Analysis of inter-quark interactions in classical chromodynamics

Transcription

1 Cent. Eur. J. Phys DOI:.478/s y Central European Journal of Physics Analysis of inter-quark interactions in classical chromodynamics Research Article Jurij W. Darewych, Askold Duviryak Department of Physics Astronomy, York University, 47 Keele St., M3J P3 Toronto, Ontario, Canada Institute for Condensed Matter Physics of the NAS of Ukraine, Svientsitskii St., UA 79 Lviv, Ukraine Received June ; accepted 3 December Abstract: PACS 8:.5.Kc,.39.Jh Keywords: The QCD gluon equation of motion is solved approximately by means of a Green function. This solution is used to reformulate the Lagrangian of QCD such that the gluon propagator appears directly in the interaction terms of the Lagrangian. The nature of the interactions is discussed. A coordinate-space form of the interactions is presented analyzed in the static, non-relativistic case. classical chromodynamics cluster interaction Versita sp. z o.o.. Introduction Classical field theory, as is well known, is the starting point basis of all fundamental theories, including quantum field theories. The use of solutions of classical field equations can simplify the approach to quantum field theories provide good approximations to problems of these theories []. A particular example is the partially reduced electrodynamics, discussed in references [ 5], the classical solution for the electromagnetic potential has been used directly in the Lagrangian in order to describe interactions between Dirac matter fields. This model, coupled darewych@yorku.ca duviryak@icmp.lviv.ua Corresponding author with the variational method in the Hamiltonian formalism the use of simple Fock-space trial states, reproduces correctly fine splitting effects in two-fermion spectra [3, 4] in three-fermion systems [5]. In the case of non-abelian theories, such as chromodynamics, the problem of solving the nonlinear field equations has been investigated for several decades [6, 7]. To date, no confining solutions of these equations have been found. However, some important classical solutions of related theories have been obtained, suggesting a possible mechanism of confinement such as the Abrikosov- Nielsen-Olesen vortex [6]. Much effort has been expended on a search for solutions to gauge field equations with point-like sources. We might mention the exact [7] the iterated [8] nonlinear generalizations of the Lienard-Wiechert potentials. These solutions, however, cannot be applied directly to the case of 336

2 Jurij W. Darewych, Askold Duviryak extended field-like sources, in contrast to linear theories the superposition principle is applicable. In other words, the chromodynamics of point-like sources is not likely an appropriate basis for approximations in QCD. In this work we are interested in classical gluon interactions among quark fermion fields. We consider the gauge field as a mediator of interactions among quarks ignore degrees of freedom of free gluons which are not observed in nature. Thus, similarly to the above-mentioned case of electrodynamics [ 5], we use the formalism of partially reduced field theory [9 ]. We solve the chromodynamic gluon field equations approximately, by use of the covariant Green function, then substitute this solution into the QCD Lagrangian. In this way we reduce the description to one that involves fermionic degrees of freedom only. The resulting reduced Lagrangian has a complicated structure including nonlocal integral terms with two-, three- four-point kernels. Some properties of the cluster quark interactions represented by these kernels are presented discussed in a static, non-relativistic approximation.. QCD Lagrangian equations of motion The QCD Lagrangian density is [] c L QCD 4 F a µ ν F a µ ν + q F a µ ν µ A ν a i q i γ µ D µ i j m q δ ij j q, ν A a µ g s f a b c A b µ A c ν D µ i j δ i j µ + i g s t a i j A µ a. 3 The QCD coupling constant is g s, f a b c are the structure constants, t a i j λa i j, the λs are the Gell- Mann matrices. As usual, repeated indices are summed over, with the colour indices i, j,, 3, the flavour indices q d, u, s, c, b, t, a, b, c,..., 8 for the gluon fields. Gluon indices will be usually indicated by round brackets to avoid confusion with vector indices; eg. A µ a or A µ a. It is convenient to write is the free gluon field tensor G a µ ν g s f a b c A b µ A c ν 6 is the non-abelian part. Upon substituting 3 into, the Lagrangian density can be written as L QCD L + L F + L I + L I3 + L I4, 7 L q with the quark current L I3 L I4 j q i γ µ µ m q j q, 8 L F 4 F a µ ν F a µ ν, 9 L I j a µ A µ a j a µ g s F a µ ν G a µ ν g s f a b c µ A a ν q i q t a i j γ µ j q ν A µ a A µ b Aν c g s f a b c A µ a Aν b µ A c ν, 4 Ga µ ν G a µ ν 4 g s f a b c f a d e A b µ A c ν A µ d Aν e. 3 Using 8 - we acquire the colour Dirac equations for quark fields: i γ µ D µ ij j q m q i q. 4 D µ ij is defined in equation 3. The equation of motion for the gluon fields is cf. ref. [3] µ F a µ ν { λ λ δ µ ν ν µ }A a µ x j a ν x, 5 the total current j ν a x j ν a x + jg ν a x + j ν a x 6 g with the quark component gluon components j ν a g g s f abc A ν c µ A µ b + Aµ b µac+a ν µ c ν A µ b, 7 F a F a µ ν F a µ ν + G a µ ν, 4 µ ν µ A a ν ν A a µ 5 is conserved, j ν a g g s f abc f bde A c µ A µ d Aν e 8 µ j a µ x, 9 due to the gauge invariance of the Lagrangian

3 Analysis of inter-quark interactions in classical chromodynamics 3. Reformulation For the study of inter-quark interactions subsequently for the study of the properties of mesons baryons it is convenient to use a formal solution of the gluon equations of motion 5 to reformulate the Lagrangian, thus the action, of QCD, so that the gluon propagator appears directly in the interaction terms. Such reformulation has been shown to be useful for the study of inter-particle forces in scalar theory with a nonlinear mediating field [9, ]. First of all we rewrite the glue" equation 5 as an integral equation, A µ a x d 4 x D µν x x j ν a x, D µν x x is the symmetric Green function of the equation 5, j ν a x is the conserved current 6. We have not included the free-gluon solution of the homogeneous equation 5 in, since free gluons do not arise so free-gluon solutions will play no role in the present considerations. At this point we consider the r.h.s. of the equation as a formal solution of the glue" equation 5. Inserting this expression into 7, neglecting unessential divergence terms taking the equation 5 into account one can reduce the QCD Lagrangian density 7 exactly to the form L L a jµ A µ a + L I3 A b ν + LI4 A b ν. The formal solution of 5 involves the use of the symmetric Green function of that equation D µν x x which is not defined uniquely, due to a degeneracy of the operator λ λ δ ν µ ν µ. A choice of the Green function reduces completely or partially the gauge freedom of the theory. Although the gauge fixing must not influence physical observables, the proof of this statement is nontrivial in different formulations of quantum chromodynamics [4, 5]. It is noteworthy that the use of different gauges in QCD is an adjuvant possibility. In particular, it may reveal various aspects of such fundamental phenomenon as confinement [6]. We shall use the Lorentz gauge, µ A µ a x. The appropriate Green function is D µν x x η µν Dx x, η µν diag+,,, is the Minkowski metrics, Dx x 4π δ x x [ δt t x x + 8π x x + δt t + x x ] is the Green function of d Alembert equation. The use of the Coulomb gauge the equivalence of both gauges will be discussed below. The expression is only a formal solution of 5, because the gluon components jg ν a + j ν a of the current g j ν a depend on A µ a. Unfortunately, it is not possible to obtain an explicit, closed-form solution of the non-linear equation i.e. of 5 for A µa in terms of the quark fields q i at least we do not know how to do so. Thus, one must resort to approximation methods. Equation can be solved as an iterative series for A µ a cf. ref. []. The first order term in this sequence is just the expression, but with j µ a replaced by j µ a only, that is A µ a x d 4 x Dx x j µ a x 3 d 4 x Dx x q g s i q x γ ν t a ij j qx. The index in 3, as in all that follows, indicates that it is a first-order iterative expression. We note that the gluon fields are expressed here explicitly in terms of the quark fields i q the propagator D only. Correspondingly, to first order, this modifies the Lagrangian density, to L L a jµ A a µ + L A ν b + L I4 A ν b, 4, in 4, A ν b is as given in 3. The reformulated Lagrangian density L, so the corresponding Hamiltonian the action, is a functional of the gluon Green function Dx x the quark fields i q only. We shall refer to the theory based on the Lagrangian density L as the reduced model". At this point we might mention that in the QED or SU case, the non-abelian terms L I3 L I4 do not arise, L corresponds to the reduced QED Lagrangian. This reduced QED Lagrangian was used previously in the quantized Hamiltonian formalism to derive relativistic few-fermion equations, from them, the relativistic energy spectra for all bound states of positronium [3], muonium [4] negative positronium muonium ions [5]. The results are exact to Oα 4, including, for positronium, the field-theoretic virtual annihilation correction. 338

4 Jurij W. Darewych, Askold Duviryak 4. Analysis of the inter-quark interactions in the reduced model including the static limit We denote the action by S d 4 x Lx shall consider, in turn, each of the interaction terms of the action in the reduced model. Thus, the term corresponding to L I, eq., is S d 4 x a jµ x A a µ x 5 d 4 x d 4 x j µ a x Dx x j a µ x, the quark-field current j µ a x is given in eq.. Since j µ a g s, we see that S corresponds to an energy contribution of Og s. We also see that this interaction term describes the quark currents interacting via the gluon Green function D, which, in the quantised theory, would correspond to a one-gluon exchange inter-quark interaction; Fig. a. To underst the physical content of the interactions corresponding to S I n n, 3, 4, it is useful to consider the case of static sources in the non-relativistic limit, as was done for the scalar model [9, ]. For the static case, j µ a x j µ a x, hence the static version of S, eq. 5, is S It follows from that dt dt d 3 x d 3 x j µ a x Dx x j a µ x. 6 dt Dx x 4π x x. 7 Thus, in the static case, S, eq. 5, can be written as S H dt L dt H d 3 x d 3 x j µ a x It is clear from 8 that H potential energy function V x, x g s 4π 4π x x ja µ x. 8 corresponds to the two-point x x g s 4π x, 9 which is the non-relativistic limit of the one-gluon exchange interaction in coordinate representation. It reflects the Og s Coulombic" contribution to the interquark potential. Note that it depends only on the distance, x, between the points x x, as expected. Similarly, the component of the action corresponding to L I3, eq., is, in first order, S d 4 x L x 3 g s f a b c d 4 x A a b ν x Aµ x µ A ν c x. This term corresponds to an Og 4 s contribution to the energy. The Green function D appears in degree 3 in this expression, which reflects the 3-gluon vertex interaction corresponding to the degree 3 in A µ a " form of L ; Fig. b. In the static case, A a µ x d 4 x Dx x j a µ x becomes A a µ x d 3 x 4π we have used 7, thus t A a ν x, k A a ν x d 3 x d 4 x Dx x j a µ x d 3 x j a µ x dt Dx x j a µ x x x, 3 4π ja d 3 x ν x x k x x 4π ja ν x x k x k x x 3, 3 k,, 3. Therefore, in the static case, S g s f a b c Thus, by 3 3, H g s 4π f 3 a b c dt H 33 [ j b d 3 k x x x x g s 4π 3 f a b c d 4 x A a b ν x Ak x k A ν c x. [ ] j ν a d 3 x d 3 x x x x ] [ d 3 x 3 j c ν x 3 xk x3 k x x 3 d 3 x d 3 x d 3 x 3 j a ν x ] j k b x j ν c x 3 U k x, x, x 3,

5 Analysis of inter-quark interactions in classical chromodynamics U k x, x, x 3 d 3 x U 3 x, x, x 3 x k x k 3 x x x x x x 3 3 U 3 x x3 k, x, x 3, 35 d 3 x x x x x x x 3 d 3 v v v x v x 3 ; 36 here x mn x nm x m x n. Note that the integral in 36 is divergent so must be regularised this is discussed in ref. [9, ]. However the derivatives 35 are well-behaved finite. We see that the interaction term 34 corresponds to a three-point potential, V k x, x, x 3 g4 s 4π 3 U kx, x, x 3 37 g4 s 4π 3 x k 3 U 3 x, x, x 3, k,, 3,, by 3, H with 4 g s f a b c f a d e 4 d 3 x A b d µ x Ac ν x Aµ x A ν e x g s 44π f 4 a b c f a d e d 3 x d 3 x d 3 x 3 d 3 x 4 j b µ x j c ν x j µ d x 3 j ν e x 4 U 4 x, x, x 3, x 4, U 4 d 3 x x, x, x 3, x 4 x x x x x x 3 x x 4 d 3 v v v x v x 3 v x 4. 4 If we include the coupling constants related factors the corresponding potential-energy function is V I4 x, x, x 3, x 4 4 g 6 s 4π 4 U4 x, x, x 3, x 4. 4 This is an Og 6 s first-order-iterative four-point potential correction to the Coulombic two-point potential 9. which is an Og 4 s first-iterative-order correction" to the two-point Coulombic interaction V x, x given in eq. 9. Lastly, the component of the action corresponding to L I4, eq. 3, is, in first-iterative-order g x x x g 4 x x 3 S d 4 x L I4 x a b 4 g s f a b c f a d e 38 d 4 x A b d µ x Ac ν x Aµ x A ν e x, x g 6 x which is of Og 6 s of degree 4 in D, corresponding to a four gluon interaction vertex; Fig. c. In the static case, this term becomes S 4 g s f a b c f a d e d 4 x A b d µ x Ac ν x Aµ x A ν e x dt H, 39 x 3 x 4 c Figure. Feynman graphs generating few-point static potentials: a -gluon exchange generating -point potential of order g ; b 3-gluon vertex generating 3-point potential of order g 4 ; c 4-gluon vertex generating 4-point potential of order g 6 All the two-, three- four-point potentials are obtained with the use of the Lorentz gauge. Let us show that, within 34

6 Jurij W. Darewych, Askold Duviryak the first-iterative-order approximation, the Coulomb gauge fixing leads to the same result. The symmetric Green function D µν x x of the equation 5, which corresponds to the Coulomb gauge condition k A k a x, can be presented in the form: D x x δt t 4π x x, D kx x, D klx x { δ kl k l } Dx x, 43 Dx x is the Green function of d Alembert equation, is the Laplacian. Let us consider the potential A a µ x the prime refers to the Coulomb gauge generated by the static quark current j a µ x. For the temporal component we have: A a x [D j a d 3 x ]x d 4 x D x x j a x dt δt t a 4π x x j x A a x, 44 " denotes the convolution. The remaining components are presented as follows: A a k D kl j l a D j a k }{{} A a k k l D j l a }{{}. 45 The last term vanishes due to the conservation law 9 which in the first-iterative-order static approximation reduces to the equality k j k a. Thereby, the vectorpotential A a µ calculated in the Coulomb gauge coincides with that A a µ in the Lorentz gauge. As a consequence, the action terms S, S, S see eqs. 6, 3, 33 thus the two-, three- four-point potentials are the same. The integrals 35, 36 4 that define the three four-point potentials cannot, in general, be evaluated explicitly, that is they cannot be expressed in terms of common analytic functions at least we do not know how to do so. Rather, the evaluation of U k x, x, x 3 can be reduced to the computation of two single quadratures that of U 4 x, x, x 3, x 4 to a double quadrature, which must be done numerically. Nevertheless various general properties of these first-iterative-order non-abelian corrections to the Coulombic inter-quark potential can be readily established, analytical expressions can be obtained for particular situations. 5. Properties evaluation of the three- four-point potentials The potentials" U 3 x, x, x 3 U 4 x, x, x 3, x 4 are, in fact, functions of the distances x mn x m x n only as might be expected of a closed system. Owing to this fact, the three four point functions U 3 U 4 can be evaluated analytically for some particular cases. Thus, when all three distances are equal, i.e. x x 3 x 3 r, the regularised three-point function of eq. 36 becomes U 3 r 4π lnr/a, a is an arbitrary distance scale [9, ]. Similarly, when points x x 3 are coincident, i.e. x 3 x x 3, U 3 x 4π lnx /a, so U k x, x, x 3 x U 3 x, x, x 3 / x3 k x3 x U3 x, x, x / x k π x x k x x. This shows that the corresponding correction to the Coulombic one-gluon exchange potential 9 due to the cubic" interaction term 34-36, in the non-relativistic limit, is of the form cf. 37 V k x, x, x 3 x g4 s 4π x x k x x. 46 Similarly, for the particular case x x 3 x x 4, there is only one distance, x, between the two pairs of coincident points, the four-point potential function U 4, eq. 4, so V I4, eq. 4, can be evaluated explicitly: U 4 x V I4 x g6 s 4 5 π d 3 v v v + x π3 x, hence x. 47 Although the expressions are only segments of the three four point potentials V k x, x, x 3, eq n 37, V I4 x, x, x 3, x 4, eq n 4, they suggest that their behaviour is Coulomb-like in general. Note that these corrections are Og 4 s Og 6 s respectively. The remaining part of this section devoted to the evaluation of the three- four-point potentials for arbitrary values of their arguments. Components of the three-point potential 35 form the vector potential: Ux, x, x 3 x 3 U 3 x, x, x 3, 48 34

7 Analysis of inter-quark interactions in classical chromodynamics the function U 3 x, x, x 3 see eq.36 is studied in [9]. Particularly useful is the following representation of this function: R cos φ + ξ + η, U 3 x, x, x 3 d 3 k d 3 x e k π 3/ x x k x x k3 x x 3 d 3 k e k k x +k k 3 x 3 +k k 3 x 3 /k k 3 d ˆk dk k e ˆk ˆk x +ˆk ˆk 3 x 3 +ˆk ˆk 3 x 3 k, 49 x mn x m x n m, n,, 3, ˆk k/k, k k k + k + k 3 d ˆk denotes integration over the unit sphere in 3D k-space. The integral 49 as well as 36 is divergent needs to be regularized. One way is to split it into two terms: U 3 Ũ + U, Ũx, x 3, x 3 is a finite function of three scalar arguments U Ua, b, c is an infinite constant" a, b, c are arbitrary constants [9]. The insertion of 49 into the r.h.s. of 48 eliminates the infinite constant U yields the formula: U U3 x 3 x 3 d ˆk dk k e ˆk ˆk x +ˆk ˆk 3 x 3 +ˆk 3 ˆk x 3 k d ˆk ˆk 3 ˆk x 3 + ˆk x 3 dk k e ˆk ˆk x + k x 3 + x 3 I, 5 I n d ˆk ˆk n ˆk 3 ˆk ˆk x + ˆk ˆk 3 x 3 + ˆk ˆk 3 x 3, n,. 5 Next, we introduce angular variables {θ, φ} on the unit sphere in 3D k-space, so that ˆk sin θ cos φ, ˆk sin θ sin φ, ˆk 3 cos θ. Then π π dφ cos φ J, I π dφ sin φ J, 5 dθ sin θ cos θ J x sin θ cos φ sin φ + cos θx3 cos φ + x3 sin φ 8 [ ] sin φ x R R arctan R 53 sin φ ξ x 3 x 3, x η [x 3 + x 3 x ][x x 3 x 3 ]. 54 x 4 Inserting 53 into 5 using the integration variable s cos φ yields the quadratures: 8 I 8 x x [ ] ds + s R s + s R arctan R, s [ ] ds s R + s s R arctan R. s 55 Note that for the special case x 3, the expressions 5 to 55 yield the result 46. In the general case, that is, for arbitrary values of x, x 3 x 3, the integrals 55 need to be evaluated numerically. The behavior of the three-point vector potential as a function of x 3 is illustrated in Fig.. We note the following symmetry properties of the vector potential: translational invariance: Ux +λ, x +λ, x 3 +λ Ux, x, x 3, λ R 3 ; rotational covariance: URx, Rx, Rx 3 RUx, x, x 3, R SO3; partial permutational invariance: Ux, x, x 3 Ux, x, x 3 ; scaling transformation: Uλx, λx, λx 3 λ Ux, x, x 3, λ R +. These properties follow from the properties of the threepoint scalar potential 36 stated in [9, ]. We also note that the scaling transformation of the potential U reflects its Coulomb-like behavior. The four-point scalar potential 4 can be treated in a similar manner: U 4 d 3 x x,..., x 4 x x x x 4 56 d 4 k d 3 x e k π x x k4 x x 4 π d 3ˆk dk e X k d 3ˆk X, 34

8 Jurij W. Darewych, Askold Duviryak This double integral can be evaluated numerically. Evaluation of 57 for the case x x 3 x x 4 gives the same result as 47. Finally we note, that the four-point potential 57 possesses the same translational invariance scaling transformation properties as the three-point one does. In addition, it is invariant under arbitrary rotation permutation of its arguments: rotational invariance: U 4 Rx, Rx, Rx 3, Rx 4 U 4 x, x, x 3, x 4, R SO3; complete permutational invariance: U 4 x, x, x 3, x 4 U 4 x, x 3, x, x 4 U 4 x, x, x 3, x 4. Figure. Two-dimensional section of the vector potential Ux, x, x 3 as a function of x 3 r for fixed x x. The arrows indicate the direction of the vector field Ux, x ; r. Note that it is invariant with respect to rotations about the axis x x. The dashed line between x x corresponds to U X ˆk ˆk x + ˆk ˆk 3 x 3 + ˆk ˆk 4 x 4 + ˆk ˆk 3 x ˆk ˆk 4 x 4 + ˆk 3 ˆk 4 x 34, ˆk n k n /k n,..., 4, k k + + k 4, d 3ˆk denotes integration over a unit hyper-sphere in 4D k- space. Next, we introduce angular variables {χ, θ, φ} on the unit hyper-sphere in 4D k-space, so that ˆk sin χ sin θ cos φ, ˆk sin χ sin θ sin φ, ˆk3 sin χ cos θ, ˆk 4 cos χ, d 3ˆk π π π dφ sin θ dθ sin χ dχ. Then using the integration variables u cos χ, v cos θ, w cos φ reduces the integral 56 to the form: I U 4 dw x,..., x 34 4 w dv I, 57 du A + u B B ln A + B + B, A A { 4 x w v + + [x 3 + x 3 + x 3 x 3w]v } v, B [x 4 + x 4 + x 4 x 4w] v + x 34v A. 6. Concluding remarks We have used an approximate, iterative solution of the non-linear classical equations of motion of QCD to derive expressions for the interaction terms corresponding to the non-abelian terms 3 of the QCD action. In first iterative order, cf. equation 3, these turn out to be expressions involving products of three four onegluon exchange Green functions, corresponding to three four-gluon interaction vertices see Fig., cf. equations 3 38 respectively. We have examined these non-abelian terms in the static, non-relativistic limit found them to be three- fourpoint static potentials, 37 4, that depend on the inter point distances only. Although we could not express these three- four-point potentials in terms of common analytic functions in general, we could do so for some restricted sections these indicate that the potentials V k x, x, x 3 V 4 x, x, x 3, x 4 are Coulomb-like in general. The derived three- four-point cluster corrections together with the one-gluon exchange interaction could be used as a short-range contribution in potential models of baryons, tetra-quarks etc. The quantized theory, based on the reduced Lagrangian 4, can be used to derive relativistic few-quark equations, as was done for the scalar theory with non-linear mediating fields [9, ]. This shall be left for future work. References [] R. Jackiw, Rev. Mod. Phys. 49, [] J. W. Darewych, A. Duviryak, Phys. Rev. A 66, 3 343

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