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In this note, we study the singular mean field equation defined on a Riemann surface with a sign-changing potential function. We prove if some singular sources happen to be placed on the zero-level curve of the potential function, a priori estimate can still be obtained. As a consequence of this estimate, existence and multiplicity results can still be obtained based on the topology of the manifold.

Possible light bosonic dark matter interactions with the Standard Model photon have been searched by microwave resonant cavities. In this paper, we demonstrate the cryogenic readout system calibration of a 7.138 GHz copper cavity with a loaded quality factor $Q_l=10^4$, operated at 22 mK temperature based on a dilution refrigerator. Our readout system consists of High Electron Mobility Transistors as cryogenic amplifiers at 4 K, plus room-temperature amplifiers and a spectrum analyzer for signal power detection. We test the system with a superconducting two-level system as a single-photon source in the microwave frequency regime and report an overall 95.6 dB system gain and -71.4 dB attenuation in the cavity's input channel. The effective noise temperature of the measurement system is 7.5 K., Comment: 7 pages, 5 figures, version to appear in CPC

The dark photon is a promising candidate for the dark matter which comprises most of the matter in our visible Universe. Via kinetic mixing with the Standard Model it can also be resonantly converted to photons in an electromagnetic cavity, offering novel experimental possibilities for the discovery and study of dark matter. We report the results of a pathfinder dark photon dark matter cavity search experiment performed at Hunan Normal University and the Institute of Physics, Chinese Academy of Sciences, representing the first stage of the APEX (Axion and dark Photon EXperiment) program. Finding no statistically significant excess, we place an upper limit on the kinetic mixing parameter $|\chi|<3.7\times 10^{-13}$ around $m_A\simeq 29.5$ $\mu$eV at 90% confidence level. This result exceeds other constraints on dark photon dark matter in this frequency range by roughly an order of magnitude., Comment: 5 pages, 4 figures. Updated to match journal version

We develop generalized no-scale supergravity models of inflation, and then study the corresponding cosmological predictions as well as the formation of primordial black holes (PBHs) and scalar-induced gravitational waves (SIGWs). With a new parameter $0

In a recent series of important works \cite{wei-zhang-1,wei-zhang-2,wei-zhang-3}, Wei-Zhang proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions., Comment: 13 pages

It is well-known that three-family supersymmetric Pati-Salam models from intersecting D6-branes, where either one or both of the ${\rm U}(2)$ gauge factors are replaced by a ${\rm USp}(2)$ group, are quite scarce. In order to construct all such kind of models with generic additional gauge symmetries, we fix the observable sectors and study all the possible hidden sectors. Thus, we are able to completely determine all types of such kind of the inequivalent models on a $\mathbb{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ orientifold from IIA string theory. We find the gauge coupling relations to be highly sensitive to the variations in the hidden sector. One of the models exhibits the gauge coupling unification for a particular solution at the string scale. In addition, we perform scan on the hidden sector variations for the models presented in arXiv:2112.09632, whose gauge coupling relations are still preserved. Thus, we fix the gap in the previous study and complete the model building for all the inequivalent models., Comment: 25+4 pages, 6 figures and 15 tables; added machine learning details; published version. arXiv admin note: text overlap with arXiv:2212.09644

We construct new three-family ${\cal N}=1$ supersymmetric Pati-Salam models from intersecting D6-branes with original gauge group ${\rm U}(4)_C \times {\rm USp}(2)_L \times {\rm U}(2)_R$ on a Type IIA $\mathbb{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ orientifold. We find that replacing a ${\rm U}(2)$ with a ${\rm USp}(2)$ group severely restricts the number of three-generation supersymmetric models such that there are only five inequivalent models. Exchanging the left and right sectors, we obtain five dual models with gauge group ${\rm U}(4)_C \times {\rm U}(2)_L \times {\rm USp}(2)_R$. These ten models have different gauge coupling relations at string scale. The highest wrapping number is 4, and one of the models contains no filler O6-planes. Moreover, we discuss in detail the particle spectra, the composite particles through strong coupling dynamics, and the exotic particle decouplings. Also, we study how to realize the string-scale gauge coupling relations in some models., Comment: 32 pages, 3 figures, published version

We have constructed all the three-family ${\cal N} = 1$ supersymmetric Pati-Salam models from intersecting D6-branes, and obtained 33 independent models in total. But how to realize the string-scale gauge coupling relations in these models is a big challenge. We first discuss how to decouple the exotic particles in these models. In addition, we consider the adjoint chiral mulitplets for $SU(4)_C$ and $SU(2)_L$ gauge symmetries, the Standard Model (SM) vector-like particles from D6-brane intersections, as well as the vector-like particles from the ${\cal N}=2$ subsector. We show that the gauge coupling relations at string scale can be achieved via two-loop renormalization group equation running for all these supersymmetric Pati-Salam models. Therefore, we propose a concrete way to obtain the string-scale gauge coupling realtions for the generic intersecting D-brane models., Comment: 40 pages, 23 figures, and 40 tables; references added, published version

In this article we study bubbling solutions of regular $SU(3)$ Toda systems defined on a Riemann surface. There are two major difficulties corresponding to the profile of bubbling solutions: partial blowup phenomenon and bubble accumulation. We prove that when both parameters tend to critical positions, if there is one fully bubbling blowup point, then under one curvature assumption, all the blowup solutions near a blowup point satisfy a spherical Harnack inequality, which completely rules out the bubble-accumulation phenomenon. This fact is crucial for a number of applications., Comment: 42 pages. arXiv admin note: text overlap with arXiv:2202.10825

We propose the generic no-scale inflation inspired from string theory compactifications. We consider the K\"ahler potentials with an inflaton field $\varphi$, as well as one, two, and three K\"ahler moduli. Also, we consider the renormalizable superpotential of $\varphi$ in general. We study the spectral index and tensor-to-scalar ratio in details, and find the viable parameter spaces which are consistent with the Planck and BICEP/Keck experimental data on the cosmic microwave background (CMB). The spectral index is $n_s\simeq 1-2/N \sim 0.965$ for all models, and the tensor-to-scalar ratio $r$ is $r\simeq12/N^2$, $ 83/N^4$ and $ 4/N^2$ for the one, two and three moduli models, respectively. The particular $r$ for two moduli model comes from the contributions of the non-negligible higher order term in potential. In the three moduli model, the scalar potential is similar to the global supersymmetry, but the K\"ahler potential is different. The E-model with $\alpha=1$ and T-model with $\alpha=1/3$ can be realized in the one modulus model and the three moduli model, respectively. Interestingly, the models with quadratic and quartic potentials still satisfy the current tight bound on $r$ after embedding into no-scale supergravity., Comment: 40 pages, 32 figures, 2 tables; v2: pubulished version, references added

All the possible three-family ${\cal N}=1$ supersymmetric Pati-Salam models constructed with intersecting D6-branes from Type IIA orientifolds on $T^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ are recently presented in arXiv: 2112.09632. Taking models with largest wrapping number $5$ and approximate gauge coupling unification at GUT scale as examples, we show string scale gauge coupling unification can be realized through two-loop renormalization group equation running by introducing seven pairs of vector-like particles from ${\cal N}=2$ sector. The number of these introduced vector-like particles are fully determined by the brane intersection numbers while there are two D6-brane parallel to each other along one two-torus. We expect this will solve the gauge coupling unification problem in the generic intersecting brane worlds by introducing vector-like particles that naturally included in the ${\cal N}=2$ sector., Comment: 3 figures

The formation of primordial black hole (PBH) dark matter and the generation of scalar induced secondary gravitational waves (SIGWs) have been studied in the generic no-scale supergravity inflationary models. By adding an exponential term to the K\"ahler potential, the inflaton experiences a period of ultraslow-roll and the amplitude of primordial power spectrum at small scales is enhanced to $\mathcal{O}(10^{-2})$. The enhanced power spectra of primordial curvature perturbations can have both sharp and broad peaks. A wide mass range of PBHs can be produced in our model, and the frequencies of the accompanied SIGWs are ranged form nanohertz to kilohertz. We show four benchmark points where the generated PBH masses are around $\mathcal{O}(10^{-16}M_{\odot})$, $\mathcal{O}(10^{-12}M_{\odot})$, $\mathcal{O}(10^{-2}M_{\odot})$ and $\mathcal{O}(10^{2}M_{\odot})$. The PBHs with masses around $\mathcal{O}(10^{-16}M_{\odot})$ and $ \mathcal{O}(10^{-12}M_{\odot})$ can make up almost all the dark matter, and the accompanied SIGWs can be probed by the upcoming space-based gravitational wave observatory. Also, the SIGWs accompanied with the formation of stellar mass PBHs can be used to interpret the stochastic GW background in the nanohertz band, detected by the North American Nanohertz Observatory for gravitational waves, and can be tested by future interferometric gravitational wave observatory., Comment: 28 pages, 5 figures, references added, typos corrected, published version

On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are \emph{isolated simple blow up points}., Comment: 41 pages

We consider the polynomial inflation with the tensor-to-scalar ratio as large as possible which can be consistent with the Quantum Gravity (QG) corrections and Effective Field Theory (EFT). To get a minimal field excursion $\Delta\phi$ for enough e-folding number $N$, the inflaton field traverses an extremely flat part of the scalar potential, which results in the Lyth bound to be violated. We get a CMB signal consistent with Planck data by numerically computing the equation of motion for inflaton $\phi$ and using Mukhanov-Sasaki formalism for primordial spectrum. Inflation ends at Hubble slow-roll parameter $\epsilon_1^H=1$ or $\ddot{a}=0$. Interestingly, we find an excellent practical bound on the inflaton excursion in the format $a+b{\sqrt r}$, where $a$ is a tiny real number and $b$ is at the order 1. To be consistent with QG/EFT and suppress the high-dimensional operators, we show that the concrete condition on inflaton excursion is $\frac{\Delta \phi}{M_{\rm Pl}} < 0.2 \times \sqrt{10}\simeq 0.632$. For $n_s=0.9649$, $N_e=55$, and $\frac{\Delta \phi}{M_{\rm Pl}} < 0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT., Comment: 12 pages

For regular $SU(3)$ Toda systems defined on Riemann surface, we initiate the study of bubbling solutions if parameters $(\rho_1^k,\rho_2^k)$ are both tending to critical positions: $(\rho_1^k,\rho_2^k)\to (4\pi, 4\pi N)$ or $(4\pi N, 4\pi)$ ($N>0$ is an integer). We prove that there are at most three formations of bubbling profiles, and for each formation we identify leading terms of $\rho_1^k-4\pi$ and $\rho_2^k-4\pi N$, locations of blowup points and comparison of bubbling heights with sharp precision. The results of this article will be used as substantial tools for a number of degree counting theorems, critical point at infinity theory in the future., Comment: 40 pages

We study the electronic structure change of yttrium trihydride $\mathrm{YH_3}$ by applying a hydrostatic pressure. At zero pressure, $\mathrm{YH_3}$ has the structure with energy favored symmetry group $P\bar{3}c1$ (165). From first principle calculation, we argue that the band crossings are caused by overlapping of an electron- and hole-like bands. Besides the space inversion symmetry ($\mathcal{I}$) and the time reversal symmetry, the band structure also exhibits an approximate particle-hole symmetry. Thus, $\mathrm{YH_3}$ can be viewed as a pseudo nodal surface semimetal belongs to class BDI of the ten-fold AZ+ $\mathcal{I}$ classifications of gapless topological matter. As pressure increases, the approximate particle-hole symmetry is gradually broken and the pseudo nodal surface turns into a nodal ring belonging to the class AI with fewer non-spatial symmetries. Also, the nodal ring is shrinking in the process. At about $31$ GPa, which is higher than the reported structure phase transition pressure $21$ GPa, the nodal ring shrinks to a nodal point. When above $31$ GPa, all band crossings are gapped out and $\mathrm{YH_3}$ becomes a trivial insulator eventually., Comment: 14 pages, 2 figures

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with bubbling sources. If the strength of the bubbling sources at blowup points are not multiple of $4\pi$ we prove that bubbling solutions are unique under non-degeneracy assumptions. This work extends a previous work of Bartolucci, et, al \cite{bart-4}., Comment: 36 pages

We discuss supergravity inflation in braneworld cosmology for the class of potentials $V(\phi)=\alpha \phi^n\rm{exp}(-\beta^m \phi^m)$ with $m=1,~2$. These minimal SUGRA models evade the $\eta$ problem due to a broken shift symmetry and can easily accommodate the observational constraints. In the high energy regime $V/\lambda\gg 1$, the numerical predictions and approximate analytic formulas are given for the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$. The models with smaller $n$ are preferred while the models with larger $n$ are out of the $2\sigma$ region. Remarkably, the $\rho^2/\lambda$ correction to the energy density in Friedmann equation results in sub-Planckian inflaton excursions $\Delta\phi <1$., Comment: 11 pages, 4 figures

We present a class of non-supersymmetric models in which so-called critical Higgs inflation ($\xi<100$) naturally can be realized without using specific values for Higgs and top quark masses. In these scenarios, the Standard Model (SM) vacuum stability problem, gauge coupling unification, neutrino mass generation and Higgs inflation mechanism are linked to each other. We adopt in our models Type I seesaw mechanism for neutrino masses. An appropriate choice of the Type I Seesaw scale allows us to have an arbitrarily small but positive value of SM Higgs quartic coupling around the inflation scale. We present a few benchmark points where we show that the scalar spectral indices are around 0.9626 and 0.9685 for the number of e-folding $N=50$ and $N=60$ respectively. The tensor-to-scalar ratios are order of $10^{-3}$. The running of the scalar spectral index is negative and is order of $10^{-4}$., Comment: typos corrected

Giving up the solutions to the fine-tuning problems, we propose the non-supersymmetric flipped $SU(5)\times U(1)_X$ model based on the minimal particle content principle, which can be constructed from the four-dimensional $SO(10)$ models, five-dimensional orbifold $SO(10)$ models, and local F-theory $SO(10)$ models. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form complete $SU(5)\times U(1)_X$ representation. Proton lifetime is around $5\times 10^{35}$ years, neutrino masses and mixing can be explained via seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and vacuum stability problem can be solved as well. In particular, we propose that inflaton and dark matter particle can be unified to a real scalar field with $Z_2$ symmetry, which is not an axion and does not have the non-minimal coupling to gravity. Such kind of scenarios can be applied to the generic scalar dark matter models. Also, we find that the vector-like particle corrections to the $B_s^0$ masses can be about 6.6%, while their corrections to the $K^0$ and $B_d^0$ masses are negligible., Comment: 5 pages, 4 figures;V2: published version

To solve the fine-tuning problem in $\mu$-Term Hybrid Inflation, we will realize the supersymmetry scenario with the TeV-scale supersymmetric particles and intermediate-scale gravitino from anomaly mediation, which can be consistent with the WMAP and Planck experiments. Moreover, we for the first time propose the $\mu$-term hybrid inflation in no-scale supergravity. With four Scenarios for the $SU(3)_C\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ model, we show that the correct scalar spectral index $n_s$ can be obtained, while the tensor-to-scalar ratio $r$ is prediced to be tiny, about $10^{-10}-10^{-8}$. Also, the $SU(2)_R\times U(1)_{B-L}$ symmetry breaking scale is around $10^{14}$ GeV, and all the supersymmetric particles except gravitino are around TeV scale while gravitino mass is around $10^{9-10}$ GeV. Considering the complete potential terms linear in $S$, we for the first time show that the tadpole term, which is the key for such kind of inflationary models to be consistent with the observed scalar spectral index, vanishes after inflation. Thus, to obtain the $\mu$ term, we need to generate the supersymmetry breaking soft term $A^{S \Phi \Phi'}_{\kappa} \kappa S \Phi \Phi'$ due to $A^{S \Phi \Phi'}_{\kappa}=0 $ in no-scale supergravity, where $\Phi$ and $\Phi'$ are vector-like Higgs fields at high energy. We show that the proper $A^{S \Phi \Phi'}_{\kappa} \kappa S \Phi \Phi'$ term can be obtained in the M-theory inspired no-scale supergravity. We also point out that $A^{S \Phi \Phi'}_{\kappa}$ around 700 GeV can be generated via the renormalization group equation running from string scale., Comment: 15 pages, 4 figures; v2: references added

We systematically study the renormalizable three-term polynomial inflation in the supersymmetric and non-supersymmetric models. The supersymmetric inflaton potentials can be realized in supergravity theory, and only have two independent parameters. We show that the general renormalizable supergravity model is equivalent to one kind of our supersymmetric models. We find that the spectral index and tensor-to-scalar ratio can be consistent with the Planck and BICEP2 results, but the running of spectral index is always out of the $2\sigma$ range. If we do not consider the BICEP2 experiment, these inflationary models can be highly consistent with the Planck observations and saturate its upper bound on the tensor-to-scalar ratio ($r \le 0.11$). Thus, our models can be tested at the future Planck and QUBIC experiments., Comment: 38 pages, 40 figures

The product landscape method has been recently proposed to solve hierarchy problems such as the cosmological constant problem. We suggest that the parameter distribution on logarithmic scales should be used as a benchmark for hierarchy, and the preferred hierarchy scales can be obtained from the distribution peak. It is shown that generating hierarchy from purely product distribution is very inefficient. To achieve a reasonably acceptable efficiency, other effects such as accumulation of weak hierarchy in the effective theory should be incorporated., Comment: 8 pages; v2: general discussion on the log plot added; v3: typos, JHEP published version