High Precision Theory For Pionic Helium Spectroscopy

Pion is one of the most important mesons as the carrier particles of the strong nuclear force.Pion decay into a muon and a muon neutrino due to the weak interaction mediated by W bosons.The exotic atoms can be produced by electromagnetic interaction between the negative pion and nucleus by the exchange of photons.Thus the pion plays an important role in Standard Model of particle physics and act the bridge and link between three fundamental forces of matter.This paper pays much attention to the accurate measurement of mass of negative pion.The validity of Standard Model is determined by the accuracy of the mass of negative pion as one of the input parameters.Currently,the most precise value of the mass of negative pion can be determined to an accuracy of 1.3 ppm(1.3 × 10-6)by x-ray wave-length measurements for transitions inπ--mesonic atoms,where the precision is mainly limited by large line widths and may stay at ppm magnitude.Fortunately,the precision for the pion mass determination of the order of ppb(10-9)would be feasible with the development of laser spectroscopy.In contrast with other π--mesonic atoms,the pionic helium can ’trap’ the pion into the orbital of metastable states.The wave functions of the π-orbitals have very little overlap with the nucleus and so the π-cannot be directly absorbed.Thus,the use of laser spectroscopy becomes possible.However,metastable state pionic helium is unstable and is difficult to be produced.The presence of pionic helium can not be verified directly all the time.With the development of low-temperature physics and laser technology,in 2020 year,the PiHe collaboration synthesized π4He+ and excited the transition(n,l)=(17,16)→(17,15).Despite no significant signals were observed for other predicted transitions,it is a good start.The accuracy of the π-mass can be greatly improved at the ppb level by searching for laser transition(17,16)→(16,15),which is predicted to be narrower.Thus the theoretical calculation is necessary.In atomic,molecular,and nuclear physics,the method of complex coordinate rotation is a widely used theoretical tool for studying resonant states.Here,we propose a novel implementation of this method based on the gradient optimization(CCR-GO).The main strength of the CCR-GO method is that it does not require manual adjustment of optimization parameters in the wave function;instead,a mathematically well-defined optimization path can be followed.Our method is proven to be very efficient in searching resonant positions and widths over a variety of few-body atomic systems,and can significantly improve the accuracy of the results.As a special case,the CCR-GO method is equally capable of dealing with bound-state problems with high accuracy,which is traditionally achieved through the usual extreme conditions of energy itself.Based on CCR-GO method,the transition frequency of(17,16)→(16,15)in pionic helium-4 is calculated to an accuracy of 4 ppb(parts per billion),including relativistic and quantum eleetrodynamic corrections up to O(R∞α5).Our calculations significantly improve the recent theoretical values[M.Hori et al.,Phys.Rev.A 89,042515(2014)].In addition,collisional effects between pionic helium and target helium on transition frequencies are estimated.Our work holds the potential to derive a much improved value of the π-mass by 2-3 orders of magnitude once measurements reach the ppb level.The fine and hyperfine structure of pionic helium metastable states is also calculated within the formalism of the Breit-Pauli Hamiltonian using variationally generated wave functions in Hylleraas coordinates.Our results not only verify the existing values of Hori et al.[M.Hori et al.,Phys.Rev.A 89,042515(2014)]for the fine structure of π4He+,but also determine the hyperfine structure of π3He+.