Isotope-shift factors with quantum electrodynamics effects for many-electron systems: A study of the nuclear charge radius of $^{26m}$Al

A method for calculating the field shift contribution to isotope shifts in many-electron atoms, incorporating quantum electrodynamics (QED) effects, is introduced. We also implement the model QED approach to incorporate QED contribution to the nuclear recoil effect at the high-order correlation effects treatment level. The proposed computational scheme is used to revise the value of the root-mean-square (rms) nuclear charge radius of the isomer of aluminium-26, $^{26m}$Al. This radius is important for the global analysis of the $V_{ud}$ element of the Cabibbo-Kobayashi-Maskawa matrix. The difference in mean-square nuclear charge radii of $^{27}$Al and $^{26m}$Al, obtained by combining the calculated atomic factors with recently measured isotope shift (IS) of the $3s^23p~^2P_{3/2} \to 3s^24s~^2S_{1/2}$ transition in Al, is $0.443(44)(19)~{\rm fm}^2$, where the first and second uncertainties are experimental and theoretical ones, respectively. The latter is reduced by a factor of 4 with respect to the previous study. Using this value and the known value of the rms charge radius of $^{27}$Al, the resultant value $R_c(^{26m}$Al) = 3.132(10)~fm is obtained. With the improved accuracy of the calculated IS factors the error in $R_c(^{26m}$Al) is now dominated by the experimental uncertainty. Similar revision of rms charge radii is made for the $^{28}$Al, $^{29}$Al, $^{30}$Al, $^{31}$Al and $^{32}$Al isotopes using existing IS measurements. Additionally, atomic factors are computed for the $3s^23p~^2P_{3/2} \to 3s^24s~^2S_{1/2}$, $3s^23p~^2P_{1/2} \to 3s^25s~^2S_{1/2}$ and $3s^23p~^2P_{3/2} \to 3s^25s~^2S_{1/2}$ transitions in Al, which can be used in future experimental studies.