Relativistic Calculations of Pionic and Kaonic Atoms Hyperfine Structure Martino Trassinelli, Paul Indelicato

Relativistic calculations of pionic and kaonic atoms hyperfine structure Martino Trassinelli, Paul Indelicato

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Martino Trassinelli, Paul Indelicato. Relativistic calculations of pionic and kaonic atoms hyperfine structure. 2007. hal-00116718v2

HAL Id: hal-00116718 https://hal.archives-ouvertes.fr/hal-00116718v2 Preprint submitted on 20 Feb 2007

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. hal-00116718, version 2 - 20 Feb 2007 nldn eaiitccretoscnb on nyfor only found HFS be for predictions can spin- in theoretical corrections found Other relativistic be 16]. including can 15, structure [14, hyperfine Ref. atom calculations be- pionic Non-relativistic interaction the for nucleus. the magnetic the the to with of moment momentum orbital due consider- kaon or atoms term pion the kaonic tween perturbation and the pionic ing in structure perfine in are endeavour Similar [2]. atoms kaonic 13]. with X-ray 12, accuracy progress high 11, using [10, mass [9] addition- spectroscopy pion standards can the X-ray evaluate low-energy atoms to new and pionic and define light few to energy used Besides, of the be precision ally 8]. a measuring 7, with by level [6, ground 5] meV the 4, interaction of width [3, strong natural the energies and study low un- pionic to an at of allows with spectroscopy hydrogen measured The kaonic been have precision. [2] precedented atoms kaonic and sddctdt h ueia acltosfrsm pionic some atoms. for kaonic calculations and IV numerical the Section to potential. dedicated electromagnetic is expan- nuclear multipole the the of using will sion we term III perturbation Sec. the In obtain equation. perturbation Klein-Gordon the a for applying method correction energy first the culate and kaon 19], the for 13, proposed 11, [2]. been [10, measurement has ppm mass that few nitrogen mass of kaonic pion accuracy for the of an which measurement at recent nitrogen, a aiming pionic for for used splitting been calcu has energy we HFS particular, the auto- In including late effects. spin used relativistic nucleus be the the can matically of it value and arbitrary case an this for to restricted not is technique † ∗ lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic 2 nti ae epeettecluaino h hy- the of calculation the present we paper this In [1] pionic in energies transition years few last the In hsatcei raie sflos nSc Iw cal- we II Sec. In follows. as organized is article This aoaor ate Brossel, Kastler Laboratoire 1 2 ulu 1,1] otayt hs ehd,our methods, these to Contrary 18]. [17, nucleus eaiitccluain fpoi n ancaoshyper atoms kaonic and pionic of calculations Relativistic ASnmes 36.m 11.p 11.d 23.j 36. 32.30.Rj, 31.15.Md, 31.15.-p, measurem recent 03.65.Pm, a numbers: PACS for calculat required is electrodynamics correction quantum a the Such in systems. struc term hyperfine additional The an potential. electromagnetic nuclear the etrainmto a enapidt h li-odne b Klein-Gordon has the operator to perturbation applied The been corrections. has relativistic method perturbation A epeetterltvsi aclto ftehprn str hyperfine the of calculation relativistic the present We .INTRODUCTION I. 1 cl oml u´rer;CR;UiestePer tMar et Université Pierre Supérieure; CNRS; Normale Ecole eelcatfu cwroefrcug amtd,Germ Darmstadt, für Schwerionenforschung, Gesellschaft ´ atn Trassinelli Martino Dtd eray2,2007) 20, February (Dated: ,2, 1, - ∗ n alIndelicato Paul and iecoordinate time where asenergy mass r ouino q 1.I hscs,w a write: can we case, this In (1). Eq. of solution ary n h clrwvfnto Ψ wavefunction scalar the and n q 1 becomes: (1) Eq. and hr ¯ where equation: operator additional an troducing W V il ihacag qa to equal charge a with par- spin-0 electromag- ticle charged negatively The a between equation. interaction netic Klein-Gordon the by scribed eda etrain ehave: magnetic we nuclear perturbation, the a with as interaction field the consider we If atcewt mass a with particle ( potential Coulomb tral p a etknit con nrdcn h ula po- nuclear the introducing account into tential taken be can W ν m 1 h etraincorrection perturbation The h eaiitcdnmco pnespril sde- is particle spinless a of dynamic relativistic The ( → oteCuobpotential Coulomb the to 10.Gv 2 r c ( 1 c si eea o-ier ntecs facorrection a of case the In non-linear. general in is ) 2 r e obtained een c 2 1 = ) I ACLTO FTEENERGY THE OF CALCULATION II. p 2 [ Ψ h ueo incadkoi tm provide atoms kaonic and pionic of ture E E A ν W [ 0 0 stePac constant, Planck the is E n ftepo mass. pion the of ent ν − ( + o fteeeg rniino these of transition energy the of ion i ( x 0 cuei incadkoi atoms. kaonic and pionic in ucture ¯ he stettleeg ftesse smo the of (sum system the of energy total the is r nteK qainvatemnmlcoupling minimal the via equation KG the in = ) qA eV + = ) uto otk noacutthe account into take to quation mc Ψ 0 eV 2 ν ( A 0 r − 2 x 2] npriua,i h aeo cen- a of case the in particular, In [20]. 0 ( )] i c via x ( ( 1 n idn energy binding and c ( = 2 r 1 2 r exp( = ) 2 CORRECTION 2, )] ) ¯ + [ ∂ i utpl xaso of expansion multipole a 2 m † ct, 2 ¯ h∂ i h ¯ + e [ + 2 2 is: t any r V ∇ V h eCrePrs6 ai,France Paris, 6, Curie-Paris ie + .W osdrhr h station- the here consider We ). 0 0 2 ∂ 2 ( ( − ∇ i eV r n structure fine r − A , iE ) ) 2 V , V 0 0 m q − 0 0 i ( ( 1 0 ( ehave: we , c r x ,teK qainfra for equation KG the ), ( t/ 2 E r = )] m r eed ntespace- the on depends ) h eoiyo h light the of velocity the c )] W 1 2 + ) ¯ h 2 2 2 ) − − c ¯ + a eddcdin- deduced be can − 2 ntezrt order zeroth the in ϕ e E W h 0 e EV eE 0 2 ( n h nucleus the and 2 ϕ ). ∇ ( r A 0 r (2) ) 2 ( ) i ( r 1 r 0 = ) ( ) ϕ Ψ r A ) ( 0 i r ( ( 0 = ) x , r ) ) , . (3) (1) (4) (6) (5) . 2

The correction to the energy due to W can be calcu- Using the properties of the spherical tensor [26, 29], we lated perturbatively with some manipulation of Eqs. (3) can show that: and (4) [21, 22], or via a linearization of the KG equation −1 C11 ∇ r using the Feeshbach-Villars formalism [23, 24]. In both q = Lq, (14) cases we have · − √2 c2 W where Lq is the dimensionless angular momentum oper- E1 = h i . (7) ator in spherical coordinates. The perturbation operator nl 2 E(0) + eV0 can be written as a scalar product in spherical coordinate h i 1 of the operator T acting on the pion wavefunction, and where we define the nuclear operator M1: ∗ 3 ϕ (r)A(r,t)ϕ(r) d r eµ ¯h A = V . (8) W (r)= 0 r−3(L M1)= T1 M1 (15) h i ϕ∗(r)ϕ(r) d3r 1 R V 2π ◦ ◦ Equation (7) is valid forR any wavefunction normalization. with eµ ¯h T 1 = 0 r−3L . (16) q 2π q III. CALCULATION OF THE HYPERFINE STRUCTURE OPERATOR The expected value of the operator W1 can be evaluated applying the scalar product properties in spherical The expression for W (r) in the HFS case is derived coordinates [29, 30]: using the multipole development of the vector potential ′ ′ ′ ′ l+I+F n l IF m W nlIFm = ( 1) δ ′ δ ′ δ ′ A(r) in the Coulomb gauge [25, 26, 27]. We neglect here h F | 1| F i − F F mF mF II × the effect due to the the spatial distribution of the nuclear F Il′ n′l′ T 1 nl I M 1 I , (17) magnetic moment in the nucleus [28] (Bohr-Weisskopf 1 l I h k k ih k k i effect), while effect due to the charge distribution (Bohr- ab c Rosenthal effect) are included in the numerical results of where represents a Wigner 6-j symbol. The Sec. IV. d e f The hyperfine structure due to the magnetic dipole reduced operator n′l′ T 1 nl is calculated from the ma- ′ h′ ′ k 1 k i interaction is obtained by taking into account the first trix elements n l m T nlm by a particular choice of h | q | i magnetic multipole term. Using the Coulomb gauge we the quantum numbers m and q applying the Wigner- have [25, 26]: Eckart theorem: µ l−1 A r 0 √ −2C11 M1 ′ ′ 1 ( 1) ′ ′ 1 ( )= i 2r . (9) n l T nl = δl0 − n l 1 T nl1 = − 4π ◦ h k k i l 1 l h | 0 | i where the symbol “ ” indicates here the general scalar 1 0 1 ◦ M1 − product between tensor operators. operates only on eµ0¯h ′ ′ −3 C11 = δl0√l√l +1√2l +1 n l 1 r Lz nl1 = the nuclear part ImI and is the vector sperical 2π h | | i harmonic [26, 29]| actingi on the pion part nlm of the eµ0¯h | i ′ ′ −3 wavefunction. We can decompose the perturbation term = δl0δll √l√l +1√2l +1 n l r nl , (18) 2π h | | i W (r) as: ab c r r r where indicates the Wigner 3-j symbol. W ( )= W1( )+ W2( ), (10) d e f The nuclear operator can be related to the magnetic where moment on the nucleus by II M 1 II = µ µ [26, 27] h | 0 | i I N i i where µ is the nuclear dipole momentum in units of the W1(r)=+i¯he 2Ai(r)∂ + [∂i, A (r)] , (11) I nuclear magneton µN = e¯h/2mpc: is the linear part and µ µ I M 1 I = I N . (19) 2 i W2(r)= e A (r)Ai(r), (12) h k k i I 1 I − I 0 I is the quadratic part. − r We study first the operator W1. We note that Considering Eq. (16), the total expression for W1( ) i [∂i, A (r)] = ∇ A(r) = 0 since we are using the becomes: Coulomb gauge.− In· this case we have: ′ ′ ′ ′ n l IF mF W1 nlIFmF = i h | | i W1(r)=+2i¯heAi(r)∂ = 2i¯heA(r) ∇ = = δ ′ δ ′ δ ′ µ µ − · F F mF mF ll I N × √2 eµ ¯h F (F + 1) I(I + 1) l(l + 1)) eµ ¯h r−2(C11 ∇) M1. (13) 0 − − n′l r−3 nl . (20) − 0 2π · ◦ 2π 2I h | | i 3 which, as expected, is equal to zero for l = 0 (then I = Applying the Wigner-Eckart theorem we obtain F ). To find the final expression of the HFS energy shift, we ′ ′ 1 nlIF F V0 nlIFF = have to evaluate the contribution of the operator W2(r)= h | | i 2 i 1 −2 1 1 e A (r)Ai(r) in the W diagonal terms. Using Eq. (9), − nl r C nl I M I 2 we− have: h i r √2h k k ih k k i× ′ ′ ′ ′ lm ImI lIF F lmImI lIFF 2 eµ0 ′ ′ h | ih | i× nlIFm W nlIFm = +2 m ,mI ,m,mI h F | 2| F i 4π × X −2C11 M1 −2C11 M1 l 1 l I 1 I nlIFmF (r ) (r ) nlIFmF . ′ ′ h | ◦ · ◦ | i m 1 m m 1 mI − (21) " − − I − We are in presence of three independent scalar products: l 1 l I 1 I ′ ′ . (28) C11 m 1 m m +1 mI two scalar products between the tensor and the vec- − − − I # tor M1, and the scalar product between the vectorial op- 11 1 −2 1 erators C M . The “ ” scalar product in W2 can be The reduced matrix element nl r C nl can be de- ◦ · decomposed using the properties of the reduced matrix composed in a radial and angularh k part k i elements of a generic operator product XK, of rank K, between non-commutating tensor operators U k and V k nl r−2C1 nl = nl r−2 nl l C1 l . (29) of rank k. For our case, this scalar product corresponds h k k i h | | ih k k i to a tensor product with K = 0, and k = 1: Due to the symmetry properties, l C1 l is equal to zero for any l [29]. This result impliesh thatk thek i reduced matrix 0 1 1 −2 11 1 −2 11 1 X = U V = (r C M ) (r C M ), (22) elements of U 1 and V 1 are always equal to zero. As a · ◦ · ◦ 1 1 −2C11 M1 consequence, the diagonal elements Ai(r)A (r) = 0 for U = V = (r ). (23) h i i ◦ any wavefunction, i.e., W2 does not contribute to the We have [29]: HFS energy shift.

We can now write the final expression for the HFS 0 1 1 0 nlIF X nlIF = ′ energy correction: h || || i ′ F F F × XF nlIF U 1 nlIF ′ nlIF ′ V 1 nlIF . (24) 2 h || || ih || || i nlF µI µN eµ0¯hc E1 = nl 4π E nl V0(r) nl × V 1 and U 1 are scalar products between commuta- 0 − h | | i F (F + 1) I(I + 1) l(l + 1) tive tensor operators, and their reduced matrix element − − nl r−3 nl . (30) F ′ V F can be calculated applying again the Wigner- 2I h | | i hEckart|| || theoremi for the component q = 0: This formula is obtained by a perturbation approach nlIF ′F ′ V 1 nlIFF of the KG equation. For this reason, all the relativistic nlIF ′ V 1 nlIF = h | 0 | i. (25) h || || i F ′ 1 F effects are automatically included in Eq. (30). In the 2 F ′ 0 F non-relativistic limit c , (E0 V )/c m and we − find the usual expression→ ∞of the HFS− h fori the→ Schödinger where [31]: equation [32] q V 1 = U 1 = r−2(C11 M1) = r−2 ( 1)q C1M 1 . 0 0 ◦ 0 − √ q −q q 2 IV. NUMERICAL RESULTS AND BEHAVIORS X (26) To evaluate the matrix element nlIF ′F ′ V 1 nlIFF We present here some calculations for a selection of h | 0 | i we can explicitly decompose nlIFF as a function of pionic and kaonic atom transitions. Such calculations are | i the eigenfunctions nlm and ImI using the Clebsch- obtained solving numerically the Klein-Gordon equation | i | i Gordan coefficients. We have using the multi-configuration Dirac-Fock code developed by one of the author (P.I.) and J.-P. Desclaux [33, 34, ′ ′ 1 nlIF F V0 nlIFF = 35, 36] that has been modified to include spin-0 particles h | | i case, even in the presence of electrons [37]. The first part 1 ′ ′ ′ ′ − lm Im lIF F lmImI lIFF 2 I is dedicated to the 5 4 and 8 7 transitions in pionic r √2 ′ ′ h | ih | i× → → m ,mXI ,m,mI and kaonic nitrogen, respectively. In the second part we ′ −2 1 ′ 1 will study the dependence of the HFS splitting against nlm r C nlm Im M− ImI h | 1 | ih I | 1| i− the nuclear charge Z to observe the role of the relativistic nlm′ r−2C1 nlm Im′ M 1 Im . (27) h | −1| ih I | 1 | I i corrections. 4

TABLE I: Energy (in eV) contribution for the selected levels TABLE II: Hyperfine transition energies and transition rate in pionic nitrogen. The first error takes into account neglected in pionic nitrogen. next order QED corrections. The second is due to the accu- Transition F-F’ Trans. rate (s−1) Trans. E (eV) Shift (eV) racy of the pion mass (±2.5 ppm). 13 5f → 4d 4-3 4.57 × 10 4057.6876 -0.00606 5g-4f 5f-4d 3-2 3.16 × 1013 4057.6970 0.00341 Coulomb 4054.1180 4054.7189 3-3 2.98 × 1013 4057.6845 -0.00910 Finite size 0.0000 0.0000 2-1 2.13 × 1013 4057.7031 0.00946 Self Energy -0.0001 -0.0003 2-2 2.25 × 1013 4057.6948 0.00112 Vac. Pol. (Uehling) 1.2485 2.9470 2-3 0.01 × 1013 4057.6822 -0.01138 Vac. Pol. Wichman-Kroll -0.0007 -0.0010 5g → 4d 5-4 7.13 × 1013 4055.3779 -0.00304 Vac. Pol. Loop after Loop 0.0008 0.0038 4-3 5.47 × 1013 4055.3821 0.00113 Vac. Pol. Källén-Sabry 0.0116 0.0225 4-4 5.27 × 1013 4055.3762 -0.00482 Relativistic Recoil 0.0028 0.0028 3-2 4.17 × 1013 4055.3852 0.00420 HFS Shift -0.0008 -0.0023 3-3 0.36 × 1013 4055.3807 -0.00029 Total 4055.3801 4057.6914 3-4 0.01 × 1013 4055.3747 -0.00624 Error ±0.0011 ±0.0011 Error due to the pion mass ±0.010 ±0.010

Coulomb term in the Table includes the non-relativistic A. Calculation of the energy levels of pionic and recoil correction using the reduced mass on the KG equa- kaonic nitrogen tion. The pion and nucleus charge distribution contribution are also included. The pion charge distribution radius contribution is included following [37, 40]. For the The precise measurement of 5g 4f transition in pi- pion charge distribution radius we take r =0.672 0.08 onic nitrogen and the related QED→ predictions allow for π [41]. For the nuclei we take values from Ref. [42].± The the precise measurement of the pion mass [10, 11, 12, 13, leading QED corrections, vacuum polarization, contribu- 19]. In the same way, the transition 8k 7i in kaonic tion is calculated self-consistently, thus taking into ac- nitrogen can be used for a precise mass measurement→ of count the loop-after-loop contribution to all orders, at the kaon [2]. For these transitions, strong interaction ef- the Uehling approximation. This is obtained by including fects between meson and nucleus are negligible, and the the Uehling potential into the KG equation [35]. Other level energies are directly dependent to the reduced mass Higher-order vacuum polarization contribution are cal- of the atom. The nuclear spin of the nitrogen isotope culated as perturbation to the KG equations: Wichman- 14N is equal to one leading to the presence of several Kroll and Källén-Sabry [43, 44]. The self-energy is cal- HFS sublevels. The observed transition is a combination culated using the expression in Ref. [45] and it includes of several different transitions between these sublevels, the recoil correction. The Relativistic recoil term has causing a shift that has to be taken into account to ex- been evaluated adapting the formulas from Refs. [18, 46] tract the pion mass from the experimental values. Tran- (more details can be found in Ref. [13]). The calcula- sition probabilities between HFS sublevels can easily be tions presented here do not take into account second or- calculated using the non-relativistic formula [32, 38] (the der recoil effects (Fig. 1 top), or higher QED corrections role of the relativistic effects is here negligible), if one as vacuum polarization and self-energy mixed diagrams neglect the HFS contribution to the transition energy: (Fig. 1 botton). The contribution from these terms has been estimated using the formula for a spin- 1 particle A ′ ′ ′ = 2 nlIF →n l IF with a mass equal to the pion’s. For the 5 4 pionic ′ 2 → (2F + 1)(2F + 1) l′ F ′ I nitrogen transitions, vacuum polarization and self-energy A ′ ′ , (31) 2I +1 F l 1 nl→n l mixed diagrams contribute in the order of 1 meV for the diagram with the vacuum polarization loop in the nuclear where photon line [47] (Fig. 1 bottom-left), and 0.0006 meV for the diagram with the vacuum polarization loop inside the ′ 3 ′ 4(Enl En l−1) α l n l−1 2 ′ ′ self-energy loop [48] (Fig. 1 bottom-right). The second Anl→n l = − 2 4 2 (Rnl ) , 3m c ¯h (Zα) 2l +1 order recoil contributions are in the order of 0.04 meV (32) [49] (Fig. 1 top). The largest contribution comes from with the unevaluated diagram with the vacuum polarization ∞ ′ ′ 1 loop in the nuclear photon line [47]. n l ∗ ′ ′ 3 Rnl = 2 φnl(r)φn l (r)r dr. (33) Assuming a statistical population distribution of the a0 0 Z HFS sublevels, we can use Eq. (31) to calculate the a0 =¯h/(mcZα) is the Bohr radius and φnl are the non- mean value of the transitions using the results in Ta- relativistic wavefunctions. ble II. Comparing this calculation with the one without For these calculations, presented in Tables I and II, we the HFS, we obtain a value for the HFS shift. For tran- used the nitrogen nuclear mass value from Ref. [39]. The sitions 5g 4f and 5f 4d we obtain shifts of 0.8 → → 5

TABLE IV: Hyperﬁne transition energies and transition rate in kaonic nitrogen. Transition F-F’ Trans. rate (s−1) Trans. E (eV) Shift (eV) 8i → 7h 7-6 1.19 × 1013 2970.4169 -0.00216 6-5 1.00 × 1013 2970.4196 0.00050 6-6 0.98 × 1013 2970.4145 -0.00453 5-4 0.84 × 1013 2970.4217 0.00265 5-5 0.03 × 1013 2970.4175 -0.00154 5-6 0.00 × 1013 2970.4125 -0.00656 8k → 7i 8-7 1.54 × 1013 2969.6365 -0.00149 7-6 1.33 × 1013 2969.6383 0.00029 7-7 1.31 × 1013 2969.6347 -0.00326 6-5 1.15 × 1013 2969.6398 0.00178 6-6 0.03 × 1013 2969.6367 -0.00126 6-7 0.00 × 1013 2969.6332 -0.00480

FIG. 1: Diagrams relative to the unevaluated QED contri- act cancellation between the weighted excited sublevels butions: second order recoil correction (top), and vacuum energy shifts as seen from Eq. (31). polarization and self-energy mixed diagrams (bottom). Their 1 eﬀects are estimated using the available formulas for spin- 2 particles. B. General behavior of the hyperﬁne structure correction over Z

TABLE III: Energy (in eV) contribution for the selected levels in kaonic nitrogen. The first error takes into account ne- For the non-relativistic case, the HFS splitting normal- glected next order QED corrections. The second is due to the ized to the binding energy and to the nuclear magnetic accuracy of the kaon mass (±32 ppm). moment, depends linearly on Zα. Any deviation from this linear dependence in the Klein-Gordon HFS can be 8k-7i 8i-7h Coulomb 2968.4565 2968.5237 attributed only to relativistic effects. To study the behavior of the normalized HFS splitting Finite size 0.0000 0.0000 9p 9p Self Energy 0.0000 0.0000 (E E )/( 0µI ) for the relativistic case, we F =3/2 − F =3/2 E Vac. Pol. (Uehling) 1.1678 1.8769 calculated the HFS for a selected choice of pionic atoms Vac. Pol. Wichman-Kroll -0.0007 -0.0008 with a stable nucleus of spin 1/2. The orbital 9p has Vac. Pol. Loop after Loop 0.0007 0.0016 been chosen to minimize the effect of the finite nuclear Vac. Pol. Källén-Sabry 0.0111 0.0152 size and strong interaction shifts, particularly for high Relativistic Recoil 0.0025 0.0025 values of Z. The results are summarized in Table V. For HFS Shift -0.0006 -0.0008 these calculations we used the nuclear mass values from Total 2969.6374 2970.4182 Error 0.0005 0.0005 Ref. [39], the nuclear radii from Refs. [42, 50] and the Error due to the kaon mass 0.096 0.096 nuclear magnetic moments from Ref. [51]. For higher Z values a non-linear dependence on Zα ap- pears as we can see in Fig. 2. This non-linearity originates and 2.2 meV, respectively. These values correspond to a in the two different parts of Eq. (30): the non-trivial de- correction to the pion mass between 0.2 and 0.6 ppm. pendency on 0 in the denominator and the expectation value nl r−3 Enl . The transition energies for the 8 7 transitions in h | | i kaonic nitrogen are presented in Tables→ III and IV. As for the pionic nitrogen, the error contribution due to the QED correction not considered is dominated by the un- V. CONCLUSIONS evaluated diagram with the vacuum polarization loop in the nuclear photon line [47], the associated correction is We presented a relativistic calculation of the hyperfine estimated in the order of 0.5 meV. For 8k 7i and structure in pionic and kaonic atoms. The precise evalu- 8i 7h transitions we have a HFS shift of→ 0.6 and ation of the specific case of pionic and kaonic nitrogen is 0.8→ meV, respectively, which correspond to a correction particularly important for the new measurement of the of the kaon mass between 0.2 and 0.3 ppm. pion and kaon mass. The small error on the theoretical As a general note, we remark that if we assume a sta- predictions, of the order of 1 meV for the 5 4 transi- tistical distribution of the initial state sublevels population, corresponds to a systematic error of > 0→.2 ppm for tions, transitions nl n′s with a s orbital as final state the pion mass evaluation, considerably smaller∼ than the have an average HFS→ shift equal to zero due to an ex- error of previous theoretical predictions [52]. 6

The formalism presented in this article can be applied TABLE V: HFS separation of the F=1/2 and F=3/2 levels 1 for other eﬀect as the quadrupole nuclear moment which for the 9p orbital for pionic atoms with spin nucleus. 2 can not be negligible for mesonic atoms with high Z. In Element Z 9p energy (eV) HFS splitting (eV) this case, HFS due to the quadrupole moment can be H 1 -39.93816 0.0001 predicted using the next multipole in the development 3He 2 -174.8370 -0.0009 13 of the electric potential of the nucleus to evaluate the C 6 -1633.402 0.0060 correspondent perturbation operator. This application is 15N 7 -2226.813 -0.0039 19 particularly important for the calculation of the atomic F 9 -3689.435 0.0767 levels in heavy pionic ions, where relativistic and nucleus 31P 15 -10286.63 0.1544 57 deformation eﬀects can be taken into account at the same Fe 26 -31023.63 0.0643 77 time. Se 34 -53146.21 0.8293 89Y 39 -69987.00 -0.3143 107Ag 47 -101681.2 -0.4266 129Xe 54 -134178.0 -4.1295 183W 74 -250634.4 1.2629 202Pb 82 -306731.6 7.8662

4.5e-05

4e-05

3.5e-05

3e-05

2.5e-05

2e-05

Relative splitting 1.5e-05 Acknowledgement 1e-05

5e-06

0 0 10 20 30 40 50 60 70 80 90 Atomic number Z We thank B. Loiseau, T. Ericson, D. Gotta and L. Simons for interesting discussion about pionic atoms. 9p 9p − E0 FIG. 2: Value of relative splitting (EF =3/2 EF =3/2)/( µI ) One of the author (M.T.) was partially sponsored by for pionic atoms with diﬀerent values of Z. Nuclear masses, the Alexander von Humbouldt Foundation. Laboratoire radii and magnetic moments values have been obtained from Kastler Brossel is Unit´eMixte de Recherche du CNRS Refs. [39, 42, 50, 51]. n◦ 8552.

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