Axial U(1) current in Grabowska and Kaplan's formulation

Recently, Grabowska and Kaplan suggested a non-perturbative formulation of a chiral gauge theory, which consists of the conventional domain-wall fermion and a gauge field that evolves by the gradient flow from one domain wall to the other. In this paper, we discuss the U(1) axial-vector current in 4 dimensions using this formulation. We introduce two sets of domain-wall fermions belonging to complex conjugate representations so that the effective theory is a 4-dimensional vector-like gauge theory. Then, as a natural definition of the axial-vector current, we consider a current that generates the simultaneous phase transformations for the massless modes in 4 dimensions. However, this current is exactly conserved and does not reproduce the correct anomaly. In order to investigate this point precisely, we consider the mechanism of the conservation. We find that this current includes not only the axial current on the domain wall but also a contribution from the bulk, which is non-local in the sense of 4-dimensional fields. Therefore, the local current is obtained by subtracting the bulk contribution from it.Comment: 25 pages, 1 figur

Solving the Naturalness Problem by Baby Universes in the Lorentzian Multiverse

We propose a solution of the naturalness problem in the context of the multiverse wavefunction without the anthropic argument. If we include microscopic wormhole configurations in the path integral, the wave function becomes a superposition of universes with various values of the coupling constants such as the cosmological constant, the parameters in the Higgs potential, and so on. We analyze the quantum state of the multiverse, and evaluate the density matrix of one universe. We show that the coupling constants induced by the wormholes are fixed in such a way that the density matrix is maximized. In particular, the cosmological constant, which is in general time-dependent, is chosen such that it takes an extremely small value in the far future. We also discuss the gauge hierarchy problem and the strong CP problem in this context. Our study predicts that the Higgs mass is 140\pm20 GeV and {\theta}=0.Comment: 35 pages, 11 figures. v2: added Section 5.3 with comments on the Wick rotation of the Lorentzian gravity. v3 some comments adde

Gravitational string-membrane hedgehog and internal structure of black holes

We investigate charged Nambu-Goto strings/membrane systems in the Einstein-Maxwell theory in 3+1 dimensions. We first construct a charged string hedgehog solution that has a single horizon and conical singularity. Then we examine a charged membrane system, and give a simple derivation of its self energy. We find that the membrane may form an extremal Reissner-Nordstrom black hole, but its interior is a flat spacetime. Finally by combining the charged strings and the membrane we construct black hole solutions that have no singularities inside the horizons. We study them in detail by varying the magnitude of the two parameters, namely, the charge times the membrane tension and the string tension. We also argue that the strings have, due to the large redshift inside the system, a fair amount of degrees of freedom that may explain the entropy of the corresponding black holes.Comment: 22 pages, 13 figures, minor revisions, version published in PT

Weak scale from Planck scale -- Mass Scale Generation in Classically Conformal Two Scalar System --

In the standard model, the weak scale is the only parameter with mass dimensions. This means that the standard model itself can not explain the origin of the weak scale. On the other hand, from the results of recent accelerator experiments, except for some small corrections, the standard model has increased the possibility of being an effective theory up to the Planck scale. From these facts, it is naturally inferred that the weak scale is determined by some dynamics from the Planck scale. In order to answer this question, we rely on the multiple point criticality principle as a clue and consider the classically conformal $\mathbb{Z}_2\times \mathbb{Z}_2$ invariant two scalar model as a minimal model in which the weak scale is generated dynamically from the Planck scale. This model contains only two real scalar fields and does not contain any fermions and gauge fields. In this model, due to Coleman-Weinberg-like mechanism, one scalar field spontaneously breaks the $\mathbb{Z}_2$ symmetry with a vacuum expectation value connected with the cutoff momentum. We investigate this using the 1-loop effective potential, renormalization group and large N limit. We also investigate whether it is possible to reproduce the mass term and vacuum expectation value of the Higgs field by coupling this model with the standard model in the Higgs portal framework. In this case, the one scalar field that does not break $\mathbb{Z}_2$ can be a candidate for dark matter, and have a mass of about several TeV in appropriate parameters. On the other hand, the other scalar field breaks $\mathbb{Z}_2$ and has a mass of several tens of GeV. These results can be verified in near future experiments.Comment: 41 pages, 9 figures, minor mistakes in Section VI.B and typos correcte

Black Hole as a Quantum Field Configuration

We describe 4D evaporating black holes as quantum field configurations by solving the semi-classical Einstein equation $G_{\mu\nu}=8\pi G \langle \psi|T_{\mu\nu}|\psi \rangle$ and quantum matter fields in a self-consistent manner. As the matter fields we consider $N$ massless free scalar fields ($N$ is large). We find a spherically symmetric self-consistent solution of the metric $g_{\mu\nu}$ and state $|\psi\rangle$. Here, $g_{\mu\nu}$ is locally $AdS_2\times S^2$ geometry, and $|\psi\rangle$ provides $\langle \psi|T_{\mu\nu}|\psi \rangle=\langle0|T_{\mu\nu}|0 \rangle+T_{\mu\nu}^{(\psi)}$, where $|0\rangle$ is the ground state of the matter fields in the metric and $T_{\mu\nu}^{(\psi)}$ consists of the excitation of s-waves that describe the collapsing matter and Hawking radiation with the ingoing negative energy flow. This object is supported by a large tangential pressure $\langle0|T^\theta{}_\theta|0 \rangle$ due to the vacuum fluctuation of the bound modes with large angular momenta. This describes the interior of the black hole when the back reaction of the evaporation is considered. The black hole is a compact object with a surface (instead of horizon) that looks like a conventional black hole from the outside and eventually evaporates without a singularity. If we count the number of self-consistent configurations $\{|\psi\rangle\}$, we reproduce the area law of the entropy. This tells that the information is carried by the s-waves inside the black hole. $|\psi\rangle$ also describes the process that the negative ingoing energy flow created with Hawking radiation is superposed on the collapsing matter to decrease the total energy while the total energy density remains positive. As a special case, we consider conformal matter fields and show that the interior metric is determined by the matter content of the theory, which leads to a new constraint to the matter content.Comment: ver4: We added a new paragraph to Sec.2.1. and made Appendix

Interior of Black Holes and Information Recovery

We analyze time evolution of a spherically symmetric collapsing matter from a point of view that black holes evaporate by nature. We first consider a spherical thin shell that falls in the metric of an evaporating Schwarzschild black hole of which the radius $a(t)$ decreases in time. The important point is that the shell can never reach $a(t)$ but it approaches $a(t)-a(t)\frac{d a(t)}{d t}$. This situation holds at any radius because the motion of a shell in a spherically symmetric system is not affected by the outside. In this way, we find that the collapsing matter evaporates without forming a horizon. Nevertheless, a Hawking-like radiation is created in the metric, and the object looks the same as a conventional black hole from the outside. We then discuss how the information of the matter is recovered. We also consider a black hole that is adiabatically grown in the heat bath and obtain the interior metric. We show that it is the self-consistent solution of $G_{\mu\nu}=8\pi G \langle T_{\mu\nu} \rangle$ and that the four-dimensional Weyl anomaly induces the radiation and a strong angular pressure. Finally, we analyze the internal structures of the charged and the slowly rotating black holes.Comment: Appear in Physical Review D. Typos fixed. References, clarifications and new appendixes adde

Phenomenological Description of the Interior of the Schwarzschild Black Hole

We discuss a sufficiently large 4-dimensional Schwarzschild black hole which is in equilibrium with a heat bath. In other words, we consider a black hole which has grown up from a small one in the heat bath adiabatically. We express the metric of the interior of the black hole in terms of two functions: One is the intensity of the Hawking radiation, and the other is the ratio between the radiation energy and the pressure in the radial direction. Especially in the case of conformal matters we check that it is a self-consistent solution of the semi-classical Einstein equation, $G_{\mu\nu}=8\pi G \langle T_{\mu\nu}\rangle$. It is shown that the strength of the Hawking radiation is proportional to the c-coefficient, that is, the coefficient of the square of the Weyl tensor in the 4-dimensional Weyl anomaly.Comment: 10 pages. Detail discussions and references added. Accepted Int. J. Mod. Phys.