Spectral functions of gauge theories with Banks-Zaks fixed points

We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalization group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a Källén-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation of complex conjugated branch cuts renders a causal representation impossible. We also derive relations for scaling exponents that determine the presence or absence of propagator nonanalyticities. Further results include spectral functions for all fields up to five loop order, bounds on the conformal window, and an algorithm to find running gauge coupling analytically at higher loops. Implications of our findings and extensions to other theories are discussed.

Figure 1

RG trajectory of a weakly coupled matter-gauge theory with coupling $a(\mu )$, interpolating between an asymptotically free fixed point in the UV and an interacting conformal fixed point $a_{*}$ in the IR with cross-over scale ${\mathrm{\Lambda }}_c$.

Figure 2

Shown are the number of branching points in the complexified ${\mu }^2$-plane of the running gauge coupling at two loop, as a function of the Veneziano parameter $ϵ$. A single branch cut at weak coupling ($0<ϵ<{ϵ}_{\text{branch cut}}$) proliferates rapidly into many more at stronger coupling (${ϵ}_{\text{branch cut}}<ϵ<{ϵ}_{\max }$) before the fixed point disappears $(ϵ={ϵ}_{\max })$.

Figure 3

Magnitude $|a|$ and phase ${\phi }_a$ of the gauge coupling $a$ in the complex plane of the RG scale ${\mu }^2$ at two-loop order. We use $ϵ=2.7$, ${\mu }_0^2=1$, and $N_c\to \infty$. In the left panel, we use the principal branch (11), and in the right panel, the modified prescription from (22). Due to the properties of the $W$-Lambert function two branch points have disappeared in this modified version of the running coupling. Branch cuts originating from the $W$-Lambert function (power law of ${\mu }^2$) are drawn as black solid (dashed) lines.

Figure 4

The resummed gluon propagator (40) and its UV/IR asymptotics (41) are shown as a function of $(-p^2{)}^{\theta }$ in the Veneziano limit ($ϵ=\frac{1}{2}$, $\xi =1$).¹

Figure 5

Magnitude $|G_A|$ and phase ${\phi }_{G_A}$ of the gluon propagator in the complex plane at two-loop order. We use $N_c\to \infty$, $\xi =1$, as well as $ϵ=0.5$ (top left), $ϵ=2.5$ (top right), and $ϵ=2.7$ (bottom). Branch cuts originating from power laws (the $W$-Lambert function) are illustrated with dashed (solid) black lines. There is only one branch cut for $ϵ=0.5$ while the other plots show additional branch cuts related to the properties of the running coupling and the $W$-Lambert function.

Figure 6

The contour integration of the propagator in the complex plane.

Figure 7

Gluon spectral function in the leading order Veneziano expansion (left) and in the loop expansion at two-loop order (right). In the left panel, we are using $ϵ=\frac{1}{10}$ and $N_c\to \infty$, while in the right panel we are using $N_c=3$ and $N_f=16$ $(ϵ=\frac{1}{6})$. We display the spectral functions as a function of $({\lambda }^2{)}^{\theta }$.¹

Figure 8

Spectral function of the quark and ghost in the Veneziano expansion (left) and the spectral function of the quark in the loop expansion (right). In the left panel, we use $ϵ=\frac{1}{10}$ and $N_c\to \infty$, while in the right panel we use $N_c=3$ and $N_f=16$ $(ϵ=\frac{1}{6})$. We display the spectral functions as a function of $({\lambda }^2{)}^{\theta }$.¹

Figure 9

Gluon spectral functions in the loop expansion at $\xi ={\xi }_A^{\mathrm{crit}}$ for $N_c\to \infty$ with $ϵ=1$ (left) and at $N_c=3$ with $N_f=16$ ($ϵ=1/6$) (right). The asterisk at the highest expansion order indicates that the highest order contribution the finite parts is missing.

Figure 10

Quark spectral functions in the loop expansion at $\xi ={\xi }_{\psi }^{\mathrm{crit}}$ for $N_c\to \infty$ with $ϵ=1/10$ (left) and at $N_c=3$ with $N_f=16$ ($ϵ=1/6$) (right). The asterisk at the highest expansion order indicates that the highest order contribution the finite parts is missing. The red color that the spectral function is not integrable in the IR.

Figure 11

Ghost spectral functions in the loop expansion at $\xi ={\xi }_c^{\mathrm{crit}}$ for $N_c\to \infty$ with $ϵ=1$ (left) and in the loop expansion at $N_c=3$ and $N_f=16$ ($ϵ=1/6$) (right). The asterisk at the highest expansion order indicates that the highest order contribution the finite parts is missing. The red color that the spectral function is not integrable in the IR.

Figure 12

Gluon propagator computed from the fRG in comparison with the perturbative result for $ϵ=\frac{5}{2}$, $N_c=100$, and $\xi =1$.