BBroken Chiral Symmetry on a Null Plane
Silas R. Beane Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie),Universit¨at Bonn, D-53115 Bonn, [email protected] +49 228 733452
Abstract:
On a null-plane (light-front), all effects of spontaneous chiral symmetry break-ing are contained in the three Hamiltonians (dynamical Poincar´e generators), while thevacuum state is a chiral invariant. This property is used to give a general proof of Gold-stone’s theorem on a null-plane. Focusing on null-plane QCD with N degenerate flavorsof light quarks, the chiral-symmetry breaking Hamiltonians are obtained, and the role ofvacuum condensates is clarified. In particular, the null-plane Gell-Mann-Oakes-Rennerformula is derived, and a general prescription is given for mapping all chiral-symmetrybreaking QCD condensates to chiral-symmetry conserving null-plane QCD condensates.The utility of the null-plane description lies in the operator algebra that mixes the null-plane Hamiltonians and the chiral symmetry charges. It is demonstrated that in a certainnon-trivial limit, the null-plane operator algebra reduces to the symmetry group SU (2 N )of the constituent quark model. Address as of 1 September, 2013: Department of Physics, University of Washington. a r X i v : . [ nu c l - t h ] J u l ontents SU ( N ) conventions 37 – 1 – Introduction
Spontaneous symmetry breaking is usually treated as a phenomenon that arises fromproperties of an asymmetric quantum mechanical vacuum state. In particular, the non-invariance of the vacuum state with respect to a symmetry is said to lead to spontaneoussymmetry breakdown. While this picture is clearly valid and useful, it is not generallyappreciated that in relativistic theories of quantum mechanics, it is strictly a matter ofconvention which arises from the (usually implicit) choice of quantization surface [1]. In-deed, the standard viewpoint —the instant form— arises from choosing to view dynamicsin Minkowski space as the evolution of families of parallel spaces at various instants of time.An alternate view of dynamics is to consider the evolution of families of parallel spacestangent to the light cone; i.e. null planes [1–6]. In this viewpoint —the front form— the mo-mentum operator has a spectrum confined to the open positive half-line and therefore thevacuum of the interacting theory may be regarded as the structureless Fock-space vacuum,which is an invariant with respect to all internal symmetries, and spontaneous symmetrybreaking must be attributed to properties of the dynamical Poincar´e generators. Thereforein the front form, spontaneous chiral symmetry breaking is a property of operators ratherthan of a complicated vacuum state. Naturally one expects that physics is independentof the choice of quantization surface. However, for theories like QCD where the detaileddynamics are largely intractable, one may suppose that the two forms of dynamics lead todistinct insights into the behavior of the theory at strong coupling. Our goal in this paperis to argue that this is indeed the case.The fundamental point which we wish to emphasize in this paper is that, in contrastto the instant form, where spontaneous symmetry breaking lives entirely in the non-trivialvacuum, in the front form, symmetry breaking is expressed entirely through the fact thatthe Hamiltonians, or dynamical Poincar´e generators, do not commute with the internalsymmetry charges. The resulting commutation relations among space-time generators andinternal symmetry generators in QCD imply powerful constraints on the spectrum and spinof the hadronic world [7–10]. There have been many studies of spontaneous chiral symmetrybreaking on null planes [7, 8, 11–41]. In many cases the emphasis has been on learningdetailed information about the dynamical mechanism of chiral symmetry breaking in QCDand in models. Here our approach is much less ambitious; we assume that chiral symmetryis broken spontaneously by complicated and not-well understood dynamics, and we thendetermine the constraints that follow from this assumption. In particular, we are interestedprimarily in formulating the model-independent consequences of chiral symmetry breakingon null-planes. A fundamental assumption we make is that physics must be independent ofthe choice of quantization surface. Nowhere in this study do we find anything resemblinga contradiction of this basic assumption. Indeed, this assumption of what one might call“form invariance” leads to various constraints which reveal a great deal about the natureand consistency of chiral symmetry breaking on null-planes. On general grounds, the null-plane chiral symmetry charges annihilate the vacuum. Therefore, in order that spontaneouschiral symmetry breaking take place, the chiral symmetry axial-vector current on the null-plane cannot be conserved [11, 12]. This property leads to a simple proof of Goldstone’s– 2 –heorem on a null-plane, which is completely decoupled from any assumptions about theformation of symmetry-breaking condensates. A second consistency condition is that thepart of the QCD vacuum energy that is dependent on the quark masses should be invariantwith respect to the choice of coordinates. This condition recovers the Gell-Mann-Oakes-Renner relation [42] in the null-plane description, and leads to a general prescription forrelating all instant-form chiral-symmetry breaking condensates to the vacuum expectationvalues of chiral singlet null-plane QCD operators.It is difficult to find a general solution of the null-plane operator algebra [7–10]. How-ever, there is a non-trivial limit in which a solution can be found. One expects that, ingeneral, the chiral symmetry breaking part of the null-plane energy has an energy scalecomparable to Λ
QCD and therefore is not parametrically small. However, assuming thatthis is small (which is the case parametrically for baryon operators at large- N c ), whilethe chiral symmetry breaking part of the spin Hamiltonians is of natural size, allows anon-trivial solution of the operator algebra which closes to the Lie brackets of SU (2 N ),thus recovering the basic group theoretical structure of the constituent quark model. Thisresult, originally found by Weinberg [10] working with current-algebra sum rules in specialLorentz frames, is shown in this context to be a general consequence of the null-plane QCDLie algebraic constraints which are valid in any Lorentz frame.The paper is organized as follows. In section 2, the null-plane coordinates and con-ventions are introduced, and the front-form Poincar´e algebra is obtained. The null-planeHamiltonians and the Lie brackets that they satisfy are identified, and the momentumeigenstates are constructed. In section 3 the null-plane internal symmetry charges areintroduced, and the commutators that mix Poincar´e and chiral generators are obtained.Using these commutators, a general proof of Goldstone’s theorem is given, and a polologyanalysis is given which elucidates the structure of the axial-vector current on the null-plane.The special case of QCD with N flavors of light quarks is considered in section 4. The QCDLagrangian is expressed in the null-plane coordinates, and the chiral symmetry breakingHamiltonians and the constraints that they satisfy are derived. The issue of condensatesin the null-plane formulation is addressed in detail; the Gell-Mann-Oakes-Renner formulais recovered in the front-form and a general method for relating instant-form condensatesto front-form condensates is presented. Section 5 explores the consequences of the QCDnull-plane operator algebra. In particular, a simple solution of the operator algebra is givenwhich contains the spin-flavor symmetries of the constituent quark model. In section 6 wesummarize our findings and conclude. Nota bene : We have made use of the many general reviews of null-plane (or light-front)quantization [43–55], as well as reviews that focus primarily on chiral symmetry relatedissues [26, 27, 56]. In order to provide a self-contained description of the subject of chiral-symmetry breaking on a null-plane, there is a significant amount of review material in thispaper. – 3 – igure 1 . A null plane is a surface tangent to the light cone. The null-plane Hamiltonians mapthe initial light-like surface onto some other surface and therefore describe the dynamical evolutionof the system. The energy P − translates the system in the null-plane time coordinate x + , whereasthe spin Hamiltonians F r rotate the initial surface about the surface of the light cone. In the front-form of relativistic Hamiltonian dynamics, one chooses the initial state of thesystem to be on a light-like plane, or null-plane, which is a hypersurface of points x inMinkowski space such that x · n = τ (see fig. 1). Here n is a light-like vector which willbe chosen below, and τ is a constant which plays the role of time. We will refer to anull-plane as Σ τn . The subgroup of the Poincar´e group that maps Σ τn to itself is calledthe stability group of the null-plane and determines the kinematics within the null-plane.The remaining three Poincar´e generators map Σ τn to a new surface, Σ τ (cid:48) n , and thereforedescribe the evolution of the system in time. The front-form is special in that it has sevenkinematical generators, the largest stability group of all of the forms of dynamics [1]. Itstands to reason that in complicated problems in relativistic quantum mechanics one wouldprefer a formulation which has the fewest number of Hamiltonians to determine.– 4 – .2 Choice of coordinates Consider the light-like vectors n µ and n ∗ µ which satisfy n = n ∗ = 0 and n · n ∗ = 1. Herewe will choose these vectors such that n µ ≡ √ (1 , , , − , n ∗ µ ≡ √ (1 , , , . (2.1)We will take the initial surface to be the null-plane Σ n . A coordinate system adapted tonull-planes is then given by x + ≡ x · n = √ ( x + x ) , x − ≡ x · n ∗ = √ ( x − x ) (2.2)which we take as the time variable and “longitudinal” position, respectively . The remain-ing coordinates, x ⊥ = ( x , x ) provide the “transverse” position. Denoting the null-planecontravariant coordinate four-vector by ˜ x µ = ( x + , x , x , x − ) = ( x + , x ⊥ , x − ), then one canwrite ˜ x µ = C µν x ν . (2.3)The matrix C µν , given explicitly in Appendix A, allows one to transform all Lorentz tensorsfrom instant-form to front-form coordinates. In particular, the null-plane metric tensor isgiven by ˜ g µν = ( C − ) αµ g αβ ( C − ) βν . (2.4)The energy, canonical to the null-plane time variable x + is p − = p + , and the mo-mentum canonical to the longitudinal position variable x − is p + = p − . Therefore, theon-mass-shell condition for a relativistic particle of mass m yields the null-plane dispersionrelation: p − = p ⊥ + m p + . (2.5)This dispersion relation reveals several interesting generic features of the null-plane for-mulation. Firstly, the dispersion relation resembles the non-relativistic dispersion relationof a particle of mass p + in a constant potential. Secondly, we see that the positivity andfiniteness of the null-plane energy of a free massive particle requires p + >
0. Only mass-less particles with strictly vanishing momentum can have p + = 0. This implies that pairproduction is subtle, and the vacuum state is in some sense simple, with the exception ofcontributions that are strictly from p + = 0 modes [43–49, 51, 52, 54, 55]. In this section we will review the Lie brackets of the Lorentz generators in the front form .The Poincar´e algebra in our convention is:[ P µ , P ν ] = 0 , [ M µν , P ρ ] = i ( g νρ P µ − g µρ P ν )[ M µν , M ρσ ] = i ( g µσ M νρ + g νρ M µσ − g µρ M νσ − g νσ M µρ ) , (2.6) This is known as the Kogut-Soper convention [5]. Our metric and other notational conventions can befound in Appendix A and in Ref. [48]. Here we follow closely the development of Refs. [4–6]. See also Ref. [57]. – 5 –here M ij = (cid:15) ijk J k and M i = K i with J i and K i the generators of rotations and boosts,respectively. Using C µν we can transform from the instant-form to the front-form giving˜ P µ = ( P + , P , P , P − ), ˜ M r + = − ˜ M + r = F r , ˜ M r − = − ˜ M − r = E r , ˜ M rs = (cid:15) rs J , and˜ M + − = − ˜ M − + = K , where we have defined P + = √ ( P + P ) , P − = √ ( P − P ) ; E r = √ ( K r + (cid:15) rs J s ) , F r = √ ( K r − (cid:15) rs J s ) . (2.7)Here P + = P − is the null-plane energy while P − = P + is the longitudinal momentum.(Note that the indices r, s, t, . . . are transverse indices that range over 1 ,
2. See Appendix A.)It is straightforward to show that P + , P r , K , E r , and J are kinematical generatorsthat leave the null plane x + = 0 intact. These seven generators form the stability group ofthe null plane. It is useful to classify the subgroups of the Poincar´e algebra by consideringthe transformation properties of the generators with respect to longitudinal boosts, whichserve to rescale the generators. Writing[ K , A ] = − iγ A (2.8)where A is a generator, one finds E r and P + have γ = 1, J , K and P r have γ = 0,and P − and F r have γ = −
1. The Poincar´e generators have subgroups G γ labeled by γ ,and there exist two seven-parameter subgroups S ± with a semi-direct product structure S ± = G × G ± . Therefore the stability group coincides with the subgroup S + . Thenon-vanishing commutation relations among these generators are:[ K , E r ] = − iE r , [ K , P + ] = − iP + ;[ J , E r ] = i(cid:15) rs E s , [ J , P r ] = i(cid:15) rs P s ;[ E r , P s ] = − iδ rs P + . (2.9)By contrast, P − and F r are the Hamiltonians which consist of the subgroup G − ; theyare the dynamical generators which move physical states away from the x + = 0 surface(see fig. 1). The non-vanishing commutators among the stability group generators and theHamiltonians are: [ K , P − ] = iP − , [ E r , P − ] = − iP r ;[ K , F r ] = iF r , [ J , F r ] = i(cid:15) rs F s ;[ P r , F s ] = iδ rs P − , [ P + , F r ] = iP r ;[ E r , F s ] = − i ( δ rs K + (cid:15) rs J ) . (2.10)This algebraic structure is isomorphic to the Galilean group of two-dimensional quantummechanics where one identifies { P − , E r , P r , J , P + } with the Hamiltonian, Galilean boosts,momentum, angular momentum, and mass, respectively. This isomorphism is responsiblefor the similarities between the front form and nonrelativistic quantum mechanics that wenoted in the dispersion relation, and was originally noted in the context of the infinitemomentum frame of instant-form dynamics [2, 3] which has a similar dispersion relation.– 6 – .4 Null-plane momentum states and reduced Hamiltonians As momentum is a kinematical observable, it is convenient to work with momentum eigen-states, such that P r | p + , p ⊥ (cid:105) = p r | p + , p ⊥ (cid:105) ; (2.11) P + | p + , p ⊥ (cid:105) = p + | p + , p ⊥ (cid:105) . (2.12)The action of the boosts on momentum states follows directly from the commutationrelations in eq. 2.9 and is given by e − iv r E r e − iωK | p + , p ⊥ (cid:105) = | e ω p + , p ⊥ + p + v ⊥ (cid:105) . (2.13)One can then define the unitary boost operator U ( p + , p r ) = e − iβ r E r e − iβ K , (2.14)with β r ≡ p r /p + and β ≡ log( √ p + /M ) which boosts the state at rest to one witharbitrary momentum: U ( p + , p r ) | M/ √ , (cid:105) = | p + , p ⊥ (cid:105) . (2.15)The action of the boosts on the momentum states is then easily found to be E r | p + , p ⊥ (cid:105) = ip + ddp r | p + , p ⊥ (cid:105) ; (2.16) K | p + , p ⊥ (cid:105) = ip + ddp + | p + , p ⊥ (cid:105) . (2.17)Unitarity of the boost operators fixes the normalization of the momentum states upto a constant. We assume the covariant normalization: (cid:104) p + (cid:48) , p (cid:48)⊥ | p + , p ⊥ (cid:105) = (2 π ) p + δ ( p + (cid:48) − p + ) δ ( p (cid:48)⊥ − p ⊥ ) , (2.18)and the corresponding completeness relation = (cid:90) dp + d p ⊥ (2 π ) p + | p + , p ⊥ (cid:105) (cid:104) p + , p ⊥ | . (2.19)We can now find angular momentum operators, J r and J , that are valid in anyframe by boosting from an arbitrary momentum state to a state at rest, acting with theangular momentum generators J r = (cid:15) rs ( F s − E s ) / √ J , and then boosting back tothe arbitrary momentum state. That is, J i | p + , p ⊥ (cid:105) = U ( p + , p r ) J i U − ( p + , p r ) | p + , p ⊥ (cid:105) . (2.20)Using this procedure one finds angular momentum operators that are valid in any frame: J = J + (cid:15) rs E r P s (cid:0) /P + (cid:1) ; (2.21) J r = (cid:15) rs (cid:2) P + F s − P − E s + (cid:15) st P t J + P s K (cid:3) (1 /M ) . (2.22)– 7 –nverting eq. 2.22 one then finds the following expressions for the Hamiltonians: P − = (cid:0) / P + (cid:1) (cid:2) P + P + M (cid:3) ; F = (cid:0) /P + (cid:1) (cid:2) − P K + P − E − P J − M J (cid:3) ; F = (cid:0) /P + (cid:1) (cid:2) − P K + P − E + P J + M J (cid:3) . (2.23)A striking feature of the null-plane formulation is that the fundamental dynamical objectsare the products M and M J r , rather than the generators themselves. Following Ref. [6],we will refer to these objects as reduced Hamiltonians. The reduced Hamiltonians, togetherwith J , commute with all kinematical generators and satisfy the algebra of U (2). This isconveniently demonstrated by making use of the Pauli-Lubanski vector W µ = ε µνρσ P ν M ρσ , (2.24)which satisfies W µ P µ = 0 and the non-trivial commutation relations:[ M µν , W ρ ] = i ( g νρ W µ − g µρ W ν ) ; (2.25)[ W µ , W ν ] = − iε µνρσ W ρ P σ . (2.26)One then finds general, compact expressions for the angular momentum operators: J = W + /P + , M J r = W r − P r W + /P + . (2.27)By considering the commutation relations among W µ , P µ and M µν one confirms that[ J , M J r ] = i (cid:15) rs M J s , [ J , M ] = 0 ;[ M J r , M J s ] = i (cid:15) rs M J , [ M , M J r ] = 0 . (2.28)Hence, the reduced Hamiltonians together with the stability group generator J satisfy thealgebra of U (2), and the problem of finding a Lorentz invariant description of a relativisticquantum mechanical system is thus equivalent to finding a representation of the threereduced Hamiltonians which satisfy this algebra . Since the essence of Lorentz invarianceresides in these Lie brackets, and they require knowledge of the reduced Hamiltonians,in theories with complicated dynamics like QCD, the formulation of the theory at weakcoupling —where QCD is defined as a continuum quantum field theory— will lack manifestLorentz invariance, which is tied up with the detailed dynamics of the theory, and is ascomplicated to achieve as finding the spectrum of the theory.We can write a general momentum eigenstate as: | p + , p ⊥ ; λ , n (cid:105) = | p + , p ⊥ (cid:105) ⊗ | λ , n (cid:105) . (2.29)Here n are additional variables that may be needed to specify the state of a system at rest,and λ is helicity, the eigenvalue of J : J | p + , p ⊥ ; λ , n (cid:105) = λ | p + , p ⊥ ; λ , n (cid:105) , (2.30) Since the mass operator, M = √ p µ p µ , commutes with the spin operators, this algebra can clearly beexpressed in the canonical form: [ J i , J j ] = i(cid:15) ijk J k and [ M , J i ] = 0. – 8 –nd therefore, using eq. 2.21, we have J | p + , p ⊥ ; λ , n (cid:105) = (cid:18) λ + i(cid:15) rs p r ddp s (cid:19) | p + , p ⊥ ; λ , n (cid:105) , (2.31)which completes the catalog of the action of the stability group generators on the momen-tum states. It is useful to write | p + , p ⊥ ; λ , n (cid:105) = U ( p + , p r ) | M/ √ , ; λ , n (cid:105) ≡ a † n (cid:0) p + , p ⊥ ; λ (cid:1) | (cid:105) , (2.32)where a † n is an operator that creates the momentum state when acting on the null-planevacuum, | (cid:105) . What is special about the null-plane description is that the kinematicalgenerators (with the exception of J ) act on states in a manner independent of the innervariables n . And the reduced Hamiltonians act exclusively on the inner variables in amanner independent of the momentum. Therefore, one may view the Poincar´e algebraby the direct sum of K and D , where K = { E r , P r , K , P + } contains all stability groupgenerators with the exception of J which is grouped with the reduced Hamiltonians, D = { J , M J r , M } [58].The structure of the Poincar´e algebra in the front-form is well suited to the study ofsystems with complicated dynamics like QCD, as the dynamical generators are directlyrelated to the most important observable quantities, namely the energy and the angularmomentum of the system, while momenta and boosts are purely kinematical and thereforeare easy to implement . The reduced Hamiltonians will have a fundamental role to playin the description of chiral symmetry breaking on null planes. Consider a Lagrangian field theory that has an SU ( N ) R ⊗ SU ( N ) L chiral symmetry. Letus assume that this system has a null-plane Lagrangian formulation which allows one,by the standard Noether procedure, to obtain the currents ˜ J µα ( x ) and ˜ J µ α ( x ), which arerelated to the symmetry currents via ˜ J µLα = ( ˜ J µα − ˜ J µ α ) / J µRα = ( ˜ J µα + ˜ J µ α ) /
2. Wewill further assume that the Lagrangian contains an operator that explicitly breaks thechiral symmetry in the pattern SU ( N ) R ⊗ SU ( N ) L → SU ( N ) F and is governed by theparameter (cid:15) χ such that as (cid:15) χ →
0, the symmetry is restored at the classical level. Thegeneral relation between currents and their associated charges is given by Q ( n · x ) = (cid:90) d y δ ( n · ( x − y ) ) n · J ( y ) , (3.1)where the vector n µ selects the initial quantization surface, which we take to be the nullplane Σ τn . Therefore, the null-plane chiral symmetry charges are˜ Q α = (cid:90) dx − d x ⊥ ˜ J + α ( x − , (cid:126)x ⊥ ) ; (3.2)˜ Q α ( x + ) = (cid:90) dx − d x ⊥ ˜ J +5 α ( x − , (cid:126)x ⊥ , x + ) , (3.3) By contrast, in the instant form of dynamics, the energy and the boosts are dynamical. As boosts arenot among the observables, one refers only to the one Hamiltonian corresponding to energy. – 9 –here the axial charges have been given explicit null-plane time dependence as they are notconserved due to the explicit breaking operator in the Lagrangian. These charges satisfythe SU ( N ) R ⊗ SU ( N ) L chiral algebra,[ ˜ Q α , ˜ Q β ] = i f αβγ ˜ Q γ , [ ˜ Q α ( x + ) , ˜ Q β ] = i f αβγ ˜ Q γ ( x + ) ; (3.4)[ ˜ Q α ( x + ) , ˜ Q β ( x + ) ] = i f αβγ ˜ Q γ . (3.5)We further assert that both types of chiral charges annihilate the vacuum. That is,˜ Q α | (cid:105) = ˜ Q α | (cid:105) = 0 . (3.6)This is the statement that the front-form vacuum is invariant with respect to the full SU ( N ) R ⊗ SU ( N ) L symmetry. In particular, this implies that there can be no vacuumcondensates that break SU ( N ) R ⊗ SU ( N ) L on a null-plane. This may seem to be an oddassumption, since the chiral charge is directly related to the axial-vector current througheq. 3.3, and in general one would expect that this current has a Goldstone boson polecontribution, in turn implying that the chiral charges acting on the vacuum state excitemassless Goldstone bosons. Below we will confirm the assertion, eq. 3.6, by using standardcurrent-algebra polology to show that indeed the Goldstone boson pole contribution to thenull-plane axial-vector current is absent. Mixed commutators among the Poincar´e generators and internal symmetry generators canbe expressed generally as [13]:[ Q α ( n · x ) , P µ ] = − i n µ (cid:90) d y δ ( n · ( x − y ) ) ∂ ν J να ( y ) ; (3.7)[ Q α ( n · x ) , M µν ] = i (cid:90) d y δ ( n · ( x − y ) ) ( n µ y ν − n ν y µ ) ∂ κ J κα ( y ) . (3.8)From these expressions one then obtains the mixed commutator between the Pauli-Lubanskivector and the internal symmetry charges:[ Q α ( n · x ) , W ν ] = i ε νδρσ (cid:90) d yδ ( n · ( x − y )) (cid:20) M δρ n σ − (cid:16) n δ y ρ − n ρ y δ (cid:17) P σ (cid:21) ∂ κ J κα ( y ) . (3.9)Using these expressions, one finds the commutation relations between null-plane chiralcharges and the reduced Hamiltonians:[ ˜ Q α ( x + ) , M ] = − i P + (cid:90) dx − d x ⊥ ∂ µ ˜ J µ α ( x − , (cid:126)x ⊥ , x + ) ; (3.10)[ ˜ Q α ( x + ) , M J r ] = i (cid:15) rs P + (cid:90) dx − d x ⊥ Γ s ∂ µ ˜ J µ α ( x − , (cid:126)x ⊥ , x + ) , (3.11)where Γ s ≡ E s − P + x s . Here and in what follows, we are assuming that SU ( N ) F is un-broken and therefore ∂ µ ˜ J µα = 0 and the reduced Hamiltonians commute with the SU ( N ) F charges: [ ˜ Q α , M ] = [ ˜ Q α , M J r ] = 0 . (3.12)– 10 – .3 Goldstone’s theorem on a null plane In the instant form, a symmetry has three possible fates in the quantum theory: thesymmetry remains exact and the current is conserved, the symmetry is spontaneouslybroken and again the current is conserved, or the symmetry is anomalous and the current isnot conserved. The front form realizes a fourth possibility: the symmetry is spontaneouslybroken and the associated current in not conserved. This fourth possibility is necessary inthe front form because the vacuum is invariant with respect to all internal symmetries. Ingeneral, we can write ∂ µ ˜ J µ α ( x − , (cid:126)x ⊥ , x + ) = (cid:15) χ ˜ P α ( x − , (cid:126)x ⊥ , x + ) , (3.13)where (cid:15) χ is the parameter that gauges the amount of chiral symmetry breaking that ispresent in the Lagrangian. Using the short hand, | h (cid:105) ≡ | p + , (cid:126)p ⊥ ; λ , h (cid:105) , (3.14)for the momentum eigenstates, we take the matrix element of eq. 3.10 between momentumeigenstates, which gives (cid:104) h (cid:48) | [ ˜ Q α ( x + ) , M ] | h (cid:105) = − i p + (cid:15) χ (cid:90) dx − d x ⊥ (cid:104) h (cid:48) | ˜ P α ( x − , (cid:126)x ⊥ , x + ) | h (cid:105) ; (3.15)If the right hand side of this equation vanishes for all h and h (cid:48) , then there can be nochiral symmetry breaking of any kind. Therefore, in order that the chiral symmetry bespontaneously broken, the chiral current cannot be conserved and we have the followingconstraint [22, 26] in the limit (cid:15) χ → (cid:90) dx − d x ⊥ (cid:104) h (cid:48) | ˜ P α ( x − , (cid:126)x ⊥ , x + ) | h (cid:105) −→ (cid:15) χ + . . . , (3.16)where the dots represent other possible terms that are non-singular in the limit (cid:15) χ → N − . Wewill assume that ˜ P α is an interpolating operator for Lorentz-scalar fields φ iα , and thereforewe can write ˜ P α ( x ) = (cid:88) i Z i φ iα ( x ) (3.17)where the Z i are overlap factors. Using the reduction formula we relate the matrix elementsof field operators between physical states to transition amplitudes. Of course here it isunderstood that there is no selection rule which would forbid these transitions. The S-matrix element for the transition h ( p ) → h (cid:48) ( p (cid:48) ) + φ iα ( q ) can be defined by (cid:104) h (cid:48) ; φ iα ( q ) | S | h (cid:105) ≡ i (2 π ) δ ( p − p (cid:48) − q ) M iα ( p (cid:48) , λ (cid:48) , h (cid:48) ; p, λ, h )= i (cid:90) d x e − iq · x (cid:16) − q + M φ i (cid:17) (cid:104) h (cid:48) | φ iα ( x ) | h (cid:105) (3.18) Note that if we took eq. 3.16 as a constraint on the operator ˜ P α rather than on its matrix elements,then this constraint would be viewed as a constraint on the zero-modes of the operator [22, 26]. Here wework entirely with matrix elements. – 11 –here M iα is the Feynman amplitude and in the second line we have used the reductionformula. It then follows that (cid:104) h (cid:48) | φ iα ( x ) | h (cid:105) = − e iq · x q − M φ i M iα ( q ) . (3.19)Using this formula together with eq. 3.17 in eq. 3.15 then gives (cid:104) h (cid:48) | [ ˜ Q α ( x + ) , M ] | h (cid:105) = 2 i p + (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − (cid:88) i (cid:15) χ Z i q + q − − (cid:126)q ⊥ − M φ i M iα ( q )= − i p + (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − (cid:88) i (cid:15) χ Z i M φ i M iα ( q − ) , (3.20)where in the second line we have used the momentum delta functions. In order that theright hand side not vanish in the symmetry limit, there must be at least one field φ iα whosemass-squared vanishes proportionally to (cid:15) χ as (cid:15) χ →
0. We will denote this field as π α ≡ φ α with M π = c p (cid:15) χ , (3.21)where c p is a constant of proportionality. There are therefore N − π α in thesymmetry limit, which we identify as the Goldstone bosons. It is noteworthy that this proofrelies entirely on physical matrix elements; i.e. there is no need to assume the existence ofa vacuum condensate that breaks the chiral symmetry. Of course, in instant-form QCD,we know that the proportionality constant in eq. 3.21 contains the quark condensate. Thisissue will be resolved below in section 4. While we have carried out this proof in the caseof SU ( N ) R ⊗ SU ( N ) L broken to SU ( N ) V , it is clearly easily generalized to other systems.We can now write ˜ P α = Z π α + . . . where the dots represent non-Goldstone bosonfields, and (cid:104) h (cid:48) | ∂ µ ˜ J µ α ( x ) | h (cid:105) = (cid:104) h (cid:48) | ¯ Z M π π α ( x ) | h (cid:105) , (3.22)where ¯ Z ≡ Z /c p . Here, as in the usual current algebra manipulations, we have assumedthat only the Goldstone bosons couple to the axial-vector current, and it is now a standardexercise to determine the overlap factor. First, define the Goldstone-boson decay constant, F π , via (cid:104) | ˜ J µ α ( x ) | π β (cid:105) ≡ − i p µ F π δ αβ e ip · x , (3.23)where | π β (cid:105) ≡ | p + , (cid:126)p ⊥ ; 0 , , π β (cid:105) . Taking the divergence of the current and raising eq. 3.22to an operator relation yields (cid:104) | ¯ Z M π π α ( x ) | π β (cid:105) = F π M π δ αβ e ip · x . (3.24)The normalization of the Goldstone-boson field, (cid:104) | π α ( x ) | π β (cid:105) = δ αβ e ip · x , (3.25)– 12 –hen gives ¯ Z = F π and we recover the standard operator relation ∂ µ ˜ J µ α ( x ) = F π M π π α ( x ) . (3.26)We can now express the mixed Lie bracket, eq. 3.20, as (cid:104) h (cid:48) | [ ˜ Q α ( x + ) , M ] | h (cid:105) = − i p + (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − F π M α ( q − ) , (3.27)where here M α ( q − ) is the Feynman amplitude for the transition h ( p ) → h (cid:48) ( p (cid:48) ) + π α ( q ).We see that while the chiral current is not conserved, its divergence is proportional to anS-matrix element. Noting that (cid:104) h (cid:48) | [ ˜ Q α ( x + ) , M ] | h (cid:105) = 2 p + q − (cid:104) h (cid:48) | ˜ Q α ( x + ) | h (cid:105) = − p + (cid:104) h (cid:48) | i ddx + ˜ Q α ( x + ) | h (cid:105) , (3.28)and from the definition of the chiral charge, eq. 3.3, (cid:104) h (cid:48) | ˜ Q α ( x + ) | h (cid:105) = (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − (cid:104) h (cid:48) | ˜ J +5 α (0) | h (cid:105) , (3.29)one finds, using eq. 3.27, M α ( q − ) = i q − F π (cid:104) h (cid:48) | ˜ J +5 α (0) | h (cid:105) , (3.30)or, in Lorentz-invariant form, M α ( q ) = i q µ F π (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) , (3.31)which is the standard current-algebra result. In order to confirm some of these propertiesin a better-known fashion, and to address the assumption we have made that the chiralcharges annihilate the vacuum, we will now consider current algebra polology on the null-plane. Our starting point is the matrix element between hadronic states h and h (cid:48) of the axial-vector current, which can be written in a general way as [59, 60] (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) = iF π q µ q − M π M α + (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) N (3.32)where as before q = p − p (cid:48) . Using translational invariance, we have (cid:104) h (cid:48) | ˜ J µ α ( x ) | h (cid:105) = e iq · x (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) . (3.33)It follows that (cid:104) h (cid:48) | ∂ µ ˜ J µ α ( x ) | h (cid:105) = i q µ (cid:104) h (cid:48) | ˜ J µ α ( x ) | h (cid:105) = e iq · x (cid:104) − F π q q − M π M α + i q µ (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) N (cid:105) , (3.34)– 13 – b e f g i c d h j k l n p q r m / Z N / Z N s i ii iii i v F i g u r e : D i ag r a m s w h i c h c o n tr i b u t e t o nu c l e o n C o m p t o n s c a tt e r i n g i n t h e ! · v = u g e a t r d ( i - i v ) a n d t h ( a - s ) o r d e r . V e rt i ce s a r e l a b e l e d a s i n F i g . . g r a p h ( a ) c o n tr i b u t e s , a n d t h i s r e p r o d u ce s t h e s p i n - i n d e p e n d e n t l o w - e n e r g y t h e o r e m f o r C o m p t o n s c a tt e r i n g f r o m a s i n g l e nu c l e o n — t h e T h o m s o n li m i t : T γ N = − " ! · " ! ! Z e M , ( ) w h e r e Z i s t h e nu c l e o n c h a r g e i n u n i t s o f | e | . A t O ( Q )t h e s - c h a n n e l p r o t o n - p o l e d i ag r a m , F i g . ( b ) , a n d i t s c r o ss e d u - c h a n n e l p a rt - n e r , t og e t h e r w i t h a γ N s e ag u ll f r o m L ( ) ( F i g . ( c )) e n s u r e t h a t H B χ P T r ec o v e r s t h e L o w , G e ll - M a n n a n d G o l d b e r g e r l o w - e n e r g y t h e o r e m f o r s p i n - d e p e n d e n t C o m p t o n s c a tt e r i n g [ ]. A t O ( Q )t h e t - c h a n n e l p i o n - p o l e g r a p h F i g . ( d ) a l s o e n t e r s ;i t s c o n tr i b u t i o n , w h i c h v a r i e s r a p i d l y w i t h e n e r g y ,i s o f t e n i n c l u d e d i n t h e d e fi n i t i o n o f t h e s p i n p o l a r i z a b ili t i e s a n d i s t h e r e a s o n t h a t t h e b a c k w a r d s p i n p o l a r i z a b ili t y i s s o m u c h l a r g e r i n m ag n i t u d e t h a n t h e f o r w a r d o n e . P i o n - l oo p g r a p h s e n t e r fi r s t a t O ( Q ) — s ee g r a p h s i - i v o f F i g . — a n d g i v e e n e r g y - d e p e n d e n t c o n tr i b u t i o n s t o t h e a m p li t u d e w h i c h i n c l u d e t h e w e ll - k n o w np r e d i c t i o n s f o rt h e s p i n - i n d e p e n d e n t p o l a r i z a b ili t i e s o f E q . ( ) , a s w e ll a s l e ss - f a m o u s p r e - d i c t i o n s f o r γ – γ [ ]. T h e t h i r d - o r d e r l oo p a m p li t u d e o f R e f s .[ , ]i s o b t a i n e d f r o m d i ag r a m s ( i ) –2 ( i v ) — t og e t h e r w i t h g r a p h s r e l a t e d t o t h e m b y c r o ss i n g a n d a l t e r n a t i v e t i m e - o r d e r i n g o f v e rt i ce s — a n d i s g i v e n i n A p p e n d i x A . A s o b s e r v e d i n R e f .[ ], t h e O ( Q ) χ P T r e s u l t f o r γ p s c a tt e r i n g i s i n goo d ag r ee m e n t w i t h t h e d a t a a t f o r w a r d a n g l e s h h π α + ˜ J µ α ( ˜ J µ α ) N ( J µ α ) N J µ α = a b e f g i c d h j k l n p q r m / Z N / Z N s i ii iii i v F i g u r e : D i ag r a m s w h i c h c o n tr i b u t e t o nu c l e o n C o m p t o n s c a tt e r i n g i n t h e ! · v = u g e a t r d ( i - i v ) a n d t h ( a - s ) o r d e r . V e rt i ce s a r e l a b e l e d a s i n F i g . . g r a p h ( a ) c o n tr i b u t e s , a n d t h i s r e p r o d u ce s t h e s p i n - i n d e p e n d e n t l o w - e n e r g y t h e o r e m f o r C o m p t o n s c a tt e r i n g f r o m a s i n g l e nu c l e o n — t h e T h o m s o n li m i t : T γ N = − " ! · " ! ! Z e M , ( ) w h e r e Z i s t h e nu c l e o n c h a r g e i n u n i t s o f | e | . A t O ( Q )t h e s - c h a n n e l p r o t o n - p o l e d i ag r a m , F i g . ( b ) , a n d i t s c r o ss e d u - c h a n n e l p a rt - n e r , t og e t h e r w i t h a γ N s e ag u ll f r o m L ( ) ( F i g . ( c )) e n s u r e t h a t H B χ P T r ec o v e r s t h e L o w , G e ll - M a n n a n d G o l d b e r g e r l o w - e n e r g y t h e o r e m f o r s p i n - d e p e n d e n t C o m p t o n s c a tt e r i n g [ ]. A t O ( Q )t h e t - c h a n n e l p i o n - p o l e g r a p h F i g . ( d ) a l s o e n t e r s ;i t s c o n tr i b u t i o n , w h i c h v a r i e s r a p i d l y w i t h e n e r g y ,i s o f t e n i n c l u d e d i n t h e d e fi n i t i o n o f t h e s p i n p o l a r i z a b ili t i e s a n d i s t h e r e a s o n t h a t t h e b a c k w a r d s p i n p o l a r i z a b ili t y i s s o m u c h l a r g e r i n m ag n i t u d e t h a n t h e f o r w a r d o n e . P i o n - l oo p g r a p h s e n t e r fi r s t a t O ( Q ) — s ee g r a p h s i - i v o f F i g . — a n d g i v e e n e r g y - d e p e n d e n t c o n tr i b u t i o n s t o t h e a m p li t u d e w h i c h i n c l u d e t h e w e ll - k n o w np r e d i c t i o n s f o rt h e s p i n - i n d e p e n d e n t p o l a r i z a b ili t i e s o f E q . ( ) , a s w e ll a s l e ss - f a m o u s p r e - d i c t i o n s f o r γ – γ [ ]. T h e t h i r d - o r d e r l oo p a m p li t u d e o f R e f s .[ , ]i s o b t a i n e d f r o m d i ag r a m s ( i ) –2 ( i v ) — t og e t h e r w i t h g r a p h s r e l a t e d t o t h e m b y c r o ss i n g a n d a l t e r n a t i v e t i m e - o r d e r i n g o f v e rt i ce s — a n d i s g i v e n i n A p p e n d i x A . A s o b s e r v e d i n R e f .[ ], t h e O ( Q ) χ P T r e s u l t f o r γ p s c a tt e r i n g i s i n goo d ag r ee m e n t w i t h t h e d a t a a t f o r w a r d a n g l e s h h a b e f g i c d h j k l n p q r m / Z N / Z N s i ii iii i v F i g u r e : D i ag r a m s w h i c h c o n tr i b u t e t o nu c l e o n C o m p t o n s c a tt e r i n g i n t h e ! · v = u g e a t r d ( i - i v ) a n d t h ( a - s ) o r d e r . V e rt i ce s a r e l a b e l e d a s i n F i g . . g r a p h ( a ) c o n tr i b u t e s , a n d t h i s r e p r o d u ce s t h e s p i n - i n d e p e n d e n t l o w - e n e r g y t h e o r e m f o r C o m p t o n s c a tt e r i n g f r o m a s i n g l e nu c l e o n — t h e T h o m s o n li m i t : T γ N = − " ! · " ! ! Z e M , ( ) w h e r e Z i s t h e nu c l e o n c h a r g e i n u n i t s o f | e | . A t O ( Q )t h e s - c h a n n e l p r o t o n - p o l e d i ag r a m , F i g . ( b ) , a n d i t s c r o ss e d u - c h a n n e l p a rt - n e r , t og e t h e r w i t h a γ N s e ag u ll f r o m L ( ) ( F i g . ( c )) e n s u r e t h a t H B χ P T r ec o v e r s t h e L o w , G e ll - M a n n a n d G o l d b e r g e r l o w - e n e r g y t h e o r e m f o r s p i n - d e p e n d e n t C o m p t o n s c a tt e r i n g [ ]. A t O ( Q )t h e t - c h a n n e l p i o n - p o l e g r a p h F i g . ( d ) a l s o e n t e r s ;i t s c o n tr i b u t i o n , w h i c h v a r i e s r a p i d l y w i t h e n e r g y ,i s o f t e n i n c l u d e d i n t h e d e fi n i t i o n o f t h e s p i n p o l a r i z a b ili t i e s a n d i s t h e r e a s o n t h a t t h e b a c k w a r d s p i n p o l a r i z a b ili t y i s s o m u c h l a r g e r i n m ag n i t u d e t h a n t h e f o r w a r d o n e . P i o n - l oo p g r a p h s e n t e r fi r s t a t O ( Q ) — s ee g r a p h s i - i v o f F i g . — a n d g i v e e n e r g y - d e p e n d e n t c o n tr i b u t i o n s t o t h e a m p li t u d e w h i c h i n c l u d e t h e w e ll - k n o w np r e d i c t i o n s f o rt h e s p i n - i n d e p e n d e n t p o l a r i z a b ili t i e s o f E q . ( ) , a s w e ll a s l e ss - f a m o u s p r e - d i c t i o n s f o r γ – γ [ ]. T h e t h i r d - o r d e r l oo p a m p li t u d e o f R e f s .[ , ]i s o b t a i n e d f r o m d i ag r a m s ( i ) –2 ( i v ) — t og e t h e r w i t h g r a p h s r e l a t e d t o t h e m b y c r o ss i n g a n d a l t e r n a t i v e t i m e - o r d e r i n g o f v e rt i ce s — a n d i s g i v e n i n A p p e n d i x A . A s o b s e r v e d i n R e f .[ ], t h e O ( Q ) χ P T r e s u l t f o r γ p s c a tt e r i n g i s i n goo d ag r ee m e n t w i t h t h e d a t a a t f o r w a r d a n g l e s h h J µ α a b e f g i c d h j k l n p q r m / Z N / Z N s i ii iii i v F i g u r e : D i ag r a m s w h i c h c o n tr i b u t e t o nu c l e o n C o m p t o n s c a tt e r i n g i n t h e ! · v = u g e a t r d ( i - i v ) a n d t h ( a - s ) o r d e r . V e rt i ce s a r e l a b e l e d a s i n F i g . . g r a p h ( a ) c o n tr i b u t e s , a n d t h i s r e p r o d u ce s t h e s p i n - i n d e p e n d e n t l o w - e n e r g y t h e o r e m f o r C o m p t o n s c a tt e r i n g f r o m a s i n g l e nu c l e o n — t h e T h o m s o n li m i t : T γ N = − " ! · " ! ! Z e M , ( ) w h e r e Z i s t h e nu c l e o n c h a r g e i n u n i t s o f | e | . A t O ( Q )t h e s - c h a n n e l p r o t o n - p o l e d i ag r a m , F i g . ( b ) , a n d i t s c r o ss e d u - c h a n n e l p a rt - n e r , t og e t h e r w i t h a γ N s e ag u ll f r o m L ( ) ( F i g . ( c )) e n s u r e t h a t H B χ P T r ec o v e r s t h e L o w , G e ll - M a n n a n d G o l d b e r g e r l o w - e n e r g y t h e o r e m f o r s p i n - d e p e n d e n t C o m p t o n s c a tt e r i n g [ ]. A t O ( Q )t h e t - c h a n n e l p i o n - p o l e g r a p h F i g . ( d ) a l s o e n t e r s ;i t s c o n tr i b u t i o n , w h i c h v a r i e s r a p i d l y w i t h e n e r g y ,i s o f t e n i n c l u d e d i n t h e d e fi n i t i o n o f t h e s p i n p o l a r i z a b ili t i e s a n d i s t h e r e a s o n t h a t t h e b a c k w a r d s p i n p o l a r i z a b ili t y i s s o m u c h l a r g e r i n m ag n i t u d e t h a n t h e f o r w a r d o n e . P i o n - l oo p g r a p h s e n t e r fi r s t a t O ( Q ) — s ee g r a p h s i - i v o f F i g . — a n d g i v e e n e r g y - d e p e n d e n t c o n tr i b u t i o n s t o t h e a m p li t u d e w h i c h i n c l u d e t h e w e ll - k n o w np r e d i c t i o n s f o rt h e s p i n - i n d e p e n d e n t p o l a r i z a b ili t i e s o f E q . ( ) , a s w e ll a s l e ss - f a m o u s p r e - d i c t i o n s f o r γ – γ [ ]. T h e t h i r d - o r d e r l oo p a m p li t u d e o f R e f s .[ , ]i s o b t a i n e d f r o m d i ag r a m s ( i ) –2 ( i v ) — t og e t h e r w i t h g r a p h s r e l a t e d t o t h e m b y c r o ss i n g a n d a l t e r n a t i v e t i m e - o r d e r i n g o f v e rt i ce s — a n d i s g i v e n i n A p p e n d i x A . A s o b s e r v e d i n R e f .[ ], t h e O ( Q ) χ P T r e s u l t f o r γ p s c a tt e r i n g i s i n goo d ag r ee m e n t w i t h t h e d a t a a t f o r w a r d a n g l e s h h = a b e f g i c d h j k l n p q r m / Z N / Z N s i ii iii i v F i g u r e : D i ag r a m s w h i c h c o n tr i b u t e t o nu c l e o n C o m p t o n s c a tt e r i n g i n t h e ! · v = u g e a t r d ( i - i v ) a n d t h ( a - s ) o r d e r . V e rt i ce s a r e l a b e l e d a s i n F i g . . g r a p h ( a ) c o n tr i b u t e s , a n d t h i s r e p r o d u ce s t h e s p i n - i n d e p e n d e n t l o w - e n e r g y t h e o r e m f o r C o m p t o n s c a tt e r i n g f r o m a s i n g l e nu c l e o n — t h e T h o m s o n li m i t : T γ N = − " ! · " ! ! Z e M , ( ) w h e r e Z i s t h e nu c l e o n c h a r g e i n u n i t s o f | e | . A t O ( Q )t h e s - c h a n n e l p r o t o n - p o l e d i ag r a m , F i g . ( b ) , a n d i t s c r o ss e d u - c h a n n e l p a rt - n e r , t og e t h e r w i t h a γ N s e ag u ll f r o m L ( ) ( F i g . ( c )) e n s u r e t h a t H B χ P T r ec o v e r s t h e L o w , G e ll - M a n n a n d G o l d b e r g e r l o w - e n e r g y t h e o r e m f o r s p i n - d e p e n d e n t C o m p t o n s c a tt e r i n g [ ]. A t O ( Q )t h e t - c h a n n e l p i o n - p o l e g r a p h F i g . ( d ) a l s o e n t e r s ;i t s c o n tr i b u t i o n , w h i c h v a r i e s r a p i d l y w i t h e n e r g y ,i s o f t e n i n c l u d e d i n t h e d e fi n i t i o n o f t h e s p i n p o l a r i z a b ili t i e s a n d i s t h e r e a s o n t h a t t h e b a c k w a r d s p i n p o l a r i z a b ili t y i s s o m u c h l a r g e r i n m ag n i t u d e t h a n t h e f o r w a r d o n e . P i o n - l oo p g r a p h s e n t e r fi r s t a t O ( Q ) — s ee g r a p h s i - i v o f F i g . — a n d g i v e e n e r g y - d e p e n d e n t c o n tr i b u t i o n s t o t h e a m p li t u d e w h i c h i n c l u d e t h e w e ll - k n o w np r e d i c t i o n s f o rt h e s p i n - i n d e p e n d e n t p o l a r i z a b ili t i e s o f E q . ( ) , a s w e ll a s l e ss - f a m o u s p r e - d i c t i o n s f o r γ – γ [ ]. T h e t h i r d - o r d e r l oo p a m p li t u d e o f R e f s .[ , ]i s o b t a i n e d f r o m d i ag r a m s ( i ) –2 ( i v ) — t og e t h e r w i t h g r a p h s r e l a t e d t o t h e m b y c r o ss i n g a n d a l t e r n a t i v e t i m e - o r d e r i n g o f v e rt i ce s — a n d i s g i v e n i n A p p e n d i x A . A s o b s e r v e d i n R e f .[ ], t h e O ( Q ) χ P T r e s u l t f o r γ p s c a tt e r i n g i s i n goo d ag r ee m e n t w i t h t h e d a t a a t f o r w a r d a n g l e s h h Figure 2 . Above shows the standard instant-form polology; the matrix element of the chiral currenthas a Goldstone-boson pole piece, and a non-pole piece. These two contributions cancel in thesymmetry limit ensuring a conserved chiral current. Below shows the standard front-form polology;the Goldstone-boson pole contribution is absent and therefore the current is not conserved butrather has a divergence which is proportional to the matrix element for the emission or absorptionof a Goldstone boson. and using (cid:104) h (cid:48) | ∂ µ ˜ J µ α ( x ) | h (cid:105) = (cid:104) h (cid:48) | F π M π π α ( x ) | h (cid:105) , (3.35)and the reduction formula, eq. 3.19, reproduces eq. 3.31. Note that in null-plane coordinateseq. 3.32 gives (cid:104) h (cid:48) | ˜ J +5 α (0) | h (cid:105) = iF π q + q + q − − (cid:126)q ⊥ − M π M α + (cid:104) h (cid:48) | ˜ J +5 α (0) | h (cid:105) N . (3.36)We therefore have lim q + ,(cid:126)q ⊥ → (cid:104) h (cid:48) | ˜ J +5 α (0) | h (cid:105) = (cid:104) h (cid:48) | ˜ J +5 α (0) | h (cid:105) N . (3.37)By comparing with eq. 3.29, it is clear that the null-plane chiral charges, by construction,do not excite the Goldstone boson states. The property, eq. 3.6, of vacuum annihilationwhich we assumed above, is therefore a general property of the null-plane chiral charges.Again consider the space-integrated current divergence in the front-form, but nowusing eq. 3.34. One finds (cid:90) dx − d x ⊥ (cid:104) h (cid:48) | ∂ µ ˜ J µ α ( x ) | h (cid:105) = (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − (cid:104) h (cid:48) | ∂ µ ˜ J µ α (0) | h (cid:105) = (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − (cid:104) − F π (cid:0) q + q − − (cid:126)q ⊥ (cid:1) q + q − − (cid:126)q ⊥ − M π M α + i q µ (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) N (cid:105) = (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − i q µ (cid:104) h (cid:48) | ˜ J µ α (0) | h (cid:105) N = (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − F π M α ( q − ) , (3.38)– 14 –here in the third line the momentum delta functions have been used, and in the last line wehave used eq. 3.31 and eq. 3.37. Now using eq. 3.10, we see that we have recovered eq. 3.27.In this derivation we see explicitly that the Goldstone-boson pole does not contribute to thedivergence of the axial-current. It is for this reason that the current cannot be conserved.For purposes of comparison, recall that in the instant form, one has (cid:90) d x (cid:104) h (cid:48) | ∂ µ J µ α ( x ) | h (cid:105) = (2 π ) δ ( (cid:126)q ) (cid:104) h (cid:48) | ∂ µ J µ α (0) | h (cid:105) ;= (2 π ) δ ( (cid:126)q ) (cid:104) − F π q q − M π M α + i q µ (cid:104) h (cid:48) | J µ α (0) | h (cid:105) N (cid:105) ; −−→ M π → (2 π ) δ ( (cid:126)q ) (cid:104) − F π M α + i q µ (cid:104) h (cid:48) | J µ α (0) | h (cid:105) N (cid:105) ;= 0 , (3.39)where in the last line, eq. 3.31 has once again been used. Here there is a cancellationbetween the pole and non-pole parts of the matrix element which ensure that the integratedcurrent divergence vanishes in the chiral limit. This analysis, which is expressed pictoriallyin fig.2, suggests that the front-form and instant-form axial-vector currents are related, atthe operator level, through ˜ J µ α = J µ α − ( J µ α ) GB pole (3.40)where the second term on the right is the purely Goldstone-boson pole part of the axial-vector current. We will see that this peculiar realization of chiral symmetry does indeedemerge in QCD.It is useful to define new objects which give a matrix-element representation of theinternal-symmetry charges [7, 8]: (cid:104) h (cid:48) | ˜ Q α ( x + ) | h (cid:105) = (2 π ) p + δ ( q + ) δ ( (cid:126)q ⊥ )[ X α ( λ ) ] h (cid:48) h δ λ (cid:48) λ ; (3.41) (cid:104) h (cid:48) | ˜ Q α | h (cid:105) = (2 π ) p + δ ( q + ) δ ( (cid:126)q ⊥ )[ T α ] h δ hh (cid:48) δ λ (cid:48) λ . (3.42)These definitions are particularly useful as they allow the preservation of the Lie-algebraicstructure of the operator algebra in the case where correlation functions are given purelyby single-particle states. The matrix element for Goldstone boson emission and absorptionis: M α ( p (cid:48) , λ (cid:48) , h (cid:48) ; p, λ, h ) = iF π ( M h − M h (cid:48) ) [ X α ( λ ) ] h (cid:48) h δ λ (cid:48) λ . (3.43)As one might expect, in the limit that chiral symmetry is restored through a second-order phase transition, the matrix [ X α ( λ ) ] h (cid:48) h becomes a true symmetry generator [61].In this limit, one also expects that the states h (cid:48) and h become degenerate. In order thatthe matrix element of eq. 3.43 not vanish in this limit, F π must approach zero in thesymmetry limit in precisely the same way [61]. The role of F π as an order parameter ofchiral symmetry breaking is then apparent in eq. 3.27, as the mixed-Lie bracket vanishes as F π →
0. Therefore, F π is an order parameter of chiral symmetry breaking on the null-plane.– 15 – .5 Broken chiral symmetry and spin Using the results of the previous two sections one finds (cid:104) h (cid:48) , λ (cid:48) | [ ˜ Q α ( x + ) , M ] | h , λ (cid:105) = δ λ (cid:48) ,λ (2 π ) p + δ ( q + ) δ ( (cid:126)q ⊥ ) (cid:0) M h − M h (cid:48) (cid:1) [ X α ( λ ) ] h (cid:48) h (3.44)and (cid:104) h (cid:48) , λ (cid:48) | [ ˜ Q α ( x + ) , M J ± ] | h , λ (cid:105) = δ λ (cid:48) ,λ ± (2 π ) p + δ ( q + ) δ ( (cid:126)q ⊥ ) × (cid:20) M h c ± ( h, λ ) [ X α ( λ ±
1) ] h (cid:48) h − M (cid:48) h c ∓ ( h (cid:48) , λ (cid:48) ) [ X α ( λ ) ] h (cid:48) h (cid:21) , (3.45)where J ± ≡ J ± i J and c ± ( h, λ ) ≡ (cid:112) j h ( j h + 1) − λ ( λ ± M h = M h (cid:48) , then chiral symmetry breaking arises solely through the transverse spinoperator, J r , which is dynamical on the null-plane. That is, (cid:104) h (cid:48) , λ (cid:48) | [ ˜ Q α ( x + ) , J ± ] | h , λ (cid:105) = δ λ (cid:48) ,λ ± (2 π ) p + δ ( q + ) δ ( (cid:126)q ⊥ ) × (cid:20) c ± ( h, λ ) [ X α ( λ ±
1) ] h (cid:48) h − c ∓ ( h (cid:48) , λ (cid:48) ) [ X α ( λ ) ] h (cid:48) h (cid:21) . (3.46)In this case, Goldstone’s theorem must be obtained from the relation (cid:104) h (cid:48) | M [ ˜ Q α ( x + ) , J r ] | h (cid:105) = − i (cid:15) rs p + (cid:15) χ (cid:90) dx − d x ⊥ (cid:104) h (cid:48) | x s ˜ P α ( x − , (cid:126)x ⊥ , x + ) | h (cid:105) , (3.47)and its corresponding constraint (cid:90) dx − d x ⊥ (cid:104) h (cid:48) | x s ˜ P α ( x − , (cid:126)x ⊥ , x + ) | h (cid:105) −→ (cid:15) χ + . . . (3.48)in the symmetry limit, (cid:15) χ →
0. Following the same steps as for the mass-squared reducedHamiltonian, we have (cid:104) h (cid:48) | [ ˜ Q α ( x + ) , J r ] | h (cid:105) = − (cid:15) rs p + M h (2 π ) δ ( q + ) δ ( (cid:126)q ⊥ ) e ix + q − (cid:88) i (cid:15) χ Z i M φ i (cid:18) ∂∂q s M iα ( q ) (cid:19) (3.49)which again leads, via the same logic presented above, to Goldstone’s theorem. Therefore,even if M commutes with the chiral charges, the chiral symmetry breaking contained inthe spin Hamiltonians implies the presence of massless states. Evaluating eq. 3.49 in therest frame, where p + → M h / √ J r → J r , and using eq. 3.31, gives (cid:104) h (cid:48) | [ ˜ J +5 α , J r ] | h (cid:105) = i √ (cid:15) rs (cid:104) h (cid:48) | ˜ J s α | h (cid:105) , (3.50)which is simply the statement that the axial current transform as a vector operator.– 16 – .6 General operator algebra and the chiral basis A physical system with an SU ( N ) R ⊗ SU ( N ) L chiral symmetry broken to the vectorsubgroup SU ( N ) F may be expressed as a dynamical Hamiltonian system which evolveswith null-plane time, whose reduced Hamiltonians satisfy the U (2) algebra of eq. 2.28,and in addition have non-vanishing Lie brackets with the non-conserved chiral charges. Inoperator form the reduced Hamiltonians satisfy:[ ˜ Q β ( x + ) , M ] (cid:54) = 0 ; [ ˜ Q β ( x + ) , M J ± ] (cid:54) = 0 , (3.51)which express the spontaneous breaking of the chiral symmetry. Eq. 3.51 has the generaloperator solution M = M + (cid:88) R M R ; M J ± = ( M J ± ) + (cid:88) R ( M J ± ) R (3.52)where denotes the singlet SU ( N ) R ⊗ SU ( N ) L representation, ( , ), and R = ( R R , R L ) isa non-trivial representation. Note that all three symmetry-breaking reduced Hamiltoniansmust transform in the same way. This follows directly from eqs. 3.10 and 3.11.It is useful to give a heuristic description of the consequences of this algebraic structure.Consider an interpolating field operator, a † h which creates a momentum state h out of thevacuum; that is, a † h | (cid:105) = | h (cid:105) . (3.53)Here and below for simplicity we will suppress the flavor indices. Because the null-planechiral charges annihilate the vacuum, ˜ Q | (cid:105) = 0, one has˜ Q | h (cid:105) = [ ˜ Q , a † h ] | (cid:105) . (3.54)Now we will assume that the interpolating field operator a † h has definite chiral transforma-tion properties with respect to the chiral charge in the sense that[ ˜ Q , a † h ] = C (cid:48) a † h (cid:48) + C (cid:48)(cid:48) a † h (cid:48)(cid:48) + . . . , (3.55)where C (cid:48) , C (cid:48)(cid:48) , . . . are group-theoretic factors. This is simply the statement that the fieldoperators { a h , a h (cid:48) , a h (cid:48)(cid:48) , . . . } are in a non-trivial SU ( N ) R ⊗ SU ( N ) L representation. It thenfollows from eq. 3.54 that ˜ Q | h (cid:105) = C (cid:48) | h (cid:48) (cid:105) + C (cid:48)(cid:48) | h (cid:48)(cid:48) (cid:105) + . . . , (3.56)and therefore the states { h, h (cid:48) , h (cid:48)(cid:48) , . . . } are also in an SU ( N ) R ⊗ SU ( N ) L representation . Note that the instant-form interpolating operators also fill out SU ( N ) R ⊗ SU ( N ) L representations.However, the instant-form charges do not annihilate the vacuum, i.e. Q | (cid:105) ≡ | ω (cid:105) , it follows that Q | h (cid:105) =[ Q , a † h ] | (cid:105) + | h ; ω (cid:105) . Therefore Q | h (cid:105) = C (cid:48) | h (cid:48) (cid:105) + C (cid:48)(cid:48) | h (cid:48)(cid:48) (cid:105) + . . . + | h ; ω (cid:105) and the utility of chiralsymmetry as a classification symmetry is lost. – 17 –ne then has, for instance, (cid:104) h (cid:48) | ˜ Q | h (cid:105) = C (cid:48) ; (3.57) (cid:104) h (cid:48)(cid:48) | ˜ Q | h (cid:105) = C (cid:48)(cid:48) , (3.58)which are, via eq. 3.41, Goldstone-boson transition matrix elements. If { h, h (cid:48) , h (cid:48)(cid:48) , . . . } are inan irreducible representation, then the C ’s are completely determined by the symmetry (i.e.are Clebsch-Gordon coefficients), while if the representation is reducible, then the C ’s willdepend on the mixing angles which mix the irreducible representations. Therefore throughthe study of Goldstone-boson transitions one learns about the chiral representations filledout by the physical states . To learn more about the chiral representations, one considersthe mixed Lie brackets, eqs. 3.44 and 3.45. Knowledge of the transformation propertiesof the chiral-symmetry breaking reduced Hamiltonians gives information about how thehadron masses and spins are related, and therefore in how the irreducible representationsmix with each other when the symmetry is broken.A natural null-plane basis can be written as | k + , (cid:126)k ⊥ ; λ , h , ( R R , R L ) (cid:105) . (3.59)While the mass eigenstates are eigenstates of helicity, they clearly are not eigenstates of SU ( N ) R ⊗ SU ( N ) L when the symmetry is spontaneously broken. Nevertheless, the chiralbasis is useful when the state h can only appear in a finite number of chiral representations,even though h may be in an infinite-dimensional reducible chiral representation, as is thecase generally in QCD at large- N c [9, 62]. In the chiral basis, the reduced Hamiltonianmatrix M is then of finite rank, even though there can be submatrices of infinite rank(and therefore the Fock expansion in the number of constituents is infinite). Ultimately,the utility of the chiral basis is determined by comparison with experiment [7–10, 63–68]. In this section, we will review the relevant symmetry properties of the instant-form QCDLagrangian for purposes of establishing conventions which will clarify the null-plane de-scription. Consider the QCD Lagrangian with N flavors of light quarks and N c colors: L QCD ( x ) = ¯ ψ ( x ) (cid:104) i (cid:16) → D µ − ← D µ (cid:17) γ µ − M (cid:105) ψ ( x ) − F aµν ( x ) F µνa ( x ) (4.1)where M is the quark mass matrix, for now taken as a diagonal N × N matrix, and thecovariant derivatives are → D µ = → ∂ µ − ig t a A aµ ( x ) , ← D µ = ← ∂ µ + ig t a A aµ ( x ) , (4.2) Here it should be stressed that the chiral multiplet structure of the states is useful only when the null-plane chiral charges mediate transitions between single-particle states [7, 8]. Multi-particle states obscurethe algebraic consequences of null-plane chiral symmetry. – 18 –here g is the strong coupling constant, and indices a, b, . . . are taken as adjoint indices ofthe SU (3)-color gauge group. The Lagrangian is invariant with respect to baryon numberand singlet axial transformations ψ → e − iθ ψ , ψ → e − iθγ ψ , (4.3)with associated currents J µ = ¯ ψγ µ ψ , J µ = ¯ ψγ µ γ ψ , (4.4)and with divergences ∂ µ J µ = 0 , ∂ µ J µ = 2 i ¯ ψ M γ ψ − N g π (cid:15) µνρσ tr ( F µν F ρσ ) , (4.5)where the singlet axial symmetry is of course anomalous. In addition, the Lagrangian isinvariant with respect to the symmetry transformations ψ → e − iθ α T α ψ , ψ → e − iθ α T α γ ψ , (4.6)where the T α are SU ( N ) generators (see appendix). By the standard Noether procedureone defines the associated currents, J µα = ¯ ψγ µ T α ψ , J µ α = ¯ ψγ µ γ T α ψ , (4.7)respectively, with divergences ∂ µ J µα = − i ¯ ψ [ M , T α ] ψ , ∂ µ J µ α = i ¯ ψ { M , T α } γ ψ . (4.8)Therefore, with N degenerate flavors the QCD Lagrangian is SU ( N ) F invariant and in thechiral limit where M vanishes, there is an SU ( N ) R ⊗ SU ( N ) L chiral symmetry generatedby the currents J µLα = ( J µα − J µ α ) / J µRα = ( J µα + J µ α ) / T µν = − g µν L QCD − F µρa F νa ρ + i ψ ↔ D µ γ ν ψ . (4.9)From the energy-momentum tensor we can form the Hamiltonian, P = (cid:90) d x T . (4.10)Here we will assume that chiral symmetry is spontaneously broken through the formationof the condensate M (cid:104) Ω | ¯ ψψ | Ω (cid:105) = M (cid:104) Ω | ∂T ∂ M | Ω (cid:105) = M ∂ E ∂ M (cid:54) = 0 , (4.11)where we have used the Feynman-Hellmann theorem, | Ω (cid:105) represents the (complicated)instant-form QCD vacuum state, and E is the QCD vacuum energy. It is straightforwardto show that the condensate transforms as the ( ¯ N , N ) ⊕ ( N , ¯ N ) representation of SU ( N ) R ⊗ – 19 – U ( N ) L . We can compute the vacuum energy in the low-energy effective field theory; i.e.chiral perturbation theory ( χ PT) [69, 70], as well. And therefore, M ∂ E ∂ M = M ∂ E χ PT0 ∂ M , (4.12)where E χ PT0 is the χ PT vacuum energy. In the non-linear realization of the chiral groupthe Goldstone boson field may be written as U ( x ) = exp ( iπ α ( x ) T α /F π ), and the leadingquark mass contribution to the χ PT Lagrangian is L χ PTQCD = v tr (cid:16) U M † + U † M (cid:17) + . . . , (4.13)with v = M π F π / M and with M π the Goldstone boson mass. One then obtains the Gell-Mann-Oakes-Renner formula [42]. − M (cid:104) Ω | ¯ ψψ | Ω (cid:105) = N M π F π + . . . . (4.14)It will be a principle goal in what follows to determine what takes the place of this relationin null-plane QCD. The QCD Lagrangian in the null-plane coordinates is obtained by generalizing the resultsgiven in Appendices B and C to the interacting case . (Note that we work in light-conegauge, A + = 0, throughout.) The QCD equations of constraint for the non-dynamicaldegrees of freedom are ψ − = 12 i → ∂ + (cid:16) − i γ r → D r + M (cid:17) γ + ψ + , ψ †− = ψ † + γ − (cid:16) i γ r ← D r − M (cid:17) i ← ∂ + (4.15)for the redundant quark degrees of freedom, and ∂ + A − a = 1 ∂ + D rab ∂ + A rb − g ∂ + ¯ ψ + γ + t a ψ + , (4.16)for the redundant gauge degrees of freedom. The null-plane QCD Lagrangian can then beexpressed in terms of the dynamical degrees of freedom as˜ L QCD = i ¯ ψ + γ + ∂ − ψ + − i ¯ ψ + γ r γ + γ s D r ∂ + D s ψ + + i ¯ ψ + γ + M ∂ + ψ + + i ¯ ψ + γ + M ( γ r g t a A ra ) 1 ∂ + ψ + − i ¯ ψ + γ + M ∂ + ( γ r g t a A ra ψ + ) − F rsa F rsa + (cid:0) ∂ + A ra (cid:1) (cid:0) ∂ − A ra (cid:1) − (cid:18) ∂ + D rab ∂ + A rb − g ∂ + ¯ ψ + γ + t a ψ + (cid:19) . (4.17)The price to pay for working with the physical degrees of freedom in the null-plane co-ordinates is a loss of manifest Lorentz covariance, as well as the appearance of operatorswhich that appear to be non-local in the longitudinal coordinate. As in the instant-form, We follow the notation and conventions of Ref. [55]. – 20 –ne should view this Lagrangian as providing a perturbative definition of QCD at largemomentum transfers, where the longitudinal zero modes play no role. Notice that in null-plane QCD there are two kinds of operators that depend on the quark-mass matrix . Oneis a kinetic term, quadratic in the quark masses, and the other is a spin-flip quark-gluoninteraction that is linear in the quark masses.Naturally we expect that null-plane QCD has the same symmetries as instant-formQCD. Consider the U (1) R ⊗ U (1) L transformations, ψ + → e − iθ ψ + , ψ + → e − iθγ ψ + . (4.19)While baryon number is unaltered in moving to the null-plane coordinates, this is clearlynot the same chiral transformation that we had in the instant form, as that transformationacts on the non-dynamical degrees of freedom, ψ − , in a distinct manner and is thereforecomplicated on the null-plane. That the chiral symmetry transformations are differentin the two forms of dynamics is essential for what follows. We will return below to therelation between the chiral symmetries in the instant-form and the front form, as thiswill be important in understanding the role of condensates on the null-plane. The U (1) A current and its divergence are [29]˜ J µ = J µ − i ¯ ψγ µ γ + γ M ∂ + ψ + ; (4.20) ∂ µ ˜ J µ = ¯ ψ + γ + γ M ∂ + ( γ r g t a A ra ) (cid:48) ψ + − N g π (cid:15) µνρσ tr ( F µν F ρσ ) . (4.21)Consider the SU ( N ) R ⊗ SU ( N ) L transformations, ψ + → e − iθ α T α ψ + , ψ + → e − iθ α T α γ ψ + . (4.22)The currents associated with eq. 4.22 are˜ J µα = J µα − i ¯ ψγ µ γ + [ M , T α ] 1 ∂ + ψ + ; (4.23)˜ J µ α = J µ α − i ¯ ψγ µ γ + γ { M , T α } ∂ + ψ + , (4.24)with divergences ∂ µ ˜ J µα = ¯ ψγ + [ M , T α ] 1 ∂ + ψ + ; (4.25) ∂ µ ˜ J µ α = ¯ ψγ + γ [ M , T α ] 1 ∂ + ψ + + ¯ ψ + γ + γ { M , T α } ∂ + ( γ r g t a A ra ) (cid:48) ψ + . (4.26)For N degenerate flavors, the quark mass matrix is proportional to the identity, the vectorcurrent is conserved, and the axial current and the divergence of the axial current are˜ J µ α = J µ α − i ¯ ψγ µ γ + γ T α M ∂ + ψ + ; (4.27) To minimize clutter, it will prove convenient to define the operator1 ∂ + ( γ r g t a A ra ) (cid:48) ψ + ≡ ( γ r g t a A ra ) 1 ∂ + ψ + − ∂ + ( γ r g t a A ra ψ + ) . (4.18) – 21 – µ ˜ J µ α = ¯ ψ + γ + γ T α M ∂ + ( γ r g t a A ra ) (cid:48) ψ + . (4.28)Here note in particular that the null-plane axial-vector current in null-plane QCD evidentlytakes the form, eq. 3.40, expected on general grounds. The null-plane singlet axial charge is defined as˜ Q = (cid:90) dx − d x ⊥ ˜ J +5 = (cid:90) dx − d x ⊥ ¯ ψ + γ + γ ψ + , (4.29)where we have used eq. 4.20. Using the momentum-space representation of ψ + , given ineq. B.20, one finds˜ Q = (cid:88) λ = ↑↓ λ (cid:90) dk + d k ⊥ k + (2 π ) (cid:110) b † λ ( k + , k ⊥ ) b λ ( k + , k ⊥ ) + d † λ ( k + , k ⊥ ) d λ ( k + , k ⊥ ) (cid:111) . (4.30)Comparison with eq. B.31 one sees that the singlet axial charge coincides (up to a factorof two) with the free-fermion helicity operator. This of course explains why the quarkmass term in the free-fermion theory is a chiral invariant; on the null-plane, chiral sym-metry breaking in the free-fermion theory implies breaking of rotational invariance in thetransverse plane.Similarly, the null-plane non-singlet vector and chiral charges are, respectively,˜ Q α = (cid:90) dx − d x ⊥ ˜ J + α = (cid:90) dx − d x ⊥ ¯ ψ + γ + T α ψ + ; (4.31)˜ Q α = (cid:90) dx − d x ⊥ ˜ J +5 α = (cid:90) dx − d x ⊥ ¯ ψ + γ + γ T α ψ + , (4.32)and using the momentum-space representation of ψ + , given in eq. B.20, one finds˜ Q α = (cid:88) λ = ↑↓ (cid:90) dk + d k ⊥ k + (2 π ) (cid:110) b † λ ( k + , k ⊥ ) T α b λ ( k + , k ⊥ ) − d † λ ( k + , k ⊥ ) T Tα d λ ( k + , k ⊥ ) (cid:111) ; (4.33)˜ Q α = (cid:88) λ = ↑↓ λ (cid:90) dk + d k ⊥ k + (2 π ) (cid:110) b † λ ( k + , k ⊥ ) T α b λ ( k + , k ⊥ ) + d † λ ( k + , k ⊥ ) T Tα d λ ( k + , k ⊥ ) (cid:111) . (4.34)One readily checks that the null-plane chiral algebra, eqs. 3.4 and 3.5, is satisfied by thesecharges. As these charge are written as sums of number operators that count the numberof quarks and anti-quarks, both chiral charges annihilate the vacuum, and we have˜ Q α | (cid:105) = ˜ Q α | (cid:105) = 0 , (4.35)as expected on the general grounds presented above. One then has[ ˜ Q α , ψ + ] = − T α ψ + ; [ ˜ Q α , ψ + ] = − γ T α ψ + . (4.36)Breaking down the fields into left- and right-handed components, ψ + R = (1 + γ ) ψ + , ψ + L = (1 − γ ) ψ + (4.37)– 22 –nd, using the results of Appendix B, one verifies the fermion transformation propertieswith respect to SU ( N ) R ⊗ SU ( N ) L : ψ + R = ψ + ↑ ∈ ( , N ) , ψ † + R = ψ † + ↓ ∈ ( , ¯ N ) ; (4.38) ψ + L = ψ + ↓ ∈ ( N , ) , ψ † + L = ψ † + ↑ ∈ ( ¯ N , ) , (4.39)and the helicity eigen-equations of the quarksΣ ψ + ↑ = ψ + ↑ ; (4.40)Σ ψ + ↓ = − ψ + ↑ , (4.41)where the helicity operator, Σ , is defined in Appendix B. Using the results of the previous section, it is straightforward to find the transformationproperties of the symmetry-breaking parts of the reduced Hamiltonians. Define the oper-ators : ˜ D α ≡ ∂ µ ˜ J µ α = ¯ ψ + γ + γ T α ∂ + M ( γ r g t a A ra ) (cid:48) ψ + ; (4.42)˜ D ≡ ¯ ψ + γ + γ ∂ + M ( γ r g t a A ra ) (cid:48) ψ + ; (4.43)˜ D α ≡ ¯ ψ + γ + T α ∂ + M ( γ r g t a A ra ) (cid:48) ψ + ; (4.44)˜ D ≡ ¯ ψ + γ + ∂ + M ( γ r g t a A ra ) (cid:48) ψ + . (4.45)It is then a textbook exercise to find[ ˜ Q α , ˜ D β ] = 1 N δ αβ ˜ D + d αβγ ˜ D γ ; (4.46)[ ˜ Q α , ˜ D ] = 2 ˜ D α ; (4.47)[ ˜ Q α , ˜ D β ] = 1 N δ αβ ˜ D + d αβγ ˜ D γ ; (4.48)[ ˜ Q α , ˜ D ] = 2 ˜ D α . (4.49)It follows that the 2 N operators ( ˜ D α , ˜ D , ˜ D α , ˜ D ) fill out the ( ¯ N , N ) ⊕ ( N , ¯ N ) represen-tation of SU ( N ) R ⊗ SU ( N ) L .The null-plane Hamiltonian P − is: P − = (cid:90) dx − d x ⊥ T − + , (4.50) From here forward we will use the definition:1 ∂ + M ≡ M ∂ + . – 23 –nd therefore the chiral-symmetry breaking part of this Hamiltonian is given by: P − ( N , N ) ≡ − i (cid:90) dx − d x ⊥ ˜ D . (4.51)One readily checks that this is consistent with eqs. 3.10 and 4.42.One then finds the symmetry breaking parts of the reduced QCD Hamiltonians: M N , N ) = − iP + (cid:90) dx − d x ⊥ ¯ ψ + γ + ∂ + M ( γ r g t a A ra ) (cid:48) ψ + ; (4.52)( M J r ) ( N , N ) = i (cid:15) rs P + (cid:90) dx − d x ⊥ Γ s ¯ ψ + γ + ∂ + M ( γ r g t a A ra ) (cid:48) ψ + , (4.53)where, in addition, we have used eqs. 3.11 and 4.28 to obtain the reduced Hamiltonian forspin. All chiral symmetry breaking in null-plane QCD is contained in these two operators.Using eqs. 3.10, 3.11 and 4.46 one finds[ ˜ Q β , [ ˜ Q α , M ]] = − iP + (cid:90) dx − d x ⊥ (cid:18) N δ αβ ˜ D + d αβγ ˜ D γ (cid:19) ; (4.54)[ ˜ Q β , [ ˜ Q α , M J r ]] = i(cid:15) rs P + (cid:90) dx − d x ⊥ Γ s (cid:18) N δ αβ ˜ D + d αβγ ˜ D γ (cid:19) . (4.55)Acting on these equations with δ αβ and d αβγ , and using the identities in Appendix D gives − iP + (cid:90) dx − d x ⊥ ˜ D = NN − Q α , [ ˜ Q α , M ]] ; (4.56) − iP + (cid:90) dx − d x ⊥ ˜ D γ = d αβγ NN − Q β , [ ˜ Q α , M ]] . (4.57)Therefore, eq. 4.54 can be written as[ ˜ Q β , [ ˜ Q α , M ]] = 1 N − δ αβ [ ˜ Q γ , [ ˜ Q γ , M ]] + NN − d αβγ d µνγ [ ˜ Q µ , [ ˜ Q ν , M ]] , (4.58)and eq. 4.55 takes the same form but with M replaced by M J ± . Defining the projectionoperator P αβ ; µν ≡ δ αν δ βµ − N − δ αβ δ µν − NN − d αβγ d µνγ , (4.59)we can express the constraints on the reduced Hamiltonians in compact notation as: P αβ ; µν [ ˜ Q µ , [ ˜ Q ν , M ]] = P αβ ; µν [ ˜ Q µ , [ ˜ Q ν , M J ± ]] = 0 . (4.60)These are quite possibly the most important equations in null-plane QCD, as they arethe mathematical expression of the specific way in which the internal symmetries andPoincar´e symmetries intersect. These equations were obtained originally in Refs. [7–9]by considering the most general form of Goldstone-boson-hadron scattering amplitudes inspecially-designed Lorentz frames, and using input from Regge-pole theory expectationsof their high-energy behavior. Note that the projection operator, P αβ ; µν , has four adjoint– 24 –ndices and is, as shown in Ref. [7] related to the interactions of Goldstone bosons (in thet-channel of Goldstone-boson-hadron scattering), which are in the adjoint of SU ( N ) F andwhose scattering amplitudes therefore transform as the product of two adjoints. In the caseof two flavors, where ⊗ = ⊕ ⊕ , it projects out the -dimensional representation( I = 2) and in the case of three flavors, where ⊗ = ⊕ ⊕ ⊕ ⊕ ¯ ⊕ , itprojects out the , ¯ , and -dimensional representations. As shown above, these arethe representations that cannot be formed from a single quark bilinear; i.e. they are notcontained in ( ¯ N , N ) ⊕ ( N , ¯ N ), as is clear from direct inspection of eqs. 4.54 and 4.55. We are now in a position to address the fate of instant-form QCD chiral-symmetry breakingcondensates in null-plane QCD. Again using the Feynman-Hellmann theorem we find M (cid:104) | ∂T − + ∂ M | (cid:105) = M ∂ ˜ E ∂ M = M ∂ E ∂ M = M ∂ E χ PT0 ∂ M , (4.61)where | (cid:105) represents the null-plane QCD vacuum state, and ˜ E is the null-plane QCDvacuum energy. In this equation we have also expressed that physics is independent of thechoice of coordinates. Therefore calculation of the leading quark-mass contribution to thevacuum energy must be independent of the quantization surface, and should be the samewhether one works with the fundamental degrees of freedom, or with the Goldstone bosonsin the infrared. One then has M ∂ ˜ E ∂ M = − M (cid:104) | i ¯ ψ + γ + ∂ + M ψ + | (cid:105) + (cid:104) | i ¯ ψ + γ + ∂ + M ( γ r g t a A ra ) (cid:48) ψ + | (cid:105) . (4.62)The second term must vanish as the chiral charges annihilate the vacuum and thereforethere can be no chiral-symmetry breaking condensates. Operationally one sees this directlyby taking the vacuum expectation value of eq. 4.46 which gives (cid:104) | i ¯ ψ + γ + ∂ + M ( γ r g t a A ra ) (cid:48) ψ + | (cid:105) = 0 . (4.63)We are then left with the null-plane expression of the Gell-Mann-Oakes-Renner relation: M (cid:104) | i ¯ ψ + γ + ∂ + M ψ + | (cid:105) = N M π F π + . . . . (4.64)Hence, a chiral-symmetry breaking condensate in the instant-form formulation of QCD hasbeen replaced by a chiral-symmetry conserving condensate in the null-plane formulation.Note that while the operator naively vanishes in the chiral limit, the matrix element isinfrared singular and therefore it need not, and indeed cannot, vanish in the chiral limit .It would be very interesting to define the relevant operator non-perturbatively and calculatethis condensate directly, perhaps using transverse lattice gauge theory methods [71–79].Note that a priori knowledge of the singlet condensate in eq. 4.64 is not very different to a priori knowledge of the symmetry-breaking quark condensate in eq. 4.14. In both cases,it is necessary to keep the quark masses finite and only at the very end take the chirallimit [80]. This expression of the Gell-Mann-Oakes-Renner formula was found previously in Ref. [29] using themethods that will be described below. – 25 – .6 Condensates on a null-plane
We will now derive the Gell-Mann-Oakes-Renner relation in a different way which willsuggest a general prescription for expressing all instant-form condensates with null-planecondensates. While the left- and right-handed components of ψ + transform irreducibly withrespect to the null-plane chiral charges, the transformation properties of ψ are complicatedby the presence of the non-dynamical component ψ − . Indeed one finds[ ˜ Q α , ψ ] = − γ T α ψ − i γ γ + T α ∂ + M ψ , (4.65)from which it follows that ψ R , ψ L ∈ ( , N ) ⊕ ( N , ) , ψ † R , ψ † L ∈ ( , ¯ N ) ⊕ ( ¯ N , ) . (4.66)Since the left- and right-handed components of the quark field transform reducibly withrespect to the chiral group, generally products of bilinear operators of the form ¯ ψ Γ ψ willhave complicated reducible chiral transformation properties. However, QCD operatorsbuilt out of these bilinears will always have a component that transforms as a chiral singlet.We will now see, for the simplest example, that this is essential to the consistency of thenull-plane formulation. Consider the transformation properties of the following set ofbilinears: D α ≡ ¯ ψ γ T α ψ , D ≡ ¯ ψ γ ψ ; (4.67) D α ≡ ¯ ψ T α ψ , D ≡ ¯ ψ ψ . (4.68)Is is again simple to check that these operators fill out the ( ¯ N , N ) ⊕ ( N , ¯ N ) representationof SU ( N ) R ⊗ SU ( N ) L with respect to the instant-form chiral charges Q α . Now consider thetransformation properties of these operators with respect to the null-plane chiral charges.One finds [ ˜ Q α , D β ] = − N δ αβ (cid:18) D + i ¯ ψ + γ + ∂ + M ψ + (cid:19) − d αβγ (cid:18) D γ + i ¯ ψ + γ + T γ ∂ + M ψ + (cid:19) ; (4.69)[ ˜ Q α , D ] = − D α ; (4.70)[ ˜ Q α , D β ] = − N δ αβ D − d αβγ D γ + f αβγ ¯ ψ + γ + γ T γ ∂ + M ψ + ; (4.71)[ ˜ Q α , D ] = − D α − i ¯ ψ + γ + T α ∂ + M ψ + . (4.72)– 26 –o close the algebra we must add, in addition, the commutation relations:[ ˜ Q α , ¯ ψ + γ + ∂ + M ψ + ] = 0 ; (4.73)[ ˜ Q α , ¯ ψ + γ + T β ∂ + M ψ + ] = i f αβγ ¯ ψ + γ + γ T γ ∂ + M ψ + ; (4.74)[ ˜ Q α , ¯ ψ + γ + γ T β ∂ + M ψ + ] = i f αβγ ¯ ψ + γ + T γ ∂ + M ψ + . (4.75)Hence the full set of operators transform as the reducible 4 N -dimensional ( , ) ⊕ ( , A ) ⊕ ( A , ) ⊕ ( ¯ N , N ) ⊕ ( N , ¯ N ) representation of SU ( N ) R ⊗ SU ( N ) L , where here A denotes the SU ( N ) adjoint representation. In particular one see that¯ ψψ ∈ ( ¯ N , N ) ⊕ ( N , ¯ N ) ⊕ ( , ) ⊕ . . . , (4.76)and therefore transforms reducibly. This is verified by direct calculation which gives M ¯ ψψ = − i M ¯ ψ + γ + ∂ + M ψ + + i ¯ ψ + γ + ∂ + M ( γ r g t a A ra ) (cid:48) ψ + . (4.77)Taking the vacuum expectation value of eq. 4.77 (or eq. 4.69) gives the general solution [29] (cid:104) | ¯ ψψ | (cid:105) = −(cid:104) | i ¯ ψ + γ + ∂ + M ψ + | (cid:105) . (4.78)Therefore only the singlet part of ¯ ψψ can acquire a vacuum expectation value on the nullplane, as must be the case since SU ( N ) R ⊗ SU ( N ) L is a symmetry of the null-plane vacuumstate. This argument readily generalizes to any chiral symmetry breaking Lorentz scalaroperator, O , that one can build out of products of quark bilinears in instant-form QCD.One can write O = (cid:88) R O R = (cid:88) ˜ R O ˜ R + O ˜ (4.79)where R is a non-trivial chiral representation with respect to the instant-form chiralcharges, Q α , and ˜ R (˜ ) is a non-trivial (the singlet) representation with respect to thefront-form chiral charges, ˜ Q α . Unless protected by another symmetry, O has a non-vanishing vacuum expectation value, which can be expressed as (cid:104) Ω | O | Ω (cid:105) = (cid:104) Ω | (cid:88) R O R | Ω (cid:105) = (cid:104) | O ˜ | (cid:105) (cid:54) = 0 . (4.80)Note that the final equality expresses an equivalence between a matrix element evaluated inthe instant form and one in the front form. This equality ensures that physics is unmodifiedin moving between the two forms of dynamics. Therefore all instant-form chiral symmetry Note that the second term, which is breaks chiral symmetry and is independent of the interaction doesnot appear in the free fermion Lagrangian as it is cancelled by a piece coming from the other kinetic term,as must be the case in order that the Lagrangian commute with the helicity operator. – 27 –reaking QCD condensates map to chiral symmetry conserving condensates in the front-form. The presence of the singlet part of the operator can always be traced to the reduciblechiral transformation property of ψ given in eq. 4.65. For the case at hand, with O = ¯ ψψ ,we have (cid:104) Ω | ¯ ψψ | Ω (cid:105) = (cid:104) | ¯ ψψ | (cid:105) , (4.81)which together with eq. 4.78, provides the desired link between the instant-form and front-form expressions of the Gell-Mann-Oakes-Renner relation.The general relation, eq. 4.80 is important for the consistency of null-plane QCD, as itdemonstrates that, as expected, the QCD vacuum energy is unaltered in moving from theinstant-form to the front-form description, and these relations must, of course, exist in orderthat the operator product expansion be independent of the choice of quantization surface.We see that a symmetry-breaking condensate can form in the instant-form coordinates withan asymmetric vacuum which is equal to a corresponding symmetry-preserving condensatein the null-plane description with a symmetric vacuum. The condensate relation eq. 4.81is one of an infinite number of relations which translates condensates which break chiralsymmetry in the instant form to null-plane condensates which transform as chiral singlets. Before considering the consequences of the null-plane QCD operator algebra, we will sum-marize the picture of chiral symmetry breaking that we have so far established. While thenull-plane QCD vacuum state is chirally invariant, chiral symmetry is spontaneously bro-ken by the three reduced Hamiltonians that have contributions, M N , N ) and ( M J r ) ( N , N ) ,which transform as ( ¯ N , N ) ⊕ ( N , ¯ N ) with respect to SU ( N ) R ⊗ SU ( N ) L . The three reducedHamiltonians satisfy the constraints, eq. 4.60. In addition to these signatures of chiral sym-metry breaking, the three reduced Hamiltonians, together with the generator of rotationson the transverse plane together generate the U (2) dynamical sub-group of the null-planePoincar´e algebra, eq. 2.28. And finally, the null-plane vector and chiral charges satisfy the SU ( N ) R ⊗ SU ( N ) L algebra, eqs. 3.4 and 3.5. The entire set of Lie-brackets provide all ofthe constraints that exist among the generators of the internal and space-time symmetriesin null-plane QCD. The consequences of chiral symmetry breaking for the spectrum andspin of QCD are contained in the symmetry-breaking parts of the reduced Hamiltonians. In searching for solutions of the algebraic system that mixes the chiral charges and thereduced Hamiltonians, one may worry about the existence of no-go theorems that forbidnon-trivial algebras that mix space-time and internal symmetries. In the null-plane for-mulation the no-go theorems are avoided because it is only the dynamical part, D , of thenull-plane Poincar´e algebra that mixes with the internal symmetry generators [58]. Unfor-tunately, a direct general solution of the null-plane QCD operator algebra in the general– 28 –ase appears difficult. However, there is a limiting case in which the algebra yields animportant non-trivial solution. Here we will treat the QCD operator algebra as an ab-stract operator algebra and consider the limit in which the chiral-symmetry breaking partof the reduced Hamiltonian M can be treated as a perturbation. However, one shouldkeep in mind that matrix elements of the operator relations between hadronic states musteventually be taken in order to extract observables. We first define[ ˜ Q α , M ] ≡ (cid:15) α , (5.1)and neglect terms of O ( (cid:15) ). This implies that all chiral symmetry breaking occurs in thespin Hamiltonians. This limit is non-trivial, as we have shown above in section 3.5 that thespin Hamiltonians alone imply the presence of Goldstone bosons. In this limit, the QCDoperator algebra reduces to [ J i , J j ] = i (cid:15) ijk J k (5.2)which generates SU (2) spin, and the SU ( N ) R ⊗ SU ( N ) L algebra,[ ˜ Q α , ˜ Q β ] = i f αβγ ˜ Q γ , [ ˜ Q α , ˜ Q β ] = i f αβγ ˜ Q γ , [ ˜ Q α , ˜ Q β ] = i f αβγ ˜ Q γ . (5.3)The remaining non-trivial mixed commutator is for the spin Hamiltonian: P αβ ; µν [ ˜ Q µ , [ ˜ Q ν , J ± ]] = 0 . (5.4)Now this simplified algebra can be put into a more familiar form. Consider an operator G αi which transforms in the adjoint of SU ( N ) and as a rotational vector in the sense that[ J i , G αj ] = i (cid:15) ijk G αk ; (5.5)[ ˜ Q α , G βi ] = i f αβγ G γi . (5.6)In general, the commutator of G αi with itself may be expressed as[ G αi , G βj ] = i f αβγ A ij,γ + i (cid:15) ijk B αβ,k , (5.7)where A ij,γ = A ji,γ and B αβ,k = B βα,k . Now we identify G α ≡ ˜ Q α . From eq. 5.3 it thenfollows that A ,α = ˜ Q α . Rotational invariance then implies A ij,α = δ ij ˜ Q α . By consideringJacobi identities of J i and ˜ Q α with the commutator in eq. 5.7 one finds, respectively,[ ˜ Q γ , B αβ,i ] = i f γβµ B αµ,i + i f γαµ B βµ,i ; (5.8)[ J i , B αβ,j ] = i (cid:15) ijk B αβ,k , (5.9)which simply indicate that B αβ,i transforms as a rank-two SU ( N ) tensor and a rotationalvector.To obtain B αβ,i we use eq. 5.4 to find: P αβ ; µν [ G α , G β ± i G β ] = 0 , (5.10)– 29 –rom which it follows that B αβ, and B αβ, have a piece proportional to δ αβ and a piece pro-portional to d αβγ . Rotational invariance then determines that B αβ,i is a linear combinationof δ αβ J i and d αβγ G γi . The coefficients of these terms are determined by considering theJacobi identity of G αi with the commutator in eq. 5.7, together with the relation among SU ( N ) structure constants given in Appendix D. Finally, one obtains[ G αi , G βj ] = i δ ij f αβγ ˜ Q γ + 2 N i δ αβ (cid:15) ijk J k + i(cid:15) ijk d αβγ G γk , (5.11)which together with[ ˜ Q α , G βi ] = i f αβγ G γi , [ J i , G αj ] = i (cid:15) ijk G αk ; (5.12)[ ˜ Q α , ˜ Q β ] = i f αβγ ˜ Q γ , [ J i , J j ] = i (cid:15) ijk J k (5.13)close the algebra of the symmetry group SU (2 N ). To find the consequences of this algebrafor observable quantities like the mass-squared matrix and the matrix elements for Gold-stone boson emission and absorption, one takes matrix elements of this algebra betweenhadron states h (cid:48) and h , and neglecting transitions from single-particle to multi-particlestates in the completeness sums over intermediate states, one recovers the same algebrabut with the replacements ˜ Q α → [ T α ] h (cid:48) h and ˜ Q α → [ X α ( λ ) ] h (cid:48) h , and corresponding re-placements for G βi and J k . This result, originally found by Weinberg [10], is here shownto be a general consequence of the null-plane QCD operator algebra, valid in any Lorentzframe.It is important to emphasize that the SU (2 N ) symmetry found here is only operativein the full interacting field theory. It is therefore unrelated to the SU (2 N ) invarianceof the QCD Lagrangian in the limit of no interaction. Indeed we have show above insection 3.5 that eq. 5.4, the main ingredient in the derivation of SU (2 N ), in itself impliesthe existence of Goldstone bosons. In addition, in a special case, this symmetry doesemerge in a well-defined limit of QCD. As (cid:104) h (cid:48) | (cid:15) α | h (cid:105) ∼ M h − M h (cid:48) , and baryons within agiven large- N c multiplet have mass splittings that scale as 1 /N c [81], the large- N c QCDscaling rules suggest that for baryons (cid:15) α ∼ /N c . Of course, as the matrix element ofchiral charges between baryon states scales as N c , the SU (2 N ) symmetry reduces to thecontracted SU (2 N ) [10, 82] for baryons in the large- N c limit, as one expects on generalgrounds [83–85].It is instructive to consider a simple example. Consider the case N = 3. Using thechiral transformation properties of the quarks, eq. 4.39, one sees that a λ = 3 / ψ + ↑ ψ + ↑ ψ + ↑ transforms as ( , ), ( , ), or ( , ) with respect to SU (3) R ⊗ SU (3) L .Therefore, if the baryon is a decuplet of SU (3) F with its λ = 3 / , ), thenone easily checks that its λ = 1 / , ) or ( , ). However, thedifferent helicity states are unrelated by chiral symmetry in itself. It is the mixed Lie-bracket, eq. 5.4, the expression of broken chiral symmetry in the spin Hamiltonian, thatrelates the helicities. Indeed taking the λ = 1 / , ) togetherwith an octet spin-1 / , ) ⊕ ( , ) togetherfill out the -dimensional representation of SU (6) as is familiar from the quark model.The difference here is that this symmetry arises from QCD symmetries and their pattern of– 30 –reaking, and, in particular, has nothing to do with the non-relativistic limit. Hence we seethat starting from the formal null-plane QCD operator algebra, the simple assumption thatthe part of the null-plane reduced Hamiltonian, M , that breaks chiral symmetry is smallimplies all of the usual consequences of the non-relativistic quark model, without the needof any further assumption like the existence of constituent quark degrees of freedom [10]. Usually one views the spontaneous breaking of a symmetry as the non-invariance of thevacuum state with respect to the symmetry. However, in relativistic theories of quantummechanics, this picture is purely a matter of convention. We have seen that the front-formvacuum is a singlet with respect to all symmetries and yet spontaneous symmetry break-ing can occur via non-conserved currents whose divergences are directly proportional toS-matrix elements for the emission and absorption of Goldstone bosons. One may viewthe null-plane description as a change of coordinates which moves dynamical informationout of the vacuum state and into the interaction operators of the theory. The primaryadvantage of working with the null-plane description is that broken chiral symmetry con-straints become manifest in the sense that there are non-trivial Lie brackets between thePoincar´e generators and the broken symmetry generators. In the instant-form, the chi-ral constraints that appear naturally in the front-form are present, but require one towork in special Lorentz frames and to make assumptions about the asymptotic behaviorof Goldstone-boson scattering amplitudes.Here we will restate the main conclusions of this paper: • In the front-form, spontaneous chiral symmetry breaking is contained entirely in the threenull-plane reduced Hamiltonians, which encode the mass spectrum and spin content of agiven theory. This must be the case as the null-plane chiral charges annihilate the vacuumstate, and therefore chiral symmetry breaking cannot be attributed to the formation ofchiral-symmetry breaking condensates. In null-plane QCD, all chiral symmetry breakingarises from the symmetry breaking parts of the reduced Hamiltonians, given explicitly ineqs. 4.52 and 4.53. • Goldstone’s theorem on the null-plane follows directly from the Lie-brackets betweenthe null-plane Hamiltonians and the chiral charges. A consistent null-plane description ofspontaneous symmetry breaking requires that a small explicit symmetry-breaking operatorbe included and that this explicit symmetry breaking be taken to zero only at the level ofmatrix elements of operators. The divergence of the axial-vector current is proportionalto the explicit symmetry breaking. Therefore, as the current cannot be conserved in thesymmetry limit, the existence of massless states arises as a consequence of the need tocancel the explicit breaking parameter that appears in its divergence. • The Gell-Mann-Oakes-Renner relation is recovered in null-plane QCD and a general– 31 –rescription exists for translating all chiral-symmetry breaking condensates in instant-form QCD to chiral-singlet condensates in null-plane QCD. It is therefore simplistic tosay that the vacuum is trivial in the front-form, since there are necessarily symmetry-preserving condensates which arise from modes with strictly zero longitudinal momentum.In particular, in contrast with claims in the literature [32–34, 39], we expect that the QCDvacuum energy is unaltered in moving from the instant-form to the front-form descriptionsof QCD, as is essential for the consistency of null-plane QCD. • A simple solution of the null-plane operator algebra recovers the spin-flavor symmetryof the constituent quark model. This result was obtained originally in Ref. [10], whichobtained the algebra of charges and Hamiltonians by working with sum rules obtainedin special Lorentz frames, and using input from Regge-pole theory expectations of theasymptotic behavior of scattering amplitudes involving Goldstone bosons. The results ofthe present work may be viewed as an attempt to clarify this original work by formulatingit in a Lorentz frame-independent manner which follows directly from null-plane QCD.In the null-plane formulation of QCD, the loss of manifest Lorentz invariance andlocality are, operationally, a result of integrating out non-dynamical degrees of freedom.Physically, it is clear that the loss of Lorentz invariance is tied to the fact that the essenceof Lorentz invariance lies in the Poincar´e Lie brackets that must be satisfied by the spingenerators, and, of course, on the null-plane spin is dynamical and therefore requires thesolution of the theory to properly implement. By contrast, the non-locality of the theorywould appear to be related to the fact that the null-plane chiral symmetry constraintson observables are properly formulated as sum rules which span many energy scales, andtherefore do not exhibit the separation of scales that allows a useful description in termsof local Lagrangian effective field theory. Indeed, it appears that, in some sense, scatteringamplitudes are the fundamental objects in the null-plane formulation. This is particularlyclear from the Lie-brackets that mix the Poincar´e and chiral symmetry generators, whichare given by the S-matrix elements for Goldstone boson emission and absorption. Froma theoretical standpoint, the most interesting consequences of the results obtained in thispaper are apparent only in the large- N c limit, which will be treated separately.I thank Ulf-G. Meißner for valuable comments on the manuscript, and T. Becher, G. Colan-gelo, H. Leutwyler, F. Niedermayer, and U. Wenger for useful discussions. I am particularlygrateful to the Institute for Theoretical Physics at the University of Bern for providing astimulating work environment during academic year 2010/2011. The Albert Einstein Cen-ter for Fundamental Physics is supported by the Innovations- und Kooperationsprojekt C-13 of the Schweizerische Universit¨atskonferenz SUK/CRUS. I gratefully acknowledge thehospitality of HISKP-theorie and the support of the Mercator programme of the DeutscheForschungsgemeinschaft during academic year 2012/2013. This work was supported in partby NSF CAREER Grant PHY-0645570 and continuing grant PHY1206498.– 32 – Null-plane conventions
We adopt the metric convention: g µν = g µν = − − − (A.1)which takes the contravariant coordinate four-vector x µ = ( x , x , x , x ) = ( t, x, y, z ) tothe covariant coordinate four-vector x µ = g µν x µ = ( x , − x , − x , − x ). With x + ≡ x · n and x − ≡ x · n ∗ , we denote the null-plane contravariant coordinate four-vector by ˜ x µ =( x + , x , x , x − ). Then we have ˜ x µ = C µν x µ , (A.2)with C µν = / √ / √
20 1 0 00 0 1 01 / √ − / √ . (A.3)This matrix transforms all Lorentz tensors in the instant-form notation to the front-formnotation. For instance, the null-plane metric tensor is given by˜ g µν = ( C − ) αµ g αβ ( C − ) βν (A.4)which gives ˜ g µν = ˜ g µν = − − . (A.5)We can now form the scalar product x · p = x µ p µ = x + p + + x − p − + x p + x p = x + p − + x − p + − x ⊥ · p ⊥ . (A.6)The indices i, j, k, . . . are spatial indices that range over 1 , ,
3, and r, s, t, . . . are transverseindices that range over 1 ,
2. We place all transverse coordinates, momenta and fields inboldface, and additionally label coordinates and momenta with the ⊥ symbol. The totallyantisymmetric symbol is (cid:15) ++12 = 1 = (cid:15) +12 − = 1 . (A.7)Note that ∂ + = ∂ − is a time-like derivative ∂/∂x + = ∂/∂x − as opposed to ∂ − = ∂ + , whichis a space-like derivative ∂/∂x − = ∂/∂x + . Many more useful relations can be found inRef. [48]. – 33 – Free fermion fields decomposed
Consider the Lagrangian of a free fermion of mass m , L ( x ) = ¯ ψ ( x ) (cid:104) i (cid:16) → ∂ µ − ← ∂ µ (cid:17) γ µ − m (cid:105) ψ ( x ) . (B.1)The Dirac equations of motion for the fermion and anti-fermion fields are: (cid:16) iγ µ → ∂ µ − m (cid:17) ψ ( x ) = 0 , ¯ ψ ( x ) (cid:16) iγ µ ← ∂ µ + m (cid:17) = 0 . (B.2)In order to express the Lagrangian in null-plane coordinates such that the null-plane dis-persion relation is recovered, the fermion field is decomposed into two components, ψ = Π + ψ + Π − ψ ≡ ψ + + ψ − , (B.3)where the projection operator is defined as Π ± = γ ∓ γ ± , with γ + ≡ γ · n = √ ( γ + γ ) , γ − ≡ γ · n ∗ = √ ( γ − γ ) . (B.4)Application of the projection operator to the Dirac equation then gives2 i → ∂ + ψ − = (cid:16) − i γ r → ∂ r + m (cid:17) γ + ψ + , i ψ †− ← ∂ + = ψ † + γ − (cid:16) iγ r ← ∂ r − m (cid:17) , (B.5)which reveals that the ψ − field is non-dynamical. One can solve for ψ − by inverting thelongitudinal coordinate derivative operator to give ψ − = 12 i → ∂ + (cid:16) − i γ r → ∂ r + m (cid:17) γ + ψ + , ψ †− = ψ † + γ − (cid:16) i γ r ← ∂ r − m (cid:17) i ← ∂ + , (B.6)where (1 /∂ + ) ∂ + = ∂ + (1 /∂ + ) = 1. An explicit representation of this operator can be takenas: (cid:18) ∂ + (cid:19) f (cid:0) x + , x − , x ⊥ (cid:1) = 14 (cid:90) + ∞−∞ dy − (cid:15) (cid:0) x − − y − (cid:1) f (cid:0) x + , y − , x ⊥ (cid:1) , (B.7)where (cid:15) ( z ) = − , , x > , = 0 , <
0, respectively. Now, using eq. B.3 and the constraintequation, eq. B.6, gives the null-plane free-fermion Lagrangian,˜ L ( x ) = − ψ † + ( x ) (cid:3) + m √ i∂ + ψ + ( x ) , (B.8)where (cid:3) ≡ ∂ + ∂ − − ∂ r ∂ r .It is useful to list the Poincar´e generators in the free fermion theory. We take T µν = − g µν L + i ψγ ν ↔ ∂ µ ψ (B.9)as the free-fermion energy-momentum tensor. The free-fermion Poincar´e generators arethen obtained via ˜ P µ = (cid:90) dx − d x ⊥ T µ + ; (B.10)˜ M µν = (cid:90) dx − d x ⊥ (cid:0) x µ T ν + − x ν T µ + + ¯ ψ { γ + , σ µν } ψ (cid:1) , (B.11)– 34 –here σ µν = i [ γ µ , γ ν ] /
2. The free-fermion stability group generators are [14]: P r = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) ∂ r ψ + ( x ) ; (B.12) P + = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) ∂ + ψ + ( x ) ; (B.13) E r = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) (cid:0) x r ∂ + − x + ∂ r (cid:1) ψ + ( x ) ; (B.14) K = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) (cid:104) − x + ∂ + (cid:0) − ∂ r ∂ r + m (cid:1) − x − ∂ + − (cid:105) ψ + ( x );(B.15) J = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) (cid:15) rs (cid:0) x r ∂ s + γ r γ s (cid:1) ψ + ( x ) , (B.16)and the Hamiltonians are: P − = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) 12 ∂ + (cid:0) − ∂ r ∂ r + m (cid:1) ψ + ( x ) ; (B.17) F r = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) (cid:104) − x r ∂ + (cid:0) − ∂ r ∂ r + m (cid:1) − x − ∂ r (B.18) − γ r ∂ + ( − γ s ∂ s + im ) (cid:105) ψ + ( x ) . (B.19)It is clear that the null-plane dispersion relation, eq. 2.5, is correctly reproduced by eq. B.17.The dynamical fermion field ψ + can be expressed in momentum space as ψ + ( x ) = (cid:88) λ = ↑↓ (cid:90) dk + d k ⊥ k + (2 π ) (cid:110) b λ ( k + , k ⊥ ) u + ( k, λ )e − ik · x + d † λ ( k + , k ⊥ ) v + ( k, λ )e ik · x (cid:111) , (B.20)where b λ ( k + , k ⊥ ) destroys a fermion and d † λ ( k + , k ⊥ ) creates an antifermion. This decom-position is meaningful only on the initial surface, x + = 0, where the fermions are free. Thecreation/destruction operators satisfy the anti-commutation relations { b λ ( k + , k ⊥ ) , b † λ (cid:48) ( k (cid:48) + , k (cid:48)⊥ ) } = 2 k + (2 π ) δ ( k + − k (cid:48) + ) δ ( k ⊥ − k (cid:48)⊥ ) δ λλ (cid:48) ; (B.21)[ d λ ( k + , k ⊥ ) , d † λ (cid:48) ( k (cid:48) + , k (cid:48)⊥ ) ] = 2 k + (2 π ) δ ( k + − k (cid:48) + ) δ ( k ⊥ − k (cid:48)⊥ ) δ λλ (cid:48) . (B.22)which in turn imply that the fermion field ψ + satisfies { ψ + ( x ) , ψ † + ( y ) }| x + = y + = √ Π + δ ( x − − y − ) δ ( x ⊥ − y ⊥ ) . (B.23)The solutions of the free Dirac equation in the chiral representation of the gammamatrices are [5]: u ( k, ↑ ) = 12 / √ k + √ k + k ⊥ m , u ( k, ↓ ) = 12 / √ k + m − ¯ k ⊥ √ k + ; (B.24) v ( k, ↑ ) = 12 / √ k + − m − ¯ k ⊥ √ k + , v ( k, ↓ ) = 12 / √ k + √ k + k ⊥ − m , (B.25)– 35 –here k ⊥ ≡ k + ik and ¯ k ⊥ ≡ k − ik . Projecting out the dynamical spinors gives u + ( k, ↑ ) = Π + u ( k, ↑ ) = 2 / √ k + = v + ( k, ↓ ) = Π + v ( k, ↓ ) ; (B.26) u + ( k, ↓ ) = Π + u ( k, ↓ ) = 2 / √ k + = v + ( k, ↑ ) = Π + v ( k, ↑ ) , (B.27)which leads to the eigenvalue equations, u † + ( k, λ ) γ u + ( k, λ ) = u † + ( k, λ )2Σ u + ( k, λ ) = 2 λ √ k + ; (B.28) v † + ( k, λ ) γ v + ( k, λ ) = v † + ( k, λ )2Σ v + ( k, λ ) = − λ √ k + , (B.29)where Σ ≡ γ γ /
2. The relation between chirality and helicity in the null-plane formula-tion arises from these relations which arise from the fact that each of the fields has only asingle non-vanishing component. Now it is a straightforward matter to express the Poincar´egenerators in the momentum-space representation. For instance, comparing eqs. 2.30, 2.31,and B.16 gives the free-fermion helicity operator, J = i √ (cid:90) dx − d x ⊥ ψ † + ( x ) Σ ψ + ( x ) , (B.30)which, using eqs. B.20 and B.29, is found to have the momentum-space representation J = (cid:88) λ = ↑↓ λ (cid:90) dk + d k ⊥ k + (2 π ) (cid:110) b † λ ( k + , k ⊥ ) b λ ( k + , k ⊥ ) + d † λ ( k + , k ⊥ ) d λ ( k + , k ⊥ ) (cid:111) . (B.31)This operator explicitly counts the helicity of the fermions and the antifermions. C Free gauge fields decomposed
Consider the Lagrangian of a free gluon field, L ( x ) = − F aµν ( x ) F µνa ( x ) . (C.1)The equation of motion is D abµ F µνb = 0 , (C.2)where D abµ = δ ab ∂ µ + g f acb A cµ , where here f acb are SU (3) structure constants. The gaugepotential can be expressed in null-plane coordinates as A µ = ( A + , A , A − ) (C.3)– 36 –here A + = n · A , A − = n ∗ · A and A = ( A , A ). Working in light-cone gauge, A + = 0,one finds ∂ + A − a = − ∂ + D abr ∂ + A rb . (C.4)Therefore A − a is non-dynamical and can be integrated out, giving L ( x ) = − F rsa F rsa + (cid:0) ∂ + A ra (cid:1) (cid:0) ∂ − A ra (cid:1) − (cid:18) ∂ + D rab ∂ + A rb (cid:19) . (C.5)The light-cone gauge does not fix the gauge entirely and therefore to eliminate all re-dundancy one should assign a boundary condition to the transverse gauge field; e.g. A ra ( x + , x ⊥ , x − = ∞ ) = 0. D SU ( N ) conventions The fundamental representation SU ( N ) generators T α with α = 1 , . . . , N − T α , T β ] = i f αβγ ; (D.1) { T α , T β } = 1 N δ αβ + d αβγ T γ , (D.2)where is the N × N unit matrix, and hence are normalized such that Tr( T α T β ) = δ αβ / f αµν f βµν = N δ αβ ; (D.3) d αµν d βµν = N − N δ αβ . (D.4)An additional useful relation is: f αβν f γµν = 2 N ( δ αγ δ βµ − δ αµ δ βγ ) + d αγν d βµν − d βγν d αµν . (D.5)– 37 – eferences [1] P. A. Dirac, Forms of Relativistic Dynamics , Rev.Mod.Phys. (1949) 392–399.[2] S. Weinberg, Dynamics at infinite momentum , Phys.Rev. (1966) 1313–1318.[3] L. Susskind,
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