Nonperturbative determination of the vacuum pion-field distribution underlying soft-pion unitarity damping in hard-hadron reactions

In 1971 Weinberg showed that the exchange of unlimited numbers of soft, massless virtual pions multiplies the component of a hard-hadron scattering amplitude belonging to the $(N/2,N/2)$ representation of chiral $SU(2)\times SU(2)$ by a damping factor $D_N$ expressible as oscillatory moments of a probability distribution $\mathscr{P}(\vec{\pi}^{\,2})$: the vacuum distribution of the (ultraviolet-cutoff) pion field, whose moments are the vacuum expectation values of products of field components at a spacetime point. All quantitative conclusions rest on assumed qualitative properties of $\mathscr{P}$ — differentiability with vanishing derivatives, smoothness, and flatness over ranges of $\vec{\pi}^{\,2}$ much larger than the $F_\pi^2$ scale of the damping integrals — motivated only by the free-field Gaussian. The open problem is a first-principles determination of $\mathscr{P}(\vec{\pi}^{\,2})$ from the nonperturbative chiral theory (today, QCD treated nonperturbatively), in order to verify or refute those assumptions, fix the parity-dependent constants $c$ in the large-$N$ law $D_N\sim c/(N+1)$ and the finite-$N$ corrections, decide the fate of the strong-damping limit $D_N\to 1/(N+1)$ ($N$ even), $0$ ($N$ odd) as the cutoff $\Lambda\to\infty$ and of the resulting multiplicative conservation law, and extend the approach to $SU(3)\times SU(3)$ and to massive pions. No published work determines or refutes this problem; it remains open.

Background

At the end of the current-algebra era, Weinberg studied how the exchange of unlimited numbers of soft (very low-momentum), massless virtual pions modifies an arbitrary 'hard' hadron reaction — a process slow enough that the pions see the hadrons as static sources. 'Chiral $SU(2)\times SU(2)$' is the symmetry algebra generated by the isospin currents and the axial-vector currents of the strong interaction; its irreducible representations are labeled $(N/2,N/2)$, and the terms of the $S$ matrix (the scattering operator) can be decomposed accordingly. In a first paper (Weinberg, 1970), using an eikonal approximation (a high-energy/short-wavelength expansion that sums repeated exchange along a fast particle's trajectory) and a pion field of the CCWZ type — the canonically preferred nonlinear pion field defined by Coleman, Wess and Zumino and by Callan, Coleman, Wess and Zumino (1969), for which the chiral Lagrangian takes its standard geometric form — he found that multipion exchange multiplies each $(N/2,N/2)$ term by a damping factor $D_N\lesssim 1$. Brown (1970) then showed that those tree-level results depended spuriously on the definition of the pion field, and Weinberg conceded the point.

The 1971 paper rederives the effect keeping all multipion interactions and loops, so that the answer is field-redefinition covariant. The vacuum expectation values that enter are reexpressed through a probability distribution $\mathscr{P}(\vec{\pi})$: by definition, the mean of any product of components of the pion field $\vec{\pi}(0)$ at a point, computed in the full interacting vacuum, equals the corresponding moment of $\mathscr{P}$; isospin conservation forces $\mathscr{P}$ to depend only on $\vec{\pi}^{\,2}$. If one keeps only tree graphs, $\mathscr{P}$ is the free-field Gaussian with width set by the ultraviolet cutoff $\Lambda$ on the virtual-pion momenta, $\langle\pi_a(0)\pi_b(0)\rangle=(\Lambda^2/8\pi^2)\delta_{ab}$; the paper deliberately does not assume this form, using it only as a qualitative guide. The damping factors are oscillatory integrals of $\mathscr{P}(\vec{\pi}^{\,2})$ against phases whose scale is set by the pion decay constant $F_\pi=190$ MeV (the normalization of the axial current). Under two qualitative assumptions — that $P(\vec{\pi}^{\,2})\,\vec{\pi}^{\,2}$ (where $P$ denotes $\mathscr{P}$ written as a function of the scalar $\vec{\pi}^{\,2}$, the source's own notation) is at least twice differentiable with derivatives vanishing at large $\vec{\pi}^{\,2}$ (infinite differentiability for the sharper remainder estimate), and, for the strong-damping limit, that $\mathscr{P}$ is smooth and flat over a range of $\vec{\pi}^{\,2}$ much larger than the $F_\pi^2$ scale of the phase, expected when $\Lambda\gg F_\pi$ — integration by parts and the Riemann–Lebesgue lemma (the theorem that the Fourier-type integral of a smooth, decaying function vanishes at high oscillation frequency) give $D_N\sim c/(N+1)$ for $N\to\infty$ through even or odd values, with parity-dependent constants $c$ given by convergent sums of the oscillatory integrals, and, in the limit $\Lambda\to\infty$, $D_N\to 1/(N+1)$ for even $N$ and $0$ for odd $N$. If the odd-$N$ terms really vanish, the whole $S$ matrix must obey a multiplicative conservation law, $\exp(i\pi\,\vec{e}\cdot\vec{B}_x)\,S=\pm S$ for any unit isospin vector $\vec{e}$, where $\vec{B}_x$ is the net chiral boost generator (the sum of all final-hadron chiral-boost matrices minus that of all initial-hadron ones, acting as a net axial charge) — a selection rule saying that amplitudes conserve a $\mathbb{Z}_2$-valued quantum number counting the parity of the chiral representation.

The paper is explicit that evaluating these vacuum expectation values with all graphs is 'a formidable requirement' that limits it to qualitative results: nothing is known about the actual shape of $\mathscr{P}(\vec{\pi}^{\,2})$ beyond the free-field suggestion. The question this raises — compute $\mathscr{P}$ from first principles, or otherwise decide whether its qualitative properties hold — remains open. Modern context bears on it: chiral perturbation theory (the systematic low-energy effective field theory of QCD founded by Weinberg's 1979 'Phenomenological Lagrangians') computes chiral vacuum expectation values order by order but is controlled only below the natural cutoff $\sim 4\pi F_\pi$; the 1990s disoriented-chiral-condensate program studied pion-field distributions, but in dynamical collision fireballs rather than the vacuum; and Banks and Fischler (2016), citing the 1970–71 papers, note that the corresponding massless-particle $S$-matrix questions 'have never been completely solved'. Real pions, of course, are massive — the source works in the chiral limit throughout.

Problem Statement

Determine, from first principles within the chiral theory of the pion (in modern terms, QCD treated nonperturbatively), the probability distribution $\mathscr{P}(\vec{\pi}^{\,2})$ introduced by Weinberg's 1971 statistical approach to soft-pion unitarity damping — the vacuum distribution, at fixed ultraviolet cutoff $\Lambda$ and for a specified (e.g. CCWZ) pion-field definition, whose moments are the vacuum expectation values of products of the components of $\vec{\pi}(0)$ — or at least the load-bearing qualitative properties named in component (1) below; and use that determination to settle the status of the damping mechanism. The components of this task, which share the single objective of replacing the paper's assumed qualitative properties of $\mathscr{P}$ by a first-principles determination, are: (1) verify or refute the assumed regularity of $P(\vec{\pi}^{\,2})\,\vec{\pi}^{\,2}$ (twice differentiability with vanishing first derivative, and infinite differentiability with all derivatives vanishing, at large $\vec{\pi}^{\,2}$) and the strong-damping assumption that $\mathscr{P}$ is smooth and flat over a range of $\vec{\pi}^{\,2}$ much larger than the $F_\pi^2$ scale entering the damping integrals, expected when $\Lambda\gg F_\pi$; (2) determine the parity-dependent constants $c$ in the large-$N$ law $D_N\sim c/(N+1)$ for $N\to\infty$ through even and odd values, and the finite-$N$ corrections to it; (3) decide whether, in the $\Lambda\to\infty$ limit, $D_N\to 1/(N+1)$ for even $N$ and $0$ for odd $N$, and hence whether hard-hadron amplitudes obey the multiplicative conservation law $\exp(i\pi\,\vec{e}\cdot\vec{B}_x)\,S=\pm S$ — including whether that limit is a well-defined limit of the underlying theory at all — and work out the phenomenological consequences for hadron production and scattering if it holds; (4) extend the statistical approach to other chiral groups such as $SU(3)\times SU(3)$ and to non-zero pion mass. A first-principles determination, at finite cutoff $\Lambda$, of $\mathscr{P}(\vec{\pi}^{\,2})$ or of the load-bearing qualitative properties named in component (1), or a refutation of a load-bearing property or of the existence of the $\Lambda\to\infty$ limit — which would refute the mechanism in its strong (infinite-cutoff) form while settling what survives at finite cutoff — resolves the task; components (2) and (3) are consequences to be drawn from that determination where it supports them, and component (4) is an optional extension of the approach to other chiral groups and to non-zero pion mass.

References

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