Effective potential between static sources in quenched light-front Yukawa theory

We compute a nonperturbative effective potential between two static fermions in light-front Yukawa theory as a Hamiltonian eigenvalue problem. Fermion pair production is suppressed, to make possible an exact analytic solution in the form of a coherent state of bosons that form clouds around the sources. The effective potential is essentially an interference term between individual clouds. The model is regulated with Pauli-Villars bosons and fermions, to achieve consistent quantization and renormalization of masses and couplings. This extends earlier work on scalar Yukawa theory where Pauli-Villars regularization did not play a central role. The key result is that the nonperturbative solution restores rotational symmetry even though the light-front formulation of Yukawa theory, with its preferred axis, appears antithetical to such a symmetry.


I. INTRODUCTION
The current understanding of quantum chromodynamics (QCD) as the theory of the strong interactions is that it provides for the confinement of quarks and gluons.Calculations within lattice QCD [1,2] have confirmed this [3][4][5] as have analytic calculations [6][7][8], 1 not to mention the experimental evidence for the absence of free quarks and gluons.What has not been established is the derivation of confinement within a light-front quantization of the theory [11][12][13][14][15][16].This is something that must be nonperturbative or at least an all-orders resummation of perturbation theory.
We wish to explore how such a nonperturbative light-front calculation might be done.Here we extend earlier work [17] 2 on quenched scalar Yukawa theory to include fermion sources.The basic approach is to consider the effective potential for two static sources as a function of their separation, computed from the change in the eigenenergy of the system relative to the energy of two well-separated sources.This involves renormalization of the source mass and, as we will show, renormalization of the coupling to bosons.Regularization is provided by the inclusion of Pauli-Villars (PV) fermions and bosons. 3The PV fermions provide a convenient simplification of the light-front quantization by eliminating what are known as instantaneous fermion terms from the light-front Hamiltonian; this is what makes the analytic solution possible.The PV bosons regulate the self-energy corrections to the fermion mass.
Some will be concerned that a method based on PV regularization cannot be extended to QCD, there being a general prejudice against PV regularization of nonabelian gauge theories.However, as shown in [16], a consistent PV regularization can be constructed.The key is that the definition of the gauge transformation must be extended when the Lagrangian is extended to include the PV fields.In other words, there does remain a gauge invariance of the QCD Lagrangian when PV fields are included.In addition, there is a BRST symmetry when the gauge is fixed covariantly.
Our definition of light-front coordinates [24] is to take x ± = t±z, with x + as the light-front time, and x ⊥ = (x, y) in the transverse.The conjugate light-front energy is p − = E − p z , and the light-front momentum is p = (p + ≡ E + p z , p ⊥ ≡ (p x , p y )).The mass-shell condition p 2 = m 2 becomes p − = ( p 2 ⊥ + m 2 )/p + .Further details can be found in the review [16].
For static sources, fixed in a lab frame, the eigenstates are no longer eigenstates of lightfront energy or momentum. 4The ordinary momentum is zero, including the z component.We therefore seek eigenstates of the ordinary energy with use of the operator E ≡ 1  2 (P − +P + ) but do so in terms of light-front coordinates.Such a choice is also motivated by the fact that the definition of an effective potential is in the dependence of the ordinary energy on the source separation.Most calculations in light-front-quantized theories need not make this distinction because they use a basis where the light-front momentum is held fixed.
The parameters of the Yukawa Lagrangian are renormalized by fixing the mass of the dressed fermion state at a physical value m and by requiring the effective potential to be of the standard Yukawa form − g 2 4πR e −µR , where R is the source separation, µ the renormalized boson mass, and g the physical coupling.For the quenched case considered here, the boson mass in the Lagrangian is not actually renormalized.The form of the effective potential is obtained, including its rotational symmetry.
In Sec.II we summarize the structure of light-front Yukawa theory with PV regularization and of the Hamiltonian eigenvalue problem for static fermions.The solution for a single source is developed in Sec.III and for a double source in Sec.IV.We summarize the results in Sec.V and leave some details to Appendices.

II. LIGHT-FRONT YUKAWA THEORY
The Lagrangian for PV-regulated Yukawa theory is [16] where k = i = j = 0 correspond to physical fields and positive integers to PV fields.The r k and s i are metric signatures, with r 0 = s 0 = 1 for the physical fields, r 1 = s 1 = −1 for the first PV fields, and the remainder (if any) determined by the constraints necessary for a consistent theory.The factors of β i and ξ k provide for adjustments of the relative couplings of the PV fields, with β 0 = ξ 0 = 1 as a definition of g 0 as the bare coupling for physical fields.The regularization of loops is provided by the constraints [16] k because one or the other of these combinations appears for each line in a loop when summed over the PV contributions.The r k and s i factors come from the propagator, and the ξ k and β i come from the vertices at the ends of the line.The leading UV divergence then involves these sums and is canceled by the constraints (2.2).For example, a boson line carrying momentum , and the leading term is zero.This cancellation will be seen explicitly in the next section.The nonperturbative calculation is effectively a resummation of the regulated loop expansion.
The mode expansions for the boson fields are with the nonzero commutation relations being The factor r k = ±1 fixes the sign of the metric for the field.The fermion field satisfies the Euler-Lagrange equation To separate the dynamical part ψ i+ ≡ 1 2 γ 0 γ + ψ i from the constrained part ψ i− ≡ 1 2 γ 0 γ − ψ i , we project this Euler-Lagrange equation using 1  2 γ 0 γ ± to yield the following two equations: and with γ µ = (β, β α).Multiplication of the constraint equation (2.7) by s i β i and a sum over i eliminates the interaction,6 such that the constrained part of the summed fermion field This is just the constraint equation for a free fermion.We can then construct the Hamiltonian from the free-fermion mode expansion with and The nonzero anticommutators are and s i = ±1 sets the metric.
Instantaneous fermion interactions do not appear.However, the physics of these interactions has not been lost; they are present implicitly and restored explicitly in the limit of infinite PV masses, which shrinks a PV-fermion exchange to a contact interaction [21].
The light-front Hamiltonian density is then (2.13) The Hamiltonian is specified by (2.17) with √ 2 ǫ 2s ≡ −(2s, i) a two-dimensional, transverse vector [29], where s is the spin index and i = √ −1.The integral over x has been left undone in the interaction terms, to accommodate the source wave packets introduced in the next sections.For the quenched theory, the term P − pair is, of course, dropped.Also, we will limit our work to the fermion sector, and antifermion terms in P − 0f and P − n.p. will play no role.The light-front momentum operator is P + = P + a + P + f with We can then define the ordinary energy operator In the next sections we explore eigenstates of E that are associated with one or two static fermion sources.

III. SINGLE STATIC SOURCE
We first consider a single source at ± R/2 to establish the renormalization of the fermion mass.The static fermion is described by a wave packet centered at p = (m, 0 ⊥ ) in momentum space and at x = (∓R z , ± R ⊥ /2) in coordinate space on the x + = 0 slice. 7The static fermion state with spin s and PV type i is then with the function F ± peaked at p = (m, 0 ⊥ ) and its Fourier transform ψ ± (x) peaked at (∓R z , ± R ⊥ /2).Here |0 f is the vacuum annihilated by b is .The physical and PV fermions are all static at the same location and at the same light-front momentum.The Fourier transform is defined as For the inverse, where the p + integration is limited to positive values, we take advantage of the narrow peak in F ± to extend the p + integral to −∞ For a static source at x = (∓R z , ± R ⊥ /2) we require that The common normalization is fixed by requiring the indefinite norm From this static-fermion state, we build a fermion state dressed by a cloud of bosons in a coherent state as the ansatz for the energy eigenstate: with and |0 a the vacuum annihilated by a k .Because the spin-flip terms in P − n.p. are proportional to p ⊥ and the static state requires p ⊥ to be zero, the eigenstate is diagonal in spin, and the coefficient C ± i and the functions G ± ki are independent of s.The Z ± i are normalization factors for the coherent state, given by This is then to be the solution to with E ± = m for the ground state, which is the state of interest.Each term in the sum over i is also an eigenstate of the boson annihilation operators a k , as is always the case for a coherent state: We begin with a projection of the eigenvalue problem onto a static fermion of type i This can be reduced with use of the following projections for individual terms in E = 1 2 (P − 0f + P + f + P − 0a + P + a + P − n.p. ): and Details of the reduction for the P − n.p. projection can be found in Appendix A. With the combination of all of these terms, the projected single-source eigenvalue problem becomes For this to hold, the coefficient of a † k (q) must be zero, to remove states with additional particles from the left-hand side: A slight rearrangement yields an implicit expression for The eigenvalue problem (3.16) reduces to On substitution of the expression for G ± ki this becomes with I the dimensionless self-energy integral and µ the physical mass of the boson.The constraint k r k ξ 2 k = 0 on the ξ k factors makes I finite.
This defines an n f × n f matrix problem where n f is the number of fermion types and given the overlap integrals ) The nontrivial ζ ± ij can be computed from nonlinear equations for self consistency with the solution for G ± ki , which arises in the projection of (3.18) onto G ± i |: The two equations (3.24) and (3.25) must then be solved simultaneously, with G ki in (3.25) given by (3.26).The only R dependence appears in the exponentials which cancel in (3.25), leaving the equations independent of R. Therefore, the ground state determines the physical fermion mass m = E ± independent of the source location ± R/2, and the eigenvalue problem (3.24) provides the renormalization of the bare mass m 0 implicitly by giving m as a function of m 0 .We then use these solutions to construct a solution for the two-source case in the next section.For this purpose, an explicit solution of the (3.24)-(3.25)system will not be needed.

IV. TWO STATIC SOURCES
To compute the effective potential between two sources a distance R apart, we place them at x = (∓R z , ± R ⊥ /2) and construct the eigenstate of the ordinary energy E. The effective potential is then the difference between the eigenvalue and the total rest mass 2m, with m specified by the single-source problem solved in the previous section.Following the case of scalar Yukawa theory [17], we construct an ansatz for the eigenstate as a product of single-source solutions: This is to be a solution of We proceed as before with a projection onto the static fermion states Cross terms between F + and F − do not contribute because they are proportional to and the second integral is zero from the lack of overlap between narrow wave packets centered apart.Again, the spin-flip terms of P − n.p. do not contribute, being proportional to the transverse fermion momentum p ⊥ for which the expectation value is zero.The right-hand side of (4.3) is The projected terms in E for the left-hand side are

Details of this last projection are again in
which is the two-source extension of (3.11), where a coherent state is an eigenstate of the annihilation operator.
The projected eigenvalue problem is The reduction of this eigenvalue problem is detailed in Appendix B.
From the reduction we obtain the effective potential V eff ≡ E − 2m as which is clearly rotationally invariant.We define the physical coupling g by a match to the standard form for the Yukawa potential − g 2 4πR e −µR , which implies We can, of course, have g = g 0 if we include 2 PV fermions and impose the additional constraint i s i β 2 i m i = 2m.

V. SUMMARY
In this work we have thus obtained the standard, rotationally invariant Yukawa potential as the effective potential between two static sources in quenched, light-front Yukawa theory.The effective potential comes from the interference between the two boson clouds that dress the individual sources and is computed nonperturbatively.The rotational invariance exists despite the special status for the z axis in light-front quantization.
The key to our approach is to recognize the ordinary energy as the relevant quantity, both because momentum is not conserved when sources are static and because the effective potential should be defined in terms of this energy.Light-front energy combines energy and a momentum component, making it only indirectly related.
To carry out the calculation, we have introduced Pauli-Villars fermions and bosons.The PV fermions eliminate instantaneous interaction terms which would otherwise interfere with the construction of analytic solutions.The PV bosons regulate the infinite self-energy of the sources.The couplings of these are adjusted to satisfy constraints that guarantee the regularization, the correct mass and coupling renormalizations, and the removal of instantaneous fermion interactions from the light-front Hamiltonian.The instantaneous interactions are restored in the limit of infinite PV mass.
Given the successful derivation of a rotationally invariant potential in quenched Yukawa theory, the next step to be taken is to introduce pair production and annihilation of free fermions and their own accompanying PV counterparts.(The PV fermions associated with the static sources need to be separate because they are themselves static.)With pairs included in the basis, a coherent state solution will no longer be available as the full solution, and Fock-space methods must be invoked.To have an eigenvalue problem of finite size will then require truncation, either explicitly in Fock space or in the operator sense of the lightfront coupled-cluster method [30].The effects of pairs will include renormalization of the boson mass, renormalization of the static fermion coupling, and modifications of the form of the effective potential.At short separations, these modifications will be due to the charge renormalization and the screening that takes place.At large separations, pairs provide for the Yukawa analog of string breaking.Completion of this work in Yukawa theory will provide a useful reference for the calculation of effective potentials in QED and QCD.

Appendix A: Projections for the no-pair contribution
We first consider the single-source case.Substitution of the definitions of the individual factors |F ± is , P − n.p. , and |G ± F ± ; s , as given in (3.1), (2.17), and (3.7), respectively, yields the Kronecker and Dirac deltas and the property s 2 i = 1 can be used to reduce the expression to The momentum-space wave function F ± is peaked at (m, 0 ⊥ ), which allows any factor of p + or p ′+ to be replaced with m.The two Fourier transforms of F ± can be written in terms of the spatial wave function ψ in (3.3) and the product replaced by the coordinate-space delta functions of (3.4): Integration over x then gives Use of (3.11) then yields the result given in (3.15).When there are two sources, the projection needed is Contraction of b ms (p 2 ) with each of the rightmost creation operators yields With use of the Kronecker and Dirac deltas, this becomes The remaining contractions produce The additional Kronecker and Dirac deltas, and the fact that s 2 i = 1, reduce this to The p ′ 1 and p ′ 2 integrations yield either unity for direct terms or zero for cross terms; the latter can be identified by the leading δ s 1 s 2 .The integral in the direct terms is the normalization integral for F ± .The integral for the cross terms is zero because of zero overlap between the wave packets of the two sources.The integrals over p 1 and p 2 can be rewritten in terms of the Fourier transforms ψ ± , with factors of p ′ 1 and p ′ 2 replaced with (m, 0 ⊥ ) at the peak of F ± .This leaves factors of |ψ ± (x)| 2 which become δ(x − ± R z )δ( x ⊥ ∓ R ⊥ /2).The integral over x can then be performed.The resulting expression is what is quoted in (4.8).

Appendix B: Reduction of the two-source eigenvalue problem
We begin from the statement of the full eigenvalue problem for two static sources as given in (4.9).The coefficient of the collected a † k (q) terms is q) is automatically zero, given the solution to the single-source case.This leaves the double-source eigenvalue problem (4.9) in the form The combination C ± i G ± ki (q)|G ± i can be replaced in each appearance with use of (3.18) The q dependent factors can be combined into a factor that is either the self-energy integral I defined in (3.21),where the exponential factors cancel, or the integral We then obtain This can be rearranged to reveal parts directly related to the single-source problem According to (3.20), with E ± = m, the first curly bracket is simply mC + i |G + i , and the second is mC − j |G − j .These two terms contribute 2mC + i C − j |G + i |G − j to the equation and, when subtracted from both sides, leave the effective potential V eff ( R) ≡ E − 2m determined by with Y ≡ Y + + Y − .To extract V eff , we define |G ± ≡ i C ± i s i β i |G ± i , multiply Eq. (B7) by s i β i s j β j , and sum over i and j To simplify the result further, we use the constraint i s i β 2 i = 0, which eliminates all terms between the square brackets except for the product of the first in each.We can then equate coefficients of |G + |G − to obtain The integrals in Y are computed in [17].They yield In the limit of infinite PV boson masses, with r 0 = 1, ξ 0 = 1, and µ 0 = µ, this reduces to e −µR /(8πR), and the effective potential is found to be This is the expression given in (4.10).

. 23 )
Now we convert the Fock-space equation (3.22) into an algebraic equation by projecting it onto Appendix A. The boson annihilation operators in (4.7) and (4.8) can be replaced with use of a k