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 To cite this version: 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 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. 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. Relativistic calculations of pionic and kaonic atoms hyperfine structure Martino Trassinelli1,2, ∗ and Paul Indelicato2, † 1Gesellschaft für Schwerionenforschung, Darmstadt, Germany 2Laboratoire Kastler Brossel, Ecole´ Normale Supérieure; CNRS; UniversitéPierre et Marie Curie-Paris 6, Paris, France (Dated: February 20, 2007) We present the relativistic calculation of the hyperfine structure in pionic and kaonic atoms. A perturbation method has been applied to the Klein-Gordon equation to take into account the relativistic corrections. The perturbation operator has been obtained via a multipole expansion of the nuclear electromagnetic potential. The hyperfine structure of pionic and kaonic atoms provide an additional term in the quantum electrodynamics calculation of the energy transition of these systems. Such a correction is required for a recent measurement of the pion mass. PACS numbers: 03.65.Pm, 31.15.-p, 31.15.Md, 32.30.Rj, 36.10.Gv I. INTRODUCTION II. CALCULATION OF THE ENERGY CORRECTION In the last few years transition energies in pionic [1] The relativistic dynamic of a spinless particle is de- and kaonic atoms [2] have been measured with an un- scribed by the Klein-Gordon equation. The electromag- precedented precision. The spectroscopy of pionic and netic interaction between a negatively charged spin-0 par- kaonic hydrogen allows to study the strong interaction ticle with a charge equal to q = e and the nucleus at low energies [3, 4, 5] by measuring the energy and can be taken into account introducing− the nuclear po- natural width of the ground level with a precision of few tential A in the KG equation via the minimal coupling meV [6, 7, 8]. Besides, light pionic atoms can addition- ν p p qA [20]. In particular, in the case of a cen- ally be used to define new low-energy X-ray standards [9] ν ν ν tral→ Coulomb− potential (V (r), 0), the KG equation for a and to evaluate the pion mass using high accuracy X-ray 0 particle with a mass m is: spectroscopy [10, 11, 12, 13]. Similar endeavour are in progress with kaonic atoms [2]. 1 m2c2Ψ (x)= [i¯h∂ + eV (r)]2 +¯h2 2 Ψ (x), (1) 0 c2 t 0 ∇ 0 In this paper we present the calculation of the hyperfine structure in pionic and kaonic atoms consider- where ¯h is the Planck constant, c the velocity of the light ing the perturbation term due to the interaction be- and the scalar wavefunction Ψ0(x) depends on the space- tween the pion or kaon orbital moment with the magnetic time coordinate x = (ct, r). We consider here the station- momentum of the nucleus. Non-relativistic calculations ary solution of Eq. (1). In this case, we can write: for the pionic atom hyperfine structure can be found in Ref. [14, 15, 16]. Other theoretical predictions for HFS Ψ0(x) = exp( iE0t/¯h) ϕ0(r) (2) including relativistic corrections can be found only for − 1 and Eq. (1) becomes: spin- 2 nucleus [17, 18]. Contrary to these methods, our technique is not restricted to this case and it can be used 1 [E + eV (r)]2 +¯h2 2 m2c2 ϕ (r)=0, (3) for an arbitrary value of the nucleus spin including auto- c2 0 0 ∇ − 0 matically the relativistic effects. In particular, we calcu- late the HFS energy splitting for pionic nitrogen, which where E0 is the total energy of the system (sum of the 2 has been used for a recent measurement of the pion mass mass energy mc and binding energy 0). E aiming at an accuracy of few ppm [10, 11, 13, 19], and The perturbation correction E1 can be deduced in- for kaonic nitrogen that has been proposed for the kaon troducing an additional operator W in the zeroth order mass measurement [2]. equation: This article is organized as follows. In Sec. II we cal- 1 [E + eV (r)]2 +¯h2 2 m2c2 W (r) ϕ(r)=0. culate the first energy correction applying a perturbation c2 0 ∇ − − method for the Klein-Gordon equation. In Sec. III we will (4) r hal-00116718, version 2 - 20 Feb 2007 obtain the perturbation term using the multipole expan- W ( ) is in general non-linear. In the case of a correction sion of the nuclear electromagnetic potential. Section IV V1 to the Coulomb potential V0, we have: is dedicated to the numerical calculations for some pionic 1 and kaonic atoms. W (r)= 2e2V (r)V (r)+2eE V (r) (5) −c2 0 1 1 If we consider the interaction with the nuclear magnetic field as a perturbation, we have: ∗Electronic address: [email protected] †Electronic address: [email protected] W (r)= i¯he 2A (r)∂i + [∂ , Ai(r)] e2Ai(r)A (r). (6) i i − i 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 evalu- ated 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.

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