Nuclear-level Effective Theory of Muon-to-Electron Conversion: Formalism and Applications

New mu-to-e conversion searches aim to advance limits on charged lepton flavor violation (CLFV) by four orders of magnitude. By considering P and CP selection rules and the structure of possible charge and current densities, we show that rates are governed by six nuclear responses. To generate a microscopic formulation of these responses, we construct in non-relativistic effective theory (NRET) the CLFV nucleon-level interaction, then embed it in a nucleus. We discuss previous work, noting the lack of a systematic treatment of the various small parameters. Because the momentum transfer is comparable to the inverse nuclear size, a full multipole expansion of the response functions is necessary, a daunting task with Coulomb-distorted electron partial waves. We perform such an expansion to high precision by introducing a simplifying local electron momentum, treating the full set of 16 NRET operators. Previous work has been limited to the simplest charge/spin operators, ignored Coulomb distortion (or alternatively truncated the partial wave expansion) and the nucleon velocity operator, which is responsible for three of the response functions. This generates inconsistencies in the treatment of small parameters. We obtain a "master formula" for mu-to-e conversion that properly treats all such effects and those of the muon velocity. We compute muon-to-electron conversion rates for a series of experimental targets, deriving bounds on the coefficients of the CLFV operators. We discuss the nuclear physics: two types of coherence enhance certain CLFV operators and selection rules blind elastic mu-to-e conversion to others. We discuss the matching of the NRET onto higher level EFTs, and the relation to mu-to-e conversion to other CLFV tests. Finally we describe a publicly available script that can be used to compute mu-to-e conversion rates in nuclear targets.Comment: 50 pages, 10 figures; a few typos fixed in v

Sulfonated 1,3-bis­(4-pyrid­yl)propane

In the title compound, 4-[3-(3-sulfonato­pyridin-1-ium-4-yl)prop­yl]pyridin-1-ium-3-sulfonate, C13H14N2O6S2, the mol­ecule is zwitterionic, with the sulfonic acid proton transfered to the basic pyridine N atom. Also, the structure adopts a butterfly-like conformation with the sulfonate groups on opposite sides of the ‘wings’. The dihedral angle between the two pyridinium rings is 83.56 (7)°, and this results in the mol­ecule having a chiral conformation and packing. There is strong inter­molecular hydrogen bonding between the pyridinium H and sulfonate O atoms of adjoining mol­ecules. In addition, there are weaker inter­molecular C—H⋯O inter­actions

Researching the use of force: The background to the international project

This article provides the background to an international project on use of force by the police that was carried out in eight countries. Force is often considered to be the defining characteristic of policing and much research has been conducted on the determinants, prevalence and control of the use of force, particularly in the United States. However, little work has looked at police officers’ own views on the use of force, in particular the way in which they justify it. Using a hypothetical encounter developed for this project, researchers in each country conducted focus groups with police officers in which they were encouraged to talk about the use of force. The results show interesting similarities and differences across countries and demonstrate the value of using this kind of research focus and methodology

Latino Officers and Their Involvement in Police Shootings

With an emphasis to examine Latino officers who have been involved in police shootings, this study analyzed twenty-one years (1990–2010) of data from one of the largest law enforcement departments in the United States. The study compared Latino population trends in the United States, the State of California, a southern California County, and focused on the representativeness of Latinos in one southern California law enforcement department. The analysis further investigated police shootings by the race of the officer, narrowing the focus to determine whether an increasing representativeness of Latino officers had any effect on police shootings. Results revealed that while the percentage of White officers in the department decreased and Latino officers increased, so too did their involvement in police shootings. Most surprisingly, Latino officer-involved shootings outpaced their growth in the department by a factor of 3.3 and in the county by a factor of more than 4

Testing a Crime Control Model: Does Strategic and Directed Deployment of Police Officers Lead to Lower Crime?

The purpose of the paper was to investigate whether implementation of a crime control model (based, in part, on the concepts of COMPSTAT) in one southern California city was effective in reducing crime. Time series regression models were fitted to data collected from the Riverside County Sheriff’s Department, city of Perris, for the years 2000 through 2010. Additional data were collected from three other cities that served as controls. Results showed that the program was effective in reducing crime rates in Perris. The effect remained significant even after taking into account time trends and control cities. Analysis also found that while the program was more effective in lessening total and property crime rates, it was less so for violent crime rates. It was concluded that strategic and directed policing models (e.g., COMPSTAT, hot spot policing, etc.) may be more effective in crime reduction efforts than reactive policing methods

Nuclear-level effective theory of μ→e conversion: Formalism and applications

Over the next decade new μ→e conversion searches at Fermilab (Mu2e) and J-PARC (COMET, DeeMe) are expected to advance limits on charged lepton flavor violation (CLFV) by more than four orders of magnitude. By considering the consequence of P and CP on elastic μ→e conversion and the structure of possible charge and current densities, we show that rates are governed by six nuclear responses and a single scale, q/mN, where q≈mμ is the momentum transferred from the leptons to the nucleus. To relate this result to microscopic formulations of CLFV, we construct in nonrelativistic effective theory (NRET) the CLFV nucleon-level interaction, pointing out the relevance of the dimensionless scales y=(qb2)2>|v→N|>|v→μ|>|v→T|, where b is the nuclear size, v→N and v→μ are the nucleon and muon intrinsic velocities, and v→T is the target recoil velocity. We discuss previous work, noting the lack of a systematic treatment of the various small parameters. Because the parameter y is not small, a proper calculation of μ→e conversion requires a full multipole expansion of the nuclear response functions, an apparently daunting task with Coulomb-distorted electron partial waves. We demonstrate that the multipole expansion can be carried out to high precision by introducing a simplifying local momentum qeff for the electron. Previous work has been limited to simple charge or spin interactions, thereby treating the nucleus effectively as a point particle. We show that such formulations are not compatible with the general form of the μ→e conversion rate, failing to generate three of the six allowed nuclear response functions. The inclusion of the nucleon velocity v→N yields an NRET with 16 operators and a rate of the general form. Consequently, in the current discovery era for CLFV, it provides the most sensible starting point for experimental analysis, defining what can and cannot be determined about CLFV from the highly exclusive process of μ→e conversion. Finally, we expand the NRET operator basis to account for the effects of v→μ, associated with the muon's lower component, generating corrections to the CLFV coefficients of the point-nucleus response functions. Using advanced shell-model methods, we compute μ→e conversion rates for a series of experimental targets, deriving bounds on the coefficients of the CLFV operators. These calculations are the first to include a general basis of CLFV operators, full evaluation of the associated nuclear response functions, and an accurate treatment of electron and muon Coulomb effects. We discuss target selection as an experimental "knob"that can be turned to probe the microscopic origins of CLFV. We describe two types of coherence that enhance certain CLFV operators and selection rules that blind elastic μ→e conversion to others. We discuss the matching of the NRET onto higher level effective field theories, such as those constructed at the light quark level, noting opportunities to build on existing work in direct detection of dark matter. We discuss the relation of μ→e conversion to μ→e+γ and μ→3e, showing how MEG II and Mu3e results will complement those of Mu2e and COMET. Finally we describe a accompanying script - in Mathematica and Python versions - that can be used to compute μ→e conversion rates in various nuclear targets for the full set of NRET operators