The Self-Energy of an Ellectron Bound in a Coulomb Field
Recent progress in calculation of the one-loop selfenergy
of the electron bound in the Coulomb field is summarized.
The relativistic multipole expansion is introduced. This expansion
is based on a single assumption: except for the part of the time
component of the electron four-momentum corresponding to the
electron rest mass, the exchange of four-momentum between the
virtual electron and photon can be treated perturbatively. For non Sstates
and normalized difference n3En −E1 of the S-states this
itself yields very accurate results after taking the method to the third
order. For the ground state the perturbation treatment of the electron
virtual states with very high three-momentum is to be avoided. For
these states one can always rearrange the pertinent expression in such
a way that free-particle approximation is allowed. Combination of
the relativistic multipole expansion and free-particle approximation
yields very accurate result after taking the method to the ninth order.
These results are in very good agreement with the previous results
obtained by the partial wave expansion and definitely exclude the
possibility that the uncertainity in determination of the proton radius
comes from the uncertainity in the calculation of the one-loop selfenergy.
<p>[1] P. J. Mohr et al. Rev. Mod. Phys. 80, 633 (2008); P. J. Mohr in Springer
Handbook of Atomic, Molecular and Optical Physics, edited by G. W.
F. Drake (Springer 2006). See also S. G. Karshenboim, Phys. Rep. 422,
1 (2005); M. I. Eides et al, Phys. Rep. 342, 63 (2001).
[2] E. Borie, Phys. Rev. A 71, 032508 (2005). See also K. Pachucki, Phys.
Rev. A 53, 2092 (1996); ibid 60, 3593 (1999).
[3] M. Niering et al, Phys. Rev. Lett. 84, 5496 (2000); T. Udem et al, ibid
79, 2646 (1997). See also S. Bourzeix et al, ibid 76, 384 (1996).
[4] R. Pohl et al, Nature 466, 213 (2010).
[5] A. Antognini et al, J. Phys.: Conf. Series 312, 032002 (2011).
[6] The influence of the non-zero proton radius on the energy levels is given
by the formula (1) where N
is the principal quantum number of the state under consideration. The
fundamental constants are denoted in the usual way and their values can
be found in (1). The proton radius rp = 0.8768fm given in (1)leads
to the the shift 1.052 MHz between 2S and 1S states, while the radius
rp = 0.84184fm found in (4) leads to the shift 0.96977 MHz.
[7] R. Karplus et al, Phys. Rev. 86, 288 (1952); G. W. Erickson and D. R.
Yennie, Ann. Phys. (NY) 35, 271 (1965); ibid 35, 447 (1965). See also
M. Baranger et al, Phys. Rev. 92, 482 (1953); A. J. Layzer, J. Math.
Phys. 2, 292 (1961); ibid 2, 308 (1961).
[8] K. Pachucki, Ann. Phys. (NY) 226, 1 (1993). See also W. Erickson,
Phys. Rev. Lett. 27, 780 (1971); J. Sapirstein, ibid 47, 1723 (1981).
[9] S. G. Karshenboim, Z. Phys. D 39, 109 (1997).
[10] U. D. Jentschura et al. Phys. Rev. A 72, 062102 (2005). See also A.
Czarnecki et al. Phys. Rev. Lett. 95, 180404 (2005); U. D. Jentschura
et al. Phys. Rev. Lett. 90, 163001 (2003); E.-O. Le Bigot et al. Phys.
Rev. A 68, 042101 (2003); U. D. Jentschura and K. Pachucki, ibid 54,
1853 (1996); U. D. Jentschura et al. ibid 56, 1739 (1997). B. J. Wundt
and U. D. Jentschura, Phys. Lett. B 659, 571 (2008).
[11] U. D. Jentschura et al, Phys. Rev. A 63, 042512 (2001). See also U. D.
Jentschura et al, Phys. Rev. Lett. 82, 53 (1999).
[12] P. J. Mohr et al., Phys. Rep. 293, 227 (1998). See also P.J. Mohr, Phys.
Rev. A 46, 4421 (1992); P. J. Mohr and Y. K. Kim, ibid 45, 2727 (1992);
P. J. Mohr, ibid 26, 2338 (1982); P. J. Mohr, Ann. Phys. (NY) 88, 26,
(1974); ibid 88, 52 (1974).
[13] K. Pachucki, Phys. Rev. A 52, 1079 (1995). See also G. Bhatt and H.
Grotch, Ann. Phys. (NY) 178, 1 (1987).
[14] The following notation is used: d4kF stands for i(2π)−2d4k and
k stands for the four-momentum of the virtual photon. We use the
summation convention A.B = AB = A0B0 − ~A.~B. In particular,
k2 = k2
0 − ω2, ω2 = ~k.~k. The Dirac γ matrices are defined by the
anticommutation relation {γμ, γ} = 2gμ, where g0μ = δ0μ and
gij = −δij . Here δ has the usual meaning of the Kronecker symbol.
When integrating over k0 in Eq. (5) we replace the electron mass m by
m−iη and take the limit η → 0 from above. This is the same notation
and the same way of integration as the one used by Feynman in his
original papers in Phys. Rev. 76, 749 (1949); ibid 76, 769 (1949). The
natural units (¯h = c = 1) are used throughout the paper. Symbol α
stands for the fine structure constant. In actual calculation we took the
value 1/137.03599911 given in (1).
[15] J. Zamastil, Ann. Phys. (NY) 327, 297 (2012).
[16] J. Zamastil and V. Patk´oˇs, Phys. Rev. A 86, 042514 (2012).
[17] J. Zamastil, Ann. Phys. (NY) 328, 139 (2013).
[18] J. Zamastil and V. Patk´oˇs, to be published.</p>
<p>[1] P. J. Mohr et al. Rev. Mod. Phys. 80, 633 (2008); P. J. Mohr in Springer
Handbook of Atomic, Molecular and Optical Physics, edited by G. W.
F. Drake (Springer 2006). See also S. G. Karshenboim, Phys. Rep. 422,
1 (2005); M. I. Eides et al, Phys. Rep. 342, 63 (2001).
[2] E. Borie, Phys. Rev. A 71, 032508 (2005). See also K. Pachucki, Phys.
Rev. A 53, 2092 (1996); ibid 60, 3593 (1999).
[3] M. Niering et al, Phys. Rev. Lett. 84, 5496 (2000); T. Udem et al, ibid
79, 2646 (1997). See also S. Bourzeix et al, ibid 76, 384 (1996).
[4] R. Pohl et al, Nature 466, 213 (2010).
[5] A. Antognini et al, J. Phys.: Conf. Series 312, 032002 (2011).
[6] The influence of the non-zero proton radius on the energy levels is given
by the formula (1) where N
is the principal quantum number of the state under consideration. The
fundamental constants are denoted in the usual way and their values can
be found in (1). The proton radius rp = 0.8768fm given in (1)leads
to the the shift 1.052 MHz between 2S and 1S states, while the radius
rp = 0.84184fm found in (4) leads to the shift 0.96977 MHz.
[7] R. Karplus et al, Phys. Rev. 86, 288 (1952); G. W. Erickson and D. R.
Yennie, Ann. Phys. (NY) 35, 271 (1965); ibid 35, 447 (1965). See also
M. Baranger et al, Phys. Rev. 92, 482 (1953); A. J. Layzer, J. Math.
Phys. 2, 292 (1961); ibid 2, 308 (1961).
[8] K. Pachucki, Ann. Phys. (NY) 226, 1 (1993). See also W. Erickson,
Phys. Rev. Lett. 27, 780 (1971); J. Sapirstein, ibid 47, 1723 (1981).
[9] S. G. Karshenboim, Z. Phys. D 39, 109 (1997).
[10] U. D. Jentschura et al. Phys. Rev. A 72, 062102 (2005). See also A.
Czarnecki et al. Phys. Rev. Lett. 95, 180404 (2005); U. D. Jentschura
et al. Phys. Rev. Lett. 90, 163001 (2003); E.-O. Le Bigot et al. Phys.
Rev. A 68, 042101 (2003); U. D. Jentschura and K. Pachucki, ibid 54,
1853 (1996); U. D. Jentschura et al. ibid 56, 1739 (1997). B. J. Wundt
and U. D. Jentschura, Phys. Lett. B 659, 571 (2008).
[11] U. D. Jentschura et al, Phys. Rev. A 63, 042512 (2001). See also U. D.
Jentschura et al, Phys. Rev. Lett. 82, 53 (1999).
[12] P. J. Mohr et al., Phys. Rep. 293, 227 (1998). See also P.J. Mohr, Phys.
Rev. A 46, 4421 (1992); P. J. Mohr and Y. K. Kim, ibid 45, 2727 (1992);
P. J. Mohr, ibid 26, 2338 (1982); P. J. Mohr, Ann. Phys. (NY) 88, 26,
(1974); ibid 88, 52 (1974).
[13] K. Pachucki, Phys. Rev. A 52, 1079 (1995). See also G. Bhatt and H.
Grotch, Ann. Phys. (NY) 178, 1 (1987).
[14] The following notation is used: d4kF stands for i(2π)−2d4k and
k stands for the four-momentum of the virtual photon. We use the
summation convention A.B = AB = A0B0 − ~A.~B. In particular,
k2 = k2
0 − ω2, ω2 = ~k.~k. The Dirac γ matrices are defined by the
anticommutation relation {γμ, γ} = 2gμ, where g0μ = δ0μ and
gij = −δij . Here δ has the usual meaning of the Kronecker symbol.
When integrating over k0 in Eq. (5) we replace the electron mass m by
m−iη and take the limit η → 0 from above. This is the same notation
and the same way of integration as the one used by Feynman in his
original papers in Phys. Rev. 76, 749 (1949); ibid 76, 769 (1949). The
natural units (¯h = c = 1) are used throughout the paper. Symbol α
stands for the fine structure constant. In actual calculation we took the
value 1/137.03599911 given in (1).
[15] J. Zamastil, Ann. Phys. (NY) 327, 297 (2012).
[16] J. Zamastil and V. Patk´oˇs, Phys. Rev. A 86, 042514 (2012).
[17] J. Zamastil, Ann. Phys. (NY) 328, 139 (2013).
[18] J. Zamastil and V. Patk´oˇs, to be published.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51050", author = "J. Zamastil and V. Patkos", title = "The Self-Energy of an Ellectron Bound in a Coulomb Field", abstract = "Recent progress in calculation of the one-loop selfenergy
of the electron bound in the Coulomb field is summarized.
The relativistic multipole expansion is introduced. This expansion
is based on a single assumption: except for the part of the time
component of the electron four-momentum corresponding to the
electron rest mass, the exchange of four-momentum between the
virtual electron and photon can be treated perturbatively. For non Sstates
and normalized difference n3En −E1 of the S-states this
itself yields very accurate results after taking the method to the third
order. For the ground state the perturbation treatment of the electron
virtual states with very high three-momentum is to be avoided. For
these states one can always rearrange the pertinent expression in such
a way that free-particle approximation is allowed. Combination of
the relativistic multipole expansion and free-particle approximation
yields very accurate result after taking the method to the ninth order.
These results are in very good agreement with the previous results
obtained by the partial wave expansion and definitely exclude the
possibility that the uncertainity in determination of the proton radius
comes from the uncertainity in the calculation of the one-loop selfenergy.
", keywords = "Hydrogen-like atoms, self-energy.", volume = "7", number = "7", pages = "1103-3", }
