Zero mode solutions of quark Dirac equations in QCD as the sources of chirality violating condensates

It is demonstrated, that chirality violating condensates in massless QCD arise from zero mode solutions of Dirac equations in arbitrary gluon fields. Basing of this idea, the model is suggested, which allows one to calculate quark condensate magnetic susceptibilities in the external constant electromagnetic field.

where G n µν is gluon field tensor, the sum is over quark flavours.
and A n µ is the gluon field. Pay attention, that in Euclidean formulation of QCD ψ is replaced by ψ + . (The review of Euclidean formulation of QCD and instantons is given in [1], see especially [2].) The Dirac equation for massless quark in Euclidean space time has the form: − iγ µ ∇ µ ψ n (x) = λ n ψ n (x) where ψ n (x) and λ n are the eigenfunctions and eigenvalues of the Dirac operator −∇ = −iγ µ ∇ µ . Expand the quark fields operators into the left and right ones where Then for nonzero λ n the Lagrangian and the action reduces to the sum of two terms completely symmetric under interchange L ←→ R. Therefore the solutions of the equations for left and right quark fields are also the same -the states, constracted from left and right quarks are completely symmetrical. This conclusion was obtained for fixed gluon field. It is evident, that the averaging over the gluon fields does not change it. Quite different situation arises in case λ 0 = 0. The contribution of this term to the Lagrangian: is equal to zero and no conclusion can be done about the symmetry of states build from left and right quark fields.
One of the consequences from the said above is that all chirality violating vacuum condensates in QCD arise from zero mode solutions of Dirac equations (3).
These general arguments are supported by the well known facts: 1. The general representation of the trace of quark propagator S(x) is expressed through the spectral function ρ(λ) as a function of eigenvalues λ (Källen-Lehmann representation): At x 2 = 0 ∆(x 2 , λ) reduces to δ(λ) and we have (in Minkowski space-time): (The Banks-Casher relation [3]).
Basing on the statements, presented above, let us formulate the model for calculation of chirality violating vacuum condensates in QCD. Suppose, that vacuum expectation value (v.e.v.) of the chirality violating operator O c.v. is proportional to matrix element ψ + 0 O c.v. ψ 0 , where ψ 0 is the zero-mode solution of Eq.(3) in Euclidean space-time: ψ 0 depends on x, on the position of the center of the solution x c , as well as on its size ρ: ψ 0 = ψ 0 (x − x c , ρ). Eq. (10) must be integrated over x c , what is equivalent to integration over x − x c . (In what follows the notation x will be used for x − x c .) We assume, that ρ =Const and find its value from comparison with the known v.e.v.'s. Finally, introduce in (10) the coefficient of proportionality n. So, our assumption has the form: Our model is similar to delute instanton gas model [6], where x c is the position of instanton center. Unlike the latter, where the instanton density has dimension 4, n has dimension 3 and may be interpreted as the density of zero-modes centers in 3-dimension space. Note, that the left-hand side of (11) is written in the Minkowski space-time, while the right-hand side in Euclidean ones. (The sign minus is put in order to have n positive.) For x and ρ-dependens of ψ 0 (x, ρ) we take the form of the zero-mode solution in the field of instanton in SU (2) colour group: where χ 0 is the spin-colour isospin (| T |= 1/2) wave function, corresponding to the total spin I + T = J equal to zero, J = 0. ψ 0 (x, ρ) is normalized to 1: Consider first the quark condensate 0 |qq | 0 , the most important chirality violating v.e.v., determining the values of baryon masses [7]- [9]. (Here q = u, d are the fields of u, d-quarks). In this case O c.v. = 1 and in accord with (The integration over SU (2) subgroup position in SU (3) colour group as well as anti-instanton contribution are included in the definition of n.) The anomous dimension of quark condensate is equal to 4/9.) According to (14) n has the same anomalous dimension. The size ρ of the zero-mode wave function can be found by calculation in the framework of our model of the v.e.v.
Working in the SU (2) colour group, substitute λ n by τ a (a = 1, 2, 3) and take for G a µν the instanton field where the parameter η aµν were defined by ′ t Hooft [12] (see also [2]). The substitution of (12) and (16) into (11) gives after simple algebra Therefore, We are now in a position to calculate less well known quantities -the magnetic susceptibilities of quark condensate, induced by external constant electromagnetic field.
The dimension 3 quark condensate magnetic susceptibility is defined by [13]: where quarks are considered as moving in external constant weak electromagnetic field F µν and e q is the charge of quark q in units of proton charge (the proton charge e is included in the definition of F µν ). The left-hand side of (19) violates chirality, so it is convenient to separate explicitly the factor 0 |qq | 0 in the right-hand side. It was demonstrated in [13] that 0 |qσ µν q | 0 F is proportional to the charge e q of the quark q. A universal constant χ is called the quark condensate magnetic susceptibility.
Let us determine the value of χ in our approach. For this goal it is necessary to consider Eq.3 in the presence of external constant electromagnetic field F µν and to find the first order in F µν correction to zero mode solution (12).
This can be easily done by representing ψ as where ψ 0 is given by (12) and ψ 1 represents the proportional to F µν correction. Substitute (20) in Eq.3 added by the term of interaction with electromagnetic field, neglect ψ 1 in this term and solve the remaining equation for ψ 1 (x, ρ)).
The result is: where σ a are Pauli matrices. The matrix element ψ + σ µν ψ appears to be equal: (The properties of η aµν symbols [12], [2] were exploited.) The v.e.v. (19) in the Minkowski space-time is given by: (The normalization condition (13) for ψ 0 (x, ρ) was used.) It is convenient to express n through quark condensate by (14), use the notation x 2 = r 2 , where r is the radius-vector in 4-dimensional space. Then according to (19) we have: The integral (24) is quadratically divergent at large r. So, the cut-off R is introduced. Its value can be estimated in following way. The volume occupied by one zero-mode in 3-dimensional space is approximately equal to 1/n (the volume of the Wigner-Seitz cell). So, for cut-off radius square R 2 in four-dimensions we put where the factor 4/3 corresponds to transition from 3 to 4 dimensions. The calculation of the integral (24) at the values of parameters ρ (18) and R 2 (25) gives The quark condensate magnetic susceptibility was previously calculated by QCD sum rule method [14]- [17] and expressed through the masses and coupling constants of mesonic resonances. The recent results are: [16]; χ(1 GeV) = −2.85 ± 0.5GeV −2 [17] (27) (The earlier results, obtained by the same method, were: χ(0.5 GeV) =−5.7 GeV −2 [14] and χ(1 GeV) = -4.4±0.4 GeV −2 [15]. The anomalous dimension of χ is equal to -16/27. It was accounted in [14]- [17], but not in the presented above calculation. (In some of these papers, the α s -corrections and continuum contribution, were also accounted.) One can believe, that the value (26) refer to 1 GeV, because the value of quark condensate (14) refer to this scale and also because the scale 1 GeV is a typical scale, where, on the one hand, the zero-modes and quark condensates are quite important (see, e.g. [10]) and, on the other, the instanton gas model is valid [6]. Since the integral is quadratically divergent it is hard to estimate the accuracy of (26). I guess, that it is not worse, than 30-50%. In the limit of this error the result (26) is in an agreement with those found in phenomenological approaches.
Turn now to quark condensate magnetic susceptibilities of dimension 5, κ and ξ defined in Ref. [13] g 0 |q Perform first the calculation of κ. In this case the expression of ψ 1 (x, ρ) (21) must be multiplyed by the additional factor: 1 2 τ b G b µν where G b µν is given by (16) and the indices µ, ν in (21) are changed to λ, σ. In the further calculation it will be taken into account, that χ 0 in (12) corresponds to total spin-colour isospin J = 0 and consequently In the relation the last term drops out after summation of zero-modes from instanton and anti-instanton configuration. The final result for κ is: where z = R 2 /ρ 2 = 12.7. Numerically, we have: The calculation of ξ is very similar to those of κ and the result is The values of κ and ξ only logarithmically depend on the cut-off. But unfortunately the logarithm in (32) is not very large and its main part is compensated by the term −13/6, appearing in (32). So, the accuracy of (33),(34) can be estimated as about 30%. The phenomenological determination of 5-dimensional quark condensate magnetic susceptibilities was performed by Kogan and Wyler [18] along the same lines, as it was done in [14], [15]. No anomalous dimensions were accounted. The results of [18] are: As can be seen, they are in a good agreement with (33),(34). The 5-dimensional quark condensate magnetic susceptibilities play a remarkable role in determination of Λ-hyperon magnetic moment [19].