Access to the kaon radius with kaonic atoms
AAccess to the kaon radius with kaonic atoms
Niklas Michel and Natalia S. Oreshkina ∗ Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: October 23, 2020)We put forward a method for determination of the kaon radius from the spectra of kaonic atoms.We analyze the few lowest transitions and their sensitivity to the size of the kaon for ions in thenuclear charge range Z = 1 − , taking into account finite-nuclear-size, finite-kaon-size, recoiland leading-order quantum-electrodynamic effects. Additionally, the opportunities of extracting thekaon mass and nuclear radii are demonstrated by examining the sensitivity of the transition energiesin kaonic atoms. Introduction.
While kaons are no elementary parti-cles, they represent themselves a very promising systemto study, because they are the lightest meson with non-zero strangeness. Despite their short lifetime of the or-der of − s, negatively charged kaons can neverthe-less be captured by a nucleus and in this way form aso-called kaonic atom. Experimental studies of kaonichydrogen and kaonic helium were performed lately bythe DA Φ NE collaboration, testing kaon-nucleon stronginteraction [1–4]. The corresponding theory lies at theinterface between atomic, nuclear and particle physics,and was presented, e.g., by accurate QED prediction forthe energies of the circular transitions for several kaonicatoms reported in Ref. [5], by the strong contribution es-timated in Ref. [6, 7], and by the analysis of scatteringamplitudes of kaonic atoms in Ref. [8].It is generally known, that experimental measurementsin combination with theoretical predictions can unveilyet unknown physical properties or constants, or im-prove those which are extremely important to know withhigh precision for further fundamental research. How-ever, kaon mass and kaon radius values have not beenupdated since they were reported more than thirty yearsago. For kaon mass determination, exotic-atom x-rayspectroscopy has been used [6, 9], whereas for the kaonradius has been measured by the direct scattering ofkaons on electrons [10]. Another exotic systems, namelymuonic atoms, have been proved to be extremely sen-sitive to nuclear parameters, and therefore their studiesallowed to retrieve information about atomic nuclei (see,e.g., Refs. [11, 12]). Being twice as heavy as muons, kaonsin kaonic atoms feature even stronger dependence on nu-clear parameters, making possible their extraction fromspectroscopic data on exotic kaonic atoms.In the current manuscript, we consider the sponta-neous decay spectra of kaonic atoms for nuclear chargesin the range Z = 1 − in order to establish how thesesystems can be used for the determination of the kaonicmass, kaonic radius and nuclear radius. This opens ac-cess to the fundamental properties of kaons and giving anew path to probe essential properties of atomic nuclei. Klein-Gordon equation.
As a spinless particle, a kaonwith a mass m K is described by the stationary Klein-Gordon equation (in the natural system of units, (cid:126) = c = 1 ) as [13]: (cid:2) ( E − V ( r )) + ∆ − m K (cid:3) ϕ ( r ) = 0 . (1)In the case of a spherically-symmetric potential V ( r ) = V ( r ) , the angular variables can be separated from theradial ones as ϕ ( r ) = R l ( r ) /r × Y lm ( ϑ, ϕ ) , where l and m are the orbital quantum number and its projection,respectively. The angular part Y lm of the bound kaonwave function consists of spherical harmonics and there-fore exactly coincides with that of a non-relativistic elec-tron, described by the Schödinger equation. The radialpart, represented as R l ( r ) = φ l ( r ) + χ l ( r ) , also satisfiesthe Schrödinger-form set of equations: (cid:20) − D l m K + m K + V ( r ) (cid:21) φ l ( r ) + D l m K χ l ( r ) = Eφ l ( r ) , (2a) D l m K φ l ( r ) + (cid:20) D l m K − m K + V ( r ) (cid:21) χ l ( r ) = Eχ l ( r ) . (2b)The differential operator D l acts as: D l ( r ) = ∂ r − l ( l + 1) r . Assuming the nucleus to be point-like and infinitelyheavy, one can describe kaon-nucleus interaction with theCoulomb potential V = − αZ/r , where α = e / (4 π ) ≈ / is the fine-structure constant. Then, Eq. (2) canbe solved analytically, resulting in the energies: (cid:15) nl ( αZ ) = m K (cid:32) αZ ) n K − / µ (cid:33) − / , (3)where n K = n − l,µ = (cid:112) ( l + 1 / − ( αZ ) , and the wavefunctions: ψ nl ( r ) = N ρ µ +1 / e − ρ/ F (1 − n K , µ + 1 , ρ ) , (4) a r X i v : . [ phy s i c s . a t o m - ph ] O c t where ρ = 2 r (cid:113) − (cid:15) /m K , and N is the normalization constant: N = (1 − (cid:15) /m K ) / Γ(2 µ + 1) × (cid:18) n K − µ + 1)( n K − n K −
1) + 2 µ + 1] (cid:19) / . Eq. (3) contains a singularity in the denominator, andfor l = 0 it breaks at αZ = 1 / , or at Z ≈ . Thisindicates that the point-like-nucleus approximation is notvalid anymore, and one has to consider a more realisticnuclear model, including finite nuclear size effects. Finite-nuclear-size effect.
One of the simplest nuclearmodels is a homogeneously charged sphere, with the cor-responding charge density of the nucleus ρ ( r ) = 3 Ze πr θ ( r − r ) . (5)Here Z is the nuclear charge and r is the effectiveradius of the nucleus, associated with a root-mean-square (RMS) radius of the nucleus as r = (cid:114) (cid:104) r (cid:105) . (6)The interaction between electron and nucleus can betherefore described by the potential V sphere ( r ) = − Zα r (cid:18) − r r (cid:19) , while r ≤ r , − Zαr , while r > r . (7)With this potential, the Eq. (2) can be in principle solvedsemi-analytically in analogy to Ref. [14, 15] for electronsand muons, described by the Dirac equation. However,even for a muon, which is more than two times lighterthan a kaon, and therefore is located on a larger distancefrom the nucleus, the semi-analytical method in a firstorder of FNS correction turns out to be not sufficient [15,16]. Finite-kaon-size effect.
Additionally to the finite-nuclear-size effect, one can take into account the finite-kaon-size (FKS) effect. To estimate the order of magni-tude of FKS, we used a comparably simple two-sphereapproach to build a potential, presented in Ref. [17].Denoting the radius of the nucleus as R N , and the ra-dius of the kaon as R K , we assume that these two spheresinteract without deformation via electromagnetic forces.Then, denoting the radii ratio as λ = R K /R N , threedifferent regions for ρ = r/R N should be considered: (i) ≤ ρ ≤ − λ , the kaon is totally inside thenucleus,(ii) − λ ≤ ρ ≤ λ , the kaon and the nucleuspartly overlap, and(iii) ρ ≥ λ , the kaon is totally outside the nu-cleus.The corresponding potential V ( ρ ) is determined as: V ( r ) = − V ( C − ρ ) , (i), − V λ (cid:18) C ρ + (cid:80) k =0 C k +2 ρ k (cid:19) , (ii), − V /ρ, (iii). (8)Here V = − αZ/R N , and the coefficients in Eq. (8) aredetermined as: C = 3 / − λ / ,C = (1 − λ + 16 λ − λ ) / ,C = ( − λ + 10 λ − λ ) / ,C = (3 + 6 λ + 3 λ ) / ,C = ( − − λ − λ ) / ,C = (1 + λ ) / ,C = 0 ,C = − / . By calculating energies of a given state with ahomogeneously-charged sphere (7) or two-spheres (8) po-tential, one can evaluate the FKS effect: δ FKS = 1 − (cid:15) nl [ V ] (cid:15) nl [ V sphere ] . (9) Quantum-electrodynamic effects.
Another importantcontribution to the energies of kaonic atoms originatesfrom the quantum-electrodynamics (QED) corrections.In the first order in α , there are self-energy (SE) andvacuum polarization (VP) corrections. For hydrogen-likeelectronic ions these two corrections are of the same or-der of magnitude. However, even for muonic atoms it isalready not so: due to the large muon-electron mass ra-tio, the VP with a virtual electron-positron pair is a feworders of magnitude larger than the VP with a virtualmuon-antimuon pair or SE correction (see, e.g., [16, 18]).The same stands also for kaonic atoms, and therefore theleading QED correction can be described by the Uehlingpotential [19]: V Uehl ( r ) = − α α π (cid:90) ∞ d r (cid:48) πρ ( r (cid:48) ) (cid:90) ∞ d t (cid:18) t (cid:19) × √ t − t exp ( − m e | r − r (cid:48) | t ) − exp ( − m e ( r + r (cid:48) ) t )4 m e rt , (10)with m e being the mass of an electron. In our calcula-tion, this potential has also been included in the Klein-Gordon equation. Therefore, the calculated energies ac-count for the leading QED effect to all orders. Recoil.
Due to the large mass of a kaon compared toa proton, the recoil effect is also extremely important forvery light ions, but becomes negligible for middle andheavy ions. To evaluate it, we used a simple reduced-mass formalism [20], replacing the nuclear mass m N with m r = m N m K m N + m K . (11)Standard atomic weights [21] have been used in the cur-rent manuscript. Sensitivities.
Taking into account FNS, FKS, the lead-ing QED and recoil effects, we calculated a kaonic atomspectrum. To characterize how values of physical observ-able change depending on the parameters of the theoryused, one can introduce sensitivity coefficients: ∆ EE = K R ∆ R N R N + K m ∆ m K m K . (12)By varying different parameters of our calculations, wecan estimate the corresponding sensitivity factors, sim-ilarly as it was done in earlier works for other physicalconstants, e.g. in Ref. [22]. We use only the most generalsensitivity coefficients K R and K m in our current work,since the kaonic atoms spectra feature complicated non-linear dependence on R N and m K . For example, themass of the kaon is a scaling factor for all energies, how-ever, it is also should be included in the nuclear potentialvia scaling of the radius and reduced mass. Analogouslywith the nuclear radius: for simple atomic systems, likeelectronic H-like ions, one can describe FNS effect viasimple term ∆ E FNS ∝ ( αZ ) [23]. For kaonic atoms theFNS correction has much higher impact, and thereforeone should take into account also higher order terms (see,e.g. [24, 25]). As an outcome, the sensitivity coefficientsare rather ion- and transition- dependent and can varyin a sizable range. Strong shift.
The strong interaction effects in the spec-tra of kaonic atoms are also extremely important and cansignificantly change the binding and transition energies.Thus, for kaonic lead Pb , the binding energy of the s state is E s [ Pb ] = − MeV. The hadronic contri-bution of ∼ MeV desreaces it to only E s [ Pb ] = − MeV [5, 6]. Analogously, the strong shift would de-crease transition energies for all other kaonic atoms. Inthe following, we will focuse only on QED calculations,however, in the future, it should be indisputably takeninto account.
Results.
Using the above described method, we calcu-lated the spectra and transition energies for Z = 1 − .In order to work with the most general expressions, weassumed the nuclear radius to be R N = 1 . Z / , andfor the mass of the nucleus the standard atomic weight Ion Transition ∆ E , [keV] δ FKS , [%] K R K m He p → s d → p f → d g → f h → g p → s d → p f → d g → f h → g p → s d → p f → d g → f h → g p → s d → p f → d g → f h → g ∆ E , finite-kaon-size effect δ FKS ,and sensitivities to the nuclear radius K R and mass of a kaonfor K m for the first few circular transitions in kaonic heliumHe , titanium Ti , xenon Xe , and uranium U . The num-ber in square parenthesis indicates the power of 10. has been used. Such simple assumptions allowed us toanalyze the general trends for our observables, however,for high precision calculation aiming the access to nu-clear or particle parameters from an experiment, one hasto use tabulated nuclear data, e.g. [26] for root-mean-square radius. The transition energies, including of FNS,FKS, the leading QED and recoil effect for the circulartransitions from p → s up to h → g , are plotted asfunctions of nuclear charge Z = 1 − in Fig. 1. InTable I, the same energies, FKS correction δ FKS and thesensitivities to the nuclear radius K R and to the mass ofa kaon K m are listed for few kaonic atoms: helium He ,titanium Ti , xenon Xe and uranium U . Our valuefor d → p transition of 6.465 keV is in a perfect agree-ment with previously reported experimental and theoret-ical values [1]. Determination of kaon’s radius.
In Fig. 2 one can seethe relative FKS correction (in percent) to the transi-tion energy as a function of nuclear charge Z . Since thepotential in Eq. (8) is a function of R K /R N , and theradius of the kaon R K = 0 . fm is smaller than anynuclear radius, one could naively expect, that the FKSeffect would be the largest when the kaon-nucleus sizeratio is minimal, e.g. for hydrogen. However, as one cansee from the Fig. 2, this is not the case. The relative FKS Figure 1. Transition energies of the first few circular transi-tions for kaonic atoms in the range Z = 1 − , in keV. effect to the p → s transition grows with Z , reachingthe maximum value of 0.8% at Z ≈ [27], and then itstarts to decrease. Also, unlike the FNS effect, which isalways maximized for the s shell, and getting smallerand finally simply negligible for the higher atomic shells,we can observe quite a different trend for the FKS effect.All other transitions exhibit the same qualitative behav-ior with respect to the FKS effect as the p → s transi-tion, however with different positions and values of theirmaxima. Thus, for d → p transition the FKS effecthas its maximum of 0.57% at Z ≈ , and f → d themaximum of 0.46% can be reached at Z ≈ . Since thestrong shift decreases the transition energies, accountingpreviously neglected strong contributions would lead tothe further enhancement in the transitions’ sensitivity tothe FKS effect. Therefore, it opens various possibilitiesfor the determination of the size of the kaon based on thedifferent transitions of the kaonic atoms with a differentnuclear charge Z . Extraction of nuclear radii.
As one can see from Ta-ble I and from Fig. 3, for all ions the transitions to thelow-lying states are sensitive to the nuclear radius, andtherefore can be used for its extraction. However, sincein the ground state the energy of a kaon is also affectedby other interactions with the nucleus, which are not soeasy to quantify (strong and weak interactions, nuclear-polarization correction, etc. ), the transition p → s can be not very suitable for this procedure. Therefore,the next transition d → p , and for heavy ions even f → d , can provide the necessary information aboutnuclear radius. Extraction of the mass of a kaon.
Due to the com-plicated dependence of the energy on kaons mass viathe nuclear radius, the sensitivity coefficient K m differsfrom unity, especially for the lowest-lying transitions, seeFig. 4. However, even for heaviest element considered, it Figure 2. Finite kaon size effect δ FKS (in percent), estimatedwithin two-spheres model, for the transition energies of thefirst few circular transitions for kaonic atoms in the range Z = 1 − . -1.4-1.2-1-0.8-0.6-0.4-0.2 0 10 20 30 40 50 60 70 80 90 1002p-1s3d-2p4f-3d5g-4f6h-5g-1.4-1.2-1-0.8-0.6-0.4-0.2 0 10 20 30 40 50 60 70 80 90 100 Figure 3. Sensitivity coefficient K R to the radius of the nu-cleus, for the transition energies of the first few circular tran-sitions for kaonic atoms in the range Z = 1 − . is close to unity for the transitions starting with h → g ,and therefore the analysis of kaonic atom spectra can beused for the determination of its mass. The fact thatthe dependence holds for any nuclei can be used to en-large the statistics and choose the system with the mostsuitable parameters for an experiment. A similar proce-dure was already used before in Ref. [9], however, withcontinuous progress in both experimental technique andtheoretical calculations one can improve the existing ac-curacy. Further improvements.
So far, we only showed theprincipal idea of using kaonic atoms for the extraction ofparticle and nuclear parameters, considering the leadingsize and QED effects. However, for high-precision the- -0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 60 70 80 90 1002p-1s3d-2p4f-3d5g-4f6h-5g-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 60 70 80 90 100
Figure 4. Sensitivity coefficient K m to the kaon mass, forthe transition energies of the first few circular transitions forkaonic atoms in the range Z = 1 − . oretical predictions to be compared with experimentaldata, one has to take into account effects already calcu-lated in this manuscript and other effects, with higheraccuracy, similarly as it was done e. g. for muonic atomsin Ref. [12]. First of all, the strong and weak interac-tion contributions can be either taken from the previ-ously reported data [6, 7] or calculated to the requiredaccuracy with current state-of-the-art methods. Then,the FNS effect can be calculated with Fermi or even de-formed Fermi nuclear potential [16], or with more realis-tic predictions based on the Skyrme-type nuclear poten-tial [28], not forgetting about nuclear deformation cor-rection [29]. There is a room for an improvement inthe evaluation of FKS as well, from a simple two-spheremodel to a more sophisticated and realistic one. Thehigher-order QED effects, such as self-energy, Wichmann-Kroll, Källén-Sabry, muonic and hadronic Uehling po-tentials [5, 29, 30] should be also included. The electronscreening effects were shown to be negligible in the spec-tra of muonic atoms [16, 31], therefore we expect them tobe even smaller for kaonic atoms, which has to be nev-ertheless checked. The recoil effect should be includedwithin a rigorous relativistic approach [5, 32], for moreprecise values for light kaonic atoms. Finally, the effectsof nuclear polarization have to be taken into account, as itwas done, e.g., in Refs. [33, 34]. For all above mentionedatomic structure effects, the 1‰ or better precision canbe reached, which makes our suggestions quite realistic. Conclusions.
We considered kaonic atoms with thenuclear charge Z = 1 − . Taking into account finite-nuclear size, finite-kaon size, leading-order quantum-electrodynamic and recoil effects, we calculated transi-tion energies and sensitivity coefficients to the nuclearradius and mass of a kaon. We analyzed the finite-kaon-size effect, showing that the value of the kaon’s radius can be extracted with almost equal efficiency from few differ-ent transitions and for few different ions. Similarly, thedecay spectra of kaonic atoms can be used for the deter-mination of nuclear radii and the mass of the kaon. Thechoice of the most suitable system for any of these pur-poses should be made based on many important param-eters, such as the accuracy of theoretical predictions, thenatural linewidths, experimental accessibility of the par-ticular nuclei, and the possibility to carry high-precisionmeasurements. Concluding, the knowledge of atomicstructure of kaonic atoms would give access to the fun-damental nuclear and particle parameters, and thereforecould motivate new experiments and high-precision cal-culations. Acknowledgements.
N. M. acknowledges support bythe IMPRS. The Authors thank Catalina Oana Curceanufor drawing our attention to kaonic atoms, and VincentDebierre and Zoltán Harman for discussion and com-ments. ∗ Email: [email protected][1] M. Bazzi, G. Beer, L. Bombelli, A. Bragadireanu,M. Cargnelli, G. Corradi, C. Curceanu (Petrascu),A. d’Uffizi, C. Fiorini, T. Frizzi, F. Ghio, B. Giro-lami, C. Guaraldo, R. Hayano, M. Iliescu, T. Ishi-watari, M. Iwasaki, P. Kienle, P. L. Sandri, A. Lon-goni, V. Lucherini, J. Marton, S. Okada, D. Pietreanu,T. Ponta, A. Rizzo, A. Romero Vidal, A. Scordo, H. Shi,D. Sirghi, F. Sirghi, H. Tatsuno, A. Tudorache, V. Tudo-rache, O. V. Doce, E. Widmann, and J. Zmeskal, Phys.Lett. B , 310 (2009).[2] M. Bazzi, G. Beer, L. Bombelli, A. Bragadireanu,M. Cargnelli, G. Corradi, C. Curceanu (Petrascu),A. d’Uffizi, C. Fiorini, T. Frizzi, F. Ghio, B. Giro-lami, C. Guaraldo, R. Hayano, M. Iliescu, T. Ishi-watari, M. Iwasaki, P. Kienle, P. Levi Sandri, A. Lon-goni, V. Lucherini, J. Marton, S. Okada, D. Pietreanu,T. Ponta, A. Rizzo, A. Romero Vidal, A. Scordo, H. Shi,D. Sirghi, F. Sirghi, H. Tatsuno, A. Tudorache, V. Tudo-rache, O. Vazquez Doce, E. Widmann, and J. Zmeskal,Phys. Lett. B , 113 (2011).[3] M. Bazzi, G. Beer, L. Bombelli, A. Bragadireanu,M. Cargnelli, G. Corradi, C. Curceanu (Petrascu),A. d’Uffizi, C. Fiorini, T. Frizzi, F. Ghio, B. Giro-lami, C. Guaraldo, R. Hayano, M. Iliescu, T. Ishiwatari,M. Iwasaki, P. Kienle, P. Levi Sandri, A. Longoni,J. Marton, S. Okada, D. Pietreanu, T. Ponta, A. Rizzo,A. Romero Vidal, A. Scordo, H. Shi, D. Sirghi, F. Sirghi,H. Tatsuno, A. Tudorache, V. Tudorache, O. VazquezDoce, E. Widmann, B. Wünschek, and J. Zmeskal, Phys.Lett. B , 199 (2011).[4] M. Bazzi, G. Beer, L. Bombelli, A. Bragadireanu,M. Cargnelli, G. Corradi, C. Curceanu (Petrascu),A. d’Uffizi, C. Fiorini, T. Frizzi, F. Ghio, C. Guar-aldo, R. Hayano, M. Iliescu, T. Ishiwatari, M. Iwasaki,P. Kienle, P. Levi Sandri, A. Longoni, V. Lucherini,J. Marton, S. Okada, D. Pietreanu, T. Ponta, A. Rizzo,A. Romero Vidal, A. Scordo, H. Shi, D. Sirghi, F. Sirghi,
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