Confinement interaction in nonlinear generalizations of the Wick-Cutkosky model
aa r X i v : . [ h e p - t h ] N ov Confinement interaction in nonlinear generalizationsof the Wick-Cutkosky model
J W Darewych † and A Duviryak ‡ † Department of Physics and Astronomy, York University, Toronto, Ontario,M3J 1P3, Canada ‡ Department for Computer Simulations of Many-Particle Systems, Institute forCondensed Matter Physics of NAS of Ukraine, Lviv, UA-79011, UkraineE-mail: † [email protected], ‡ [email protected] Abstract.
We consider nonlinear-mediating-field generalizations of the Wick-Cutkosky model. Using an iterative approach and eliminating the mediating fieldby means of the covariant Green function we arrive at a Lagrangian density containingmany-point time-nonlocal interaction terms. In low-order approximations of ϕ + ϕ theory we obtain the usual two-current interaction as well as a three-current interactionof a confining type. The same result is obtained without approximation for aversion of the dipole model. The transition to the Hamiltonian formalism andsubsequent canonical quantization is performed with time non-locality taken intoaccount approximately.A relativistic three-particle wave equation is derived variationally by using a three-particle Fock space trial state. The non-relativistic limit of this equation is obtainedand its properties are analyzed and discussed.PACS numbers: 11.10Ef, 11.10Lm Submitted to:
J. Phys. A: Math. Gen. onfinement interaction in nonlinear Wick-Cutkosky model
1. Introduction
Confinement is evidently related to the nonlinearity of chromodynamics. Since confiningsolutions of classical non-Abelian field equations are not known at present [1], it isbelieved that confinement is an essentially quantum effect. This is supported bynumerical computations of QCD on the lattice [2, 3]. However, the analytical studyof confinement, particularly in gauge field theory like QCD, remains a challenging task[3]. Thus the study of simpler field-theoretical models that simulate the characteristicfeatures of confinement remains relevant.In this regard, it is worth mentioning earlier models, such as the dipole model [4] andthe related higher derivative model [5] with its subsequent non-Abelian generalization[6]. They indicate a 1 /k infrared behavior of the “gluon” propagator, and thus alinear interaction potential, even at the classical level. In spite of some quantizationinconsistencies, these phenomenological models treat the confinement interaction asan elementary process, i.e., a two-particle interaction arising from the lowest-orderapproximation of perturbative dynamics of the models.More realistic models are the Dual Abelian Higgs model [3] and non-Abelianversions [7, 3] in which the spontaneous symmetry breaking mechanism is usedto generate a vacuum condensate with confining properties. In this approach theconfinement interaction is a kind of collective effect similar to that in condensed matterphysics.The two classes of models mentioned above represent quite different points of viewon the confinement mechanism. The purpose of the present study is to shed light onthe question: is an intermediate mechanism possible, in which confinement cannot bereduced to an elementary processes but is governed by cluster interactions involvingfinite numbers of particles?To investigate such a possibility, we utilize the variational method, in a reformulatedHamiltonian formalism of Quantum Field Theory (QFT), which has been demonstratedto be a promising and powerful approach to the relativistic bound state problem [8]–[13]. In particular, this approach has been used to derive (and solve approximately)relativistic equations for two and three fermion systems, such as Positronium (Ps) andMuonium (Mu) [14], and also Ps − and Mu − [15], and it was shown that the derivedbound state energies agree with conventional perturbation theory and with experimentalresults (where available).The use of many-particle Fock-space components in the variational trial states leadsto wave equations with systematically improvable bound state energy levels, as has beenshown, for example, on the simple scalar Yukawa model [12, 13].In this paper, we analyze the interactions that arise from the non-linear terms in themediating-field sector of the QFT Lagrangian. In particular we consider the ( ϕ + ϕ )-generalization of the Wick-Cutkosky (i.e. massless scalar Yukawa) model [16] as well asa version of the dipole model [4, 11].We note that the models being considered are not of a non-Abelian gauge-field type. onfinement interaction in nonlinear Wick-Cutkosky model
2. Partially reduced Wick-Cutkosky model
The Wick-Cutkosky model [16] is based on the classical action integral: I = Z d x L ( x ) , (2.1)with the Lagrangian density ( ~ = c = 1) L = ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ − gφ ∗ φ χ + ∂ µ χ∂ µ χ, (2.2)where φ ( x ) is a complex scalar “matter” field with rest mass m , and χ ( x ) is a realmassless scalar field interacting with φ via the Yukawa term gφ ∗ φ χ (here g is aninteraction constant).The stationary property of the action (2.1)-(2.2), i.e. δI = 0, leads to the coupledset of the Euler-Lagrange equations,( (cid:3) + m ) φ = − gφχ, (2.3)( (cid:3) + m ) φ ∗ = − gφ ∗ χ, (2.4) (cid:3) χ = ρ, (2.5)which determine the field dynamics; here ρ ≡ − gφ ∗ φ .Equation (2.5) can be solved exactly: χ = D ∗ ρ + χ , (2.6)where “ ∗ ” denotes the convolution [ D ∗ ρ ] ( x ) ≡ R d x ′ D ( x − x ′ ) ρ ( x ′ ) and D ( x ) = π δ ( x ) is the symmetric Green function of the d’Alembert equation. If the free χ field plays no role in the investigation the arbitrary solution χ of the homogeneousd’Alembert equation can be omitted. Then the use of the formal solution (2.6) (with χ = 0) in the Lagrangian (2.2) leads to a self-contained variational principle for theinteracting fields φ ( x ) and φ ∗ ( x ). The modified Lagrangian ¯ L which we shall refer toas the partially-reduced Lagrangian, is an important basis for the quantization of themodel; cf. refs. [9, 11].We demonstrate here how to derive the partially-reduced Lagrangian for the Wick-Cutkosky model without the use of the condition χ = 0. For this purpose we considerthe equality (2.6) as a change of variable χ → χ where the new field χ is not a’priori subjected to any field equation. The substitution of (2.6) directly in the Lagrangian(2.2) gives L = ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ + ρ ( D ∗ ρ + χ ) + [ ∂ µ ( D ∗ ρ + χ ) ∂ µ ( D ∗ ρ + χ )] ≃ ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ + ρ ( D ∗ ρ + χ ) − ( D ∗ ρ + χ ) (cid:3) ( D ∗ ρ + χ ) ≃ ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ + ρD ∗ ρ | {z } ¯ L + ∂ µ χ ∂ µ χ | {z } ∆ L free , (2.7) onfinement interaction in nonlinear Wick-Cutkosky model ≃ denotes equality modulo surface terms. In this form the system is effectivelysplit into two independent subsystems: the interacting φ matter field and the free χ field. From this point on the physically trivial χ -dependent ∆ L free term can be ignored(as indicated above). ‡ The partially-reduced Lagrangian ¯ L is non-local in space-time coordinates. Thetreatment of non-local theories of this type is a conceptually intricate, but practicallyrealisable procedure. In particular, partially-reduced versions of Yukawa-like modelsare worked out in [11]. In the next section we consider a non-linear generalization ofWick-Cutkosky model within the partially-reduced formulation.
3. Nonlocal Lagrangian from a nonlinear Wick-Cutkosky model
We proceed from the Lagrangian density L = ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ − gφ ∗ φ χ − λ ( φ ∗ φ ) + ∂ µ χ∂ µ χ − V ( χ ) , (3.1)where λ > V ( χ ) is an arbitrary potential(all other quantities are the same as in (2.2)).The new terms, λ ( φ ∗ φ ) and V ( χ ), modify the Euler-Lagrange equations (2.3)-(2.5).In particular, the equation (2.5) becomes the non-linear inhomogeneous d’Alembertequation (cid:3) χ = ρ − V ′ ( χ ) , (3.2)where V ′ ( χ ) ≡ d V ( χ ) / d χ . It can be formally solved by iteration (cf. ref. [17]). In the1st-order approximation we have: χ = D ∗ [ ρ − V ′ ( D ∗ ρ )] + χ , (3.3)where χ includes an arbitrary solution of the homogeneous equation.Similarly to the case of the linear Wick-Cutkosky model, we use the replacement(3.3) (where χ is a new field variable) in the Lagrangian (3.1). In 1st order this gives, L ≃ ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ + ρD ∗ ρ − λ ( φ ∗ φ ) + ∂ µ χ ∂ µ χ − V ( D ∗ ρ + χ ) + χ V ′ ( D ∗ ρ ) . (3.4)Unlike the Lagrangian (2.7), this functional is not completely split in the φ and χ variables. The Euler-Lagrange equation for χ , (cid:3) χ = −V ′ ( D ∗ ρ + χ ) + V ′ ( D ∗ ρ ) , (3.5)is a free-field one only in zero-order approximation. Nevertheless, it possesses thesolution χ = 0 which, upon substitution into (3.4), gives the reduced Lagrangian:¯ L ≃ ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ + ρD ∗ ρ − λ ( φ ∗ φ ) − V ( D ∗ ρ ) ≡ L free + L (2)int + L ( > (3.6) ‡ It is noteworthy that, within the variational problem based on (2.7), the primary meaning of χ in(2.6) as general solution of the homogeneous d’Alembert equation is restored. onfinement interaction in nonlinear Wick-Cutkosky model > L = L − ¯ L , i.e., the χ -dependent part of the total Lagrangian(3.4), is at least quadratic in the χ variable:∆ L = ∆ L free + ∆ L int where ∆ L free = ∂ µ χ ∂ µ χ , ∆ L int = V ( D ∗ ρ ) + χ V ′ ( D ∗ ρ ) − V ( D ∗ ρ + χ )= − χ V ′′ ( D ∗ ρ ) − χ V ′′′ ( D ∗ ρ ) − . . . (3.7)This structure shows that the term ∆ L is not important in the present work, as will beexplained in more detail in Section 7.The non-local Lagrangian (3.6) is the 1st-order approximate result of the reductionprocedure applied to nonlinear generalizations of the Wick-Cutkosky model. In theAppendix we construct another local model, a kind of dipole model (with a pair ofmediating fields), that can be reduced to the Lagrangian (3.6) exactly.
4. Quantization
In order to proceed farther we need to specify the interaction potential V ( χ ). We choose V ( χ ) = κχ + κ χ , (4.1)where κ and κ > H = H free + H (2)int + H (3)int + H (4)int , (4.2)where H (2)int ( x ) = − Z d x ′ ρ ( x ) D ( x − x ′ ) ρ ( x ′ ) + λ ( φ ∗ ( x ) φ ( x )) ≡ − Z d x ′ ρ ( x ) (cid:20) D ( x − x ′ ) − λ g δ ( x − x ′ ) (cid:21) ρ ( x ′ ) , (4.3) H (3)int ( x ) = κ Z Z Z d x ′ d x ′′ d z D ( z − x ) D ( z − x ′ ) D ( z − x ′′ ) ρ ( x ) ρ ( x ′ ) ρ ( x ′′ ) , (4.4) H (4)int ( x ) = κ ZZ Z Z d x ′ d x ′′ d x ′′′ d z D ( z − x ) D ( z − x ′ ) D ( z − x ′′ ) D ( z − x ′′′ ) ×× ρ ( x ) ρ ( x ′ ) ρ ( x ′′ ) ρ ( x ′′′ ) . (4.5)The total interaction Hamiltonian density (4.2) is then expressed in terms of the Fourieramplitudes A k , B k and A † k , B † k , of the field φ ( x ) (see eq. (2.14) in [13]; actually, theprocedure is somewhat more intricate [11] but the result is the same). Upon quantization onfinement interaction in nonlinear Wick-Cutkosky model H = Z d x : H ( t =0 , x ) : , (4.6)where “: :” denotes the normal ordering of operators. Other canonical generators, suchas linear and angular momentum, can be easily obtained.The term H free is the standard Hamiltonian of the free complex scalar field. Theexplicit form of the pair interaction term H (2)int is known (see [9, 11]) and so we shallconcentrate on the H (3)int term. It has the following somewhat cumbersome form: H (3)int = − κg π ) Z d k . . . d k √ k . . . k X η ± ......η ± ˜ D ( η k + η k ) ˜ D ( η k + η k ) ˜ D ( η k + η k ) ×× δ ( η k + . . . + η k ) : η B k η A k η B k η A k η B k η A k : , (4.7)where + B = B , − B = A † , + A = A , − A = B † and the Fourier transform, ˜ D ( k ) = −P /k ,of the symmetric Green function of the d’Alembert equation depends on the on-shell4-momentum k = { k , k } , where k = √ m + k . The expression (4.7) includes 2 = 64terms. The term H (4)int is of similar but more cumbersome form. We do not exhibitit explicitly, since, as will be seen below, it makes no contribution to the three-bodyequation derived in this work.
5. Variational three-particle wave equations
In the variational approach to QFT the trial state of the system is built offew particle channel components [12, 13] such as the two-particle state vector | i = √ R d p d p F ( p , p ) A † p A † p | i , the particle-antiparticle one | i = R d p d p G ( p , p ) A † p B † p | i , and so on. The three-particle component has the form | i = 1 √ Z d p d p d p F ( p , p , p ) A † p A † p A † p | i , (5.1)where the channel wave function F , which is to be determined variationally, iscompletely symmetric under the permutation of the variables p , p , p . In thevariational method the channel components, | ψ N i , are used to determine thematrix elements of the Hamiltonian, namely h ψ N | H | ψ N ′ i , where N, N ′ stand for1 , ¯1 , , , ¯2 , , , , . . . We are interested here in the matrix element of the interaction H int = H (2)int + H (3)int + H (4)int of the Hamiltonian. We note that h | H (3)int | i = 0, h | H (3)int | i = 0. In otherwords, purely two-particle trial states, and so the resulting variational wave equations,do not sample the term H (3)int . Thus we first consider the three-particle case and calculatethe matrix element h | H int | i = Z d p ′ ... d p ′ d p ... d p F ∗ ( p ′ ... p ′ ) F ( p ... p ) K ( p ′ ... p ′ , p ... p ) , (5.2) onfinement interaction in nonlinear Wick-Cutkosky model K = K (2)33 + K (3)33 consists of the following components: K (2)33 ( p ′ ... p ′ , p ... p ) = − π ) δ ( p ′ + p ′ + p ′ − p − p − p ) ×× δ ( p ′ − p ) p p ′ p ′ p p h g ˜ D ( p ′ − p ) − λ/ i , (5.3) K (3)33 ( p ′ ... p ′ , p ... p ) = − κg π ) δ ( p ′ + p ′ + p ′ − p − p − p ) ×× ˜ D ( p ′ − p ) ˜ D ( p ′ − p ) ˜ D ( p ′ − p ) p p ′ ...p ′ p ...p , (5.4)and p i = p m + p i and similarly for p ′ j ( i, j = 1 , , H (4)int does notcontribute in K , i.e., K (4)33 = 0.The kernel K determines the interaction in the relativistic three-particle waveequation that follows from the variational principle δ h | H − E | i = 0, namely { p + p + p − E } F ( p , p , p )+ Z d p ′ d p ′ d p ′ K ( p , p , p , p ′ , p ′ , p ′ ) F ( p ′ , p ′ , p ′ ) = 0 (5.5)where the kernel is understood to be the completely symmetrized expression (withrespect to the variables p ′ , p ′ , p ′ and p , p , p ) of (5.3) and (5.4).The term K (2)33 of the kernel corresponds to the attractive interaction via masslessboson exchange and repulsive contact interaction between each pair of particles while K (3)33 describes a cluster three-particle interaction.From the mathematical viewpoint the three-body wave-equation (5.5) is an integralequation with a singular kernel. Even in simpler (say, two-particle) cases such equationsare usually solved approximately (variationally, numerically, perturbatively), and it isnot easy to get a general qualitative characteristic of the solutions, or to estimate therole of different terms of the kernel.In order to have some understanding of the properties of the cluster interaction weconsider the non-relativistic limit of the equation (5.5), in which case the kernels simplifyconsiderably, and then perform the Fourier transformation into coordinate space. Inthis representation the equation is simply a Schr¨odinger equation for the three-particleeigenfunction Ψ( x , x , x ) (see [12]) and eigenenergy ǫ = E − m : (cid:26) m ( p + p + p ) + V ( x , x , x ) − ǫ (cid:27) Ψ( x , x , x ) = 0 , (5.6)where p i = − i ∇ i ( i = 1 , , V ( x , x , x ), like the relativistic kernel K , consists of two parts, V = V (2)33 + V (3)33 : V (2)33 ( x , x , x ) = − g πm (cid:26) | x − x | + 1 | x − x | + 1 | x − x | (cid:27) + λ m { δ ( x − x ) + δ ( x − x ) + δ ( x − x ) } , (5.7) V (3)33 ( x , x , x ) = 2 κg (8 πm ) U ( x , x , x ) . (5.8) onfinement interaction in nonlinear Wick-Cutkosky model U ( x , x , x ) ≡ − Z d z | z − x || z − x || z − x | . (5.9)The integral in r.h.s. of (5.9) is a divergent quantity and thus equation (5.6) mayseem to be meaningless. However, the gradients ∂U ( x , x , x ) /∂ x i ( i = 1 , ,
3) whichdetermine the forces in the classical background of this problem, are well defined andfinite. Thus the “function” (5.9) can be presented in the form U ( x , x , x ) = ˜ U ( x , x , x ) + U (5.10)where ˜ U ( x , x , x ) in a regular (finite) function and U is an infinite negative constant(independent of the variables x , x , x ). This constant can be absorbed by theeigenenergy ǫ so that the wave equation (5.6) gets reformulated as follows: V (3)33 ( x , x , x ) → ˜ V (3)33 ( x , x , x ) = 2 κg (8 πm ) { U ( x , x , x ) − U }≡ κg (8 πm ) ˜ U ( x , x , x ) , (5.11) ǫ → ˜ ǫ = E − m − κg (8 πm ) U (5.12)where the eigenenergy ˜ ǫ is finite (as is the potential ˜ V (3)33 ).In order to perform this reformulation explicitly, we need to resort to regularizationof the integral (5.9) which we consider in the next section.The problem of divergences is expected in the relativistic case too. But the analysisof the integral equation (5.5) is a more subtle problem which shall not be undertakenin this work.
6. Properties and evaluation of the 3-point potential
Various regularization procedures are possible. In essence, one introduces some cut-offparameter which finally is put to 0 (or ∞ ). We enumerate some possibilities:1) We could consider the case where the mediating χ field is massive, whereupon therewould be a mass term − µ χ in the Lagrangian (3.1). In that case the gravity-like1 r factors would be replaced by the Yukawa forms e − µr r . Thus we could regard U of eq.(5.9) as the massless-mediating-field limit of the massive-mediating-field case, U µ ( x , x , x ) = − Z d z e − µ | z − x | | z − x | e − µ | z − x | | z − x | e − µ | z − x | | z − x | , (6.1)which is well defined and finite for any µ > z to v = z − x in eq.(6.1), we can write U µ as U µ ( x , x , x ) = − Z d v e − µv v e − µ | v + x | | v + x | e − µ | v + x | | v + x | = ¯ U µ ( x , x ) , (6.2) onfinement interaction in nonlinear Wick-Cutkosky model x ij = x i − x j and v = | v | .2) Another way would be to regard U of eq. (5.9) as a limiting case, as R → ∞ , of¯ U R ( x , x ) = − Z R d v v Z d ˆ v | v + x || v + x | , (6.3)where ˆ v = v /v and R is an arbitrarily large, but finite, “radius of space”.3) We could, also, regard U of eq. (5.9) as a limiting case, as Λ → + , of¯ U Λ ( x , x ) = − Z ∞ d v v e − Λ v Z d ˆ v | v + x || v + x | . (6.4)Evidently, any other suitable and convenient cut-off function can be used in place of e − Λ v .Of course, physical results would be meaningful to the extent that they wereindependent of the choice of the regularization procedure.Below we establish some general properties of the regularized ˜ U function. Wediscuss in detail a convenient method of its evaluation and show that it possesses alogarithmic confining property when µ → U µ ( x , x , x ) (6.1). It obviously obeys thefollowing symmetry properties:(i) translational invariance: U µ ( x + λ , x + λ , x + λ ) = U µ ( x , x , x ), where λ ∈ R ;(ii) rotational invariance: U µ (R x , R x , R x ) = U µ ( x , x , x ), where R ∈ SO(3);(iii) permutational invariance: U µ ( x , x , x ) = U µ ( x , x , x ) = U µ ( x , x , x );(iv) scaling transformation: U µ ( λ x , λ x , λ x ) = U λµ ( x , x , x ), where λ ∈ R + .These properties have implications for the structure of the regularized potential.The properties (i)–(iii) hold for arbitrary values of the cut-off parameter µ , includingthe formal limiting case µ →
0. Moreover, these are fundamental symmetries inherent toany interaction potential of a closed (nonrelativistic) system of three identical particles.Thus the regularized potential must possess the properties (i)–(iii) of necessity.The scaling property (iv) has specific implication for the regularization (6.1). Inthe formal limit µ → U ≡ U µ =0 is scale invariant:(iv)’ scale invariance: U ( λ x , λ x , λ x ) = U ( x , x , x ), where λ ∈ R + .However, as is shown below, the scaling property of the regularized potential˜ U ( x , x , x ) is different.We note that an important property of the potential U ( x , x , x ), with any ofthe regularizations (6.1)–(6.4), follows from the symmetries 1–3, namely that it actuallydepends only on the three inter-point distances x , x , x , where x ij = | x ij | . Explicitly,this is readily seen if the factors e − µ | v + x ij | | v + x ij | in equations (6.1)–(6.4) are expanded inspherical harmonics ( µ ≡ R d ˆ v ... are carriedout, and the orthogonality properties of the spherical harmonics are used, then (after the onfinement interaction in nonlinear Wick-Cutkosky model v ), the result is seen to depend only on the lengths of thetwo vectors x , x and on the angle between them (or, equivalently, on x , x , x ).The direct calculation of the regularized potential, with any of the regularizations(6.1)–(6.4), is complicated. Instead, we propose a representation for the function (5.9)in which its dependence on scalar arguments is manifest. This greatly simplifies theregularization and evaluation of U . Let us apply the well known formula:1 r = 1 √ π Z ∞−∞ d k e − k r to each factor of the integrand of the expression (5.9) (which shall be treated formally).Then changing the order of integration we have: U ( x , x , x ) = − π / Z d k Z d z e − k ( z − x ) − k ( z − x ) − k ( z − x ) = − Z d kk e − ( k k x + k k x + k k x ) /k = − Z d ˆ k ∞ Z d kk e − (ˆ k ˆ k x +ˆ k ˆ k x +ˆ k ˆ k x ) k (6.5)where ˆ k = k /k . It is obvious in this form that U ( x , x , x ) = U ( x , x , x ) and,in addition, that the internal integral in the last line of (6.5) is divergent at its lowerboundary k = 0.The potential difference: U ( x , x , x ) − U ( y , y , y ) = − Z d ˆ k ∞ Z d kk h e − X k − e − Y k i , (6.6)where X = ˆ k ˆ k x + ˆ k ˆ k x + ˆ k ˆ k x , Y = ˆ k ˆ k y + ˆ k ˆ k y + ˆ k ˆ k y , will be finitesince infinite constants U (see (5.10)) from the first and second terms of (6.6) mutuallycancel. Indeed, using the cut-off parameter ε in the internal integral in r.h.s. of (6.6)yields: ∞ Z ε d kk h e − Y k − e − X k i = ∞ Z Y ε − ∞ Z Xε d tt e − t = Xε Z Y ε d tt e − t −→ ε → ln XY , i.e., the integral is convergent.Next, we introduce angular variables { ϑ, ϕ } on the unit sphere in k -space, so thatˆ k = sin ϑ cos ϕ , ˆ k = sin ϑ sin ϕ , ˆ k = cos ϑ . Then U ( x , x , x ) − U ( y , y , y ) = W (¯ x , ¯ x , ¯ x ) − W (¯ y , ¯ y , ¯ y ) , (6.7)where W (¯ x , ¯ x , ¯ x ) =
12 2 π Z d ϕ π Z sin ϑ d ϑ ln (cid:2) (¯ x sin ϑ cos ϕ sin ϕ ) + (¯ x cos ϑ sin ϕ ) + (¯ x cos ϑ cos ϕ ) (cid:3) (6.8) onfinement interaction in nonlinear Wick-Cutkosky model x ij = x ij /a . The arbitrary constant a (with dimension of length) is introducedso that the argument of the logarithm will be dimensionless. Actually, the potentialdifference (6.7) does not depend on a while the function (6.8) itself does. Since thisfunction is well defined and finite, it can be considered, up to some additive constant,as the regularized potential:˜ U ( x , x , x ) = W (¯ x , ¯ x , ¯ x ) − W . (6.9)The choice of the constant W is a matter of taste; it can be canceled by an appropriaterescaling of the constant a : W ( x /a, ... ) = W ( x /b, ... ) + 4 π ln( b/a ). Thus anarbitrariness of the regularized potential arises due to the scale constant a .We note that the regularized function (6.9) obeys the following scaling property:( e iv) scale invariance: ˜ U ( λ x , λ x , λ x ) = ˜ U ( x , x , x ) + 4 π ln λ , where λ ∈ R + .The inner integration (over ϑ ) in (6.8) can be performed explicitly. Then the changeof variable ϕ → s = cos ϕ yields:˜ U ( x , x , x ) = 4 π ln x + x a + I ( ξ, η ) , (6.10)where I ( ξ, η ) = 4 Z − d s p ( s + ξ ) + η arctan r ( s + ξ ) + η − s , (6.11) ξ = x − x x , η = [( x + x ) − x ][ x − ( x − x ) ] x , (6.12)and we have chosen for convenience: W = 4 π (ln 2 − x + x ≥ x , x + x ≥ x and x + x ≥ x .The regularized potential (6.10)–(6.12) possesses the permutational invariance (iii)implicitly. This is evident from the fact that any particle permutation is equivalentto some renumbering of k -variables in the integrals (6.5), (6.6) and, finally, to anotherchoice of angular variables in the integral (6.8).In the particular cases where the points x , x and x lie on a straight line theintegral (6.11) can be calculated analytically:˜ U ( x , x , x ) = 4 π ln x > a , where x > = max( x , x , x ) . (6.13)Another analytically solvable case is that of equidistant points, x = x = x = r ,whereupon in (6.12), ξ = 0 and η = 3, so that I of (6.11) is a finite constant independentof r . Thus ˜ U ( r, r, r ) = 4 π ln( r/a ) + c , where c is a finite constant, which we can ignore(it does not affect energy differences). For convenience we shall use “atomic units”, thatis energies will be in units of mα , and lengths in units of a = mα , where α = g πm is the dimensionless “fine structure constant”. The total potential V = V (2)33 + V (3)33 (cf.eqs. (5.7) and (5.8)) is (with λ = 0), in atomic units, V ( r ) = − r + γ ln r, ( r is rmα, and V is V /mα ) (6.14) onfinement interaction in nonlinear Wick-Cutkosky model γ = 4 κ/g . We see that V ( r ), in this equidistant-points subspace, is a uniformlyincreasing, logarithmically confining potential (for γ > V ( r ) ≃ − r forsmall r ( r → + ) but V ( r ) ≃ γ ln r for large r . Recall that if κ = γ = 0, the bound-state eigenvalue spectrum (in atomic units) is the Rydberg spectrum ǫ n = −
32 1 n , where n = 1 , , , ... , and there are no bound states for ǫ >
0. However, for γ >
0, thelogarithmic confining potential stretches out this Rydberg spectrum, so that there isa purely bound-state spectrum for ǫ >
0. Using various approximations [18, 19] onecan estimate ǫ n ≃ γ ln n for n ≫
1. (The repulsive contact (delta-function) potentials,which we have ignored by taking λ = 0, are of little consequence, since such repulsivecontact potentials have an insignificant effect on the energy spectrum.)Other regularization methods lead, basically, to the same results. For example,if we use the cut-off regularization of (6.3), then for the case x = 0, we obtain¯ U R = − π [1 + ln( R/x )] = 4 π ln( x /a ) − c , where a is an arbitrary length parameter(length unit), and c = 4 π [1+ln( R/a )] is a very large constant, which has to be absorbedinto a redefined (shifted) energy, as in (5.12). This result is the same as eq. (6.13).In the general case, a numerical integration of (6.11) is required. We illustrate thebehavior of the potential in Figure 1 for the particular case x = a , x = − a as afunction of x = r . The value of potential for arbitrary configuration can be obtainedfrom it using the symmetry properties (i)–(iii) and ( e iv). ~ Uz/a /a ρ Figure 1.
The potential ˜ U ( a , − a , r ) as a function of r = { x, y, z } ; ρ = p x + y ; a = | a | . The function is symmetric under the inversion z → − z and rotation around0 z . In particular, ˜ U = 4 πθ ( | z | − a ) ln ( | z | /a + 1) if ρ = 0. In the case where one of the points is far from the others, the equality (6.13) isvalid asymptotically. Thus the regularized potential reveals logarithmic confinementproperties.A detailed analysis of the (non-relativistic) bound-state spectrum for the generalcase requires the solution of the three body equation (5.6). This is a quite challengingtask in itself. However, from the confining nature of the three-point potential, we can see onfinement interaction in nonlinear Wick-Cutkosky model
7. Concluding remarks
We have considered generalizations of the Wick-Cutkosky (massless scalar Yukawa)model that include nonlinear mediating fields. Covariant Green functions were usedto eliminate the mediating field, thus arriving at a Lagrangian that contains nonlocalinteraction terms.In the case of a massless mediating field χ , with a κχ + κ χ nonlinearity, weevaluate the corresponding interaction term explicitly and show that the kernel has theform of a three- and four-point “cluster potential”, cf. (4.4), (4.5).We consider the quantized version of this model in the Hamiltonian formalism,and use the variational method, with trial states built from Fock-space components, toderive a relativistic integral wave equation for the three-particle system. The kernels(relativistic potentials) are shown to contain one-quantum exchange terms and a three-point cluster term. In the non-relativistic limit we evaluate the explicit coordinate-spaceform of the interaction potentials and show that they consist of attractive pairwiseCoulombic potentials and a cluster three-point confining potential. The three-pointpotential, which arises from the κχ term in the Hamiltonian, is divergent (and soneeds regularization), but the potential differences are finite. The regularized three-point potential is shown to be logarithmically confining, and dependent only on thethree inter-point distances. Its evaluation, for arbitrary values of its arguments, isshown to be reducible to a single quadrature.The three-body wave-equation derived in this paper is quite complicated and mustbe solved using approximation methods. This will be the subject of forthcoming work.The three-particle trial state (5.1) is found to be the simplest variational ansatzwhich manifests the confinement properties of the model. However, other sectors ofthe Fock space in the variational problem are also of interest. For example, an openproblem is the role of the three-point interaction in the particle-antiparticle problem.It was pointed out in the section 5 that the simple variational particle-antiparticle trialstate | i does not sample the H (3)int term (4.7) of the Hamiltonian. Thus, this termdoes not influence the variational wave equation derived by using only | i (see[9, 11, 12]), in which case the only Coulomb-like interaction arise. But the inclusionof both the | i and | i sectors leads to a coupled set of two many-body wave-equations [13] in which the effects of H (3)int and H (4)int are present. Whether these effectsare confining is a question that needs to be investigated.Lastly, we comment on the role of “chion” Fock-space sector in the variationalbound state problem within the reduced Hamiltonian formalism of QFT used in thiswork. This role can be examined by taking into account the χ -dependent extra terms∆ L of the total non-local Lagrangian (3.4). They are at least quadratic in χ includingthe free-field term ∂ µ χ ∂ µ χ and interaction terms; see eq.(3.7). Thus the additional onfinement interaction in nonlinear Wick-Cutkosky model H , has no effect on variational states | Ψ i without free “chions” (i.e., quanta of the field χ ), since h Ψ | ∆ H | Ψ i = 0 for suchstates. A non-trivial contribution to a variational bound-state problem may arise fromstates with two or more virtual “chions” but this is a higher-order effect in the couplingconstants ( κ , κ or others) of the potential V . Acknowledgments
The authors are grateful to V. Tretyak, T. Krokhmalskii and Yu. Yaremko for helpfuldiscussion of this work.
Appendix. Nonlocal Lagrangian from a nonlinear dipole model
In this section we consider a model which is built in analogy to the linear “dipole model”[4, 11] that simulates the confinement interaction of quarks in mesons. This model isnonlinear and gives Yukawa + cluster interactions. It is specified by the Lagrangian L = ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ − λ ( φ ∗ φ ) + ρ ( χ + ϕ ) + ∂ µ χ ∂ µ ϕ − V ( ϕ ) , (A.1)where both the χ ( x ) and ϕ ( x ) are real massless scalar fields and ρ = − gφ ∗ φ as in (3.1).The variation of the action (2.1), (A.1) leads to the coupled set of the Euler-Lagrange equations,( (cid:3) + m ) φ = − g φ ( χ + ϕ ) − λφ ( φ ∗ φ ) , (A.2)( (cid:3) + m ) φ ∗ = − g φ ∗ ( χ + ϕ ) − λφ ∗ ( φ ∗ φ ) , (A.3) (cid:3) ϕ = ρ, (A.4) (cid:3) χ = ρ − V ′ ( ϕ ) , (A.5)which determine the field dynamics.Equations (A.4) and (A.5) possess the exact formal solution: ϕ = D ∗ ρ, (A.6) χ = D ∗ (cid:8) ρ − V ′ ( ϕ ) (cid:9) = D ∗ (cid:8) ρ − V ′ ( D ∗ ρ ) (cid:9) , (A.7)which can immediately be used in the r.h.s. of eqs. (A.2), (A.3):( (cid:3) + m ) φ = − gφD ∗ { ρ − V ′ ( D ∗ ρ ) } − λφ ( φ ∗ φ ) , (A.8)and similarly for φ ∗ . These equations can be derived from δ I = 0, with a Lagrangianidentical to (3.6) (but note that no iterative expansion, like that in eq. (3.6), needs tobe made in this case). References [1] Actor A 1979
Rev. Mod. Phys. Introduction to the classical theory of particles and fields (Heidelberg: Springer)[2] Greensite J 2003
Prog. Part. Nucl. Phys. onfinement interaction in nonlinear Wick-Cutkosky model [3] Swanson E S 2004 AIP Conference Proceedings
Phys. Rev. D Phys. Rev. D Theor. Math. Phys. Nucl. Phys. B
J. Phys. A Canadian J. Phys. Cond. Mat. Phys. Cond. Mat. Phys. J. Phys. G Phys. Rev. D Phys. Rev. A J. Phys. A J. Phys. G J. Phys. G J. Math. Phys. J. Math. Phys. J. Phys. B Phys. Rev. Phys. Rev. J. Phys. Studies Phys. Rep. J. Phys. A41