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Brodsky, Stanley J. Gauge Theories on the Light-Front Brazilian Journal of Physics, vol. 34, núm. 1A, march, 2004, pp. 157-165 Sociedade Brasileira de Física Sâo Paulo, Brasil

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Gauge Theories on the Light-Front

Stanley J. Brodsky Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, USA

Received on 10 February, 2003.

The light-front of gauge theories in light-cone gauge provides a frame-independent wavefunction representation of relativistic bound states, simple forms for current matrix elements, explicit unitary, and a trivial . The light-front Hamiltonian form of QCD provides an alternative to lattice for the computation of nonperturbative quantities such as the hadronic spectrum and the corresponding eigenfunctions. In the case of the electroweak theory, spontaneous symmetry breaking is represented by the appearance of zero modes of the Higgs field. Light-front quantization then leads to an elegant ghost-free theory of massive gauge particles, automatically incorporating the Lorentz and ’t Hooft conditions, as well as the Goldstone boson equivalence theorem.

+ − 2 1 Introduction The HLC = P P −P⊥, the “light-cone Hamilto- nian”, is frame-independent. This can in principle be solved One of the challenges of relativistic quantum field theory by diagonalizing the matrix hn|HLC |mi on the free Fock is to compute the wavefunctions of bound states such as basis: [2] the amplitudes which determine the and sub- X 2 structure of in . In light- hn|HLC |mi hm|ψi = M hn|Ψi . (2) front quantization [1], one fixes the initial boundary con- m ditions of a composite system as its constituents are inter- 2 0 cepted by a single light-wave evaluated on the hyperplane The eigenvalues {M } of HLC = HLC + VLC give the x+ = t + z/c. The light-front quantization of QCD pro- squared invariant of the bound and continuum spec- vides a frame-independent, quantum-mechanical represen- trum of the theory. The light-front Fock space is the eigen- tation of a at the amplitude level, capable of en- states of the free light-front Hamiltonian; i.e., it is a Hilbert coding its multi-quark, hidden-color and gluon , space of non-interacting and , each of which m2+k2 helicity, and flavor correlations in the form of universal 2 2 − ⊥ satisfy k = m and k = k+ ≥ 0. The projections process-independent and frame-independent hadron wave- {hn|Ψi} of the eigensolution on the n-particle Fock states functions [2]. Remarkably, quantum fluctuations of the vac- provide the light-front wavefunctions. Thus solving a quan- uum are absent if one uses light-front time to quantize the tum field theory is equivalent to solving a coupled many- system, so that matrix elements such as the electromagnetic body quantum mechanical problem: form factors only depend on the currents of the constituents " # Z described by the light-cone wavefunctions. The degrees of Xn 2 2 X 2 m + k⊥ 0 freedom associated with vacuum phenomena such as spon- M − ψn = hn|VLC |n i ψn0 (3) xi taneous symmetry breaking in the Higgs model have their i=1 n0 counterpart in light-front k+ = 0 zero modes of the fields. where the convolution and sum is understood over the Fock In Dirac’s “Front Form” [1], the generator of light-front number, transverse momenta, plus momenta, and helicity of time translations is P − = i ∂ . Boundary conditions are set ∂τ the intermediate states. Light-front wavefunctions are also x x− = z − ct on the transverse plane labelled by ⊥ and . related to momentum-space Bethe-Salpeter wavefunctions P − Given the Lagrangian of a quantum field theory, can be by integrating over the relative momenta k− = k0 − kz constructed as an operator on the Fock basis, the eigenstates since this projects out the dynamics at x+ = 0. of the free theory. Since each particle in the Fock basis is The light-front quantization of gauge theory can be most k2 +m2 − 0 3 ⊥ + on its shell, k ≡ k − k = k+ , and its conveniently carried out in the light-cone gauge A = 0 1 + − 0 z − k = 2 (k + k ) is positive, only particles with positive A + A = 0. In this gauge the A field becomes a depen- momenta k+ ≡ k0 + k3 ≥ 0 can occur in the Fock basis. dent degree of freedom, and it can be eliminated from the + P + Since the total plus momentum P = n kn is conserved, Hamiltonian in favor of a set of specific instantaneous light- the light-cone vacuum cannot have any particle content. front time interactions. In fact in QCD(1 + 1) theory, this The Heisenberg equation on the light-front is instantaneous interaction provides the confining linear x− interaction between quarks. In 3 + 1 dimensions, the trans- 2 ⊥ HLC | Ψi = M | Ψi . (1) verse field A propagates massless -one gluon quanta 158 Stanley J. Brodsky with polarization vectors [3] which satisfy both the gauge orbital angular momenta + condition ²λ = 0 and the Lorentz condition k · ² = 0. Ã ! LF quantization is especially useful for quantum chro- ∂ ∂ lz = −i k1 − k2 (5) modynamics, since it provides a rigorous extension of j j ∂k2 j ∂k1 many-body to relativistic bound states: j j the quark, and gluon momenta and spin correlations of a derives from the n − 1 relative momenta. This excludes hadron become encoded in the form of universal process- the contribution to the orbital due to the independent, Lorentz-invariant wavefunctions [4]. For ex- motion of the center of mass, which is not an intrinsic prop- ample, the eigensolution of a in QCD, projected on erty of the hadron. The numerator structure of the light- the eigenstates {| ni} of the free Hamiltonian HQCD(g = LC front wavefunctions is in large part determined by the angu- 0) τ = t − z/c at fixed with the same global quantum num- lar momentum constraints. Thus wavefunctions generated bers, has the expansion: by perturbation theory provide a template for the numerator ¯ E structure of nonperturbative light-front wavefunctions. ¯ + ~ ¯ΨM ; P , P⊥, λ = Dae Sung Hwang, Bo-Qiang Ma, Ivan Schmidt, and I X Z 2 [5] have shown that the light-front wavefunctions generated n d k⊥idxi Πi=1 √ 3 by the radiative corrections to the in QED provide xi16π n≥2,λi a simple system for understanding the spin and angular mo-   Ã ! mentum decomposition of relativistic systems. This pertur- Xn Xn 3 (2) ~ bative model also illustrates the interconnections between ×16π δ 1 − xj δ k⊥` (4) Fock states of different particle number. The model is pat- j ` terned after the quantum structure which occurs in the one- ¯ E ¯ + ~ ~ ~ loop Schwinger α/2π correction to the electron magnetic × ¯n; xiP , xiP⊥ + k⊥i, λi ψn/M (xi, k⊥i, λi). 1 moment [6]. In effect, we can represent a spin- 2 system 1 as a composite of a spin- 2 fermion and spin-one vector The set of light-front wavefunctions {ψn/M } rep- boson. A similar model has been used to illustrate the ma- resent the ensemble of quark and gluon states possible when trix elements and evolution of light-front helicity and orbital the meson is intercepted at the light-front. The light-front angular momentum operators [7]. This representation of a momentum fractions x = k+/P + = (k0 + kz)/(P 0 + P z) P i i πP i composite system is particularly useful because it is based n ~ n ~ ~ with i=1 xi = 1 and k⊥i with i=1 k⊥i = 0⊥ represent on two constituents but yet is totally relativistic. We can then 2 2 the relative momentum coordinates of the QCD constituents explicitly compute the form factors F1(q ) and F2(q ) of and are independent of the total momentum of the state. the electromagnetic current and the various contributions to Remarkably, the scalar light-front wavefunctions the form factors A(q2) and B(q2) of the energy-momentum ~ ψn/p(xi, k⊥i, λi) are independent of the ’s momen- tensor. + 0 z tum P = P + P , and P⊥. (The light-cone spinors and Recently Ji, Ma, and Yuan [8] have derived perturbative polarization vectors multiplying ψn/p are functions of the QCD counting rules for light-front wavefunctions with gen- absolute coordinates.) Thus once one has solved for the eral values of orbital angular momentum which constrain light-front wavefunctions, one can compute hadron matrix their form at large transverse momentum. elements of currents between hadronic states of arbitrary momentum. The actual physical transverse momenta are ~ ~ z ~p⊥i = xiP⊥ + k⊥i. The λi label the light-front spin S pro- 3 Applications of Light-Front Wave- jections of the quarks and gluons along the quantization z direction. The spinors of the light-front formalism automat- functions ically incorporate the Melosh-Wigner rotation. The physical gluon polarization vectors ²µ(k, λ = ±1) are specified in Matrix elements of spacelike currents such as spacelike elec- light-cone gauge by the conditions k ·² = 0, η ·² = ²+ = 0. tromagnetic form factors have an exact representation in The parton degrees of freedom are thus all physical; there terms of simple overlaps of the light-front wavefunctions in are no ghost or negative metric states. momentum space with the same xi and unchanged parton number n [9, 10, 6]. The Pauli form factor and anomalous moment are spin-flip matrix elements of j+ and thus connect 2 Properties of Light-Front Wave- states with ∆Lz = 1. Thus, these quantities are nonzero only if there is nonzero orbital angular momentum of the functions quarks in the proton. The Dirac form factor is diagonal in 2 Lz and is typically dominated at high Q by highest states An important feature of the light-front formalism is that the with the highest orbital angular momentum. projection Jz of the total is kinematical and conserved. Each The formulas for electroweak current matrix elements of light-front Fock state component thus satisfies the angular j+ can be easily extended to the T ++ coupling of gravi- z Pn z Pn−1 z momentum sum rule: J = i=1 Si + j=1 lj . The sum- tons. In, fact, one can show that the anomalous gravito- z mation over Si represents the contribution of the intrinsic magnetic moment B(0), analogous to F2(0) in electro- spins of the n Fock state constituents. The summation over magnetic current interactions, vanishes identically for any Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 159 system, composite or elementary [5]. This important fea- target spectators, which is usually assumed to be an irrele- ture, which follows in general from the equivalence princi- vant gauge artifact, actually affects the leading-twist struc- ple [11, 12, 13, 14, 15], is obeyed explicitly in the light-front ture functions in a profound way. The diffractive scattering formalism [5]. of the fast outgoing quarks on spectators in the target in turn The light-front Fock representation is specially advan- causes shadowing in the DIS cross section. Thus the de- tageous in the study of exclusive B decays. For example, pletion of the nuclear structure functions is not intrinsic to we can write down an exact frame-independent representa- the of the nucleus, but is a coherent effect tion of decay matrix elements such as B → D`ν from the arising from the destructive interference of diffractive chan- overlap of n0 = n parton conserving wavefunctions and the nels induced by final-state interactions. Thus the shadowing overlap of n0 = n − 2 from the of a quark- corrections related to the Gribov-Glauber mechanism, the antiquark pair in the initial wavefunction [16]. The off- interference effects of leading twist diffractive processes in diagonal n+1 → n−1 contributions give a new perspective nuclei are separate effects in deep inelastic scattering, are for the physics of B-decays. A semileptonic decay involves not computable from the wavefunctions of the not only matrix elements where a quark changes flavor, but target nucleon or nucleus. Similarly, the effective pomeron also a contribution where the leptonic pair is created from distribution of a hadron is not derived from its light-front the annihilation of a qq0 pair within the Fock states of the wavefunction and thus is not a universal property. initial B wavefunction. The semileptonic decay thus can Measurements from the HERMES and SMC collabo- occur from the annihilation of a nonvalence quark-antiquark rations show a remarkably large single-spin asymmetry in pair in the initial hadron. semi-inclusive pion leptoproduction γ∗(q)p → πX when The “handbag” contribution to the leading-twist off- the proton is polarized normal to the photon-to-pion pro- forward parton distributions measured in deeply virtual duction plane. Recently, Hwang, Schmidt, and I [22] have has a similar light-front wavefunction shown that final-state interactions from gluon exchange be- representation as overlap integrals of light-front wavefunc- tween the outgoing quark and the target spectator system tions [17, 18]. lead to single-spin asymmetries in deep inelastic - proton scattering at leading twist in perturbative QCD; i.e., The distribution amplitudes φ(xi,Q) which appear in the rescattering corrections are not power-law suppressed at factorization formulae for hard exclusive processes are the 2 valence LF Fock wavefunctions integrated over the relative large photon virtuality Q at fixed xbj. The existence of such single-spin asymmetries requires a phase difference between transverse momenta up to the resolution scale Q [3]. These z 1 quantities specify how a hadron shares its longitudinal mo- two amplitudes coupling the proton target with Jp = ± 2 to mentum among its valence quarks; they control virtually all the same final-state, the same amplitudes which are neces- exclusive processes involving a hard scale Q, including form sary to produce a nonzero proton anomalous magnetic mo- factors, Compton scattering and photoproduction at large ment. The single-spin asymmetry which arises from such momentum transfer, as well as the decay of a heavy hadron final-state interactions does not factorize into a product of into specific final states [19, 20]. distribution function and fragmentation function, and it is not related to the transversity distribution δq(x, Q) which The quark and gluon probability distributions q (x, Q) i correlates transversely polarized quarks with the spin of the and g(x, Q) of a hadron can be computed from the ab- transversely polarized target nucleon. These effects high- solute squares of the light-front wavefunctions, integrated light the unexpected importance of final- and initial-state in- over the transverse momentum. All helicity distributions are teractions in QCD observables—they lead to leading-twist thus encoded in terms of the light-front wavefunctions. The single-spin asymmetries, diffraction, and nuclear shadow- DGLAP evolution of the structure functions can be derived ing, phenomena not included in the wavefunction of the tar- from the high k properties of the light-front wavefunctions. ⊥ get. Thus given the light-front wavefunctions, one can compute [3] all of the leading twist helicity and transversity distribu- tions measured in polarized deep inelastic lepton scattering. Similarly, the transversity distributions and off-diagonal he- 4 Measurements of Light-Front licity convolutions are defined as a density matrix of the Wavefunctions light-front wavefunctions. However, it is not true that the leading-twist structure It is possible to measure the light-front wavefunctions of a 2 functions Fi(x, Q ) measured in deep inelastic lepton scat- relativistic hadron by diffractively dissociating it into jets tering are identical to the quark and gluon distributions. in high-energy hadron-nucleus collisions such as πA → For example, it is usually assumed, following the parton jetjetA0. Only the configurations of the incident hadron model, that the F2 structure function measured in neutral which have small transverse size and minimal color dipole current deep inelastic lepton scattering is at leading or- moments can traverse the nucleus with minimal interactions 2 2 P 2 2 der in 1/Q simply F2(x, Q ) = q eqxq(x, Q ), where and leave it intact. The forward diffractive amplitude is thus 2 x = xbj = Q /2p · q and q(x, Q) can be computed from coherent over the entire nuclear volume and proportional to the absolute square of the proton’s light-front wavefunction. nuclear number A. The fractional momentum distribution Recent work by Hoyer, Marchal, Peigne, Sannino, and my- of the jets is correlated with the valence quarks’ light-cone self shows that this standard identification is wrong [21]. momentum fractions xi. [23, 24, 25, 26]. The QCD mecha- Gluon exchange between the fast, outgoing partons and the nisms for hard diffractive dissociation can be more compli- 160 Stanley J. Brodsky cated in the case of proton targets. A review and references tensor is related in the heavy-quark limit to the forward +α +β 2 µν is given in Ref. [27]. matrix element hp|trc(G G Gαβ)/mQ|pi, where G The fact that Fock states of a hadron with small particle is the gauge field strength tensor. Diagrammatically, this number and small impact separation have small color dipole can be described as a heavy quark loop in the proton self- moments and weak hadronic interactions is a remarkable energy with four gluons attached to the light, valence quarks. manifestation of the gauge structure of QCD. It is the basis Since the non-Abelian commutator [Aα,Aβ] is involved, the for the predictions for “color transparency” in hard quasi- heavy quark pairs in the proton wavefunction are necessarily exclusive [28, 29] and diffractive reactions [24, 25, 26]. The in a color-octet state. It follows from dimensional analysis E791 experiment at has verified the nuclear num- that the momentum fraction carried by the QQ pair scales 2 2 ber scaling predictions and have thus provided a remark- as k⊥/mQ where k⊥ is the typical momentum in the hadron able confirmation of this consequence of QCD color trans- wave function. In contrast, in the case of Abelian theo- parency [23]. The new EVA spectrometer experiment E850 ries, the contribution of an intrinsic, heavy lepton pair to 4 at Brookhaven has also reported striking effects of color the bound state’s structure first appears in O(1/mL). transparency in quasi-elastic proton-proton scattering in nu- The presence of intrinsic charm quarks in the B wave clei [30]. function provides new mechanisms for B decays. For ex- The CLEO collaboration [31] has verified the scaling ample, Chang and Hou have considered the production of and angular predictions for the photon-meson to meson form final states with three charmed quarks such as B → J/ψDπ 2 ∗ 0 ∗ factor Fγπ0 (q ) which is measured in γ γ → π reac- and B → J/ψD [41]; these final states are difficult to re- tions. The results are in close agreement with the scaling alize in the valence model, yet they occur naturally when and normalization of the PQCD predictions [32], provided the b quark of the intrinsic charm Fock state | bucci decays that the pion distribution amplitude φπ(x, Q) is close to the via b → cud. Susan Gardner and I have shown that the x(1−x) form, the asymptotic solution to the evolution equa- presence of intrinsic charm in the hadrons’ light-front wave tion. The pion light-front momentum distribution measured functions, even at a few percent level, provides new, compet- in diffractive dijet production in pion-nucleus collisions by itive decay mechanisms for B decays which are nominally the E791 experiment [23] has a similar√ form [33]. Data [34] CKM-suppressed [42]. Similarly, Karliner and I [43] have for γγ → π+π+ + K+K− at W = s > 2.5 GeV are also shown that the transition J/ψ → ρπ can occur by the rear- in agreement with the perturbative QCD predictions. More- rangement of the cc from the J/ψ into the | qqcci intrinsic over, the angular distribution shows the expected transition charm Fock state of the ρ or π. On the other hand, the over- to the predicted QCD form as W is raised. A compilation of lap rearrangement integral in the decay ψ0 → ρπ will be the two-photon data has been given by Whalley [35]. Mea- suppressed since the intrinsic charm Fock state radial wave- surements of the reaction γγ → π0π0 are highly sensitive to function of the light hadrons will evidently not have nodes in the shape of the pion distribution amplitude. The perturba- its radial wavefunction. This observation provides a natural tive QCD predictions [32] for this channel contrast strongly explanation of the long-standing puzzle [44] why the J/ψ with model predictions based on the QCD Compton hand- decays prominently to two-body pseudoscalar-vector final bag diagram [36]. states, breaking hadron helicity conservation [45], whereas the ψ0 does not. 5 Higher Particle-Number Fock 6 Light-Front Quantization of QCD States Quantum field theories are usually quantized at fixed “in- The light-front Fock state expansion of a hadron contains stant” time t. The resulting Hamiltonian theory is com- fluctuations with an arbitrary number of quark and gluon plicated by the dynamical nature of the vacuum state and partons. The higher Fock states of the light hadrons describe the fact that relativistic boosts are not kinematical but in- the sea quark structure of the deep inelastic structure func- volve interactions. The calculation of even the simplest cur- tions, including “intrinsic” strangeness and charm fluctua- rent matrix elements requires the computation of amplitudes tions specific to the hadron’s structure rather than gluon sub- where the current interacts with particles resulting from the structure [37, 38]. The maximal contributionP of an intrinsic fluctuations of the vacuum. All of these problems are dra- x ' m / m m = pheavy quark occurs at Q ⊥Q i ⊥ where ⊥ matically alleviated when one quantizes quantum field theo- 2 2 m + k⊥; i.e. at large xQ, since this minimizes the in- ries at fixed light-cone time τ. A review of the development 2 variant mass Mn. The measurements of the charm structure of light-front quantization of QCD and other quantum field function by the EMC experiment are consistent with intrin- theories is given in Ref. [2]. sic charm at large x in the nucleon with a probability of order Prem Srivastava and I [46] have presented a new sys- 0.6±0.3% [39] which is consistent with the recent estimates tematic study of light-front-quantized gauge theory in light- based on instanton fluctuations [40]. Franz, Polyakov, and cone gauge using a Dyson-Wick S-matrix expansion based Goeke have analyzed the properties of the intrinsic heavy- on light-front-time-ordered products. The Dirac bracket quark fluctuations in hadrons using the operator-product ex- method is used to identify the independent field degrees of pansion [40]. For example, the light-cone momentum frac- freedom [47]. In our analysis one imposes the light-cone tion carried by intrinsic heavy quarks in the proton xQQ as gauge condition as a linear constraint using a Lagrange mul- measured by the T ++ component of the energy-momentum tiplier, rather than a quadratic form. We then find that the Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 161

LF-quantized free gauge theory simultaneously satisfies the where covariant gauge condition ∂ ·A = 0 as an operator condition as well as the LC gauge condition. The gluon propagator has + i + ij j cµ the form j a = ψ γ (ta) ψ + fabc(∂−Abµ)A . (10) Z ­ ® iδab D (k) 0| T (Aa (x)Ab (0)) |0 = d4k e−ik·x µν µ ν (2π)4 k2 + i² (6) The constants in the non-Abelian theory where we have defined have been shown [46] to satisfy the identity Z1 = Z3 at one- loop order, as expected in a theory with only physical gauge 2 nµkν + nν kµ k degrees of freedom. The renormalization factors in the D (k) = D (k) = −g + − n n . µν νµ µν (n · k) (n · k)2 µ ν light-cone gauge are independent of the reference direction (7) nµ. The QCD β function computed in the noncovariant LC Here nµ is a null four-vector, gauge direction, whose com- gauge agrees with the conventional theory result [48, 49]. + µ µ ponents are chosen to be nµ = δµ , n = δ −. Note also Dimensional regularization and the Mandelstam-Leibbrandt prescription [51, 50, 52] for LC gauge were used to define λ ⊥ Dµλ(k)D ν (k) = Dµ⊥(k)D ν (k) = −Dµν (k), (8) the Feynman loop integrations [53]. There are no Faddeev- µ µ Popov or Gupta-Bleuler ghost terms. k Dµν (k) = 0, n Dµν (k) ≡ D−ν (k) = 0, µν 0 µρ 0 Dλµ(q) D (k) Dνρ(q ) = −Dλµ(q)D (q ). The running coupling constant and QCD β function have also been computed at one loop in the doubly-transverse 2 The gauge field propagator i Dµν (k)/(k + i²) is trans- light-cone gauge [46]. It is also possible to effectively quan- verse not only to the gauge direction nµ but also to kµ, i.e., tize QCD using light-front methods in covariant Feynman it is doubly-transverse. Thus D represents the polarization gauge [54]. sum over physical propagating modes. The last term propor- tional to nµnν in the gauge propagator does not appear in It is well-known that the light-cone gauge itself is not the usual formulations of light-cone gauge. However, in tree completely defined until one specifies a prescription for graph calculations it cancels against instantaneous gluon ex- the poles of the gauge propagator at n · k = 0. The change contributions. Mandelstam-Liebbrandt prescription has the advantage of The remarkable properties of (the projector) Dνµ greatly preserving causality and analyticity, as well as leading to simplifies the computations of loop amplitudes. For exam- proofs of the renormalizability and unitarity of Yang-Mills ple, the coupling of gluons to propagators carrying high mo- theories [55]. The ghosts which appear in association with menta is automatic. In the case of tree graphs, the term pro- the Mandelstam-Liebbrandt prescription from the single portional to nµnν cancels against the instantaneous gluon poles have vanishing residue in absorptive parts, and thus exchange term. However, in the case of loop diagrams, the do not disturb the unitarity of the theory. separation needs to be maintained so that one can identify A remarkable advantage of light-front quantization is the correct one-particle-irreducible contributions. The ab- that the vacuum state | 0i of the full QCD Hamiltonian ev- sence of collinear divergences in irreducible diagrams in the idently coincides with the free vacuum. The light-front light-cone gauge greatly simplifies the leading-twist factor- vacuum is effectively trivial if the interaction Hamiltonian ization of soft and hard gluonic corrections in high momen- applied to the perturbative vacuum is zero. Note that all tum transfer inclusive and exclusive reactions [3] since the particles in the Hilbert space have positive energy k0 = numerators associated with the gluon coupling only have 1 (k+ + k−), and thus positive light-front k±. Since the transverse components. 2 P plus momenta k+ is conserved by the interactions, the The interaction Hamiltonian of QCD in light-cone gauge i perturbative vacuum can only couple to states with particles can be derived by systematically applying the Dirac bracket in which all k+ = 0; i.e., so called zero-mode states. Bas- method to identify the independent fields [46]. It contains i setto and collaborators [56] have shown that the computa- the usual Dirac interactions between the quarks and glu- tion of the spectrum of QCD(1 + 1) in equal time quantiza- ons, the three-point and four-point gluon non-Abelian inter- tion requires constructing the full spectrum of non perturba- actions plus instantaneous light-front-time gluon exchange tive contributions (instantons). In contrast, in the light-front and quark exchange contributions quantization of gauge theory, where the k+ = 0 singular- i µ ij j ity of the instantaneous interaction is defined by a simple Hint = −g ψ γ Aµ ψ infrared regularization, one obtains the correct spectrum of g abc a a bµ cν + f (∂µA ν − ∂ν A µ)A A QCD(1+1) without any need for vacuum-related contribu- 2 tions. g2 + f abcf adeA AdµA Aeν 4 bµ cν Zero modes of auxiliary fields are necessary to distin- 2 guish the theta-vacua of massless QED(1+1) [57, 58, 59], g i + ⊥0 ij 1 ⊥ jk k − ψ γ (γ A⊥0 ) (γ A⊥) ψ or to represent a theory in the presence of static external 2 i∂− boundary conditions or other constraints. Zero-modes pro- 2 g + 1 + vide the light-front representation of spontaneous symmetry − j a 2 j a (9) 2 (∂−) breaking in scalar theories [60]. 162 Stanley J. Brodsky

(3) (3) (3) 2 7 Light-Front Quantization of the Eeff µ = Eµ − kµ k · E /k which is identical to the usual polarization vector of a massive vector with norm Standard Model (3) (3) Eeff · Eeff = −1. Thus, unlike the conventional quanti- Prem Srivastava and I have also shown how light-front quan- zation of the Standard Model, the Goldstone particle only tization can be applied to the Glashow, Weinberg and Salam provides part of the physical longitudinal mode of the elec- (GWS) model of electroweak interactions based on the non- troweak particles. abelian gauge group SU(2)W × U(1)Y [61]. This theory In the LC gauge LF framework, the free massive gauge contains a nonabelian Higgs sector which triggers spon- fields in the electroweak theory satisfy simultaneously the ’t taneous symmetry breaking (SSB). A convenient way of Hooft conditions as an operator equation. The sum over the implementing SSB and the (tree level) three physical polarizations is given by Kµν in the front form theory was developed earlier by Srivas- X M 2 tava [62, 63, 64]. One separates the quantum fluctuation K (k) = E(α)E(α) = D (k) + n n(15) fields from the corresponding dynamical bosonic conden- µν µ ν µν (k+)2 µ ν (α) sate (or zero-longitudinal-momentum-mode) variables, be- n k + n k (k2 − M 2) fore applying the Dirac procedure in order to construct the = −g + µ ν ν µ − n n Hamiltonian formulation. The canonical quantization of LC µν (n · k) (n · k)2 µ ν gauge GWS electroweak theory in the front form can be µ 2 + derived by using the Dirac procedure to construct a self- which satisfies: k Kµν (k) = (M /k ) nν and µ ν 2 consistent LF Hamiltonian theory. This leads to an attractive k k Kµν (k) = M . The free propagator of the massive new formulation of the Standard Model of the strong and gauge field Aµ is electroweak interactions which does not break the physical vacuum and has well-controlled ultraviolet behavior. The h0|T (Aµ(x)Aν (y)) |0i = (16) Z only ghosts which appear in the formalism are the n · k = 0 i K (k) d4k µν e−i k·(x−y). modes of the gauge field associated with regulating the light- (2π)4 (k2 − M 2 + i²) cone gauge prescription. The massive gauge field propa- gator has good asymptotic behavior in accordance with a It does not have the bad high energy behavior found in the renormalizable theory, and the massive would-be Goldstone (Proca) propagator in the unitary gauge formulation, where fields can be taken as physical degrees of freedom. the would-be Nambu-Goldstone boson is gauged away. For example, consider the Abelian Higgs model. The In the limit of vanishing mass of the vector boson, the interaction Lagrangian is gauge field propagator goes over to the doubly transverse µ µ 1 gauge, (n Dµν (k) = k Dµν (k) = 0), the propagator L = − F F µν + |D φ|2 − V (φ†φ) (11) found [46] in QCD. The numerator of the gauge prop- 4 µν µ agator Kµν (k) also has important simplifying properties, where similar to the ones associated with the projector Dµν (k). Dµ = ∂µ + ieAµ, (12) The transverse polarization vectors for massive or mass- less vector boson may be taken to be Eµ (k) ≡ −Dµ (k), and (⊥) ⊥ 2 † † 2 whereas the non-transverse third one in the massive case is V (φ) = µ φ φ + λ(φ φ) , (13) (3) found to be parallel to the LC gauge direction Eµ (k) = 2 + with µ < 0, λ > 0. The complex scalar field φ is decom- −(M/k ) nµ. Its projection along the direction trans- posed as verse to kµ shares the spacelike vector property carried by Eµ (k). The Goldstone boson or electroweak equivalence 1 1 (⊥) φ(x) = √ v + ϕ(x) = √ [v + h(x) + iη(x)] (14) theorem becomes transparent in the LF formulation. 2 2 The interaction Hamiltonian for the Abelian Higgs + where v is the k+ = 0 zero mode determined by the min- model in LC gauge A = 0, is found to be 2 imum of the potential: v2 = − µ , h(x) is the dynamical λ − Hint = Lint Higgs field, and η(x) is the Nambu-Goldstone field. The 2 µ e mh 2 2 quantization procedure determines ∂·A = MR, the ’t Hooft = e M AµA h − (η + h ) h condition. One can now eliminate the zero mode component 2 M 2 of the Higgs field v which gives masses for the fundamental µ e 2 2 µ + e(h ∂µη − η ∂µh) A + (h + η ) AµA quantized fields. The A⊥ field then has mass M = ev and 2 2 2 2 2 µ ¶ µ ¶ the Higgs field acquires mass mh = 2λv = −2µ . λ 2 2 2 e 1 + 1 + A new aspect of LF quantization, is that the third polar- − (η + h ) − j j (17) 4 2 ∂− ∂− ization of the quantized massive vector field Aµ with four µ (3) momentum k has the form Eµ = nµM/n · k. Since where jµ = (h ∂µη − η∂µ h). The last term here is the n2 = 0, this non-transverse polarization vector has zero additional quartic instantaneous interaction in the LF theory norm. However, when one includes the constrained in- quantized in the LC gauge No new instantaneous cubic inter- teractions of the Goldstone particle, the effective longitu- action terms are introduced. The massive gauge field, when dinal polarization vector of a produced vector particle is the mass is generated by the Higgs mechanism, is described Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 163

in our LC gauge framework by the independent fields A⊥ 8 Discretized Light-Front Quantiza- and η; the component A− is a dependent field. The interaction Hamiltonian of the Standard Model can tion be written in a compact form by retaining the dependent − components A and ψ− in the formulation. Its form closely If one imposes periodic boundary conditions in x− = resembles the interaction Hamiltonian of covariant theory, + except for the presence of additional instantaneous four- t + z/c, then the plus momenta become discrete: ki = 2π + 2π P point interactions. The resulting Dyson-Wick perturba- L ni,P = L K, where i ni = K [65, 66]. For a given tion theory expansion based on equal-LF-time ordering has “harmonic resolution” K, there are only a finite number of also been constructed, allowing one to perform higher-order ways positive integers ni can sum to a positive integer K. computations in a straightforward fashion. Thus at a given K, the dimension of the resulting light-front The singularities in the noncovariant pieces of the field Fock state representation of the bound state is rendered finite propagators may be defined using the causal ML prescrip- without violating Lorentz invariance. The eigensolutions of tion for 1/k+ when we employ dimensional regularization, a quantum field theory, both the bound states and continuum as was shown in our earlier work on QCD. The power- solutions, can then be found by numerically diagonalizing counting rules in LC gauge then become similar to those a frame-independent light-front Hamiltonian HLC on a fi- found in covariant gauge theory. nite and discrete momentum-space Fock basis. Solving a Spontaneous symmetry breaking is thus implemented in quantum field theory at fixed light-front time τ thus can be a novel way when one quantizes the Standard Model at fixed formulated as a relativistic extension of Heisenberg’s matrix light-front time τ = x+. In the general case, the Higgs field mechanics. The continuum limit is reached for K → ∞. This formulation of the non-perturbative light-front quan- φi(x) can be separated into two components: tization problem is called “discretized light-cone quanti- − − zation” (DLCQ) [66]. The method preserves the frame- φi(τ, x , ~x⊥) = ωi(τ, ~x⊥) + ϕ(τ, x , ~x⊥), (18) independence of the Front form. + where ωi is a classical k = 0 zero-mode field and ϕ is The DLCQ method has been used extensively for solv- the dynamical quantized field. Here i is the weak-isospin ing one-space and one-time theories [2], including applica- index. The zero-mode component is determined by solving tions to supersymmetric quantum field theories [67] and spe- the Euler-Lagrange tree-level condition: cific tests of the Maldacena conjecture [68]. There has been progress in systematically developing the computation and V 0(ω) − ∂ ∂ ω = 0. (19) i ⊥ ⊥ i renormalization methods needed to make DLCQ viable for QCD in physical spacetime. For example, John Hiller, Gary A nonzero value for ω corresponds to spontaneous symme- i McCartor, and I [69, 70, 71] have shown how DLCQ can be try breaking. The nonzero ω couples to the gauge boson and i used to solve 3+1 theories despite the large numbers of de- Fermi fields through the Yukawa interactions of the Standard grees of freedom needed to enumerate the Fock basis. A key Model. It can then be eliminated from the theory in favor of feature of our work is the introduction of Pauli Villars fields mass terms for the fundamental matter fields in the effec- to regulate the UV divergences and perform renormaliza- tive theory. The resulting masses are identical to those of tion while preserving the frame-independence of the theory. the usual Higgs implementation of spontaneous symmetry A recent application of DLCQ to a 3+1 quantum field the- breaking in the Standard Model. ory with Yukawa interactions is given in Ref. [69]. One can The generators of isospin rotations are defined from the also define a truncated theory by eliminating the higher Fock dynamical Higgs fields: states in favor of an effective potential [73, 72, 74]. Sponta- Z neous symmetry breaking and other nonperturbative effects ⊥ − Ga = −i dx dx (∂−ϕ)i(ta)ij ϕj . (20) associated with the instant-time vacuum are hidden in dy- namical or constrained zero modes on the light-front. An Note that the weak-isospin charges and the currents corre- introduction is given in Refs. [75, 57]. A review of DLCQ and its applications is given in Ref. [2]. sponding to Ga are not conserved if the zero mode ωi is nonzero since the cross terms in ϕ, and ω are missing. Thus The pion distribution amplitude has been computed us- [HLF ,Ga] 6= 0. Nevertheless, the charges annihilate the ing a combination of the discretized DLCQ method for vacuum: Ga|0 >LF = 0, since the dynamical fields ϕi have the x− and x+ light-front coordinates with a spatial lattice no support on the LF vacuum, and all quanta have positive + [76, 77, 78, 79] in the transverse directions. A finite lattice k . Thus the LF vacuum remains equal to the perturbative spacing a can be used by choosing the parameters of the ef- vacuum; it is unaffected by the occurrence of spontaneous fective theory in a region of renormalization group stability symmetry breaking. ´ + to respect the required gauge, Poincare, chiral, and contin- In effect one can interpret the k = 0 zero mode field uum symmetries. − ωi as an x -independent external field, analogous to an ap- plied constant electric or magnetic field in atomic physics. Dyson-Schwinger models [80] of hadronic Bethe- In this interpretation, the zero mode is a remnant of a Higgs Salpeter wavefunctions can also be used to predict light- field which persists from early cosmology; the LF vacuum front wavefunctions and hadron distribution amplitudes by however remains unchanged and unbroken. integrating over the relative k− momentum. 164 Stanley J. Brodsky

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