Photons in dense nuclear matter: Random-phase approximation

We present a comprehensive and pedagogic discussion of the properties of photons in cold and dense nuclear matter based on the resummed one-loop photon self-energy. Correlations among electrons, muons, protons, and neutrons in $\beta$ equilibrium that arise as a result of electromagnetic and strong interactions are consistently taken into account within the random phase approximation. Screening effects, damping, and collective excitations are systematically studied in a fully relativistic setup. Our study is relevant to the linear response theory of dense nuclear matter, calculations of transport properties of cold dense matter, and investigations of the production and propagation of hypothetical vector bosons such as the dark photons.

Figure 1

Identical one-loop resummations of the photon propagator. Left: direct resummation according to Eq. (3). Right: resummation of the polarization function according to Eq. (10) and subsequent insertion into the photon propagator.

Figure 2

Real parts of ${\mathrm{\Pi }}_{00}$ and ${\mathrm{\Pi }}_{\perp }$. Purple lines correspond to the case where the fermions in the plasma are electrons, blue lines correspond to the case where they are protons, and dashed lines correspond to the respective HDL approximations. Thin vertical lines indicate the values $q_0=v_f\left|\mathit{q}\right|$ for each particle species. Effective chemical potential and mass of the proton as well as the chemical potential of the electron are determined at nuclear saturation density in $\beta$ equilibrium, using NRAPR (non-relativistic Akmal, Pandharipande and Ravenhall) Skyrme-type interactions (see Sec. 3a for details): $m_p^{★}=575\mathrm{MeV}$, $m_e=0.5\mathrm{MeV}$, ${\mu }_p^{★}=589\mathrm{MeV}$, ${\mu }_e=122\mathrm{MeV}$. The momenta $\left|\mathit{q}\right|$ in both cases are fixed at $25\%$ of the chemical potential and $q_0$ is plotted in between values of $\pm 2\left|\mathit{q}\right|$. The real parts of the polarization functions are even functions of the frequency. Despite the relatively large chosen value of $\left|\mathit{q}\right|$, the HDL approximations work very well for electrons. In the presence of protons, one finds the typical behavior of nonrelativistic polarization functions (“Lindhard functions”), which display a much smaller and smoother peak. Hard dense loops do not capture this feature. The transverse component is in general less well approximated by HDL.

Figure 3

Zeros of the real parts of the longitudinal (blue) and transverse (red) photon propagator in a degenerate plasma composed of electrons (left) and protons (right). Same parameters as in Fig. 2. Dashed lines correspond to $q_0=v_f\left|\mathit{q}\right|$ (the gray dashed line in the right figure corresponds to the light cone); dot-dashed lines correspond to $q_0=c{v}_f\left|\mathit{q}\right|$ where the constant $c$ is obtained by numerically solving Eq. (29). In the longitudinal case, one encounters a characteristic “thumblike” structure that has been reported in several references [15, 32]. Only the upper branch corresponds to a collective mode for as long as it resides outside the Landau damped region; see also Fig. 6. The electron plasma qualitatively resembles the ultrarelativistic case ($v_f\sim 1$) in which the two longitudinal branches approach the light cone asymptotically. In the proton plasma, both longitudinal branches merge at significantly lower momenta. This feature is not captured by the HDL approximation which predicts the existence a longitudinal mode up to much larger momenta, as can be seen from the proton peak in Fig. 2. The transverse mode obtains a finite thermal mass and remains always timelike. The spectral functions in Fig. 7 are plotted along the thin gray lines.

Figure 4

Physical interpretation of the various contributions to the imaginary part of the photon self-energy according to the cutting rules. The first diagram corresponds to the creation of a fermion-antifermion pair while the latter two correspond to Landau damping: particles (and antiparticles) are emitted from or absorbed by the thermal bath. The latter two contributions are particle-hole processes; particle-antiparticle processes are exponentially suppressed in degenerate matter by the large chemical potentials.

Figure 5

Top left: domains where Landau damping (gray shaded area) and pair creation (blue shaded area) are kinematically allowed. The boundaries are $q_0=\left|\mathit{q}\right|$ and $q_0=\sqrt{{\left|\mathit{q}\right|}^2+4m^2}$. Top right: effect of degeneracy on the boundaries. Landau damping is confined in between the two thick solid lines $q_0=-\mu +\sqrt{{\mu }^2+2k_f\left|\mathit{q}\right|+{\left|\mathit{q}\right|}^2\phantom{{}^2}}$ and $q_0=-\mu +\sqrt{{\mu }^2-2k_f\left|\mathit{q}\right|+{\left|\mathit{q}\right|}^2\phantom{{}^2}}$. The thin vertical line indicates the position of $\left|\mathit{q}\right|=2k_f$. At $q_0=0$, there is no more Landau damping for $\left|\mathit{q}\right|>2k_f$. Pair creation in degenerate matter is determined by the interplay of the two boundaries $q_0=\mu +\sqrt{{\mu }^2-2k_f\left|\mathit{q}\right|+{\left|\mathit{q}\right|}^2\phantom{{}^2}}$ (dashed) and $q_0=\sqrt{{\left|\mathit{q}\right|}^2+4m^2\phantom{{}^2}}$ (dotted): For $\left|\mathit{q}\right|<2k_f$ (despite being kinematically allowed), pair creation is completely suppressed for values of $q_0$ larger than the dotted line and only sets in above the dashed line. At $\left|\mathit{q}\right|=2k_f$, both boundaries assume the value $q_0=2\mu$ and for $\left|\mathit{q}\right|>2k_f$ pair creation becomes possible for values of $q_0$ above the dotted line. Finally, for values of $q_0$ above the dot-dashed line $q_0=\mu +\sqrt{{\mu }^2+2k_f\left|\mathit{q}\right|+{\left|\mathit{q}\right|}^2\phantom{{}^2}}$, the degeneracy suppression has completely faded away and pair creation assumes its vacuum value. The analytic calculation of the imaginary part for each of these regions is presented in Appendix pp2-s1. The thin blue lines indicate the position of the light cone and of $q_0=v_f\left|\mathit{q}\right|$. It is easy to see that the energy $q_0$ required to create a fermion-antifermion pair is minimal when the photon has momentum $\left|\mathit{q}\right|=k_f$. Bottom left: imaginary part of ${\mathrm{\Pi }}_{00}$ for $\left|\mathit{q}\right|<2k_f$ (left) and $\left|\mathit{q}\right|>2k_f$ (right). A convenient choice to illustrate the imaginary parts of both regions is to use the muon mass with a muon chemical potential at saturation density of about 122 MeV. In this case, the separation of both regions is clearly visible, yet not too large. The value of $2k_f$ calculates to roughly 124 MeV. The momenta are set to $\left|\mathit{q}\right|=80\mathrm{MeV}$ (left) and 200 MeV (right). Solid lines represent the total magnitude of the imaginary part, and dashed lines represent the vacuum values. The outline in both cases can be understood from the top right diagram: For $\left|\mathit{q}\right|<2k_f$, a nonzero imaginary part sets in immediately at $q_0=0$ until the boundary of Landau damping is reached. Upon increasing $q_0$, one traverses through a dissipation free region until the threshold for pair creation is reached. Degeneracy effects suppress the imaginary part due to pair creation until one eventually crosses the dot-dashed line in the top right diagram, above which pair creation assumes its unsuppressed vacuum value. For $\left|\mathit{q}\right|>2k_f$, the situation is similar except that Landau damping is absent for sufficiently small $q_0$. With increasing momentum, the imaginary part due to pair creation outgrows the imaginary part due to Landau damping.

Figure 6

Top left and right: imaginary parts of ${\mathrm{\Pi }}_{00}$ and ${\mathrm{\Pi }}_{\perp }$. Same parameters as in Fig. 2. Purple lines correspond to electrons, blue lines correspond to protons, and dashed lines correspond to the respective HDL approximations. Thin vertical lines indicate the positions of $q_0=v_f\left|\mathit{q}\right|$. Imaginary parts of polarization functions are odd functions of $q_0$. Despite the fact that we have chosen a (relatively) large value of $\mathit{q}$, the HDL approximation works very well for electrons but fails for protons. Analogously to the real parts, the large proton mass leads to a sizable reduction of the magnitude of $\mathrm{Im}{\mathrm{\Pi }}_{00}$ in the vicinity of $q_0=v_f\left|\mathit{q}\right|$. Note further that the imaginary parts of ${\mathrm{\Pi }}_{00}$ and ${\mathrm{\Pi }}_{\perp }$ in the presence of protons do not go to zero at $q_0=v_f\left|\mathit{q}\right|$ [as predicted by (36) and (37)] but at a larger value. The correct domain boundaries are worked out in the top right diagram of Fig. 5. Using the full RPA results, one finds that the transition to the Landau-damped region is always smooth though very rapid, in particular in the electron case. This phenomenon is further investigated in the bottom two figures: The solutions to (25) are plotted over the magnitude of the imaginary part in an electron plasma (left) and proton plasma (right), where darker colors indicate larger (negative) values. Dashed and dot-dashed lines correspond to $q_0=v_f\left|\mathit{q}\right|$ and $q_0=c{v}_f\left|\mathit{q}\right|$ respectively. In a region where $q_0$ is slightly smaller than $v_f\left|\mathit{q}\right|$, excitations are maximally damped. The position where the undamped mode ${\omega }_L$ “dives” into the Landau-damped region and loses energy rapidly defines a cutoff momentum ${\left|\mathit{q}\right|}_{\max }$. Gapless solutions ${\omega }_{<}$ including the “tip of the thumb” (the position where $\partial \omega /\partial \left|\mathit{q}\right|$ diverges) are confined to the Landau-damped region and hence should not be regarded as actual collective modes. The plasmon mode shares the same fate in a plasma comprised of electrons or protons, but due to the small mass of the electrons the transition to the Landau-damped region occurs at much larger momenta.

Figure 7

Ļongitudinal (left) and transverse (right) low-energy photon spectrum in a plasma composed of electrons (purple) and protons (blue). Same parameters as in Fig. 2. Horizontal axis are normalized over $v_f\left|\mathit{q}\right|$, where $v_f$ is the respective Fermi velocity of each particle species. Dashed lines correspond to the positions of the collective modes ${\omega }_L$ and ${\omega }_{\perp }$; dot-dashed lines correspond to positions of the gapless solutions ${\omega }_{<}$. The value of $\left|\mathit{q}\right|$ is set to $0.2m_{D,e}\sim 2.4$ MeV (top) and $0.9m_{D,p}\sim$ 23.8 MeV (bottom). These positions are indicated by thin horizontal lines in Fig. 3 and the spectral functions are plotted along those lines. At sufficiently low momenta, the solutions ${\omega }_{<}$ in the electron and proton cases are both located at a value of roughly $q_0=0.83v_f\left|\mathit{q}\right|$ and hence appear on top of each other in the top right plot. Transverse spectral functions are roughly an order of magnitude larger than longitudinal ones. At higher momenta (second line), the plasmon modes ${\omega }_L$ are located just outside the Landau-damped region. Note further that the longitudinal spectrum in the proton plasma appears unusually large at the edge. This is due to the fact that the magnitude of the imaginary part is reduced in the vicinity of ${\omega }_{<}$ at larger momenta (see Fig. 6, bottom right). If the plasma is charge neutralized by an additional particle species, Landau damping due to the other constituent takes over and reduces the magnitude of the spectrum in this region.

Figure 8

Ļongitudinal (solid) and transverse (dashed) spectral functions in a muon plasma including regions subject to Landau damping and pair creation. Identical parameters as in Fig. 5: The muon chemical potential is set to ${\mu }_{\mu }=122\text{MeV}$ and the momenta to $\left|\mathit{q}\right|=80$ (left) and 200 MeV (right). At these momenta, plasmon modes are longer present in the spectrum, and dashed vertical lines correspond to the position of the (transverse) photon modes located at the light cone in the dissipation-free region. With increasing momenta $\left|\mathit{q}\right|$, the spectrum due to pair creation outgrows the spectrum due to Landau damping. Since the chosen values for $\left|\mathit{q}\right|$ are rather large, the corresponding spectral functions are relatively tiny.

Figure 9

Ştatic screening ${\overline{m}}_{D,p}={\tilde{m}}_{D,p}/m_{D,p}^{\prime }$ in nuclear matter in $\beta$ equilibrium based on various Skyrme forces according to Eq. (55), normalized over the Debye masses in the noninteracting system. Given the considerable differences of effective masses or proton fractions at higher densities (see Appendix pp2-s2), the results for the screening mass is fairly consistent. The screening diverges close to the spinodal instability and is slightly reduced at higher densities. The plot on the right-hand side displays an enlargment of the one on the left-hand side.

Figure 10

Dyson equation of the photon propagator resumming contributions from all constituents of the plasma interacting via single-photon exchange. Vertices carry a flavor index, and the one-loop polarization tensor is diagonal in flavor space. The photon propagator naturally remains a scalar in flavor space, and summing over the indices $a$ and $b$ simply returns the trace $\mathrm{Tr}\mathrm{\Pi }$. As in the single-species case of Fig. 1, one may alternatively utilize the dressed polarization tensor, Eq. (62). To leading order, one recovers the (diagonal) one-loop polarization tensor, and higher order terms describe how the interactions between the various species in the plasma are screened.

Figure 11

Solutions to Eqs. (67) and (68) in a plasma composed of electrons, protons, and muons at saturation density using NRAPR Skyrme forces. Electron and muon chemical potentials are ${\mu }_e={\mu }_{\mu }=122\mathrm{MeV}$; for protons one finds ${\mu }_p=589\mathrm{MeV}$ and $m_p^{*}=575\mathrm{MeV}$. Blue, purple, and green lines correspond to longitudinal proton-like, electron-like, and muon-like solutions respectively. Dashed lines correspond to $q_0=v_f\left|\mathit{q}\right|$ for each particle species; the black line corresponds to the transverse mode. In the collisionless limit, the various longitudinal solutions appear completely decoupled from one another. The electron-like solution is the only gapped solution in the spectrum and consequently identified with the plasmon mode of the combined electron, muon, and proton system. Muon-muon interactions are screened by electrons and proton-proton interactions are screened by electrons and muons. As a result, their respective plasmon modes are effectively reduced to sound-like modes.

Figure 12

Regions in which Landau damping and pair creation of electrons (purple shaded), muons (green shaded), and protons (blue shaded) operate. White areas indicate dissipation-free regions, thick lines denote the outermost boundaries where dissipation sets in, and thin lines mark the boundaries where dissipation of a particular constituent sets in. Analytic expressions for the boundaries are listed in the caption of Fig. 5. Dashed lines display the positions of $v_f\left|\mathit{q}\right|$ for each species (which in the case of electrons overlaps with the thick purple line). The labels are assigned as follows: ($A$) Landau damping due to all three constituents, ($B$) Landau damping due to electrons and protons, ($C$), Landau damping due to electrons and muons, ($D$) Landau damping due to protons, ($E$) Landau damping due to electrons, ($\overline{E}$) electron pair creation, and ($\overline{C}$) electron and muon pair creation. Proton pair creation appears for much higher values of $q_0$ and is not displayed in the plot. As long as the fermions in the plasma are massive, there is always a finite gap in between the area where pair creation and Landau damping operate, due to their small mass this gap becomes tiny for electrons.

Figure 13

Evolution of the total imaginary part for different values of $\left|\mathit{q}\right|$. Thin vertical lines (with the usual color coding of each constituent) mark the positions of $q_0=-{\mu }_a+\sqrt{{\mu }_a^2+2k_{f,a}\left|\mathit{q}\right|+{\left|\mathit{q}\right|}^2\phantom{{}^2}}$ below which Landau damping operates and thin dashed lines the positions $q_0={\mu }_a+\sqrt{{\mu }_a^2-2k_{f,a}\left|\mathit{q}\right|+{\left|\mathit{q}\right|}^2\phantom{{}^2}}$ above which pair creation sets in. At low $\left|\mathit{q}\right|$, the imaginary part due to Landau damping (gray shaded) is huge compared to the imaginary part due to pair creation. The situation is reversed at high $\left|\mathit{q}\right|$. Owing to the small mass of electrons, the dissipation-free area squeezed in between Landau damping and $e^{+}{e}^{-}$ pair creation becomes tiny for $\left|\mathit{q}\right|>k_{f,e}$.

Figure 14

Evolution of the longitudinal [(a), (c), (e)] and transverse [(b), (d), (f)] low-momentum spectrum of the photon in a degenerate electron, muon, and proton plasma in $\beta$ equilibrium at saturation density. The spectra are plotted as functions of $q_0$ for fixed momenta; compare as well to Fig. 11. The solutions to Eq. (67) are indicated as follows: thick dashed lines show the positions of the sound-like solutions $u_a$ and the real plasmon mode ${\omega }_L$, and thin dot-dashed lines indicate the positions of the solutions ${\omega }_{<,a}$. While the solutions $u_a$ are indeed located at the maxima of the spectrum, their corresponding peaks do not stand out too much from the bulk of the spectrum at lower momenta. Only the proton mode develops a well-defined peak at larger momenta.

Figure 15

Photon propagator (83) expanded in powers of the electromagnetic coupling. QED vertices are indicated by small dots, and short-range strong interactions are shown by small squares. At leading order, one obtains the insertions of bare loops corresponding to each particle species. Protons are themselves polarized in the nuclear medium and unfold an infinite series of (proton and neutron) loops whenever they appear. Neutrons appear in the resummation at order ${\alpha }_f{f}_{pn}^2$.

Figure 16

Real and imaginary parts of the proton contribution to the photon polarization functions ${\tilde{\mathrm{\Pi }}}_{00}$, Eq. (81), resummed in the subspace of protons and neutrons interacting via strong forces [see Eq. (81)]. In each plot, the momenta $\left|\mathit{q}\right|$ are fixed at 10 MeV. Since $\beta$ equilibrium has to be determined separately for each Skyrme parameter set, the Fermi momenta of protons and neutrons vary in each case. NRAPR parameters (thick black line) which predict the largest proton fraction are chosen as reference and $q_0$ is normalized over $v_{f,p}\left|\mathit{q}\right|$ (where the Fermi velocity of the protons consequently also corresponds to the NRAPR set). The blue dashed line displays the polarization effects from protons in the absence of nuclear interactions, again using NRAPR parameters. The peaks of real and imaginary parts of all other parameter sets are hence shifted to the left of the NRAPR peak. Various densities are shown; the last row shows the corresponding transverse polarization at $1.6n_0$ for comparison. The impact of induced interactions is completely negligible for transverse components at any density but sizable for longitudinal ones, in particular for densities below $n_0$. At saturation density, the results with and without induced interaction are roughly comparable. At higher densities, results are fairly model dependent. The qualitative tendencies of the impact of induced interactions are, however, consistent in all tested models.

Figure 17

Poles of the real part of the longitudinal and transverse photon propagator in the usual color coding, using the NRAPR parameter set, with and without induced interactions. At saturation density, both scenarios lead to almost equal results (i.e., to the spectrum displayed in Fig. 11), mainly because there bare and induced screening are roughly of equal magnitude; see Fig. 9. Dashed lines correspond to $v_f\left|\mathit{q}\right|$ for each particle species. For comparison, we additionally show the dot-dashed line in the bottom two figures below which the proton contribution to Landau damping sets in. The “tip of the thumb” is always located below this line.

Figure 18

Evolution of the longitudinal spectral function with increasing momenta using the NRAPR parameter set for $n=0.65n_0$ (first row, muons are absent), $n=0.85n_0$ (second row), and $n=1.6n_0$ (third row). Thick black lines correspond to the case with induced interactions; thin black lines correspond to the case without. Vertical dashed lines indicate the positions of the electron plasmon mode ${\omega }_L$ and the muon and proton sound modes $u_p$ (the overdamped lower branches ${\omega }_{<}$ are not displayed).

Figure 19

Comparison of the various Skyrme parameter sets at $n=0.65n_0$ in the absence of muons (left) and $n=1.6n_0$ (right) at fixed photon momentum $\left|\mathit{q}\right|=10\mathrm{MeV}$. Thick black lines correspond to the NRAPR parameter set which we use as a reference. The blue dashed line serves as a reference to the NRAPR result without induced interactions. The suppression of the proton peak at lower densities is to a reasonable degree independent of the chosen parameters; the (small) enhancement at higher densities is much more model dependent.

Figure 20

Retarded photon polarization tensor with loop momentum $k$. In the given momentum assignment, the propagator carrying momentum $k$ in Eq. (A1) is advanced while the propagator carrying momentum $k+q$ is retarded.

Figure 21

Properties of homogeneous nuclear matter derived from the NRAPR Skyrme parameter set. Homogeneous matter is stable on the right-hand side of the black horizontal line. Top left: comparison of relativistic and original nonrelativistic (thin gray lines) symmetry energies in symmetric nuclear matter (dashed) and in $\beta$ equilibrium (solid). In both cases, small deviations due to relativistic corrections appear only at high densities. Top right: proton (solid), electron (dashed), and muon (dot-dashed) fractions. Using NRAPR parameters, muons appear around 0.75 $n_0$. Thin gray lines depict the nonrelativistic results in $\beta$ equilibrium which indicate a larger proton fraction at higher densities. Bottom left: residual density-density interaction potentials of the quasiparticles. Bottom right: current-current interaction potentials of the quasiparticles.

Figure 22

Comparison of different Skyrme models. The critical densities are slightly different for each model; see Table 1. We are specifically interested in properties relevant to the induced interactions: The proton fractions (top left) agree fairly well at lower densities among the tested models. The reduction of proton the fractions at higher densities is in part due to relativistic corrections; see Eq. (51). The same is true for the residual density-density interactions between neutrons and protons (bottom left). Both quantities are derived from $l=0$ contributions of the energy functional (41). The $l=1$ contributions are far less concurring.