Ground state hyperfine structure in muonic lithium ions

Journal of Physics B: Atomic, Molecular and Optical Physics PAPER Related content Ground state hyperfine structure in muonic lithium  Ab initio calculation of nuclearstructurecorrections in muonic atoms C Ji, S Bacca, N Barnea et al. ions  Breit equation with form factors in the hydrogen atom ynenko and A A Ulybin 2015 J. Phys. B: At. Mol. Opt. Phys. F García Daza, N G Kelkar and MTo cite this article: A P Mart 48 195003 Nowakowski  Vacuum polarization and quadrupole corrections to the hyperfine splitting of P states in muonic deuterium View the ticle online for updates and enhancements. A P Martynenko and V V Sorokin ar This content was downloaded from IP address 124.16.154.193 on 15/03/2019 at 04:31 Journal of Physics B: Atomic, Molecular and Optical Physics J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 (16pp) doi:10.1088/09534075/48/19/195003 Ground state hyperﬁne structure in muonic lithium ions A P Martynenko1,2 and A A Ulybin1 1 Samara State University, Pavlov Str. 1, 443011, Samara, Russia 2 Samara State Aerospace University named after S.P. Korolyov, Moskovskoye Shosse 34, 443086, Samara, Russia Email: a.p.martynenko@samsu.ru Received 27 April 2015, revised 9 July 2015 Accepted for publication 16 July 2015 Published 18 August 2015 Abstract On the basis of perturbation theory in the ﬁne structure constant α and the mass ratio of the electron and muon, we calculate the oneloop vacuum polarization, electron vertex corrections, nuclear structure and recoil corrections of the hyperﬁne splitting of the ground state in muonic lithium ions (m e 63Li)+ and (m e 7 +3Li) . We obtain complete results for small hyperﬁne splittings of the ground state in (m e 6 +3Li) of Dn1 = 21572.16 MHz and Dn2 = 14152.56 MHz and in (m e 73Li)+ Dn1 = 21733.06 MHz and Dn2 = 13994.35 MHz, which can be regarded as a reliable estimates for comparison with future experimental data. Keywords: hyperﬁne structure, light muonic atoms, quantum electrodynamics 1. Introduction precise than the 2010 CODATA value which was derived using different methods including hydrogen spectroscopy. It Muonic lithium ions (m e 6Li)+3 and (m e 73Li)+ are the sim differs from the CODATA value by 7s. Note that the Zemach plest threebody atoms, consisting of one electron, a nega radius of the proton rZ = 1.045(16) fm and magnetic radius tively charged muon and a positively charged nucleus, 63Li or rM = 0.778(29) fm were previously obtained more accurately 7 3Li. The lifetime of muonic atoms is determined by the muon by comparing experimental data with the predictions for lifetime tm = 2.19703(4) · 106 s. It is longer than the time hydrogen hyperﬁne splitting [3]. Similar measurements are for atomic processes, so the muon has the time to make a also being performed in the case of muonic deuterium and number of transitions between energy levels which are ions of muonic helium, the results of which are intended for accompanied by γradiation. These threeparticle systems publication. Light muonic atoms are important for checking have a complicated ground state hyperﬁne structure which the standard model and bound state theory in quantum elec arises due to the interaction of the magnetic moments of the trodynamics, and in the search for exotic interactions of ele electron, muon and nucleus. Light muonic atoms are unique mentary particles. Thus, for example, muonic systems can be laboratories for the precise determination of nuclear proper used in the search for Lorentz and CPT symmetry viola ties such as the nuclear charge radius [1, 2]. In the last few tions [4]. years we have observed the essential progress achieved by the HFS of the ground state of the muonic helium atom Charge Radius Experiment with Muonic Atoms (CREMA) (m e 32He) was measured many years ago with sufﬁciently collaboration in the study of the energy structure in muonic high accuracy by [5]. This is the only experiment to date on hydrogen. The measurement of the Lamb shift (2P–2S) and muonic threeparticle systems. In turn, the theoretical hyperﬁne splitting (HFS) of the 2Sstate allows the more investigation of the energy spectrum of the muonic helium precise determination of the value of the proton charge radius atom and other threeparticle systems has achieved much rp = 0.84087(39) fm, the Zemach radius rZ = 1.082(37) fm success under two approaches [6–13]. The ﬁrst approach and the magnetic radius rM = 0.87(6) fm. The obtained value [6, 7, 11] is based on the perturbation theory (PT) for the for the proton charge radius rp is an order of magnitude more Schrödinger equation. In this case there is an analytical 09534075/15/195003+16$33.00 1 © 2015 IOP Publishing Ltd Printed in the UK J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin solution for the threeparticle bound state wave function in They are ﬁrst presented in an analytical integral form and then the initial approximation. Using this, a calculation of dif calculated analytically or numerically. ferent corrections to the HFS can be performed. Another The basic contribution to the hyperﬁne interaction in the approach [9, 10, 12, 14, 15] is built on the variational ground state of (m e 6,7Li)+3 is determined by the following method which allows the numerical calculation of the Hamiltonian: energy levels in threeparticle systems with high accuracy. g HFS 2pa mgN In order to ﬁnd an arrangement of the low lying energy DH0 = (Sm · I)d (xm)3 m m levels with high precision we should take into account dif m pg g ferent corrections to the particle interaction operator. First of  2pa e m (Se · Sm)d (xm  xe) all these corrections are related to recoil effects, vacuum 3 memm polarization (VP) and nuclear structure effects. The aim of + 2pa gegN S · I d (x ), (4) determining an analytical calculation of HFS in a muonic ( e ) e3 memp helium atom including an excited state was realized in [6, 7, 10–12, 16]. It allowed one to present HFS in an where ge, gm and gN are the gyromagnetic factors of the analytical form as a series into small parameters existing in electron, muon and nucleus. The total spin of the three spin the task. In this work we aim to extend this approach to particles can be either 2, 1 or 0 for (m e 6Li)+3 and 5/2, 3/2 or 7 muonic lithium ions, which are of potential interest for 1/2 for (m e +3Li) . experimental study. So, the purpose of this paper is to HFS of the energy levels in muonic lithium ions is provide a detailed calculation of the HFSs for the systems determined by the following matrix elements: (m e 63Li)+, (m e 73Li)+. n = DHHFS = a I · S  b S · S + c S · I , The bound particles in muonic lithium ions have different 0 m m e e masses me mm mLi. As a result the muon and Li nucleus (5) comprise the pseudonucleus (m 6,7Li)++3 and the muonic lithium ion looks like a twoparticle system in the rst where the spinspace expectation values can be calculatedﬁ approximation. The threeparticle bound system (m e 6,7Li)+ using the following basic transformation [17]:3 is described by the Hamiltonian: Y = å(1)Sm+I+SS SS e+SNm z (2SNm + 1)(2SNe + 1) H = H0 + DH + DHrec + DHVP + DHstr + DHvert, SNe 1 H = 2  1 2  3a  2a , ´⎨⎧ Se S Sm N Ne⎫0 e ⎬Y . (6) 2Mm 2Me xm x ⎩Sm S S m⎭ SNeSSze (1) N DH = a  a D =  1 SNm is the spin in the muon–nucleus subsystem, S is the, Hrec m · e, ( Ne2) x x M spin in the electron–nucleus subsystem and S is totalme e Li angular momentum. The properties of the 6jsymbols where xm and x are the muon and electron coordinates can also be found in [17]. As follows from (4) and (5),e relative to the lithium nucleus, and M = m M (m + M ) the basic contributions to coefﬁcients a, b and c are thee e Li e Li and Mm = mmMLi (mm + MLi ) are the reduced masses of the following: subsystems (e 6,7 ++3 Li) and (m 6,7 ++ g gD D D 3 Li) . The Hamiltonian 2pa N m terms HVP, Hstr and Hvert, which describe the VP, a0 = d (xm) , 3 mpmm structure and vertex corrections, are constructed below. In the 2pa gmge initial approximation the wave function of the ground state b0 = d (xm  xe) , has the form: 3 mmme Y 2pa g g0( e Nxe, xm) = ye0 (xe)ym0 (xm) c0 = d (xe) , (7)3 memp = 1p (6a2MeMm) 3 2e3aMmxme2aMexe. (3) where á...ñ denotes the expectation value in coordinate space over wave functions (3). We have to take into As follows from the structure of the Hamiltonian presented in account the numerical values of the gyromagnetic factors (1)–(2), we include in the basic Hamiltonian H0 only part of ge = 2 for the coefﬁcient b, ge = 2(1 + ke ) = 2(1+ the Coulomb electron–nucleus interaction. The remainder is 1.15965218111(74) · 103), and for the coefﬁcient c, considered as a perturbation as is the the Coulomb muon– gm = 2(1 + km) = 2 · (1 + 1.16592069(60) · 103), g (6N 3Li)= electron interaction. In this way we can explore an analytical 0.822047, g (7N 3Li) = 2.170951. method for the calculation of hyperﬁne structure based on PT. The expectation value (5) is the 4 ´ 4 matrix corre An analytical solution for the wave function (3) allows us to sponding to different values of total spin and muon–nucleus obtain the perturbation contributions in two small parameters spin: (S = 0, SNm = 1 ), (S = 1, S = 1Nm ), (S = 1, S 3Nm = ), α and M 2 2 2e Mm as demonstrated below. The corrections due to 3 electron–muon interaction and mass polarization term (2) are (S = 2, SNm = ) for the ion (m e 6 + 13Li) ; and (S = , SNm = 1),2 2 considered in the second order of PT in subsequent sections. (S = 3 , S 3Nm = 1), (S = , SNm = 2), (S = 5 , SNm = 2) for2 2 2 2 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin (m e 7Li)+3 ): = 2pa gegmb0 ò Y(xe, xm)d (xe  xm)Y(xe, xm3 memm ´ gegm 1dxedxm = nF ⎛ ⎞ (10)4 32Me ⎝⎜1 + ⎟3Mm ⎠ ⎢⎡ ⎛ ⎤= n + k + ( + k )⎜⎜ M 8 M 2 ⎞ 1 m 1 m 2 e + eF ⎣⎢ ⎟ ⎝ Mm 3 M2 ⎠⎟ ⎥, m ⎦⎥ 3 n = 64Me a 4 = { 36140.290 MHzF ,3memm 36141.701 MHz c0 = 2pa gegN ò Y(xe, xm)d (xe)Y(xe, xm)dxedxm3 memp = n mm gegN = { 1674.700 MHzF ,mp 4 4422.900 MHz (11) where we have extracted in square brackets the Fermi energy nF, muon anomalous magnetic moment correction kmnF and Figure 1. Schematic HFSs of the ground state in muonic lithium recoil terms. Their corresponding numerical values for two ions. Numerical values of angular momenta are presented in the case m + lithium ions are presented in table 1.( e 63Li) . Note that, as we determine contributions to the energy spectrum numerically, corresponding results are presented m + with an accuracy of 0.001MHz. We express further the HFSthe ion ( e 73Li) . After its diagonalization we obtain four contributions in the frequency unit using the relation energy eigenvalues ni. In the case of muonic lithium ions we DEHFS = 2pDnHFS. Modern numerical values of funda have relations a b and a c. So, small HFS intervalsDni mental physical constants are taken from [18–20]: the related to the experiment can be written with good accuracy in electron mass me = 0.510998928(11) · 103 GeV, the the simple form: muon mass mm = 0.1056583715(35) GeV, the ﬁne structure constant a1 = 137.035999074(44), the proton mass m DnHFS(m 6 )= 2(b  p 2c) e Li + ⎛ b c ⎞O⎜ , ⎟, = 0.938272046(21) GeV, the magnetic moments of the1 3 3 ⎝ a a ⎠ Li nucleus in nuclear magnetons m (63Li) = 0.8220473(6) and 7 Dn (m )= b + 4c + ⎛ b c ⎞ m (3Li) = 3.256427(2), the masses of the Li nucleus M ( 6Li) HFS e 6 3 2 3Li O⎜⎝ ,3 a a ⎠⎟, (8) = 5.60152 GeV and M (73Li) = 6.53383 GeV, the muon anomalous magnetic moment km = 1.16592091(63) · 103, (  ) ⎛ ⎞ and the electron anomalous magnetic moment ke =DnHFS(m 7 )= 5 b 3c1 e 3Li + ⎜ b cO⎝ , ⎠⎟, 1.15965218076(27) · 103.8 a a In the following sections we calculate different correc DnHFS(m 7 )= 3(b + 5c) ⎛ b c ⎞e Li + O⎜ , ⎟. (9) tions to coefﬁcients b0 and c⎝ 0 over two small parameters α2 3 8 a a ⎠ and Me Mm. For the angular momentum of the muon–nucleus subsystem SmN = 3 2 and SmN = 1 2 (m e 63Li)+ the HFS intervals (8) between states with total angular momentum S = 2, 1 and 2. Recoil corrections S = 1, 0 arise from magnetic interaction between the electron and pseudonucleus (m 6Li)++. The same situation Let us consider a calculation of important recoil corrections of3 a M a M2 M a M24 e 4 e e 4 eis valid for HFS intervals (9) for (m e 7Li)+. A schematic orders , 2 ln and 2 . Using the basic relations3 Mm Mm Mm Mm diagram of the HFS in muonic lithium ions is presented in obtained in [6] for the muonic helium atom we present here ﬁgure 1. corresponding results for muonic lithium ions. Some of these In ﬁrst order PT the basic contributions to the coefﬁcients corrections have already appeared in the equation (10). In b and c (7) can be calculated analytically using (3) (hereafter second order PT (SOPT) we also have the contribution to the upper and lower values correspond to (m e 63Li)+ and HFS which contains the necessary order corrections. The 3 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin Table 1. HFS of the ground state in muonic lithium ions (m e 6,7 +3 Li) . Contribution to HFS b, MHz c, MHz Equation The Fermi splitting 36140.290 36141.701 1674.700 4422.900 (10)–(11) Recoil correction –512.303 –511.012 8.467 22.302 (25), (28) of order a4 (me mm ) Correction of muon anomalous 42.137 42.138 — — (10) magnetic moment of order a5 Relativistic correction of order a6 5.774 5.774 0.535 1.413 [7] Oneloop VP correction in 1g 0.701 0.706 0.066 0.175 (37), (38) interaction of orders a5, a6 Oneloop VP correction 0.723 0.724 0.103 0.271 (40), (41), in the SOPT (44), (48), (52), (53), (57), (61), (63) Nuclear structure correction — — –0.283 –0.707 (65) in 1ginteraction of order a6 Nuclear structure correction — — –0.195 –0.486 (67) in 2ginteractions of order a5 Nuclear structure correction –0.525 –0.470 –0.153 –0.385 (69), (71), of order a6 in SOPT (73), (75), Recoil correction of order 6.430 6.431 — — (77) a5 (me mm )ln(me mm ) Recoil correction of order 0.593 0.507 — — (82)–(84) a4 (Me MLi ) Me Mm Electron vertex correction 40.956 40.958 — — (86) of order a5 in 1ginteraction Electron vertex correction –0.056 –0.056 — — (89), (91), of order a5 in SOPT (92) Summary contribution 35724.720 35727.401 1683.240 4445.483 correction to the coefﬁcient b is the following: For the subsequent integration over the coordinates in (14) we = ò Y D  ˜ ( ¢ ¢ ) use the compact expression of the electron reduced Coulombb1 2 (x , xm) HHFSe 0 (xe xm)G xe, xm; x e, x m Greenʼs function obtained in [21]: ´ DH (x¢e, x¢m)Y(x¢e, x¢m)dxedxmdx¢e dx¢m, ¥ yen (x3)y (x1) (12) Ge (x1, x3) = å en n = 0 Ee0  Een where the reduced Coulomb Greenʼs function has the form: = 2aM 2 e e2aMe (x1+x3)p ⎢ ⎡ 1 ˜ ( ¢ ¢ ) ⎣ 4aMex>G xe, xm; x e, x m = å ymn (xm)yen¢ (xe)ymn (x¢m)yen¢ (x¢e) . (13)  ln(4aMex>)  ln(4aMex<) + E i(4aMex<) n,n¢ = 0 Em0 + Ee0  Emn  Een¢ 7 1  e4aMex+   a < ⎤2C 2 Me (x1 + x3) + ⎥, (16) Dividing the sum over muon states into two parts with n = 0 2 4aMex< ⎦ and n = 0 we obtain for the ﬁrst part: 4pa g gm where x< = min(x1, xò 3 ), x> = max(x1, x3), C = 0.577216¼ e b1(n = 0) = ym0 (x3) 2ye0 (x3) is the Eulerʼs constant and Ei(x) is the exponentialintegral 3 memm function. The result of coordinate integration in (14) can be ´ å¥ yen¢ (x3)yen¢ (x1) ( )y written as an expansion in M M : Vm x e m 1 e0 (x1)dx1dx3, (14) n¢ =0 Ee0 Een¢ ⎡ 2 ò ⎢⎡ a a ⎥⎤ b1(n = 0) = n (1 + k )⎢ 11 Mm e + 1 MeF 2 Vm (x1) = ym0 (x2)⎣⎢   ⎦⎥ym0 (x ) x ⎢ d ⎣ 24 Mm 72 Mm2 2 x2 x1 x1 ⎤ = a ⎛ ⎞ ( + a ) 6ax Mm ( ) ´ ⎜ M ⎝ 64 ln e  7  128 ln 2 + 64 ln 3⎟⎥. (17) 1 3 x1Mm e 1 . 15 M ⎥ x m ⎠⎦1 4 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin The second contribution to b corresponding to muon excited where we deﬁne states is equal to ⎛ ⎞1 2 2 pa g g = å⎜ Emn  Em0⎟ ám ∣ x4 e mb1(n=0) = ò ym0 (x3)y ( ) S1 2 ⎝ ⎠ 0 ∣mnñ . (23)e0 x33 m mm n Rm ame ´ åymn (x3)ymn (x2)Ge (x3, x1, z) n=0 ⎡ ⎤ Discrete and continuum state contributions to (23) are ´ ⎢ a  a ⎥y (x )y (x )dx dx dx , (18) correspondingly equal [22, 23]:⎢⎣  ⎥⎦ m0 2 e0 1 1 2 3x2 x1 x1 d 28n6 (n  1)2n 92 where the electron Coulomb Greenʼs function S1 2 = å = 1.90695 ..., (24)( 9n n + 1)2n+ 2 ¥ Ge (x3, x1, z) = å yen¢ (x3)yen¢ (x1) n¢=0 z  E ¥en¢ c = ò 28kdk ⎛ 1 + ik ⎞i kS ⎜ ⎟¥ y 1 2¢ (x )y (x ) 0 9 2 2på ⎝ ⎠= en 3 en¢ 1 2 1  ik. (19) (k + 1) (1  e k ) n¢=0Em0 + Ee0  Emn  Een¢ = 1.03111 ..., The term (a x1) does not contribute due to the orthogon 9 2 ality of muon wave functions. In order to perform an where Rm = Mma . Summing the corrections in the ﬁrst and2 analytical integration in (18) we use a replacement of G by SOPT we obtain the total recoil correction to the coefﬁcient be 4 the free electron Greenʼs function [6]: in order a : Ge (x3, x1, Em0 + Ee0  Em ) ⎢⎡ M 8 M 2 M 4 ⎜⎛ ⎞ 3 2 n b = n 1 + k 3 e  e Mln e + e ⎟ ( rec F ( m )⎢ 2Ge0 x3  x1, Em0 + Ee0  Emn) ⎣ Mm 9 Mm Mm 9 ⎝Mm ⎠ = Me e b x3x1 8 M 2 ⎛ 185 ⎤ p  , (20) ´ S e 1 2 + ⎜⎝  2 ln 2 + ln 3⎟ ⎞ ⎠⎥⎥2 x3 x .1 9 Mm2 64 ⎦ (25) where b = 2Me (Emn  Ee0  Em0 ) . Moreover, we replace the electron wave functions in (18) with their values at the origin ye0 (0). The omitted terms in this approximation can There are similar contributions to the coefﬁcient c in give contributions of second order in Me . The results of SOPT. In order to obtain these we have to use Mm DHHFS (x ) = 2pa gegN d (x ) in the general expression (12). numerical integration presented in [6] for muonic helium 0 e 3 mem ep show that these corrections are numerically small. After used After evident simpliﬁcations the recoil correction to c can be approximations an analytical integration over coordinate x written as1 gives the result: 4pa gegm c1 = mp ò ye0 (0)G˜e (0, x1)Vm (x1)ye0 (x1)dx1. ò eb x3x1 dx 3 m1 ex3  x1 x2  x1 (26) = p ⎡⎢ 1  1  + 1 b  2 Appearing here, the electron reduced Greenʼs function with4 ⎣ b x3 x2 x3 x22 6 one zero argument has the form: b2 x3  3 + ¼ ⎤ x2 ⎦⎥, (21) å¥ y 2G˜ (0, x) = en (0)yen (x) 2aMe 2aM x24 e =  e e n = 0 Ee0  Een p where an expansion of the exponent eb x2x3 over b ⎡ 1 5 ⎤∣x2  x3∣ is used. It is equivalent to an expansion in powers ´ ⎣⎢  ln 4aMex +  C  2aMex⎥⎦. (27) of Me Mm . Whereas the ﬁrst term b1 does not contribute, 4aMex 2 the second term in (19) yieldsn 35MeF . In addition, the third24Mm The result of an analytical integration is presented as an term in (21) leads to the following integral: expansion in Me Mm: ò ym0 (x3)å 2Me (Emn  Em0) ⎡= Me 8 M 2 ⎛ 1 3 M ⎞⎤c c e en 1 0⎢ ´ y ( )y ( )( · )y ( ) ⎣⎢ + ⎜ + ln  ln ⎟⎥ Mm 9 Mm 2 ⎝ 4 2 Mm ⎠⎥ mn x3 mn x2 x2 x ⎦ 3 m0 x2 dx2dx3 ⎛ ⎞3 2 ={ 8.467 MHz= 1 ⎜ M . (28)e ⎟ S1 2, (22) 22.302 MHz 3aMe ⎝Mm ⎠ 5 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin b: 3 32 = 8a (2aMe ) (3aMm )bVP 9memm p3 ´ ò ¥ r (x)dx ò dx ò dxme6aMmxme4aMe exe1 ⎡ 2 2 ⎤ ´ ⎢pd (  mxm x e x 2m x x x⎢ e)  e e m e ⎥⎥. (34)⎣ xm  xe ⎦ Two integrals over the muon and electron coordinates in (34) can be evaluated analytically: Figure 2. The VP effects. The dashed line represents the Coulomb photon. The wave line represents the hyperﬁne part of the Breit I1= ò dx 6aM x 4aM xe ò dxme m me e epd (x  x )potential. G˜ is the reduced Coulomb Greenʼs function. m e = p 2 ⎛ ⎞ , (35)3. The effects of VP 3( a )3⎜ + 2M3 Mm 1 e⎝ ⎟3Mm ⎠ The VP effects lead to the appearance of new terms in the Hamiltonian which we denote by DHVP in (1). The ratio of I = dx dx e6aMmxme4aMexe the electron Compton wave length to the Bohr radius in the 2 ò e ò m subsystem (m6,73 Li)++ : Zmma me = 2.96185¼ is not a small m 2 2e x 2mex xmxe value. So, we cannot use for the calculation of VP effects an ´ xm  e xe expansion over α. In this section we present a calculation of VP corrections to hyperﬁne structure in the ﬁrst and second ⎡ 2 ⎛ ⎞2 ⎛ ⎞⎤ orders of PT. A modiﬁcation of the Coulomb potentials due to ⎢ 4Me⎢ + ⎝⎜1 + mex ⎟ + Me ⎜6 + 4mex ⎟⎥ VP effects is described by the following relations [24, 25]: p2m 2x2 ⎣ 9Mm2 3Mma ⎠ 3Mm ⎝ 3Mma ⎠⎥e ⎦ D = a ò ¥ r x ⎛⎜ 3a ⎞ = (3aM ) .5m ⎛ ⎞3⎛ ⎞2⎛ ⎞2 V eN (x ) ( ) ⎟e2m xx dx, ⎜1 + 2MeVP e e ep ⎝ ⎠ ⎝ ⎟⎠ ⎜⎝1 + mex ⎟ ⎜ 2Me + mex ⎟ 3 1 xe 3Mm 3Mma ⎠ ⎝ 3Mm 3Mma ⎠ x2r x =  1( ) (2x 2 + 1) ( (36)x , 29)4 They are separately divergent in the subsequent integration a ò ¥ ⎛ 3a ⎞ over the spectral parameter ξ. But their sum is ﬁnite and canDVmNVP (xm ) = p r (x)⎝⎜ ⎟e2mexxm⎠ dx, (30) be written as follows:3 1 xm 2aM ¥e DV emVP ( xe  xm ) = a ¥p ò r (x) a e2m xx m x ( ) bVP = n r (x dxe e d , 31 F ⎛ 3 ò )3 1 xem 9p ⎜ 2MMm 1 + e⎝ ⎟ ⎞ 1 3M where xem = m ⎠ ∣xe  xm∣. These terms give contributions to the ⎡ ⎤ hyperﬁne structure in the SOPT and are discussed below. The ⎢ 2M ⎛ ⎞e + mex 2Me + m x2 e ⎜ + m2 ex ⎟⎥ VP correction in the ﬁrst order PT is connected with the ⎢⎣ 3Mm 3Mma 3Mm 3Mma ⎝ 3Mma ⎠⎥⎦ modiﬁcation of HFS part of the Hamiltonian (4) (the ´ ⎛⎜ m x ⎞ 2⎛ ⎞2 amplitude in ﬁgure 2(a)). It can be written in integral form 2M m x ⎝1 + e e e in the coordinate representation [26]: 3Mma ⎟⎠ ⎝⎜ + ⎟3Mm 3Mma ⎠ D HFS,em ( 8aVVP xem) =  (Se · Sm) a ={ 0.701 MHz ,3memm 3p 0.706 MHz ⎡ ⎤ (37)2 2 ´ ò ¥ r (x)dx ⎢⎣pd ( me xx 2m xxem )  e e em⎥⎦, (32) Two small parameters α and Me Mm determine the order of1 xem this contribution and are written explicitly in (37). The 8ag a correction bVP has the ﬁfth order in α and the ﬁrst order inDVHFS,eNVP (xe) = N (Se · I) Me Mm. The muon VP contribution to HFS is negligibly6memp 3p ⎡ ⎤ small. The twoloop VP contribution to the hyperﬁne2 2´ ò ¥ r (x)dx ⎢pd (x )  me x e2m xx ⎥ ( ) structure is suppressed relative to the oneloop VP contribu⎣ e e e⎦. 331 x tion by the factor a p. Thus at the present level of accuracye we can neglect this correction because its numerical value is The matrix element of the potential (32) over the wave small. Higher orders of PT which contain oneloop VP and function (3) gives the necessary contribution to the coefﬁcient the Coulomb interaction (2) give recoil corrections of order 6 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin n aM2e Mln m . Such contributions are included in the theore The contribution (40) has the same order of magnitudeF Mm2 Me O (a5Me ) as the previous correction (37) in the ﬁrst order PT. tical error. Mm A similar correction to the coefﬁcient c of order a6 is The same evaluation can be performed in the case of calculated analytically using the potential (33) (a1 = 2aMe me): the muon–nucleus Coulomb VP potential (30). The electron ag m is in the intermediate 1Sstate and the reduced Coulombm c NVP = nF p Greenʼs function of the system transforms to the Greenʼs6 mp function of the muon. For this case the correction of the 1  a2 (6a + a3  3p) + (6  3a2 + 6a4)arccosa operator (30) to the HFS (the coefﬁcient b) is obtained in the´ 1 1 1 1 1 1 form: 3a31 1  a21 { m a ¥ ¥ N = 0.066 MHz bVP, SOPT = nF p ò r (x)dx x 2dx3. 33 1 ò0 0.175 MHz (38) ¥  ⎛⎜ + 2M ⎟⎞  ⎛´ ò x e3⎝1 ⎠ x2 ⎝⎜1+ mex ⎞⎟ x 3M 3M2dx2e m e ma ⎠ 0 ⎡ The electron VP effect (the potentials (29)–(31)) gives ´ ⎣⎢ 1  ln x>  ln x< + E i(x<) the corrections in SOPT (the amplitude in ﬁgure 2(b)). The x> contribution of the electron–nucleus Coulomb interaction (29) 7 x x ⎤2 + x3 1  e < to the HFS can be presented in the form: +  2C  +2 2 x< ⎦⎥  = 4pag gm ò ò ={ 0.694 MHzeb e N dx dx . (41)VP, SOPT 1 23m mm 0.693 MHze ´ ò a ¥dx3 p ò r (x)dxym0 (x3)ye0 (x3) The most difﬁcult aspect for the computation is the VP3 1 ¥ y y y y correction to HFS which is determined by the operator (31) in´ å mn (x3) en¢ (x3) mn (x2) en¢ (x1) SOPT. In this case we should consider the intermediate , ¢ =0 Em0 + Ee0  Emn  Een¢ excited states both for the muon and the electron. Thisn n ⎛ a ⎞ contribution is divided into two parts. The ﬁrst part with the´ ⎜ 3 ⎟e2mexx⎝ ⎠ 1ym0 (x2)ye0 (x1), muon in the intermediate 1Sstate has the form:x1 (39) me = = 256a 2 (2aM )3(3aM )3( e mb n 0) where the indices on the coefﬁcient b indicate the VP VP, SOPT 9memm contribution in SOPT when the electron–nucleus Coulomb ¥ 2 a (2Me+6Mm)x3 VP potential is considered. The summation in (39) is carried ´ ò x3 dx3e0 out over the complete system of the eigenstates of the electron ¥  a ¥2 2 Mex1 and muon excluding the state with n, n¢ = 0. The evaluation ´ ò x1 dx1e ò r (x)dxDVVP m (x1)Ge (x1, x3),0 1 of (39) can be carried out using the orthogonality condition (42) for the muon wave functions: 2aM 2 ò ¥ ò ¥ where the auxiliary function VVP m (x1) is equalb eN eVP, SOPT = nF p r (x)dx x 22 3 dx39 Mm 1 0 (3aMm )3 ¥ ⎛ x ⎞ DV (x ) = dx e 6aMmx2 ò  2Mx e ⎜1+ me´ 1 ⎝ a ⎟⎠  ⎛ + 2M ⎞ VP m 1 2x ⎜1 e ò3 ⎟ p x dx e 3Mm 2 Me e ⎝ 3M1 1 m ⎠ a 0 ´ e2mex x1x2 ⎡ ⎛ ⎞ ⎛ x1  x2 ´ ⎢ 3Mm⎢  ln ⎜ 2Me ⎝ x<⎠⎟  ln ⎝⎜ 2M ⎞e ⎟ 4 3 ⎣ x>⎠ = 108aM2Mex> 3Mm 3Mm m 2 ⎛ ⎞ x 2 2 2 21(36a Mm  4me x ) + 2M 7 ME i⎜ e⎝ x<⎠⎟ +  2C  e (x1 + x3) 3Mm 2 3M ´ ⎡⎣12aMm (e2mm exx1  e6aMmx1) ⎤⎥ + x (4m 2x2 2 2 6aMmx1⎤2M 1 e  36a Mm )e ⎦. (43)e x 1  e3Mm <+ ⎥ = {1.136 MHz . (40)2Me ⎥x< 1.137 MHz Substituting (43) into (42) we obtain the result after numerical 3Mm ⎥⎦ integration: It is necessary to emphasize that the transformation of the b me (n = 0) = 0.310 MHz . (44) expression (39) into (40) is performed by means of (16). VP, SOPT {0.311 MHz 7 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin The second part of the correction to HFS with VP coming sequence of transformations in the coordinate representation: from the electron–muon interaction can be written as follows: å¥ Emn ò dx2 ò dx3ym0 (x2)ymn (x3) m 2= =  4a gegm n=0b eVP, SOPT (n 0) 9 m mm ò dx3 ´ ymn (x2 ) x3  x2 y (xe m0 2) ´ ò dx2 ò ¥ r (x)dxym (x3)y (x3) = dx0 e0 ò 2 ò dx3d (x3  x2)1 b x x ⎡ 2⎢ ⎤´ åy ( )y ( ) M e 3 1 3m x em x ⎣ x3  x2 yn 3 n 2 p  m0 (x3)⎥⎦ym0 (x2). (50)2Mn=0 2 x3 x1 m ´ a e2mex x2x1 ym0 (x2 )ye0 (x1), (45) We have the divergent expression in (50) due to the presencex2 x1 of the δfunction. The same divergence occurs in the other term with b2 entering in the square brackets of (46). But their where we have replaced the exact electron Coulomb Greenʼs sum is ﬁnite and can be evaluated analytically with the function with the free electron Greenʼs function. Also following result: neglecting the higher order recoil corrections we replace the 2 ⎛ 2 2 ⎞ electron wave functions by their values at the origin. Then the = n 3a 2M bb e2 F ⎜⎜1 + 5 a Mm ⎟⎟. (51)integration over x can be performed analytically: 8m Mm ⎝ 8 m 21 e e ⎠ ò eb x3x1 e2mex x2xJ = 1dx Numerically this correction is essentially smaller than the1 x3  x1 x2  x1 leading order term. The other terms in (46) give negligibly =  4p 1 small contributions. x  x b2  4m 2x2 The potential (29) with VP does not contain the muon3 2 e ⎡ ⎤ coordinate. A corresponding contribution to the coefﬁcient c in´ ⎣eb x3x2  e2mex x3x2 ⎦ SOPT can be determined by setting n = 0 for the muon state in ⎡ (   x  ) the Coulomb Greenʼs function. Moreover, the d (xe) function in⎢ 1 e 2me x3 x2= p  b (4) leads to the appearance of the electron Greenʼs function with2 ⎣⎢ 2m 2x2 x  x 2m 2x2 one zero argument. The value of HFS in this case is equal toe 3 2 e (1  e2mex x3x2 )b2 b2 x  x c eNVP, SOPT = n ammgegNF+ 3 2 4pmp 8m 4e x + 4 x 2 23  x2 4me x ´ ò ¥ 2r (x) x2a1 + 3a1 + 2a1 ln ad 1  2b3 3 2 ⎤ 3  b (x3  x1) + 1 2ax ...⎥, (46) 18m 4 4e 12m 2x2e ⎦ ={ 0.104 MHz . (52)0.274 MHz where an expansion of the ﬁrst exponent in square brackets in powers of b ∣x3  x2∣ is carried out. Further transformation is The VP in the Coulomb muon–nucleus (m  N) interaction based on the completeness condition: does not contribute to c in SOPT because of the muon wave å y ( )y ( ) = d (  )  y ( )y ( ) ( ) functionʼs orthogonality. Let us calculate the correction to themn x3 mn x2 x3 x2 m0 x3 m0 x2 . 47 = coefﬁcient c arising from (31) in SOPT. Only an intermediaten 0 muon state with n = 0 in the Greenʼs function gives the The orthogonality of wave functions leads to the zero results necessary contribution. By means of (27) we perform for the second and ﬁfth terms in the square brackets of (46). coordinate integration and express this correction in the form The ﬁrst term in (46) gives the leading order contribution in g = x a (g = mex 3aMm, g1 = 2Me 3Mm):two small parameters α and Me Mm ( me 3 Mm): 2 b me (n=0) = b + b c em = n 2ammgNMeVP, SOPT 11 12 VP, SOPT F 27pm 2pMm = ⎨⎧0.432 MHz =  3a2M, b e⎩ 11 nF, (48) ¥ r (x)dx ¥ 0.431 MHz 8m ´ ò ò xeg1xe ( ) dx1a ¥ r x 1  g2 2 02= n Me ò ( )dx [16 + g (5g (g + 4) + 29)b ]12 F .24pme 1 x (1 + g)4 ´ ⎡ x ⎤⎣⎢egx  ex + ex (g2  1)2 ⎦⎥ (49) ´ ⎡⎢ 1  5ln g x +  C  1 g ⎤x⎥ The summary numerical value for b11 + b12 is included in ⎣ g 1 11x 2 2 ⎦ table 1. The calculation of other terms of the expression (46) in 0.018 MHz the HFS is also important. Taking the fourth term in (46), which ={ . (53) is proportional to b2 = 2M (Em  Em ), we perform the 0.047 MHze n 0 8 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin Let us evaluate the terms in the coefﬁcient b with n = 0. The δlike term of the potential (33) gives the following contribution to the HFS: ¥ ⎛(21) ( = ) = n a ò r (x) x ⎜ 35M ⎞b eVP, SOPT n 0 F p d ⎝ ⎟. (58)3 1 24Mm ⎠ Another correction from the second term of (33) can be simpliﬁed by means of the replacement of the exact electron Greenʼs function by the free electron Greenʼs function: (22) 16a3Mem 2e Figure 3. The effects of VP in SOPT. The dashed line represents the bVP, SOPT(n=0) =  ﬁrst part of the potential DH (3). The wavy line represents the 9pmemm hyperﬁne part of the Breit potential. ´ ò ¥ r (x)x2dx ò dx2 ò dx31 ò e2m´ ex x3x4dx4ym0 (x4)There is another contribution of SOPT in which the x3  x4 hyperﬁne VP potential enters as a perturbation (32)–(33) (see ¥ ﬁgure 3). The other perturbation potential in this case is ´ åymn (x4)ymn (x2) x3  x2 ym0 (x2). (59) determined by the ﬁrst term of equation (2). Dividing the HFS n=0 correction of (33) into two parts we write the ﬁrst part with The analytical coordinate integration in (59) gives the result: n = 0 for the muon ground state. The second part with n = 0 contains excited muon states. The term with the δfunction in b (22) ( = ) = n 2aMeVP, SOPT n 0 F (32) gives the following contribution to HFS at n = 0: 9pMm ( ¥ ¥ ⎡b 11) a 11Me 1 1VP, SOPT(n = 0) = nF p ò r (x)dx . (54) ´ ò r (x)dx ⎢ 3 1 24Mm 1 ⎣ g (1 + g)4 The integral in the spectral variable ξ is divergent. So, we ´ ⎛⎜ + 1 + g + 215g 2 + 35g 3 + 35g 4 ⎞ 4 10 ⎟⎥ ⎤ . (60) should consider the contribution of the second term of the ⎝ g 16 4 16 ⎠⎦ potential (32) to HFS which is determined by the following formula: The sum of contributions (58) and (60) is ﬁnite: a ¥ b (21)16 2m 2 ò VP, SOPT(n=0) + b (22) VP, SOPT(n=0) b (12) e 2VP, SOPT(n = 0) = p r (x)x dx = n 2aM ò ¥ 35 + 76g + 59g2 39 memm 1 eF p r (x)dx + 16g´ ò 9 Mm 1 16(1 + g)4dx3ye0 (x3)DV1(x3) = 0.432 MHz . ´ ò å yen¢ (x3)y {en¢ (x1) DV2 (x1)ye0 (x1)dx1, (55) 0.431 MHz (61)n¢ = 0 Ee0 Een¢ D D The absolute values of the calculated VP corrections (38),where V1(x3) is deﬁned in (43) and V2 (x1) in (15). (42), (44), (45), (47), (57) and (61) are sufﬁciently large, but Integrating in (55) over all coordinates we obtain the their summary contribution to the HFS (see table 1) is small following result in the leading order in the ratio (Me Mm ): because they have different signs. 2 = = n m M The HFS interaction (33) gives the contributions to theb (12)VP, SOPT(n 0) e eF M p 2 coefﬁcient c in SOPT. Since the muon coordinate does note 216 Mm enter into the potential (33), we set n = 0 for the muon ´ ò ¥ 2 3r (x)x x32 + 63g + 44g + 11gd . (56) intermediate states in the Greenʼs function. The basic formula1 (1 + g)4 for this correction is This integral also has the divergence at large values of the 8a3g ¥N parameter ξ. But the sum of integrals (54) and (56) is ﬁnite: cVP, SOPT = p ò r (x)dx dx1 dx39 mem 1 ò òp b (11)VP, SOPT(n = 0) + b (12)VP, SOPT(n = 0) ´ ò dx 24 ym0 (x3) ye0 (x4)ye0 (x1) = n aMe ò ¥ r (x) x11 + 12g + 3g2F d ⎡ 1 1 ⎤72pMm 1 (1 + g)4 ´ ⎢  ⎥ { ⎢⎣ x3  x4 x4 ⎥⎦ Ge (x4, x1) = 0.067 MHz . (57) ⎛ 2 2 ⎞ 0.067 MHz ´ ⎜pd (x )  me x e2m1 exx1⎝ ⎟. (62)x1 ⎠ 9 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin Integrating over x3 analytically as in (15) we divide (62) into the last integral in analytical form: two parts. The coordinate integration in the ﬁrst term with the 33aMemmgegN 0.195 MHz δfunction is performed by means of (28). In the second term cstr, 2g = nF = { . (67) of (62) we use the electron Greenʼs function in the form ( ) 16mpL 0.486 MHz16 . The summary result can be written in the integral form in the Other parts of the iteration contribution áV f HFS1g ´ G ´ V1gñstr leading order in Me Mm: are used in SOPT (see ﬁgure 6). a Two different types of nuclear structure corrections togNmmMecVP, SOPT = nF p another coefﬁcient c in SOPT are presented in ﬁgure 6. The18 mpMm ﬁrst contribution is determined by amplitudes in ﬁgure 6(a), + mex (b) when the hyperﬁne part of the ﬁrst perturbation is deter3 2 ´ ò ¥ r x x 3aMm = { 0.017 MHz mined by magnetic form factor GM and the second pertur( )d ⎛ ⎞ . (63)x 2 0.044 MHz bation is connected to the nucleus charge radius r1 N:⎜ m⎝1 + e3a ⎟ 2Mm ⎠ DV Cstr,eN (r) = pZar2Nd (r). (68) 3 This correction is described by the following general integral expression and has the numerical value (a2 = 4aMe L): eN = n a 2r2M 2g g mm 4. Nuclear structure and recoil effects c N e e N1,str, SOPT F mp Another important type of correction to HFS of muonic ´ ò ¥ 2 ⎛ 5 1 ⎞x dxex (1+a2)⎝⎜ln a2x +  C  a2x⎟lithium ions which we investigate in this work is determined 0 2 2 ⎠ by the nuclear structure and recoil [27–31]. We describe the 0.0003 MHz charge and magnetic moment distributions of the Li nucleus ={ . (69)0.0008 MHz by means of two form factors GE (k2) and GM (k2) for which eN we use the dipole parameterization: Numerically, the contribution c1,str, SOPT is obtained by means of the charge radii of nucleus 6,73 Li r (63Li) = 2.589(39) fm G 2E (k )= 1⎛ ⎞ , G (k2M ) and r ( 7 3Li) = 2.444(42) fm [33]. The second type of nuclear 2 2 ⎜ + k1 ⎟ structure correction from amplitudes in ﬁgure 6(c), (d) is⎝ L2 ⎠ calculated by means of the potential DH (2) and the nucleus = G (0) = mN magnetic form factor. In the case of the amplitude in⎛ ⎞ , G (0) gN , (64)2 2 Zm ﬁgure 6(c) we perform the integration over the muon⎜ + k ⎟ p⎝1 L ⎠ coordinate in the muon state with n = 0 and present the2 correction to the coefﬁcient c as follows (a3 = 6aMm L): where the parameter Λ is related to the nucleus charge radius 2 c eN 2a2Me mmgegN r : L = 12 r . In 1ginteraction the nuclear structure 2,str, SOPT + c1 = nFN N mpL2 correction to the coefﬁcient c is determined by the amplitudes ¥ ¥ 2 shown in ﬁgure 4. The purely point contribution in ﬁgure 4(b) ´ ò x1 dx1ex1(1+a2) ò x2dx20 0 leads to the HFS value (11). Then the nuclear structure ⎛ 1 ⎞ correction is given by ´ ⎝⎜1 + a3x ⎠⎟ex2 (a2+a )2 32 g g mm ⎡ G (x) ⎤ ⎡ c e Nstr, 1g = nF ⎢ò M e4aMex⎣ dx  1( ) ⎥ 14m G 0 ⎦ ´ ⎢⎣  ln(a2x>)  ln(a2x<) + E i(a2x<)p M a2x> ={0.283 MHz a x . (65)0.707 MHz + 7  2C  1 2 < ⎤a2 (x1 + x2 ) + 1  e ⎦⎥2 2 a2x< = 8.314 MHz . (70) Twophoton amplitudes of the electron–nucleus (e–N) { 21.918 MHz interaction (see ﬁgure 5) give the contribution to HFS of order a After the subtraction of the point contribution c1 (28) we5. This can be presented in integral form in terms of the form obtain factors GE and GM taking into account the subtraction term [30, 32]: c eN 0.153 MHz2,str, SOPT = { . (71)3aM mmg g dp ( ) 0.384 MHz= n e e N Gc M pstr, 2g F 2p ò G (p  1 , (66)2mp p4 GM (0) [ E ) ] There is a nuclear structure contribution to the coefﬁcient where the subtraction term contains the magnetic form factor b in SOPT, which is presented in ﬁgure 7. For the Coulomb GM (p). Using the dipole parameterization (64) we can present muon–nucleus interaction the structure correction takes the 10 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin Figure 4. Nuclear structure correction to coefﬁcient c in 1ginteraction. The bold dot represents the nuclear vertex operator. The wavy line represents the hyperﬁne part of the Breit potential. Figure 5. Nuclear structure corrections to coefﬁcient c in 2ginteractions. The bold dot represents the nuclear vertex operator. The wavy line represents the hyperﬁne part of the Breit potential. The dashed line corresponds to the Coulomb potential. Figure 6.Nuclear structure corrections to coefﬁcient c in SOPT. The bold dot represents the nuclear vertex operator. The wavy line represents the hyperﬁne part of the Breit potential. The dashed line corresponds to the Coulomb potential. G˜ is the reduced Coulomb Greenʼs function. form: m 32p2N = a 2 2 1b 3 2str rN (3aMm ) 3memm p ´ ò dx3ym0 (x3) ye0 (x 23) Gm (x3, 0, Em0). (72) An analytical integration over the coordinate x3 in (72) can be performed using the expression for the muon Greenʼs function similar to (27). Expanding the result of the integration of order O (a6) in the ratio Me Mm we obtain: ⎛ ⎞ Figure 7. Nuclear structure correction to coefﬁcient b in SOPT. The m 2N = n M 22 M wavy line represents the hyperﬁne (e–μ) interaction. is the reducedbstr F24a2Mm2r2N ⎜⎜ e  e + ¼⎟ G˜⎝Mm 9 Mm2 ⎠⎟ Coulomb Greenʼs function. ={0.416 MHz . (73)0.372 MHz the formula: A similar approach can be used in the calculation of the 32p2a2 2 structure correction to the electron–nucleus interaction. The b eN 2str = rN ò dx1ò dx3 ym0 (x3) electron also feels the distribution of the nucleus electric 3memm charge. The corresponding contribution of the nuclear ´ ye0 (x3)Ge (x3, x1, Ee0)ye0 (x1)d (x1). (74) structure effect on the hyperﬁne structure is determined by 11 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin electron are excited to Pstates: 32Pa3D =  MeMmbrec,SOPT dx3 m mmM òe Li ´ ò dx2 ò dx1Ym0(x3)Ye0(x3) ´ å Ymn(x3)Yen¢(x3)Ymn(x2)Yen¢(x1) n,n¢ =0 Em0 + Ee0  Emn  Een¢ Figure 8. Twophoton exchange diagrams for the electron muon ´ (n1 · n2)Ym0(x2)Ye0(x1). (78)– hyperﬁne interaction. In order to present an analytical estimate of this correc After an analytical integration in (74) we obtain the following tion we transform (78) as in section 2, introducing the free expansion: electron Greenʼs function: b eN = nF6a 22 3M 2 16a M MeMmr D e mstr N b ⎡ ⎤ rec,SOPT = ò dx3 2 ⎛ ⎞ memmMLi ´ ⎢1  4Me 2Mln e + 4Me ⎜ 2M⎢ 6 ln e  4⎟ + ¼⎥ ⎣ 3Mm 3Mm 9Mm2 ⎝ 3Mm ⎠ ⎦⎥ ´ ò dx2 ò dx1Ym0(x3)Ye0(x3) ={0.109 MHz b x x´ Y Y 3 1 . mn(x3) ( ) ex20.098 MHz å mn(75) n=0 x3  x1´ (n1 · n2)Ym0(x2)Ye0(x1). (79) The total nuclear structure contribution to the coefﬁcient b After that the integration over x1 and expansion in b (or in which is equal to the sum of numerical values (73) and (75) is Me Mm ) give the result: included in table 1. b x3x1 The twophoton electron–muon interaction shown in ò ( edx1 n1 · n2) ﬁgure 8 provides large recoil corrections. They were inves x3  x1 tigated in quantum electrodynamics in [10, 24, 34]. The ⎡ 4x 2 3 ⎤3 x3 2bx3 leading order recoil contribution to the electron–muon inter = 2p (n2 · n3)⎣⎢  + + ¼⎥. (80)3b 2 15 ⎦ action operator is determined by the following expression: Taking the ﬁrst term in square brackets in (80) we perform an 2 D HFS =  a angular integration and introduce the dimensionless variablesVrec,me (xme) 8 mm2  m 2 in integrals with radial wave functions:e ´ mmln (smse)d (xme). (76) d = n 64Mb e Me nme rec,SOPT F å9MLi Mm 2n>1 n  1 ¥ 3 After averaging DVHFS ´ x3 R10 (x3)Rn1(x3)dx3rec,me over the wave functions (3) we ò0 obtain the recoil correction to the coefﬁcient b: ´ ò ¥ x 22 R10 (x2 )Rn1(x2 )dx2. (81) me = n 3a memm mm 0 1 brec F p  lnmm2 m 2e m 3e ⎛⎜ 2M ⎞ The two contributions of the discrete and continuous spectra1 + e⎝ ⎟⎠ are the following:3Mm { 6.430 MHz 2n 9= 11 6. (77) dbdisc 2 Me Me n (n  1) 26.431 MHz rec,SOPT = nF å9MLi Mm n>1 (n + 1)2n+ 92 0.392 MHz There are also another twophoton interactions between the ={ . (82) bound particles in muonic lithium ions. So, for example, one 0.336 MHz hyperﬁne photon transfers the interaction from the electron to 4 arctg(k the muon and another Coulomb photon from the electron to 11d ¥ ) b cont = n 2 Me Me ke k dk the nucleus (or from the muon to the nucleus). Assuming that rec,SOPT F 9M Mm òLi 0 (1  e2p k)(k2 + 1)3 2 these amplitudes give a smaller contribution to HFS we included them in the theoretical error. ={ 0.212 MHz . The ﬁrstorder recoil correction O (Me MLi ) has a con 0.182 MHz (83) tribution from intermediate states in which the muon and 12 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin present the electron vertex correction to HFS in the form: 3 = n (1 + km )me Mebvert, 1g F 81p2a2Mm4 ´ ò ¥ k2dk ⎡⎣G (e)(k2)  1⎤M ⎦0 ⎪⎧⎡⎢ ⎛ ⎞ 2 ⎤2⎡ ⎛ 2 2 2 ´ ⎨ + ⎜ me ⎟ ⎥ ⎢ ⎤ ⎜ 2M ⎟⎞ ⎛e + ⎜ me ⎟ ⎞ ⎥ ⎪ ⎫1 ⎪ 1 k 2 k2 ⎬ ⎩⎣⎢ ⎝ 6aMm ⎠ ⎥⎦ ⎣⎢ ⎝ 3Mm ⎠ ⎝ 6aMm ⎠ ⎦⎥ ⎪⎭ ={ 40.956 MHz .40.958 MHz (86) Figure 9. Correction of the electron vertex in ﬁrst and second orders The contribution (86) has the order a5. The numerical value of PT. The Coulomb photon is shown by the dashed line. The wavy (86) is obtained after numerical integration with the oneloop line represents the hyperﬁne part of the Breit potential. G˜ is the expression of the electron magnetic form factor G (e) (k2). reduced Coulomb Greenʼs function. M When using the value G (e)M (k2 = 0) we obtain the electron vertex correction 41.959MHz. Thus, the electron form factor The calculation of the second term in square brackets in (80) in the oneloop approximation leads to the 1MHz decrease of is essentially simpler and gives the result the vertex correction to the HFS in the 1ginteraction. Taking (85) as an additional perturbation potential we have to 2M 2d (2) = n e evaluate its contribution to HFS in SOPT (the diagram in ﬁgbrec,SOPT F 3MmMLi 9(b)). The dashed line represents the Coulomb Hamiltonian { 0.011 MHz DH (2). As previously, we can divide the total contribution of= (84) the amplitude in ﬁgure 9(b) into two parts which correspond0.011 MHz to the muon ground state (n = 0) and muon excited intermediate states (n = 0). The ﬁrst contribution with n = 0 takes the form: 8a2 bvert, SOPT (n = 0) = 2 5. Correction of the electron vertex function 3p memm a4 ´ ¥ k ⎡G (e)(k2)  1⎤dk The leading order contribution to the hyper ⎣ M ⎦ﬁne structure is ò0 related to the interaction operator (4) as discussed above. ´ dx1 dx3ye0 (x3) Among the many corrections to (4) there is a contribution of ò ò the electron vertex function, which is presented in gure 9(a). ´ DV˜ﬁ 1(k, x3)Ge (x1, x3)Vm (x1)ye0 (x1), (87) First it is convenient to write this correction in the momentum where Vm (x1) is deﬁned by (15) andrepresentation: sin 2 (k x3  x4 ) DVHFS (k2) =  8a ⎛s⎜ esm ⎞⎟⎡⎣G (e)(k2)  1⎤⎦ DV˜( ) 1(k, x3)= ò dx4ym0 (x4 ) y, 85  m0 (x4)vertex 3m mm ⎝ 4 ⎠ M x3 x4e = sin(kx3) 1 . 2 where we take the factor a p of the bracket in the expression x3 ⎡ ⎤ ( )  ( ) ⎢ + k 2 e (88)[GM (k2) 1] ⎥ containing the magnetic form factor G eM (k2) ⎢1 2 ⎥ of the electron. A commonly used approximation for the ⎣ (6aMm ) ⎦ magnetic form factor of the electron G (e) (k2M ) » G (e)M (0)=+ k After the substitution of the electron Green function (28) to1 e is not applicable in this task. Since the typical photon ~ a (88) we transform this expression to integral form:momentum exchange is k Mm we cannot neglect it in G (e) (k2M ) compared to the mass of the electron me. Therefore, ⎛ ⎞2⎛ ⎞2 we must use the exact expression for the Pauli form factor ( 2a m Mbvert, SOPT n = 0) = n eF ⎜ ⎟ ⎜ e ⎟ g (k2) (G (e)M (k2)  » 21 g (k2)) [25] trying to improve the 81p ⎝aMm ⎠ ⎝Mm ⎠ estimate of the correction due to the electron anomalous ¥ k ⎡⎣G (e)M (k2)  1⎤⎦dk magnetic moment. ´ ò 2 Using the Fourier transform of the potential (85) we 0 ⎢⎡ 2 2 ⎤me k average the obtained expression over wave functions (4) and ⎣⎢ 1 + ⎥(6aM )2m ⎦⎥ 13 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin ´ ò ¥  2Me x ⎛ m k ⎞ ⎛ ⎞ 3 x3e 3Mm 3 sin ⎜ e⎝ a x3⎟⎠dx3 b2,vert, SOPT (n=0) = n a me Me 6 M F ⎜ ⎟ 0 m 162p2 ⎝aMm ⎠ Mm ⎛ 2M ⎞ ¥e ´ ò ¥ ⎛ x (e)x ⎜1 + 1⎞ x1⎜⎝1+ ⎟⎠ ´ ò k2 ⎡⎟e 3Mm dx ⎣GM (k2)  1⎤⎦dk1⎝ 2 ⎠ 1 00 ⎡ ⎛ ⎞ ⎢⎡´ ⎢ 3Mm  2M 1 2⎢ ln⎜ e x<⎟ ´ ( ) ⎢ (g  (g1 + 1) ⎣ 2Mex> ⎝ 3Mm ⎠ 1 + g2 2 ⎢⎣ 2 2 ⎡ 2 2⎤ 2 2 1 + g2) ⎣ (1 + g1) + g2⎦  ln⎜ ⎛ 2M ⎞ ⎛ ⎞e ⎟ + ⎜ 2Me 7x> E i x<⎟ +  2C 2 g2  3g2 ⎤⎝ 3Mm ⎠ ⎝ 3Mm ⎠ 2   2 1 ⎥(g + 1)2 + g2 (g2 + g2)3 ⎥⎤ 1 2 1 2 ⎦ 2M  e x ⎥ M + + 1 e3Mm < ⎥ ={0.582 MHze (x1 x3) 3Mm 2Me ⎥  . (92) 0.582 MHz x< 3Mm ⎥⎦ { 0.054 MHz It is useful to emphasize that the theoretical error in the= . (89) summary contribution b0.054 MHz 1,vert, SOPT (n = 0) + b2,vert, SOPT(n = 0) is determined by the factor Me Mm connected with the omitted terms of the used expansion. It can amount to10% All integrations over the coordinate x1, x3 are carried out of the total results of (91)–(92) which is a value near analytically and ﬁnal integration in k is performed numeri 0.010MHz. cally. Here we omit the intermediate expression before the The electron vertex corrections investigated in this integration in k because of its bulky form. The second part of section have the order a5 in the HFS interval. The summary the vertex contribution (ﬁgure 9(b)) with n = 0 can be value of all obtained contributions in SOPT is equal to –0.056 converted to the following form after several simpliﬁcations MHz (63Li) and (73Li). Summing this number with the cor which are discussed in section 2: rection (86) we obtain the value 40.900MHz. It differs sig a niﬁcantly by 1.059MHz from the result 41.959MHz which27 4M 3eMm b 3aM x was obtained in the approximation of vertex correction by thevert, SOPT (n=0) = nF p ò e m 2dx3 2 electron anomalous magnetic moment. ´ ò e2aMex3dx ò e3aMmx3 4dx4 ´ ò ¥ k sin(k x3  x4 )(G (e)M (k2)  1) 6. Summary and conclusion0 ´ x3  x2 ⎡ ⎣d (x4  x2)  ym0 (x4)ym0 ( )⎦ ⎤ ( ) In this work we have carried out analytical and numericalx2 . 90 x x computation of HFS intervals in muonic lithium ions3 4 (m e 6,7Li)+3 on the basis of the PT method suggested pre viously in the case of muonic helium in [6]. To increase the Two terms in square brackets in (90) give two different accuracy of the calculations we take into account several contributions. Then the integration in (90) over coordinates important corrections to HFS of the ground state of orders a5 x1 and x3 is carried out analytically. We obtain and a6 connected with the VP, nuclear structure, recoil effects (g2 = mek 6aMm ): and electron vertex corrections. The numerical values of ⎛ ⎞ different contributions to hyperﬁne structure are presented ina 3( = ) = n ⎜ me ⎟ Me table 1.b1,vert, SOPT n 0 F 162p2 ⎝aMm ⎠ Mm Let us list a number of basic features of the calculations. ´ ò ¥ 2 ⎣⎡ (e)( 2)  ⎦⎤ 1 1. Muonic lithium atoms have a complicated hyperﬁnek GM k 1 dk0 (g2  1)3 structure which appears due to the interaction of the1 ⎡ magnetic moments of the three particles. We investigate⎢ 4g 2 21(g  1) g1(3 + g ) small HFSs which can be important in experimental´ 1⎢ ( )  1 3 ( )2 studies.⎣ 1 + g22 1 + g22 2. In this problem there are two small parameters, the ﬁne g (g  ) ⎤ structure constant and the ratio of particle masses,+ 4 2 2 1 1 1 + 1 + 3g 2 1 ⎥ which can be used for the construction of the (g2 + g2)3 (g2 + g2)2 ⎥⎦ perturbation interactions. The basic contributions1 2 1 2 appear in orders a4, a5 and a6 taking into account of ={ 0.472 MHz , (91) ﬁrst and second order recoil effects.0.472 MHz 3. The VP effects are important in order to obtain theoretical splittings with high accuracy. They give rise 14 J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 195003 A P Martynenko and A A Ulybin to the modiﬁcation of the twoparticle interaction the difference between our work and [13], which lies in the M potential which provides the a5 e order corrections region 0.7 ¸ 1.5 MHz for separate HFSs, remains. We con Mm sider that this is related to terms of order nFM 2 Mm2e which are to the hyperﬁne structure. We take into account the VP not taken into account exactly in our work. We included this corrections in the ﬁrst and second orders of PT. term in the total theoretical error. Further improvement of the 4. The electron vertex corrections to the coefﬁcient b results obtained in this work can be achieved in the calcula should be considered with the exact account of the one tion of second order corrections in two small parameters α loop magnetic form factor of the electron because the and Me Mm. characteristic momentum incoming into the electron The estimate of theoretical uncertainty can be performed in vertex operator is of order of the electron mass. terms of the Fermi energy nF and small parameters α and the 5. Nuclear structure corrections to the ground state HFS ratio of the particle masses. In our opinion, there exist several are expressed in terms of electromagnetic form factors main sources for the theoretical errors. First of all, as we and the charge radius of two Li nuclei. mentioned above in section 2, the recoil corrections of order 6. Relativistic correction is obtained by means of the M 2 Mm2e are not taken into account exactly because of a repla expression from [7]: cement of the electron Greenʼs function by free one. Numeri ⎛ 3 1 ⎞ cally this contribution can give 0.88MHz. The second source ofDnrel = n 2 2F ⎝⎜1 + (Z1a)  (Z2a) ⎟⎠, (93) the error is related to contributions of order a2nF which appear2 3 both from QED amplitudes and in higher orders of PT. In the case of twoparticle bound states these corrections were calcu which gives contributions to both coefﬁcients b and c lated in [31, 35–37]. Considering that they should be studied (see table 1). more carefully for threeparticle bound states we included a correction a2nF » 1.92 MHz in the theoretical error. Another Using the total numerical values of coefﬁcients b and part of the theoretical error is determined by the twophoton c, presented in table 1, we ﬁnd the following HFSs for Dn m + threebody exchange amplitudes mentioned above. They are ofmuonic lithium ions: ( e 61 3Li) = 21572.160 MHz Dn m + Dn m + the ﬁfth order over α and contain the recoil parameterand 62 ( e 3Li) = 14152.560 MHz; 1( e 73Li) + (me ma )ln(me ma ), so that their possible numerical value can= 21733.056MHz and Dn (m e 72 3Li) = 13994.345MHz. be equal to ±0.22MHz. Thereby, the total theoretical uncer The calculation of hyperﬁne structure in threeparticle muonic tainty does not exceed ±2.13MHz. To obtain this estimate we atoms (muonic helium, ions of muonic lithium) was per add the above mentioned uncertainties in quadrature. formed in [12, 13] using a variational method. The second paper in [12] is devoted to muonic helium and the ﬁrst paper in [12] contains an estimate of HFSs in muonic lithium ion. Later it this estimate was corrected in [13]. So, we make our Acknowledgments comparison namely with the values obtained in [13]: Dn (m e 6Li)+1 3 = 21567.112 MHz and Dn (m e 6 +2 3Li) + We are grateful to A M Frolov for sending us paper [13] and= 14148.678 MHz; and Dn 71(m e 3Li) = 21729.22 MHz useful communications. The work is supported by the Russian Dn 72 (m e 3Li)+ = 13989.19MHz. Foundation for Basic Research (grant 140200173) and the An analysis of the separate contributions to the hyperﬁne Ministry of Education and Science of Russia under Compe structure coefﬁcients b and c in table 1 shows that relativistic titiveness Enhancement Program 20132020. and electron vertex corrections have large values. So, for example, the difference of our calculation from the results in [13] for the electron vertex corrections is a value of order 1 MHz and the relativistic corrections amount to 6MHz. The References recoil contribution from 2g exchange amplitudes has a similar value of ∼6MHz. 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