The Search for the Electric Dipole Moment of the Electron (eEDM) in Mercury Halides Using the Relativistic Coupled Cluster Theory

Abstract

The electric dipole moment of the electron (eEDM) could arise due to the simultaneous violations of parity (P) and time reversal (T) symmetries. If observed, it would be an intrinsic property of the electron, like its mass and spin. If the CPT theorem holds, then T violation would imply CP violation. The Standard Model of particle physics (SM) predicts an extremely small value for the eEDM; less than 10−38 e cm. Extensions of SM, like several variants of supersymmetry, however, predict much larger ranges of eEDM. Therefore, upper bounds on eEDMs from experiments are useful in constraining theories beyond the SM. Since one of the necessary conditions for the matter-antimatter asymmetry is CP violation, and since CP violation is also necessary for the eEDM, the latter can throw light into the baryon asymmetry problem. Since an electron accelerates away in the presence of an external electric field, therefore making accurate measurements impossible, neutral systems like atoms or molecules, typically with one unpaired electron, are used to measure eEDM. Heavy polar diatomics are preferred over atoms, due to their experimental sensitivity being much larger. A shift in the energy of a diatomic in some state, due to the outer electron having an eEDM, is equal to the product of the negative of the eEDM and an effective electric field that the unpaired electron experiences, due to the other electrons and nuclei. This effective electric field, Eef f , cannot be measured, and has to be calculated using relativistic quantum many-body theory. A molecule with a large Eef f is preferred, for better sensitivity in experiment, among other factors. It is also important that the molecule itself has a reasonably large molecular electric dipole moment (PDM). In this thesis, mercury monohalides (HgF, HgCl, HgBr, and HgI) have been considered, and identified as promising candidates for future eEDM search experiments. A fully relativistic coupled cluster theory was employed, in the singles and doubles approximation (CCSD), to calculate the effective fields and the PDMs of these molecules. None of the occupied orbitals were frozen, in our calculations. Only the linear terms in the expansion of the CCSD wave function were included, in the expectation value expressions for the properties. The effective electric fields are about 100 GV/cm, for all of these molecules, which is one and IX a half times that in ThO, the molecule that provides the current best limit on eEDM. In our work, we had also identified that the mercury monohalides are experimentally very attractive, and among the HgX, HgBr may hold a lot of promise. The alkaline earth monofluorides (AEMs), a class of molecules that are useful to probe several aspects of fundamental physics, were also studied, to try understand the interplay between relativistic and correlation effects. The PDMs were calculated, at the double, triple, and the quadruple zeta levels of basis. The accuracy of our calculations were also established in this work, with the calculated PDMs (at the quadruple zeta level) agreeing very well with experiment. It was observed that the correlation effects increase as the molecule becomes progressively heavier. We also performed an in-depth analysis of the terms that make the PDM. In subsequent works, the effective electric fields and the PDMs of HgX were also analyzed. The expressions for both the properties, at the Dirac-Fock (DF) as well as the correlation levels, have been broken down into its constituent terms, to identify where the leading contributions come from, and thereby attempt to explain why some terms are larger than the others. The PDMs in HgX follow an unusual trend; the effect of correlations is to reduce the PDM. In fact, in HgI, the DF value is reduced by more than half, because of the correlation effects! It was observed from our analysis of the effective electric fields of HgX that at the DF level, the mixing between the s and p1/2 orbitals, of the heavier atom contribute the most, at the DF level. The mixing from the lighter atom is almost always negligible, even in the case of HgI! We then choose HgF as a representative candidate, and look at the various correlation terms, to get insights into what type of matrix elements contribute, and also to understand which terms cancel each other out, and to what extent. Finally, we compared our work with earlier ones (on HgF), in a detailed manner, and show how our approach is very accurate, as compared to earlier works. The thesis chapters are structured as follows: Chapter one gives an introduction to the eEDM. It briefly discusses why P and T need to be simultaneously violated for a non-zero eEDM. It also discusses how the eEDM can be measured, and illustrates that a combination of theory and experiment is necessary, to determine the eEDM. The chapter also discusses briefly about higher order electric and magnetic moments. Chapter two is a review of many-electron theory. It starts with the Born-Oppenheimer approximation, and proceeds to introduce the use of second quantization in many-body theory. The Hartree-Fock equations, their relativistic version, and the matrix formulation of the theory are discussed. The chapter then discusses electron correlation, and theories that include this effect, like configuration interaction and many-body perturbation theory. The third chapter is a detailed X discussion on the relativistic coupled cluster method. The chapter, towards the end, also covers details of how we tested the accuracy of our coupled cluster method in the SrF molecule. It then explains the interplay between the relativistic and correlation effects in the alkaline earth monofluoride systems. Chapter four and five are the core chapters of the thesis. In chapter four, mercury monohalides are identified as promising eEDM search candidates. We do this by calculating the effective electric fields of the molecules, and show that they are extremely large. We also discuss the experimental advantages that they offer, briefly. Chapter five elaborates on the previous one, where we perform an in-depth analysis of the effective fields. Chapter six discusses the PDMs of HgX, since the property also plays a role in determining the sensitivity of the experiment. Chapter seven mentions the future work, and includes basis set dependence, the finite field method and our preliminary results using this approach, as well as the mercury alkalis as possible candidates for eEDM search experiments.

The electric dipole moment of the electron (eEDM) could arise due to the simultaneous violations of parity (P) and time reversal (T) symmetries. If observed, it would be an intrinsic property of the electron, like its mass and spin. If the CPT theorem holds, then T violation would imply CP violation. The Standard Model of particle physics (SM) predicts an extremely small value for the eEDM; less than 10−38 e cm. Extensions of SM, like several variants of supersymmetry, however, predict much larger ranges of eEDM. Therefore, upper bounds on eEDMs from experiments are useful in constraining theories beyond the SM. Since one of the necessary conditions for the matter-antimatter asymmetry is CP violation, and since CP violation is also necessary for the eEDM, the latter can throw light into the baryon asymmetry problem. Since an electron accelerates away in the presence of an external electric field, therefore making accurate measurements impossible, neutral systems like atoms or molecules, typically with one unpaired electron, are used to measure eEDM. Heavy polar diatomics are preferred over atoms, due to their experimental sensitivity being much larger. A shift in the energy of a diatomic in some state, due to the outer electron having an eEDM, is equal to the product of the negative of the eEDM and an effective electric field that the unpaired electron experiences, due to the other electrons and nuclei. This effective electric field, Eef f , cannot be measured, and has to be calculated using relativistic quantum many-body theory. A molecule with a large Eef f is preferred, for better sensitivity in experiment, among other factors. It is also important that the molecule itself has a reasonably large molecular electric dipole moment (PDM). In this thesis, mercury monohalides (HgF, HgCl, HgBr, and HgI) have been considered, and identified as promising candidates for future eEDM search experiments. A fully relativistic coupled cluster theory was employed, in the singles and doubles approximation (CCSD), to calculate the effective fields and the PDMs of these molecules. None of the occupied orbitals were frozen, in our calculations. Only the linear terms in the expansion of the CCSD wave function were included, in the expectation value expressions for the properties. The effective electric fields are about 100 GV/cm, for all of these molecules, which is one and IX a half times that in ThO, the molecule that provides the current best limit on eEDM. In our work, we had also identified that the mercury monohalides are experimentally very attractive, and among the HgX, HgBr may hold a lot of promise. The alkaline earth monofluorides (AEMs), a class of molecules that are useful to probe several aspects of fundamental physics, were also studied, to try understand the interplay between relativistic and correlation effects. The PDMs were calculated, at the double, triple, and the quadruple zeta levels of basis. The accuracy of our calculations were also established in this work, with the calculated PDMs (at the quadruple zeta level) agreeing very well with experiment. It was observed that the correlation effects increase as the molecule becomes progressively heavier. We also performed an in-depth analysis of the terms that make the PDM. In subsequent works, the effective electric fields and the PDMs of HgX were also analyzed. The expressions for both the properties, at the Dirac-Fock (DF) as well as the correlation levels, have been broken down into its constituent terms, to identify where the leading contributions come from, and thereby attempt to explain why some terms are larger than the others. The PDMs in HgX follow an unusual trend; the effect of correlations is to reduce the PDM. In fact, in HgI, the DF value is reduced by more than half, because of the correlation effects! It was observed from our analysis of the effective electric fields of HgX that at the DF level, the mixing between the s and p1/2 orbitals, of the heavier atom contribute the most, at the DF level. The mixing from the lighter atom is almost always negligible, even in the case of HgI! We then choose HgF as a representative candidate, and look at the various correlation terms, to get insights into what type of matrix elements contribute, and also to understand which terms cancel each other out, and to what extent. Finally, we compared our work with earlier ones (on HgF), in a detailed manner, and show how our approach is very accurate, as compared to earlier works. The thesis chapters are structured as follows: Chapter one gives an introduction to the eEDM. It briefly discusses why P and T need to be simultaneously violated for a non-zero eEDM. It also discusses how the eEDM can be measured, and illustrates that a combination of theory and experiment is necessary, to determine the eEDM. The chapter also discusses briefly about higher order electric and magnetic moments. Chapter two is a review of many-electron theory. It starts with the Born-Oppenheimer approximation, and proceeds to introduce the use of second quantization in many-body theory. The Hartree-Fock equations, their relativistic version, and the matrix formulation of the theory are discussed. The chapter then discusses electron correlation, and theories that include this effect, like configuration interaction and many-body perturbation theory. The third chapter is a detailed X discussion on the relativistic coupled cluster method. The chapter, towards the end, also covers details of how we tested the accuracy of our coupled cluster method in the SrF molecule. It then explains the interplay between the relativistic and correlation effects in the alkaline earth monofluoride systems. Chapter four and five are the core chapters of the thesis. In chapter four, mercury monohalides are identified as promising eEDM search candidates. We do this by calculating the effective electric fields of the molecules, and show that they are extremely large. We also discuss the experimental advantages that they offer, briefly. Chapter five elaborates on the previous one, where we perform an in-depth analysis of the effective fields. Chapter six discusses the PDMs of HgX, since the property also plays a role in determining the sensitivity of the experiment. Chapter seven mentions the future work, and includes basis set dependence, the finite field method and our preliminary results using this approach, as well as the mercury alkalis as possible candidates for eEDM search experiments.