Spinning pairs: Supporting ${}^3{P}_0$ quark-pair creation from Landau-gauge Green’s functions

Abundant phenomenology suggests that strong decays from relatively low-excitation hadrons into other hadrons proceed by the creation of a light quark-antiquark pair with zero total angular momentum, the so called ${}^3{P}_0$ mechanism originating from a scalar bilinear. Yet the quantum chromodynamics (QCD) interaction is perturbatively mediated by gluons of spin one, and QCD presents a chirally symmetric Lagrangian. Such scalar decay term must be spontaneously generated upon breaking chiral symmetry. We attempt to reproduce this with the help of the quark-gluon vertex in Landau gauge, whose nonperturbative structure has been reasonably elucidated in the last years, and insertions of a uniform, constant chromoelectric field. This is akin to Schwinger pair production in quantum electrodynamics (QED), and we provide a comparison with its two field-insertions diagram. We find that, the symmetry being cylindrical, the adequate quantum numbers to discuss the production are rather ${{}^3\mathrm{\Sigma }}_0$, ${{}^3\mathrm{\Sigma }}_1$, and ${{}^3\mathrm{\Pi }}_0$ as in diatomic molecules, and we indeed find a sizeable contribution of the third decay mechanism, which may give a rationale for the ${}^3{P}_0$ phenomenology, as long as the momentum of the produced pair is at or below the scale of the bare or dynamically generated fermion mass. On the other hand, ultrarelativistic fermions are rather ejected with ${{}^3\mathrm{\Sigma }}_1$ quantum numbers. In QED, our results suggest that ${{}^3\mathrm{\Sigma }}_0$ dominates, whereas the constraint of producing a color singlet in QCD leads to ${{}^3\mathrm{\Pi }}_0$ dominance at sub-GeV momenta.

Figure 1

Depiction of an OZI-allowed decay by the creation of a valence $q\overline{q}$ pair via a multilegged transition amplitude.

Figure 2

Minimum $A_0$ field insertions coupling to a fermion pair in quantum electrodynamics.

Figure 3

A resummation of all ladder diagrams with electric field insertions in QED continues reproducing the zero in Eq. (4).

Figure 4

Comparison of the spin-triplet orbital-scalar vs. spin-triplet orbital-vector QED pair production in the uniform electric field. Left: three-dimensional rendering comparing ${{}^3\mathrm{\Pi }}_0$ and ${{}^3\mathrm{\Sigma }}_1$ production against the electron-positron momentum and the collinear fraction $\chi$ that the first field insertion provides. Right: bidimensional projection with $\chi =0$ as function of the fermion pair momentum. In both cases, the momentum ($x$-axis) is in units of the fermion mass, that is, $m=1$.

Figure 5

Representation of a meson and its chromoelectric flux tube, the blue cylinder (of radius $R$ and length $L$) depicts the region where the electric field can be approximated as constant. We will study the creation of quark pairs in this cylindrical region.

Figure 6

The electric (blue colored) and magnetic field (gold colored) corresponding to the potentials in Eq. (41) for $\kappa =1$ (left) and $\kappa =7$ (right). The more intense the fields, the bigger the arrowheads of the vectors are.

Figure 7

Lattice data for the quark mass function $M(p_E^2)$ from [51]. The green line is the continuum parametrization of Eq. (45), normalized so that $M(0)=300\mathrm{MeV}$. In this way, it nicely captures the qualitative behavior of the numeric computation.

Figure 8

Quark-gluon vertex and our choice of momentum flow.

Figure 9

Evaluation of the absolute values of the vertex model functions $g_i(x)$. These form factors have poles at the value $x_i^{*}$ with $x_1^{*}\simeq -1.46$ (outside the graph and irrelevant for light quark physics), $x_{2,3}^{*}\simeq -6.48\times {10}^{-2}$ and $x_{4,7}^{*}\simeq -2.88\times {10}^{-1}$, seen as the mountainlike enhancements in the three-dimensional views, and as the asymptotes in the two-dimensional cut taken along the real axis.

Figure 10

The perturbative octet-channel quark-gluon scattering kernel that gives rise to pair creation is chirally invariant. However, in the skeleton expansion, both the propagator and the two vertices which appear in its skeleton expansion [82] acquire chiral-symmetry breaking parts conveniently generated by their respective Dyson-Schwinger equations assisted by lattice data.

Figure 11

The perturbative singlet-channel quark-gluon scattering kernel that gives rise to pair creation, analogous to the Breit-Wheeler process, is chirally invariant save for the small current quark mass in the fermion propagator. However, in the skeleton expansion, both the propagator and the two vertices acquire chiral-symmetry breaking parts conveniently generated by their respective Dyson-Schwinger equations assisted by lattice data.

Figure 12

Comparison of the ${{}^3\mathrm{\Pi }}_0$ and ${{}^3\mathrm{\Sigma }}_1$ angular momentum components in the QCD skeleton approximation here developed, in four kinematic sections. The $q\overline{q}$ pair is produced in its center of mass frame. The ${{}^3\mathrm{\Pi }}_0$ vertex is seen to be dominant for small quark momentum ($OX$ axis, in units of the quark mass) and the ${{}^3\mathrm{\Sigma }}_1$ dominates for large momenta. $|\mathit{p}|$ is in units of $m$, except for the last plot in which arbitrary units are employed (with the scale ultimately controlled by ${\mathrm{\Lambda }}_A$, ${\mathrm{\Lambda }}_B$, etc. from the Schwinger-Dyson obtained parametrization from Sec. 5).

Figure 13

Absolute values (in log scale, and up to factors common to both) of the ${{}^3\mathrm{\Pi }}_0$ and ${{}^3\mathrm{\Sigma }}_1$ contributions to our computed pair production kernel at threshold (where $\mathit{p}=0$) as function of $m^2$ on the physical Minkowski side. The ${{}^3\mathrm{\Pi }}_0$ structure dominates for all plotted values. This is in spite and independently of the (perhaps artificial) pole of the propagator parametrization (that induces a pole of the amplitudes), when $x_{2,3}^{*}\simeq -6.48\times {10}^{-2}$. Close to $x=-0.0418857$ both amplitudes almost vanish at the $\mathit{p}=0$ threshold.