Some remarks on the spectral functions of the Abelian Higgs model

We consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows us to disentangle what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. We highlight the role of the tadpole graphs and the gauge choices to get sensible results. We also introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry. We clearly illustrate its nonunitary nature directly from the spectral function viewpoint. This provides a functional analogue of the Ojima observation in the canonical formalism: there are ghost states with nonzero norm in the BRST-invariant states of the Curci-Ferrari model.

Figure 1

Contributions to one-loop photon self-energy in the Abelian Higgs model, including tadpole contributions in the second line. Wavy lines represent the photon field, dashed lines the Higgs field, solid lines the Goldstone boson, and double lines the ghost field.

Figure 2

Contributions to the one-loop Higgs self-energy. Line representations are as in Fig. 1.

Figure 3

Gauge dependence of the residue of the pole for the Higgs field, for the parameter values $m=2\mathrm{GeV}$, $m_h=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$, $e=\frac{1}{10}$.

Figure 4

The reduced spectral function of the photon, with $t$ given in ${\mathrm{GeV}}^2$, for the parameter values $m=2\mathrm{GeV}$, $m_h=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$, $e=\frac{1}{10}$. The first-order pole mass lies at $t=4.08286{\mathrm{GeV}}^2$, and the two-particle state of one photon field and one Higgs field starts at $t^{*}=(m_h+m{)}^2=6.25{\mathrm{GeV}}^2$.

Figure 5

The reduced spectral function of the Higgs boson, with $t$ given in ${\mathrm{GeV}}^2$, for $\xi =2$ (green, solid), $\xi =3$ (red, dotted), $\xi =4$ (yellow, dashed) and the parameter values $m=2\mathrm{GeV}$, $m_h=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$, $e=\frac{1}{10}$. The first-order pole mass lies at $t=0.251665{\mathrm{GeV}}^2$. The threshold for the Higgs two-particle state is given at $t^{*}=(m_h+m_h{)}^2=1{\mathrm{GeV}}^2$ and for the photon two-particle state at $t^{*}=(m+m{)}^2=16{\mathrm{GeV}}^2$. We find a (negative) Goldstone two-particle state at $t^{*}=(\sqrt{\xi }m+\sqrt{\xi }m{)}^2$, corresponding to $t^{*}=(32{\mathrm{GeV}}^2,48{\mathrm{GeV}}^2,64{\mathrm{GeV}}^2)$ for $\xi =(2,3,4)$.

Figure 6

Gauge dependence of the Higgs pole mass obtained iteratively to first order (green) and the approximated pole mass (red), for the parameter values $m=2\mathrm{GeV}$, $m_h=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$, $e=\frac{1}{10}$. The right panel shows the real part; the left panel shows the imaginary part.

Figure 7

Behavior of the one-loop correction of the Higgs propagator ${\mathrm{\Pi }}_{hh}(p^2)$ around the pole mass, for the values $a=-{10}^{-6}$ (yellow, dashed), $a=0$ (red, dotted), $a={10}^{-6}$ (green, solid). The value $x$ is a small imaginary variation of the argument in ${\mathrm{\Pi }}_{hh}(p^2)$. Only for $a=0$ do we find a continuous function at $x=0$, meaning that for any other value, we are on the branch cut.

Figure 8

Contributions to one-loop CF photon self-energy. Wavy lines represent the photon field, and dashed lines represent the scalar field.

Figure 9

Contributions to the one-loop CF scalar self-energy. Line representation as in Fig. 8.

Figure 10

The reduced spectral function of the photon field in the Abelian CF model, with $t$ given in ${\mathrm{GeV}}^2$, for the parameter values $m=2\mathrm{GeV}$, $m_h=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$, $e=\frac{1}{10}$. The first-order pole mass lies at $t=0.25151{\mathrm{GeV}}^2$. The photon two-particle state starts at $t^{*}=(m_{\phi }+m_{\phi }{)}^2=16{\mathrm{GeV}}^2$.

Figure 11

Gauge dependence of the first order pole mass for the scalar field. The left panel shows the real part; the right panel shows the imaginary part. The chosen parameter values are $m=2\mathrm{GeV}$, $m_{\phi }=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$, $e=\frac{1}{10}$.

Figure 12

The spectral function of the composite operator $\frac{b^2}{2}+m^2\overline{c}c$, for $\alpha =2$ (green, dotted), $\alpha =3$ (red, solid), $\alpha =5$ (yellow, dashed). The chosen parameter values are $m=\frac{1}{2}\mathrm{GeV}$, $\mu =10\mathrm{GeV}$. The threshold for the branch cut of the propagator is given by $t^{*}=(\sqrt{\alpha }m+\sqrt{\alpha }m{)}^2$, with $t^{*}=(2{\mathrm{GeV}}^2,3{\mathrm{GeV}}^2,5{\mathrm{GeV}}^2)$ for $\xi =(2,3,5)$.