Hamiltonian, Path Integral and BRST Formulations of Large N Scalar QC D 2 on the Light-Front and Spontaneous Symmetry Breaking
aa r X i v : . [ h e p - t h ] M a r Hamiltonian, Path Integral and BRST Formulations of Large N Scalar
QCD on the Light-Front and Spontaneous Symmetry Breaking Usha Kulshreshtha [a , b]1 , Daya Shankar Kulshreshtha [a , c] ,and James P. Vary [a] 2 a Department of Physics and Astronomy,Iowa State University, Ames, Iowa 50011, USAb Department of Physics, Kirori Mal College,University of Delhi, Delhi-110007, India.c Department of Physics and Astrophysics,University of Delhi, Delhi-110007, India Abstract
Recently Grinstein, Jora, and Polosa have studied a theory of large- N scalarquantum chromodynamics in one-space one-time dimension. This theory admits aBethe-Salpeter equation describing the discrete spectrum of quark-antiquark boundstates. They consider gauge fields in the adjoint representation of SU ( N ) and scalarfields in the fundamental representation. The theory is asymptotically free and lin-early confining. The theory could possibly provide a good field theoretic frameworkfor the description of a large class of diquark-antidiquark (tetra-quark) states. Re-cently we have studied the light-front quantization of this theory without a Higgspotential. In the present work, we study the light-front Hamiltonian, path integraland BRST formulations of the theory in the presence of a Higgs potential. Thelight-front theory is seen to be gauge-invariant, possessing a set of first-class con-straints. The explicit occurrence of spontaneous symmetry breaking in the theoryis shown in unitary gauge as well as in the light-front ’t Hooft gauge. Corresponding Author Email Addresses: [email protected] (U. Kulshreshtha), [email protected] (D. S. Kulshreshtha),[email protected] (J. P. Vary) Introduction
Study of multi-quark states in quantum chromodynamics (QCD) has been a subject ofwide interest [1]-[27]. Their interpretation remains a challenging task, and a number ofphenomenological models [1]-[26] have been proposed to understand the various experi-mental observations. Some of the notable heavier states [9]-[19] which do not fit into thestandard classification of mesons (quark-antiquark ( q ¯ q ) states) and baryons (three-quarkstates) [1]-[3] are the exotic charmonium-like X, Y, Z resonances [4], [9]-[19]. Even somerelatively lighter states [20]-[26] do not find proper interpretation within the standardclassification of mesons and baryons [20]-[26].Various possibilities for understanding hadron structure beyond the usual mesons andbaryons [3, 4] have been considered in the literature [1]-[26]. Some exotic states find anatural interpretation in terms of the four-quark or tetra-quark ( q ¯ qq ¯ q ) states [3], [9]-[26].By now it is widely perceived that not only heavy states such as the X, Y, Z states havean exotic structure as tetra-quark states or diquark( Q )-antidiquark( ¯ Q ) states [3], [9]-[19],but even some light scalar mesons could also be identified as diquark-antidiquark ( Q ¯ Q )or tetra-quark systems [20]-[25].In the first approximation, even the nonet formed by f (980), a (980), κ (900), σ (500)is interpreted as the lowest Q ¯ Q multiplet [20]-[25], and the decuplet of scalar mesonswith masses above 1 Gev, formed by f (1370), f (1500), f (1710), a (1450), K (1430), isinterpreted as the lowest q ¯ q scalar multiplet (cf. Refs.[20]-[25]).The multi-quark hadron states can be extremely broad [9]-[25], and thus they couldescape experimental identification. In this context the diquark-antidiquark structures [3],[9]-[25] have been suggested to explain several decay patterns of light scalar mesons [20]-[26], heavy-light diquarks have also been introduced to study the X, Y, Z spectroscopy[9]-[19].Further, ’t Hooft, Isidori, Maiani, Polosa and Riquer [24] and others [20]-[25] haveshown how one could explain the decays of the light scalar mesons by assuming a dominantdiquark-antidiquark structure for the lightest scalar mesons [20]-[25], where the diquark isbeing taken to be a spin zero anti-triplet color state [20]-[25]. Grinstein, Jora and Polosa[25] have studied a model of large- N scalar QCD [20]-[25] in one-space and one-timedimension. Their model admits [25] a Bethe-Salpeter equation describing the discretespectrum of q ¯ q bound states [20]-[25].The work of Grinstein et al. [25] is seen to further support this hypothesis. In thework of Grinstein et al. [25], the gauge fields have been considered [25] in the adjointrepresentation of SU ( N ) and the scalar fields in the fundamental representation. Thetheory is asymptotically free and linearly confining [25]. Different aspects of this theoryhave been studied by several authors in various contexts [20]-[26].Also, because there is no spin-statistics connection in one space and one-time dimen-sion, the spinor QCD is structurally similar to the scalar QCD [5]-[8]. It is thereforeenough to consider the scalar QCD for a study of several aspects of QCD [5]-[8]. Thelarge-N behavior of scalar QCD has been studied in details by ’t Hooft and others [4]-[8].In view of the above, the motivations for our present studies could be easily highlighted.In the first place, the work of ’t Hooft, Isidori, Maiani, Polosa and Riquer [24] and others[20]-[25] has clearly shown as to how one could achieve a satisfactory explanation of2ight scalar meson decays by assuming a dominant diquark-antidiquark structure for thelightest scalar mesons [20]-[25] (where the diquark is being taken to be a spin zero anti-triplet color state). In this work, a coherent picture of scalar mesons as a mixture oftetra-quark states (dominating the lightest mesons) and heavy quark-antiquark states(dominating the heavier mesons) emerges [24].The studies of Grinstein, Jora and Polosa [25] on the large- N scalar QCD [20]-[25],further support the hypothesis of ’t Hooft, Isidori, Maiani, Polosa and Riquer [24] andothers [20]-[25], about the assumption of a dominant diquark-antidiquark structure for thescalar mesons. The work of Grinstein, Jora and Polosa is based on the assumption thatscalar QCD with a large number of colors could be used to compute the mass spectrum aswell as to estimate the mass of the first radial excitation of the lowest diquark-antidiquarkscalar meson. They have applied a numerical procedure to solve the Bethe-Salpeterequations and compute the bound state discrete spectrum of this confining theory [25].They have even obtained the possible masses of the spinor and scalar quarks by imposingthat the ratio of the ground state eigenvalues of the spinor and scalar Bethe-Salpeterequations respectively, is equal to the ratio of the physical masses m π /m σ (cf. Ref. [25],for further details). They have even extended their discussion to the case of spin-onediquarks.The above studies of Grinstein, Jora and Polosa [25], based on the scalar QCD witha large number of colors in one-space one-time dimension , clearly point towards somedefinite possibilities of gaining some insight, at least at the qualitative level, about thephysical tetra-quark states in three-space one-time dimension . In addition to this, it mayalso be possible to study this theory, in three-space one-time dimension , at a somewhatlater point of time.In view of the above, it seems reasonable to pursue these studies further. In fact,in a recent paper [26], we have studied the light-front (LF) quantization (LFQ) [28]-[36] of this theory (with a mass term for the complex scalar (diquark) field but withoutthe Higgs potential) on the hyperplanes defined by the equal light-cone time τ = x + =( x + x ) / √ φ in the definition of our Higgspotential [25], [26], and then we study the action of the theory.One of the important motivations for introducing the Higgs potential is to studythe aspects related to the spontaneous symmetry breaking (SSB)[32]-[34] in the theory.Another important motivation for introducing the Higgs potential in the theory is relatedto our long-term goal related to the study of this theory using the discrete light-cone(LC) quantization (DLCQ) along with the coherent state formalism [40]-[48], where wewish not only to study the aspects of the spontaneous symmetry breaking (SSB) but wealso wish to make contact with the experimentally observational aspects of this theoryusing the LF Hamiltonian approach to study the two- and three- body relativistic boundstate problems [25], [40]-[48]. This work therefore constitutes a part of our bigger project3hich involves a study of some aspects related to the spontaneous symmetry breaking aswell as to a study of its DLCQ using the coherent state formalism [40]-[48], in the LFHamiltonian approach to study the two- and three-body relativistic bound state problems[40]-[48].In this sense one could think that the theory under our present consideration couldperhaps provide a good basic field theoretic framework for a study of a large class ofdiquark-antidiquark or the tetra-quark states [2, 3], [9]-[26] which have been investigatedin various experiments. These are some of the motivations that necessitate our presentstudies.Now, because the theory is GI, we also study its BRST quantization [37]-[39] underappropriate BRST light-cone gauge-fixing. Usually in the Hamiltonian and the pathintegral quantization of a theory the gauge-invariance of the theory gets broken becausethe procedure of gauge-fixing converts the set of first-class constraints of the theory into aset of second-class constraints. A possible way to achieve the quantization of a GI theory,such that the gauge-invariance of the theory is maintained even under gauge-fixing is touse a generalized procedure called the BRST quantization [37]-[39], where the extendedgauge symmetry, called the BRST symmetry, is maintained even under gauge-fixing.In the next section, we briefly recap some basics of the instant-form (IF) quantization(IFQ) of this theory in the presence of a Higgs potential. Its LFQ in the presence of aHiggs potential is then considered in Sec. 3, using the Hamiltonian and path integralformulations. The light-front BRST formulation of the theory is studied in Sec. 4, underthe appropriate BRST light-cone gauge-fixing. Finally, the summary and discussion aregiven in Sec. 5. In this section we recap some of the basics of this theory of large- N scalar QCD in thepresence of a Higgs potential, studied earlier by Grinstein, Jora and Polosa without aHiggs potential (but with a mass term for the complex scalar (diquark) field φ ) [25] (themass term for the complex scalar (diquark) field φ is absorved in the definition of ourHiggs potential [25, 26]). The theory of large- N scalar QCD that we propose to study4s defined by the action [25]: S = Z L ( φ, φ † , A µ ) d x L = (cid:20) − F µν F µν + ∂ µ φ † ∂ µ φ + [ iρ ( φA µ ∂ µ φ † − φ † A µ ∂ µ φ ) + ρ φ † φA µ A µ ] − V ( | φ | ) (cid:21) | φ | = φ † φ , V ( | φ | ) = (cid:20) µ ( φ † φ ) + λ φ † φ ) (cid:21) , φ = 0 , ( − µ > , λ > g := (4 πα s ) , ρ = g √ Ng µν = g µν := (cid:18) − (cid:19) , µ, ν = 0 , IF ) g µν = g µν := (cid:18) (cid:19) , µ, ν = + , − ( F F ) (1)Here α s is the QCD coupling constant. The covariant derivative in our considerations isdefined as: D µ = ( ∂ µ + iρA aµ T a ) = ( ∂ µ + iρA µ ) (2)where A µ ( ≡ A aµ T a ) are the gluon gauge fields and T a are the generators of Lie algebracorresponding to the group SU ( N c ) obeying the commutation relations:[ T a , T b ] = if abc T c ; a, b, c = 1 , , ......, ( N c −
1) (3)with N c = 2 for SU (2) and N c = 3 for SU (3) . The structure constants f abc areantisymmetric in all indices. The gluon gauge field strength F aµν is defined as: F aµν = [( ∂ µ A aν − ∂ ν A aµ ) + ρ ( A µ × A ν ) a ]= [( ∂ µ A aν − ∂ ν A aµ ) + ρf abc A bµ A cν ] (4)Here ( A µ × A ν ) a = f abc A bµ A cν defines the cross product for any two “isotopic” vectors: A aµ and A aν [8].Further, the scalar fields φ and φ † transform as the N and ¯ N representations of the U ( N ) color group respectively [2]. Also, following the work of Grinstein, Jora and Polosa[25], we ignore all gluon self-coupling terms that arise from our chosen Lagrangian.In the Lagrangian density of our theory (defined by Eq.(1)), the first term representsthe kinetic energy of the gluon field, the second term represents the kinetic energy termfor the scalar (diquark) field, the third term represents the interaction term for the scalar (diquark) field with the gluon field, and the last term represents the Higgs potential whichis kept rather general, without making any specific choice for the parameters µ and λ .However, they are chosen such that the potential remains a double well potential withthe vacuum expectation value φ = < | φ ( x ) | > = 0, so as to allow for the spontaneoussymmetry breaking in the theory. Also, the mass term for the scalar (diquark) field hasbeen absorbed in the definition of the Higgs potential. The values: µ = m and λ = 0reproduce the theory of Grinstein, Jora and Polosa [25].5he Euler-Lagrange equations of motion of the theory (with µ , ν = 0 , µ , ν = + , − for LFQ) are obtained as:[ ∂ µ F µν + iρ ( φ∂ ν φ † − φ † ∂ ν φ ) + 2 ρ φ † φA ν ] = 0 (cid:20) − µ φ † − λ φ † φ ) φ † + ρ φ † A µ A µ + iρA µ ∂ µ φ † + iρ∂ µ ( φ † A µ ) − ∂ µ ∂ µ φ † (cid:21) = 0 (cid:20) − µ φ − λ φ † φ ) φ + ρ φA µ A µ − iρA µ ∂ µ φ − iρ∂ µ ( φA µ ) − ∂ µ ∂ µ φ (cid:21) = 0 (5) We now consider the instant-form (IF) quantization (IFQ) of the theory. The action ofthe above theory in the IF of dynamics (with A ≡ A a T a , A ≡ A a T a ) reads [25]: S = Z L dtdx L = (cid:20)
12 ( ∂ A − ∂ A ) + ( ∂ φ † ∂ φ − ∂ φ † ∂ φ ) + ρ φ † φ ( A − A )+ iρ ( φA ∂ φ † − φA ∂ φ † − φ † A ∂ φ + φ † A ∂ φ ) − µ ( φ † φ ) − λ φ † φ ) (cid:21) (6)Here t = x = x and x = x = − x . Canonical momenta obtained from the above actionare: π := ∂ L ∂ ( ∂ φ ) = ( ∂ φ † − iρA φ † ) , π † := ∂ L ∂ ( ∂ φ † ) = ( ∂ φ + iρA φ )Π := ∂ L ∂ ( ∂ A ) = 0 , E (= Π ) := ∂ L ∂ ( ∂ A ) = ( ∂ A − ∂ A ) (7)Here π, π † , Π ( ≡ Π a T a ) and E := Π ( ≡ Π a T a ) are the momenta canonically conjugaterespectively to φ, φ † , A and A . The above equations however, imply that the theorypossesses only one primary constraint: χ = Π ≈ ≈ here denotes a weak equality in the sense of Dirac [28], and it impliesthat the constraints hold as a strong equality only on the reduced hyper surface of theconstraints and not in the rest of the phase space of the classical theory (and similarlyone can consider it as a weak operator equality for the corresponding quantum theory).The canonical Hamiltonian density corresponding to L is: H c := (cid:20) π∂ φ + π † ∂ φ † + Π ∂ A + E∂ A − L (cid:21) = (cid:20)
12 ( E ) − A ∂ E + π † π + ∂ φ † ∂ φ + ρ A φ † φ − iρA ( φπ − φ † π † ) − iρA ( φ † ∂ φ − φ∂ φ † ) + µ ( φ † φ ) + λ φ † φ ) (cid:21) (9)6fter including the primary constraint χ in the canonical Hamiltonian density H c withthe help of the Lagrange multiplier field u , the total Hamiltonian density H T could bewritten as : H T = (cid:20) Π u + 12 ( E ) − A ∂ E + π † π + ∂ φ † ∂ φ + ρ A φ † φ − iρA ( φπ − φ † π † ) − iρA ( φ † ∂ φ − φ∂ φ † ) + µ ( φ † φ ) + λ φ † φ ) (cid:21) (10)Hamilton’s equations of motion of the theory that preserve the constraints of the theoryin the course of time could be obtained from the total Hamiltonian: H T = R H T dx (andare omitted here for the sake of brevity). Demanding that the primary constraint χ bepreserved in the course of time, one obtains the secondary Gauss-law constraint of thetheory as: χ = [ ∂ E + iρ ( φπ − φ † π † )] ≈ χ for all times gives rise to one further constraint: χ = [2 ρ A π † φ † + iρA ( φ∂ φ † + φ † ∂ φ )] ≈ χ i (with i = 1,2,3). The matrix R αβ of the Poisson brackets among the set of constraints χ i with ( i = 1 , ,
3) is seen to besingular (the other details of the matrix R αβ are omitted here for the sake of brevity). Thisimplies that the set of constraints χ i is first-class and that the theory under considerationis gauge-invariant (GI). Consequently the theory is seen to possess the local vector gaugesymmetry defined by the local vector gauge transformations: δφ = iρβφ , δφ † = − iρβφ † , δA = ∂ β , δA = ∂ β (13)where β ≡ β ( x , x ) is an arbitrary real function of its arguments. This theory could nowbe quantized under some appropriate gauge-fixing conditions, e.g., under the time-axialor temporal gauge: A ≈
0. The details of this IFQ are however, outside the scope ofthe present work (what actually happens is that one of the matrix elements of the matrix R αβ involves a linear combination of a Dirac distribution function δ ( x − y ) and its firstderivative and finding its inverse is a rather non-trivial task). We now proceed with theLFQ of this theory in the next section. In this section we study the LF Hamiltonian and path integral formulations [25]-[31] ofthe above theory [25] under appropriate LC gauge-fixing. The action for the scalar theory7n LF coordinates x ± := ( x ± x ) / √ A + ≡ A + a T a , A − ≡ A − a T a ) reads: S = Z L dx + dx − L = (cid:20)
12 ( ∂ + A + − ∂ − A − ) + ( ∂ + φ † ∂ − φ + ∂ − φ † ∂ + φ ) − µ ( φ † φ ) − λ φ † φ ) + iρA + ( φ∂ + φ † − φ † ∂ + φ ) + iρA − ( φ∂ − φ † − φ † ∂ − φ ) + 2 ρ φ † φA + A − (cid:21) (14)In the work of Ref.[25], the authors have studied the above action, after implementingthe gauge-fixing condition (GFC) A + ≈ A + ≈
0) as one of the gauge constraints [28]-[31]which becomes strongly equal to zero only on the reduced hyper surface of the constraintsand remains non-zero in the rest of the phase space of the theory.It may be important to note here that one of the salient features of Dirac quantizationprocedure [28] is that in this quantization the gauge-fixing conditions should be treatedon par with other gauge-constraints of the theory which are only weakly equal to zeroin the sense of Dirac[28], and they become strongly equal to zero only on the reducedhyper surface of the constraints of the theory and not in the rest of the phase space ofthe classical theory (in the corresponding quantum theory these weak equalities becomethe weak operator equalities).Another thing to be noted here is that we have introduced the Higgs potential inour present work and we have absorbed the mass term for the scalar (diquark) field inthe definition of our Higgs potential [25, 26], [32]-[34]. We now proceed to study the LFHamiltonian and path integral formulations of the theory defined by the above action.The LF Euler-Lagrange equations of motion of the theory are: (cid:20) ( ∂ + ∂ − A − − ∂ + ∂ + A + ) + iρ ( φ∂ + φ † − φ † ∂ + φ ) + 2 ρ φ † φA − (cid:21) = 0 (cid:20) ( ∂ + ∂ − A + − ∂ − ∂ − A − ) + iρ ( φ∂ − φ † − φ † ∂ − φ ) + 2 ρ φ † φA + (cid:21) = 0 (cid:20) − µ φ † − λ φ † φφ † − ∂ + ∂ − φ † + 2 iρ ( A + ∂ + φ † + A − ∂ − φ † )+ iρφ † ( ∂ + A + − ∂ − A − ) + 2 ρ φ † A + A − (cid:21) = 0 (cid:20) − µ φ − λ φ † φφ − ∂ + ∂ − φ − iρ ( A + ∂ + φ + A − ∂ − φ ) − iρφ ( ∂ + A + − ∂ − A − ) + 2 ρ φA + A − (cid:21) = 0 (15)In the following, we consider the Hamiltonian formulation of the theory described by the8bove action. The canonical momenta obtained from the above action are: π := ∂ L ∂ ( ∂ + φ ) = ( ∂ − φ † − iρA + φ † ) , π † := ∂ L ∂ ( ∂ + φ † ) = ( ∂ − φ + iρA + φ )Π + := ∂ L ∂ ( ∂ + A − ) = 0 , Π − := ∂ L ∂ ( ∂ + A + ) = ( ∂ + A + − ∂ − A − ) (16)Here π, π † , Π + ( ≡ Π + a T a ) and Π − ( ≡ Π − a T a ) are the momenta canonically conjugaterespectively to φ, φ † , A − and A + . The above equations however, imply that the theory possesses three primary con-straints: χ = Π + ≈ , χ = [ π − ∂ − φ † + iρA + φ † ] ≈ , χ = [ π † − ∂ − φ − iρA + φ ] ≈ L is: H c = (cid:20) π∂ + φ + π † ∂ + φ † + Π + ∂ + A − + Π − ∂ + A + − L (cid:21) = (cid:20)
12 (Π − ) + Π − ( ∂ − A − ) + µ ( φ † φ ) + λ φ † φ ) − iρA − ( φ∂ − φ † − φ † ∂ − φ ) − ρ φ † φA + A − (cid:21) (18)After including the primary constraints χ , χ and χ in the canonical Hamiltonian density H c with the help of the Lagrange multiplier fields u, v and w , the total Hamiltonian density H T could be written as : H T = (cid:20) (Π + ) u + ( π − ∂ − φ † + iρA + φ † ) v + ( π † − ∂ − φ − iρA + φ ) w + µ ( φ † φ )+ λ φ † φ ) + 12 (Π − ) + Π − ∂ − A − − iρA − ( φ∂ − φ † − φ † ∂ − φ ) − ρ φ † φA + A − (cid:21) (19)The Hamilton’s equations of motion of the theory that preserve the constraints of thetheory in the course of time could be obtained from the total Hamiltonian (and areomitted here for the sake of brevity): H T = R H T dx − . Demanding that the primaryconstraint χ be preserved in the course of time, one obtains the secondary Gauss-lawconstraint of the theory as: χ = [ ∂ − Π − + iρ ( φ∂ − φ † − φ † ∂ − φ ) + 2 ρ φ † φA + ] ≈ χ , χ and χ , for all times does not give rise to any further constraints.The theory is thus seen to possess only four constraints χ i (with i = 1,2,3,4). Theconstraints χ , χ and χ could however, be combined in to a single constraint: ψ = [ ∂ − Π − + iρ ( φπ − φ † π † )] ≈ set of constraints of the theory could be written as:Ω = χ = Π + ≈ , Ω = ψ = [ ∂ − Π − + iρ ( φπ − φ † π † )] ≈ i , with ( i = 1 ,
2) isseen to be a singular matrix implying that the set of constraints Ω i is first-class and thatthe theory under consideration is gauge-invariant. Expressions for the components of thevector gauge current density of the theory are obtained as: j + = [ − iρβφ∂ − φ † + iρβφ † ∂ − φ − ρ βA + φ † φ − β ( ∂ − ∂ + A + − ∂ − ∂ − A − )] j − = [ − iρβφ∂ + φ † + iρβφ † ∂ + φ − ρ βA − φ † φ + β ( ∂ + ∂ + A + − ∂ + ∂ − A − )] (23)The divergence of the vector gauge current density of the theory could now be easily seento vanish satisfying the continuity equation: ∂ µ j µ = 0 , implying that the theory possessesat the classical level, a local vector-gauge symmetry. The action of the theory is indeedseen to be invariant under the local vector gauge transformations: δφ = − iρβφ , δφ † = iρβφ † , δA − = ∂ + β , δA + = ∂ − β (24a) δπ = [ ρ βφ † A + + iρβ∂ − φ † ] , δπ † = [ ρ βφA + − iρβ∂ − φ ] (24b) δu = δv = δw = δ Π + = δ Π − = δ Π u = δ Π v = δ Π w = 0 (24c)where β ≡ β ( x + , x − ) is an arbitrary real function of its arguments and Π u , Π v and Π w arethe momenta canonically conjugate to the Lagrange multiplier fields u, v and w respec-tively, which are treated here as dynamical fields. Using the Euler-Lagrange equations ofmotion of the theory and the expressions for the components of the vector gauge currentdensity of the theory, one could now easily show that: j + = β (1 + ρ )[ ∂ − ∂ − A − − ∂ + ∂ − A + ] (25)It may be important to point out here that Grinstein, Jora and Polosa [25], have ob-tained an equation (under the gauge A + = 0) analogous to the above equation connecting ∂ − ∂ − A − and j + (cf. Eq. (5) of Ref. [25]) which has been shown [25], to admit a solution(in the absence of background fields) [25], which when substituted in to the Lagrangiandensity of the theory implies a linear potential between charges (for further details, werefer to the work of Ref.[25]).In order to quantize the theory using Dirac’s procedure we now convert the set offirst-class constraints of the theory η i into a set of second-class constraints, by impos-ing, arbitrarily, some additional constraints on the system called gauge-fixing conditions(GFC’s) or the gauge-constraints [28]-[31]. For the present theory, we could choose, forexample, the following set of GFC’s: ζ = A + ≈ ζ = A − ≈
0. Here the gauge A + ≈ A − ≈ ξ = Ω = χ = Π + ≈ ξ = Ω = ψ = [ ∂ − Π − + iρ ( φπ − φ † π † )] ≈ ξ = ζ = A + ≈ ξ = ζ = A − ≈ R αβ of the Poisson brackets among the set of constraints ξ i with ( i = 1 , , , (cid:20) || det ( R αβ ) || (cid:21) = (cid:20) ∂ − δ ( x − − y − ) δ ( x − − y − ) (cid:21) (27)The other details of the matrix R αβ are omitted here for the sake of brevity. Finally,following the Dirac quantization procedure, the nonvanishing equal light-cone-time com-mutators of the theory, under the GFC’s: A + ≈ A − ≈ φ ( x + , x − ) , π ( x + , y − )] = i δ ( x − − y − ) (28a)[ φ † ( x + , x − ) , π † ( x + , y − )] = i δ ( x − − y − ) (28b)[ φ ( x + , x − ) , Π − ( x + , y − )] = 12 ρφ ǫ ( x − − y − ) (28c)[ φ † ( x + , x − ) , Π − ( x + , y − )] = − ρφ † ǫ ( x − − y − ) (28d)[ π ( x + , x − ) , Π − ( x + , y − )] = 12 ρ π ǫ ( x − − y − ) (28e)[ π † ( x + , x − ) , Π − ( x + , y − )] = − ρ π † ǫ ( x − − y − ) (28f)[Π − ( x + , x − ) , φ ( x + , y − )] = 12 ρφ ǫ ( x − − y − ) (28g)[Π − ( x + , x − ) , φ † ( x + , y − )] = − ρφ † ǫ ( x − − y − ) (28h)[Π − ( x + , x − ) , π ( x + , y − )] = − ρπ ǫ ( x − − y − ) (28i)[Π − ( x + , x − ) , π † ( x + , y − )] = 12 ρπ † ǫ ( x − − y − ) (28j)The first-order Lagrangian density L I of the theory is: L I := (cid:20) π ( ∂ + φ ) + π † ( ∂ + φ † ) + Π + ( ∂ + A − ) + Π − ( ∂ + A + )+ Π u ( ∂ + u ) + Π v ( ∂ + v ) + Π w ( ∂ + w ) − H T (cid:21) = (cid:20)
12 (Π − ) + ∂ + φ † ∂ − φ + ∂ − φ † ∂ + φ + 2 ρ φ † φA + A − − iρA − ( φ † ∂ − φ − φ∂ − φ † ) − iρA + ( φ † ∂ + φ − φ∂ + φ † ) − µ φ † φ − λ φ † φ ) (cid:21) (29)In the path integral formulation [29]-[31], the transition to quantum theory is made bywriting the vacuum to vacuum transition amplitude for the theory called the generatingfunctional Z [ J k ]. For the present theory [25] under the GFC’s: ζ = A + ≈ ζ = A − ≈ J k it reads: Z [ J k ] = Z [ dµ ] exp (cid:20) i Z d x (cid:20) J k Φ k + π∂ + φ + π † ∂ + φ † + Π + ∂ + A − + Π − ∂ + A + + Π u ∂ + u + Π v ∂ + v + Π w ∂ + w − H T (cid:21)(cid:21) (30)11ere, the phase space variables of the theory are: Φ k ≡ ( φ, φ † , A − , A + , u, v, w ) with thecorresponding respective canonical conjugate momenta: Π k ≡ ( π, π † , Π + , Π − , Π u , Π v , Π w ).The functional measure [ dµ ] of the generating functional Z [ J k ] under the above gauge-fixing is obtained as :[ dµ ] = [ ∂ − δ ( x − − y − ) δ ( x − − y − )][ dφ ][ dφ † ][ dA + ][ dA − ][ du ][ dv ][ dw ][ dπ ][ dπ † ][ d Π − ][ d Π + ][ d Π u ][ d Π v ][ d Π w ] δ [Π + ≈ δ [ A − ≈ δ [( ∂ − Π − + iρ ( φπ − φ † π † )) ≈ δ [ A + ≈
0] (31)The LF Hamiltonian and path integral quantization of the theory under the set of GFC’s: A + ≈ A − ≈ In this section, we consider the spontaneous symmetry breaking (SSB) in the theory in (i)the so-called unitary gauge and (ii) in the ’t Hooft gauge and show explicitly the existenceof SSB [32]-[34] in the theory in both the cases.Our Higgs potential possesses a local maximum at φ ( x ) = φ = s(cid:18) − µ λ (cid:19) e iθ , ≤ θ < π (32)where the phase angle θ defines a direction in the complex φ − plane. Here the vacuumstate (or the ground state) of the system is clearly non unique, and the SSB will occur forany particular choice of the value of θ . In our considerations we however, choose θ = 0which in turn implies: φ = s(cid:18) − µ λ (cid:19) = v √ v >
0. We now parametrize the field φ ( x ) in terms of its deviationsfrom its vacuum expectation value (VEV): < | φ ( x ) | > = φ = ( v/ √ > σ ( x )and η ( x ), which measure the deviations of the field φ ( x ) from theequilibrium ground state configuration φ ( x ) = φ . For this we expand our complex scalarfield φ ( x ), in terms of two real fields σ ( x ) and η ( x ) as: φ ( x ) = [ ϕ + ϕ ( x )] = 1 √ v + σ ( x )) + iη ( x )] (34)with ϕ = v √ , ϕ ( x ) = 1 √ σ ( x )) + iη ( x )] (35)such that the real fields σ ( x ) and η ( x ) have vanishing vacuum expectation values. In fact,the term ϕ here could be interpreted as the zero mode of the theory [32]-[34] and thefluctuation field ϕ ( x ) could be interpreted as the normal mode of the theory [32]-[34].The Lagrangian density of our LF theory in terms of the real fields σ ( x ) and η ( x ) (afterdropping the terms which are irrelevant for our discussions namely, a constant term, a12erm linear in the field σ ( x ) and all the quartic interaction terms in the fields) (with m σ = p (2 µ + λv ) / m v = | vρ | ) becomes: L = (cid:20)
12 ( ∂ + A + − ∂ − A − ) + 2 ∂ + σ∂ − σ + 2 ∂ + η∂ − η − m σ σ + vρ [ A + ( x ) ∂ + η ( x ) + A − ( x ) ∂ − η ( x )] −
112 (6 µ + λv ) η + m v A + A − + ρσ [ A + ( x ) ∂ + η ( x ) + A − ( x ) ∂ − η ( x )] − ρη [ A + ( x ) ∂ + σ ( x ) − A − ( x ) ∂ − σ ( x )]+ 2 vρ σA + A − − λvσ ( σ + η ) (cid:21) (36)The first term in the above Lagrangian density represents the kinetic energy of the elec-tromagnetic field; the second term represents the kinetic energy of the real scalar field σ ( x ); the third term represents the kinetic energy of the real scalar field η ( x ); the fourthterm represents the mass term for the real scalar field σ ( x ); the fifth term, which involvesthe product of the fields A + ( x ) and A − ( x ) with the derivatives of the field η ( x ), representsa quadratic interaction term involving the fields A + ( x ) , A − ( x ) and η ( x ), and it impliesthat the fields A + ( x ) , A − ( x ) and η ( x ) are not independent normal coordinates and aretherefore not free fields; consequently the sixth and seventh terms cannot be interpretedas the mass terms for the real scalar field η ( x ) and the electromagnetic field respectively.It also implies that the above Lagrangian density contains an unphysical field; we willeliminate it under some suitable gauge. The last four terms in the above Lagrangiandensity represent simply the cubic interaction terms of the theory which will be neededfor our later discussions. We now consider this theory in the so-called unitary gauge. In fact, for any complex field φ ( x ), a gauge transformation can be found which transforms φ ( x ) into a real field suchas: φ ( x ) = 1 √ v + σ ( x )] (37)The gauge in which the transformed field has this form is called as the unitary gauge.With this substitution the transformed Lagrangian density in the so-called unitary gauge(after dropping as before, the terms which are irrelevant for our discussions, namely, aconstant term, a term linear in the field σ ( x ) and the quartic interaction terms) becomes: L U = L U + L intU L U = (cid:20)
12 ( ∂ + A + − ∂ − A − ) + 2 ∂ + σ∂ − σ − m σ σ + m v A + A − (cid:21) L intU = (cid:20) vρ A + A − σ − λ vσ (cid:21) (38)The interaction part of the above Lagrangian density however, does not contain anyquadratic coupling terms involving the coupling of the fields σ ( x ), A + ( x ) and A − ( x ).13ence treating the interaction part of the Lagrangian density L intU ( x ), in perturbationtheory, one could interpret L U as the free-field Lagrangian density of a real Klein-Gordonfield σ ( x ) and a real massive vector field A µ ( x ). Upon quantizing the theory, the field σ ( x ) gives rise to neutral scalar bosons of mass m σ = p (2 µ + λv ) / A µ ( x )gives rise to neutral vector bosons of mass m v = | vρ | . This is an explicit demonstrationof the SSB in the theory through the Higgs mechanism where the massive spin 0 bosonassociated with the field σ ( x ) is the Higgs boson (or Higgs Scalar) of the theory. Here thevector field A µ ( x ) has become massive in the process of SSB through the Higgs mechanism. We consider the LF ’t Hooft gauge defined by:[ ∂ + A + + ∂ − A − − ρvη ( x )] ≈ L :˜ L = [ L + L ′ tH ] (40)by adding the LF ’t Hooft-gauge- fixing term L ′ tH : L ′ tH = (cid:20) −
12 [ ∂ + A + + ∂ − A − − ρvη ( x )] (cid:21) (41)to the Lagrangian density of the theory expressed in terms of the real scalar fields σ and η given by Eq.(33) and ignoring the terms irrelevant for our discussion as explained inthe forgoing. The LF ’t Hooft gauge-fixed action of the theory ˜ S could now be writtenafter a partial integration (and with m η = p (6 µ + λv ) / S = Z ˜ L dx + dx − ˜ L = (cid:20)
12 ( ∂ + A + − ∂ − A − ) + 2 ∂ + σ∂ − σ + 2 ∂ + η∂ − η − m σ σ − m η η + m v A + A − (cid:21) (42)The fields σ ( x ), η ( x ) and A µ ( x ), could now be treated in perturbation theory as threeindependent fields which could be quantized in the usual manner. The LF ’t Hooft gaugehere reintroduces the field η ( x ) which gets eliminated in the so-called unitary gauge.However, there are no real particles corresponding to the quantized η ( x ) field and theyappear in a manner akin to the longitudinal and scalar photons of QED theory. For the BRST formulation of the model, we rewrite the theory as a quantum systemthat possesses the generalized gauge invariance called BRST symmetry. For this, we firstenlarge the Hilbert space of our gauge-invariant theory and replace the notion of gauge-transformation, which shifts operators by c-number functions, by a BRST transformation,14hich mixes operators with Bose and Fermi statistics. We then introduce new anti-commuting variable c and ¯ c (Grassman numbers on the classical level and operators inthe quantized theory) and a commuting variable b such that [37]-[39]:ˆ δφ = − iρcφ , ˆ δφ † = iρcφ † , ˆ δA − = ∂ + c , ˆ δA + = ∂ − c (43a)ˆ δπ = [ ρ cφ † A + + iρc∂ − φ † ] , ˆ δπ † = [ ρ cφA + − iρc∂ − φ ] (43b)ˆ δu = ˆ δv = ˆ δw = ˆ δ Π + = ˆ δ Π − = ˆ δ Π u = ˆ δ Π v = ˆ δ Π w = 0 (43c)ˆ δc = 0 , ˆ δ ¯ c = b , ˆ δb = 0 (43d)with the property ˆ δ = 0. We now define a BRST-invariant function of the dynamicalphase space variables of the theory to be a function f such that ˆ δf = 0. Now theBRST gauge-fixed quantum Lagrangian density L BRST for the theory could be obtainedby adding to the first-order Lagrangian density L I , a trivial BRST-invariant function,e.g. as follows: L BRST = (cid:20)
12 (Π − ) + ∂ + φ † ∂ − φ + ∂ − φ † ∂ + φ − iρA − ( φ † ∂ − φ − φ∂ − φ † ) − λ φ † φ ) − µ φ † φ + 2 ρ φ † φA + A − − iρA + ( φ † ∂ + φ − φ∂ + φ † ) + ˆ δ [¯ c ( ∂ + A − + 12 b )] (cid:21) (44)The last term in the above equation is the extra BRST-invariant gauge-fixing term. Afterone integration by parts, the above equation could now be written as: L BRST = (cid:20)
12 (Π − ) + ∂ + φ † ∂ − φ + ∂ − φ † ∂ + φ − iρA − ( φ † ∂ − φ − φ∂ − φ † ) − µ φ † φ − λ φ † φ ) + 2 ρ φ † φA + A − − iρA + ( φ † ∂ + φ − φ∂ + φ † ) + ∂ + A − + 12 b + ( ∂ + ¯ c )( ∂ + c ) (cid:21) (45)Proceeding classically, the Euler-Lagrange equation for b reads: − b = ( ∂ + A − ) (46)the requirement ˆ δb = 0 then implies − ˆ δb = [ˆ δ ( ∂ + A − )] (47)which in turn implies ∂ + ∂ + c = 0 (48)The above equation is also an Euler-Lagrange equation obtained by the variation of L BRST with respect to ¯ c . In introducing momenta one has to be careful in defining thosefor the fermionic variables. We thus define the bosonic momenta in the usual manner sothat Π + := ∂∂ ( ∂ + A − ) L BRST = b (49)but for the fermionic momenta with directional derivatives we setΠ c := L BRST ←− ∂∂ ( ∂ + c ) = ∂ + ¯ c , Π ¯ c := −→ ∂∂ ( ∂ + ¯ c ) L BRST = ∂ + c (50)15mplying that the variable canonically conjugate to c is ( ∂ + ¯ c ) and the variable conjugateto ¯ c is ( ∂ + c ). For writing the Hamiltonian density from the Lagrangian density in theusual manner we remember that the former has to be Hermitian so that: H BRST = (cid:20) π∂ + φ + π † ∂ + φ † + Π + ∂ + A − + Π − ∂ + A + + Π b ∂ + b + Π u ∂ + u + Π v ∂ + v + Π w ∂ + w + Π c ( ∂ + c ) + ( ∂ + ¯ c )Π ¯ c − L BRST (cid:21) = (cid:20)
12 (Π − ) + Π − ( ∂ − A − − ρ φ † φA + A − + µ φ † φ + λ φ † φ ) − iρA − ( φ∂ − φ † − φ † ∂ − φ ) −
12 (Π + ) + Π c Π ¯ c (cid:21) (51)The consistency of the last two equations could now be easily checked by looking at theHamilton’s equations for the fermionic variables. Also for the operators c, ¯ c, ∂ + c and ∂ + ¯ c ,one needs to satisfy the anticommutation relations of ∂ + c with ¯ c or of ∂ + ¯ c with c , butnot of c , with ¯ c . In general, c and ¯ c are independent canonical variables and one assumesthat [37]-[39]: { Π c , Π ¯ c } = { ¯ c, c } = ∂ + { ¯ c, c } = 0 , { ∂ + ¯ c, c } = ( − { ∂ + c, ¯ c } (52)where { , } means an anticommutator. We thus see that the anticommulators in theabove equation are non-trivial and need to be fixed. In order to fix these, we demandthat c satisfy the Heisenberg equation:[ c, H BRST ] = i∂ + c (53)and using the property c = c = 0 one obtains[ c, H BRST ] = { ∂ + ¯ c, c } ∂ + c (54)The last three equations then imply : { ∂ + ¯ c, c } = ( − { ∂ + c, ¯ c } = i (55)Here the minus sign in the above equation is nontrivial and implies the existence ofstates with negative norm in the space of state vectors of the theory. The BRST chargeoperator Q is the generator of the BRST transformations. It is nilpotent and satisfies Q = 0. It mixes operators which satisfy Bose and Fermi statistics. According to itsconventional definition, its commutators with Bose operators and its anti-commutatorswith Fermi operators for the present theory satisfy:[ φ, Q ] = − iρφc , [ φ † , Q ] = iρφ † c (56a)[ π, Q ] = − iρcπ , [Π † , Q ] = iρcπ † (56b)[ A + , Q ] = ∂ − c , [ A − , Q ] = ∂ + c , [Π + , Q ] = [Π − , Q ] = 0 (56c) { ∂ + ¯ c, Q } = [ − ∂ − Π − − iρ ( φπ − φ † π † )] , { ¯ c, Q } = ( − Π + ) (56d)16ll other commutators and anti-commutators involving Q vanish. In view of this, theBRST charge operator of the present theory can be written as: Q = Z dx − (cid:20) ic ∂ − Π − − ρc ( φπ − φ † π † ) − i∂ + c Π + (cid:21) (57)This equation implies that the set of states satisfying the conditions:Π + | ψ i = 0 , [ ∂ − Π − + iρ ( φπ − φ † π † )] | ψ i = 0 (58)belong to the dynamically stable subspace of states | ψ > satisfying Q | ψ > = 0, i.e., itbelongs to the set of BRST-invariant states. In order to understand the condition neededfor recovering the physical states of the theory we rewrite the operators c and ¯ c in termsof fermionic annihiliation and creation operators. For this purpose we consider Eulerlagrange equation for the variable c derived earlier. The solution of this equation gives(for the light-cone time x + ≡ τ ) the Heisenberg operators c ( τ ) and correspondingly ¯ c ( τ )in terms of the fermionic annihilation and creation operators as: c ( τ ) = G ( τ ) + F ( τ ) , ¯ c ( τ ) = G † ( τ ) + F † ( τ ) (59)Which at the light-cone time τ = 0 imply c ≡ c (0) = F, ¯ c ( τ ) ≡ ¯ c (0) = F † (60a) ∂ + c ( τ ) ≡ ∂ + c (0) = G, ∂ + ¯ c ( τ ) ≡ ∂ + ¯ c (0) = G † (60b)By imposing the conditions (obtained earlier): c = ¯ c = { ¯ c, c } = { ∂ + ¯ c, ∂ + c } = 0 (61a) { ∂ + ¯ c, c } = ( − { ∂ + c, ¯ c } = i (61b)we then obtain F = ( F † ) = { F † , F } = { G † , G } = 0 , { G † , F } = ( − { G, F † } = i (62)Now let | > denote the fermionic vacuum for which G | > = F | > = 0 (63)Defining | > to have norm one, the last three equations imply < | F G † | > = i , < | GF † | > = − i (64)so that G † | > = 0 , F † | > = 0 (65)The theory is thus seen to possess negative norm states in the fermionic sector. Theexistence of these negative norm states as free states of the fermionic part of H BRST is17owever, irrelevant to the existence of physicsl states in the orthogonal subspace of theHilbert space. In terms of annihilation and creation operators H BRST is: H BRST = (cid:20)
12 (Π − ) + Π − ( ∂ − A − ) + µ φ † φ + λ φ † φ ) − ρ φ † φA + A − − iρA − ( φ∂ − φ † − φ † ∂ − φ ) −
12 (Π + ) + G † G (cid:21) (66)and the BRST charge operator is: Q = Z dx − (cid:20) iF ∂ − Π − − ρF ( φπ − φ † π † ) − iG Π + (cid:21) (67)Now because Q | ψ > = 0, the set of states annihiliated by Q contains not only the set forwhich the constraints of the theory hold but also additional states for which F | ψ > = G | ψ > = 0 , Π + | ψ i 6 = 0 , [ ∂ − Π − + iρ ( φπ − φ † π † )] | ψ i 6 = 0 (68)The Hamiltonian is also invariant under the anti-BRST transformation given by:¯ˆ δφ = iρ ¯ cφ , ¯ˆ δφ † = − iρ ¯ cφ † , ¯ˆ δA − = − ∂ + ¯ c , ¯ˆ δA + = − ∂ − ¯ c (69a)¯ˆ δπ = [ − ρ ¯ cφ † A + − iρ ¯ c∂ − φ † ] , ¯ˆ δπ † = [ − ρ ¯ cφA + + iρ ¯ c∂ − φ ] (69b)¯ˆ δu = ¯ˆ δv = ¯ˆ δw = ¯ˆ δ Π + = ¯ˆ δ Π − = ¯ˆ δ Π u = ¯ˆ δ Π v = ¯ˆ δ Π w = 0 (69c)¯ˆ δc = − b , ¯ˆ δ ¯ c = 0 , ¯ˆ δb = 0 (69d)with generator or anti-BRST charge¯ Q = Z dx − (cid:20) − i ¯ c ∂ − Π − − ρ ¯ c ( φπ − φ † π † ) + i∂ + ¯ c Π + (cid:21) (70)which in terms of annihilation and creation operators reads:¯ Q = Z dx − (cid:20) − iF † ∂ − Π − − ρF † ( φπ − φ † π † ) + iG † Π + (cid:21) (71)We also have ∂ + Q = [ Q, H
BRST ] = 0 , ∂ + ¯ Q = [ ¯ Q, H
BRST ] = 0 (72)with H BRST = Z dx − H BRST (73)and we further impose the dual condition that both Q and ¯ Q annihilate physical states,implying that: Q | ψ > = 0 and ¯ Q | ψ > = 0 (74)The states for which the constraints of the theory hold, satisfy both of these conditionsand are in fact, the only states satisfying both of these conditions, since although with G † G = ( − GG † (75)18here are no states of this operator with G † | ψ > = 0 and F † | ψ > = 0, and hence no freeeigenstates of the fermionic part of H BRST that are annihilated by each of G , G † , F , and F † . Thus the only states satisfying Q | ψ > = 0 and ¯ Q | ψ > = 0 are those that satisfy theconstraints of the theory.Now because Q | ψ > = 0, the set of states annihilated by Q contains not only theset of states for which the constraints of the theory hold but also additional states forwhich the constraints of the theory do not hold. This situation is, however, easily avoidedby additionally imposing on the theory, the dual condition: Q | ψ > = 0 and ¯ Q | ψ > = 0.By imposing both of these conditions on the theory simultaneously, one finds that thestates for which the constraints of the theory hold are the only states satisfying both ofthese conditions. This is traced to the conditions on the fermionic variables c and ¯ c whichconstrain the solutions such that one cannot have simultaneously c , ∂ + c and ¯ c , ∂ + ¯ c , appliedto | ψ > giving zero. Thus the only states satisfying Q | ψ > = 0 and ¯ Q | ψ > = 0 are thosethat satisfy the constraints of the theory and they belong to the set of BRST-invariant aswell as to the set of anti-BRST-invariant states.Alternatively, one can understand the above point in terms of fermionic annihiliationand creation operators as follows. The condition Q | ψ > = 0 implies the that the set ofstates annihilated by Q contains not only the states for which the constraints of the theoryhold but also additional states for which the constraints do not hold. However, ¯ Q | ψ > = 0guarantees that the set of states annihilated by ¯ Q contains only the states for which theconstraints hold, simply because G † | ψ > = 0 and F † | ψ > = 0. This completes the BRSTformulation of the theory. Theoretical and experimental studies of multi-quark states are challenging and a numberof phenomenological models [1]-[26] have been proposed in order to provide interpretationand gain understanding.Some of the states [2, 3], [9]-[19] which do not fit in to the standard classificationof mesons (two quark states) and baryons (three quark states) [1]-[3] find a rather morenatural interpretation in terms of the tetra-quark states or the diquark-antidiquark states[2, 3], [9]-[25].In particular, as mentioned in the foregoing, Grinstein, Jora and Polosa [25] have stud-ied a model of large- N scalar QCD [25]. This theory of Grinstein et al. [25] admits aBethe-Salpeter equation describing the discrete spectrum of q ¯ q bound states [20]-[25]. Inthe their work, the gauge fields have been considered [25] in the adjoint representation of SU ( N ) and the scalar fields in the fundamental representation. The theory is asymptot-ically free and linearly confining [25]. Different aspects of this theory have been studiedby several authors in various contexts [20]-[25].In Ref.[26], we have studied the LFQ of the theory of large- N scalar QCD studiedby Grinstein, Jora and Polosa [25], without Higgs potential[20]-[25] on the LF using theHamiltonian [28] and path integral [29]-[31] formulations. In in the present work, wehave studied this theory in the presence of a Higgs potential and we have studied itsLFQ using the Hamiltonian, path integral and BRST formulations [37]-[39]. We have also19hown explicitly the occurrence of the SSB in the theory in the unitary gauge as well asin the LF ’t Hooft gauge [32]-[34].In the Hamiltonian and path integral quantization of the theory the gauge-invarianceof the theory gets broken because of the gauge-fixing. In view of this, we go to a moregeneralized quantization procedure called the BRST quantization [37]-[39], where theextended gauge symmetry of the theory is maintained even under gauge-fixing.In the present work, we have studied the LF-BRST quantization of the theory undersome specific LF-BRST gauge-fixing (where a particular but non-unique gauge has beenchosen). In this procedure, we embed the original GI theory into a BRST system, thequantum Hamiltonian H BRST (which includes the gauge-fixing contribution) commuteswith the BRST charge as well as with the anti-BRST charge. The new extended gaugesymmetry which replaces the gauge invariance is maintained (even under the BRST gauge-fixing) and projecting any state onto the sector of BRST and anti-BRST invariant statesyields a theory which is isomorphic to the original GI theory.
The authors thank Stan Brodsky for the motivations and continuous support and for hiscollaboration in the early stages of this work and especially for his contributions on theSSB aspects of this work and for several clarifying suggestions and discussions as well asfor providing useful references on the subject. This work was supported in part by theUS Department of Energy under Grant No. DE-FG02-87ER40371.20 eferences [1] M. Gell-Mann, Phys. Lett. , (1964) 214; E. Witten, Nucl. Phys. B160 , (1979) 57;for a good review of the subject, see: S. J. Brodsky, H. C. Pauli and S.S. Pinsky,Phys. Rep. , (1998) 299.[2] R. L. Jaffe, Phys. Rev.
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