Search

We present "Custodial Naturalness" as a new mechanism to explain the separation between the electroweak (EW) scale and the scale of potential ultraviolet completions of the Standard Model (SM). We assume classical scale invariance as well as an extension of the SM scalar sector custodial symmetry to $\mathrm{SO}(6)$. This requires a single new complex scalar field charged under a new $\mathrm{U}(1)_\mathrm{X}$ gauge symmetry which partially overlaps with $B-L$. Classical scale invariance and the high-scale scalar sector $\mathrm{SO}(6)$ custodial symmetry are radiatively broken by quantum effects that generate a new intermediate scale by dimensional transmutation. The little hierarchy problem is solved because the Higgs boson arises as an elementary (i.e. non-composite) pseudo-Nambu-Goldstone boson (pNGB) of the spontaneously broken $\mathrm{SO}(6)$ custodial symmetry. The minimal setting has the same number of parameters as the SM and predicts new physics in the form of a heavy $Z'$ with fixed couplings to the SM and a mass of $m_{Z'}\approx4-100\,\mathrm{TeV}$, as well as a light but close-to invisible dilaton with a mass $m_{h_\Phi}\approx75\,\mathrm{GeV}$., Comment: 6 pages, 6 figures, 2 tables

Only four $\mathbb{T}^2/\mathbb{Z}_K$ orbifold building blocks are admissible in heterotic string compactifications. We investigate the flavor properties of all of these building blocks. In each case, we identify the traditional and modular flavor symmetries, and determine the corresponding representations and (fractional) modular weights of the available massless matter states. The resulting finite flavor symmetries include Abelian and non-Abelian traditional symmetries, discrete $R$ symmetries, as well as the double-covered finite modular groups $(S_3\times S_3)\rtimes\mathbb{Z}_4$, $T'$, $2D_3$ and $S_3\times T'$. Our findings provide restrictions for bottom-up model building with consistent ultraviolet embeddings., Comment: 27 pages, 9 tables

Finding out if neutrinos are Dirac or Majorana particles is known to be extremely difficult due to the smallness of neutrino mass and the fact that in the limit $m_\nu=0$ both Dirac and Majorana neutrinos become Weyl particles, i.e. are indistinguishable. There have been suggestions in the literature that in the case of processes with production of a neutrino-antineutrino pair (if neutrinos are Dirac particles) or two neutrinos (if they are of Majorana nature) quantum statistics may be of help. This is because for Majorana neutrinos quantum indistinguishability of identical particles requires the amplitude of the process to be antisymmetrized with respect to the interchange of the final-state neutrinos, whereas no such antisymmetrization must be done for Dirac neutrinos. It has been claimed that the resulting differences between the cross sections for Dirac and Majorana neutrinos persist even for arbitrarily small but not exactly vanishing neutrino mass. We demonstrate that, at least in the framework of the Standard Model, this is not the case. We also give a general proof that within the Standard Model quantum statistics does not help tell Dirac and Majorana neutrinos apart in the limit of negligibly small $m_\nu/E$., Comment: LaTeX, 20 pages, 2 figures. v2: Notes, references and acknowledgements added. Results and conclusions unchanged

We discuss an extra-dimensional braneworld with a 5th dimension compactified on a circle. As a characteristic feature, the warp factor is hyperbolic and separates the hidden and visible branes by a bulk horizon without a singularity. The two most widely separated scales of 4D physics - the 4D Planck mass and 4D cosmological constant - are determined by two physical scales in the extra dimension, namely: $(i)$ the proper size of the extra dimension, $R$, and, $(ii)$ the distance between the visible brane and the horizon, $R_0$. A realistic scale hierarchy between 4D Planck mass and 4D cosmological constant is obtained for $R/R_0\sim2.34$. The usual fine tuning is not reduced but promoted to a fine tuning of two separate brane energy densities that must approach the fundamental scale of the model with very high precision. Our scenario is based on an exact solution to the 5D Einstein equations with a strictly empty bulk and Friedmann-Lema\^itre-Robertson-Walker metric on the 4D branes. This requires positive 4D brane energy densities and describes an adiabatic runaway solution in agreement with the de Sitter swampland conjecture. The Kaluza-Klein (KK) graviton states are solutions of a modified P\"oschl-Teller potential which permits a discrete graviton spectrum of $\textit{exactly two}$ modes. In addition to the usual massless graviton, our scenario predicts an extra massive spin-2 graviton with a mass gap of $m_1=\sqrt{2}H_0\approx2\times10^{-33}\,\mathrm{eV}$ which might be detectable in the foreseeable future. A KK tower of gravitons, or a possible continuum of massive graviton states, is prohibited by unitarity with respect to the horizon. We discuss hurdles in turning this model into a realistic cosmology at all times, which points us towards 4D brane tensions that that must be raising towards the fundamental scale of the model, while the observable 4D expansion rate is decreasing., Comment: 33 pages, 2 figures

The flavor puzzle of the Standard Model quark sector is formulated in a non-perturbative way, using basis invariants that are independent of the choice of quark field basis. To achieve this, we first derive the algebraic ring of 10 CP even (primary) and 1 CP odd (secondary) basis invariants, using the Hilbert series and plethystic logarithm. An orthogonal basis in the ring of basis invariants is explicitly constructed, using hermitian projection operators derived via birdtrack diagrams. The thereby constructed invariants have well defined CP transformation behavior and give the most direct access to the flavor symmetric alignments of basis covariants. We firstly "measure" the orthogonal basis invariants from experimental data and characterize their location in the available parameter space. The experimentally observed orthogonal basis invariants take very close to maximal values and are highly correlated. Explaining the location of the invariants at close to maximal points, including the associated miniscule and highly correlated deviations, corresponds to solving the flavor puzzle in the invariant language. Once properly normalized, the orthogonal basis invariants are close to scale (RGE) invariant, hence, provide exquisite targets for fits of both, low- and high-scale (bottom-up and top-down) flavor models. Our result provides an entirely new angle on the flavor puzzle, and opens up ample opportunities for its ultimate exploration., Comment: 23+13 pages, 7 figures, 1 table; v2: version accepted by JHEP

The framework of compactified heterotic string theory offers consistent ultraviolet (UV) completions of the Standard Model (SM) of particle physics. In this approach, the existence of flavor symmetries beyond the SM is imperative and the flavor symmetries can be derived from the top down. Such a derivation uncovers a unified origin of traditional discrete flavor symmetries, discrete modular flavor symmetries, discrete R symmetries of supersymmetry, as well as charge-parity (CP) symmetry - altogether dubbed the eclectic flavor symmetry. I will illustrate how the eclectic flavor symmetry is unambiguously computed from the top-down construction, discuss the different arising sources of spontaneous flavor symmetry breaking, and expose possible lessons for bottom-up flavor model building. Finally, I will focus on one explicit example model that provides a successful fit to all available experimental data while giving rise to concrete predictions for so-far undetermined parameters., Comment: 14 pages+references, 5 figures, 6 tables; Contribution to the Proceedings of the 8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022), 7-11 November 2022, Baden-Baden, Germany; extended version of: arXiv:2204.06581

We investigate charge-parity (CP) and non-CP outer automorphism of groups and the transformation behavior of group representations under them. We identify situations where composite and elementary states that transform in exactly the same representation of the group, transform differently under outer automorphisms. This can be instrumental in discriminating composite from elementary states solely by their quantum numbers with respect to the outer automorphism, i.e. without the need for explicit short distance scattering experiments. We discuss under what conditions such a distinction is unequivocally possible. We cleanly separate the case of symmetry constrained (representation) spaces from the case of multiple copies of identical representations in flavor space, and identify conditions under which non-trivial transformation in flavor space can be enforced for composite states. Next to composite product states, we also discuss composite states in non-product representations. Comprehensive examples are given based on the finite groups $\Sigma(72)$ and $D_8$. The discussion also applies to $\mathrm{SU}(N)$ and we scrutinize recent claims in the literature that $\mathrm{SU}(2N)$ outer automorphism with antisymmetric matrices correspond to distinct outer automorphisms. We show that outer automorphism transformations with antisymmetric matrices are related by an inner automorphism to the standard $\mathbb{Z}_2$ outer automorphism of $\mathrm{SU}(N)$. As a direct implication, no non-trivial transformation behavior can arise for composite product states under the outer automorphism of $\mathrm{SU}(N)$., Comment: 29+10 pages, 1 table. Incl. rebuttal of claims of existence of non-standard outer automorphism transformation of $\mathrm{SU}(2N)$, as recently proposed by 1903.06662, 2111.04728, 2212.02413

We consider a mechanism which allows to decrease attenuation of high energy gamma ray flux from gamma ray burst GRB 221009A. The mechanism is based on the existence of a heavy $m_N\sim0.1\,\mathrm{MeV}$ mostly sterile neutrino $N$ which mixes with active neutrinos. $N$'s are produced in GRB in $\pi$ and $K$ decays via mixing with $\nu_\mu$. They undergo the radiative decay $N\rightarrow \nu \gamma$ on the way to the Earth. The usual exponential attenuation of gamma rays is lifted to an attenuation inverse in the optical depth. Various restrictions on this scenario are discussed. We find that the high energy $\gamma$ events at $18\,\mathrm{TeV}$ and potentially $251\,\mathrm{TeV}$ can be explained if (i) the GRB active neutrino fluence is close to the observed limit, (ii) the branching ratio of $N\rightarrow \nu \gamma$ is at least of the order 10%., Comment: 6 pages, 1 figure; v2: typos fixed, references added

Eclectic flavor groups arising from string compactifications combine the power of modular and traditional flavor symmetries to address the flavor puzzle. This top-down scheme determines the representations and modular weights of all matter fields, imposing strict constraints on the structure of the effective potential, which result in controlled corrections. We study the lepton and quark flavor phenomenology of an explicit, potentially realistic example model based on a T6/Z3xZ3 orbifold compactification of the heterotic string that gives rise to an Omega(2) eclectic flavor symmetry. We find that the interplay of flavon alignment and the localization of the modulus in the vicinity of a symmetry-enhanced point leads to naturally protected fermion mass hierarchies, favoring normal-ordered neutrino masses arising from a see-saw mechanism. We show that our model can reproduce all observables in the lepton sector with a small number of parameters and deliver predictions for so far undetermined neutrino observables. Furthermore, we extend the fit to quarks and find that Kaehler corrections are instrumental in obtaining a successful simultaneous fit to the quark and lepton sectors., Comment: 28 pages + appendices, 7 figures, 7 tables; v2: references added, discussion extended, typos in app. C corrected, results unchanged, matches JHEP version

We show that vector-like quarks in the fundamental or higher-dimensional representations of QCD can generate the electro-weak scale in a phenomenologically viable way by chiral symmetry breaking condensates. The thereby generated scales are determined by numerically solving the Dyson-Schwinger equation and these scales are sizable, because they grow with the hard vector-like mass. Communicating such a scale to the Standard Model via a conformally invariant scalar sector can dynamically generate the electro-weak scale without a naturalness problem, because all non-dynamical mass scales are protected by chiral symmetry. We present a minimal setup which requires only a new neutral scalar with mass not too far above the electro-weak scale, as well as vector-like quarks at the (multi-)TeV scale. Both are consistent with current bounds and are attractive for future experimental searches at the LHC and future colliders. Depending on the hypercharge of the vector-like quarks, hadrons made of them are color-neutral bound states which would be interesting Dark Matter candidates., Comment: 19 pages, 9 figures, 1 table

The framework of compactified heterotic string theory offers consistent ultraviolet (UV) completions of the Standard Model (SM) of particle physics. In this approach, the existence of flavor symmetries beyond the SM is imperative and the flavor symmetries can be derived from the top-down. Such a derivation uncovers a unified origin of traditional discrete flavor symmetries, discrete modular flavor symmetries, discrete R symmetries of supersymmetry, as well as CP symmetry - altogether called the eclectic flavor symmetry. I will show a specific example of such a top-down derived eclectic flavor symmetry, discuss the different sources of breaking of the eclectic flavor symmetry, as well as the possible lessons for bottom-up flavor model building., Comment: 6 pages+references, 3 tables, 1 figure, Proceedings of DISCRETE 2020-2021

A top-down approach to the flavor puzzle leads to eclectic flavor groups which include modular and traditional flavor symmetries. Based on examples of semirealistic T2/Z3 orbifold compactifications of heterotic string theory, we discuss the breakdown patterns of the eclectic flavor group via the interplay of vacuum expectation values (vevs) of moduli and flavon fields. This leads to an attractive flavor scheme with various possibilities to obtain "flavor hierarchies" through the alignment of these vevs. Despite the fact that the top-down approach gives strong restrictions for bottom-up flavor model building, it seems to be well suited to provide a realistic flavor pattern for quarks and leptons., Comment: 19 pages + citations, 3 figures, 6 tables; v2: statements in introduction clarified, matches version published in PRD

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond $\mathrm{SL}(2,\mathbb Z)$. In the simplest case with one Wilson line $Z$, K\"ahler modulus $T$ and complex structure modulus $U$, we are led to the Siegel modular group $\mathrm{Sp}(4,\mathbb Z)$. It includes $\mathrm{SL}(2,\mathbb Z)_T\times\mathrm{SL}(2,\mathbb Z)_U$ as well as $\mathbb Z_2$ mirror symmetry, which interchanges $T$ and $U$. Possible applications to flavor physics of the Standard Model require the study of orbifolds of $\mathrm{Sp}(4,\mathbb Z)$ to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of $\mathrm{Sp}(4,\mathbb Z)$. Some cases correspond to symmetric orbifolds that extend previously discussed cases of $\mathrm{SL}(2,\mathbb Z)$. Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of $\mathrm{Sp}(4,\mathbb Z)$ reveal exciting new aspects of modular symmetries with promising applications to flavor model building., Comment: 27 pages + appendices, 2 tables

The transfer function of the baryon power spectrum from redshift $z\approx 1100$ to today has recently been, for the first time, determined from data by Pardo and Spergel. We observe a remarkable coincidence between this function and the transport function of a cold ideal Fermi gas at different redshifts. Guided by this, we unveil an infinite set of critical temperatures of the relativistic ideal Fermi gas which depend on a very finely quantized long-distance cutoff. The sound horizon scale of Baryon Acoustic Oscillations (BAO) seems to set such a cutoff, which dials a critical temperature that is subsequently reached during redshift. At the critical point the Fermi gas becomes scale invariant and may condense to subsequently undergo gravitational collapse, seeding small scale structure. We mention some profound implications including the apparent quantization of Fermi momentum conjugate to the cutoff and the corresponding "gapping" of temperature., Comment: 10 pages, 8 figures; v3: Added introduction and references, matches version published in PLB

It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute a transfer matrix which realizes the simultaneous block diagonalization of unitary matrices whose decomposition in irreducible blocks (common invariant subspaces) is known from elsewhere. Our main motivation lies in particle physics, where the resulting transfer matrix must be known explicitly in order to unequivocally determine the action of outer automorphisms such as parity, charge conjugation, or time reversal on the particle spectrum., Comment: 6 pages, accepted by J.Phys.A

In the quest for unification of the Standard Model with gravity, classical scale invariance can be utilized to dynamically generate the Planck mass $M_\mathrm{Pl}$. Then, the relation of Planck scale physics to the scale of electroweak symmetry breaking $\mu_H$ requires further explanation. In this paper, we propose a model that uses the spontaneous breaking of scale invariance in the scalar sector as a unified origin for dynamical generation of both scales. Using the Gildener-Weinberg approximation, only one scalar acquires a vacuum expectation value of $v_S \sim (10^{16-17})\,\mathrm{GeV}$, thus radiatively generating $M_\mathrm{Pl} \approx \beta_S^{1/2} v_S$ and $\mu_H$ via the neutrino option with right handed neutrino masses $m_N = y_M v_S \sim 10^7 \,\mathrm{GeV}$. Consequently, active SM neutrinos are given a mass with the inclusion of a type-I seesaw mechanism. Furthermore, we adopt an unbroken $Z_2$ symmetry and a $Z_2$-odd set of right-handed Majorana neutrinos $\chi$ that do not take part in the neutrino option and are able to produce the correct dark matter relic abundance (dominantly) via inflaton decay. The model also describes cosmic inflation and the inflationary CMB observables are predicted to interpolate between those of $R^2$ and linear chaotic inflationary model and are thus well within the strongest experimental constraints., Comment: 21+10 pages, 11 figures, 1 table, v2: matches published version

We derive necessary and sufficient conditions for all global symmetries of the most general two Higgs doublet model (2HDM) scalar potential entirely in terms of reparametrization independent, i.e. basis invariant, objects. This culminates in what we call a "Symmetry Map" of the parameter space of the model and the fundamental insight that there are, in general, two algebraically distinct ways of how symmetries manifest themselves on basis invariant objects: either, basis invariant objects can be non-trivially related, or, basis covariant objects can vanish. These two options have different consequences on the resulting structure of the ring of basis invariants and on the number of remaining physical parameters. Alongside, we derive for the first time necessary and sufficient conditions for CP conservation in the 2HDM entirely in terms of CP-even quantities. This study lays the methodological foundation for analogous investigations of global symmetries in all other models that have unphysical freedom of reparametrization, most notably the Standard Model flavor sector., Comment: 27 pages, 2 tables, 1 figure

The XENON1T collaboration recently reported an excess in electron recoil events in the energy range between $1-7\,\mathrm{keV}$. This excess could be understood to originate from the known solar neutrino flux, if neutrinos couple to a light vector-mediator with strength $g_{\nu N}$ that kinetically mixes with the photon with strength $\chi$, and $g_{\nu N}\chi\sim10^{-13}$. Here, we show that such coupling values can naturally arise in a renormalizable model of long-range vector-mediated neutrino self-interactions. The model could be discriminated from other explanations of the XENON1T excess by the characteristic $1/T^2$ energy dependence of the neutrino-electron scattering cross section. Other signatures include invisible Higgs and $Z$ decays and lepto-philic charged Higgses at a few $100\,\mathrm{GeV}$. ALPS II will probe part of the viable parameter space., Comment: 7 pages, 3 figures; v2: references and a note added; v3: typos fixed, PRL version

We present a simple extension of the Standard Model that leads to renormalizable long-range vector-mediated neutrino self-interactions. This model can resolve the Hubble tension by delaying the onset of neutrino free-streaming during recombination, without conflicting with other measurements. The extended gauge, scalar and neutrino sectors lead to observable signatures, including invisible Higgs and $Z$ decays, thereby relating the Hubble tension to precision measurements at the LHC and future colliders. The model has a new neutrinophilic gauge boson with $m_{Z'}\sim\mathcal{O}(10~\mathrm{eV})$ and charged Higgses at a few $100~\mathrm{GeV}$. It requires hidden neutrinos with active-hidden mixing angles larger than $5\times10^{-4}$ and masses in the range $1\div300\mathrm{eV}$, which could also play a role for short baseline neutrino oscillation anomalies., Comment: 9 pages, 2 figures, 1 table; v2: matches published version

We describe a new, generally applicable strategy for the systematic construction of basis invariants (BIs). Our method allows one to count the number of mutually independent BIs and gives controlled access to the interrelations (syzygies) between mutually dependent BIs. Due to the novel use of orthogonal hermitian projection operators, we obtain the shortest possible invariants and their interrelations. The substructure of non-linear BIs is fully resolved in terms of linear, basis-covariant objects. The substructure distinguishes real (CP-even) and purely imaginary (CP-odd) BIs in a simple manner. As an illustrative example, we construct the full ring of BIs of the scalar potential of the general Two-Higgs-Doublet model., Comment: 8 pages, to be published in JPCS

Asymmetric tri-bi-maximal mixing is a recently proposed, grand unified theory (GUT) based, flavor mixing scheme. In it, the charged lepton mixing is fixed by the GUT connection to down-type quarks and a $\mathcal{T}_{13}$ flavor symmetry, while neutrino mixing is assumed to be tri-bi-maximal (TBM) with one additional free phase. Here we show that this additional free phase can be fixed by the residual flavor and CP symmetries of the effective neutrino mass matrix. We discuss how those residual symmetries can be unified with $\mathcal{T}_{13}$ and identify the smallest possible unified flavor symmetries, namely $(\mathbb{Z}_{13}\times\mathbb{Z}_{13})\rtimes \mathrm{D}_{12}$ and $(\mathbb{Z}_{13}\times\mathbb{Z}_{13})\rtimes \mathrm{S}_4$. Sharp predictions are obtained for lepton mixing angles, CP violating phases and neutrinoless double beta decay., Comment: 16 pages, 7 figures, 2 tables

In this paper we present an in-depth analysis of a recently proposed Standard Model extension with a complete fourth generation of quarks and leptons, which are vector-like with respect to the Standard Model gauge group and charged under a new spontaneously broken vector-like $\mathrm{U(1)^\prime}$ gauge symmetry. The model is designed to explain the known muon anomalies, i.e. the observed deviations from Standard Model predictions in the anomalous magnetic moment of the muon, $\Delta a_\mu$, and in $b \rightarrow s \ell^+ \ell^-$ processes. We perform a global $\chi^2$ analysis of the data with $65$ model parameters and including $98$ observables. We find many points with $\chi^2$ per degree of freedom $\leq 1$. The vector-like leptons and the new heavy $Z^\prime$ are typically much lighter than a TeV and would, thus, be eminently visible at the HL-LHC., Comment: 80 pages, 9 figures and 21 tables

Modular transformations of string theory (including the well-known stringy dualities) play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They are at the origin of CP transformations and provide a unification of CP with traditional flavor symmetries. Here, we present a novel, fully systematic method to reliably compute the unified flavor symmetry of the low-energy effective theory, including enhancements from the modular transformations of string theory. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of "Local Flavor Unification" where different sectors of the theory can be subject to different flavor structures., Comment: 38 pages, 2 figures

We consider the Standard Model (SM) with the addition of a $\mathrm{U(1)^\prime}$ gauge symmetry and a complete fourth family of quarks and leptons which are vector-like with respect to the full $\mathrm{SU(3)_C}\times \mathrm{SU(2)_L} \times \mathrm{U(1)_Y}\times \mathrm{U(1)^\prime}$ gauge symmetry. The model provides a unified explanation of experimental anomalies in $(g - 2)_\mu$ as well as $b \rightarrow s \ell^+ \ell^-$ decays. We find good fits to the deviations from the SM, while at the same time fitting all other SM observables. The model includes a new $Z^\prime$ gauge boson, a $\mathrm{U(1)^\prime}$-breaking scalar, and vector-like leptons all with mass of order a few $100$ GeV. It is consistent with all currently released high energy experimental data, however, it appears imminently testable with well designed future searches. Also precision flavor experiments, especially more accurate direct determinations of CKM matrix elements, would allow to probe the best fit points., Comment: 40 pages, 2 figures and 14 tables; Figure 2 replaced, references added

Physical observables cannot depend on the basis one chooses to describe fields. Therefore, all physically relevant properties of a model are, in principle, expressible in terms of basis-invariant combinations of the parameters. However, in many cases it becomes prohibitively difficult to establish key physical features exclusively in terms of basis invariants. Here, we advocate an alternative route in such cases: the formulation of basis-invariant statements in terms of basis-covariant objects. We give several examples where the basis-covariant path is superior to the traditional approach in terms of basis invariants. In particular, this includes the formulation of necessary and sufficient basis-invariant conditions for various physically distinct forms of CP conservation in two- and three-Higgs-doublet models., Comment: 20 pages, no figures

Flavor symmetry plays a crucial role in the standard model of particle physics but its origin is still unknown. We develop a new method (based on outer automorphisms of the Narain space group) to determine flavor symmetries within compactified string theory. A picture emerges where traditional (discrete) flavor symmetries, CP-like symmetries and modular symmetries (like T-duality) of string theory combine to unified flavor symmetries. The groups depend on the geometry of compact space and the geographical location of fields in the extra dimensions. We observe a phenomenon of "local flavor groups" with potentially different flavor symmetries for the various sectors of quarks and leptons. This should allow interesting connections to existing bottom-up attempts in flavor model building., Comment: 18 pages, 3 figures; v2: minor changes, version accepted by PLB

A new systematic method for the explicit construction of (basis-)invariants is introduced and employed to construct the full ring of basis invariants of the Two-Higgs-Doublet-Model (2HDM) scalar sector. Co- and invariant quantities are obtained by the use of hermitian projection operators. These projection operators are constructed from Young tableaux via birdtrack diagrams and they are used in two steps. First, to extract basis-covariant quantities, and second, to combine the covariants in order to obtain the actual basis invariants. The Hilbert series and Plethystic logarithm are used to find the number and structure of the complete set of generating invariants as well as their interrelations (syzygies). Having full control over the complete ring of (CP-even and CP-odd) basis invariants, we give a new and simple proof of the necessary and sufficient conditions for explicit CP conservation in the 2HDM, confirming earlier results by Gunion and Haber. The method generalizes to other models, with the only foreseeable limitation being computing power., Comment: 19+11 pages; Young tableaux, birdtracks, syzygies, color; v2: version published in JHEP

Three-Higgs-doublet models (3HDM) allow for a novel, physically distinct form of CP invariance: CP symmetry of order 4 (CP4). Due to the large basis change freedom in 3HDM, it is imperative to recognize the presence of a possibly hidden CP4 in a basis-invariant way. In the present work, we solve this problem and establish basis-invariant necessary and sufficient conditions for a 3HDM to possess a CP4 symmetry. We also derive a basis-invariant criterion to decide whether or not a CP4 symmetric 3HDM possesses any additional CP symmetry, as well as a criterion to decide whether or not CP4 is spontaneously broken., Comment: 32 pages

We identify a natural way to embed $\mathcal{CP}$ symmetry and its violation in string theory. The $\mathcal{CP}$ symmetry of the low energy effective theory is broken by the presence of heavy string modes. $\mathcal{CP}$ violation is the result of an interplay of $\mathcal{CP}$ and flavor symmetry. $\mathcal{CP}$ violating decays of the heavy modes could originate a cosmological matter-antimatter asymmetry., Comment: 7 pages, 4 figures

The Standard Model (SM) is amended by one generation of quarks and leptons which are vector-like (VL) under the SM gauge group but chiral with respect to a new $\mathrm{U}(1)_{3-4}$ gauge symmetry. We show that this model can simultaneously explain the deviation of the muon $g-2$ as well as the observed anomalies in $b\rightarrow s\mu^+\mu^-$ transitions without conflicting with the data on Higgs decays, lepton flavor violation, or $B_s-\bar{B}_s$ mixing. The model is string theory motivated and GUT compatible, i.e. UV complete, and fits the data predicting VL quarks, leptons and a massive $Z'$ at the $\mathrm{TeV}$ scale, as well as $\tau\to3\mu$ and $\tau\to\mu\gamma$ within reach of future experiments. The Higgs couplings to SM generations are automatically aligned in flavor space., Comment: 15 pages + Appendix, 4 figures, 4 tables; v2: added references

Predictions of inflationary schemes can be influenced by the presence of extra dimensions. This could be of particular relevance for the spectrum of gravitational waves in models where the extra dimensions provide a brane-world solution to the hierarchy problem. Apart from models of large as well as exponentially warped extra dimensions, we analyze the size of tensor modes in the Linear Dilaton scheme recently revived in the discussion of the "clockwork mechanism". The results are model dependent, significantly enhanced tensor modes on one side and a suppression on the other. In some cases we are led to a scheme of "remote inflation", where the expansion is driven by energies at a hidden brane. In all cases where tensor modes are enhanced, the requirement of perturbativity of gravity leads to a stringent upper limit on the allowed Hubble rate during inflation., Comment: 29 pages, 7 figures; v2: added discussion on the emergence of curvature singularities and removed discussion on the NKKK case with horizon in the bulk, conclusions unaltered, matches the published version

It is explained that the Standard Model combined charge conjugation and parity transformation (CP) is a simultaneous complex conjugation outer automorphism transformation of gauge and space-time symmetries. Simple examples are given for the general concept of outer automorphisms ("symmetries of symmetries"), as well as for their possible actions on physical theories. It is highlighted that complex conjugation outer automorphisms do not, in general, exist for all symmetries. Examples are given for cases in which the physical CP transformation is violated as a consequence of requiring another symmetry. A toy model is illustrated in which CP is spontaneously violated in the broken phase of a continuous gauge symmetry, while an unbroken outer automorphism protects the topological vacuum angle at $\theta=0$., Comment: Talk given at DISCRETE 2016: Fifth Symposium on Prospects in the Physics of Discrete Symmetries, Warsaw, 28 Nov - 3 Dec, 2016, 10 pages

A $\mathcal{CP}$ conserving $\mathrm{SU}(3)$ gauge theory is spontaneously broken to $\mathrm{T}_7$ by the vacuum expectation value (VEV) of a $\boldsymbol{15}$-plet. Even though the $\mathrm{SU}(3)$-$\,\mathcal{CP}$ transformation is not broken by the VEV, the theory exhibits physical $\mathcal{CP}$ violation in the broken phase. This is because the $\mathrm{SU}(3)$-$\,\mathcal{CP}$ transformation corresponds to the unique order-two outer automorphism of $\mathrm{T}_7$, which is not a physical $\mathcal{CP}$ transformation for the $\mathrm{T}_7$ states, and there is no other possible $\mathcal{CP}$ transformation. We explicitly demonstrate that $\mathcal{CP}$ is violated by calculating a $\mathcal{CP}$ odd decay asymmetry in the broken phase. This scenario provides us with a natural protection for topological vacuum terms, ensuring that $\theta\,G_{\mu\nu}\widetilde{G}^{\mu\nu}$ is absent even though $\mathcal{CP}$ is violated for the physical states of the model., Comment: 19 pages, 2 figures, 2 tables; v2: considering a different contribution to the decay asymmetry, conclusion unchanged

The behavior of a decoupled ideal Fermi gas in a homogeneously expanding three-dimensional volume is investigated, starting from an equilibrium spectrum. In case the gas is massless and/or completely degenerate, the spectrum of the gas can be described by an effective temperature and/or an effective chemical potential, both of which scale down with the volume expansion. In contrast, the spectrum of a decoupled massive and non-degenerate gas can only be described by an effective temperature if there are strong enough self-interactions such as to maintain an equilibrium distribution. Assuming perpetual equilibration, we study a decoupled gas which is relativistic at decoupling and then is red-shifted until it becomes non-relativistic. We find expressions for the effective temperature and effective chemical potential which allow us to calculate the final spectrum for arbitrary initial conditions. This calculation is enabled by a new expansion of the Fermi-Dirac integral, which is for our purpose superior to the well-known Sommerfeld expansion. We also compute the behavior of the phase space density under expansion and compare it to the case of real temperature and real chemical potential. Using our results for the degenerate case, we also obtain the mean relic velocity of the recently proposed non-thermal cosmic neutrino background., Comment: 19 pages, 16 figures; v2:references added, matches published version

Outer automorphisms of symmetries ("symmetries of symmetries") in relativistic quantum field theories are studied, including charge conjugation (C), space-reflection (P) , and time-reversal (T) transformations. The group theory of outer automorphisms is pedagogically introduced and it is shown that CP transformations are special outer automorphisms of the global, local, and space-time symmetries of a theory. It is shown that certain discrete groups allow for a group theoretical prediction of parameter independent CP violating complex phases with fixed geometrical values. The remainder of this thesis pioneers the study of outer automorphisms which are not related to C, P, or T. It is shown how outer automorphisms, in general, relate symmetry invariants and, in theories with spontaneous symmetry breaking, imply relations between different vacuum expectation values. Thereby, outer automorphisms can give rise to emergent symmetries. An example model with a discrete symmetry and three copies of the Standard Model Higgs field is discussed, in which the rich outer automorphism structure completely fixes the Higgs VEVs in their field space direction, including CP violating relative phases. This underlies the phenomenon of spontaneous geometrical CP violation. The possible relevance of outer automorphisms for a wide field of future studies is highlighted., Comment: PhD thesis, 130 pages, http://mediatum.ub.tum.de?id=1307248

We point out that, for Dirac neutrinos, in addition to the standard thermal cosmic neutrino background (C$\nu$B) there could also exist a non-thermal neutrino background with comparable number density. As the right-handed components are essentially decoupled from the thermal bath of standard model particles, relic neutrinos with a non-thermal distribution may exist until today. The relic density of the non-thermal (nt) background can be constrained by the usual observational bounds on the effective number of massless degrees of freedom $N_\mathrm{eff}$, and can be as large as $n_{\nu_{\mathrm{nt}}}\lesssim 0.5\,n_\gamma$. In particular, $N_\mathrm{eff}$ can be larger than 3.046 in the absence of any exotic states. Non-thermal relic neutrinos constitute an irreducible contribution to the detection of the C$\nu$B, and, hence, may be discovered by future experiments such as PTOLEMY. We also present a scenario of chaotic inflation in which a non-thermal background can naturally be generated by inflationary preheating. The non-thermal relic neutrinos, thus, may constitute a novel window into the very early universe., Comment: 6 pages, 2 figures

We show that there is a class of finite groups, the so-called perfect groups, which cannot exhibit anomalies. This implies that all non-Abelian finite simple groups are anomaly-free. On the other hand, non-perfect groups generically suffer from anomalies. We present two different ways that allow one to understand these statements., Comment: 11 pages

We investigate transformations which are not symmetries of a theory but nevertheless leave invariant the set of all symmetry elements and representations. Generalizing from the example of a three Higgs doublet model with $\Delta(27)$ symmetry, we show that the possibility of such transformations signals physical degeneracies in the parameter space of a theory. We show that stationary points only appear in multiplets which are representations of the group of these so-called equivalence transformations. As a consequence, the stationary points are amongst the solutions of a set of homogeneous linear equations. This is relevant to the minimization of potentials in general and sheds new light on the origin of calculable phases and geometrical CP violation., Comment: 20+9 pages, 1 figure; v1: minor changes, added clarification, matches the published version

We discuss the origin of CP violation in settings with a discrete (flavor) symmetry $G$. We show that physical CP transformations always have to be class-inverting automorphisms of $G$. This allows us to categorize finite groups into three types: (i) Groups that do not exhibit such an automorphism and, therefore, in generic settings, explicitly violate CP. In settings based on such groups, CP violation can have pure group-theoretic origin and can be related to the complexity of some Clebsch-Gordan coefficients. (ii) Groups for which one can find a CP basis in which all the Clebsch-Gordan coefficients are real. For such groups, imposing CP invariance restricts the phases of coupling coefficients. (iii) Groups that do not admit real Clebsch-Gordan coefficients but possess a class-inverting automorphism that can be used to define a proper (generalized) CP transformation. For such groups, imposing CP invariance can lead to an additional symmetry that forbids certain couplings. We make use of the so-called twisted Frobenius-Schur indicator to distinguish between the three types of discrete groups. With $\Delta(27)$, $T^{\prime}$, and $\Sigma(72)$ we present one explicit example for each type of group, thereby illustrating the CP properties of models based on them. We also show that certain operations that have been dubbed generalized CP transformations in the recent literature do not lead to physical CP conservation., Comment: 45 pages, 3 figures

We discuss non-Abelian discrete R symmetries which might have some conceivable relevance for model building. The focus is on settings with N=1 supersymmetry, where the superspace coordinate transforms in a one-dimensional representation of the non-Abelian discrete symmetry group. We derive anomaly constraints for such symmetries and find that novel patterns of Green-Schwarz anomaly cancellation emerge. In addition we show that perfect groups, also in the non-R case, are always anomaly-free. An important property of models with non-Abelian discrete R symmetries is that superpartners come in different representations of the group. We present an example model, based on a semidirect product of a Z_3 and a Z_8^R symmetry, to discuss generic features of models which unify discrete R symmetries, entailing solutions to the mu and proton decay problems of the MSSM, with non-Abelian discrete flavor symmetries., Comment: 21 pages