Fermion states localized
on a self-gravitating non-Abelian monopole

For the gauge and Higgs field we employ the usual static spherically symmetric hedgehog Ansatz [25, 26]

The spherically symmetric Ansatz with a harmonic time dependence for the isospinor fermion field localized by the monopole can be written in terms of two $2\times 2$ matrices $\chi$ and $\eta$ [46, 53] as

Here $u(r)$ and $v(r)$ are two real functions of the radial coordinate only and $\omega$ is the eigenvalue of the Dirac operator.

For the line element we employ Schwarzschild-like coordinates, following closely the usual consideration of gravitating monopole (see, e.g., Refs. [11, 12])

The metric function $N(r)$ can be rewritten as $N(r)=1-\frac{2G\mu(r)}{r}$ with the mass function $\mu(r)$; the ADM mass of the configuration is defined as $M=\mu(\infty)$. The above metric implies the following form of the orthonormal tetrad:

such that $ds^{2}=\eta_{ab}(e^{a}_{\mu}dx^{\mu})(e^{b}_{\nu}dx^{\nu})$, where the Minkowski metric $\eta_{ab}=(+1,-1,-1,-1)$ and $\gamma^{\mu}=e^{\mu}_{\phantom{a}a}\hat{\gamma}^{a}$ with $\hat{\gamma}^{a}$ being the usual flat space Dirac matrices.

Substitution of the Ansatz (6)-(7) into the general system of equations (3) and (5) yields the following set of six coupled ordinary differential equations for the functions $W,H,u,v,N,\sigma$ (here the prime denotes differentiation with respect to the radial coordinate, $\sigma^{\prime}=\frac{d\sigma}{dr}$, etc. ):

Here we define a new dimensionless radial coordinate, $x=e\phi_{0}r$, and three rescaled effective coupling constants $\alpha^{2}=4\pi G\phi_{0}^{2}\,,\beta^{2}=\frac{\lambda}{e^{2}}\,,\tilde{h}=\frac{h}{e}$. The scaled bare mass parameter and the eigenfrequency of the fermion field are $\tilde{m}=\frac{m}{e\phi_{0}}$ and $\tilde{\omega}=\frac{\omega}{g\phi_{0}}$, respectively. The fermion field scales as $\psi\to\psi/(\sqrt{e}\phi_{0}^{3/2})$. To simplify the formulas, we will drop the tilde notation henceforth. Also, in what follows, we restrict ourselves to the case of fermions with zero bare mass setting $m=0$. Hence, the solutions depend essentially on three dimensionless parameters given by the mass ratios

where $M_{Pl}=G^{-1/2}$ is the Plank mass and $M_{v}=e\phi_{0}$, $M_{s}=\sqrt{\lambda}\phi_{0}$ and $M_{f}=h\phi_{0}/2$ are the masses of the gauge field, Higgs field and fermion field, respectively.

The system of equations (8)-(13) is supplemented by the normalization condition of the localized fermion mode¹¹1In our numerical calculations we fix $e=0.689$.

Note that, as $\omega\neq 0$, the metric field $\sigma$ cannot be eliminated from the system (8)-(13), as is done, for example, for a self-gravitating monopole (see, e.g., Ref. [1]).

The system (8)-(13) admits embedded Reissner-Nordström (RN) solution [57, 58]; for the case of unit magnetic charge it reads

A horizon occurs when $N(x)\to 0$, in the Schwarzschild-like parametrization it happens at some finite critical value of $x=x_{\text{cr}}=\alpha_{\text{cr}}$.

The system (1) possesses two limits. The flat space monopole corresponds to the case $\alpha=0$; further, setting $\beta=0$, yields the familiar self-dual BPS solution [54, 55] (see also Ref. [56] for a review),

There is a remarkable flat space solution for the background isospin fermion zero ($\omega=0$) mode [46, 53]. Indeed, in this case the last pair of equations (8)-(13) is decoupled, and it is reduced to

Using the vacuum boundary conditions, we can see that the linearized asymptotic equations for the spinor components approaching the vacuum are

Therefore, gravitationally localized fermion modes with exponentially decaying tail may exist if $\omega^{2}<h^{2}/4$.

The normalizable solution for the localized zero mode is

and it exists only for non-zero negative values of the scaled Yukawa coupling $h$. For example, setting $h=-2$ and making use of the exact BPS monopole solution (16), we obtain

In our numerical calculations we used these closed form BPS solutions as a input.

Another limit $h\to 0$ while $\beta$ is kept fixed, corresponds to the decoupled fermionic sector. In such a case the well known pattern of evolution of the self-gravitating monopole is recovered, a branch of gravitating solutions emerges smoothly from the flat space monopole as the effective gravitational coupling $\alpha$ increases from zero and $\beta$ remains fixed [10, 11, 12]. Along this branch the metric function $N(x)$ develops a minimum, which decreases monotonically. The branch terminated at a critical value $\alpha_{\text{cr}}$ at which the gravitating monopole develops a degenerate horizon and configuration collapses into the extremal Reissner-Nordström black hole, as displayed in the left panel of Fig. 1. A short backward branch of unstable solutions arises in the BPS limit $\beta=0$ at $\alpha=\alpha_{\text{max}}$, it bends backwards and bifurcates with the branch of extremal RN solutions of unit magnetic charge at $\alpha_{\text{cr}}<\alpha_{\text{max}}$ [10]. Note that the ADM mass of the monopole coupled to the fermion zero mode remains the same as the mass of the pure self-gravitating monopole; this is because the non-zero spinor component $u(x)$ is decoupled and there is no backreaction of the fermions, see below.

Generally, the system of mixed order differential equations (8)-(13) can be solved numerically together with constraint imposed by the normalization condition (14). The boundary conditions are found by considering the asymptotic expansion of the solutions on the boundaries of the domain of integration together with the assumption of regularity and asymptotic flatness. Explicitly, we impose

Consider first the evolution of the fermion zero mode localized on the self-gravitating BPS monopole. Note that since both the bare mass of the fermion field and the eigenvalue of the Dirac operator are zero, there is no backreaction of the fermions on the monopole, the system of equations (8)-(13) becomes decomposed into 3 familiar coupled equations for self-gravitating monopole [10, 11, 12] and two decoupled equations for the components of the localized fermion mode.

The fundamental branch of gravitating BPS monopoles with bounded fermionic zero mode smoothly arise from the flat space configuration (16),(18) as the effective gravitational constant $\alpha$ is increased above zero. This branch reaches a limiting solution at maximal value $\alpha_{\text{max}}=1.403$, here it bifurcates with the short backward branch which leads to the extremal RN black hole with unit magnetic charge, see Fig. 1.

In Fig. 2 we displayed the corresponding solutions for some set of values of the effective gravitational coupling $\alpha$ at $h=-1$ and $\beta=0$. With increasing $\alpha$ the size of the configuration with localized modes is gradually decreasing. As the critical value of $\alpha$ is approached, the minimum of the metric function $N(x)$ tends to zero at $x=x_{\text{cr}}$. The metric becomes splitted into the inner part, $x<x_{\text{cr}}$ and the outer part, $x>x_{\text{cr}}$ separated by the forming hozizon. The Higgs field is taking the vacuum expectation value in exterior of the black hole while the gauge field profile function $W(x)$ trivializes there, so the limiting configuration corresponds to the embedded extremal RN solution (15) with a Coulomb asymptotic for the magnetic field. At the same time, the fermion field becomes absorbed into the interior of the black hole, see Fig. 2.

Apart from the zero mode, the system of equations (8)-(13) supports a tower of regular normalizable solutions for fermionic modes with $\omega\neq 0,~{}~{}|\omega|<|h/2|$. Here, both components $u$ and $v$ are non-zero, and for $h<0$ they posses at least one node while for $h>0$ they are nodeless. These solutions can be obtained numerically, now we have to solve the full system of coupled differential equations (8)-(13) imposing the boundary conditions (19). Note that this system is not invariant with respect to inversion of the sign of $\omega$. Indeed, it is seen in Figs. 3 and 4, which display the metric components $N(x)\,,\sigma(x)$ and the fields $u(x)\,,v(x)\,,W(x)\,,H(x)$ for some set of values of the gravitational coupling $\alpha$ and fixed $\beta=1$ and $h=1$, that, as $\omega\to+0$ and $\omega\to-0$, the configurations approach the RN limit in a different way.

In the flat space limit the fermion mode becomes delocalized as $|\omega|\to|h/2|$, while increasing of the gravitational coupling stabilizes the system. Both the ADM mass of the configuration and the eigenvalue $\omega$, which is defined from the numerical calculations, are decreasing as $\alpha$ increases, see the right panel of Fig. 1. The evolution scenario depends generically on the values of the parameters of the model. For example, setting $\beta=1$ and $h=1$, we observe that there are two branches of solutions which are linked to the negative and positive continuum: they end at the critical value $\alpha_{\text{cr}}\approx 1.095$ as $\omega\to+0$, and at $\alpha_{\text{cr}}\approx 1.193$ as $\omega\to-0$ (cf. Fig. 5 and Table 1). In both cases the configuration reaches the embedded extremal RN solution (15) in a way which is qualitatively similar to that of the BPS monopole with localized fermion zero mode discussed above. As $\alpha$ tends to the critical value, the eigenvalue $\omega$ approaches zero and the fermion field is fully absorbed into interior of the forming black hole.

In Fig. 5 we plot the normalized energy of the localized fermionic states as a function of the Yukawa coupling constant $h$. Having constructed some set of solution for different values of $\alpha$, the following scenario becomes plausible. As the Yukawa coupling increases from zero, while both $\beta$ and $\alpha$ are kept fixed, a branch of normalizible non-zero fermion modes emerges smoothly from the self-gravitating monopole. The energy of the localized fermionic states is restricted as $|\omega|<|h/2|$, as the gravitational coupling remains relatively weak, the modes remain close to the continuum threshold.

The spectral flow is more explicit as the coupling $\alpha$ becomes stronger, see Fig. 5. Increase of the Yukawa coupling, which yields the mass of the fermionic states, leads to increase of eigenvalues $\omega$. However, an interesting observation is that at some critical value of the parameter $h$, the energy of the localized mode approaches some maximal value. As the Yukawa coupling continue to grow, the corresponding eigenvalue starts to decrease, it tends to zero as some maximal value $h_{\text{cr}}$. Again, in this limit the configuration approaches the embedded RN solution (15) and the fermion fields are again fully absorbed into interior of the forming black hole. The pattern is illustrated in Fig. 5, where two blue curves display the spectral flow of both positive and negative Dirac eigenvalues $\omega$ for $\alpha=1.09$ and $\alpha=1.185$, respectively. In the limiting case $\omega\to+0$ one has $h_{\text{cr}}\approx 1.5$ (for $\alpha=1.09$) and when $\omega\to-0$ we have $h_{\text{cr}}\approx 0.25$ (for $\alpha=1.185$).

The general scenario is that, depending on the value of the Yukawa coupling constant $h$, there exist a critical value of the gravitational coupling $\alpha_{\text{cr}}$ at which the spectral flow approaches the limit $\omega\to\pm 0$ and the configuration runs to the embedded RN solution (15). Some corresponding values are given in Table 1, and they are also displayed by the bold red dots in Fig. 5. Once again, each particular value of the Yukawa coupling gives rise to two distinct spectral flows approaching the embedded RN solution as $\omega\to\pm 0$ at two different values of $\alpha_{\text{cr}}$.

Finally, we note that the system of equations (8)-(13) possesses two characteristic limiting cases, $h\to\infty$ and $\beta\to\infty$. First, for a fixed value of $\beta$ and increasing Yukawa coupling, the backreaction of the localized fermions becomes stronger, the energy of the gravitating bounded fermionic mode increases and the profile functions of the monopole are significantly deformed, see Fig. 6. We observe that an increase of the Yukawa coupling moves the configuration closer to the RN solution (see the bottom plots of Fig. 6). Note that deformations of the configuration caused by its coupling with massive fermion modes may produce a number of interesting effects related with backreaction of the fermions [59, 60, 61, 62].

Secondly, as the scalar field becomes very massive, the core of the monopole shrinks and in the limit $\beta\to\infty$ the Higgs field is taking its vacuum expectation value everywhere in space apart the origin. One can expect that, for the intermediate range of values of $\beta$, the scenario reported above for the $\beta=1$, should persist. Our numerical results confirm that an increase of the scalar coupling $\beta$ decreases the critical value of the Yukawa coupling $h$ at which the configuration approaches the extremal Reissner-Nordström solution. However, for relatively large values of $\beta$, the pattern of evolution of the self-gravitating monopole becomes different [10, 11, 12, 13, 63]. One might expect also that the behavior of the fermion field could be different in the large-$\beta$ regime.