# Ab initio studies of electron correlation effects in the atomic parity violating amplitudes in Cs and Fr

###### Abstract

We have studied the correlation effects in Cs and Fr arising from the interplay of the residual Coulomb interaction to all orders and the neutral weak interaction which gives rise to the parity violating electric dipole transition to first order, within the framework of the relativistic coupled-cluster theory which circumvents the constrain of explicitly summing over the intermediate states. We observe that, the contributions arising from the perturbed doubly excited states are quite significant and hence, any calculation should not be considered accurate unless it includes the perturbed double excitations comprehensively. In this article, we have reported a comparative study of various results related to the parity violation in Cs and Fr.

###### pacs:

31.15.Ar, 31.15.Dv, 31.25.Jf, 32.10.Dk^{†}

^{†}preprint: July 20, 2021

## I Introduction

One of the most challenging contemporary problems in physics is the search for possible new physics beyond the well known standard model (SM) of elementary particles mohapatra ; barr ; liu ; ginges . Apart from using gigantic accelerators at high-energy scales, it is also possible to use high precision, albeit, low-energy table top atomic experiments, such as, the measurement of atomic parity violation, in combination with accurate relativistic many-body calculations of the atomic parity nonconserving transition amplitude, to achieve this goal ginges ; bouchiat . Some of the prominent signatures of physics beyond the SM which can be inferred from these atomic experiments are: a tight limit on the mass of extra Z-bosons, precise value of the Weinberg angle, limit to the radiative corrections for the electron-nuclear weak interactions etc ginges . As some of these SM results are known to high precision, they demand similar sub-one percent accuracies in both the measurements and the atomic calculations. As the interaction Hamiltonian for the atomic parity violation (APV) due to the nuclear spin-independent (NSI) electron-nucleus interactions is proportional to where being the atomic number bouchiat , heavy atomic systems are chosen for the study of APV effects. A series of APV experiments on a number of atomic systems, including those of Cs wood and Tl vetter have been carried out. However, the high accuracy of has been achieved only for Cs. Furthermore, a number of ab initio calculations of APV amplitudes in Cs have been carried out using a variety of many-body approaches. Some results based on the relativistic coupled-cluster (RCC) theory are also available for Cs, however, their accuracies are somewhat uncertain, since most of them have used the sum-over-states approach which considers contributions from the core orbitals approximately and accounts for only a selected number of excited states whose contributions are dominant. In addition, the doubly excited intermediate atomic states and the normalization of the RCC wave functions are treated only approximately. We have developed a technique in the frame work of RCC theory that circumvents these drawbacks and it has been employed earlier in the calculation of the amplitudes in Ba bijayaba and Ra bijayara in which we have demonstrated that the accuracies of and , respectively, were possible.

In this work, we employ the new RCC approach, mentioned above, to study various correlation effects in the parity violating amplitudes in Cs and Fr. We report a comparative study of their results along with those reported previously.

## Ii Theory of APV

The dominant interaction in an atom is the electromagnetic interaction which, as well known, conserves parity. However, there is a non-zero probability of the interaction between the electrons and the nucleus of an atom due to the weak force with the exchange of a boson, as shown in Fig. 1, which violates parity. The interaction Hamiltonian between the electrons and the nucleus due to the weak interaction can have two components: one, vector–axial-vector and the other, axial-vector–vector currents. The latter depends on the nuclear spin and most of its contribution cancels out except from the odd nucleon and hence, it is relatively smaller in magnitude bouchiat than the former; the NSI component. In this work, we shall consider the APV effect due to the NSI component alone.

The APV interaction Hamiltonian due to the NSI component is given by,

(1) |

where is the Fermi constant, is the electron density over the nucleus and , is the product of the four Dirac matrices that involve electron spin, and is the nuclear weak charge, which is equal to where and denote the electron-proton and the electron-neutron coupling constants for the atomic () and neutron () numbers, respectively. The values of these coupling constants predicted by the SM, in the lowest order of electroweak interaction (at the tree level), are given by

(2) |

where is the Weinberg angle and its measured value is anthony . Substituting these values in , we get which is proportional to . Hence, the perturbation due to is generally expressed in the scale of . Since does not commute with the parity operator, its inclusion with the atomic Hamiltonian of the electromagnetic interaction, which commutes with the parity operator, mixes the opposite parity states of same angular momentum. The strength of this interaction is sufficiently weak, which justifies its consideration as a first-order perturbation.

The Dirac-Coulomb (DC) Hamiltonian used, here, in the calculation of the atomic wave functions of definite parity is given by

where is the velocity of light, and are the Dirac matrices (note that ) and is the nuclear potential.

The atomic wave functions () corresponding to can be considered as the unperturbed wave functions. The total wave function of a system including the first order correction due to the interaction Hamiltonian is given by

(4) |

where is the first order perturbed wave function of its unperturbed valence state and is used as a coupling constant.

Electric dipole (E1) transitions between the states of same parity are forbidden due to the electromagnetic selection rules. However, an E1 transition between the states of mixed parity, mixed due to interaction, is possible and the corresponding transition amplitude can be expressed as

(5) |

where is the E1 operator, the subscripts and denote initial and final valence orbitals, respectively.

Expanding the total wave function as given in Eq. (4) and retaining the terms only up to first order in , we get

(6) | |||||

where the subscripts and represent the intermediate unperturbed states. We have used, here, the explicit form for the first order wave function given by

(7) |

An important question we address in this paper is: How significant are the contributions from those states which were considered approximately in the sum-over-states approach and how they vary with the size of the systems? We would address this by carrying out a comparative study of results in two systems, namely Cs and Fr, of different atomic sizes. In order for the contributions of these higher excited states to be included, it is necessary to solve the first order perturbation equation directly. In other words, it is necessary to solve the equation following equation

(8) |

where is the first order correction to which, however, vanishes in the present case.

## Iii Application of RCC theory to APV

The RCC method, which is equivalent to all order perturbation theory, has been used in the recent past and accurate results have been reported for many single valence systems bijayaba ; bijayara ; csur ; sahoo . In the RCC framework, the wave function of a single valence atom can be expressed as

(9) |

where is the reference state constructed from the Dirac-Fock (DF) wave function of the closed-shell configuration by appending the valence electron n, that is, where represents a creation operator which creates the valence electron . Here and are the RCC excitation operators which excite electrons from and , respectively, due to the residual Coulomb interactions. The corresponding excitation amplitudes are obtained by solving the following equations

(10) | |||||

(11) |

with the superscript representing the singly and doubly excited states from the corresponding reference states and the wide-hat symbol over represent the linked terms of normal order atomic Hamiltonian and RCC operator . In the CCSD (CC with single and double excitations) approximation, the corresponding RCC operators are defined by

(12) |

and

(13) |

for the closed-shell and single valence open-shell systems, respectively. The quantity in the above expression is the electron affinity energy (or negative of the ionization potential (IP)) for the valence electron which is evaluated by

(14) |

In addition to having considered full singles and doubles in the CCSD equations given in Eq. (11), we have also included the contributions from the important triple excitations perturbatively (known in the literature as CCSD(T) method) by defining

(15) |

where the superscript denotes the perturbation and their contributions to are evaluating as

(16) |

After solving for the amplitudes of , we solve Eqs. (11) and (14) simultaneously and obtain the amplitudes of operators.

Now, the total atomic wave function in the presence of is expressed, in the RCC ansatz, as

(17) |

where and are the first order perturbed amplitudes corresponding to the unperturbed RCC operators and , respectively. On expanding the above equation keeping the terms only up to first order in yields

(18) |

Comparing the above equation with Eq. (4), we get

(19) |

In order to calculate as a solution of Eq. (8) in the RCC theory, we solve the excitation operator amplitudes of and using the following equations

(20) |

and

(21) | |||||

after solving Eq. (10) and Eq. (11), respectively. In the above expression, notation is used for the connecting terms between and . To keep the level of approximation uniform through out, both and are truncated at single and double excitations by defining

(22) |

and

(23) |

where and correspond to the perturbed single excitations and and correspond to the perturbed double excitations, from closed- and open-shells, respectively. Since both the perturbed single and double excitation amplitudes are solved simultaneously, certain correlation effects due to the perturbed double excitations also reflect indirectly in the contributions of the perturbed single excitations.

After obtaining the unperturbed and the perturbed RCC operator amplitudes in both the closed-shell and one-valence open-shell atoms, we proceed to calculate the amplitude as

(24) | |||||

where we define and . The non-truncative series for and are expanded using the Wick’s generalized theorem and are truncated when the terms are below fifth order of the Coulomb interaction. The core-valence and valence correlation contributions are obtained from and , respectively, along with their conjugate terms.

Corrections due to the normalization of the wave functions are accounted by evaluating

where .

Although, the CCSD(T) method described here, accounts for the contributions from the important unperturbed triple excitations it fails to include the direct triple excitation contributions to the calculations. To account for, at least, the lowest order direct triple excitation contributions (minimum up to fourth order in Coulomb interaction), we construct them with the open-shell RCC operators perturbatively as follows

(26) |

and

(27) |

where is the single particle energy of an orbital . These operators are finally considered as parts of and in Eq. (III). This can be called as lo-CCSDvT approximation, which also accounts important lower order valence triple excitation effects in the final property calculations.

## Iv Results and Discussions

### iv.1 Orbitals generation

We have used Gaussian type functions

(28) |

to construct the DF orbitals where is an arbitrary parameter which has to be chosen and represents a radial grid given by

(29) |

where the step size is taken to be , the radial grid is increased up to , is in atomic units and is the radial quantum number of the orbitals. Here s are chosen to satisfy the even tempering condition

(30) |

and we have chosen different and values for different symmetries () (known as even tempered basis) and they are given in Table 1.

\backslashbox | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

0.00190 | 0.001825 | 0.00183 | 0.00185 | 0.00187 | |

2.91 | 2.94 | 2.94 | 3.05 | 3.09 |

The finite size of the nucleus in these systems is accounted by assuming a two-parameter Fermi-nuclear-charge distribution for evaluating the electron density over the nucleus as given by

(31) |

where is the density for the point nuclei, and are the half-charge radius and skin thickness of the nucleus. These parameters are chosen as

(32) |

and

(33) |

where is the root mean square radius of the corresponding nuclei which is determined by

(34) |

in for the atomic mass .

Cs | Fr | Method | Reference |

DF | This work | ||

CCSD | This work | ||

CCSD(T) | This work | ||

lo-CCSDvT | This work | ||

LCCSD(T) SL SS | blundell | ||

BO GFCP | dzuba | ||

CI | shabaev | ||

CCSD(T) | das | ||

BO GFCP | dzuba1 | ||

CCSDvT SS | porsev | ||

BO GFCP | dzuba2 | ||

LCCSD RPA ExpEng SS | safronova |

NOTE: Contributions from Breit interaction, QED correction and nuclear effects are not considered here. |

Abbreviations | SS | : | sum-over-states |

SL | : | scaling | |

BO | : | Brueckner orbitals | |

GFCP | : | Green function technique for all order correlation potential | |

CI | : | configuration interaction method | |

ExpEng | : | experimental energy | |

RPA | : | random phase approximation |

(A) Cs | |||

Initial pert. terms | Final pert. terms | ||

() | () | ||

DF result | |||

0.3224 | |||

0.0054 | |||

0.0301 | |||

(B) Fr | |||

Initial pert. terms | Final pert. terms | ||

() | () | ||

DF result | |||

5.9836 | |||

0.0769 | |||

0.6043 |

### iv.2 Results

In Table 2, we give the amplitude results obtained from various calculations. From this work, we present the results using the DF, CCSD, CCSD(T) and lo-CCSDvT methods for both Cs and Fr. In the same table, we also compare our results with previously reported results using various many-body methods. Our Cs result matches reasonably well with the other calculations, but our Fr result differs significantly. We give below individual contributions from various RCC terms and express them in terms of correlation diagrams and level of excitations in order to facilitate the readers to understand the role of various correlation effects and intermediate states. Briefly, we discuss here the methods used in the other calculations. The difference between our DF and lo-CCSDvT results gives an idea about the amount of total correlation effects through the present method in these calculations. Considering the same DF wave functions, we have also employed CCSD and CCSD(T) methods where we find that the CCSD results are larger in magnitude than the CCSD(T) results. However, the result increases in magnitude for Cs in the lo-CCSDvT approximation, but it decreases for Fr; indicating that the triple excitation effects are stronger in Fr. From a comparison between these results, we observe that the dominant triple excitations effects arise through the CCSD(T) method.

### iv.3 Discussions

Here we discuss briefly different reported calculations of the above results at the DC approximation. About two decades ago, Blundell et al blundell had employed the linearized CCSD(T) method (LCCSD(T) method) to evaluate the unperturbed wave functions in Cs and then they had used a sum-over-states approach to evaluate the amplitude for the transition in Cs. However, they had scaled their wave functions to fit the calculated energies of different states with the experimental results which reproduced many atomic properties quite accurately, but that does not show that the method they had used is capable of producing accurate ab initio results. Shabaev et al shabaev have also obtained results using a CI method with a local form of the DF wave functions. Their Cs result matches with the result of Blundell et al. In our previous work on Cs das , we had also calculated this quantity by considering the same and parameters for orbitals of all symmetries (known as universal basis). With the new parameters, the Gaussian basis orbitals produce better wave functions in the nuclear region which are verified by studying the hyperfine interactions that will be reported elsewhere. The convergence of the RCC amplitudes are better than in the present case than in das and it gives a slightly different result. Dzuba et el have carried out a few calculations of these quantities using Brueckner orbitals using a Green function technique (Feynman diagram approach) that takes into account various classes of correlation effects to all orders and avoids the sum-over-states approach dzuba ; dzuba1 ; dzuba2 . Their results also differ from each other and in some cases with others as can be seen in Table 1. The most recent calculation on Cs is reported by Porsev et al porsev using the RCC method that includes all single and double excitations with all valence triple excitations (CCSDvT method). However, they have finally used a sum-over-states approach to calculate the amplitude of the transition. They give contributions from to singly excited states as ”Main”, which contributes , and the remaining contributions as ”Tail”, which is obtained using other many-body methods as , (results are always given in here onwards) at the DC approximation. In contrast to their approach, we have used the lo-CCSDvT method, but have included contributions from the core and doubly excited states in a manner similar to that of the singly excited states.

Results | Results | ||||
---|---|---|---|---|---|

### iv.4 Cs vs. Fr

In Table 3, we present the individual contributions from different RCC terms in our CCSD(T) method to the calculations for the and transitions in Cs and Fr, respectively. It is evident that the trends of the contributions from the different RCC terms are similar for both Cs and Fr. The important contributions come from three terms: , and and their corresponding conjugate terms where is the effective one-body terms of and bare operator is its lowest order term. Other terms correspond to higher order RCC terms, but they are not small. Contributions given as represent the RCC terms that come from the effective two-body terms of after contracting with the open-shell RCC terms. We give diagrammatic representations of the above three RCC terms in Fig 2 (without their conjugate terms) and their lowest order terms. From this figure, it can be noticed that and contain the lowest order DF contributions from the core (hole) and virtual orbitals, respectively; hence always gives the largest contribution. Again, the perturbed states arising through the ground state contribute predominantly while contributions from the perturbed excited states are comparatively smaller than those corresponding to the ground state but with the opposite signs. The final results are the outcome of these cancellations. The most core correlation effects are coming from and its conjugate terms while there are small contributions that come from and its conjugate terms which are included in ”Others”. As seen these correlation effects mostly cancel o