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On the radiative corrections α2 ln α to the positronium decay rate
Volume 246, number 3,4
PHYSICS LETTERS B
On the radiative corrections
OL2
30 August 1990
In a to the positronium decay rate
I.B. K h r i p l o v i c h a n d A.S. Y e l k h o v s k y
Institute of Nuclear Physics, SU-630 090 Novosibirsk, USSR Received 8 May 1990
The radiative corrections ~ a 2In c~to the positronium decay rate are calculated in the Breit approximation which is shown to be quite adequate for the problem. For orthopositronium the result coincides with the previous one, for parapositronium it differs from the old results.
1. The problem of radiative corrections to the posi t r o n i u m decay rate is quite serious due to the strong disagreement between the theoretical value for orthopositronium (o-Ps) that includes the corrections of the order a and a 2 In a [ 1-4]
F?(~']r = m°~ 6 2 ( ~
9 ) ( l - l O'2 8 ~~z -13 c~21n
= 7.03830 ~ts- ~ ,
') ( 1)
and the recent experimental value [ 5 ] o-es = 7 . 0 5 1 6 ( 1 3 ) las 1. Fexp
(2)
The measurements of the p-Ps decay rate will also become sensitive enough in the near future to feel the corrections ~ a 2 In a. In the present paper we rederive in a way, which looks (at least to us) simple and transparent, the corrections ~ a 2 In a to the ortho- and parapositronium (p-Ps) decay rates. In the case of o-Ps our result coincides with the known one [3] [see eq. ( 1 ) ] . So, unfortunately, we cannot say anything substantial about the m e n t i o n e d disagreement between ( 1 ) and (2), although some remarks, hopefully pertinent to the problem, will be made below. On the other hand, our corresponding result for the p-Ps decay rate is new in the sense that it differs from the previous ones [ 3,6 ]. 2. As distinct from the previous calculations [ 2 4,6], our approach is not based on any fully relativistic two-body wave equation. It has been noted al520
ready [ 2 ], that corrections non-analytic in a arise due to the range of m o m e n t a between m and m a . Our observation is that the logarithmic nature of the correction allow us to restrict to the range m a < p < < m, where we can treat relativistic effects as perturbation. Hence we can consider the corrections ~ a 2 In a as such corrections to the wave function of positronium, that logarithmically diverge at small distances and are cut off at r ~ 1 / m where the a n n i h i l a t i o n occurs. These effects can be described as ( v / c ) 2 corrections. The theoretical status of the corresponding Breit h a m i l t o n i a n (BH) is absolutely safe. The part of the BH that corresponds to the relativistic corrections to the dispersion law of the particles and to their Coulomb interaction (see e.g. ref. [ 7 ] ), p4 l/'l-
~z~
4m 3 + ~ 5 6 ( r )
,
(3)
can be easily transformed to og2
Vt ~ - 4mr2
oL
2m2r i Or .
(4)
In (4) we retain only those terms which are sufficiently singular at r-~0 to influence the behavior of the wave function at small distances. Let us note that 0rRnj:o Ir~o - - a1 RnJ=° r~o
(5)
where Rnj=o is the radial wave function of the Coulomb state with the principal q u a n t u m n u m b e r n and vanishing orbital angular m o m e n t u m l. The Bohr
0370-2693/90/$ 03.50 © 1990 - Elsevier SciencePublishers B.V. ( North-Holland )
Volume 246, number 3,4
PHYSICS LETTERS B
radius a in positronium is a = 2/mot, so that with our accuracy Vl (r) vanishes. The conclusion that this part of relativistic effects does not work in the ot2 In ot corrections to the Ps decay rate, was made already in ref. [31. The next spin-independent term in the BH, ot
2
30 August 1990
action, and from the one-quantum annihilation contribution which does not vanish in the triplet case only. We shall solve the corresponding wave equation
( - m1 A - a - Er o + A ~ a ~56(r) ) A=~s(s+l)-2=-2,
s=0,
= 38,
s=l,
q/(r)=0,
(12)
describes the magnetic electron-positron interaction due to the orbital motion. At l = 0 it can be easily transformed to
by iterations. In its turn in the inhomogeneous equation
172=
( A + mot
~ ot U r ( P 2 + r1 0 r ) .
(7)
m4ot2 )
r
Using the same substitution [see (5) ], Or-+ - ½mot, and retaining only the terms singular as r-2, we get Ol 2
V2--* 2mr2 .
(8)
Now one can show in a straightforward way that the s-wave radial function, instead of being constant at r-+0, behaves as / . . ' X --Ct 2 / 2
qJ[ . . . . ~ / a )
~ 1 - ½ot21n(motr) "
(9)
The corresponding relative correction to the particle density at distances r ~ 1/m, where the annihilation takes place, and therefore to the decay rate itself is ot21n 1 .
(10)
ot
The spin-orbital part of the BH is irrelevant to our problem since it does not work at all in s-states. The part of the BH that describes the tensor spin-spin interaction does not contribute to the decay rate to the accuracy considered. Indeed, applying it to the s-state, this interaction either annihilates it (in the singlet case) or transforms it (in the triplet one) into the dstate with the same total angular m o m e n t u m . But the annihilation from a d-state is strongly hampered. So, we are left with the contact spin-spin interaction V3=~5~zot [] s(s+
l ) - 2 ]c~(r) .
aot q/l(r)=A--6(r)~o(O),
(13)
m
we omit the term - ¼m2ot 2, regular at r--,0, and treat mot~r, which is less singular than A, as a perturbation. Simultaneously this procedure is also an expansion in ot. The solution of the zeroth order in ot, ot
~?> = - A a-mmr~'o(0), at distances r ~ 1 / m of interest to us leads evidently to the correction ~ ot to the decay rate. The calculation of the corresponding numerical factor is beyond our accuracy. The first-order solution, ~u} ') =A" l a21n(mar) ~#o(0) ,
(14)
gives evidently at r ~ l / m the relative correction A. ~ot 2 In ot to ~o (r) Ir- l/m and the correction -- A - o t 2 1 n l = o t 2 1 n l X1 , 2 ot ot
X(-4),
s=O, s=l,
(15)
to I ~o(r) I~-l/m and to the decay rate. The total correction to the decay rate includes both (10) and ( 15 ) and thus constitutes otZln 1 X 2 ,
s=0,
19l
x(-l),
s=l.
(16)
(11)
It originates from both the magnetic spin-spin inter-
3. One may feel dissatisfied with the above treatment of the spin-spin interaction ( 1 1 ). Have we not 521
Volume 246, number 3,4
PHYSICS LETTERS B
buried some contribution ~ a 2 In a in the linearly divergent, at r-~0, term ~,}o)? To take away the suspicion we shall calculate the discussed contribution o f 1/3 in a straightforward way. We shall follow a line o f reasoning close to that ofrefs. [8,9]. Let us consider the diagram o f fig. 1. Its left vertex corresponds to the interaction (11); in the m o m e n t u m representation this a m p l i t u d e is -Ano~/m 2. The right vertex is the annihilation one. In the m o m e n t u m representation it is, to our accuracy, a constant, correction to which we are looking for. The C o u l o m b attraction (the dashed vertical line) supplements the imaginary part o f the corresponding one-loop diagram by the well-known C o u l o m b factor na/2v where 2v is the relative velocity of the particles. So, the imaginary part o f the diagram becomes constant: I m M = - A ~ nO/ 1@~ 4m 2 7~OL - - lTroL2A . 2v
~ /4
A o~21n 1 - ~ a (18)
is in fact the correction to the annihilation a m p l i t u d e we are looking for. The correction to the decay rate is evidently twice as large and coincides with ( 15 ). 4. Our result for o-Ps coincides with that found earlier. As for the disagreement with earlier results for p-Ps, its origin can be easily elucidated in the case of ref. [3]. The prescription o f the latter p a p e r is stated there explicitly: " F o r p a r a p o s i t r o n i u m , the lifetime calculation is the same as for orthopositronium except that the single-photon annihilation kernels are exactly zero". Indeed, under such a prescription we would get for p-Ps constant A the value 3a 2 = ~ instead of - 2, and for the factor at the relative
Fig. 1. Diagrammatic representation of the 0:2In 0: contribution due to the spin-spin contact interaction.
522
correction a2 In a to the decay rate the value 1 -- ] = 3, 2 presented in ref. [ 3 ]. Our value 2 for this factor arises at the transition from o-Ps to p-Ps, if one not only omits the annihilation kernel, but also modifies correspondingly the contact magnetic interaction. Concerning the strong disagreement between the theoretical ( 1 ) and experimental ( 2 ) results for oPs, it m a y be due, to our opinion, to a large and still uncalculated numerical coefficient at ( a / n ) 2 correction to the decay rate. Some hint to it can be found in the large, ~ 10, factor at the a / n correction. By the way, some large (but far from sufficient) contribution to the ( a / n ) 2 correction can be found directly from this number. Indeed, it arises from the - 5a/n correction to the annihilation amplitude which, being squared, gives also a 25 ( a / n ) 2 correction to the decay rate.
(17)
The dispersion integral 1 i dEIm2~/~ ReM=no E - E o I~o. . . . .
30 August 1990
Our attention to the p r o b l e m has been attracted by A. Rich. G r e a t stimulating interest has been shown together with him by A. A1 R a m a d a n , R. Conti and D. Gidley. In particular, they informed us o f the plans to measure precisely the p-Ps decay rate. We also had useful theoretical discussions with S. Brodsky, V.L. Chernyak, M.I. Eides, V.S. F a d i n and Yu. Tomozawa. To all of them we owe our sincere gratitude. One o f us (I.Kh.) highly appreciates the kind hospitality extended to him at the Theoretical Physics Institute, University of Minnesota, where part of this work was done.
References [ 1] A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696. [2] W.E. Caswell, G.P. Lepage and J.R. Sapirstein, Phys. Rev. Len. 38 (1977) 488. [3 ] W.E. Caswell and G.P. Lepage, Phys. Rev. A 20 (1979) 36. [4] G.S. Adkins, Ann. Phys. 146 (1983) 78. [5] C.I. Westbrook, D.W. Gidley, R.S. Conti and A. Rich, Phys. Rev. Len. 58 (1978) 1328. [6] Yu. Tomozawa, Ann. Phys. 128 (1980) 463. [ 7 ] V.B. Berestetsky, E.M. Lifshitz and L.P. Pitaevsky, Quantum electrodynamics (Nauka, Moscow, 1980). [8] R. Barbieri, P. Christillin and E. Remiddi, Phys. Rev. A 8 (1973) 2266. [9] A. Owen, Phys. Rev. Lett. 30 (1973) 887.
PHYSICS LETTERS B
On the radiative corrections
OL2
30 August 1990
In a to the positronium decay rate
I.B. K h r i p l o v i c h a n d A.S. Y e l k h o v s k y
Institute of Nuclear Physics, SU-630 090 Novosibirsk, USSR Received 8 May 1990
The radiative corrections ~ a 2In c~to the positronium decay rate are calculated in the Breit approximation which is shown to be quite adequate for the problem. For orthopositronium the result coincides with the previous one, for parapositronium it differs from the old results.
1. The problem of radiative corrections to the posi t r o n i u m decay rate is quite serious due to the strong disagreement between the theoretical value for orthopositronium (o-Ps) that includes the corrections of the order a and a 2 In a [ 1-4]
F?(~']r = m°~ 6 2 ( ~
9 ) ( l - l O'2 8 ~~z -13 c~21n
= 7.03830 ~ts- ~ ,
') ( 1)
and the recent experimental value [ 5 ] o-es = 7 . 0 5 1 6 ( 1 3 ) las 1. Fexp
(2)
The measurements of the p-Ps decay rate will also become sensitive enough in the near future to feel the corrections ~ a 2 In a. In the present paper we rederive in a way, which looks (at least to us) simple and transparent, the corrections ~ a 2 In a to the ortho- and parapositronium (p-Ps) decay rates. In the case of o-Ps our result coincides with the known one [3] [see eq. ( 1 ) ] . So, unfortunately, we cannot say anything substantial about the m e n t i o n e d disagreement between ( 1 ) and (2), although some remarks, hopefully pertinent to the problem, will be made below. On the other hand, our corresponding result for the p-Ps decay rate is new in the sense that it differs from the previous ones [ 3,6 ]. 2. As distinct from the previous calculations [ 2 4,6], our approach is not based on any fully relativistic two-body wave equation. It has been noted al520
ready [ 2 ], that corrections non-analytic in a arise due to the range of m o m e n t a between m and m a . Our observation is that the logarithmic nature of the correction allow us to restrict to the range m a < p < < m, where we can treat relativistic effects as perturbation. Hence we can consider the corrections ~ a 2 In a as such corrections to the wave function of positronium, that logarithmically diverge at small distances and are cut off at r ~ 1 / m where the a n n i h i l a t i o n occurs. These effects can be described as ( v / c ) 2 corrections. The theoretical status of the corresponding Breit h a m i l t o n i a n (BH) is absolutely safe. The part of the BH that corresponds to the relativistic corrections to the dispersion law of the particles and to their Coulomb interaction (see e.g. ref. [ 7 ] ), p4 l/'l-
~z~
4m 3 + ~ 5 6 ( r )
,
(3)
can be easily transformed to og2
Vt ~ - 4mr2
oL
2m2r i Or .
(4)
In (4) we retain only those terms which are sufficiently singular at r-~0 to influence the behavior of the wave function at small distances. Let us note that 0rRnj:o Ir~o - - a1 RnJ=° r~o
(5)
where Rnj=o is the radial wave function of the Coulomb state with the principal q u a n t u m n u m b e r n and vanishing orbital angular m o m e n t u m l. The Bohr
0370-2693/90/$ 03.50 © 1990 - Elsevier SciencePublishers B.V. ( North-Holland )
Volume 246, number 3,4
PHYSICS LETTERS B
radius a in positronium is a = 2/mot, so that with our accuracy Vl (r) vanishes. The conclusion that this part of relativistic effects does not work in the ot2 In ot corrections to the Ps decay rate, was made already in ref. [31. The next spin-independent term in the BH, ot
2
30 August 1990
action, and from the one-quantum annihilation contribution which does not vanish in the triplet case only. We shall solve the corresponding wave equation
( - m1 A - a - Er o + A ~ a ~56(r) ) A=~s(s+l)-2=-2,
s=0,
= 38,
s=l,
q/(r)=0,
(12)
describes the magnetic electron-positron interaction due to the orbital motion. At l = 0 it can be easily transformed to
by iterations. In its turn in the inhomogeneous equation
172=
( A + mot
~ ot U r ( P 2 + r1 0 r ) .
(7)
m4ot2 )
r
Using the same substitution [see (5) ], Or-+ - ½mot, and retaining only the terms singular as r-2, we get Ol 2
V2--* 2mr2 .
(8)
Now one can show in a straightforward way that the s-wave radial function, instead of being constant at r-+0, behaves as / . . ' X --Ct 2 / 2
qJ[ . . . . ~ / a )
~ 1 - ½ot21n(motr) "
(9)
The corresponding relative correction to the particle density at distances r ~ 1/m, where the annihilation takes place, and therefore to the decay rate itself is ot21n 1 .
(10)
ot
The spin-orbital part of the BH is irrelevant to our problem since it does not work at all in s-states. The part of the BH that describes the tensor spin-spin interaction does not contribute to the decay rate to the accuracy considered. Indeed, applying it to the s-state, this interaction either annihilates it (in the singlet case) or transforms it (in the triplet one) into the dstate with the same total angular m o m e n t u m . But the annihilation from a d-state is strongly hampered. So, we are left with the contact spin-spin interaction V3=~5~zot [] s(s+
l ) - 2 ]c~(r) .
aot q/l(r)=A--6(r)~o(O),
(13)
m
we omit the term - ¼m2ot 2, regular at r--,0, and treat mot~r, which is less singular than A, as a perturbation. Simultaneously this procedure is also an expansion in ot. The solution of the zeroth order in ot, ot
~?> = - A a-mmr~'o(0), at distances r ~ 1 / m of interest to us leads evidently to the correction ~ ot to the decay rate. The calculation of the corresponding numerical factor is beyond our accuracy. The first-order solution, ~u} ') =A" l a21n(mar) ~#o(0) ,
(14)
gives evidently at r ~ l / m the relative correction A. ~ot 2 In ot to ~o (r) Ir- l/m and the correction -- A - o t 2 1 n l = o t 2 1 n l X1 , 2 ot ot
X(-4),
s=O, s=l,
(15)
to I ~o(r) I~-l/m and to the decay rate. The total correction to the decay rate includes both (10) and ( 15 ) and thus constitutes otZln 1 X 2 ,
s=0,
19l
x(-l),
s=l.
(16)
(11)
It originates from both the magnetic spin-spin inter-
3. One may feel dissatisfied with the above treatment of the spin-spin interaction ( 1 1 ). Have we not 521
Volume 246, number 3,4
PHYSICS LETTERS B
buried some contribution ~ a 2 In a in the linearly divergent, at r-~0, term ~,}o)? To take away the suspicion we shall calculate the discussed contribution o f 1/3 in a straightforward way. We shall follow a line o f reasoning close to that ofrefs. [8,9]. Let us consider the diagram o f fig. 1. Its left vertex corresponds to the interaction (11); in the m o m e n t u m representation this a m p l i t u d e is -Ano~/m 2. The right vertex is the annihilation one. In the m o m e n t u m representation it is, to our accuracy, a constant, correction to which we are looking for. The C o u l o m b attraction (the dashed vertical line) supplements the imaginary part o f the corresponding one-loop diagram by the well-known C o u l o m b factor na/2v where 2v is the relative velocity of the particles. So, the imaginary part o f the diagram becomes constant: I m M = - A ~ nO/ 1@~ 4m 2 7~OL - - lTroL2A . 2v
~ /4
A o~21n 1 - ~ a (18)
is in fact the correction to the annihilation a m p l i t u d e we are looking for. The correction to the decay rate is evidently twice as large and coincides with ( 15 ). 4. Our result for o-Ps coincides with that found earlier. As for the disagreement with earlier results for p-Ps, its origin can be easily elucidated in the case of ref. [3]. The prescription o f the latter p a p e r is stated there explicitly: " F o r p a r a p o s i t r o n i u m , the lifetime calculation is the same as for orthopositronium except that the single-photon annihilation kernels are exactly zero". Indeed, under such a prescription we would get for p-Ps constant A the value 3a 2 = ~ instead of - 2, and for the factor at the relative
Fig. 1. Diagrammatic representation of the 0:2In 0: contribution due to the spin-spin contact interaction.
522
correction a2 In a to the decay rate the value 1 -- ] = 3, 2 presented in ref. [ 3 ]. Our value 2 for this factor arises at the transition from o-Ps to p-Ps, if one not only omits the annihilation kernel, but also modifies correspondingly the contact magnetic interaction. Concerning the strong disagreement between the theoretical ( 1 ) and experimental ( 2 ) results for oPs, it m a y be due, to our opinion, to a large and still uncalculated numerical coefficient at ( a / n ) 2 correction to the decay rate. Some hint to it can be found in the large, ~ 10, factor at the a / n correction. By the way, some large (but far from sufficient) contribution to the ( a / n ) 2 correction can be found directly from this number. Indeed, it arises from the - 5a/n correction to the annihilation amplitude which, being squared, gives also a 25 ( a / n ) 2 correction to the decay rate.
(17)
The dispersion integral 1 i dEIm2~/~ ReM=no E - E o I~o. . . . .
30 August 1990
Our attention to the p r o b l e m has been attracted by A. Rich. G r e a t stimulating interest has been shown together with him by A. A1 R a m a d a n , R. Conti and D. Gidley. In particular, they informed us o f the plans to measure precisely the p-Ps decay rate. We also had useful theoretical discussions with S. Brodsky, V.L. Chernyak, M.I. Eides, V.S. F a d i n and Yu. Tomozawa. To all of them we owe our sincere gratitude. One o f us (I.Kh.) highly appreciates the kind hospitality extended to him at the Theoretical Physics Institute, University of Minnesota, where part of this work was done.
References [ 1] A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696. [2] W.E. Caswell, G.P. Lepage and J.R. Sapirstein, Phys. Rev. Len. 38 (1977) 488. [3 ] W.E. Caswell and G.P. Lepage, Phys. Rev. A 20 (1979) 36. [4] G.S. Adkins, Ann. Phys. 146 (1983) 78. [5] C.I. Westbrook, D.W. Gidley, R.S. Conti and A. Rich, Phys. Rev. Len. 58 (1978) 1328. [6] Yu. Tomozawa, Ann. Phys. 128 (1980) 463. [ 7 ] V.B. Berestetsky, E.M. Lifshitz and L.P. Pitaevsky, Quantum electrodynamics (Nauka, Moscow, 1980). [8] R. Barbieri, P. Christillin and E. Remiddi, Phys. Rev. A 8 (1973) 2266. [9] A. Owen, Phys. Rev. Lett. 30 (1973) 887.