Form factors of the gauge-invariant three-gluon vertex

The gauge-invariant three-gluon vertex obtained from the pinch technique is characterized by 13 nonzero form factors, which are given in complete generality for unbroken gauge theory at one loop. The results are given in $d$ dimensions using both dimensional regularization and dimensional reduction, including the effects of massless gluons and arbitrary representations of massive gauge bosons, fermions, and scalars. We find interesting relations between the functional forms of the contributions from gauge bosons, fermions, and scalars. These relations hold only for the gauge-invariant pinch-technique vertex and are $d$-dimensional incarnations of supersymmetric nonrenormalization theorems which include finite terms. The form factors are shown to simplify for $\mathcal{N}=1$, 2, and 4 supersymmetry in various dimensions. In four-dimensional nonsupersymmetric theories, eight of the form factors have the same functional form for massless gluons, quarks, and scalars, when written in a physically motivated tensor basis. For QCD, these include the tree-level tensor structure which has prefactor ${\beta }_0=(11N_c-2N_f)/3$, another tensor with prefactor $4N_c-N_f$, and six tensors with $N_c-N_f$. In perturbative calculations our results lead naturally to an effective coupling for the three-gluon vertex, $\tilde{\alpha }(k_1^2,k_2^2,k_3^2)$, which depends on three momenta and gives rise to an effective scale $Q_{\mathrm{eff}}^2(k_1^2,k_2^2,k_3^2)$ which governs the behavior of the vertex. The effects of nonzero internal masses $M$ are important and have a complicated threshold and pseudothreshold structure. A three-scale effective number of flavors $N_F(k_1^2/M^2,k_2^2/M^2,k_3^2/M^2)$ is defined. The results of this paper are an important part of a gauge-invariant dressed skeleton expansion and a related multiscale analytic renormalization scheme. In this approach the scale ambiguity problem is resolved since physical kinematic invariants determine the arguments of the couplings.

Figure 1

The notation and loop momentum routing used throughout this paper. The internal particle could be a gauge boson, ghost, quark, or scalar.

Figure 2

The tree-level skeleton graphs dressed with pinch-technique vertices and self-energies, which are used to define effective charges both for the quark-quark-gluon vertex (depending on a single gluon virtuality) and for the three-gluon vertex, which depends on three different momenta.

Figure 3

The effective scale $Q_{\mathrm{eff}}^2(10{\mathrm{GeV}}^2,10{\mathrm{GeV}}^2,p^2)$ is the lower blue curve, while $Q_{\mathrm{eff}}^2(-10{\mathrm{GeV}}^2,-10{\mathrm{GeV}}^2,p^2)=Q_{\mathrm{eff}}^2(10{\mathrm{GeV}}^2,10{\mathrm{GeV}}^2,-p^2)$ is the upper red curve. These both asymptote to zero, although very slowly for the upper curve.

Figure 4

The effective scale $Q_{\mathrm{eff}}^2(10{\mathrm{GeV}}^2,p^2,p^2)$ is the lower blue curve, while $Q_{\mathrm{eff}}^2(-10{\mathrm{GeV}}^2,p^2,p^2)=Q_{\mathrm{eff}}^2(10{\mathrm{GeV}}^2,-p^2,-p^2)$ is the upper red curve. These both asymptote to $10e^{\Omega }{\mathrm{GeV}}^2\approx 31.25{\mathrm{GeV}}^2$.

Figure 5

A contour plot of $Q_{\mathrm{eff}}^2(1,x,y)$. The contours, from red to blue, are at 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Figure 6

A three-dimensional contour plot of $Q_{\mathrm{eff}}^2(1,x,y)$.

Figure 7

$L_{\mathrm{MQ}}(a/M^2,a/M^2,a/M^2)$ vs $a/M^2$ for timelike $a>0$. The solid line is the real part and the dashed line is the imaginary part.

Figure 8

$L_{\mathrm{MQ}}(-Q^2/M^2,-Q^2/M^2,-Q^2/M^2)$ vs $Q/M$ for spacelike $-Q^2<0$.

Figure 9

$L_{\mathrm{MQ}}(a/M^2,a/M^2,-a/M^2)$ vs $a/M^2$. The solid line is the real part and the dashed line is the imaginary part. The real thresholds are at $a=\pm 4M^2$ while the pseudothreshold is at $a=5M^2$.

Figure 10

The effective number of quark flavors. The lower blue solid curve is $N_F(Q^2/M^2)$ for the symmetric spacelike ($a=b=c=-Q^2$) three-gluon vertex, while the upper dashed red curve is the fermion number of flavors $N_{1/2}(Q^2/M^2)$ for the single-scale effective charge.

Figure 11

The hadro-production of heavy quarks and two associated transverse jets, as a direct probe of the three-gluon vertex.