THE IMPACT OF NEUTRINO MAGNETIC MOMENTS ON THE EVOLUTION OF MASSIVE STARS

The search for the neutrino magnetic moment goes back to the neutrino discovery experiment by Cowan & Reines (1957), who put the first bound on the magnetic moment, μ_ν ≲ 10⁻⁹ μ_B.

On the theoretical front, it was realized many decades ago that a neutrino with a nonzero mass should have a nonvanishing magnetic moment even with the SM interactions (Marciano & Sanda 1977; Lee & Shrock 1977; Fujikawa & Shrock 1980). Due to the specific nature of the SM interactions, however, namely, coupling of W to left-handed currents only, the induced value of the magnetic moment turns out to be very small, μ_ν/ μ_B ≃ 3 × 10⁻¹⁹ × (m_ν/1 eV). In extension of the SM, this limitation need not apply and much larger values of μ_ν can be obtained.

There are several classes of models beyond the SM predicting a large magnetic moment (as large as μ_ν/μ_B ∼ 10⁻¹⁰). These scenarios typically manage to simultaneously have a small neutrino mass and a large magnetic moment by one of the following two mechanisms: (1) requiring that the new physics dynamics obeys some (approximate) symmetry that forces m_ν = 0 while allowing nonvanishing μ_ν (Voloshin 1988; Barbieri & Mohapatra 1989; Georgi & Randall 1990; Babu & Mohapatra 1990; Grimus & Neufeld 1991); (2) engineering a spin suppression mechanism that keeps m_ν small (Barr et al. 1990).

It is important to stress that the value of the magnetic moment in SM extensions cannot be arbitrarily large. It can be shown in a model-independent way that a large value of μ_ν would induce, via electroweak radiative corrections, a neutrino mass above the current experimental limit (conservatively, ∼1 eV). Quite interestingly, the resulting "naturalness" upper limits are substantially different for Dirac and Majorana neutrinos, due to the different nature and breaking of the approximate symmetries imposed to simultaneously ensure m_ν ≃ 0 and μ_ν ≠ 0. For the Dirac case, a general effective theory analysis leads to (Barbieri & G. Fiorentini 1988; Bell et al. 2005)

$$\begin{equation} \frac{\mu _\nu }{\mu _{\mathrm{B}}} \lesssim \ 3 \times 10^{-15} \times \left(\frac{\delta m_\nu }{1\,{\mathrm{e}\!\mathrm{V}}} \right) \, \left(\frac{1\,{\mathrm{T}{\mathrm{e}\!\mathrm{V}}}}{\Lambda } \right)^2, \end{equation} \tag{ 2 }$$

where δm_ν is the induced mass term and Λ is the (unknown) new physics mass scale at which the magnetic moment originates. For the Majorana transition magnetic moments, one finds (Davidson et al. 2005; Bell et al. 2006)

$$\begin{equation} \frac{\mu _\nu ^{\alpha \beta }}{\mu _{\mathrm{B}}} \lesssim 4 \times 10^{-9} \left(\frac{\left[\delta m_\nu \right]_{\alpha \beta }}{1\,{\mathrm{e}\!\mathrm{V}}}\right) \left(\frac{1\,{\mathrm{T}{\mathrm{e}\!\mathrm{V}}}}{\Lambda }\right)^2 \left| \frac{m_\tau ^2}{m_\alpha ^2 - m_\beta ^2} \right|, \end{equation} \tag{ 3 }$$

where (δm_ν)_αβ is the induced contribution to the neutrino mass matrix and m_α, m_β represent the charged lepton masses for the flavors α, β. Therefore, the naturalness limit on the neutrino magnetic moment is considerably weak for Majorana neutrinos than for Dirac neutrinos. Note that in both cases there is a strong quadratic dependence on the new physics scale Λ, for which we have taken the reference value of Λ = 1 TeV.

In summary, evidence of μ_ν > 10⁻¹⁹ μ_B would indicate the need for a radical extension of the weak sector of the SM. Moreover, evidence of a magnetic moment in the window 10⁻¹⁵ μ_Bμ_ν10⁻¹⁰ μ_B would provide strong evidence for the Majorana nature of neutrinos. Therefore, astrophysical probes of the neutrino magnetic moment with sensitivity right below μ_ν ∼ 10⁻¹⁰ μ_B are of considerable theoretical value.

The first experimental bound on the magnetic moment of the electron neutrino, μ_ν ≲ 10⁻⁹ μ_B, was obtained by Cowan & Reines (1957), by analyzing electron recoil spectra in (anti)-neutrino electron scattering. The same technique has led, over time, to more stringent bounds, μ_ν < (5–10) × 10⁻¹¹ μ_B, at 90 % C.L. (Daraktchieva et al. 2005; Beda et al. 2007). A similar bound (μ_ν < 5.4 × 10⁻¹¹ μ_B, at 90 % C.L.) was recently confirmed by the Borexino collaboration (Galbiati 2008; Galbiati et al. 2008) from the analysis of solar neutrinos.

Another set of constraints comes from the analysis of energy loss in stars. This method was pioneered by Bernstein et al. (1963) who considered the effect of the addition cooling due to the neutrino magnetic moment on the lifetime of the Sun and estimated that μ_ν ≲ 10⁻¹⁰ μ_B. A recent bound from the Sun is μ_ν ≲ 4 × 10⁻¹⁰ μ_B (Raffelt 1999). A more stringent bound, μ_ν ≲ 3 × 10⁻¹² μ_B, comes from the analysis of the RG branch, where the extra cooling could delay the onset of the helium flash (Raffelt 1990a, 1990b; Raffelt & Weiss 1992; Haft et al. 1994; Castellani & degl'Innocenti 1993; Catelan et al. 1996). A comparable bound also follows from the analysis of WD cooling (Blinnikov & Dunina-Barkovskaya 1994; the original idea of considering white dwarfs to probe the neutrino electromagnetic properties is found in Sutherland et al. 1976). In contrast, the effect of a nonzero neutrino magnetic moment on the cooling of a neutron star, studied by Iwamoto et al. (1995), leads to the considerably less stringent constraint μ_ν ≲ 5 × 10⁻⁷ μ_B, though a value as small as ∼10⁻¹⁰ μ_B would change the cooling of the crust and could be observable in the future. Finally, HB stars are also quite sensitive to a nonvanishing magnetic moment. A value of μ_ν of about 1.5 × 10⁻¹¹ μ_B would shorten the lifetime of the He-burning stage of those stars, and consequently change the number ratio of HB stars versus RG stars in globular clusters above the observational bound (Raffelt et al. 1989; Raffelt 1996). An important feature of these constraints is that they do not depend on the nature—Dirac or Majorana—or flavor of the neutrinos.

For Dirac neutrinos with a nonvanishing magnetic moment, the left-handed states could be transformed into right-handed states in an external electromagnetic field. For a supernova (SN), where left-handed neutrinos are trapped and right-handed neutrinos are not, this would mean a very efficient cooling mechanism. This leads to the bound μ_ν ≲ 3 × 10⁻¹² μ_B (Barbieri & Mohapatra 1988; Lattimer & Cooperstein 1988; Ayala et al. 1999). A comparable bound comes from the observation that some of the right-handed neutrinos could transform back into the active ones in the galactic magnetic field, and should have been observed by the detectors which measured the neutrino signal from SN 1987A (Nötzold 1988).

Similarly, the analysis of big bang nucleosynthesis puts limits on the magnetic moment for Dirac neutrinos. A large magnetic moment would bring right-handed neutrinos in thermal equilibrium before the cosmological nucleosynthesis, spoiling the prediction of light elements abundances. The current bound is approximately μ_ν ≲ 10⁻¹¹ μ_B (see Dolgov 2002 and references therein).

Lastly, a Majorana neutrino with a nonvanishing magnetic moment could be transformed into an antineutrino of a different flavor in an external electromagnetic field. Miranda et al. (2004) inferred the bound of a few 10⁻¹² μ_B from the lack of the observation by KamLAND of the antineutrino flux from the Sun. Using the same data, Friedland (2005), however, argued that the bound is actually an order of magnitude weaker.

To conclude this section, we point out that the bounds discussed above are sensitive to different effective magnetic moments, namely to different combinations of the elements of the magnetic moment matrix μ^αβ_ν. The terrestrial experiments using reactor sources (electron antineutrinos) are sensitive to (μ_ν)²_e = ∑_α|μ^eα_ν|². On the other hand, the bounds based on solar neutrinos analyses are sensitive to a weighted average of (μ_ν)²_e, (μ_ν)²_μ, (μ_ν)²_τ, with roughly equal weights determined by the oscillation probabilities. Finally, the bounds based on energy loss in stars are sensitive to μ²_ν = ∑_α,β|μ^αβ_ν|².

In this subsection, for the convenience of the reader, we briefly review the main features of the standard evolution of massive stars relevant to our subsequent discussion of the effects of the neutrino magnetic moment. We collect some of the technical details in Appendix A. For a standard introduction to the evolution of massive stars, see, e.g., Clayton (1983), Arnett (1996), and Woosley et al. (2002).

We consider here stars in the mass range of 7 M_☉ ≲ M ≲ 25 M_☉. The basic information about the physical conditions in the core of these stars can be obtained from simple considerations of hydrostatic equilibrium and balance between energy generation and loss. Assuming for the moment that the interiors of these stars can be described as ideal, nondegenerate gas, from the consideration of hydrostatic equilibrium, one can find a simple relation among the mass M, central temperature T_c, and central density ρ_c (Clayton 1983; Arnett 1996)

$$\begin{equation} \left(\frac{{T_{\mathrm{c}}}}{1\,{\mathrm{K}}}\right) \simeq 4.6\times 10^6 \, \mu \, \left(\frac{\! M}{\, {M_{\odot }}}\right)^{2/3}\,\left(\frac{{\rho _{\mathrm{c}}}}{1\,{\mathrm{g}\,\mathrm{cm}^{-3}}}\right)^{1/3} \,, \end{equation} \tag{ 4 }$$

where μ = (∑_iY_i + Y_e)⁻¹, with Y_e the mean number of electrons per baryon, and Y_i = X_i/A_i with X_i and A_i the mass fraction and atomic number of species i, respectively.

The curves in Figure 1 represent the evolution of the central temperature and density for stars of three different masses, as found with our numerical code. Once one fuel is exhausted, the star follows a period of compression, which lasts until the temperature and density satisfy the required condition for the ignition of another nuclear reaction. We can see that these stars indeed approximately follow Equation (4): in particular, the more massive ones have higher central temperatures. This simple fact will have important implications later.

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The approximation of ideal nondegenerate gas is valid for a relatively limited set of densities and temperatures, as seen in Figure 2 which shows the ratio of the combined electron and photon pressure P_e + P_rad ≡ P_e+γ to the corresponding electron ideal gas pressure, P_e,i.g. ≡ n_ekT. At high temperatures and low densities, the pressure is dominated by radiation (photons and relativistic electron–positron plasma). At lower temperatures and high densities, the pressure is set by electron degeneracy.

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Comparing Figures 1 and 2, we see that the evolutionary trajectory of the 25 M_☉ star indeed lies almost entirely in the region where the ideal gas equation of state applies, except at the very last stages of the evolution which are on the edge of being degenerate. The lower mass stars considered in Figure 1, having lower central temperatures, are more subject to the effects of degeneracy which set in after the start of the carbon burning (Arnett 1996).

Two important observations need to be made at this point. First, so long as there is steady nuclear burning at the center, the star has to be outside, or at least on the edge, of the degenerate region. Indeed, a strongly degenerate system does not have the necessary negative feedback mechanism to regulate the rate of burning: energy production does not lead to expansion and cooling of the central region if the pressure is independent of temperature, as is the case for strongly degenerate systems (see, e.g., Raffelt 1996). The detour of the 10 M_☉ star through the region of high degeneracy seen in Figure 1 indicates that the nuclear burning at this stage happens in a shell rather than in the core (Woosley et al. 2002). When the burning reaches the center, the temperature jumps and the feedback is enabled. As we will see, this behavior will be even more pronounced in the presence of the neutrino magnetic moment.

Second, there is an important physical distinction between the regimes of nonrelativistic and relativistic degeneracy. The former is characterized by an adiabatic exponent γ_ad = 5/3 (polytropic index n = 3/2)⁶ while the latter has γ_ad = 4/3 (n = 3). A nonrelativistic degenerate gas responds more strongly to compression than a relativistic one. In fact, γ_ad = 4/3 corresponds to the edge of stability: if a sufficient fraction of the stellar core has this adiabatic exponent, the star becomes unstable to collapse (Zeldovich & Novikov 1971). In the case of white dwarfs, this leads to the well-known upper limit on the mass, the Chandrasekhar mass. In our case of extra cooling, this will lead to the appearance of a new type of supernova.

We now turn to the mechanisms of energy loss. Energy generated by nuclear reactions is lost either through the surface of the star as photon radiation, or through the volume via neutrino emission. Photon energy loss per unit mass is about 10⁴–10⁵ erg g⁻¹ s⁻¹ for stellar masses considered here (see, e.g., Arnett 1996, Section 6.4). Neutrino cooling becomes important when the emission rate per unit mass exceeds this value.

At the temperatures and densities of interest for our discussion, neutrinos are dominantly produced by the following processes (see Beaudet et al. 1967; Dicus 1972 and reference therein):

• 1.  
  photo process, $\gamma \, e^-\rightarrow e \, \nu \, \overline{\nu }$;
• 2.  
  pair process, $e^+\,e^- \rightarrow \nu \, \overline{\nu }$;
• 3.  
  plasma process, $\gamma \rightarrow \nu \, \overline{\nu }$,also known as plasmon decay;
• 4.  
  bremsstrahlung, $e^-(Ze)\rightarrow (Ze)\,e^-\, \nu \, \overline{\nu }$.

The diagrams for the first three processes are depicted in Figure 3.

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Which of these processes dominates depends on the density and temperature, as seen in Figure 4(a). The contour lines show the rate of energy loss, and these are computed using the expressions for the neutrino cooling rates given by Itoh et al. (1996). These rates are used as a starting point in our numerical analyses to compute the reference models with standard particle physics.

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Superimposing Figure 1 to Figure 4(a), we see that in the early burning stages (H- and He-burning), the neutrino production is dominated by the photo process. The energy loss (see Equation (A1) or contour lines in Figure 4(a)), however, is still small compared to the stellar luminosity, 10⁴–10⁵ erg g⁻¹ s⁻¹. The neutrino energy loss starts dominating over the radiation only after central helium depletion, for temperatures above ∼5 × 10⁸ K. When carbon is ignited, T ≃ 7 × 10⁸ K, the star is already losing most of its energy through neutrinos. For the more massive stars (see the curve corresponding to 25 M_☉ and 15 M_☉ in Figure 1), neutrinos will be mainly produced by the pair process. For lower mass stars, however, there will be a competition between the pair and plasma processes. As we see, the bremsstrahlung process is never important for a massive star.

Since the neutrino emission rate increases rapidly with temperature (contour lines in Figure 4(a)), the fuel is consumed faster in the late stages of stellar evolution. For this reason, these later stages are characterized by progressively shorter timescales. In fact, a massive star spends about 90% of its life burning H, and almost all of the remaining 10% burning He. The timescale for the other burning stages is thousands of years for C, years for Ne and O, and only days for Si (Woosley et al. 2002).

In our numerical model (see details in Section 4), stars whose initial mass is above ∼9 M_☉ can ignite all burning phases (H, He, C, Ne, O, and Si burning).⁷ They end up with a typical "onion structure" with iron in the center and several layers around, in which fusion of lighter elements occurs. No energy can be gained by fusion reactions in the iron core, and once its mass exceeds ∼1.4 M_☉, the star collapses producing a CCSN.

Stars less massive than about 9 M_☉ have a different destiny. If the initial mass is less than about 7.5 M_☉, they never reach the stage of carbon burning and end their life as carbon–oxygen white dwarfs (CO-WDs, see the bottom row of Figure 7). If their initial mass is in the range 7.5 ≲ M/M_☉ ≲ 9, they do ignite carbon, but not neon, and end their lives as oxygen–neon–magnesium white dwarfs (ONeMg WDs). These numbers are important to us because they will be shifted by the presence of the additional cooling. Moreover, a new evolution possibility will be opened up.

We also note that the stars that we just classified as making ONeMg WDs could yet have another fate. They are close to the Chandrasekhar mass, and, after a typical "deep third dredge-up" that mixes the envelope with helium layer of the core, the core may continue to grow by hydrogen/helium shell burning. Eventually, collapse of the core may occur due to electron capture and a supernova can result (electron capture supernova, ECSN, Miyaji et al. 1980). Such a supernova typically would produce only very little I56Ni. The details of whether the core can grow highly depend on the dredge-up (mixing) after and between the thermal pulses of the hydrogen/helium shell burning, competing with the mass loss of star that terminates core growth when the envelope has been lost. The historical developments on this topic are summarized by Woosley et al. (2002), while some recent developments are discussed by, e.g., Poelarends et al. (2008). Due to the uncertainty of the mass range of ECSN, we will consider them as part of our ONeMg core "bin" and just note that they comprise some fraction of the upper end of the ONeMg WD mass range.

Also note that we only consider single star evolution here. Special evolution paths that may occur in binary star systems, such as type I SNe, are not considered in this paper.

If neutrinos have a nonvanishing magnetic moment, additional processes shown in Figure 5 must be considered. It turns out that the main effect of the neutrino magnetic moment is to increase the plasmon decay rate. For example, for μ_ν/μ_B = 10⁻¹⁰, density ρ ∼ 10⁴ g cm⁻³ and temperature T = 10⁸ K, the nonstandard plasma process energy loss is about 10³ larger than the standard one (see Equation (B1)). On the other hand, for the same value of μ_ν the nonstandard pair production rate never exceeds 30% of the standard one (see Figure 9). Details of the analysis of the modified energy loss rates are provided in Appendix B.

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Graphically, the effects of the magnetic moment are shown in panels (b) and (c) of Figure 4. We see that the plasmon-dominated region enlarges considerably. Comparing Figure 4 with Figure 1, we see that the 25 M_☉ star "misses" the plasmon-dominated region, while the 15 M_☉ and 10 M_☉ stars, having lower central temperatures, enter it. Therefore, for them the effects of the anomalous cooling should be important.

The above argument does not take into account possible feedback effects of the extra cooling on the evolutionary trajectory. Indeed, such feedback might be anticipated on general grounds: extra cooling, if sufficiently strong, can lead to the decrease of temperature, which would in turn push the star deeper in the plasmon-dominated degenerate region. Once the effects of degeneracy become important, cooling does not change the pressure consistently and, so, there is not much compressional heating as a reaction (Arnett 1996). Thus, a qualitative change in the evolutionary trajectory might be expected.

We will now see how these qualitative expectations are borne out by our detailed numerical simulations.

Our numerical modeling is based on KEPLER, a one-dimensional implicit hydrodynamics stellar evolution code (Weaver et al. 1978). The physical ingredients of this code—such as the treatment of convection and opacities—in the case of the standard physics are well documented in the literature (Woosley et al. 2002; Woosley & Heger 2007; Heger & Woosley 2008).

In order to simulate the effects of nonstandard cooling in the evolution of massive stars, we modified the neutrino cooling routine of the code. Specifically, we computed the energy loss induced by the electromagnetic pair production (see Appendix B.2) and derived an interpolating function which is accurate at the level of 5% in all the region of interest. The bremsstrahlung and photo processes were left unchanged since they do not play any role in our analysis.⁸ For the plasma process, we used the fitting function given by Haft et al. (1994) which is shown to be better than 5% in all the region of interest for our purpose (see Figure 8(e) of Haft et al. 1994).

We used the modified code to compute a two-dimensional grid of data points, varying the mass of the star, 4 M_☉ < M < 25 M_☉, and the value of the neutrino magnetic moment, 0 < μ_ν < 0.5 × 10⁻¹⁰ μ_B. The results are discussed below.

Figure 6 displays the evolution of the central temperature and density for stars of four representative masses: 11.5 M_☉, 15 M_☉, 17 M_☉, and 20 M_☉. These simulations include our largest extra cooling of μ_ν = 0.5 × 10⁻¹⁰ μ_B.

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As expected, we see that the evolution of the most massive of these stars, 20 M_☉, shows no visible departure from the standard trajectory (Figure 1). The lower mass stars, in contrast, deviate from their standard trajectories into the region of lower temperature. The curves start to bend between the He- and the C-burning stages, that is as soon as the neutrino cooling becomes more important than the radiation energy loss. The deviation becomes more pronounced the lower the mass of the star, again as anticipated.

The physics behind this phenomenon is easy to understand. As the CO core contracts following He burning, the heat that would otherwise increase the temperature to the point of carbon ignition is instead lost due to the additional cooling. The core is unable to stay on the "ideal gas" trajectory and instead collapses to a degenerate configuration. The center of core is too cold to ignite carbon burning, which is instead ignited off-center. The center becomes a relatively "cold" white dwarf (T∼ (3–5) ×10⁸ K) surrounded by a hotter burning shell (T ∼ 1 × 10⁹ K). Notice that while off-center ignition is observed in the standard case for lower mass stars (∼10 M_☉, see Figure 1), here it is seen for stars of 15 M_☉.

What happens next depends on the mass of the star. In the case of the 15 M_☉ star, the burning eventually reaches the center. At this point, the corresponding curve in Figure 6 abruptly jumps back up to the standard "ideal gas" trajectory. The silicon burning stage in the center ensues and the star eventually undergoes iron core collapse. The remarkable detour into the degenerate region results in a somewhat different composition of the supernova progenitor (see Section 4.5).

A dramatically different fate awaits stars of lower mass, as represented by the 11.5 M_☉ curve in Figure 6. In this case, the detour into the degenerate region is more pronounced, with the central temperature dropping down to T∼ (1–2) × 10⁸ K. The shell burning of CO extinguishes before reaching the center. As a result of the outer shell burning, a relativistic degenerate core with a central CO region builds up until it reaches the Chandrasekhar limit. The core then undergoes a thermonuclear runaway, consuming the CO fuel in its center (CO explosion) and possibly igniting the surrounding layer as well. This is indicated by the vertical line in Figure 6. In effect, the extra cooling leads to a new kind of supernova—a type Ia-like thermonuclear explosion inside a massive star—which perhaps could be viewed as new "type I.7." Our simulation stops at this point.

The fates of the stars of different masses as a function of the neutrino magnetic moment are shown in Figure 7. We see that neutrino magnetic moment appreciably shifts the mass thresholds between the different outcomes. In the presence of the extra cooling, more and more massive stars are unable to ignite carbon burning. The CO window expands as a result and the ONeMg window shrinks. The CO explosion window first opens up at μ_ν = 0.2 × 10⁻¹⁰ μ_B and for μ_ν = 0.5 × 10⁻¹⁰ μ_B covers stars with masses 9.8 M_☉ ≲ M ≲ 11.5 M_☉. A complete overview of all the models computed is given in Table 1 of Appendix C.

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As mentioned earlier, one of the effects of the extra cooling is to make more difficult the ignition of various reactions, in particular carbon, oxygen, and neon. To burn those nuclei now requires a larger stellar mass, meaning that the threshold mass separating stars that end up as white dwarfs from those that end up as supernovae increases, as seen in Figure 7.

This may have interesting observational implications. In particular, in recent years there has been important progress identifying progenitors of observed supernovae. Hints are accumulating that stars of initial mass 8 M_☉–9 M_☉ do, in fact, end up as supernova explosions. A particular case is supernova SN 2003gd (type II-P), for which the observations favor a red supergiant progenitor of mass ∼8 M_☉–9 M_☉ (Van Dyk et al. 2003; Smartt et al. 2004; Van Dyk et al. 2006).

The situation with observations continues to improve at a rapid pace. For example, very recently a progenitor of supernova SN 2008bk (also a type II-P) was identified to be a red supergiant with the mass of 8.5 ± 1.0 M_☉ (Mattila et al. 2008). Likewise, a recent study of 20 progenitors of type II-P supernovae finds that the minimum stellar mass for these stars is 8⁺¹_−1.5 M_☉ (Smartt et al. 2008). Referring to Figure 7, we see that values μ_ν ≳ 0.35 × 10⁻¹⁰ μ_B are already disfavored. It seems highly likely that more observational progress is to follow.

The interpretation of the observations relies, among many factors, on the stellar models used to infer the mass of the progenitor from its color and luminosity. Hence, further refinement of the stellar models would also be beneficial. To this end, it must be mentioned that the magnetic moment cooling by itself does not change the relationship between the surface characteristics and the mass: our simulation shows changes in the central region of the star at late stages of the evolution, but not in the surface characteristics.

The appearance of a new type of supernova as a consequence of the additional cooling is perhaps the most remarkable finding of our simulations. Should such objects have distinct characteristics not observed in actual supernova explosions, a robust bound on the magnetic moment, μ_ν < 0.2 × 10⁻¹⁰ μ_B, would result.

We again stress that we were unable to follow the development of the explosion beyond the ignition of the degenerate CO burning. Such calculations would be highly desirable. Provisionally, we could consider a plausible model of a type Ia supernova exploding inside a ∼9 M_☉–11 M_☉ red supergiant. A simple estimate we performed shows that, especially for stars in the low mass end of the "CO explosions" bin, the amount of energy available upon burning of the entire CO core is enough to disrupt the star, though a detailed hydrodynamic study to confirm this assessment has yet to be performed. The resulting explosion would leave no neutron star remnant, and no neutrino signal would be observed. The amount of ⁵⁶Ni and ⁵⁶Co could be close to what is produced in a type Ia explosion, ∼0.4 M_☉–0.7 M_☉ (Blinnikov et al. 2006). This would significantly exceed the amounts ejected in a standard core collapse explosion, ∼0.1 M_☉–0.15 M_☉, or in possible ECSNe from our ONeMg WD bin. As discussed by Woosley et al. (2002), the decay of ⁵⁶Co to ⁵⁶Fe (half-life 77 days) is observationally very significant, giving a "radioactive tail" to the light curve. Thus, excessive amounts of ⁵⁶Co could be seen. For the current accuracy of the determination of ⁵⁶Ni mass from observations see, e.g., Figure 9 in Smartt et al. (2008).

Notice that in the previous argument we concentrated on the low-mass end of the "CO explosions" bin. In fact, the pre-explosion composition of the stars in the "CO explosions" bin varies with the mass of the star. Two typical examples are shown in Figure 8. We see that only the inner fraction of the core has a CO composition. This CO inner core is surrounded by either an ONeMg layer at lower masses, or a silicon–sulfur (SiS) layer (from ONeMg shell burning) at higher masses. We find that the size of the central CO "nugget" decreases as the mass of the star increases. This will affect densities, nucleosynthesis, and energetics of the explosion.

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At the upper mass limit of the "CO explosions" bin, the energy may eventually become insufficient to disrupt the star; in this case, a burning pulse may eject the outer layer just before the final core collapse SN, making it rather bright at the peak due to the interaction of the core collapse SN ejecta and the previously ejected layers. This mechanism is similar to the model for pair-instability supernovae presented by Woosley et al. (2007). The resulting SN would have little I56Ni to power the tail of the light curve.

In a few cases at the upper mass limit we also observe the formation of an "iron" layer from silicon shell burning; these cases have only a tiny (∼0.1 M_☉) CO core left in the center and collapse as core collapse SNe before igniting the central CO nugget; they would probably be indistinguishable from regular CCSNe.

The detailed information from our simulations is collected in Table 1. We can see that a significant fraction of the supernovae in the "CO explosions" bin do have sizable CO cores, enough to disrupt the stars and give the characteristic excess ⁵⁶Co observational signature.

Lastly, we note recent observations of the Kepler's supernova remnant (Reynolds et al. 2007).⁹ The abundance of iron in the shocked ejecta together with no evidence for oxygen-rich ejecta indicates that this object underwent a thermonuclear explosion. Curiously, a detailed analysis described in that paper suggests the possibility of a type Ia explosion in a more massive progenitor. That the explosion occurred in the environment that is not low in metallicity is especially interesting. Clearly, further observational studies are very desirable.

There are several other effects of the extra cooling that are worth mentioning. For example, the resulting composition of the core-collapse supernova progenitor is somewhat modified. Our calculations of the 12 M_☉ star show the size of the CO core decreases by 8% with μ_ν = 0.5 × 10⁻¹⁰ μ_B. This small change could be of interest, since the explosion mechanism is a sensitive function of the chemical composition.

Another effect is the change of the presupernova neutrino signal (Odrzywolek et al. 2004). This occurs because the last stage before the iron core collapse—Si burning—proceeds significantly faster since the energy escapes the core more easily. For 12 M_☉ and μ_ν = 0.35 × 10⁻¹⁰ μ_B, the Si shell burning time is only about 8 hr, instead of 15 hr for μ_ν = 0. The presupernova neutrino signal at a future megaton-sized neutrino detector could be shorter with higher luminosity.

Other effects may be revealed upon a more detailed investigation. For example, the extra cooling would modify the nucleosynthetic yields of various elements.

The standard and electromagnetically induced plasmon emission rates (ε_plasma and ε^μ_plasma) depend differently on the plasma frequency, and hence on the density. This behavior can be anticipated on the grounds of dimensional analysis. The energy loss rate can be schematically written as (decayrate) × (photonenergy) × (numberdensityofphotons). Therefore, the ratio of the standard ε_plasma and the additional nonstandard ε^μ_plasma cooling rates is equal to the corresponding ratio of the plasmon decay rates, Γ^μ/Γ (see Friedland & Giannotti 2008). The latter, in turn, can be estimated by dimensional analysis. The rates are proportional to the squares of the corresponding coupling constants, μ_ν or G_F/e. The remaining dimension should be fixed by ω_pl alone (in the rest frame of the plasmon, T cannot enter). Therefore, Γ^μ/Γ ∼ μ²_ν/[G²_F/α]ω⁻²_pl. In the region of temperature and density of interest for us, a detailed calculation gives (Haft et al. 1994)

$$\begin{equation} \frac{\varepsilon _{\mathrm{plasma}}^\mu }{\varepsilon _{\mathrm{plasma}}}\simeq 0.318 \, \left(\frac{10\,{\mathrm{k}{\mathrm{e}\!\mathrm{V}}}}{\omega _{\rm pl}} \right)^2 \mu _{12}^2, \end{equation} \tag{ B1 }$$

where the plasma frequency is numerically given by, e.g., Raffelt (1996),

$$\begin{equation} \omega _{\rm pl}=28.7\,{\mathrm{e}\!\mathrm{V}} \frac{({Y_{\mathrm{e}}} \rho)^{1/2}}{[1+(1.019\times 10^{-6}\,{Y_{\mathrm{e}}} \rho)^{2/3}]^{1/4}}, \end{equation} \tag{ B2 }$$

with ρ in g cm⁻³. Therefore, at low densities, ρ ≪ 10⁶g cm⁻³, the ratio between the electromagnetic-induced plasmon emission rate and the standard one is inversely proportional to the density, whereas at high density ρ ≫ 10⁶g cm⁻³, it is inversely proportional to ρ^2/3. Finally, taking for example μ_ν/μ_B = 10⁻¹⁰, Equation (B1) shows that, for density sufficiently low, the nonstandard plasma process can be much larger than the standard one. Therefore, when the nonstandard neutrino interactions are turned on, the region where the plasma process is dominant becomes larger, especially at low densities (see panels (b) and (c) of Figure 3).

The energy loss per unit time and volume in neutrino pair production is given by the transition probability for $e^+ e^- \rightarrow \nu \bar{\nu }$ annihilation multiplied by the neutrino pair energy and integrated over the density of both initial and final states. This reads

$$\begin{eqnarray} &&Q^{\mathrm{pair}}=\frac{4}{(2 \pi)^6}\int \!\! \frac{d^3p_1}{e^{(E_1-\mu)/T}+1}\frac{d^3p_2}{e^{(E_2 +\mu)/T}+1} (E_1+E_2) v_{\mathrm{rel}}\,\sigma ^{\mathrm{pair}},\nonumber\\ \end{eqnarray} \tag{ B3 }$$

where μ is the chemical potential, v_rel is the electron–positron relative velocity, and σ^pair represents the spin-averaged cross-section for $e^+ e^- \rightarrow \nu \bar{\nu }$ annihilation. The quantity E₁E₂v_relσ^pair is Lorentz invariant and is given by (p_i = (E_i, p_i), q_i = (ω_i, q_i))

$$\begin{eqnarray} E_1 E_2 v_{\mathrm{rel}}\, \sigma ^{\mathrm{pair}}&=&\frac{1}{4} \int \frac{d^3q_1\,d^3q_2}{(2\pi)^3 2\omega _1\,(2\pi)^3 2 \omega _2} |M|^2 \, (2\pi)^4\,\delta ^{4}\nonumber\\ &&\times (p_1+p_2-q_1-q_2) \ \ \ \end{eqnarray} \tag{ B4 }$$

in terms of the spin-averaged squared amplitude |M|². If the pair production proceeds via neutrino magnetic moment, we find (s = (p₁ + p₂)², t = (p₁ − q₁)², u = (p₁ − q₂)²)

$$\begin{equation} |M_{\mu _\nu }|^2= \frac{e^2\mu _\nu ^2}{s} [ s^2 - (u - t)^2 ], \end{equation} \tag{ B5 }$$

where μ²_ν represents the effective dipole moment

$$\begin{equation} \mu _\nu ^2 = \sum _{i,j=1}^{3} \ |\mu _{ij} |^2. \end{equation} \tag{ B6 }$$

Integrating over the invariant phase space leads to

$$\begin{equation} E_1 E_2 v_{\mathrm{rel}}\sigma ^{\mathrm{pair}}_{\mu _\nu }= \frac{\alpha \, \mu _\nu ^2}{12} \left(s+2m_{\mathrm{e}}^2\right), \end{equation} \tag{ B7 }$$

or equivalently $v_{\mathrm{rel}}\sigma ^{\mathrm{pair}}_{\mu _\nu } = (\alpha \mu _\nu)^2/3 \, (1 + 2 m_{\mathrm{e}}^2/s)$. Performing the angular integrations in Equation (B4) we then find

$$\begin{eqnarray} &&Q^{\mathrm{pair}}_{\mu }=\frac{\alpha \, \mu _\nu ^2}{6 \pi ^4}\!\!\int _{m_{\mathrm{e}}}^{\infty } dE_1\; dE_2 \ p_1 \, p_2 \frac{(E_1+E_2)\big(2m_{\mathrm{e}}^2+E_1 E_2\big)}{(e^{(E_1-\mu)/T}+1)(e^{(E_2+\mu)/T}+1)},\nonumber\\ \end{eqnarray} \tag{ B8 }$$

showing that the expression for the luminosity factorizes in the product of two one-dimensional integrals.

Following Beaudet et al. (1967), we now define the dimensionless variables

$$\begin{equation} \lambda =T/m_{\mathrm{e}}, \,\qquad \nu =\mu /T, \end{equation} \tag{ B9 }$$

and the dimensionless integrals

$$\begin{equation} G^{\pm }_{n}(\lambda,\nu)= \lambda ^{3+2n} \int _{\lambda ^{-1}}^{\infty } dx \frac{x^{2n+1}(x^2-\lambda ^{-2})^{1/2}}{e^{(x\pm \nu)}+1}. \end{equation} \tag{ B10 }$$

The luminosity due to nonstandard pair production reads

$$\begin{eqnarray} Q^{\mathrm{pair}}_{\mu }&=&\frac{\alpha \, \mu _\nu ^2 \,m_{\mathrm{e}}^7}{6 \pi ^4} \big[ 2G^-_{-1/2}G^+_{0}+2G^-_{0}G^+_{-1/2}\nonumber\\ &&+G^-_{0}G^+_{1/2}+G^-_{1/2}G^+_{0}\big], \end{eqnarray} \tag{ B11 }$$

and is a function of the temperature and the chemical potential. In order to have a function of temperature and density, the chemical potential must be solved in terms of temperature and density. This requires the solution of the following equation(Beaudet et al. 1967):

$$\begin{equation} \rho = 2.922\times 10^6\,{\mathrm{g}\,\mathrm{cm}^{-3}} \, (G^{-}_{0}-G^{+}_{0}) \,, \end{equation} \tag{ B12 }$$

which cannot, in general, be inverted analytically in terms of μ. The numerical solutions are plotted in Figure 1 of Beaudet et al. (1967).

In order to assess the impact of nonstandard cooling, the above result has to be compared to the SM emission rate (Dicus 1972)

$$\begin{eqnarray} Q^{\mathrm{pair}}_{\mathrm{SM}}({\nu _{{I{}{e}}}}) &=&\frac{G_{\mathrm{F}}^2 m_{\mathrm{e}}^9}{18\pi ^5}\big\lbrace\big(7 C_{\mathrm{V}}^2\,{-}\,2C_{\mathrm{A}}^2\big)\big[G^-_{0}G^+_{-1/2}+G^-_{-1/2}G^+_{0}\big] \nonumber\\ &&{+}\,9C_{\mathrm{V}}^2\big[G^-_{1/2}G^+_{0}+G^-_{0}G^+_{1/2}\big]+\big(C_{\mathrm{V}}^2\,{+}\,C_{\mathrm{A}}^2\big) \big[4G^-_{1}G^+_{1/2}\nonumber\\ &&{+}\,4G^-_{1/2}G^+_{1}\,{-}\,G^-_{1}G^+_{-1/2} -G^-_{1/2}G^+_{0}-G^-_{0}G^+_{1/2}\nonumber\\ &&{-}\,G^-_{-1/2}G^+_{1}\big] \big \rbrace. \end{eqnarray} \tag{ B13 }$$

Here C_V = 1/2 + 2sin²θ_W and C_A = 1/2, with the weak mixing angle numerically given by sin²θ_W = 0.23. The energy loss due to ν_μ and ν_τ is obtained from Q^pair_SM(ν_Ie) by replacing C_V → C_V − 1 and C_A → −C_A.

A simple comparison of the standard and nonstandard pair production amplitudes suggests that the two contributions are of comparable size if μ_ν ∼ 10⁻¹⁰ μ_B. Using the full expressions of Equations (B11) and (B13), we plot in Figure 9 the ratio Q^pair_μ/Q^pair_SM for μ_ν = 10⁻¹⁰ μ_B in the region of interest in the ρ–T plane. The results show that indeed the nonstandard loss never exceeds 30% of the electro-weak one, being most important in the nonrelativistic and nondegenerate region. From the behavior of G^±_n(λ, ν) in the relativistic and/or degenerate limit one can simply verify that Q^pair_μ/Q^pair_SM → 0 in these physical regimes, as shown in Figure 9.

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For the numerical implementation in the stellar evolution code, we have used the simple fitting formula

$$\begin{eqnarray} \frac{Q^{\mathrm{pair}}_{\mu }}{Q^{\mathrm{pair}}_{\mathrm{SM}}} (\bar{\rho }, \bar{T}) &=& \ \mu _{10}^2 \times \big [ n(\bar{\rho }) + a(\bar{\rho }) (\log _{10} \bar{T} - 7.5)^2 \nonumber\\ &&+ b(\bar{\rho }) (\log _{10} \bar{T} - 7.5)^{10} \big ]^{-1}, \end{eqnarray} \tag{ B14 }$$

$$\begin{eqnarray} n(\bar{\rho }) &=& 2.959105 + 0.0021515 (\log _{10}\bar{\rho } - 3)^5, \end{eqnarray} \tag{ B15 }$$

$$\begin{eqnarray} a(\bar{\rho }) &=& 0.362925 + 0.000408304 \, \bar{\rho }^{0.436811}, \end{eqnarray} \tag{ B16 }$$

$$\begin{eqnarray} b(\bar{\rho }) &=& 0.00998357 + 2.76174 \cdot 10^{-7} \, \bar{\rho }^{0.5018889}, \end{eqnarray} \tag{ B17 }$$

where μ₁₀ is defied by μ_ν = μ₁₀ 10⁻¹⁰ μ_B, $\bar{T} = T/{\mathrm{K}}$, and $\bar{\rho } = (\rho /\mu _e)/({\mathrm{g}\,\mathrm{cm}^{-3}})$. The fitting formula is accurate at the level of 5% or better in the region defined by $10^{7.5} < \bar{T} < 10^{10}$ and $10^{3} < \bar{\rho } < 10^{10}$.

Finally, in the region where the pair process dominates neutrino emission, the additional energy loss per unit time and mass is approximately given by

$$\begin{equation} \varepsilon _{\mathrm{pair}}^\mu \simeq 1.6 \times 10^{11}\, {\mathrm{erg}\,\mathrm{g}^{-1}\,\mathrm{s}^{-1}} \times \frac{ \mu _{10}^2}{ \rho _4}\, e^{-118.5/T_8}\,, \end{equation} \tag{ B18 }$$

where μ₁₀ = μ_ν/10¹⁰ μ_B. Comparing with Equations (A2) and (A3) we see that this energy loss is never more than ∼30% of the standard model value, if μ_ν is below the experimental bound.