(2+1)-Dimensional Yang-Mills Theory and Form Factor Perturbation Theory
aa r X i v : . [ h e p - t h ] S e p (2 + 1) -Dimensional Yang-Mills Theory and Form Factor Perturbation Theory Axel Cort´es Cubero ∗ Baruch College, The City University of New York, 17 Lexington Avenue, New York, NY 10010, U.S.A. andThe Graduate School and University Center, The City University of New York, 365 Fifth Avenue, New York, NY 10016, U.S.A.
We study Yang Mills theory in 2+1 dimensions, as an array of coupled (1+1)-dimensional principal chiralsigma models. This can be understood as an anisotropic limit where one of the space-time dimensions isdiscrete and the others are continuous. The SU ( N ) × SU ( N ) principal chiral sigma model in 1+1 dimensionsis integrable, asymptotically free and has massive excitations. New exact form factors and correlation functionsof the sigma model have recently been found by the author and P. Orland. In this paper, we use these new resultsto calculate physical quantities in (2+1)-dimensional Yang-Mills theory, generalizing previous SU (2) results byOrland, which include the string tensions and the low-lying glueball spectrum. We also present a new approachto calculate two-point correlation functions of operators using the light glueball states. The anisotropy of thetheory yields different correlation functions for operators separated in the x and x -directions. PACS numbers: 2.30.IK, 03.65.Ge, 11.10.Kk, 11.55.Bq, 11.15.-q
I. INTRODUCTION
In this paper, we study an anisotropic version of (2+1)-dimensional Yang-Mills theory. The anisotropy is given by a longi-tudinal rescaling of coordinates of the form x , → λx , , and x → x . The gauge fields transform as A , → (1 /λ ) A , , A → A . The strength of the interactions is different in different directions. We explore the highly anisotropic regime, where λ → .We realize this rescaling by starting with the Kogut-Susskind Hamiltonian formulation of lattice gauge theory, with latticespacing a . The rescaling of coordinates amounts to taking the continuum limit in the x and x -directions, with the latticespacing rescaling as λa .We view the anisotropic model as an array of two-dimensional field theories, coupled together to form a higher-dimensionaltheory. The strength of the coupling between these two-dimensional models depends on the rescaling parameter λ . The two-dimensional theory is the principal chiral sigma model (PCSM)[1]. The PCSM is known to be integrable, and this property hasbeen exploited to find exact results [2], [3]. The main goal of our program is to use exact results from the PCSM to calculatephysical quantities in anisotropic QCD, finding corrections for small λ .This anisotropic regime of Yang-Mills theory has been studied extensively by P. Orland. In Ref. [1] it was established thatthe anisotropic theory is equivalent to an array of coupled PCSM’s, and it was shown that the model confines quarks and has amass gap. In references [4], and [5] the string tensions for quark-antiquark pairs was found for the SU (2) gauge group. In Ref.[6] the low-lying glueball spectrum was found for SU (2) . In this paper we generalize all these results to all N . This is doneusing new form factors of the PCSM that were found in Ref. [7]. We also use the light glueball states to calculate long-distancecorrelation functions of gauge-invariant operators.The longitudinal rescaling of coordinates is inspired by a similar investigation in 3+1 dimensions by Verlinde and Verlinde [8],in the context of heavy ion collisions. A similar anisotropic limit was used by McLerran and Venugopalan in their derivation ofthe Color Glass Condensate picture [9]. An anisotropic theory has been explored in Ref. [10], where the anisotropy is producedby an external magnetic field.This approach is especially interesting for (2 + 1) -dimensional QCD, since there are two different coupling constants for thegluon field, but they are both small compared to the cutoff. This makes our approach fundamentally different from other analyticstudies of (2 + 1) -dimensional QCD (which are generally at large dimensionless coupling) [11],[12]. Recently, Karabali, Nairand Yelnikov [13] have computed corrections to the results in [11], in powers of the coupling constant. Their approach couldeventually be used to study confinement at weak coupling.Physical quantities in the anisotropic gauge theory can be evaluated in the context of form-factor perturbation theory [14],[15], [16]. The gauge theory with λ = 0 is integrable. The S-matrix, some form factors and correlation functions of the PCSMare known. We do a perturbative expansion in powers of λ , rather than the Yang-Mills coupling constant. The perturbationtheory starts from an integrable, rather than a free theory.A very similar approach has been used by Konik and Adamov [17], and James and Konik [18] to examine the 3-dimensionalIsing model as an array of 2-dimensional chains. Here they have successfully computed critical exponents and the entanglement ∗ Electronic address: acortes˙[email protected] entropy and spectra using an improved version of the truncated conformal spectrum approach.In the next section, we present a discussion of the longitudinally-rescaled Yang-Mills theory. We show how the rescaled theoryis equivalent to an array of integrable models. This equivalence is shown in the axial gauge on the Kogut-Susskind Hamiltonian.In Section III, we compute the string tension for a static quark-antiquark pair separated in the x -direction only. In Section IV,we calculate the string tension of a quark-antiquark pair separated in the x -direction. These string tensions are different becausethe theory is anisotropic. In Section V we compute the mass spectrum of the lightest glueball states. These results generalizeOrland’s SU (2) results to SU ( N ) .In Section VI, we calculate the long-distance two-point correlation function of two gauge-invariant operators separated in the x -direction. This calculation is inspired by a similar calculation for the 2-dimensional Ising model in an external magnetic field,by Bhaseen and Tsvelik [19].In Section VII, we propose a method for calculating correlation functions in the x -direction. This is done by defining a trans-fer matrix that describes the evolution of the system along the x -direction. The partition function and correlation functions canbe found, in principle, by diagonalizing the transfer matrix in the basis of physical states. We are only able to find an expressionfor the transfer matrix using the light glueball states from Section V. However, this matrix is very difficult to diagonalize. Theproblem is reduced to an integral eigenvalue equation, which we leave unsolved.We present our conclusions in the last section. A short summary of the S-matrix and form factors of the PCSM is given in theappendix. II. LONGITUDINALLY RESCALED YANG-MILLS HAMILTONIAN IN THE AXIAL GAUGE
In the Kogut-Susskind lattice Hamiltonian formulation [20], there are SU ( N ) -valued gauge fields U ( x ) j , and electric-fieldoperators l ( x ) bj in the adjoint representation of SU ( N ) , at every space link ( x, j ) , for j = 1 , and b = 1 , , . . . , N − . Thesesatisfy the commutation relations (cid:2) l ( x ) bj , l ( y ) ck (cid:3) = iδ x y δ j k f dbc l ( x ) j d , (cid:2) l ( x ) bj , U ( y ) k (cid:3) = − δ x y δ j k t b U ( x ) j . The gauge fields in the x direction are fixed by the temporal gauge condition U = 1 . The Hamiltonian is obtained by takingthe continuum limit of the Wilson action in the time direction. The Kogut-Susskind Hamiltonian, inside a box of size a L × L ,is H = L , L X x ,x = − L , L X j =1 N − X b =1 g a (cid:2) l ( x ) bj (cid:3) − L , L X x ,x = − L , L g a Tr (cid:2) U ( x ) U ( x + ˆ1) U † ( x + a ˆ2) U † ( x ) + C.C. (cid:3) . (2.1)where L , L are even integers.In temporal gauge, physical states, Ψ , are those which satisfy Gauss’s law: X j =1 [ D j l j ( x )] b Ψ = 0 , (2.2)where [ D j l j ( x )] b = l j ( x ) − R j ( x − ˆ ja ) cb l j ( x − ˆ ja ) c , where R j ( x ) cb t c is the adjoint representation of the gauge field, R j ( x ) cb t c = U j ( x ) t b U † j ( x ) . We find the electric field component l by solving Gauss’s law (2.2); and then impose the axial gauge U ( x ) = , yielding l ( x , x ) b = x X y = − L [ D l ( y , x )] b . (2.3)There is a global invariance left after the axial gauge fixing: L X x = − L [ D l ( x , x )] b Ψ = 0 . (2.4)The lattice Hamiltonian in axial gauge is found by substituting the new nonlocal expression for the electric field (2.3) into(2.1): H = L X x = − L L X x = − L g a [ l ( x )] − L X x = − L L X x = − L g a [Tr U ( x , x ) † U ( x + a, x ) + c.c. ] − g a L X x = − L L X y = − L L X x = − L | x − y | [ l ( x , x ) − R ( x , x − a ) l ( x , x − a )] × [ l ( y , x ) − R ( y , x − a ) l ( y , x − a )] . (2.5)The Hamiltonian (2.5) is nonlocal in x , and depends only on the transverse degrees of freedom U , l .We now explore anisotropic Yang-Mills theory by longitudinally rescaling the coordinates. This is a summary of the calcula-tion done in [1], [4]. The longitudinally-rescaled lattice has spacing λa in the x , directions and spacing a in the x -direction.In the λ → limit, it is sensible to treat x and x as continuous directions, and x discrete.Longitudinally rescaling the lattice Hamiltonian (2.5), gives H = H + λ H , where H = L X x = − L L X x = − L g a [ l ( x )] − L X x = − L L X x = − L g a [Tr U ( x , x ) † U ( x + a, x ) + c.c. ] ,H = − λ g a L X x = − L L X y = − L L X x = − L | x − y |× [ l ( x , x ) − R ( x , x − a ) l ( x , x − a )][ l ( y , x ) − R ( y , x − a ) l ( y , x − a )] . Henceforth we drop the Lorentz index from U , l .We treat H as a perturbation. In the interaction representation, U satisfies the Heisenberg equation of motion, ∂ U = i [ H , U ] . The solution of this equation of motion is l ( x , x ) b = iag Tr t b ∂ U ( x , x ) U ( x , x ) † , R ( x , x ) cb l ( x , x ) c = iag Tr t b U ( x , x ) † ∂ U ( x , x ) . (2.6)Substituting (2.6) into H , and taking the continuum limit in the x direction, we find H = X x H ( x ) = X x Z dx g n(cid:2) j L ( x , x ) b (cid:3) + (cid:2) j L ( x , x ) b (cid:3) o = X x Z dx g n(cid:2) j R ( x , x ) b (cid:3) + (cid:2) j R ( x , x ) b (cid:3) o , where j Lµ ( x ) b = i Tr t b ∂ µ U ( x ) U ( x ) † , j Rµ ( x ) b = i Tr t b U ( x ) † ∂ µ U ( x ) , (2.7)where µ = 0 , .We now note that H ( x ) is the Hamiltonian of a (1+1)-dimensional PCSM located at x . The PCSM has the action L PCSM = Z d x g η µν Tr ∂ µ U † ∂ ν U. (2.8)This model has a global SU ( N ) × SU ( N ) symmetry given by the transformation U ( x ) → V L U ( x ) V R , where V L,R ∈ SU ( N ) .The Noether currents corresponding to these global symmetries are j L,R given in (2.7). The Hamiltonian corresponding to theaction (2.8) of a single PCSM at fixed x is H ( x ) . The unperturbed Hamiltonian, H , is an array of PCSM’s, one at each valueof x , H = X x H ( x ) = X x H PCSM ( x ) . It is important to note that the PCSM is known to be integrable and to have a mass gap. We call m the mass of the elementaryparticles of the sigma model.The residual Gauss’s law, (2.4) becomes Z dx (cid:2) j L ( x , x ) b − j R ( x , x − a ) b (cid:3) Ψ = 0 , (2.9)for each value of x , when x is continuous.Using (2.6), we write the interaction Hamiltonian H in the continuous x limit: H = X x Z dx Z dy g a | x − y | (cid:2) j L ( x , x ) − j R ( x , x − a ) (cid:3) (cid:2) j L ( y , x ) − j R ( y , x − a ) (cid:3) . (2.10)The Hamiltonian (2.10) couples adjacent sigma models, which allows particles to propagate in the x -direction. The coupling issuppressed in the λ → limit.There are several important points to mention about the Hamiltonian H = H + λ H . It has been shown that this anisotropicmodel confines quarks. The string tensions are different if there is a quark-antiquark pair separated in the x or the x -direction.We call these the horizontal string tension, σ H and the vertical string tension σ V , respectively. To lowest order in λ , these aregiven by [4], [5], σ H = λ g a C N , σ V = ma , (2.11)where C N is the smallest eigenvalue of the Casimir operator of SU ( N ) . In Sections III and IV, we compute quantum correctionsto the string tensions (2.11) using the exact form factors of the sigma model (shown in the appendix). This calculation is ageneralization of the results computed by Orland in References [4] and [5] for the gauge group SU (2) . Orland’s results werecomputed using the form factors of the O (4) -symmetric nonlinear sigma model [21], by virtue of SU (2) × SU (2) ≃ O (4) .Recently some form factors of the PCSM for general N > have been found [7], which allow us to generalize Orland’s resultto the gauge group SU ( N ) .The anisotropic Hamiltonian has a mass gap. The lightest gauge invariant excitation is a glueball composed of a sigma-modelparticle-antiparticle pair. The light gluon mass spectrum was calculated by Orland for the gauge group SU(2) in Ref. [6]. Theglueball masses are of the form M n = 2 m + E n , where E n is the binding energy of the particle-antiparticle pair. The determination of the spectrum of energies, E n , involvedknowledge of the exact S-matrix of the O(4) sigma model [22]. We generalize this calculation for N > in Section V, using theexact S-matrix of the PCSM found by Wiegmann [3]. III. THE HORIZONTAL STRING TENSION
In this section we compute quantum corrections to the string tension σ H . This calculation has been done before, in Reference[4], for N = 2 using the form factors of the O (4) sigma model. In this section we generalize these results for N > .It is convenient to rewrite the Hamiltonian (2.10) by reintroducing the auxiliary field Φ = − A , such that H = X x Z dx (cid:26) g a ∂ Φ( x , x ) ∂ Φ( x , x ) − j L ( x , x )Φ( x , x ) − j R ( x , x )Φ( x , x + a ) (cid:27) . (3.1)By integrating out the auxiliary field, Φ , we see the Hamiltonians, (3.1) and (2.10) are equivalent.We can easily introduce static quarks into the Hamiltonian (3.1) by coupling them to the auxiliary field, Φ . Our goal is to findthe potential energy of a quark-antiquark pair separated only in the x -direction. By integrating out the sigma model degrees offreedom, we can find the quantum corrections to the string tension σ H . The Hamiltonian with a static quark of charge q at thespace point ( u , u ) , and an antiquark of charge q ′ at the space-time point ( v , v ) , is H = X x Z dx (cid:26) g a ∂ Φ( x , x ) ∂ Φ( x , x ) − j L ( x , x )Φ( x , x ) − j R ( x , x )Φ( x , x + a )+ g q Φ( u , u ) − g q ′ Φ( v , v ) (cid:9) . (3.2)With these static quarks, the residual gauss law on physical states is modified to: Z dx (cid:2) j L ( x , x ) b − j R ( x , x − a ) b + q b δ ( x − u ) δ x u − q ′ b δ ( x − v ) δ x v (cid:3) Ψ = 0 . (3.3)To find the string tension, σ H , we set u = v , and integrate out the sigma model field, U . We obtain an effective action, S eff (Φ) , by e iS eff (Φ) = h |T e i R dx λ H | i , (3.4)where T stands for time ordering. The field Φ in (3.4) is treated as a background classical field. Expanding (3.4) in powers of λ ,up to quartic order, we find S eff ( A ) ≈ − iλ X x Z d x g a ∂ Φ + iλ S (2) (Φ) + O ( λ ) − λ X x Z d x (cid:2) g q ( x ) Φ( x , u , u ) − g q ′ ( x )Φ( x , v , v ) (cid:3) , (3.5)where iS (2) ≡ − X x Z d xd y D ( x , x , y , y , x ) acef Φ( x , x , x ) ac Φ( y , y , x ) ef , where D ( x , x , y , y , x ) acef ≡ h |T j L ( x , x , x ) ac j L ( y , y , x ) ef | i . (3.6)We compute the correlation function (3.6) by introducing a complete set of intermediate states between the two operators.The non-time-ordered correlation function is given by h | j L ( x ) ac j L ( y ) ef | i = ∞ X M =1 N ( M !) Z dθ . . . dθ M (2 π ) M e − i ( x − y ) · [ P Mj =1 p j ] ×h | j Lµ (0) a c | A, θ , b , a ; . . . ; A, θ M , b M a M ; P, θ M +1 , a M +1 , b M +1 ; . . . ; P, θ M , a M , b M i× (cid:2) h | j Lν (0) e f | A, θ , b , a ; . . . ; A, θ M , b M a M ; P, θ M +1 , a M +1 , b M +1 ; . . . ; P, θ M , a M , b M i (cid:3) ∗ . The correlation function (3.6) can be found exactly at large N using the form factors from Ref. [23]. For general N < ∞ , we canonly calculate a large-distance approximation, using the two-particle form factor (shown in the appendix). At large distances, itis sufficient to compute only the first intermediate state, with one particle and one antiparticle.The sigma-model form factor with one particle and one antiparticle is (see the appendix) h | j Lµ ( x ) ac | A, θ , b , a ; P, θ , a , b i = ( p − p ) µ (cid:18) δ a a δ c a δ b b − N δ a c δ a a δ b b (cid:19) e − ix · ( p + p ) × πi ( θ + πi ) exp Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin [ ξ ( πi − θ ) / π ]sinh ξ . (3.7)Inserting (3.7) into (3.6) and time ordering, we find D ( x, y ) acef = Z dθ dθ (2 π ) m (cosh θ − cosh θ ) (cid:18) δ aa δ ca − N δ ac δ a a (cid:19) (cid:18) δ ea δ fa − N δ ef δ a a (cid:19) × exp (cid:8) − im sgn( x − y ) (cid:2) ( x − y )(cosh θ + cosh θ ) − ( x − y )(sinh θ + sinh θ ) (cid:3)(cid:9) × πi ( θ + πi ) exp Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin [ ξ ( πi − θ ) / π ]sinh ξ . (3.8)The color factor in (3.8) is (cid:18) δ aa δ ca − N δ ac δ a a (cid:19) (cid:18) δ ea δ fa − N δ ef δ a a (cid:19) = δ ae δ ef − N δ ac δ ef . (3.9)The term in the right-hand side of (3.9) proportional to N does not contribute when we substitute (3.8) back into (3.5), becausethe field Φ is traceless, so we will ignore this term from now on.We evaluate iS (2) (Φ) using coordinates X µ , r µ , defined by x µ = X µ + r µ , and y µ = X µ − r µ . We then use the derivativeexpansion for X ≫ r : Φ( x ) = Φ( X ) + r µ ∂ µ Φ( X ) + r µ r ν ∂ µ ∂ ν Φ( X ) + . . . , Φ( y ) = Φ( X ) − r µ ∂ µ Φ( X ) + r µ r ν ∂ µ ∂ ν Φ( X ) ± . . . , (3.10)where ∂ µ denotes ∂/∂X µ . This derivative expansion is valid at large distances. The quadratic contribution to the effective actionis iS (2) = − i Z d Xd r D (cid:16) X + r , X − r (cid:17) acef Φ (cid:16) X + r (cid:17) ac Φ (cid:16) X − r (cid:17) ef . (3.11)We substitute (3.10) into (3.11) and find iS (2) = − i Z d Xd r Z dθ dθ (2 π ) m (cosh θ − cosh θ ) δ ae δ cf × exp (cid:8) − im sgn( r ) (cid:2) ( r )(cosh θ + cosh θ ) − ( r )(sinh θ + sinh θ ) (cid:3)(cid:9) × πi ( θ + πi ) exp Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin [ ξ ( πi − θ ) / π ]sinh ξ × (cid:18) Φ( X ) ac + r µ ∂ µ Φ( X ) ac + r µ r ν ∂ µ ∂ ν Φ( X ) ac (cid:19) (cid:18) Φ( X ) ef − r µ ∂ µ Φ( X ) ef + r µ r ν ∂ µ ∂ ν Φ( X ) ef (cid:19) . (3.12)We keep only terms quadratic in r in (3.12) and then integrate out the r variable. Only the terms proportional to ( r ) give anon-vanishing contribution in (3.12). Integration yields the effective action: S eff (Φ) = Z d x
12 Φ ∂ Φ ( − λ N m π ) Z dθ dθ sinh (cid:0) θ + θ (cid:1) sinh (cid:0) θ − θ (cid:1) cosh (cid:0) θ + θ (cid:1) cosh (cid:0) θ − θ (cid:1) × δ ′′ (cid:20) m cosh (cid:18) θ + θ (cid:19) sinh (cid:18) θ − θ (cid:19)(cid:21) π ( θ − θ ) + π × exp 2 Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin [ ξ ( πi − ( θ − θ )) / π ]sinh ξ − λ X x Z d x (cid:2) g q ( x ) Φ( x , u , u ) − g q ′ ( x )Φ( x , v , v ) (cid:3) . We read off the renormalized string tension σ H , after integrating out the auxiliary field Φ : σ H = λ g a C N ( − " λ N m π ) Z dθ dθ sinh (cid:0) θ + θ (cid:1) sinh (cid:0) θ − θ (cid:1) cosh (cid:0) θ + θ (cid:1) cosh (cid:0) θ − θ (cid:1) × δ ′′ (cid:18) m cosh (cid:18) θ + θ (cid:19) sinh (cid:18) θ − θ (cid:19)(cid:19) π ( θ − θ ) + π × exp 2 Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin [ ξ ( πi − ( θ − θ )) / π ]sinh ξ − . After the integration over θ and θ , the string tension is σ H = λ g a C N − λ N m (2 π ) exp 2 Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin (cid:16) iξ (cid:17) sinh ξ − . (3.13)The string tension (3.13) generalizes the result from Refence [4] from N = 2 , to general N > .In the next section we compute the string tension for a quark-antiquark pair separated in the x -direction, rather than the x -direction. IV. THE VERTICAL STRING TENSION
In this section we calculate the string tension, σ V , for a quark-antiquark pair separated only in the x -direction. This calcula-tion has been done before in Reference [5] for the SU (2) gauge group. We show here how to generalize this result for N > using the form factors from Reference [7].If we place a static quark at the space point u , u , and an antiquark at u , v , with u > v , The residual Gauss’s Law (3.3)requires that there be at least one sigma-model particle in each x layer, for u > x > v . The left-handed color index ofa particle at x is contracted with the right-handed color of the particle at x + a . The left-handed color index of the particleat u − a and the right-handed color of the particle at v + a are contracted with the color indices of the quark at u , andthe antiquark at v , respectively. The physical state is a color-singlet string of sigma-model particles, whose endpoints are thequarks. The vertical string tension is obtained by calculating the energy of this string, σ V = lim | u − v |→∞ E string | u − v | . The first approximation is to assume the energy of the string is the total mass of the sigma-model particles, such that E string = ma | u − v | , so σ V = m/a .Corrections to the vertical string tension are found by calculating the contributions to the energy of the string from theHamiltonian λ H . As in Reference [5], we will use a nonrelativistic approximation, where the sigma-model particles havemomenta much smaller than their masses. We ignore any creation or annihilation of particles.The projection of the Hamiltonian onto the nonrelativistic string state is H = u X x = v (cid:26) m + Z dp π p m A † P ( p ) ab A P ( p ) ab (cid:27) + λ H , where A † P ( p ) ab , and A P ( p ) ab are the sigma-model particle creation and annihilation operators, respectively, and H = X x Z dx Z dy g a | x − y |× (cid:2) j L ( x , x ) − j R ( x , x − a ) + q b δ ( x − u ) δ x u − q ′ b δ ( x − u ) δ x v (cid:3) × (cid:2) j L ( y , x ) − j R ( y , x − a ) + q b δ ( y − u ) δ x u − q ′ b δ ( y − u ) δ x v (cid:3) , (4.1)where we have again eliminated the auxiliary field, A .We now need to find the expectation value h string | H | string i , (4.2)where the state | string i has a sigma-model particle for every x , whose center of mass is located at x = z ( x ) . To evaluate(4.2), we need the matrix elements of the form h P, z , a , b | j C ( x ) ac | P, z , a , b i = Z dp π √ E Z dp π √ E × e − ip · ( z − x )+ ip · ( z − x ) h P, θ , a , b | j C ( x ) ac | P, θ , a , b i , (4.3)where the matrix element on the right hand side of (4.3) is the two particle form factor found in the appendix (with the incomingantiparticle crossed to an outgoing particle), and C = L, R . By applying crossing symmetry on the form factor (A.2), we find h P, θ , a , b | j C ( x ) ac | P, θ , a , b i = ( p + p ) D Ca c a a b b πiθ + 2 πi exp Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:16) e ξN − (cid:17) − e − ξ sin [ ξθ/ π ]sinh ξ , where D La c a a b b = δ a a δ c a δ b b − N δ a c δ a a δ b b , D Ra c a a b b = δ a b δ c b δ a a − N δ a c δ a a δ b b . Taking the nonrelativistic limit, we find √ E √ E h P, θ , a , b | j C ( x ) a c | P, θ , a , b i ≈ D Ca c a a b b exp − A N m ( p − p ) . where A N = Z ∞ dξ π ξ sinh ξ (cid:20) sinh (cid:18) ξN (cid:19) − (cid:16) e ξ/N − (cid:17)(cid:21) = 116 π (cid:20) π − ψ (1) (cid:18) − N (cid:19) − ψ (1) (cid:18)
12 + 1 N (cid:19)(cid:21) , for N > , where ψ ( n ) ( x ) = d n +1 ln Γ( x ) /dx n +1 is the n -th polygamma function.The matrix element (4 . is then h P, z , a , b | j C ( x ) ac | P, z , a , b i = s m πA N D Ca c a a b b exp " − m A N (cid:18) z + z − y (cid:19) δ ( z − z ) . (4.4)This means that the color density of a particle is a Gaussian distribution in the nonrelativistic limit. In this sense, particles arenot point-like, but the color is smeared over space.We now use (4.4) to write the effective Hamiltonian of the nonrelativistic string. This is given by the projection of theHamiltonian (4.1) onto the state | string i , which has a sigma-model particle at each x layer, located at the point z ( x ) , for u > x > v , a static quark at u , u , and an antiquark at u , v . The string Hamiltonian is H string = ma ( v − u ) − m u − a X x = v ∂ ∂z ( x ) + λ V bulk + λ V ends , where V bulk = − m πA N g a u − a X x = v + a Z dx dy | x − y |× (cid:26) e − m AN [ z ( x ) − x ] D L ( x ) a c a a b b − e − m AN [ z ( x − a ) − x ] D R ( x − a ) a c a a b b (cid:27) × (cid:26) e − m AN [ z ( x ) − y ] D L ( x ) a c a a b b − e − m AN [ z ( x − a ) − y ] D R ( x − a ) a c a a b b (cid:27) , (4.5)and V ends = − g a Z dx dy | x − y | s m πA N e − m AN [ z ( v ) − x ] D R ( v ) a c a a b b + δ ( x − v ) q ′ a c πδ a a δ b b × s m πA N e − m AN [ z ( v ) − y ] D R ( v ) a c a a b b + δ ( y − u ) q ′ a c πδ a a δ b b − g a Z dx dy | x − y | s m πA N e − m AN [ z ( u − a ) − x ] D L ( u − a ) a c a a b b + δ ( x − u ) q ′ a c πδ a a δ b b × s m πA N e − m AN [ z ( u − a ) − y ] D L ( u − a ) a c a a b b + δ ( y − u ) q ′ a c πδ a a δ b b . (4.6)Imposing the residual Gauss’s law (3.3) on (4.5) and (4.6), implies Z dx − s m πA N e − m AN [ z ( x ) − x ] D L ( x ) a c a a b b + s m πA N e − m AN [ z ( x − a ) − x ] D R ( x − a ) a c a a b b Ψ = 0 , (4.7)for u > x > v , and Z dx s m πA N (cid:26) e − m AN [ z ( v ) − x ] D R ( v ) a c a a b b − q ′ a c δ ( x − u )4 πδ a a δ b b (cid:27) Ψ = 0 , Z dx s m πA N (cid:26) e − m AN [ z ( u − a ) − x ] D L ( u − a ) a c a a b b − q a c δ ( x − u )4 πδ a a δ b b (cid:27) Ψ = 0 , (4.8)respectively. The constraint (4.7) is satisfied by identifying D L ( x ) a c a a b b = D R ( x − a ) a c a a b b . The constraint (4.8) issatisfied by identifying D R ( v ) a c a a b b = q ′ a c πδ a a δ b b , and D L ( u − a ) a c a a b b = q a c πδ a a δ b b . Using this, wecan eliminate the color degrees of freedom from (4.5) and (4.6).Next we integrate out the variables x and y from equations (4.5) and (4.6). The integrals involved are: Z dx dy | x − y | e − m AN [ ( x ) +( y ) ] = 4 √ πA / N m , Z dx dy | x − y | e − m AN [ ( x + r ) +( y ) ] = 4 √ πA / N m P ( r ) , Z dx | x − u | e − m AN [ x − z ( U ) ] = 2 A N m P h √ z ( u ) − √ u i , Where P ( r ) is a function for which we do not have an exact analytic expression, but its behavior for small and large r is P ( r ) = ( m r A N , r << m , q π A N m | r | , r >> m . (4.9)After integrating out x , and y , the string Hamiltonian is H string = ma ( u − v ) − m u − a X x = v ∂ ∂z ( x ) − λ N ( N − m g a r A N π u X x = v + a (cid:8) − P (cid:2) z ( x ) − z ( x − a ) (cid:3)(cid:9) − λ N ( N − m g a r A N π (cid:16) P n √ (cid:2) z ( v ) − u (cid:3)o + P n √ (cid:2) z ( u − a ) − u (cid:3)o(cid:17) , (4.10)where we have used (cid:0) D C (cid:1) = N (cid:0) N − (cid:1) . The potential energy between a static quark-antiquark pair is then determined by finding the ground state of the Hamiltonian(4.10).We further simplify the Hamiltonian (4.10) using the small-gradient approximation. That is, in the nonrelativistic limit (whenthe sigma model mass gap is taken to be very large), we expect that the sigma-model particles in two adjacent x layers are closeto each other in the x -direction. Specifically, we assume | z ( x ) − z ( x − a ) | << m − . At the endpoints of the string, wealso assume | z ( v ) − u | << m − , and | z ( u − a ) − u | << m − . Using Eq. (4.9), the small-gradient approximation givesthe Hamiltonian H string = λ N ( N − m g a r A N π + ma ( u − v ) − m v − a X x = v ∂ ∂z ( x ) + λ N ( N − mg a r πA N u − a X x = v + a (cid:2) z ( x ) − z ( x − a ) (cid:3) + λ N ( N − mg a r πA N n(cid:2) z ( v ) − u (cid:3) + (cid:2) z ( u − a ) − u (cid:3) o . (4.11)0The first term in the Hamiltonian (4.11) is just a constant with no physical significance, so we will ignore it from now on. TheHamiltonian (4.11) is equivalent Q = (cid:0) u − v (cid:1) /a coupled harmonic oscillators. The ground-state energy is then given by E = mQ − λ p N ( N − g a (cid:18) πA N (cid:19) Q X q =0 sin πq Q . (4.12)Using the Euler summation formula, for large Q : Q X q =0 F (cid:18) qQ (cid:19) = Q Z dxF ( x ) −
12 [ F (1) − F (0)] + 112 Q [ F ′ (1) − F ′ (0)] + O (cid:18) Q (cid:19) , the ground-state energy (4.12) becomes (dropping any constants that do not depend on Q ) E = " ma − λ p N ( N − πg a (cid:18) πA N (cid:19) L + π λ p N ( N − g (cid:18) πA N (cid:19) L + O (cid:18) L (cid:19) . (4.13)where the distance between the quark and antiquark is L = Qa .We can easily read the vertical string tension off (4.13): σ V = ma − λ p N ( N − πg a (cid:18) πA N (cid:19) . (4.14)There is also a Coulomb-like term in the quark-antiquark potential, which is proportional to /L . V. THE LOW-LYING GLUEBALL MASS SPECTRUM
The constraint (2.9) requires that in the absence of quarks, there be an equal number of sigma-model particles and antiparticlesin each x layer. Furthermore, it requires that the excitations form left- and right-color singlets. If the sigma model at x has aparticle with a left color index, a , this index has to be contracted with either the left-color index of an antiparticle in the x layer,or the right color index of a particle in the ( x + a ) layer. A glueball in this theory consists of several sigma-model excitations,forming a color-singlet bound state.The simplest and lightest glueball is one composed of only one particle and one antiparticle, at the same value of x . Thegauss law constraint requires that their left and right handed color indices be contracted. The interaction Hamiltonian (2.10)provides a confining linear potential, with string tension σ = 2 σ H . (5.1)The factor of comes the fact that both the left and right color charges are confined.The problem is now essentially (1+1)-dimensional. The low-lying gluon spectrum has been found before by P. Orland inReference [6] for the SU (2) gauge group. A similar analysis was used to find the massive spectrum of (1+1)-dimensionalmassive Yang-Mills theory for all N , in Ref. [24]. This method is in turn inspired by the determination of the spectrum of thetwo-dimensional Ising model in an external magnetic field [25], [19].The low-lying glueball masses are M n = 2 m + E n , where m is the mass of a sigma model excitation, and E n is the binding energy. The goal of this section is to compute thebinding energies E n for N > . This is done by finding the wave function of an unbound sigma-model particle-antiparticle pair.There is a possibility of these two excitations scattering which is accounted by the exact particle-antiparticle S-matrix. We laterfind the wave function of a particle-antiparticle pair, confined by a linear potential. We obtain a quantization condition for thebinding energy by requiring that the two wave functions agree when the particles are close to each other. We are able to do thiscalculation only in the nonrelativistic limit, where we take the momenta of the excitations to be much smaller than their masses.The particle-antiparticle S-matrix is found in the appendix. The S-matrix has an incoming antiparticle with rapidity θ andcolor indices a , b and a particle with rapidity θ and color indices a , b . There is an outgoing antiparticle with color indices c , d and a particle with indices c d . The S-matrix is S ( θ ) d c ; c d a b ; b a = S ( θ ) (cid:20) δ c a δ c a − π i N ( π i − θ ) δ a a δ c c (cid:21) (cid:20) δ d b δ d b − π i N ( π i − θ ) δ b b b d d (cid:21) , S ( θ ) = exp 2 Z ∞ dξξ sinh ξ h e ξ/N − − sinh(2 ξ/N ) i sinh ξθπ i , (5.2)for N > .The constraint (2.9) requires that the particle-antiparticle pair form a left- and right-handed color singlet. The S-matrix of thispair, S ( θ ) , is obtained by contracting the color indices of the excitations: S ( θ ) = 1 N δ a a δ b b δ c c δ d d S ( θ ) d c ; c d a b ; b a = (cid:18) θ + πiθ − πi (cid:19) S ( θ ) . In the nonrelativistic limit ( θ << m ) the color-singlet S-matrix becomes S ( θ ) = exp (cid:18) − ih N πm | p − p | (cid:19) , where h N = 2 Z ∞ dξ sinh ξ h e ξ/N − − sinh(2 ξ/N ) i = − γ − ψ (cid:18)
12 + 1 N (cid:19) − ψ (cid:18) − N (cid:19) − , (5.3)and γ is the Euler-Mascheroni constant, and ψ ( x ) = d ln Γ( x ) /dx is the digamma function.We find the wave function of an antiparticle at x , and a particle at y , with momenta p , p , respectively, in the nonrelativisticlimit. It is convenient to switch to center-of-mass coordinates, X, x and their respective momenta
P, p . These are defined by X = x + y , y − x , P = p + p , and p = p − p . The nonrelativistic wave function is Ψ p ( x ) singlet = cos( px + ω ) , for x > , cos[ − px + ω − χ ( p )] , for x < , (5.4)where χ ( p ) = − h N πm | p | . We now calculate the nonrelativistic wave function for a linearly-bound particle-antiparticle pair. In center-of-mass coordi-nates, the wave function satisfies the Schroedinger equation − m d dx Ψ( x ) + σ | x | Ψ( x ) = E Ψ( x ) , (5.5)where E is the binding energy [25], and σ = 2 λ g a C N , is the string tension. The solution of Eq. (5.5) is Ψ( x ) = C Ai (cid:20) ( mσ ) (cid:18) | x | − Eσ (cid:19)(cid:21) , (5.6)where Ai( x ) is the Airy function of the first kind, and C is a normalization constant.We require that the wave functions (5.4) and (5.6) agree as | x | → . We identify | p | = ( mE ) . For small | x | , the function(5.6) is approximated by Ψ( x ) b b = C ( x + Eσ ) cos h ( mσ ) (cid:0) − x + Eσ (cid:1) − π i A b b , for x > ,C ( x + Eσ ) cos h − ( mσ ) (cid:0) x + Eσ (cid:1) + π i A b b , for x < . (5.7)By comparing (5.4) and (5.7) as x ↓ , we fix C = (cid:18) Eσ (cid:19) , ω = 23 ( mσ ) (cid:18) Eσ (cid:19) − π . x ↑ , gives the quantization condition.
43 ( mσ ) (cid:18) Eσ (cid:19) + h N πm ( mE ) − (cid:18) n + 14 (cid:19) π = 0 , (5.8)with n = 0 , , , ... .The solution to (5.8) is E n = (cid:26)h ǫ n + (cid:0) ǫ n + β N (cid:1) i + h ǫ n − (cid:0) ǫ n + β N (cid:1) i (cid:27) , (5.9)where ǫ n = 3 π (cid:16) σm (cid:17) (cid:18) n + 14 (cid:19) , β N = h N σ πm . (5.10) VI. HORIZONTAL CORRELATION FUNCTIONS
In this section we compute the long-distance correlation function of two gauge-invariant operators separated in the x -directiononly. This is D A ( x ) = h |A ( x , x ) A (0 , x ) | i . (6.1)The correlation function (6.1) can be evaluated by inserting a complete set of physical states between the two operators: D A ( x ) = X Ψ h |A ( x , x ) | Ψ ih Ψ |A (0 , x ) | i . The physical, gauge invariant excitations of the theory are glueball bound states of sigma-model particles. At large separations( x → ∞ ), the function D A ( x ) can be approximated by inserting only one-glueball states. The lightest glue balls are thosecomposed of a sigma-model particle and antiparticle, whose masses where calculated in the previous section.We denote the state with one glueball with rapidity φ , and rest energy M n , by | B, φ, n i . The long-distance correlation functionis D A ( x ) = n s X n =1 Z dφ π h |A ( x , x ) | B, φ, n ih B, φ, n |A (0 , x ) | i , where n s is the energy level of the heaviest stable glueball, defined by M n s ≤ m ≤ M n s +1 .We need a way to compute the one-glueball form factor of the operator A . One approach was proposed by Fonseca andZamolodchikov [26] in the Ising model perturbed by a weak external magnetic field. In the nonrelativistic limit, the glueballstate is given by the so-called two-quark approximation: | B, , n i = 1 √ m Z ∞−∞ dθ π Ψ n ( θ ) | A, θ, a , b ; P, − θ, a , b i , where Ψ n ( θ ) is the Fourier transform of the glueball wave function calculated in last section: Ψ n ( θ ) = Z dze izm sinh θ (cid:18) E n σ H (cid:19) Ai (cid:20) ( mσ H ) (cid:18) | z | − E n σ H (cid:19)(cid:21) . If the operator A has spin s , the one-glueball form factor is h |A ( x , x ) | B, φ, n i = e sφ e ix M n sinh φ Z dz Z dθ π e izm sinh θ √ m (cid:18) E n σ H (cid:19) Ai (cid:20) ( mσ H ) (cid:18) | z | − E n σ H (cid:19)(cid:21) ×h |A (0 , x ) | A, θ, a , b ; P, − θ, a , b i . For the rest of this section we will assume s = 0 .3The two-point correlation function is then D A ( x ) = h |A ( x , x ) | ih |A (0 , x ) | i + n s X n =1 Z dφ π e ix M n sinh φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dz Z dθ π e izm sinh θ √ m (cid:18) E n σ H (cid:19) Ai (cid:20) ( mσ H ) (cid:18) | z | − E n σ H (cid:19)(cid:21) F (2 θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where F ( θ ) = C A N θ + πi ) exp Z ∞ dξξ − (cid:16) ξN (cid:17) sinh ξ + 4 e − ξ (cid:0) e ξ/N − (cid:1) − e − ξ sin [ ξ ( πi − θ ) / π ]sinh ξ , and C A is a normalization constant for the form factor of the operator A .The integral over φ gives Z dφ π e ix M n sinh φ = 12 π K ( M n x ) , (6.2)where K α ( x ) is the modified Bessel function of the second kind. For x → ∞ , this modified Bessel function is approximatedby K ( M n x ) → r π M n x e − M n x . In the anisotropic limit where λ → , the glueball masses become similar to each other ( M n ≈ M n +1 ). Following [19], weassume that the form factors for any n give approximately the same contribution. All the n dependence is contained in the Besselfunction (6.2). In this limit, the sum over n is approximated by a continuous integral, so we evaluate n x X n =1 r M n x e − M n x = n s X n =1 r mx + E n x e − mx e − E n x ≈ e − mx √ mx Z ∞ dn e − E n x ≡ e − mx √ mx I ( x , σ H , m ) . (6.3)The integral I ( x , σ H , m ) from Eq. (6.3) is in general quite complicated. One particularly simple case is when N → ∞ , where I ( x , σ H , m ) = Z ∞ dn exp " π (cid:18) σ H m (cid:19) n = 720 π (cid:16) σ H m (cid:17) x . The horizontal correlation function at large distances and small λ is D A ( x ) = = h |A ( x , x ) | ih |A (0 , x ) | i + e − mx √ m x I ( x , σ H , m ) (cid:18) E σ H (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)Z dz Z dθ π e izm sinh θ Ai (cid:20) ( mσ H ) (cid:18) | z | − E n σ H (cid:19)(cid:21) F (2 θ ) (cid:12)(cid:12)(cid:12)(cid:12) . VII. VERTICAL CORRELATION FUNCTIONS
In this section we present a method to evaluate correlation functions of two gauge-invariant operators separated in the x -direction only. This problem is significantly harder than calculating horizontal correlation functions, and we are able to makeprogress only in the large- N limit. For large separation in x , the problem is reduced to solving an integral eigenvalue equation.We want to calculate the correlator D A ( aR ) = h |A ( x , x ) A ( x , x + aR ) | i . (6.1)Our strategy is to define a transfer matrix operator, T x ,x + a , that describes the evolution of the system in the x -direction.We impose periodic boundary conditions in the x -direction. We call size of the x -dimension L . The partition function andcorrelation functions can be computed by diagonalizing this transfer matrix.The transfer matrix is defined by T x ,x + a = e − H x ,x a , H x ,x + a = 12 H PCSM ( x ) + 12 H PCSM ( x + a ) + λ H ( x , x + a ) and H ( x , x + a ) = Z dx dy g a | x − y | (cid:2) j L ( x , x ) j L ( y , x ) + j R ( x , x ) j R ( y , x ) (cid:3) + Z dx dy g a | x − y | (cid:2) j L ( x , x + a ) j L ( y , x + a ) + j R ( x , x + a ) j R ( y , x + a ) (cid:3) − Z dx dy g a | x − y | (cid:2) j L ( x , x + a ) j R ( y , x ) + j L ( y , x + a ) j R ( x , x ) (cid:3) . (6.2)We can now compute the matrix elements in of T x ,x + a between the particle states of the sigma models at x and x + a .We label a state with a two-particle bound state, of rapidity φ and energy level n on the sigma model at x by | B, φ, n, x i . Forlarge separations in x , it is sufficient to compute the matrix elements with just one bound state in each x -layer.We define the functions T = h | T x ,x + a | i ,T n ( φ ) = h B, φ, n, x | T x ,x + a | B, φ, n, x i ,T nn ′ ( φ, φ ′ ) = h B, φ, n, x ; B, φ ′ , n ′ , x + a | T x ,x + a | B, φ, n, x ; B, φ ′ , n ′ , x + a i . (6.3)In the basis of one-glueball states, the transfer matrix is τ nn ′ ( φ, φ ′ ) = T + T n ( φ ) + T n ′ ( φ ′ ) + T nn ′ ( φ, φ ′ ) . (6.4)The partition function and correlation functions can be found, in principle, by finding the eigenvalues of the matrix τ nn ′ ( φ, φ ′ ) .This means one has to solve the integral equation n s X n ′ =1 Z dφ ′ π τ nn ′ ( φ, φ ′ ) ψ ( l ) n ′ ( φ ′ ) = λ ( l ) ψ ( l ) n ( φ ) . (6.5)If the eigenvalues λ ( l ) and eigenfunctions ψ ( l ) n ( φ ) are known, the transfer matrix may be diagonalized as τ nn ′ ( φ, φ ′ ) = X l λ ( l ) ψ ( l ) n ( φ ) ψ ( l ) n ′ ( φ ′ ) , (6.6)where the eigenfunctions are normalized by X n Z dφ π ψ ( l ) n ( φ ) ψ ( m ) n ( φ ) = δ lm . The partition function is then given by Z = X l h λ ( l ) i N , where N = L a . In the thermodynamic limit, N → ∞ , the partition function is Z = [ λ (0) ] N , where λ (0) is the largest eigenvalue of the transfer matrix.5The operators in the correlation function (6.1) are expressed in the one-glueball basis as the functions A nn ′ ( φ, φ ′ ) . We assumethat the functions A and τ are not simultaneously diagonalizable. The two-point correlation function is D A ( aR ) = 1 (cid:2) λ (0) (cid:3) N X l,l ,l X n ,n ,n R ,n R Z dφ dφ dφ R dφ R +1 (4 π ) × n ψ ( l ) n ( φ ) A n n ( φ , φ ) ψ ( l ) n ( φ )[ λ ( l ) ] R − ψ ( l ) n R ( φ R ) A n R n R ( φ R , φ R +1 ) ψ ( l ) n R ( φ R +1 ) o . (6.7)In the limit of large separation R , Eq. (6.7) becomes D A ( aR ) = C + (cid:18) λ (1) λ (0) (cid:19) R C , (6.8)where λ (1) is the second largest eigenvalue, and C = X l ,l X n ,n ,n R ,n R Z dφ dφ dφ R dφ R +1 (4 π ) × n ψ ( l ) n ( φ ) A n n ( φ , φ ) ψ ( l ) n ( φ ) ψ ( l ) n R ( φ R ) A n R n R ( φ R , φ R +1 ) ψ ( l ) n R ( φ R +1 ) o . We define the inverse correlation length M as D A ( aR ) ∼ e −M aR . From Eq. (6.8), the inverse correlation length is M = − a ln (cid:18) λ (1) λ (0) (cid:19) . The rest of this section is dedicated to finding an expression for τ nn ′ ( φ, φ ′ ) , though we are never able to solve the integraleigenvalue equation (6.5). This is left as an open problem.The contribution to T nn ′ ( φ, φ ′ ) which couples two adjacent x layers involves the two-bound state form factor of the currentoperator. This means that we need the four-excitation form factors of the PCSM. The functions T, T n ( φ ) , T nn ′ ( φ, φ ′ ) involvetwo-point functions of current operators. These correlation functions will be computed keeping only terms proportional to thetwo-and four-particle form factors. Form factors of more than two excitations are only known in ’t Hooft’s large- N limit. Forthe rest of this section we work exclusively in the large- N limit. The form factors of the sigma model at large N were found inreferences [7], [23], [27], and are reviewed in the appendix.We first calculate the constant T = h | e − H PCSM ( x ) − H PCSM ( x + a )+ λ H ( x ,x + a ) | i≈ exp (cid:26) h | − H PCSM ( x ) − H PCSM ( x + a ) − λ H ( x , x + a ) | i (cid:27) . (6.9)The constant T only has a contribution from H ( x , x + a ) . This contribution is h | H ( x , x + a ) | i = Z dx dy | x − y | g a (cid:8) h | j L ( x , x ) j L ( y , x ) | i + h | j R ( x , x ) j R ( y , x ) | i (cid:9) + Z dx dy | x − y | g a (cid:8) h | j L ( x , x + a ) j L ( y , x + a ) | i + h | j R ( x , x + a ) j R ( y , x + a ) | i (cid:9) . (6.10)We now examine the correlation functions in the right hand side of Eq. (6.10) using up to two-glueball form factors. This is,for the left-handed current, h | j L ( x , x ) j L ( y , x ) | i = X n Z dφ π h | j L ( x , x ) | B, φ, n, x ih B, φ, n, x | j L ( y , x ) | i + X n ,n Z dφ dφ (4 π ) (cid:2) h | j L ( x , x ) | B, φ , n , x ; B, φ , n , x i× h B, φ , n , x ; B, φ , n , x | j L ( y , x ) | i (cid:3) . X n Z dφ π h | j L ( x , x ) | B, φ, n, x ih B, φ, n, x | j L ( y , x ) | i = X n Z dφ π M n sinh φe − i ( x − y ) M n sinh φ (cid:18) E n σ (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12)Z dz Z dθ π Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) πi tanh θ θ + πi (cid:12)(cid:12)(cid:12)(cid:12) . The contribution to T from the one-glueball form factor is T (2) = exp ( − λ Z dx dy | x − y | g a X n Z dφ π M n sinh φe − i ( x − y ) M n sinh φ (cid:18) E n σ (cid:19) N F (2) ) , (6.11)where F (2) = (cid:12)(cid:12)(cid:12)(cid:12)Z dz Z dθ π Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) πi tanh θ θ + πi e izm sinh θ (cid:12)(cid:12)(cid:12)(cid:12) . Using the large-N, four-excitation form factors of the sigma model, we find X n ,n Z dφ dφ (4 π ) h | j L ( x , x ) | B, φ , n , x ; B, φ , n , x ih B, φ , n , x ; B, φ , n , x | j L ( y , x ) | i = X n ,n Z dφ dφ (4 π ) (cid:18) E n E n σ (cid:19) p M n M n ( M n sinh φ + M n sinh φ ) e − i ( x − y )( M n sinh φ + M n sinh φ ) N F (4) , where F (4) = (cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) tanh θf ( θ, θ ′ )[( θ + θ ′ ) + π ] (2 θ ′ + πi ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) tanh (cid:16) θ + θ ′ (cid:17) f ( θ, θ ′ )(2 θ + πi )( θ ′ + θ − πi )(2 θ ′ + πi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) tanh (cid:16) θ + θ ′ (cid:17) f ( θ, θ ′ )( θ + θ ′ + πi )(2 θ ′ + πi )(2 θ + πi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) tanh θ ′ f ( θ, θ ′ )[( θ + θ ′ ) + π ] (2 θ + πi ) (cid:12)(cid:12)(cid:12)(cid:12) , and f ( θ, θ ′ ) = Z dz dz πi e iz m sinh θ + iz sinh θ ′ Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) . The contribution to T from the two-glueball form factor is T (4) = exp ( − λ Z dx dy | x − y | g a X n ,n Z dφ dφ (4 π ) (cid:18) E n E n σ (cid:19) × p M n M n ( M n sinh φ + M n sinh φ ) e − i ( x − y )( M n sinh φ + M n sinh φ ) N F (4) ) , (6.12)such that T = T (2) T (4) . (6.13)We now calculate the function T n ( φ ) from Eq. (6.3). The contribution to T n ( φ ) from the sigma model in the x + a layer isjust √ T (2) T (4) . There is a contribution to the function T n ( φ ) from the unperturbed Hamiltonian, given by h B, φ, n, x | H PCSM ( x ) | B, φ, n, x i ≡ T (0) n ( φ )= 1 M n Z dz dz (cid:18) E N σ (cid:19) Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) × Z dθ π m cosh θe i ( z − z ) m sinh θ e i ( z − z ) M n cosh φ . (6.14)7There are contributions to T n ( φ ) from the current correlation function h B, φ, n, x | j L ( x , x ) j L ( y , x ) | B, φ, n, x i = h B, φ, n, x | j L ( x , x ) | ih | j L ( y , x ) | B, φ, n, x i + ′ X n Z dφ ′ π h B, φ, n, x | j L ( x , x ) | B, φ ′ , n ′ , x ih B, φ ′ , n ′ , x | j L ( y , x ) | B, φ, n, x i + . . . . Using the two-excitation form factors of the sigma model, we find h B, φ, n, x | j L ( x , x ) | ih | j L ( y , x ) | B, φ, n, x i = M n sinh φ e − i ( x − y ) M n sinh φ (cid:18) E n σ (cid:19) F (2) . The contribution to T n ( φ ) from the one-glueball form factors of the sigma model at x is T (2) n ( φ ) = exp ( − λ Z dx dy | x − y | g a M n sinh φ e − i ( x − y ) M n sinh φ (cid:18) E n σ (cid:19) F (2) ) . (6.15)Using the form factor of the sigma model with two incoming and two outgoing excitations (also found in the appendix), wefind the contribution T n ( φ ) from the two-glueball form factor T (4) n ( φ ) = exp ( − λ Z dx dy | x − y | g a X n ′ Z dφ π (cid:18) E n E n ′ σ (cid:19) N × √ M n M n ′ ( M n sinh φ + M n ′ sinh φ ′ ) e − i ( x − y )( − M n sinh φ + M n ′ sinh φ ′ ) F ′ (4) (cid:27) , (6.16)where F ′ (4) = (cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) tanh θf ′ ( θ, θ ′ )( θ + θ ′ + 2 πi )( θ ′ + θ − πi )(2 θ ′ + πi ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) coth (cid:16) θ + θ ′ (cid:17) f ′ ( θ, θ ′ )(2 θ + πi )( θ ′ + θ − πi )(2 θ ′ + πi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) coth (cid:16) θ + θ ′ (cid:17) f ′ ( θ, θ ′ ) θ + θ ′ + 2 πi )(2 θ + πi )(2 θ ′ + πi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z dθdθ ′ (4 π ) tanh θ ′ f ′ ( θ, θ ′ )( θ + θ ′ + 2 πi )(2 θ + πi )( θ + θ ′ − πi ) (cid:12)(cid:12)(cid:12)(cid:12) , and f ′ ( θ, θ ′ ) = 8 πi Z dz Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) e iz m sinh θ (cid:26)Z dz Ai (cid:20) ( mσ ) (cid:18) | z | − E n σ (cid:19)(cid:21) e iz m sinh θ ′ (cid:27) ∗ . We can combine the results from Equations (6.13), (6.14), (6.15), and (6.16) to write T n ( φ ) = √ T T (0) n ( φ ) T (2) n ( φ ) T (4) n ( φ ) . (6.17)We now evaluate the function T nn ′ ( φ, φ ′ ) . This function has only one new contribution, which couples between the x and x + a layers. This is K nn ′ ( φ, φ ′ ) = exp (cid:26) − λ Z dx dy | x − y | g a h B, φ, n, x | j R ( x , x ) | B, φ, n, x i× h B, φ ′ , n ′ , x + a | j L ( y , x + a ) | B, φ ′ , n ′ , x + a i (cid:9) = exp ( − λ Z dx dy | x − y | g a (cid:18) E n E n ′ σ (cid:19) N × M n M n ′ sinh φ sinh φ ′ e − ix M n sinh φ + iy M n ′ sinh φ ′ F ′ (4) o . (6.18)With Eq. (6.18) and Equations (6.13), (6.14), (6.15), and (6.16), we write T nn ′ ( θ, θ ′ ) = T (0) n ( φ ) T (2) n ( φ ) T (4) n ( φ ) T (0) n ′ ( φ ′ ) T (2) n ′ ( φ ′ ) T (4) n ′ ( φ ′ ) K nn ′ ( φ, φ ′ ) . (6.19)The transfer matrix τ nn ′ ( φ, φ ) is given in Eq. (6.4), by combining Equations (6.13), (6.17), (6.19). The problem of findingthe vertical correlation functions is now reduced to diagonalizing the function τ nn ′ ( φ, φ ) , and expressing it in the form of Eq.(6.6).8 VIII. CONCLUSION
We have used new exact results from the principal chiral sigma model to compute physical quantities in anisotropic Yang-Mills theory. The two-particle form factors of the sigma model are now known for general
N > . This allowed us to generalizeOrland’s results for the SU (2) gauge group, to SU ( N ) . These results include the string tensions for quarks separated in the x and x -directions, and the spectrum of the lightest glueball masses.Once we found the glueball states, we used them to calculate correlation functions of gauge invariant operators. For twooperators separated in the x -direction only, the correlation function is calculated at long distances by summing over a completeset of intermediate one-glueball states.The correlation functions of operators separated in the x -direction are much harder to calculate. We proposed a method forhow these correlation functions may be calculated, though we do not solve the problem completely. We compute the elements ofa transfer matrix which evolves the system in the x -direction. These elements are computed in the basis of one-glueball states.The problem of calculating correlation functions is reduced to solving an integral eigenvalue equation for the transfer matrix.An obvious problem for the future is to find a solution to the eigenvalue equation , Eq. (6.5). This would allow us to calculateexplicitly the partition function and correlation functions in the x -direction. The rapidities of the glueballs, φ, φ ′ can bediscretized by placing the sigma models in a finite box of size L . One can impose an energy cutoff by discarding glueball statesabove some maximum rapidity. The transfer matrix then becomes discrete and finite, and can thus be diagonalized numericallyon a computer. This computation would be similar to that done for the Ising model by Konik and Adamov [17]. We wouldlike to point out that the methods of Ref. [17] can, in principle, be used to find results applicable to the fully isotropic (2+1)-dimensional theory. In this reference, the authors studied the three-dimensional Ising model as an array of two-dimensionalchains, for different values of the interchain coupling (corresponding to our parameter λ ), up to the fully isotropic value. Theirtransfer matrix was obtained by an improved version of the truncated spectrum approach [29]. One difficulty for the Yang-Mills-theory case is that gauge invariance needs to be imposed on the states of the truncated spectrum, making the construction of thetransfer matrix non trivial. This numerical diagonalization is the most promising approach that we know with which we couldstudy the fully isotropic (2+1)-dimensional theory.It would be interesting to extend our methods to 3+1 dimensions. It has been shown that longitudinally rescaled (3+1)-dimensional Yang-Mills theory can also be expressed as an array of sigma models [28]. There is an additional interaction termgiven by the additional components of the magnetic field. It would be interesting to see what is the effect of this additionalinteraction on the quantities calculated in this paper. Acknowledgments
I wish to thank Peter Orland for our many discussions about his earlier work that were so essential to this project, andfor carefully reading this manuscript. I also thank Adrian Dumitru for some interesting discussions about the applications ofanisotropic gauge theories. Lastly, I thank Robert Konik, for some very useful conversations we had over a year ago, whichinspired some of this work.
Appendix: The S-matrix and Form Factors of the Principal Chiral Sigma Model
For the purposes of this paper, we only need to know the two-and four-particle form factors of the Noether current operatorsof the sigma model. These were found in the ’t Hooft limit in Ref. [7]. For finite N , only the two-particle form factor is foundin the same paper. These results were later generalized to form factors of an arbitrary number of particles, at large N , if Ref.[23]. These form factors were used to calculate two-point correlation functions. It is worth mentioning that the form factors andcorrelation functions of other operators have also been found in the ’t Hooft limit. The renormalized field operator was studiedin Reference [27], and the energy-momentum tensor was studied in [23].This appendix is not meant to be a review of form factors of integrable theories. We merely present results without a meticulousderivation. For a complete derivation of the results in this appendix, see Ref. [7]. A modern review of the integrable bootstrapprogram for calculating form factors are found in References [30].The derivation of the form factors makes use of the two-particle S-matrix of the sigma model. This S-matrix has been found inRefs. [3] [2]. The S-matrix, S P P ( θ ) c d ; c d a b a b of two incoming particles with rapidities θ , and θ and left and right color indices a , b , and a , b respectively, and two outgoing particles with rapidities θ ′ and θ ′ , and left and right color indices c , d , and c , d , respectively, is given by out h P, θ ′ , c , d ; A, θ ′ , d , c | P, θ , a , b ; A, θ , b , a i in = S P P ( θ ) c d ; c d a b ; a b π δ ( θ ′ − θ ) 4 π δ ( θ ′ − θ ) , θ = θ − θ . The result from [3], [2] is S P P ( θ ) c d ; c d a b ; a b = χ ( θ ) S CGN ( θ ) c ; c a ; a S CGN ( θ ) d d b ; b , where S CGN is the S-matrix of two elementary excitations of the SU ( N ) chiral Gross-Neveu model [31], [32]: S CGN ( θ ) c c a a ; a = Γ( iθ/ π + 1)Γ( − iθ/ π − /N )Γ( iθ/ π + 1 − /N )Γ( − iθ/ π ) (cid:18) δ c a δ c a − πiN θ δ c a δ c a (cid:19) , and χ ( θ ) = sinh (cid:0) θ − πiN (cid:1) sinh (cid:0) θ + πiN (cid:1) . The particle-antiparticle S-matrix is related to the particle-particle S-matrix by crossing symmetry, i.e. θ → ˆ θ = πi − θ . Itwas found in Ref. [7], that the particle-antiparticle S-matrix can be written in the exponential form: S ( θ ) d c ; c d a b ; b a = S ( θ ) (cid:20) δ c a δ c a − π i N ( π i − θ ) δ a a δ c c (cid:21) (cid:20) δ d b δ d b − π i N ( π i − θ ) δ b b b d d (cid:21) , where S ( θ ) = exp 2 Z ∞ dξξ sinh ξ h e ξ/N − − sinh(2 ξ/N ) i sinh ξθπ i , (A.1)for N > .The two-particle form factor of the left-handed Noether current was found using Eq. (A.1). This is [7] h | j Lµ ( x ) a c | A, θ , b , a ; P, θ , a , b i in = ( p − p ) µ e − ix · ( p + p ) (cid:18) δ a a δ c a δ b b − N δ a c δ a a δ b b (cid:19) . × πi ( θ + πi ) exp Z ∞ dxx " − (cid:0) xN (cid:1) sinh x + 4 e − x (cid:0) e x/N − (cid:1) − e − x sin [ x ( πi − θ ) / π ]sinh x . (A.2)The form factor with one incoming and one outgoing antiparticle can be found by crossing the particle in (A.2) into an outgoingparticle, shifting the rapidity θ → θ − πi . The right-handed current has a very similar expression, but the color indices of theoperator are contracted with the right-handed color indices of the particle and antiparticle.Next we show the four-excitation form factor at large N . The form factor is nonzero only if two of the excitations are particlesant two are antiparticles. The form factor is h | j Lµ (0) a c | A, θ , b , a ; A, θ , b , a ; P, θ , a , b ; P, θ , a , b i = − ǫ µν ( p + p + p + p ) ν π iN × ( tanh (cid:0) θ (cid:1) ( θ + πi )( θ + πi )( θ + πi ) (cid:18) δ a a δ a c δ a a δ b b δ b b − N δ a c δ a a δ a a δ b b δ b b (cid:19) + tanh (cid:0) θ (cid:1) ( θ + πi )( θ + πi )( θ + πi ) (cid:18) δ a a δ a c δ a a δ b b δ b b − N δ a c δ a a δ a a δ b b δ b b (cid:19) + tanh (cid:0) θ (cid:1) ( θ + πi )( θ + πi )( θ + πi ) (cid:18) δ a a δ a a δ a c δ b b δ b b − N δ a c δ a a δ a a δ b b δ b b (cid:19) + tanh (cid:0) θ (cid:1) ( θ + πi )( θ + πi )( θ + πi ) (cid:18) δ a a δ a a δ a c δ b b δ b b − N δ a c δ a a δ a a δ b b δ b b (cid:19)(cid:27) , (A.3) It is important to mention that the four-particle form factor from Ref. [7] has been found to not be completely correct as written. The momentum-vectorprefactor chosen in [7] is ( p + p − p − p ) µ instead of − ǫ µν ( p + p + p + p ) ν , as we have written in Eq. (A.3). 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