( − (cid:19) | (cid:105) , n = 3 : (cid:16) B † ji (1) B † lk (1) B † nm (1) − traces (cid:17) | (cid:105) , etc. (4.10)They all involve products of n factors of B † (1) acting on the Fock vacuum, with all theSU( N ) traces removed. Even though they are all constructed from B † ji (1), one can obtaindescendants where no B † ji (1)’s appear because the currents can contain B ji (1)’s which mayannihilate the B † ji (1)’s. It is straightforward to check that these states are annihilated byall J ij ( n ) with n <
0. Note that the states of even n are bosons while those of odd n arefermions.For n >
1, the primaries in (4.10) transform in reducible representations of SU( N ). Toform SU( N ) irreps, we should further symmetrize and/or anti-symmetrize in the fundamental For the n = 3 state, the traces that need to be subtracted are:1 N (cid:20) δ kj J il ( − B † nm (1) − δ il J kj ( − B † nm (1) − δ mj J in ( − B † lk (1) + δ in J mj ( − B † lk (1) − δ kn J ml ( − B † ji (1) + δ ml J kn ( − B † ji (1) (cid:21) . (4.9) i , k , m , . . . , and then because the B † ’s anti-commute, the states in (4.10) will havethe opposite symmetry properties in the anti-fundamental indices j , l , n , etc. The numberof SU( N ) irreps one can construct is equal to the number of Young diagrams with n boxes,with the first few representations being the ones marked as appearing in the anti-symmetricproduct of n adjoints in Table 1.At finite N , each block can be diagonalized separately, and one expects no relation be-tween the eigenvalues of blocks whose primaries are in different SU( N ) representations, withone exception. Because the adjoint representation is real, each time a complex representa-tion appears in the product of several adjoint representations, its conjugate representationmust appear too. For example, at n = 2, we have R and R , and at n = 3 we have R and R , etc. Since our gauge theory has charge conjugation symmetry, defined in (2.7), theeigenvalues of P − for a current block with primary in representation R must be equal tothose of the conjugate current block whose primary is in R . Thus, one expects two-folddegeneracies for these states in the finite N spectrum of P − . Such degeneracies were noticedin the numerical study of [39], but no clear explanation was presented. In particular, thedouble degeneracies in Table 4 of [39] correspond to the n = 2 degeneracies between the R and R blocks and the double degeneracy discussed after Eq. (12) of [39] correspond to the n = 3 degeneracy between states in the R and R blocks. N when N f = 0 At large N , the states in each block further split into single-trace and multi-trace sectors,and more exact degeneracies appear between single-trace and multi-trace states, as we nowexplain. In particular, at leading order in large N , the single-trace eigenvalues from a sectorlabeled by n ∗ > n ∗ eigenvalues of single-trace states from the n = 1 sector. Thus, in the current block labeled by n ∗ there will bemulti-trace states that are exactly degenerate with single-trace ones.The proof of this fact relies on the fact that the theory with one Majorana adjoint andno fundamentals has the same value of the KM level k KM = N as the theory of N f = N fundamental fermions with no adjoints. For brevity, let us denote the former theory as T adj and the latter as T fund . As mentioned above, this means that the spectrum of P − in currentblocks whose primaries transform in the same representation of SU( N ) must be identical inthe two theories. The two theories are not equivalent, and indeed T fund has more currentblocks than T adj for a given n , but the description in terms of fundamental fermions in T fund will make it easier to determine the eigenvalues in the n > /N .18n particular, instead of computing the P − eigenvalues in the current blocks in (4.10) in T adj , we can equivalently compute the P − eigenvalues in the following blocks of T fund withprimaries given by n = 0 : | (cid:105) , n = 1 : (cid:16) C † αi (1) D † jβ (1) − flavor and gauge trace (cid:17) | (cid:105) , n = 2 : (cid:16) C † αi (1) D † jβ (1) C † γk (1) D † lδ (1) − flavor and gauge traces (cid:17) | (cid:105) , etc. (4.11)We will set the flavor indices to be all distinct, in which case there is no flavor trace sub-traction necessary.At finite N the two problems are equivalent, but at large N there are two simplificationsthat occur. The first is that the SU( N ) trace subtractions are subleading in 1 /N bothin (4.11) and (4.10), so one can ignore the corresponding terms. The second is that anycommutators that break any of the strings are also subleading. In the n = 0 block, we havethe gauge-invariant single-trace states n = 0 : 1 N p tr( J ( − n ) · · · J ( − n p )) | (cid:105) (4.12)in both the T adj and T fund theories. For n = 1, we have the gauge-invariant single-trace states n = 1 : 1 N p + ( J ( − n ) · · · J ( − n p ) B ( − | (cid:105)∼ = 1 N p + ( J ( − n ) · · · J ( − n p )) ji C † αi (1) D † jβ (1) | (cid:105) , (4.13)for any α (cid:54) = β . The “ ∼ =” sign means that as far as the matrix elements and spectrum of P − are concerned, we can identify (up to normalization) the states on the left in the T adj theory with the states on the right in the T fund theory. While the “single-trace” terminologyis appropriate in the T adj theory, in T fund the state (4.13) would be better referred to as“single-string,” because it consists of a sequence of C † ’s and D † ’s that alternate betweenhaving color indices and flavor indices contracted except at the endpoints of the string. For a given n , these are not the only KM primaries of the T fund theory. n = 2, we have the gauge-invariant single-trace states n = 2 : [ J ( − n ) · · · J ( − n p )] jk [ J ( − m ) · · · J ( − m p )] li N p + p +1 B † ji (1) B † lk (1) | (cid:105)∼ = [ J ( − n ) · · · J ( − n p )] jk [ J ( − m ) · · · J ( − m p )] li N p + p +1 C † αi (1) D † jβ (1) C † γk (1) D † lδ (1) | (cid:105) , (4.14)for any pairwise distinct α, β, γ, δ . Again, these states are single-trace only in T adj ; in T fund they are “double-string” states. A similar construction holds for n > (cid:81) p k =1 J ( − n ,k )) i i · · · ( (cid:81) p n k =1 J ( − n n ,k )) i n i N p + ··· + p n + n B † i i (1) · · · B † i n i n − (1) | (cid:105)∼ = ( (cid:81) p k =1 J ( − n ,k )) i i · · · ( (cid:81) p n k =1 J ( − n n ,k )) i n i N p + ··· + p n + n C † α i (1) D † i β (1) · · · C † α n i n − (1) D † i n β n (1) | (cid:105) , (4.15)with { α i , β i } all distinct. Neglecting terms suppressed in 1 /N , the state on the RHS of (4.15)can also be written as( (cid:81) p k =1 J ( − n ,k )) i i C † α i (1) D † i β (1) N p + · · · ( (cid:81) p n k =1 J ( − n ,k )) i n i C † α i (1) D † i n β n (1) N p n + | (cid:105) , (4.16)which has the form of a multi-string state, with n factors as in (4.13).Thus, single-trace states of the current block labeled by n in the T adj theory are in factmulti-string states in the T fund theory, with the single-string factors coming from the n = 1block. Because for multi-string states large N factorization works similarly as for multi-tracestates, it follows that the eigenvalues of single-trace states in the n th block of the T adj theoryare in one-to-one correspondence sums of n eigenvalues from the 1st current block.The formula (4.15) explains how to construct the eigenstates. Suppose in the n = 1sector the eigenstate of P − eigenvalue P − = E a is (cid:88) { n ,...,n p } c an ,...,n p tr( J ( − n ) · · · J ( − n p ) B ( − | (cid:105) , (4.17)then the state in the n th block with eigenvalue E a + · · · + E a n is (cid:88) { n i, ,...,n i,p } c a n , ,...,n ,p · · · c a n n n , ,...,n n ,p ( J ( − n , ) · · · J ( − n ,p )) i i · · · ( J ( − n n , ) · · · J ( − n n ,p n )) i n i × B i i ( − · · · B i n − i n ( − | (cid:105) . (4.18)20ne can obtain a multi-trace state in T adj with the same P − eigenvalue E a + · · · + E a n bycontracting the indices in the first line with those in the second line of (4.18) in a differentway. Of course, some of these states may vanish because of Fermi statistics.The discussion above is restricted to the large N spectrum of gluinoballs in T adj . Thediscussion also applies to the gluinoball spectrum of the theory with a Majorana adjoint and N f fundamental fermions that we are interested in here, in the limit N → ∞ at fixed N f . N for N f > Let us now go back to the SU( N ) gauge theory with a Majorana adjoint fermion and N f Dirac fermions of primary interest in this paper, and let us list the first few KM blocks.Unlike in the pure adjoint theory whose primaries are listed in (4.10), we now haveseveral current blocks for each n . For n = 0 (SU( N ) singlets), we have the Fock vacuum | (cid:105) , the equal linear combinations of two-bit mesons of the form C † αi ( n ) D † iβ ( n ) | (cid:105) with anytotal even value of K ≥ the equal linear combination of three-bit mesons of the form C † αi ( n ) B † ij ( n ) D † jβ ( n ) | (cid:105) with any total odd value of K ≥
3, as well as infinitely many othermore complicated singlet KM primaries with more than three-bit components for which wedo not have a concise expression: n = 0 : | (cid:105) , | ζ αβ,K (cid:105) = (cid:88) m + m = Km , > C † αi ( m ) D † iβ ( m ) | (cid:105) , | ξ αβ,K (cid:105) = (cid:88) m + m + m = Km , , > C † αi ( m ) B † ij ( m ) D † jβ ( m ) | (cid:105) , . . . (4.19)As per the discussion above, each such block has the exact same P − spectrum, so every P − eigenvalue in the n = 0 sector is highly degenerate. In particular, all KM primaries inthe n = 0 sector have P − = 0 just like the Fock vacuum | (cid:105) . As we will explain in thenext section, this theory has a finite but growing number of P − = 0 states at each K , andtheir presence is due to the fact that the infrared is controlled by a nontrivial CFT. Each P − = 0 state generates its own n = 0 current block. Note that as emphasized in [19], inlight-cone quantization one can see all massive states and the right-moving massless ones,but not the left-moving massless states. Thus, the P − = 0 states seen in DLCQ do notprovide a complete description of the massless sector. The state | ζ αβ,K (cid:105) is the unique exactly massless state of the ’t Hooft model [5]. The expression for P − in (4.7) makes manifest that P − is a non-negative-definite operator and that P − | χ (cid:105) = 0 if and only if J ij ( n ) | χ (cid:105) = 0 for all n > n = 1 (SU( N ) adjoints), we have the KM primary from the pure adjoint theory in(4.10), as well as primaries constructed using the fundamental fermions: n = 1 : B † ji (1) | (cid:105) , (cid:18) D † jβ (1) C † αi (1) − δ αβ N B † jk (1) B † ki (1) − gauge traces (cid:19) | (cid:105) , (cid:18) D † jβ (1)[ C † (1) B † (1)] αi − δ αβ N [ B † (1) B † (1) B † (1)] ji − δ αβ B † ji (3) − gauge traces (cid:19) | (cid:105) , (cid:18) [ B † (1) D † (1)] jβ (1) C † αi (1) − δ αβ N [ B † (1) B † (1) B † (1)] ji + δ αβ B † ji (3) − gauge traces (cid:19) | (cid:105) ,. . . (4.20)As we can see, the state in the first line has K = 1; the state on the second line has K = 2;and the states on the third and fourth lines have K = 3. The pattern continues: we can find p KM primaries with K = p + 1 that start with (cid:0) [ B † (1) q D † (1)] jβ [ C † (1) B † (1) p − − q ] αi + · · · (cid:1) | (cid:105) , q = 0 , , . . . , p − . (4.21)Thus, for every eigenstate of P − with some value of K belonging to the block on the firstline of (4.20), we will have a whole family of states degenerate with it: a K + 1 state from theblock on the second line, two K + 2 states from the blocks on the third and fourth lines, etc.In general, at resolution parameter K + p + 1 we will have p eigenstates of P − degeneratewith the eigenstate of P − belonging to the first block in (4.20) at resolution parameter K .For n = 2, we can construct KM primaries from the U( N f ) singlet B † ji (1) and the U( N f )adjoint D † jβ (1) C † αi (1), as well as factors such as the first term in (4.21). Unlike the N f = 0case, we can now obtain all SU( N ) representations that appear in the product of two adjoints,namely R , R and also R (cid:48) and R (cid:48)(cid:48) —see Table 1. While we leave a full analysis of theserepresentations to future work, let us point out that the first couple n = 2 primaries in R ⊕ R are n = 2 primaries in R ⊕ R : (cid:16) B † ji (1) B † lk (1) − traces (cid:17) | (cid:105) , (cid:18) B † ji (1) D † lβ (1) C † αk (1) + δ αβ N B † ji (1) B † lm (1) B † mk (1) − traces (cid:19) | (cid:105) ,. . . (4.22)22here the first line is as in (4.10). As in the n = 1 sector, we see that for every P − eigenstateat K coming from the first block we will have a P − eigenstate at K + 1, and so on. N for N f > As discussed in Section 4.2, at large N the states in each current block split into single-trace and multi-trace, and there are additional degeneracies between the single-trace andmulti-trace ones.Of the n = 0 current blocks in (4.19), the block whose primary is the Fock vacuum | (cid:105) contains single-trace gluinoball states of the form (4.12) (as well as multi-trace gluinoballstates) that are in one-to-one correspondence with and have the exact same P − spectrum asthe n = 0 sector of the pure adjoint theory T adj at leading order in 1 /N . Each of the other n = 0 blocks in (4.19) whose KM primaries are gauge-invariant single-trace states such as | ζ αβ,K (cid:105) or | ξ αβ,K (cid:105) contains only one single-trace state (namely the KM primary itself) as wellas multi-trace states.Of the n = 1 current blocks in (4.20), the block whose primary is B † ji (1) | (cid:105) containssingle-trace gluinoball states of the same form as on the LHS of (4.13) (as well as multi-trace gluinoball states) that are in one-to-one correspondence with and have the exact same P − spectrum as the n = 1 sector of the pure adjoint theory. The other n = 1 blocks in(4.20)–(4.21) contain single-trace mesons with the exact same P − spectrum as the n = 1gluinoballs (at leading order in large N we should only keep the first terms in (4.20)–(4.21)when constructing gauge-invariant states because the remaining terms are subleading), witha degeneracy pattern described after (4.21): for every single-trace fermionic n = 1 gluinoballat K , we will have p single-trace meson at K + p , with p = 1 , , , . . . . In Section 7 we willalso “explain” this degeneracy pattern of single-trace mesons through the existence of an osp (1 |
4) symmetry.The structure of states in the n = 2 sector is similar to that in the n = 1 sector. Inparticular, we have the block whose primary is (cid:16) B † ji (1) B † lk (1) − traces (cid:17) | (cid:105) , which containssingle-trace gluinoball states with the same P − spectrum at large N as the n = 2 single-tracegluinoball of the pure adjoint theory. For each such gluinoball state, there is then a towerof single-trace meson states, namely for every gluinoball state at K there are p single-tracemeson states at K + p that are degenerate with it. For p = 1, this state belongs to the blockon the second line of (4.22). We did not analyze the case n >
2, but we expect a similarstructure of single-trace states for these sectors too.23
Numerical results for gluinoball spectrum
Let us now study numerically the spectrum of single-trace gluinoballs and mesons in thelarge N limit. The two decouple at leading order in large N , so in this section we will focusonly on the single-trace gluinoballs. As already mentioned, in the limit where N f is keptfixed while N is taken to infinity, the spectrum of gluinoballs is exactly the same as in the N f = 0 case. The gluinoball spectrum was studied using the DLCQ procedure in [16, 25].As was done in [16, 25], working at fixed K and considering the basis of gluinoball statesof the form (3.11), the operators M and P − can be written as finite-dimensional matricesthat can be diagonalized numerically. For y adj = 0, Refs. [16, 25] obtained the spectrum of M up to K = 25. With moderncomputers, one can do much better. We redid this analysis, and using the “Scalable Libraryfor Eigenvalue Problem Computations” (SLEPc) [40–43], we are able to obtain the massesof the lowest-lying states up to K = 41, which represents more than a thousandfold increasein the number of states in the discretized basis.In Figure 1, we show the masses of gluinoball states as a function of 1 /K . We splitthe states according to their statistics (states with odd K are fermions, and those witheven K are bosons), and their quantum numbers with respect to the Z charge conjugationsymmetry (2.7). For the fermionic states, we find that the first two “single-particle states”converge to M F ≈ . g N/π and M F ≈ . g N/π in the continuum limit K → ∞ . Athigher mass-squared values, we see what appears to be the beginning of a continuum in thespectrum, as was noticed in [25–28]. Extrapolating the lowest masses in this region, we findthat they converge to around M ≈ . g N/π in both the Z -even and odd sectors. Amongthe bosonic states, we find the first single-particle state converging to M B ≈ . g N/π inthe continuum limit, as well as evidence for a continuous spectrum starting around M ≈ . g N/π in both the Z -even and odd sectors. That the values of M where the continuumstarts in both sectors is very close to 4 M F ≈ . g N/π suggests that the continuum iscomposed of two-particle states [25].A striking feature of these plots are the exact relations stating that some of the eigenvaluesof P − ( K ) at some given K are sums of P − eigenvalues for fermionic states at K i with K = (cid:80) i K i , i.e. there are states obeying P − ( K ) = (cid:80) i P − ( K i ). Such relations were explainedin Section 4.3 as arising at large N in the n > .00 0.02 0.04 0.06 0.08 0.1017.1922.951.3851030354045 (a) The squared masses of fermionic gluinoball states with odd Z parity.A low-lying trajectory appears to converge towards M F ≈ . g N/π . (b) The squared masses of fermionic gluinoball states with even Z parity.A low-lying trajectory appears to converge towards M F ≈ . g N/π . .00 0.02 0.04 0.06 0.08 0.1025.5222.93510153035404550 (c) The squared masses of bosonic gluinoball states with odd Z par-ity. The lowest single-particle state appears to converge towards M B ≈ . g N/π . (d) The squared masses of bosonic gluinoball states with even Z parity.A low-lying trajectory appears to converge towards M B ≈ . g N/π . Figure 1: The masses of gluinoball states in DLCQ with m adj = 0 as a function of 1 /K , upto K = 41. The spectrum was first described in [16, 20] up to K = 25. The orange pointsare single-trace gluinoball states that are exactly degenerate with multi-trace states.26 − ( K ) = (cid:80) n i =1 P − ( K i ) implies M ( K ) K = n (cid:88) i =1 M ( K i ) K i . (5.1)The states whose masses obey these exact relations are shown in orange in Figure 1. Inparticular, the orange dots at the top of the plots in the fermionic sectors are n = 3 stateswith P − written as a sum of three fermionic eigenvalues. For the bosonic states in Figure 1,the orange dots visible in the plots correspond to n = 2 states. States with n > M ≈ . g N/π in the bosonic Z -odd sector is marked by statesexactly degenerate with the double-trace states, we observe similar thresholds in the otherthree sectors. It appears that, in the continuum limit K → ∞ , every sector therefore exhibitsthreshold bound states. The reason why the other three sectors, for example, the fermionicones, do not exhibit exact thresholds is related to the fact that a pair of fermionic boundstates cannot simply bind to form another fermionic state. To make the threshold single-trace state a fermion, it needs to contain another fermionic insertion, such as B † ij (1) (sincewe are using the anti-periodic boundary conditions, this is the lowest frequency mode). Thisinsertion would explain the breaking of exact degeneracy, but would have negligible effectin the limit K → ∞ . As noted in [27], this provides an argument for the existence of theapproximate thresholds when the DLCQ boundary conditions are anti-periodic.Note that in the continuum limit K → ∞ , the relation (5.1) implies that for any group of n trajectories whose masses asymptote to { M i } ≤ i ≤ n as K → ∞ , we will have a continuumof states with masses starting at M threshold = n (cid:88) i =1 M i . (5.2)Indeed, at large enough K , we can approximate M ( K i ) ≈ M i , and we can effectively varycontinuously the momentum fractions x i ≡ K i /K . Thus, (5.1) becomes, at large K , M ≈ n (cid:88) i =1 M i x i , n (cid:88) i =1 x i = 1 . (5.3)We see from (5.3) that M can be made arbitrarily large by taking (at least) one of the x i to be small. The lowest value of M is attained when x i = M i / (cid:80) n j =1 M j and it is givenby M . Since the x i can be varied continuously, we thus find a continuum of states27 .00 0.02 0.04 0.06 0.08 0.1023.7135.6442.4510153050 (a) (b) .00 0.02 0.04 0.06 0.08 0.1030.9833.4541.0151015202550 (c) (d) Figure 2: The masses of gluinoball states in DLCQ as a function of 1 /K , up to K = 41,with the adjoint mass parameter y adj = m πg N = 0 . M ≥ M threshold as K → ∞ . For any finite mass window ∆ M above the threshold, thenumber of states that approximate the continuum grows as K n − at large but finite K . Atfinite K , we will in general not find states with masses given by M threshold , partly because M ( K i ) varies slightly with K for single particle states, and partly because we will not findrational x i = K i /K that come arbitrarily close to the optimal value x i = M i / (cid:80) n j =1 M j .As we will see shortly, the gluinoball states in Figure 1 are related to the spectrum ofsingle-trace meson states. For a fixed m adj >
0, we expect the continua in the gluinoball spectrum at large K todisappear and be replaced by a discrete spectrum. Since we have diagonalized the discretized P − matrix up to higher values of K than was achieved in previous work, we present heresome additional evidence for this phenomenon.Figure 2 shows the squared masses of DLCQ states up to K = 41 with the adjoint massparameter fixed at y adj = 0 .
1. Indeed, the apparent continua in Figure 1 are absent here. Ineach sector, we extrapolate to estimate the masses of the lowest few states in the continuum.
Let us now proceed to a similar analysis for the single-trace meson spectrum. In the theorywith N f fundamental quarks, single-trace meson states transform in the adjoint represen-tation of the U( N f ) flavor symmetry and thus have degeneracy N f . At leading order inlarge N with fixed N f , the meson masses are independent of N f . Thus, in the numericalcomputations that follow we set N f = 1 without loss of generality.In this section we restrict to the case y adj = y fund = 0 where the fermions are massless;we examine the case where the fundamental fermions are massive in Section 8. As in thegluinoball case, to find the single-trace meson spectrum we can again write M and P − asfinite-dimensional matrices if we work in the basis of states (3.10) at fixed K . The basisstates (3.10) are in one-to-one correspondence with the ordered partitions of the integer K into any number of odd integers. It can be shown that the number of meson states at given K equals F K if K is even and F K − K is odd, where { F n } ∞ n =1 is the Fibonacci sequencewith F = F = 1—see Table 2 for the first few examples. As written, the states in (3.10)are orthonormal at leading order in large N .30 K . Diagonalizing M and P − numerically as in the gluinoball case, we find that for y fund = y adj = 0, the spectra of the discretized M and P − operators are highly degenerate. First,unlike in the gluinoball case, we now find meson states with M = 0 whose number growswith K , leading to an infinite number of such states in the continuum limit. The first fewmassless states were presented around Eq. (4.19), where it was pointed out that they arenecessarily the Kac-Moody primaries of their respective current blocks.The presence of an infinite number of massless states in the continuum limit is notunexpected, because in the deep IR, our theory is governed by a non-trivial CFT. Tosupport this conclusion analytically, let us calculate the central charge of the gauged WZWmodel which describes the far infrared limit of the theory. Since the free UV theory has N − N N f Majorana fermion fields, it may be bosonized [44] into the SO( N − N N f ) WZW model with central charge c UV = N N f + ( N − N )currents satisfy the Kac-Moody algebra at level k KM = N + N f . Therefore, the IR limit ofthe gauge theory is described by theSO( N − N N f ) SU( N ) N + N f (6.1)coset model. The central charge of the SU( N ) k WZW model is well-known [24] to be ( N − kN + k . Therefore, we find that the coset model has central charge c IR = c UV − ( N − N + N f N + N f = N f (3 N + 2 N f N + 1)2(2 N + N f ) . (6.2)For large N this grows as 3 N N f / y fund = y adj = 0 there is no justification fordiscarding the components of the fermions moving along x − . The precise description of thefar IR limit of the theory is provided by the coset model (6.1), but we will not discuss itfurther in this paper. For the massive states, the spectrum of P − has degeneracies that are even more striking thanin the gluinoball case. The eigenvalues of P − for mesonic states are shown up to K = 35in Figure 3, where at each value of K we also colored the eigenvalues according to theirdegeneracies. We notice that all the P − eigenvalues we encounter at some K are also P − eigenvalues at K + 1. More precisely, we find that an eigenvalue λ > n at K will have degeneracy at least n + 1 at K + 1 (and usually exactly n + 1, except in rarecases to be described later). Moreover, if the eigenvalue λ first appears in the spectrum at K = K λ (usually with degeneracy 1 but sometimes with higher degeneracy), we find that,in most cases, at K = K λ − λ is a P − eigenvalue in the single-trace gluinoball spectrum.Let us first comment on the pattern of degeneracies between the mesonic states, anddiscuss the degeneracy between mesons at K λ and gluinoballs at K λ − P − spectrum agrees for anytwo (or more) KM blocks whose KM primaries transform in the same SU( N ) representation.Indeed, for the n = 1 blocks, we found in (4.20)–(4.21) a number of KM blocks that increaseswith K in precisely the same way as the pattern of degeneracies we observed above. Whilewe have not constructed all the Kac-Moody primaries explicitly, we expect that a similarexplanation would hold for n >
1. This construction would provide a complete explanation ofthe degeneracies in the single-trace massive meson spectrum at large N , because all massivestates belong to n ≥ osp (1 |
4) symmetry enjoyedby the discretized theory at large N . In particular, the massive spectrum splits into infinite-dimensional unitary irreducible representations of osp (1 | λ , the meson states at K = K λ where that eigenvalue firstappears are referred to as “ osp (1 |
4) primary states,” whereas the states with the same λ at K > K λ are osp (1 |
4) descendants.From the P − eigenvalues, we can immediately recover the M = 2 P + P − eigenvalues.32 Figure 3: The eigenvalues P − up to K = 35. States are colored according to their de-generacies, with the darkest states being singlets. Along the horizontal trajectories, thedegeneracies increase in steps of 1 moving from right to left, except for the series of masslessstates, which have degeneracy (cid:98) K/ (cid:99) . 33 ●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●● ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ���� ���� ���� ���� ���� ��������������� ● ����� ▲ ���������� (a) The states approaching M ≈ . g N/π . ●● ●●●●● ●●●●● ●●●●●● ●●●●●● ●●●●●●● ●●●●●●●● ●●●●●●●● ●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●● ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ���� ���� ���� ���� ���� ��������������������� ● ����� ▲ ���������� (b) The states approaching M ≈ . g N/π . Figure 4: The masses of the states in Figure 3, along with the gluinoball states described inSection 6.3, and the trajectories along which they approach their K → ∞ values.34he latter organize into different groups of multiple trajectories of consecutive degeneraciesthat converge to the same M as K → ∞ , as a consequence of the degeneracy of the P − eigenvalues at different values of K . Indeed, if M ( K ) are the squared masses of a trajectoryof singlet M eigenvalues (arising from distinct P − eigenvalues at each K ), then there existtrajectories of degeneracy 2 j + 1 with j = 0 , , , , · · · . As a consequence of the osp (1 | j + 1 are related to thosein the singlet trajectory through M j ( K ) = KK − j M ( K − j ). This relation implies that aswe take K → ∞ , we find that all trajectories converge to the same M :lim K →∞ M j ( K ) = lim K →∞ M ( K ) . (6.3)We show the lowest two groups of such trajectories in Figure 4. The lowest one convergesto M ≈ . g N/π as K → ∞ and the next one converges to M ≈ . g N/π . Theseare precisely the lowest two squared masses in the fermionic gluinoball spectrum shown inthe top panel of Figure 1, which is a consequence of the relation between single-trace mesoneigenvalues and gluinoball ones mentioned above.
Let us now discuss the relation between the P − eigenvalues for mesons and gluinoballs inmore detail. As already mentioned, a given eigenvalue λ for a osp (1 |
4) primary meson at K λ also appears, in most cases, in the gluinoball spectrum at K λ −
1. The converse is, however,not true: there are (many) bosonic gluinoball eigenvalues that do not appear in the fermionicmeson spectrum at larger K . In particular, looking at the lowest 50 P − eigenvalues of meson osp (1 |
4) primary states up to K = 35, we find the following pattern of degeneracies betweenmesons at K and gluinoballs at K − • The set of P − eigenvalues of single-trace bosonic primary mesons at any fixed even K is identical to the set of P − eigenvalues of single-trace fermionic gluinoballs at K − • The set of P − eigenvalues of single-trace fermionic primary mesons at any fixed odd K is identical to the set of sums of two single-trace fermionic gluinoball eigenvaluesat odd K and odd K , with K − K + K . In most cases, the sum of fermionicgluinoball eigenvalues at K and K is also a bosonic gluinoball eigenvalue at K + K .For a concrete example of the pattern in the first bullet point, we refer to Table 3. Here,we give all the eigenvalues of P − for mesons at K = 8 and for gluinoballs at K = 7. All of35esons, K = 8 Gluinoballs, K = 7 P − Degeneracy P − Degeneracy
13 2 (cid:0)
59 + √ (cid:1) (cid:0)
59 + √ (cid:1)
32 1 2 1
51 3 (cid:0) − √ (cid:1) (cid:0) − √ (cid:1)
10 4Table 3: All of the eigenvalues of P − in the meson sector at K = 8 and in the gluinoballsector at K = 7. Cyan rows are meson osp (1 |
4) primary states, and gray rows are descendantstates. The set of primary state eigenvalues of mesons at K = 8 is identical to the set ofeigenvalues of gluinoballs at K = 7.the eigenvalues at K = 8 are either osp (1 |
4) descendant states of osp (1 |
4) primary mesons atlower K , or are osp (1 |
4) primary states and occur in the gluinoball spectrum at K = 7. Fora concrete example of the pattern in the second bullet point above, see Table 4. Here, we firsttabulate the P − eigenvalues of osp (1 |
4) primary mesons at K = 11. We then write each ofthese eigenvalues as sums of gluinoball eigenvalues at ( K , K ) = (3 ,
7) and ( K , K ) = (5 , n Kac-Moody block that contains single-trace gluinoball states(for instance the first line of (4.20) in the n = 1 case), one can also construct Kac-Moodyblocks in the same representation of SU( N ) that contain single-trace mesons and that haveone extra unit of K relative to the gluinoball blocks (for instance the second of (4.20) inthe n = 1 case). Similarly, the second bullet point is explained by the fact that for everyeven n ≥ n = 2 case), one can also construct Kac-Moody blocks in the samerepresentation of SU( N ) that contain single-trace mesons and that have one extra unit of K relative to the gluinoball blocks (for instance the second of (4.22) in the n = 2 case).36 sp (1 |
4) primary mesons, K = 11 Gluinoballs at( K , K ) = (3 ,
7) Gluinoballs at( K , K ) = (5 , P − Degeneracy P − P − +
72 52 + (cid:0)
113 + √ (cid:1) + (cid:0)
59 + √ (cid:1) + 2 1 + (cid:0) − √ (cid:1) + (cid:0) − √ (cid:1) P − eigenvalues of osp (1 |
4) primary meson states at K = 11. They agreewith sums of eigenvalues of fermionic gluinoballs at K and K with K + K = K − (cid:8) (cid:9) at K = 3, (cid:8) , (cid:9) at K = 5, and (cid:8) (cid:0) − √ (cid:1) , , (cid:0)
59 + √ (cid:1) , (cid:9) at K = 7.The degeneracies between mesons and gluinoballs at finite K give rise to degeneraciesbetween the continuum spectra of mesons and gluinoballs. For instance, at every odd valueof K there is a state in the gluinoball spectrum on a trajectory approaching M ≈ . g Nπ .Each of these fermionic gluinoballs corresponds to a bosonic meson at one higher K , withthe same eigenvalue of P − . Their mass-squared values are thus related by M ( K + 1) K + 1 = M ( K ) K , (6.4)It follows that there is a series of meson primary states also approaching M ≈ . g Nπ .These states form the lowest meson trajectory in Figure 4a. The other meson trajectoriesapproaching the same value are explained by the pattern of degeneracies among meson stateswhich we have already described. N Let us now provide another explanation for the degeneracy in the P − spectrum observedin Figure 3. As already mentioned, this degeneracy can be traced to the existence of an osp (1 |
4) symmetry algebra that commutes with P − .37 .1 osp (1 | algebra The osp (1 |
4) generators can be constructed from the four basic “supercharges” of the dis-cretized model q L ± = 1 √ N (cid:88) n ,n (cid:16) C † j ( n + n ± B ij ( n ) C i ( n ) + C † j ( n ) B † ji ( n ) C i ( n + n ∓ (cid:17) ,q R ± = 1 √ N (cid:88) n ,n (cid:16) D † i ( n + n ± B ij ( n ) D j ( n ) + D † i ( n ) B † ji ( n ) D j ( n + n ∓ (cid:17) . (7.1)The first two supercharges, q L ± , act on the left end of the mesonic string, with q L + raising thevalue of K by one unit and q L − lowering it by one unit. Similarly, q R ± act on the right endof the mesonic string, and q R + raises K by one unit while q R − lowers it by the same amount.That K changes in this fashion under the action of (7.1) means that[ P + , q L ± ] = ± L q L ± , [ P + , q R ± ] = ± L q R ± . (7.2)The left supercharges and the right supercharges each separately generate an osp (1 | osp (1 |
4) algebra. The osp (1 | q L − , q L + ] = [ q R − , q R + ] = 12 . (7.3)To see this, let us first compute [ q L − , q L + ] by plugging in the definitions (7.1):[ q L − , q L + ] = 12 N (cid:88) n ,n p ,p (cid:32)(cid:104) C † j ( n + n − B ij ( n ) C i ( n ) , C † l ( p ) B † lk ( p ) C k ( p + p − (cid:105) + (cid:104) C † j ( n ) B † ji ( n ) C i ( n + n + 1) , C † l ( p + p + 1) B kl ( p ) C k ( p ) (cid:105)(cid:33) . (7.4)There are several terms in the commutators on the right hand side, but most of them areeither suppressed by factors of 1 /N or annihilate all single-trace mesonic states. From the The quantities denoted by q in this section are unrelated to the fundamental quarks q iα appearing in(1.1). N limit in the summand are N δ n ,p δ n ,p C † i ( n + n − C i ( p + p −
1) + δ n + n ,p + p C † i ( p ) B † ij ( p ) C k ( n ) B kj ( n ) − N δ n ,p δ n ,p C † i ( p + p + 1) C i ( n + n + 1) − δ n + n ,p + p C † i ( n ) B † ij ( n ) C k ( p ) b kj ( p )(7.5)where the first line comes from the first commutator in (7.4) while the second line comesfrom the second commutator in (7.4). The C † B † CB terms cancel under the summation.The C † C terms telescope, and so the full commutator (7.4) at leading order in large N is[ q L − , q L + ] = 12 (cid:88) n C † i ( k ) C i ( k ) = 12 , (7.6)where the second equality holds only on the single-trace mesonic states that involve only one C † operator. An analogous computation shows that [ q R − , q R + ] = when acting on single-tracemesonic states at leading order in 1 /N , thus concluding the proof of (7.3).Having established (7.3), it is straightforward to show that q L ± generate an osp (1 |
2) alge-bra and similarly for q R ± . In particular, focusing on q L ± , let Q ≡ q L − , Q ≡ q L + , (7.7)and define the symplectic form ω (2) AB ≡ ( iσ ) AB = ( − ), with A, B = 1 ,
2. The osp (1 |
2) isgenerated by the Q A as well as the sl (2 , R ) generators M AB ≡ { Q A , Q B } . (7.8)Eq. (7.3) can be written as [ Q A , Q B ] = ω (2) AB , which can be used to show that[ Q A , M BC ] = [ Q A , Q B Q C ] + [ Q A , Q C Q B ] = ω (2) AB Q C + ω (2) AC Q B , (7.9)as well as [ M AB , M CD ] = ω (2) AD M BC + ω (2) BC M AD + ω (2) AC M BD + ω (2) BD M AC . (7.10)Eqs. (7.8)–(7.10) are the defining equations of the osp (1 |
2) algebra. We will denote thisalgebra generated by { Q , Q , M , M , M } by osp (1 | L because it acts at the left end of39he mesonic string. Similarly, defining Q ≡ ( − F q R + , Q ≡ ( − F q R − , (7.11)where F is the fermion number operator, we find that { Q , Q , M , M , M } (with M AB defined as in (7.8)) also obey the commutation relations of an osp (1 |
2) algebra that we denoteby osp (1 | R because it acts at the right end of the mesonic string. , Furthermore, as we now show, the { Q A , M AB } where now A, B = 1 , . . . , osp (1 |
4) algebra when acting on the massive states. To begin, let us compute the commutatorbetween Q , Q and Q , Q . With α, β = ± , we have[ q Lα , ( − F q Rβ ] = − ( − F { q Lα , q Rβ } , (7.12)so let us then compute the anticommutator of q Lα with q Rβ : (cid:8) q Lα , q Rβ (cid:9) = 12 N (cid:88) n ,n p ,p (cid:34) (cid:110) C † j ( n + n + α ) B ij ( n ) C i ( n ) , D † k ( p ) B † lk ( p ) D l ( p + p − β ) (cid:111) + (cid:110) C † j ( n ) B † ji ( n ) C i ( n + n − α ) , D † k ( p + p + β ) B kl ( p ) D l ( p ) (cid:111) (cid:35) . (7.13)We only have to anti-commute the B and B † operators, which gives (cid:8) q Lα , q Rβ (cid:9) = − N (cid:88) n ,n ,n ,n δ n + n ,n + n + α + β × (cid:18) C † i ( n ) D † i ( n ) C j ( n ) D j ( n ) − N C † i ( n ) D † j ( n ) C i ( n ) D j ( n ) (cid:19) . (7.14)The matrix elements of C † i D † i C j D j are only of order N between two 2-bit states, and thematrix elements of C † i D † j C i D j are always of order 1 or smaller. Thus, for two single-trace We could’ve defined Q = q R − and Q = q R + and then { Q , Q , M , M , M } would’ve also definedan osp (1 |
2) algebra. We used the definition (7.11) instead because with this definition we can extend thesymmetry algebra to osp (1 |
4) when acting on the massive states. Note that the existence of an osp (1 | L algebra acting on the left end of the string and an osp (1 | R algebra acting on the right end of the string does not imply that there is a osp (1 | L ⊕ osp (1 | R symmetryalgebra acting on the single-trace mesonic states in the large N limit. We did not show, for instance, that M AB , with A, B = 1 , M CD , with C, D = 3 ,
4, so the two osp (1 |
2) algebras may not beindependent. We will show shortly that the two osp (1 |
2) act independently only on the massive states. | ψ (cid:105) and | χ (cid:105) , the matrix elements are (cid:104) χ | (cid:8) q Lα , q Rβ (cid:9) | ψ (cid:105) = − (cid:104) χ | ψ (cid:105) × | ψ (cid:105) and | χ (cid:105) are 2-bit0 otherwise , (7.15)at leading order in large N . This relation means that { q Lα , q Rβ } is a rank one operator. Itannihilates all the states except for the equal linear combination of all two-bit states of agiven K (if K is even) and it outputs an equal linear combination of two-bit states with K + α + β units of P + momentum. These states are massless and were defined in (4.19).Thus, we can write (cid:8) q Lα , q Rβ (cid:9) = − (cid:88) K even K (cid:114) K + α + βK | ζ K + α + β (cid:105) (cid:104) ζ K | . (7.16)Importantly, if we restrict ourselves to the massive sector, we simply have (cid:8) q Lα , q Rβ (cid:9) = 0 atleading order in 1 /N .From now on, let us restrict to the massive states only. Since (cid:8) q Lα , q Rβ (cid:9) = 0, we have from(7.12) that Q , commute with Q , , so[ Q A , Q B ] = 12 ω (4) AB , (7.17)where ω (4) AB = ( ⊗ ω (2) ) AB is a 4 × denotes the 2 × Q A , M BC ] = ω (4) AB Q C + ω (4) AC Q B , [ M AB , M CD ] = ω (4) AD M BC + ω (4) BC M AD + ω (4) AC M BD + ω (4) BD M AC , (7.18)which are the commutation relations defining the osp (1 |
4) algebra. Thus, { Q A , M AB } gen-erate an osp (1 |
4) algebra when acting on the massive states.When restricted to the massive sector, the osp (1 |
4) generators commute with P − atleading order in 1 /N . While we do not have a proof of this fact in full generality, we checkedit numerically for fixed K up to K = 35. 41 .2 osp (1 | representations of P − eigenstates Let us now connect the discussion of osp (1 |
4) with the degeneracies in the spectrum of mesonspresented in Figure 3. The massive states are acted on by osp (1 | osp (1 | N = 1 supersymmetry, because osp (1 |
4) is the N = 1 superconformal algebra in three dimensions. Given that the bosonicpart of osp (1 |
4) is sp (4 , R ) ∼ = so (3 , P − degeneracies occur both between states at the same P + andbetween states at different values of P + , so the first question we should ask is which osp (1 | P + . As already seen in (7.2), the Q A do not commute with P + ,but some of the bosonic M AB generators, namely those that are anti-commutators of Q A ’s ofopposite P + eigenvalues, do commute with P + . These are { M , M , M , M } , and theyform an su (2) × u (1) algebra. We can exhibit this algebra more clearly by defining J ≡ M + M , J ≡ i M + M J ≡ M − M ,D ≡ M − M . (7.19)They satisfy [ J i , J j ] = i(cid:15) ijk J k , [ D, J i ] = 0 , (7.20)so the J i generate su (2) while D generates the u (1). In the language of 3d CFTs, thegenerator D measures the scaling dimension while J = j ( j + 1) measures the spin j .Because J i and D commute with P + , each multiplet represented by a dot in Figure 3forms a representation of this su (2) × u (1). For instance, the P − = 3 / K = 6 is an su (2) triplet that can be split into three J eigenstates (where ( n , n , . . . , n N ) is a shortcutnotation for the state C † i ( n ) B † ij ( k ) · · · D † j ( k N ) | (cid:105) ):42igenvector P − D J − (1 , − (5 ,
1) + 2(3 , − , , ,
1) + 3(1 , , ,
32 32 -1(1 , − (5 ,
1) + 2(1 , , , − (1 , , , − (1 , , ,
1) + 2(3 , , ,
32 32 − (1 , − (5 ,
1) + 2(3 ,
3) + (1 , , , − (1 , , ,
1) + 4(1 , , , , ,
32 32 osp (1 |
4) that are familiar from N = 1 SCFTs are lowest-weightrepresentations that have the following tree-like structure. There is a unique “(superconfor-mal) primary” state with lowest D eigenvalue ∆ and some value of j . All the other statesin the irrep are generated by acting with a string of Q ’s on the primary state. The action ofeach Q increases ∆ by 1 / / K by 1. Indeed, from thecommutator between Q A and M BC in (7.9), one can see that D/L has the same commutationrelations with Q A as P + . It follows that P + − D/L commutes with the osp (1 |
4) algebra,and consequently for a given osp (1 |
4) representation, P + − D/L is a constant. This impliesthat increasing ∆ by 1 / K = 2 LP + by 1. Thus, the primarystate in each irrep of a given P − eigenvalue must be the state with lowest K . Explicitly acting with D on the states of smallest K for the various P − eigenvalues ofmassive states in Figure 3 reveals that these states have ∆ = 1 /
2, and they also have su (2)spin j = 0. Such osp (1 |
4) irreducible representations for which the primary has ∆ = 1 / j = 0 are usually referred to as “ osp (1 |
4) singletons” and are shorter than generic irreps.In SCFT language, a singleton correspond to the irrep consisting of a free massless scalar anda free massless Majorana fermion, together with their superconformal descendants. Underthe bosonic so (3 ,
2) algebra, an osp (1 |
4) singleton decomposes into a direct sum of two irreps,namely a scalar singleton (which corresponds to the free real scalar with ∆ = 1 / j = 0)and a fermionic singleton (which corresponds to the free Majorana fermion with ∆ = 1 and j = 1 / We can give a more refined description of the state counting in terms of the scaling dimension. Onecan show that the number of osp (1 |
4) primary states at a given K is F K − + ( − K , where { F n } ∞ n =1 is theFibonacci sequence with F = F = 1. Using the tree-like structure described above, this implies that at agiven K there are 2∆ (cid:0) F K − − + ( − K − − (cid:1) massive states of scaling dimension ∆. Summing over ∆and adding the (cid:98) K/ (cid:99) massless states, we recover the total of F K states when K is even and F K − K is odd. Most of these states are singlets, and for them it is obvious that j = 0. We checked explicitly that evenin cases in which the first time a P − eigenvalue appears in the spectrum as a doublet, the J eigenvalue stillvanishes. osp (1 |
4) singleton represen-tations by computing the eigenvalues of the so (3 ,
2) and osp (1 |
4) quadratic Casimir. The so (3 ,
2) quadratic Casimir is C so (3 , = − ω AC ω BD M AB M CD , (7.21)where ω AB = − ω AB , while the osp (1 |
4) one is C osp (1 | = − ω AC ω BD M AB M CD − ω AB Q A Q B . (7.22)(It is straightforward to check that C so (3 , commutes with all M AB and that C osp (1 | com-mutes with the Q A .) The eigenvalue of C so (3 , for an so (3 ,
2) irrep whose primary state has D = ∆ and J = j ( j + 1) is λ so (3 , (∆ , j ) = ∆(∆ −
3) + j ( j + 1) , (7.23)while the eigenvalue of C osp (1 | for an osp (1 |
4) irrep whose primary state has D = ∆ and J = j ( j + 1) is λ osp (1 | (∆ , j ) = ∆(∆ −
2) + j ( j + 1) . (7.24)For the osp (1 |
4) singleton representation we have C so (3 , = λ so (3 , ( ,
0) = λ so (3 , (1 , ) = 5 / C osp (1 | = λ osp (1 | ( ,
0) = 3 /
4. We checked that we obtain these valueswhen acting with (7.21) and (7.22) explicitly on all the massive states.While the osp (1 |
4) symmetry explains the degeneracies in the meson spectrum, it doesnot explain the degeneracies pointed out in Section 6.3 between the osp (1 |
4) primary mesonstates and gluinoball states. As described in Section 6.3, these latter degeneracies can beseen from the current algebra approach. For a partial alternative explanation that involvesoperators related to the osp (1 |
4) charges, see Appendix A.44
Making the quarks massive
In order to further probe our model, we would like to study the meson spectrum as a functionof the fundamental fermion mass. If we make y fund > y adj = 0, we expectthe theory to retain some of the non-trivial dynamical properties of the massless adjointQCD . We will note that some of the DLCQ degeneracies between the single-trace andmulti-trace states are not lifted and that the meson spectrum is continuous above a certainthreshold. This provides new quantitative evidence, along the lines of Footnote 4 of [20],that the fundamental string tension vanishes in the massless adjoint QCD .For y fund = y adj = 0, we have found an infinite series of single-trace mesonic states withthe same value of P − as some fermionic gluinoballs at an odd value of K (see Figures 3 and4). These include the bosonic mesons with resolution parameters K + 1 , K + 3 , K + 5 , . . . andthe fermionic mesons with resolution parameters K + 2 , K + 4 , K + 4 , . . . . Since the spectrumof the model contains massless mesons, both bosonic and fermionic, this is consistent withthe pattern of degeneracies between single-trace and multi-trace states. In particular, somemassive bosonic mesons are degenerate with double-trace states of a fermionic gluinoballand a massless fermionic meson, and some massive fermionic mesons are degenerate withdouble-trace states of a fermionic gluinoball and a massless bosonic meson. As explained inSection 6.3, when a meson is degenerate with a bosonic gluinoball, it is because the bosonicgluinoball is in turn degenerate with a multi-trace state formed from fermionic gluinoballs.This follows from the Kac-Moody approach reviewed in Section 4. We may thus think ofthe degeneracy as being between a fermionic meson and a triple-trace state formed from amassless fermionic meson and two fermionic gluinoballs.After we make the fundamental fermions massive, there are no more massless mesons inthe spectrum; yet, some of the degeneracies survive. We continue to find that some bosonicmesons are degenerate with double-trace states of the fermionic gluinoballs and massivefermionic mesons. These degeneracies hold for any y fund >
0, and an example at y fund = 1 isshown in Figure 5c and 5d, where the degenerate states are marked in orange. In fact, wefind that every double-trace state formed from a fermionic meson and a fermionic gluinoballis degenerate with a bosonic meson.Likewise, we continue to find that some fermionic mesons are degenerate with triple-tracestates built from a fermionic meson and two fermionic gluinoballs. Almost all of the triple-trace states of this form are degenerate with single-trace fermionic mesons with the same45 .00 0.02 0.04 0.06 0.08 0.108.0115202530354045505560 (a) (b) .00 0.02 0.04 0.06 0.08 0.106.558.0315202530354045505560 (c) (d) Figure 5: The squared masses of single-trace meson states in theory T (the theory with anadjoint and a fundamental fermion) with y adj = 0 and y fund = 1. The blue points are thestates which are degenerate with multi-string states in the theory T (cid:48) defined in Section 8.2.The orange points are the states which additionally are degenerate with a multi-trace stateformed from a meson and one or more gluinoballs in theory T . The dashed lines show thethreshold at M ≈ g Nπ above which the extrapolated spectrum is continuous. There is abosonic Z -odd state shown with green dots that lies below this threshold. Its extrapolatedsquared mass is M ≈ . g Nπ . 47 = ∗∗ (a) q q ∼ = q q ∗∗ (b) Figure 6: The degeneracies among different states in theory T , which survive at y fund > P + . Examples at y fund = 1 are shown in Figure 5a and 5b, with the degenerate statesbeing marked in orange.Qualitatively, we may think of the degeneracies we have found in terms of adjoint flux lineswith vanishing energy. Figure 6 shows schematically how an adjoint flux line contributingzero mass to a state could lead to degeneracies between a single-trace gluinoball and adouble-trace gluinoball, or between a single-trace meson and a double-trace state composedof a meson and a gluinoball. Similar pictures could be drawn for multi-trace states withmore components. The relations between the eigenvalues at y fund > P − eigenvalues of n > P − eigenvalues of n = 1gluinoballs can be traced back to the fact that the massive spectrum of the adjoint QCD theory T adj is part of the massive spectrum of the theory T fund whose matter content consistsof N massless Dirac fermions, as explained in [19]. In particular, the P − eigenvalues of n > T adj at resolution parameter K correspond to T fund states at resolution parameter K + n − B † ij (1)in the construction of the states in T adj is replaced by C † αi (1) D † jβ (1) in T fund . Intuitively, each48 † ij (1) serves as a breaking point of the closed gluinoball string, and thus if B † ij (1) appears n − T adj , then in T fund we end up with n − T adj and T fund to a massive fundamental Dirac fermionand compare the two resulting theories. One of them, denoted by T , has a Dirac fermionwith mass m q coupled to a massless adjoint and it is the theory we studied in the previoussubsection; the other, denoted by T (cid:48) , has N + 1 Dirac fermions, the first N of which aremassless and the ( N + 1)st with mass m q . The mass spectrum of the theory T is again partof the mass spectrum of the theory T (cid:48) , generalizing the result of [19]. At large N , it is easierto see various relations between the P − eigenvalues in the theory T (cid:48) .While we leave a careful analysis for the future, let us describe how the correspondencebetween the meson states in the theory of interest T and states in T (cid:48) works at a qualitativelevel. We can construct the states by considering KM primaries only with respect to theadjoint contribution to the SU ( N ) current J ij . Thus, in the current algebra construction,each meson state in T will have some number of B † ij (1)’s which in T (cid:48) will be replaced by C † αi (1) D † jβ (1), thus increasing K by one unit and breaking the string.Thus, a meson with m B † (1)’s in T becomes an m + 1-string state in T (cid:48) , which we canwrite schematically asmeson with m B † (1)’s in T ←→ [H − L][L − L] · · · [L − L][L − H] (cid:124) (cid:123)(cid:122) (cid:125) m + 1 factors in T (cid:48) , (8.1)where [L − L] denotes a string where the quarks at both ends are light (i.e. massless), while[H − L] and [L − H] denote strings where the quark at the end marked with H is heavy(i.e. of mass m q ) while the one at the other end is light. If the state on the LHS of (8.1) isat resolution parameter K in T , the state on the RHS is at resolution parameter K + m in T (cid:48) . Note that with the same notation, we have n = 1 gluinoball in T ←→ [L − L] in T (cid:48) , (8.2)where if the resolution parameters are K and K + 1 on the LHS and RHS, respectively. SeeFigure 7 for a diagramatic representation of the relations (8.1)–(8.2).In (8.1), the states in T with even m are bosons and those with odd m are fermions.Because for multi-string states in T (cid:48) the values of P − add, based on (8.1) and (8.2) weexpect the following relations between the eigenvalues of P − within theory T : • There are bosonic mesons (namely with m = 0) whose P − eigenvalues are unrelated to49 q ∼ = qq (a) ∼ = (b) Figure 7: The degeneracies among different states can be interpreted as splitting a string. In(a), we show an m = 1 meson in theory T which splits into a two-string state [H − L][L − H]in theory T (cid:48) . In (b), we show an n = 1 gluinoball in T which splits into a one-string state[L − L] in theory T (cid:48) . Diagram (a) illustrates the origin of the continuous spectrum of mesonsin theory T .other states in theory T . These states are marked in black in Figures 5c and 5d, andthey correspond to states of the form [H − H] in T (cid:48) . • There are fermionic mesons (namely with m = 1) whose P − eigenvalues are unrelatedto other states in theory T . They are marked in blue in Figures 5a and 5b, and theycorrespond to states of the form [H − L][L − H] in T (cid:48) . • The remaining bosonic mesons (namely with m = 2 , , , . . . ) are degenerate with asum of n = 1 fermionic mesons and an odd number m − n = 1 gluinoballs, withthe same total K . They are marked in orange in Figures 5c and 5d. • The remaining fermionic mesons (namely with m = 1 , , , . . . ) are degenerate with asum of n = 1 fermionic mesons and an even number m − n = 1 gluinoballs, withthe same total K . They are marked in orange in Figures 5a and 5b.The last two bullet points explain the degeneracies noticed in the previous section.The second bullet point above has remarkable implications. It states that the P − eigen-values of all fermionic mesons with m = 1 at resolution parameter K in theory T can bewritten as sums of two eigenvalues of [H − L] states in theory T (cid:48) with total resolution pa-rameter K + 1. From this information, we can reconstruct the [H − L] spectrum of P − eigenvalues in theory T (cid:48) .For example, at K = 3 in T we have only meson state, C † (1) B † (1) D † (1) with P − eigenvalue 2 y fund . The only possibility is that this is the sum of two K = 2 [H − L] states50n T (cid:48) each with eigenvalue p = y fund . (8.3)At K = 5, in T we have the following eigenvalues9 + 20 y fund ± (cid:112)
81 + 24 y fund + 16 y . (8.4)These must be the sum of one K = 2 eigenvalue and one K = 4 eigenvalue in T (cid:48) . Since weknow that the only K = 2 eigenvalue is given by p , we should subtract it from (8.4) to findthe K = 4 eigenvalues in T (cid:48) : p = 9 + 8 y fund + (cid:112)
81 + 24 y fund + 16 y , p = 9 + 8 y fund − (cid:112)
81 + 24 y fund + 16 y . (8.5)We can already have a check on these equations: the quantities 2 p , 2 p , and p + p wouldbe P − eigenvalues of two-string states in T (cid:48) at K = 8, so they must appear in the mesonspectrum in T at K = 7. One can check that this is indeed the case. Continuing tohigher K analytically is not feasible because the P − eigenvalues are roots of high orderpolynomials, but one can continue this procedure and reconstruct the spectrum of [H − L]states in the T (cid:48) theory numerically for given y fund . For y fund = 1, we show this reconstructedspectrum in Figure 8. It would be interesting to provide a confirmation of this plot bydirectly diagonalizing P − in the theory T (cid:48) .An immediate implication of these results is that the spectrum of the fermionic mesons in T is continuous, and the threshold at which the continuum starts is 4 times the squared massof the lightest [H − L] state in T (cid:48) . As shown in Figure 8, for y fund = 1 the latter extrapolatesto ≈ g Nπ in the continuum limit. The lowest threshold at M ≈ g Nπ is clearly seen notonly in the fermionic meson sectors of T , but also in the bosonic ones (see Figure 5). Also,there appears to be a bosonic Z -odd meson with squared mass ≈ . g Nπ , which lies belowthis threshold.For y fund (cid:29) theory. One implication of the vanishing fundamental stringtension in massless adjoint QCD is that the spectrum of single-meson states at large N intheory T becomes continuous above a certain threshold [20]. Now we have established thisresult quantitatively for any y fund (and in particular for y fund (cid:29) .00 0.02 0.04 0.06 0.08 0.102.11.72025303540 Figure 8: The inferred masses of bosonic single-string states in the theory T (cid:48) which containexactly one massive quark. The massive quark has m = g Nπ , corresponding to y fund = 1.52f the fact that, at large N , the P − eigenvalues of every fermionic meson in T are sumsof eigenvalues of [H − L] states in T (cid:48) . This provides a new confirmation of the screeningphenomenon directly through the studies of meson spectra. In this paper, we used DLCQ to study a 2d SU( N ) gauge theory with a Majorana fermionin the adjoint representation of SU( N ) and N f quarks in the fundamental representation.With anti-periodic boundary conditions for the fermions, we diagonalized the light-conecomponents of the momentum P + and P − in the large N limit, and extracted the massspectrum of the single-trace gluinoball and meson states of this theory. When the adjoint andfundamental fermions are massless, we observed an intricate pattern of exact degeneraciesof the P − eigenvalues, both between mesons with different P + eigenvalues, between mesonsand gluinoballs, and between single-trace and multi-trace states. We provided two seeminglyindependent explanations for these degeneracies, one building on the Kac-Moody approachof [19] and the other based on an osp (1 |
4) symmetry present in this model at large N .Under the osp (1 |
4) symmetry, we found that the single-trace meson states transform ininfinite-dimensional unitary representations referred to as osp (1 |
4) singletons.Lastly, we noticed that when the fundamental quark mass parameter y fund > y adj = 0, some of the degeneracies between the single-tracemesons and double-trace states comprised of a single-trace gluinoball and a single-tracemeson still survive. These are thus non-trivial examples of threshold states in our 2d modelfor QCD. Moreover, by relating the P − eigenvalues of meson states to eigenvalues of statesin a theory with N massless fundamental fermions and a massive one, we showed that themeson spectrum is continuous above a certain threshold, with the fermionic mesons havingno discrete states below the bottom of the continuum. The presence of this continuumprovides a direct confirmation of the screening of the fundamental flux line by the masslessadjoint fermions.There are various unanswered questions which we hope to return to in the future. Whileour numerical analysis was limited to the large N limit, it would be interesting to performa similar analysis at finite N generalizing the study of [39]. The Kac-Moody analysis ofSection 4 indicates that the same pattern of degeneracies in the meson sector persists atfinite N as well. On the other hand, the osp (1 |
4) symmetry we observed was constructedonly in the large N limit. Thus, it would be interesting to explore whether the osp (1 | N as well and, if so, construct the symmetry generators. Moregenerally, one can aim to combine the osp (1 |
4) analysis with the Kac-Moody approach.It would also be interesting to consider various generalizations of the model presentedhere. For instance, one can consider quarks in different representations of the gauge groupor gauge theories with different gauge groups. Another generalization would be to considerperiodic boundary conditions for the fermions in the light-cone direction parameterized by x − . The current algebra approach of Section 4 should generalize to this case, suggesting thata pattern of degeneracies similar to the one we noticed would still be present. Acknowledgments
We are grateful to Thomas Dumitrescu, Zohar Komargodski, Fedor Popov, and AndreiSmilga for useful discussions. RD and SSP were supported in part by the Simons Foun-dation Grant No. 488653, and by the US NSF under Grant No. PHY-1820651. RD wasalso supported in part by an NSF Graduate Research Fellowship. The research of IRK wassupported in part by the US NSF under Grant No. PHY-1914860.
A Degeneracy between mesons and gluinoballs
In this Appendix, we provide an alternative partial explanation for the degeneracies betweengluinoballs and osp (1 |
4) primary mesons observed in Section 6.3. As discussed there, themeson primary states have P − eigenvalues equal to a sum of one or two P − eigenvaluesof fermionic gluinoballs with one less unit of total P + . Clearly q L ± and q R ± cannot be usedto construct the relevant gluinoball states from the meson primaries, since each term inboth operators contains a fundamental creation operator and a fundamental annihilationoperator. However, we note that the terms of q L ± + q R ± can be found in the discretization ofthe continuum operator ˜ q ± = 1 √ N (cid:90) πL dx − e ± ix − / (2 L ) v † i ψ ij v j . (A.1)In addition, the discretization of this operator contains the terms˜ q ± ⊃ − π ) / √ N (cid:88) n ,n (cid:16) B † ij ( n + n ± C i ( n ) D j ( n ) + C † i ( n ) D † j ( n ) B ij ( n + n ∓ (cid:17) , (A.2)54hich can couple our mesonic states to the gluinoball states. We define the following string-closing and string-opening operators based on terms of ˜ q − : q close = (cid:88) n ,n B † ij ( n + n − C i ( n ) D j ( n ) , (A.3) q open = (cid:88) n ,n C † i ( n ) D † j ( n ) B ij ( n + n − . (A.4)The operator q close converts a meson at K to a single-trace gluinoball at K −
1. In fact,if we act on a bosonic meson singlet state at K with ˜ q − , we obtain the corresponding single-trace fermionic gluinoball at K − P − . For instance, at K = 6, thereis a meson primary state with P − = 1, | ψ (cid:105) = 12 √ N (cid:18) C † i (1) D † i (5) − C † i (5) D † i (1) + 1 N C † i (1) (cid:16) B † ij (1) B † jk (3) + B † ij (3) B † jk (1) (cid:17) D † k (1) (cid:19) | (cid:105) . (A.5)Acting with q close , we find q close | ψ (cid:105) ∝ √ N B † ij (1) B † jk (1) B † ki (3) | (cid:105) , (A.6)which is a gluinoball eigenstate with P − = 1.For fermionic meson primaries which are degenerate with double-trace gluinoball states,acting with q close gives a single-trace bosonic gluinoball state with the same P − eigenvalue.For instance, at K = 9, there are two meson primary states with P − = 4. These are bothdegenerate with a double-trace gluinoball state at K = 8,Tr (cid:0) B † (1) (cid:1) Tr (cid:0) B † (1) (cid:1) | (cid:105) . (A.7)This double-trace state is in turn degenerate with two single-trace bosonic gluinoballs at K = 8. We can choose a basis for the two meson primaries at K = 9 and these two bosonicgluinoballs at K = 8 such that one of the meson primaries is annihilated by q close , and theother gives one of the two degenerate bosonic gluinoballs.Finally, fermionic meson primaries which are not degenerate with double-trace gluinoballstates are annihilated by q close . The P − eigenvalues of these meson primaries are equal todouble the P − eigenvalues of a fermionic gluinoball, and thus the corresponding double-tracestate would be null by the fermion statistics.In summary, we see that the commutator [ q close , P − ] annihilates all meson primary states.55f we project onto meson primary states before acting with q close , then we will have anoperator which commutes with P − . We can construct such an operator using the observationthat all meson primaries have ∆ = . Since the eigenvalues of D are all half-integers, O ≡ ∞ (cid:89) n =1 (cid:18) − D − / n (cid:19) = sin(2 π ( D − / π ( D − /
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