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Quasipotential equation for hydrogen isotopes. Muonic atoms. Ground state energy levels
Volume 93B, number 3
PHYSICS LETTERS
16 June 1980
QUASIPOTENTIAL EQUATION FOR HYDROGEN ISOTOPES.
MUONIC ATOMS. GROUND STATE ENERGY LEVELS D. BAKALOV
Laboratory of Theoretical Physics, Dubna, USSR Received 5 February 1980
The quasipotential for the electromagnetic interaction of two particles of spin ~1 or 1 with arbitrary electromagnetic interaction of two particles of spin g1 or 1 with arbitrary electromagnetic structure is constructed in the one-photon approximation. Todorov's quasipotential equation is applied to calculate the ground state energy levels of the muonic atoms p/~, d# and tu with accuracy 10-3 eV.
1. Introduction. Many interesting phenomena take place when negative muons are stopped in a mixture of hydrogen isotopes. The fast muons loose their energy in collisions with molecules and form muonic atoms in highly excited states that deexcitate to the ground state 1so through radiative transitions and ejection of Auger electrons [1]. All observed reactions (such as ortho-para transitions in/j-atoms, transition of the muon from the lighter to the heavier nucleus and formation of mesic molecules, or/J-capture by nuclei protons and/J-decay, as well as nuclear fusion in mesic molecules) start from this state on. A crucial point in the attempt of describing the kinetics of the ensemble of processes just referred is the need of knowing the energy levels of the mesic molecules and of the muonic atoms with an accuracy ~10 - 3 eV. When solving this problem one should keep in mind that relativistic and nuclear structure effects are expected to prove significant at this level of accuracy, because the muon is about 200 times heavier than the electron. In the present work Todorov's quasipotential approach [2] is applied to get a Schr6dinger-type equation for 1 1 1 the system of two charged particles of spins s 1, s 2 for (s 1, s2) = (5, ~), (1, 5) or (1, 1) with electromagnetic (e.m.) interaction, that accounts for their e.m. structure and reproduces the relativistic effects in the binding energy upto order c~4. Numerical results are obtained for the energy levels of the muonic atoms p/J, d/J and t/J ground state, and a local energy-independent operator is extracted from the quasipotential to be used in many-particle calculations as a correction term to the pairwise Coulomb potential, describing nuclear structure and O(a 4) relativistic effects (other than recoil) in the mesic molecules binding energy. 2. Quasipotential equation for a system of two particles o f spin ~ or 1 with e.m. interaction. The approximate quasipotential for the e.m. interaction of two particles (labeled by i = 1, 2) of masses mi, spins si, charges Zie (where e is the proton charge), magnetic dipole moment #i (in units eh/2mic ) and electric quadrupole moment Qi (fro-2) has been constructed following the general prescription [2] from the Born term T B of the perturbative expansion for the scattering amplitude of particles 1 and 2: T B 2,~a~2(pl, P2;ql, q2)=(Pl~llj(1)[qlr?l )
_gUU
(P2~21j~2)[q2:?2),
(P 1 - q 1)2 where (Pill [J(i)lqir~i)is the matrix element of the ith particle e.m. current operator between states of momenta Pi(qi) and space fixed Z-axis projection of spin ~i(~li), I~i[ ~ si, 1~/iI ~< si. The general relativistic-covariant expressions have been used for these matrix elements (for p2 = q2 = m2). For s i = 1 we have [3]: 265
Volume 93B, number 3
PHYSICS LETTERS
16 June 1980
4mi [yv,')'~]/t'2~(i)( k 2) ( # i -
( Pi ~i ]j(i)[ qi ~7i} = Zi e ff~ i (Pi) F~Fli)(k2)+~ L
1)1 u~i(qi) ,
k=Pl-ql
(Ur~i(qi) and u~i(Pi) being the Dirac spinors of the initial and final states), while for s i = 1 the result is [4] : o * (Pi~i[4i)lqi~Ti) = Zie(e~i(Pi))
o,
_
+(~-+~F~t)(k2)( tap
o
X (Pi + qi), + I~iF(20(k2)(gV°5~ - g 8•)k
(e~i(qi))o ,
kok°
k2
k = Pl - ql ,
e~i(pi) and e~i(qi) being the final (initial) state polarization vectors; the formfactors are normalized by: F~(.i)(0) = 1, k = 1,2, (3), i = 1,2. The two-particle quasipotential equation can be represented [2] in the form of a local Schr6dinger-type equation for the (2s 1 + 1) × (2s 2 + 1)-component wave function ~ , ~2(r, w): S1
S2
nl=_Sl ~2=_s2 - 2E(w) (At + b2(w))~ln16~2~2 + U~l~2,nln2(r' w) ~l~2(r, w) = 0 ,
(1)
with E(w) = (w 2 - m 2 - rn2)/2w, b2(w) = E2(w) - (mlrn2/w) 2, and with w = x/-(q 1 + q2) 2 being the two-particle total energy in the center-of-mass frame. In order to write down U(r, w) explicitly, some notations should be introduced first. The Fourier-transform of the particle formfactors:
fij(r)
=
/~ d3k F~l)(_k 2)F/(2)(_k2) 47r exp(ik" r) r a (2rr) 3 ~ ,
and the functions
]cii=fi/-rf'} _ _ _r3 . .= . ' ~./ = f0.+ ~1f i ; '., . . ~'/ -]if/r + 47rfi](0)~i(3)(r) , appear in Uinstead of the usual factors 1/r, 1/r 3 or 5(3)(r). The symbols: e i = ( - 1 ) 2si+1 and 8 i = 1 forsi = ~, and 0 otherwise, permit one to unify the expressions for U for all values of the spins under consideration (~ ~< s i <~ 1, i = 1, 2) in the following formula: U = U (0) + U (1) , U}I)-
U~I)<1)_
Z1Z2°e r
U (0) = Z1Z2a/r
(fll-
1)
,
U2(1)
(a=e2/4rr),
U (1) = U} l) + U (1) + U~1) + U4(1) ,
ml +m2 (Z1Z2~fll) 2 2mlm 2
2s 1 m2 (elJ~ll + gl --/51 J~21) + 2s 2 ml -6 +(2~1+1)ml Sl (2~2+1)-m22 ( e2"/~11+~/2 s2 2 f12)]
Z1Z2a[f.11
~
Z1Z2~ 2 Flai-6i ( 1 + m 3 - i ~ ) 3
2~lm2i~=lk .=
S--~l
mi ] 3-i,i
+
(ei m3-i mi
+~t
s i ] 711] ( L ' s i )
ZlZ2°~ SlS2mlm2
X { g [#l/a2f22 + 6 l~t2(.f12 - -~22) + ~ 2~1(-f21- f22) + 6162(711- f12 - J~21 + J~22)] ($1Ts2) + 6 [/'tl~2k2 + 81~/2(f12
k2) + ~2//l(kl-
J~22) + 8 162(fll - ¢12 - k l + f22)1 (SlS2)}
Z 1 Z i o e F ~ 6i - 1 2 Li= 1 ~ (J~ll - ttif3-i i + (m2Qi + lai- 1)f5-2i, ai-1)($iTsi)], l
266
(2)
Volmnc 93B, number 3
PHYSICS LETTERS
16 June 1980
where : ( a T b ) = 3(a" r ) ( b " r)/r 2 - ( a "
L = -ir X Vr,
b).
If U (1) is treated as a perturbation to the Coulomb potential U (°), the following interpretation o f the various terms in (2) looks natural: U} 1) describes the effects of particle finite size and causes a shift Aefs z of the Coulomb energy level eNR when solving the bound state problem;finite size effects modify all the other terms in (2) too. U~1)_ arises when squaring the Coulomb potential [2]; the correction AeSQ attached to it is always negative; U~ 1) describes a contact-type interaction and the corresponding AeCONT practically vanishes for all but s-states. U~1) gives the usual s p i n - o r b i t , s p i n - s p i n and tensor terms that determine the hyperfine splitting AeitFS. For particles of spin 1 a spin self-interaction appears too. Finally the dependence of the coefficients of eq. (1) on the energy w describes the recoil effect and gives rise to a correction ACREc. It is worth mentioning that for formfactors F ( k 2) that vanish fast enough when k 2 ~ ~ (which is the realistic case) U(r, w ) is singularity-free at r = 0. Of course, eq. (1) remains unchanged for the suitably (anti) symmetrized wave function in the case of identical particles. u ( l ) ( r ) is energy independent up to order c~2. This fact enables the use of U (1) as a correction of order ~o~2 to the zeroth order pairwise potential depending on internal variables only when constructing the many-particle interaction operator in Foldy Krajcik's approach [5]. The contribution of U (1) then will describe all the effects listed above but recoil; the latter is however taken into account in a proper way too [5,6]. 3. N u m e r i c a l results f o r m u o n i e atoms PlY, dtt and t~ ground state energy levels. In numerical calculations data for proton (i = 1) [7], deuteron (i = 2) [8] and triton (i = 3) [9] formfactors have been approximated by functions of the type:
F/~PP(-q 2) = ~
C~
i
1
(1
+
•
q2/(a~)2)t~
.
(3)
The numerical values of the parameters, obtained by means of A]exandrov's method [10] are listed in table 1; the muon was assumed structureless with magnetic moment 1.0012 [11].
(eh/2muc)
Table 1 Numerical values of the parameters Ok, i A ki and l~ of the approximants (3) for the e.m. formfactors for proton, deuteron and triton.
i
Proton formfactors F k, k = 1, 2
Triton formfactors Fk, k = 1, 2
electric (k = 1)
magnetic (k = 2)
electric (k = 1)
magnetic (k = 2)
1
2
1
2
1
ck
0.7891
0.2109
0.4368
0.5632
1.6448
Aik (MeV/c) i
851.7
2095
769.9
1194
701.1
847.8
522.2
695.1
2
3
3
3
5
5
3
3
i
Ik
2 -0.6448
1 1.5326
2 -0.5326
Deuteron formfactors Fk, k = 1,2, 3 electric (k = 1) i i
ck A ik
l ki
(MeV/c)
magnetic (k = 2)
1
2
0.3504
0.6625
3 -0.0129
quadrupole (k = 3)
1
2
1
2
0.3890
0.6110
0.5606
0.4394
373.5
706.6
1980
380.8
755.5
444.4
798.4
5
5
5
5
5
5
5
267
Volume 93B, number 3
PHYSICS LETTERS
16 June 1980
Table 2 Numerical values of the relativistic (of order c~4) and finite size corrections to the ground state energy levels of muonic atoms p#, d~z and t/~ with accuracy 10 -3 eV (with data about nuclei from refs. [9,11,13]). Muonic atom
p~
d~
t~
Nuclear mass (MeV/c 2) Coulomb binding energy eNR (eV) -+ 8eNR (eV) Recoil correction AeRE C (eV) Finite size correction AeFS z (eV) Coulomb squared correction AeSQ (eV) Contact term correction AeCONT (eV) Vacuum polarization [ 11 ] AeVp (eV) Total relativistic + finite size shift Hyperfine splitting AeHFS (eV)
938.2796 -2528.517 -+0.011 +0.098 +0.022 -0.268 +0.139 -1.896 -1.905 0.182
1875.628 -2663.226 -+0.011 +0.105 +0.215 -0.282 +0.140 -2.196 -2.018 0.049
2808.943 -2711.268 +-0.011 +0.107 +0.146 -0.287 +0.146 -2.212 -2.100 0.240
Table 2 contains the numerical values o f the corrections to the nonrelativistic C o u l o m b binding energy eNR of the l s o state o f p/a, d/a and t/a atoms, evaluated in first order of p e r t u r b a t i o n theory. T o g e t h e r with the correction A e v p for the v a c u u m polarization [ 12] these results provide the energy levels of the ortho- and para- 1se state o f the m u o n i c atoms with an accuracy ~ 1 0 - 2 eV. (Notice that the accuracy o f our present knowledge of/1 mass rnu = 2 0 6 . 7 6 8 5 9 ( 2 9 ) m e = 105.65946(24) MeV/c 2 and R y = meC2a2/2= 13.605804(36) eV does not allow for any better results [ 11 ] .) In conclusion the author w o u l d like to thank Professor L.I. P o n o m a r e v , Professor I.T. T o d o r o v and Dr. S.I. Vinitsky for helpful discussions.
References [1 ] S.S. Gerstein and L.I. Ponomarev, in: Muon physics, Vol. III, eds. V.W. Hughes and C.S. Wu (Academic Press, New York). [2] I.T. Todorov, Phys. Rev. D3 (1971) 2351; Nucl. Phys. B98 (1975) 447; in: Properties of fundamental interactions, Vol. 9C, ed. A. Zichichi (Editrice Compositori, Bologna, 1973) pp. 951-979. [3] J. Bjorken and S. Drell, Relativistic quantum mechanics (McGraw-Hill, New York). [4] V. Glaser and B. Jaksic, Nuovo Cimento 5 (1957) 1197. [5] L.L. Foldy and R.A. Krajcik, Phys. Rev. D12 (1975) 1700. [6] D.D. Bakalov, Zh. Eksp. Teor. Fiz., to be published. [7] S. Bylenkaya et al., Zh. Eksp. Teor. Fiz. 61 (1971) 2225. [8] V.N. Mouzafarov and V.E. Trotzky, Pis'ma Zh. Eksp. Teor. Fiz. 30 (1979) 78. [9] T. Griffi and L. Schiff, in: High energy physics, Vol. 1 (New York, 1967) p. 341. [10] L. Alexandrov, Communications JINR B1-5-9969 (in Russian). [11] Particle data group, Review of particle properties, Phys. Lett. 75B (1978) 1. [12] V.S. Melezhik and L.I. Ponomarev, Phys. Lett. 77B (1978) 217. [13] A.H. Wapstra and K. Bos, At. Data Nucl. Data Tables 19 (1977) 177; G.H. Fuller and V.W. Cohen, Nuclear moments, App. 1 to Nucl. Data Sheets (May 1965).
268
PHYSICS LETTERS
16 June 1980
QUASIPOTENTIAL EQUATION FOR HYDROGEN ISOTOPES.
MUONIC ATOMS. GROUND STATE ENERGY LEVELS D. BAKALOV
Laboratory of Theoretical Physics, Dubna, USSR Received 5 February 1980
The quasipotential for the electromagnetic interaction of two particles of spin ~1 or 1 with arbitrary electromagnetic interaction of two particles of spin g1 or 1 with arbitrary electromagnetic structure is constructed in the one-photon approximation. Todorov's quasipotential equation is applied to calculate the ground state energy levels of the muonic atoms p/~, d# and tu with accuracy 10-3 eV.
1. Introduction. Many interesting phenomena take place when negative muons are stopped in a mixture of hydrogen isotopes. The fast muons loose their energy in collisions with molecules and form muonic atoms in highly excited states that deexcitate to the ground state 1so through radiative transitions and ejection of Auger electrons [1]. All observed reactions (such as ortho-para transitions in/j-atoms, transition of the muon from the lighter to the heavier nucleus and formation of mesic molecules, or/J-capture by nuclei protons and/J-decay, as well as nuclear fusion in mesic molecules) start from this state on. A crucial point in the attempt of describing the kinetics of the ensemble of processes just referred is the need of knowing the energy levels of the mesic molecules and of the muonic atoms with an accuracy ~10 - 3 eV. When solving this problem one should keep in mind that relativistic and nuclear structure effects are expected to prove significant at this level of accuracy, because the muon is about 200 times heavier than the electron. In the present work Todorov's quasipotential approach [2] is applied to get a Schr6dinger-type equation for 1 1 1 the system of two charged particles of spins s 1, s 2 for (s 1, s2) = (5, ~), (1, 5) or (1, 1) with electromagnetic (e.m.) interaction, that accounts for their e.m. structure and reproduces the relativistic effects in the binding energy upto order c~4. Numerical results are obtained for the energy levels of the muonic atoms p/J, d/J and t/J ground state, and a local energy-independent operator is extracted from the quasipotential to be used in many-particle calculations as a correction term to the pairwise Coulomb potential, describing nuclear structure and O(a 4) relativistic effects (other than recoil) in the mesic molecules binding energy. 2. Quasipotential equation for a system of two particles o f spin ~ or 1 with e.m. interaction. The approximate quasipotential for the e.m. interaction of two particles (labeled by i = 1, 2) of masses mi, spins si, charges Zie (where e is the proton charge), magnetic dipole moment #i (in units eh/2mic ) and electric quadrupole moment Qi (fro-2) has been constructed following the general prescription [2] from the Born term T B of the perturbative expansion for the scattering amplitude of particles 1 and 2: T B 2,~a~2(pl, P2;ql, q2)=(Pl~llj(1)[qlr?l )
_gUU
(P2~21j~2)[q2:?2),
(P 1 - q 1)2 where (Pill [J(i)lqir~i)is the matrix element of the ith particle e.m. current operator between states of momenta Pi(qi) and space fixed Z-axis projection of spin ~i(~li), I~i[ ~ si, 1~/iI ~< si. The general relativistic-covariant expressions have been used for these matrix elements (for p2 = q2 = m2). For s i = 1 we have [3]: 265
Volume 93B, number 3
PHYSICS LETTERS
16 June 1980
4mi [yv,')'~]/t'2~(i)( k 2) ( # i -
( Pi ~i ]j(i)[ qi ~7i} = Zi e ff~ i (Pi) F~Fli)(k2)+~ L
1)1 u~i(qi) ,
k=Pl-ql
(Ur~i(qi) and u~i(Pi) being the Dirac spinors of the initial and final states), while for s i = 1 the result is [4] : o * (Pi~i[4i)lqi~Ti) = Zie(e~i(Pi))
o,
_
+(~-+~F~t)(k2)( tap
o
X (Pi + qi), + I~iF(20(k2)(gV°5~ - g 8•)k
(e~i(qi))o ,
kok°
k2
k = Pl - ql ,
e~i(pi) and e~i(qi) being the final (initial) state polarization vectors; the formfactors are normalized by: F~(.i)(0) = 1, k = 1,2, (3), i = 1,2. The two-particle quasipotential equation can be represented [2] in the form of a local Schr6dinger-type equation for the (2s 1 + 1) × (2s 2 + 1)-component wave function ~ , ~2(r, w): S1
S2
nl=_Sl ~2=_s2 - 2E(w) (At + b2(w))~ln16~2~2 + U~l~2,nln2(r' w) ~l~2(r, w) = 0 ,
(1)
with E(w) = (w 2 - m 2 - rn2)/2w, b2(w) = E2(w) - (mlrn2/w) 2, and with w = x/-(q 1 + q2) 2 being the two-particle total energy in the center-of-mass frame. In order to write down U(r, w) explicitly, some notations should be introduced first. The Fourier-transform of the particle formfactors:
fij(r)
=
/~ d3k F~l)(_k 2)F/(2)(_k2) 47r exp(ik" r) r a (2rr) 3 ~ ,
and the functions
]cii=fi/-rf'} _ _ _r3 . .= . ' ~./ = f0.+ ~1f i ; '., . . ~'/ -]if/r + 47rfi](0)~i(3)(r) , appear in Uinstead of the usual factors 1/r, 1/r 3 or 5(3)(r). The symbols: e i = ( - 1 ) 2si+1 and 8 i = 1 forsi = ~, and 0 otherwise, permit one to unify the expressions for U for all values of the spins under consideration (~ ~< s i <~ 1, i = 1, 2) in the following formula: U = U (0) + U (1) , U}I)-
U~I)<1)_
Z1Z2°e r
U (0) = Z1Z2a/r
(fll-
1)
,
U2(1)
(a=e2/4rr),
U (1) = U} l) + U (1) + U~1) + U4(1) ,
ml +m2 (Z1Z2~fll) 2 2mlm 2
2s 1 m2 (elJ~ll + gl --/51 J~21) + 2s 2 ml -6 +(2~1+1)ml Sl (2~2+1)-m22 ( e2"/~11+~/2 s2 2 f12)]
Z1Z2a[f.11
~
Z1Z2~ 2 Flai-6i ( 1 + m 3 - i ~ ) 3
2~lm2i~=lk .=
S--~l
mi ] 3-i,i
+
(ei m3-i mi
+~t
s i ] 711] ( L ' s i )
ZlZ2°~ SlS2mlm2
X { g [#l/a2f22 + 6 l~t2(.f12 - -~22) + ~ 2~1(-f21- f22) + 6162(711- f12 - J~21 + J~22)] ($1Ts2) + 6 [/'tl~2k2 + 81~/2(f12
k2) + ~2//l(kl-
J~22) + 8 162(fll - ¢12 - k l + f22)1 (SlS2)}
Z 1 Z i o e F ~ 6i - 1 2 Li= 1 ~ (J~ll - ttif3-i i + (m2Qi + lai- 1)f5-2i, ai-1)($iTsi)], l
266
(2)
Volmnc 93B, number 3
PHYSICS LETTERS
16 June 1980
where : ( a T b ) = 3(a" r ) ( b " r)/r 2 - ( a "
L = -ir X Vr,
b).
If U (1) is treated as a perturbation to the Coulomb potential U (°), the following interpretation o f the various terms in (2) looks natural: U} 1) describes the effects of particle finite size and causes a shift Aefs z of the Coulomb energy level eNR when solving the bound state problem;finite size effects modify all the other terms in (2) too. U~1)_ arises when squaring the Coulomb potential [2]; the correction AeSQ attached to it is always negative; U~ 1) describes a contact-type interaction and the corresponding AeCONT practically vanishes for all but s-states. U~1) gives the usual s p i n - o r b i t , s p i n - s p i n and tensor terms that determine the hyperfine splitting AeitFS. For particles of spin 1 a spin self-interaction appears too. Finally the dependence of the coefficients of eq. (1) on the energy w describes the recoil effect and gives rise to a correction ACREc. It is worth mentioning that for formfactors F ( k 2) that vanish fast enough when k 2 ~ ~ (which is the realistic case) U(r, w ) is singularity-free at r = 0. Of course, eq. (1) remains unchanged for the suitably (anti) symmetrized wave function in the case of identical particles. u ( l ) ( r ) is energy independent up to order c~2. This fact enables the use of U (1) as a correction of order ~o~2 to the zeroth order pairwise potential depending on internal variables only when constructing the many-particle interaction operator in Foldy Krajcik's approach [5]. The contribution of U (1) then will describe all the effects listed above but recoil; the latter is however taken into account in a proper way too [5,6]. 3. N u m e r i c a l results f o r m u o n i e atoms PlY, dtt and t~ ground state energy levels. In numerical calculations data for proton (i = 1) [7], deuteron (i = 2) [8] and triton (i = 3) [9] formfactors have been approximated by functions of the type:
F/~PP(-q 2) = ~
C~
i
1
(1
+
•
q2/(a~)2)t~
.
(3)
The numerical values of the parameters, obtained by means of A]exandrov's method [10] are listed in table 1; the muon was assumed structureless with magnetic moment 1.0012 [11].
(eh/2muc)
Table 1 Numerical values of the parameters Ok, i A ki and l~ of the approximants (3) for the e.m. formfactors for proton, deuteron and triton.
i
Proton formfactors F k, k = 1, 2
Triton formfactors Fk, k = 1, 2
electric (k = 1)
magnetic (k = 2)
electric (k = 1)
magnetic (k = 2)
1
2
1
2
1
ck
0.7891
0.2109
0.4368
0.5632
1.6448
Aik (MeV/c) i
851.7
2095
769.9
1194
701.1
847.8
522.2
695.1
2
3
3
3
5
5
3
3
i
Ik
2 -0.6448
1 1.5326
2 -0.5326
Deuteron formfactors Fk, k = 1,2, 3 electric (k = 1) i i
ck A ik
l ki
(MeV/c)
magnetic (k = 2)
1
2
0.3504
0.6625
3 -0.0129
quadrupole (k = 3)
1
2
1
2
0.3890
0.6110
0.5606
0.4394
373.5
706.6
1980
380.8
755.5
444.4
798.4
5
5
5
5
5
5
5
267
Volume 93B, number 3
PHYSICS LETTERS
16 June 1980
Table 2 Numerical values of the relativistic (of order c~4) and finite size corrections to the ground state energy levels of muonic atoms p#, d~z and t/~ with accuracy 10 -3 eV (with data about nuclei from refs. [9,11,13]). Muonic atom
p~
d~
t~
Nuclear mass (MeV/c 2) Coulomb binding energy eNR (eV) -+ 8eNR (eV) Recoil correction AeRE C (eV) Finite size correction AeFS z (eV) Coulomb squared correction AeSQ (eV) Contact term correction AeCONT (eV) Vacuum polarization [ 11 ] AeVp (eV) Total relativistic + finite size shift Hyperfine splitting AeHFS (eV)
938.2796 -2528.517 -+0.011 +0.098 +0.022 -0.268 +0.139 -1.896 -1.905 0.182
1875.628 -2663.226 -+0.011 +0.105 +0.215 -0.282 +0.140 -2.196 -2.018 0.049
2808.943 -2711.268 +-0.011 +0.107 +0.146 -0.287 +0.146 -2.212 -2.100 0.240
Table 2 contains the numerical values o f the corrections to the nonrelativistic C o u l o m b binding energy eNR of the l s o state o f p/a, d/a and t/a atoms, evaluated in first order of p e r t u r b a t i o n theory. T o g e t h e r with the correction A e v p for the v a c u u m polarization [ 12] these results provide the energy levels of the ortho- and para- 1se state o f the m u o n i c atoms with an accuracy ~ 1 0 - 2 eV. (Notice that the accuracy o f our present knowledge of/1 mass rnu = 2 0 6 . 7 6 8 5 9 ( 2 9 ) m e = 105.65946(24) MeV/c 2 and R y = meC2a2/2= 13.605804(36) eV does not allow for any better results [ 11 ] .) In conclusion the author w o u l d like to thank Professor L.I. P o n o m a r e v , Professor I.T. T o d o r o v and Dr. S.I. Vinitsky for helpful discussions.
References [1 ] S.S. Gerstein and L.I. Ponomarev, in: Muon physics, Vol. III, eds. V.W. Hughes and C.S. Wu (Academic Press, New York). [2] I.T. Todorov, Phys. Rev. D3 (1971) 2351; Nucl. Phys. B98 (1975) 447; in: Properties of fundamental interactions, Vol. 9C, ed. A. Zichichi (Editrice Compositori, Bologna, 1973) pp. 951-979. [3] J. Bjorken and S. Drell, Relativistic quantum mechanics (McGraw-Hill, New York). [4] V. Glaser and B. Jaksic, Nuovo Cimento 5 (1957) 1197. [5] L.L. Foldy and R.A. Krajcik, Phys. Rev. D12 (1975) 1700. [6] D.D. Bakalov, Zh. Eksp. Teor. Fiz., to be published. [7] S. Bylenkaya et al., Zh. Eksp. Teor. Fiz. 61 (1971) 2225. [8] V.N. Mouzafarov and V.E. Trotzky, Pis'ma Zh. Eksp. Teor. Fiz. 30 (1979) 78. [9] T. Griffi and L. Schiff, in: High energy physics, Vol. 1 (New York, 1967) p. 341. [10] L. Alexandrov, Communications JINR B1-5-9969 (in Russian). [11] Particle data group, Review of particle properties, Phys. Lett. 75B (1978) 1. [12] V.S. Melezhik and L.I. Ponomarev, Phys. Lett. 77B (1978) 217. [13] A.H. Wapstra and K. Bos, At. Data Nucl. Data Tables 19 (1977) 177; G.H. Fuller and V.W. Cohen, Nuclear moments, App. 1 to Nucl. Data Sheets (May 1965).
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