Janus-Facedness of the Pion: Analytic Instantaneous Bethe-Salpeter Models

Inversion enables the construction of interaction potentials underlying - under fortunate circumstances even analytic - instantaneous Bethe-Salpeter descriptions of all lightest pseudoscalar mesons as quark-antiquark bound states of Goldstone-boson nature.


Introduction: quark-antiquark bound states of Goldstone-boson identity
Within quantum chromodynamics, the pions or, as a matter of fact, all light pseudoscalar mesons must be interpretable as both quark-antiquark bound states and almost massless (pseudo) Goldstone bosons related to the spontaneously (and, to a minor extent, also explicitly) broken chiral symmetries of QCD.
Relativistic quantum field theory describes bound states by their Bethe-Salpeter amplitudes, Φ(p), controlled by the homogeneous Bethe-Salpeter equation defined (for two bound particles of individual and relative momenta p 1,2 and p) by their full propagators S 1,2 (p 1,2 ) and the integral kernel K(p, q) that encompasses their interactions (notationally suppressing dependences on the total momentum p 1 +p 2 ): The application of suitably adapted inversion techniques [1] allows us to retrieve all the underlying interactions -rooted, of course, in QCD -analytically in the form of a (configuration-space) central potential V(r), r ≡ |x|, from presumed solutions to the Bethe-Salpeter equation [2]. By that, we are put in a position to construct exact analytic Bethe-Salpeter solutions for all massless pseudoscalar mesons [3] in the sense of establishing in a rigorous manner the analytic relationships between interactions and resulting solutions: all analytic findings [4] can be confronted with associated numerical outcomes [5].

Sequence of simplifying assumptions crucial for the inversion formalism
By a few steps, we cast the Bethe-Salpeter equation into a shape that allows us to talk about potentials.
1. Assuming, for each involved quark, both instantaneous interactions and free propagation, with a mass dubbed as constituent, simplifies the Bethe-Salpeter equation to a bound-state equation for the Salpeter amplitude φ( p), obtained from the Bethe-Salpeter amplitude by integration over p 0 : Generically, for a spin-1 2 fermion and a spin-1 2 antifermion of equal constituent masses m, bound to a spin-singlet state (which, for instance, clearly is the case for any such pseudoscalar state), its three-dimensional wave function involves just two independent components, here called ϕ 1,2 ( p): 2. Upon supposing that the quark interactions in the kernel respect spherical and Fierz symmetries, our bound-state equation for φ( p) collapses to the system of coupled radial eigenvalue equations for the bound-state mass eigenvalue M [6]. Therein, V(r) enters via its Fourier-Bessel transform 3. In the strictly massless (Goldstone) case M = 0, the system decouples: one Salpeter component, Denoting the Fourier-Bessel transform of the kinetic term E(p) ϕ 2 (p) by T (r), the potential V(r) may be simply read off from the configuration-space representation of this bound-state equation:

Constraints on lightest-pseudoscalar-meson Bethe-Salpeter amplitudes
Information on the input Salpeter component ϕ 2 (p) can be gained from the full quark propagator S (p), which is determined by its mass function M(p 2 ) and its wave-function renormalization function Z(p 2 ): Studies of S (p) within the Dyson-Schwinger framework, preferably done in Euclidean space signalled by underlined quantities, allow for pivotal insights. In the chiral limit, a Ward-Takahashi identity links [7] this quark propagator to the flavour-nonsinglet pseudoscalar-meson Bethe-Salpeter amplitude [3]: First, in order to devise analytically accessible scenarios, we exploit two crucial pieces of information: 1. In the chiral limit, phenomenologically sound Dyson-Schwinger studies [8] imply, for the quark mass function M(k 2 ), at large Euclidean momenta k 2 a decrease essentially proportional to 1/k 2 .
2. From axiomatic quantum field theory, we may deduce [9] that the presence of an inflection point at finite space-like momenta k 2 > 0 in the quark mass function M(k 2 ) entails colour confinement.
Of course, any imposition of such kind of requirements on M(k 2 ) has to be reflected by Φ(k). An ansatz for Φ(k) compatible with both constraints, involving a mass parameter, µ, and a mixing parameter, η, is An integration of this Φ(k) with respect to the time component of the Euclidean momentum k results in in configuration space expressible in terms of modified Bessel functions of the second kind K σ (z) [10]: For η values satisfying η < −1 or η > 0, ϕ 2 (r) has one zero, which clearly induces a singularity in V(r).

Analytic outcomes [3, 4] for interquark potentials exhibiting confinement
For a few particular values of the dimensionless ratio m/µ, the analytic expression of V(r) can be found [3,4]. (Throughout this section, any quantity has to be understood in units of the adequate power of µ.) As a consequence of our ansatz for Φ(k), giving rise to the particular form (1) of ϕ 2 (r), for η −1 each extracted V(r) will develop, at the spatial origin r = 0, a logarithmically softened Coulomb singularity:

Analytically manageable scenario of massless quarks, i.e., of constituent mass m = 0
For our choice of ϕ 2 (r), V(r) involves both modified Bessel (I n ) and Struve (L n ) functions [10] (n ∈ N), and rises in a confinement-betraying manner to infinity either at the zero of ϕ 2 (r) or for r → ∞ (Fig. 1):

Analytically expressible observation for quarks with common constituent mass m = µ
For m = µ, the kinetic term T (r) is a mixture of Yukawa and exponential behaviour, whence (cf. Fig. 2)

Reliability check of findings: numerical determination of the potential [5]
Our findings may be scrutinized by use of the chiral-limit quark mass function's pointwise form M(k 2 ), provided graphically in Ref. [8] and shown in Fig. 3 as M(k) with k ≡ (k 2 ) 1/2 , which we parametrize by Note that the product of the two exponents in the second term above yields 1.48×0.752 ≈ 1.1, which is pretty close to unity, as demanded by the large-k constraint. Feeding this M(k) parametrization into our inversion procedure, we obtain potentials that are finite at r = 0 and, for sufficiently small m, rise with r to infinity but, for large m, remain negative, as illustrated in Fig. 3 for selected constituent mass values.