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Atomic hydrogen energy levels calculated to high precision are required to assist experimental researchers working on spectroscopy in the pursuit of testing quantum electrodynamics (QED) and probing for physics beyond the Standard Model. There are two important parts to the problem of computing these levels: an accurate evaluation of contributions from QED and using an accurate value for the proton charge radius as an input. Recent progress on QED corrections to the fine structure, as well as increasing evidence that a proton charge radius in the range of 0.84 fm is favored over the previously adopted larger value in the 0.88 fm range, has advanced the field, yet several state-of-the-art measurements remain in contradiction with this smaller value. Motivated by on-going and future work in this area, we present here a simple parameterization for the energy levels of hydrogen at the level of hyperfine structure using the so-called relativistic Ritz approach. The fitting of a finite sample of QED-generated levels at low to intermediate principal quantum number, $n$, gives a generally applicable formula for \emph{all} values of $n$ for each distinct angular momentum channel, given in this work up to orbital angular momentum number $\ell=30$. We also provide a simple linear parameterization for the shift in hydrogen energy levels as a function of the proton radius, providing a useful cross check for extant and future measured energy intervals., Comment: 6 pages of main text, 3 figures. Accepted by J. Phys. B

A long-distance effective theory of hydrogen-like atoms, dubbed the relativistic Ritz approach was recently introduced and some its theoretical consequences were explored. In this article, the relativistic Ritz approach is used to fit extant measurements of atomic hydrogen and deuterium transitions using information-theoretic analyses. As a result, the fine-structure constant ($\alpha$), a fundamental parameter of the Standard Model, may be determined simultaneously with the ionization energies of hydrogen and deuterium, $E_I^{\text{(H)}}$ and $E_I^{\text{(D)}}$. The best hydrogen analysis yields $\alpha^{-1} = 137.035\,999\,185(25)$, in good agreement with the value obtained by other methods and without relying on a separately determined Rydberg constant. From the same analysis, I find that $E_I^{\text{(H)}} = 13.598\,434\,599\,684(25)\,\text{eV}$, an improvement of two orders of magnitude in precision compared to previous determinations and in agreement with the Standard Model prediction at 1.8 parts per trillion. The best deuterium analysis yields $E_I^{\text{(D)}} = 13.602\,134\,636\,543(31)\,\text{eV}$, agreeing with the Standard Model at 2.3 parts per trillion. This study demonstrates how the relativistic Ritz approach can be used for testing the Standard Model with the spectra of hydrogen-like atoms., Comment: v2:14 pages of main text, 4 figures, no appendices; v1: 5 pages of main text, 5 pages of appendices

The Rydberg formula along with the Ritz quantum defect ansatz has been a standard theoretical tool used in atomic physics since before the advent of quantum mechanics, yet this approach has remained limited by its non-relativistic foundation. Here I present a long-distance relativistic effective theory describing hydrogen-like systems with arbitrary mass ratios, thereby extending the canonical Ritz-like approach. Fitting the relativistic theory to the hydrogen energy levels predicted by bound-state QED indicates that it is superior to the canonical, nonrelativistic approach. An analytic analysis reveals nonlinear consistency relations within the bound-state QED level predictions that relate higher-order corrections to those at lower order, providing guideposts for future perturbative calculations as well as insights into the asymptotic behavior of Bethe logarithms. Applications of the approach include fitting to atomic spectroscopic data, allowing for the determination the fine-structure constant from large spectral data sets and also to check for internal consistency of the data independently from bound-state QED., Comment: v2: 11 pages of main text, 14 figures, 2 appendices

An effective theory of the excited states of positronium is derived and some of its consequences are explored. At large physical separation, the binding of the electron and positron is assumed to be described completely by QED, whereas all short-ranged phenomena, including those within and beyond QED, can be accounted for with energy-dependent quantum defects. This theory has at least two practical applications. First, it provides an accurate and economical, yet largely QED-independent, means to fit the positronium spectrum in order to predict and compare the outcome of experiments. Second, matching the spectrum in this effective theory to that predicted by QED reveals nontrivial relationships that exist \emph{within} bound-state QED; some higher order contributions to the spectrum may be obtained from lower order contributions. These relations are verified up to order $m\alpha^{6}$, and predictions are made for the order $m\alpha^{7}$ and $m\alpha^{8}$ corrections. This theory and its extensions to other hydrogenic systems may provide a useful complement to bound-state QED., Comment: 7 pages, 3 figures. Phys Rev A accepted

We apply the ideas of effective field theory to nonrelativistic quantum mechanics. Utilizing an artificial boundary of ignorance as a calculational tool, we develop the effective theory using boundary conditions to encode short-ranged effects that are deliberately not modeled; thus, the boundary conditions play a role similar to the effective action in field theory. Unitarity is temporarily violated in this method, but is preserved on average. As a demonstration of this approach, we consider the Coulomb interaction and find that this effective quantum mechanics can predict the bound state energies to very high accuracy with a small number of fitting parameters. It is also shown to be equivalent to the theory of quantum defects, but derived here using an effective framework. The method respects electromagnetic gauge invariance and also can describe decays due to short-ranged interactions, such as those found in positronium. Effective quantum mechanics appears applicable for systems that admit analytic long-range descriptions, but whose short-ranged effects are not reliably or efficiently modeled. Potential applications of this approach include atomic and condensed matter systems, but it may also provide a useful perspective for the study of blackholes.

Long-range effective methods are ubiquitous in physics and in quantum theory, in particular. Furthermore, the reliability of such methods is higher when the nature of short-ranged interactions need not be modeled explicitly. This may be necessary for two reasons: (1) there are interactions that occur over a short range that cannot be accurately modeled with a potential function and/or (2) the entire Hamiltonian loses its reliability when applied at short distances. This work is an investigation of the utility and consequences of omitting a finite region of space from quantum mechanical analysis, accomplished by imposition of an artificial boundary behind which obscured short-ranged physical effects may operate. With this method, a free function of integration that depends on momentum is interpreted as a function encoding information needed to match a long-distance wavefunction to an appropriate state function on the other side of the boundary. Omitting part of the space from analysis implies that the strict unitarity requirement of quantum mechanics must be relaxed, since particles can actually propagate beyond the boundary. Strict orthogonality of eigenmodes and hermiticity of the Hamiltonian must also be relaxed in this method; however, all of these canonical relations are obeyed when averaged over sufficiently long times. What is achieved, therefore, appears to be an effective long-wavelength theory, at least for stationary systems. As examples, the quantum defect theory of the one-dimensional Coulomb interaction is recovered, as well as a new perspective of the inverse-square potential and the free particle, as well as the Wigner time delay associated with contact interactions. Potential applications of this method may include three-dimensional atomic systems and two-dimensional systems, such as graphene., Comment: 13 pages, 7 figures, Phys. Rev. A accepted (12/02/19)

Robin (or mixed) boundary conditions in quantum mechanics have received considerable attention in the last two decades, in particular, for applications to nanoscale systems. However, their utility has remained obscure to the larger physics community; in fact, one may find extant claims in the literature about the supposed naturalness of the specific Dirichlet (or vanishing) boundary condition. Here we demonstrate that not only are Dirichlet boundary conditions unnatural, but that Robin boundary conditions have utility in the long-wavelength approximation of short-ranged potentials. For illustration, we consider a non-relativistic particle in one dimension under the influence of the multi-step and Morse potential. We derive the scattering and bound states for these potentials and determine the parameters that describe those states in an effective system, namely one in which there is a free particle with a boundary. This method appears to be generically applicable and also generalizable to many other real systems., Comment: 8 pages, 3 figures. v3: submitted to Phys. Rev. A

Real physical systems are only understood, experimentally or theoretically, to a finite resolution so in their analysis there is generally an ignorance of possible short-range phenomena. It is also well-known that the boundary conditions of wavefunctions and fields can be used to model short-range interactions when those interactions are expected, a priori. Here, a real-space approach is described wherein an artificial boundary of ignorance is imposed to explicitly exclude from analysis the region of a system wherein short-distance effects may be obscure. The (artificial) boundary conditions encode those short-distance effects by parameterizing the possible UV completions of the wavefunction. Since measurable quantities, such as spectra and cross sections, must be independent of the position of the artificial boundary, the boundary conditions must evolve with the radius of the boundary in a particular way. As examples of this approach, an analysis is performed of the non-relativistic free particle, harmonic oscillator, and Coulomb potential, and some known results for point-like (contact) interactions are recovered, however from a novel perspective. Generically, observables differ from their canonical values and symmetries are anomalously broken compared to those of idealized models. Connections are made to well-studied physical systems, such as the binding of light nuclei and cold atomic systems. This method is arguably more physically transparent and mathematically easier to use than other techniques that require the regularization and renormalization of delta-function potentials, and may offer further generalizations of practical use., Comment: 7 pages, 2 figures, v2: abstract modified for clarity, minor typographic errors fixed, v3: text modified for clarity and Coulomb scattering results added. v4: matches published version

At present, the best model for the evolution of the cosmos requires that dark matter make up approximately $25\%$ of the energy content of the Universe. Most approaches to explain the microscopic nature of dark matter, to date, have assumed its composition to be of intrinsically weakly interacting particles; however, this need not be the case to have consistency with all extant observations. Given decades of inconclusive evidence to support any dark matter candidate, there is strong motivation to consider alternatives to the standard particle scenario. One such example is macro dark matter, a class of candidates (macros) that could interact strongly with the particles of the Standard Model, have large masses and physical sizes, and yet behave as dark matter. Macros that scatter completely inelastically could have altered the primordial production of the elements, and macro charge-dependent constraints have been obtained previously. Here we reconsider the phenomenology of inelastically interacting macros on the abundance of primordially produced $^4\text{He}$ and revise previous constraints by also taking into account improved measurements of the primordial $^4\text{He}$ abundance. The constraints derived here are limited in applicability to only leptophobic macros that have a surface potential $V \gtrsim 0.5\,$MeV. However, an important conclusion from our analysis is that even neutral macros would likely affect the abundance of the light elements. Therefore, constraints on that scenario are possible and are currently an open question., Comment: v2: 5 pages; substantial improvement over the first version now includes consideration of Debye screening of the primordial plasma during BBN. Accepted by Phys. Rev. D

The current standard model of cosmology, $\Lambda$CDM, requires dark matter to make up around $25\%$ of the total energy budget of the Universe. Yet, quite puzzlingly, there appears to be no candidate particle in the current Standard Model of particle physics. Assuming the validity of the cold dark matter (CDM) paradigm, dark matter has evaded detection thus far either because it is intrinsically a weakly interacting substance or because its interactions are suppressed by its high constituent mass and low number density. Most approaches to explain dark matter to date assume the former and therefore require beyond-the-Standard-Model particles that have yet to be observed directly or indirectly. Given the dearth of evidence for this class of candidates it is timely to consider the latter possibility, which allows for candidates that may or may not arise from the Standard Model. In this work we extend a recent study of this general class of so-called macro dark matter--candidates with characteristic masses of grams and geometric cross sections of cm$^2$. We consider new bounds that can be set using existing data from the resonant bar gravitational wave detectors NAUTILUS and EXPLORER., Comment: 5 pages, 1 figure. Revision matches published version

Dark matter is a vital component of the current best model of our universe, $\Lambda$CDM. There are leading candidates for what the dark matter could be (e.g. weakly-interacting massive particles, or axions), but no compelling observational or experimental evidence exists to support these particular candidates, nor any beyond-the-Standard-Model physics that might produce such candidates. This suggests that other dark matter candidates, including ones that might arise in the Standard Model, should receive increased attention. Here we consider a general class of dark matter candidates with characteristic masses and interaction cross-sections characterized in units of grams and cm$^2$, respectively -- we therefore dub these macroscopic objects as Macros. Such dark matter candidates could potentially be assembled out of Standard Model particles (quarks and leptons) in the early universe. A combination of Earth-based, astrophysical, and cosmological observations constrain a portion of the Macro parameter space. A large region of parameter space remains, most notably for nuclear-dense objects with masses in the range $55 - 10^{17}$ g and $2\times10^{20} - 4\times10^{24}$ g, although the lower mass window is closed for Macros that destabilize ordinary matter., Comment: 13 pages, 1 table, 4 figures. Submitted to MNRAS. v3: corrected small errors and a few points were made more clear, v4: included CMB bounds on dark matter-photon coupling from Wilkinson et al. (2014) and references added. Final revision matches published version

Extra dimensions are a common feature of beyond the Standard Model physics. In a braneworld scenario, local physics on the brane can depend strongly on the brane's location within the bulk. Generically, the relevant properties of the bulk manifold for the physics on/of the brane are neither local nor global, but depend on the structure of finite regions of the bulk, even for locally homogeneous and isotropic bulk geometries. In a recent work, various mechanisms (in a braneworld context) were considered to stabilize the location of a brane within bulk spaces of non-trivial topology. In this work we elaborate on and generalize that work by considering additional bulk and brane dimensionalities as well as different boundary conditions on the bulk scalar field that provides a Casimir force on the brane, providing further insight on this effect. In D=2+1 (D=5+1) we consider both local and global contributions to the effective potential of a 1-brane (4-brane) wrapped around both the 2-dimensional hyperbolic horn and Euclidean cone, which are used as toy models of an extra-dimensional manifold. We calculate the total energy due to brane tension and elastic energy (extrinsic curvature) as well as that due to the Casimir energy of a bulk scalar satisfying both Dirchlet and Neumann boundary conditions on the brane. In some cases stable minima of the potential are found that result from the competition of at least two of the contributions. Generically, any one of these effects may be sufficient when the bulk space has less symmetry than the manifolds considered here. We highlight the importance of the Casimir effect for the purpose of brane stabilization., Comment: 50 pages, 9 figures. Updated to match the PRD version

Extra-dimensional scenarios have become widespread among particle and gravitational theories of physics to address several outstanding problems, including cosmic acceleration, the weak hierarchy problem, and the quantization of gravity. In general, the topology and geometry of the full spacetime manifold will be non-trivial, even if our ordinary dimensions have the topology of their covering space. Most compact manifolds are inhomogeneous, even if they admit a homogeneous geometry, and it will be physically relevant where in the extra-dimensions one is located. In this letter, we explore the use of both local and global effects in a braneworld scenario to naturally provide position-dependent forces that determine and stabilize the location of a single brane. For illustrative purposes, we consider the 2-dimensional hyperbolic horn and the Euclidean cone as toy models of the extra-dimensional manifold, and add a brane wrapped around one of the two spatial dimensions. We calculate the total energy due to brane tension and bending (extrinsic curvature) as well as that due to the Casimir energy of a bulk scalar satisfying a Dirchlet boundary condition on the brane. From the competition of at least two of these effects there can exist a stable minimum of the effective potential for the brane location. However, on more generic spaces (on which more symmetries are broken) any one of these effects may be sufficient to stabilize the brane. We discuss this as an example of physics that is neither local nor global, but regional., Comment: 4 pages, 2 figures. PRL submitted

We construct the most general effective Lagrangian coupling gravity and electromagnetism up to mass dimension 6 by enumerating all possible non-minimal coupling terms respecting both diffeomorphism and gauge invariance. In all, there are only two unique terms after field re-definitions; one is known to arise from loop effects in QED while the other is a parity violating term which may be generated by weak interactions within the standard model of particle physics. We show that neither the cosmological propagation of light nor, contrary to earlier claims, solar system tests of General Relativity are useful probes of these terms. These non-minimal couplings of gravity and electromagnetism may remain a mystery for the foreseeable future., Comment: 9 pages. Minor corrections made. To appear in Phys. Rev. D