††thanks: The idea elaborated in this paper was outlined earlier in a less
comprehensive paper [1].

# The Lamb shift in muonic hydrogen and the electric rms radius of the proton.

Thomas Walcher  Institut für Kernphysik, Johannes Gutenberg-Universität Mainz,
55128 Mainz, Germany.

(April 12, 2023)

###### Abstract

The ”proton radius puzzle” is the 7-standard-deviations difference of the
charge radius of the proton as determined from the Lamb shift in electronic
hydrogen and elastic electron scattering off the proton on the one side and
the high precision determination from the Lamb shift in muonic hydrogen on the
other side. So far the explanation of this difference has been mostly searched
for in the limitations of the non-muonic experiments as the extrapolation to
$Q^{2}\rightarrow 0$ GeV2 for electron scattering. Since the time scale of the
vacuum polarization $e^{+}e^{-}$-pairs, causing the bulk contribution of the
Lamb shift in muonic hydrogen, is very much shorter than that of the photon
exchanges, causing the Coulomb interaction, it is argued that the muon on its
orbit around the proton has to be considered as quasi particle dressed by
$e^{+}e^{-}$ pairs and may not be treated as a bare particle in an external
Uehling potential. The proper realization of this view makes the proton radius
puzzle disappear. The value for the rms charge radius of the proton determined
from the muonic Lamb shift, taking this distinction into account, is
$r_{p}=0.87455(48)\,\textrm{fm}$ in agreement with the CODATA-2010 value.

###### pacs:

14.20.Dh, 31.30.jr, 36.10.Ee , 25.30.Bf

##  I Introduction

About a decade ago a 7-standard-deviations difference of the root-mean-square
(rms) charge radius of the proton $r_{p}=\langle r_{p}^{2}\rangle^{1/2}$
derived from experiments with electrons and from the Lamb shift of muonic
hydrogen was observed and had caused considerable attention in the physics
community and beyond. Since the Lamb shift is a corner stone of the tests of
Quantum Electrodynamics QED this difference requires indeed a convincing
clarification. There was a hope that further experiments and their analysis
would finally establish the correct value as summarized in the recent review
paper of Gao and Vanderhaeghen [2] and in Fig. 1. However, it shows that the
values did not converge over time but still scatter around the values of
CODATA-2010 and CODATA-2018.

Figure 1: Collection of $r_{p}$ values, see [2] for references.
(a) 7-std jump, (b) electron scattering, (c) muonic hydrogen (error bars
$\times 5$), (d) electronic hydrogen, (e) reanalyses: mostly data from (b),
open circles: extrapolation variations.

The precise value of the rms radius derived from the Lamb shift in muonic
hydrogen is $r_{p}=0.84087(39)\,$fm [3, 4] which has to be compared to the old
CODATA-2010 value of 0.8775(51) fm [5], the 7-standard-deviations difference.
The CODATA-2010 value is based on a measurement of the Lamb shift in
electronic hydrogen [6] and on electron scattering experiments [7, 8, 9, 10,
11]. Due to the excellent accuracy of the muonic experiment the reason for the
deviation is mostly searched for in the electronic experiments having a lesser
precision whether in electron scattering or in optical experiments with
electronic hydrogen [2]. Therefore the new CODATA-2018 value of 0.8414(19) fm
[12] is based on a different weighting of the different experiments, however,
there exist no good reasons to doubt and even to exclude the older experiments
and the controversy stays.

With electron scattering experiments, the rms-radius is determined from the
slope of the form factor as function of $Q^{2}$ at $Q^{2}\rightarrow 0$ GeV2.
In many attempts for explaining the deviation the validity of this
extrapolation is questioned [13], but this is already since some time
incompatible with the detailed analysis of Sick [14, 11]. Questioning the low
$Q^{2}$ expansion of the form factor with its well defined and constrained
parameters $\langle r^{2}\rangle,\langle r^{4}\rangle,\dots$, basically means
questioning the validity of QED for this scattering [15].

However, the calculation of the muonic Lamb shift is not as ironclad as it is
mostly assumed. Weinberg ([16], chpt. 11.2) argues, after his calculations of
the radiative and vacuum polarization effects causing the Lamb shift in
electronic atoms, that in muonic atoms one has to be careful by saying:
”However, in this case the muonic atomic radius is not much larger than the
electron Compton wave length, so the approximate result [in equation]
(11.2.39) only gives the order of magnitude of the energy shift due to vacuum
polarization.” Though his rough equation has little to do with the
sophisticated calculations [17, 18, 19, 20, 21, 22], used for the analysis of
the muonic measurement, the remark hints to an overlooked issue.

Weinbergs conjecture can be extended by considering the order of magnitude
parameters of the Bohr model of hydrogen. The Compton wave length of the
$e^{+}e^{-}$ pair of the vacuum polarization is $\lambda_{C}=(\hbar
c)/(2m_{e}c^{2})\approx 193$ fm, the muonic Bohr radius for $n=1$
$r_{\mu,\,B}=(1/\alpha)(\hbar
c)/(m_{\mu}c^{2})=(2/\alpha)(m_{e}/m_{\mu})\lambda_{C}\approx 256$ fm. This
means one has up to $(2\pi r_{\mu,\,B})/\lambda_{C}\approx 10$ separate and
independent spatial interaction regions for the muon mediated by the
$e^{+}e^{-}$ pair fluctuations on its way around the proton. From this follows
that according to quantum mechanical scattering theory the muon experiences
consecutive independent interactions via $e^{+}e^{-}$ pairs. These
interactions happen on a time scale of $\tau_{C}\approx 10^{-21}s$ which is
close to zero compared to the time for one muon-orbit round of
$t_{\textrm{round}}\approx 10^{-18}s$. One has to imagine the ”asymptotic” in-
out states in the scattering mediated by the $e^{+}e^{-}$ pairs, not as plane
waves, but as the wave function of the bound muon in the external Coulomb
field. The interactions are independent and produce a time-dependence of the
interaction on the mentioned scales.

On the other hand the orbits of the muon are produced by the Coulomb
interaction which one had actually to consider for consistency reasons also in
the QED frame work. This interaction is equivalent to the exchange of many
photons having infinite interaction ranges. Since no one knows how to put the
sum of photon exchange diagrams in a wave equation used for the description of
atoms, one just uses the external Coulomb potential, which can be justified by
summing up all ladder diagrams of the multi-photon exchanges as has been shown
by Weinberg ([16], chpt. 13.6). This Coulomb potential is evidently time-
independent in contrast to the potential one has to assign to the multi
$e^{+}e^{-}$ pair exchanges for which the proof of Weinberg does not work. The
Coulomb potential may be viewed as a time-independent external potential,
whereas the $e^{+}e^{-}$-pair potential is time-dependent. The approximation
used so far in the calculations of the muonic Lamb shift is the Uehling
potential, due to the exchange of $e^{+}e^{-}$ pairs, assumed to act on the
same time scale as the photon interaction and consequently is taken to be
time-independent [17, 18, 19, 20, 21, 22]. This means that the $e^{+}e^{-}$
pair potential is taken also as an external potential just as the Coulomb
potential in the wave equations, forgetting about its time dependence due to
the series of separate scatterings. The Uehling effect is not a potential as a
correction to the Coulomb potential as said in ref. [19] before their equation
(200).

If one wants to take the correct approximations of the interaction in muonic
atoms with the external Coulomb potential und the time-dependent $e^{+}e^{-}$
pair potential one encounters the problem that one has to stay in the frame
work of time-dependent wave equations since the time dependency cannot be
separated in the usual way by assuming an exponential with a constant energy
([23], chpt. 26). This applies also for more sophisticated formulations of the
scattering theory as e.g. with the Lippmann-Schwinger equation ([24], chpt. 6
and 7). Fortunately, there exist solutions for an analog problem, namely the
passing of electrons through a crystal: time-dependent corrections, described
by Feynman diagrams, have to be applied to the solution with the time-
independent potential created by the crystal-atoms.

Mattuck mentions in ”A Guide to Feynman Diagrams in the Many-Body Problem”
[25] how the quantum mechanical ”pinball” idea can be used to solve the
problem not only for many body problems but also for atoms ([25], chpt. 4.7,
p. 91). The analogy is in the many independent interactions, not in the many
particles of the crystal or the one proton, producing the external potential
confining the electrons or the one muon, respectively. In the following it
will be shown how this works and that the modification of the Lamb shift
calculation realizing time-dependent $e^{+}e^{-}$ pair exchange instead of
using the time-independent external Uehling potential solves the proton-
charge-radius problem.

##  II Basic theory

For a more detailed discussion of the arguments presented in the introduction
we summarize some basic theory. The Lamb shift is due to the ”self-energy”
diagrams, i.e. the radiative corrections and the vacuum polarization. For the
electronic hydrogen the radiative corrections dominate and the vacuum
polarization is small whereas the reverse holds for muonic hydrogen (see e.g.
Weinberg [16], chpt. 14.3). The two diagrams of the self-energy interactions
are depicted in Fig. 2.

Figure 2: The muon self energy $\Sigma$ consisting of radiative corrections
and the vacuum polarization loop. Double lines are a short hand for the
influence of the external Coulomb field on the propagators and the thick lines
indicate the proton.

The thick line indicates the proton and double lines are a short hand for the
influence of the external Coulomb field on the propagators via photon
exchange. The task for describing the muonic hydrogen correctly in the frame
work of QED is now to sum up all possible Feynman diagrams allowed with the
proper time ordering to all orders as shown in Fig. 3. The sum of the diagrams
had to be calculated via the sum of the G-Matrix iterations and its poles gave
the energies of the bound states including all self-energy corrections. This
approach is of course not feasible and one has to find an approximation
reflecting the situation outlined. We first remark that the proton and the
muon are not relativistic and only the ladder diagrams ordered by
$L(\alpha^{1})$, $L(\alpha^{2})$, $L(\alpha^{3})$, $\dots$ are important. As
mentioned Weinberg proofs ([16], chpt. 13.6) that the sum of the uncrossed
photon ladder diagrams gives the familiar external Coulomb potential. With
”external” one signifies an interaction which is not represented by a Feynman
diagram in analogy to e.g. the external electric field of the Stark effect. It
does not depend on propagators or wave functions of the particles interacting.

Figure 3: A selection of Feynman diagrams contributing to the interaction
between muon and proton.

The spin-orbit coupling is an automatic consequence of the Dirac theory with
the time-independent Coulomb potential and is therefore also an external
interaction. The same is true for the hyperfine interaction.

Next one has to consider the radiative corrections which, however, are small
in muonic hydrogen compared to the vacuum polarization [16]. The radiative
corrections have the same range as the photon exchanges producing the Coulomb
potential and therefore they are approximated as time-independent. This
approximation might explain why some results with the electronic Lamb shift
scatter. The radiative corrections are not further considered in this paper.

The diagrams containing $e^{+}e^{-}$ pairs, loops or ”bubbles” [25] plus
additional photons are at least by order $\alpha$ smaller than the ones
containing the same number of $e^{+}e^{-}$ bubbles and may be neglected.

The sum of the vacuum polarization ladder diagrams produce the major effect of
the muonic Lamb shift and has to be treated differently than the sum of photon
exchanges for the reasons sketched in the introduction. A more detailed
picture is given in Fig. 4. The $e^{+}e^{-}$ loops have a life time of
$\tau_{C}\approx 10^{-21}s$ and are independent one from the other.
Consequently they will have to be summed up as products of probability
amplitudes with $(k_{\textrm{in}},E)=(k_{\textrm{out}},E)$. These interactions
represent ”forward scattering” (see [25], chpt. 4.5, after eq. (4.53) and p.
82) just changing the unperturbed wave function to the perturbed one resulting
in the momentum distribution of the bound state-wave function.

Since the scattering state is a bound shell state the in-state wave function
has to equal the out-state wave function on its closed orbit and, therefore,
one may set $G^{-}(k_{\textrm{in}},E)=G^{+}(k_{\textrm{out}},E)=G(k,E)$ . We
shall see in the next section how this peculiar state can be obtained.

Figure 4: Diagram of the pinball scattering in a bound state of muonic
hydrogen. The forward scattering happens through in- and out-states with
$G^{-}(k_{\textrm{in}},t)=G^{+}(k_{\textrm{out}},t)=G(k,t)$. $\tau_{C}\approx
10^{-21}$ s is the lifetime of the $e^{+}e^{-}$ pairs of the vacuum
polarization and $\approx 0$ compared to the characteristic time scales as
required for interactions mediated by bubbles.

The diagrams of next to leading order $\alpha^{4}$ shown in Fig. 5 have to be
distinguished from those of Fig. 4 since they are coherent within $\tau_{C}$.
Fig. 5(a) and Fig. 5(b) represent loops internally off-mass shell and their
momenta have to be integrated and summed over. But, as will be shown in
section IV, the influence of the perturbed wave function is very small so that
its change compared to the unperturbed wave function can be neglected.

Figure 5: Order $\alpha^{4}$ diagrams: (a) ”double vacuum polarization
correction” [18], also ”polarization insertions in two Coulomb lines” [21],
(b) muon-hole excitations of the shell, (c) ”two loop vacuum polarization”
[18].

##  III Theoretical details and approximations

After outlining the basic idea one has to show how one can sum up all diagrams
forming effectively a generalized Dyson series as depicted in Fig. 6(a). We
shall neglect the effect of the size of the proton which is a separate issue
since the searched for difference is in the Lamb shift for a point charge.
This issue is discussed in the literature (see e.g [17, 18, 20, 21, 22]). Due
to the multiplicative property of the forward scattering amplitudes a
geometric series emerges which can be summed up as symbolically shown in Fig.
6(b).

(a) expanded form (b) geometric sum (c) reiterated form Figure 6: The
generalized Dyson series in the presence of the external Coulomb potential in
three different forms. (See Mattuck [25], chpt. 10.2 for the convention of the
diagrams.)

The diagram in Fig. 6(b) can be transformed into the analytical form of the
G-matrix:

$G(\vec{k},E)=\frac{1}{E-E^{(0)}_{n}-\langle\widetilde{\psi}_{n}|\Sigma(\vec{k},E)|\widetilde{\psi}_{n}\rangle}$
(1)

with $\langle\widetilde{\psi}_{n}|\widetilde{\psi}_{n}\rangle=1$. The pole at

$\Delta
E_{LS}=\widetilde{E}_{n}-E^{(0)}_{n}=\langle\widetilde{\psi}_{n}|\Sigma(\vec{k},\widetilde{E}_{n})|\widetilde{\psi}_{n}\rangle$
(2)

represents the Lamb shift in leading order given by $\Sigma$ with the up to
here unknown exact wave functions $\widetilde{\psi}_{n}$. In order to arrive
at this formula we have integrated over the coordinates of the proton wave
function being in good approximation a $\delta$-function only (see [25], eq.
(4.73)).

The diagram in Fig. 6(c) reads in the form of the G-matrix:

$G(\vec{k},E)=G_{0}(\vec{k},E)+G_{0}(\vec{k},E)\,\Sigma(\vec{k},E)\,G(\vec{k},E)$
(3)

as a Lippmann-Schwinger equation for forward scattering amplitudes. This
equation has to be solved iteratively starting with the unperturbed
$G_{0}(\vec{k},E)$ for $G(\vec{k},E)$ on the right hand side, calculating
$G_{1}(\vec{k},E)$, inserting this again on the right hand side calculating
$G_{2}(\vec{k},E)$ and continuing until nothing changes any more. Since eq.
(3) is an operator equation it is easier to take advantage of the observation
that it is fully equivalent to the Hartree-Fock equation ([25] chpt. 11.1,
[26]). Since we have only one muon orbiting we do not have to anti-symmetrize
and the Hartree equation is appropriate.

From our consideration above we know the Lamb shift as the self-energy eq.
(2). Adapting the considerations of Mattuck ([25], chpt. 4.7, pp. 90-91) on
Hartree quasi particles we arrive at the time-independent Schrödinger
equation:

$(T+V_{\textrm{Coulomb}})\,\widetilde{\psi}_{n}+\langle\widetilde{\psi}_{n}|\Sigma|\widetilde{\psi}_{n}\rangle\widetilde{\psi}_{n}=\widetilde{E}_{n}\,\widetilde{\psi}_{n}$
(4)

The $\widetilde{\psi}_{n}$ are the exact solutions of the Hartree equation
with the condition
$\langle\widetilde{\psi}_{n}|\widetilde{\psi}_{n}\rangle=1$. This equation is
solved iteratively until a self consistent solution $\widetilde{\psi}_{n}$ is
achieved. This procedure is called ”self-consistent renormalization in an
external field” ([25] chpt. 11.1). The self-consistent procedure implied by
eq. (4) can be easily performed in the following way. We begin with the
unperturbed wave function $\psi^{(0)}_{n}$ and the usual ansatz:

$\psi_{n}=\psi^{(0)}_{n}+\delta\psi_{n},~{}\delta\psi_{n}\bot\psi^{(0)}_{n}~{}\textrm{and}~{}\langle\psi^{(0)}_{n}|\psi^{(0)}_{n}\rangle=1.$
(5)

this means that no orthonormal basis for expanding $\psi_{n},$ as in the time-
independent perturbation theory is chosen. Since $\psi_{n}$ is not normalized
we set $\widetilde{\psi}_{n}=\psi_{n}/\langle\psi_{n}|\,\psi_{n}\rangle^{1/2}$
and eq. (4) reads

$(T+V_{\textrm{Coulomb}})\,\psi_{n}+\frac{\langle\psi_{n}|\Sigma|\psi_{n}\rangle}{\langle\psi_{n}|\,\psi_{n}\rangle}\,\psi_{n}=\widetilde{E}_{n}\,\psi_{n}.$
(6)

Multiplying from left with $\langle\psi^{(0)}_{n}|$ one gets an integral
equation which lends itself well for the iterative solution:

$\frac{\langle\psi^{(0)}_{n}|T+V_{\textrm{Coulomb}}|\,\psi_{n}\rangle}{\langle\psi^{(0)}_{n}|\psi_{n}\rangle}+\frac{\langle\psi_{n}|\Sigma|\psi_{n}\rangle}{\langle\psi_{n}|\psi_{n}\rangle}=\widetilde{E}_{n}$
(7)

From this equation follows in accord with eq. (2) derived for normalized wave
functions:

$\Delta
E_{n\,LS}=\widetilde{E}_{n}-E^{(0)}_{n}=\frac{\langle\psi_{n}|\Sigma|\,\psi_{n}\rangle}{\langle\psi_{n}|\,\psi_{n}\rangle}.$
(8)

If one dismisses the request that the Lamb shift is the self-energy and
identifies the operator $\Sigma$ with an external potential we get a wave
equation with the external ”Uehling potential” $V_{\textrm{Uehling}}$ [17, 18,
20, 21, 22]. This is just the step from the ”self energy” point of view to the
”external potential” point of view. The solution of this wave equation gives
somewhat different $\psi_{n}$ and $E_{n}$, denoted by $\psi^{\prime}_{n}$ and
$E^{\prime}_{n}$:

$(T+V_{\textrm{Coulomb}})\,\psi^{\prime}_{n}+V_{\textrm{Uehling}}\,\psi^{\prime}_{n}=E^{\prime}_{n}\,\psi^{\prime}_{n}.$
(9)

If one solves this wave equation numerically with and without
$V_{\textrm{Uehling}}$ the eigenvalues are independent of the normalization of
$\psi^{\prime}_{n}$, and $\Delta E^{\prime}_{n\,LS}$ can be calculated
directly [21].

However, more insight is gained if one multiplies eq. (9) with
$\langle\psi_{n}^{(0)}|$ from the left. One obtains for the Lamb shift in this
approximation:

$\Delta
E^{\prime}_{n\;LS}=E^{\prime}_{n}-E^{(0)}_{n}=\frac{\langle\psi_{n}^{(0)}|V_{\textrm{Uehling}}\,|\psi^{\prime}_{n}\rangle}{\langle\psi_{n}^{(0)}|\psi^{\prime}_{n}\rangle}.$
(10)

Eq. (10) differs from eq. (8) by the appearance of the unperturbed wave
function $\langle\psi_{n}^{(0)}|$ instead of the perturbed
$\langle\psi^{\prime}_{n}|$ on the left side of the matrix elements. It is
this difference which is responsible for the fact that the proton radius
determined from muonic hydrogen deviates from that determined with electron
scattering.

One may now ask what the difference of eq. (10) to the classic perturbation
theory used in the key calculations of [17, 18, 20, 22] is. One has for
increasing order of the expansion parameter $\lambda$ [27]:

$\displaystyle{\cal{O}}(\lambda^{1}):\Delta_{n}^{(1)}=\langle\psi_{n}^{(0)}|V|\psi_{n}^{(0)}\rangle$
$\displaystyle{\cal{O}}(\lambda^{2}):\Delta_{n}^{(2)}=\langle\psi_{n}^{(0)}|V|\psi_{n}^{(1)}\rangle$
$\displaystyle{\cal{O}}(\lambda^{3}):\Delta_{n}^{(3)}=\langle\psi_{n}^{(0)}|V|\psi_{n}^{(2)}\rangle$
$\displaystyle\cdots$ (11)

where $V=\Sigma$ is the perturbing potential, $|\psi_{n}^{(0)}\rangle$ the ket
of the unperturbed state and the next kets of the indicated order. It is
evident that
$\sum_{i=0}^{m}|\psi_{n}^{(i)}\rangle\rightarrow|\psi^{\prime}_{n}\rangle$ for
$m\rightarrow\infty$ and in this way one realizes that
$\sum_{i=1}^{m}\Delta_{n}^{(i)}\rightarrow\Delta E^{\prime}_{n\;LS}$. So the
higher order energy shift of time-independent perturbation theory converges to
the value of the numerical solution of eq. (9).

In time-independent perturbation theory, which takes the Uehling potential as
an external potential, one expands in the basis of unperturbed eigenstates and
gets:

$\Delta E_{n\;LS}=\langle\psi^{(0)}_{n}|\Sigma|\psi^{(0)}_{n}\rangle\;+\\\
\sum_{k\neq n}\int_{E_{k}\geqslant
0}\frac{\langle\psi^{(0)}_{n}|\Sigma|\psi^{(0)}_{k}\rangle\langle\psi^{(0)}_{k}|\Sigma|\psi^{(0)}_{n}\rangle}{E^{(0)}_{n}-E^{(0)}_{k}}.$
(12)

This equation reminds that the perturbed state $|\psi^{(1)}_{n}\rangle$ is
interpreted as the sum over intermediate excitations shown in Fig. 5(b) as
part of the time-dependent perturbation theory. So these intermediate
excitations also appear in the external potential approximation for $\Sigma$
and the numerical solution of eq. (9) contains them through
$|\psi^{\prime}_{n}\rangle$. One can account for these higher order
contributions either summing over the intermediate states of eq. (12) or by
solving eq. (9) numerically.

The reference point of the calculations of [17, 18, 20, 22] is the first order
term in eq. (12), i.e. first order Lamb shift, to which all corrections are
added. The most important correction due to the second term $\Delta_{n}^{(2)}$
is the ”double vacuum polarization correction” [18]. We shall use this
reference point as well and add our correction determined from the solution of
eq. (8) to it. It is important to realize that these solutions contain no
higher order contributions as they are shown in Fig. 5. The pinball mechanism
does not allow these higher order coherent diagrams. Therefore, no higher
order contributions equivalent to the expansion in eq. (11) are produced by
solving eq. (7) numerically. This means that the $\Delta E_{n\;LS}$ in eq. (8)
does not contain the ”double vacuum polarization correction”.

The higher order contributions and further corrections have been discussed and
calculated in detail in [17, 18, 28, 20, 22] and the question arrises whether
one has to redo them in the self consistency framework used here. However, as
will be seen in the next section IV, the self consistent wave functions change
only very little compared to the unperturbed ones, just enough to explain the
small Lamb shift difference producing the radius difference. Therefore, one
may use for higher order corrections the unperturbed wave functions and stay
with the corrections available in the literature. In principle one should
include the coherent higher order contributions as in Fig. 5 as part of the
$\Sigma$ to the geometric sum of the Dyson series and then solve eq. (7) self
consistently. This is actually done in many body problems [25], but is not
needed here.

On the other hand Carroll et al. [21] solved the adaption of eq. (9) as a
time-independent Dirac equation with the external Uehling potential
numerically, effectively including the higher order contributions of the time-
independent perturbation theory according to
$\sum_{i=1}^{m}\Delta_{n}^{(i)}\rightarrow\Delta E^{\prime}_{n\;LS}$. In this
way the ”double vacuum polarization correction” is already included in their
numerical solution of the wave equation. Therefore, they correctly did not add
this correction (”polarization insertion in two Coulomb lines” [21]) to the
reference point. This point will be discussed further at the end of section
IV.

The time-dependent and the time-independent perspectives are very different
and reflect different physics ideas. However, it should be clear that no
double counting is done.

As shown with the derivation of eq. (2), one has to use eq. (8) for
calculating $\Delta E_{n\,LS}$ with the self-consistent renormalization which
can only be done by iteration. As the first step of the iteration we solve eq.
(9) which is according to eq. (10) already a good estimate. The resulting
$\psi^{\prime}_{n}$ we insert as the first approximation $\psi_{n}^{(1)}$ for
$\psi_{n}$ in eq. (8) yielding $\Delta E_{n\,LS}^{(1)}$. It turns out that
this is already good enough. In order to show this we calculated the second
step of the iteration by again using a perturbative ansatz:

$\psi^{(2)}_{n}=\psi^{(1)}_{n}+\kappa\,\delta\psi^{(1)}_{n},~{}\delta\psi^{(1)}_{n}\bot\psi^{(1)}_{n}$
(13)

where the parameter $\kappa$ is varied until eq. (7) is satisfied with
$\widetilde{E}_{n}=E^{(1)}_{n}$. We do not assume that $\psi^{(1)}_{n}$ is
normalized. The perpendicular change $\delta\psi^{(1)}_{n}$ can be gained
from:

$\delta\psi^{(1)}_{n}=\psi_{n}^{(0)}-\frac{\langle\psi_{n}^{(0)}|\psi_{n}^{(1)}\rangle}{\langle\psi_{n}^{(1)}|\psi_{n}^{(1)}\rangle}\psi_{n}^{(1)}$
(14)

where the choice of $\psi_{n}^{(0)}$ is somewhat arbitrary. One could have
taken any Hilbert vector, but for a fast convergence the choice of
$\psi_{n}^{(0)}$ is advantageous. The $\psi^{(2)}_{n}$ derived in this way is
inserted in eq. (8) giving $\Delta E_{n\,LS}^{(2)}$ which one compares to
$\Delta E_{n\,LS}^{(1)}$. It proves that the second iteration is by far good
enough considering the errors of the radius determination from the muonic
hydrogen.

##  IV Results

In distinction to Carroll et al. in ref. [21] who dealt with the relativistic
Dirac equation we restrict ourselves to the non relativistic Schrödinger
equation with the external Coulomb potential of a point charge. The
Schrödinger equation suffices for showing the correction of the Lamb shift
without considering fine structure splitting, finite size effect, etc..
Relativistic effects on the Lamb shift are small [20] and do not change
anything for this discussion. As the first step we calculate the exact
solutions of the Schrödinger equation with the external point Coulomb
potential and the external ”Uehling potential” according to eq. (9) and then
use eq. (8) for calculating the Lamb shift.

A compact representation of the ”Uehling potential” $V_{\textrm{Uehling}}$,
well suited for our calculation, is the representation provided by Pachucki
[18]. (We have to be careful distinguishing between the ”Uehling interaction”
as the effect of the $e^{+}e^{-}$ bubbles on the muon as a ”quasi particle”,
i.e. its ”self energy”, and the ”Uehling interaction” taken as an external
potential. Since the analytical form is the same we shall use the notion
”Uehling potential” for both situations. The respective meaning is clear from
the context.) For the numerical calculations of the Schrödinger equation we
have used Mathematica. As Carroll et al. [21] we have made extensive tests to
guaranty the quality of the solutions. All calculations have been made with an
internal precision of 64 digits and an accuracy goal of 20 digits. The optimal
method is the ”Explicit Runge-Kutta” integration for the S-State, and for the
P-state the change between various methods provided ”Automatic”. The numerical
eigenvalues of the unperturbed $2S$ and $2P$ states, i.e. without the ”Uehling
potential”, are compared to the non relativistic exact solution (Bohr
energies) and found to be good to a few neVs for different boundary conditions
at small ($\approx 0.1$ fm) and large ($\approx 10000$ fm) radii. Of course,
since we do not take the difference of the large energy eigenvalues as Carroll
et al. [21] but calculate the Lamb shift with eq. (8), we do not really need
this extreme accuracy.

When using our numerical wave functions $\psi^{(0)}$ from the numerical
integration of eq. (9) without the perturbation $V_{Uehling}$, i.e. the wave
equation for unperturbed states, and insert them in eq. (8) we get $\Delta
E_{Lamb}=205.00463502$ meV in complete agreement with the calculation using
the exact analytical wave functions [18]. Additionally, as a further check, we
have used the values $E_{n}$ and $E^{(0)}_{n}$, both obtained from the
numerical integration of eq. (9) with and without $V_{Uehling}$, we get
$\Delta E_{Lamb}=E_{n}-E^{(0)}_{n}=205.00463506$ meV. This value for the
proton with point charge is the mentioned reference point and the analyses of
the muonic Lamb shift are resting on it so far.

Figure 7 and Fig. 8 show the difference of the density of the unperturbed
state $|n\,l;0\rangle^{2}r^{2}=(\psi^{(0)}_{n\,l})^{2}r^{2}$ minus the density
of the normalized perturbed state
$|n\,l\rangle_{N}^{2}r^{2}=(\psi_{n\,l})^{2}r^{2}/\langle\psi_{n\,l}|\psi_{n\,l}\rangle$
for $n\,=\,2$ and $l\,=\,0$, 1, respectively.

Figure 7: The difference $(|2\,S;0\rangle^{2}-|2\,S\rangle_{N}^{2})r^{2}$
showing the polarization charge density due to the ”Uehling effect” for the
$2\,S$ state divided by the negative elementary charge of the muon .

Figure 8: The difference $(|2\,P;0\rangle^{2}-|2\,P\rangle_{N}^{2})r^{2}$
showing the polarization charge density due to the ”Uehling effect” for the
$2\,P$ state divided by the negative elementary charge of the muon

.

One nicely sees the polarization charge induced in the vacuum by the muon. The
wiggles at small radii in Fig. 7 are no numerical artifacts but due to the
double bump structure of the 2S state. In order to get an idea of the scales
we note that the Bohr radius for the muon is $r_{\mu,\,B}=268$ fm, the scale
of the Compton wave length of the electron-positron pair of the vacuum
polarization $\lambda_{C}=193$ fm, the rms radius of the 2S state $\langle
2\,S|r^{2}|2\,S\rangle^{1/2}=1854$ fm and of the 2P state $\langle
2\,P|r^{2}|2\,P\rangle^{1/2}=1560$ fm. As expected some positive charge is
pushed to larger radii compensated by negative charge at small radii
indicating the induction of the polarization cloud in the vacuum. The argument
of the importance considering the interaction scale in the introduction by
comparing just to the Bohr radius is amplified by realizing the about 5 times
larger radii of the $n=2$ orbits.

Calculating the difference of the Lamb shifts with the perturbed
$\psi^{(2)}_{2S}$ and $\psi^{(2)}_{2P}$ states according to eq. (8) one gets
the salient result of this paper:

$\Delta E^{(2)\,\textrm{point charge}}_{2P\rightarrow
2S}=E^{(2)}_{2P}-E^{(2)}_{2S}=205.30658(100)\textrm{meV}$ (15)

where the error indicates the variation of the value with different
integration boundaries. The first iteration as described in the previous
section yields:

$\Delta E^{(1)\,\textrm{point charge}}_{2P\rightarrow
2S}=E^{(1)}_{2P}-E^{(1)}_{2S}=205.30664(100)\,\textrm{meV}$ (16)

The result differs from the reference point for the unperturbed wave functions
$\Delta E^{(0)\,\textrm{point charge}}_{\text{Lamb}}=205.005(1)$ meV by

$\delta(\Delta E_{2P\rightarrow 2S})=0.302(1)\,\textrm{meV}.$ (17)

The radius of the muonic hydrogen is derived from a combined measurement of
the singlet transition $2\,S_{1/2}^{F=0}\rightarrow 2\,P_{3/2}^{F=1}$ and the
triplet transition $2\,S_{1/2}^{F=1}\rightarrow 2\,P_{3/2}^{F=2}$. From these
two measurements one can deduce both, the Lamb shift and the 2S-HF splitting
independently [3, 4]. From this determination of the Lamb shift one obtains
the rms radius by equating it to the theoretical shift:

$\Delta E^{\textrm{theory}}_{2P\rightarrow
2S}=(206.0336(15)-5.2275(10)\,r_{p}^{2}/\textrm{fm}^{2}+\\\
0.0332(20))\,\textrm{meV}$ (18)

The first term contains the Lamb shift with a multitude of corrections whereas
the following terms represent the finite size effect due to the position
probability of the muon at the proton [29, 18, 3]. Some details of the finite
size dependence of the Lamb shift [21] are neglected in [3]. For comparability
we follow [3]. Using the two different radii given in the introduction and
their errors as well as those of the constants in the theoretical formula
given above one gets for the deviation of the Lamb shift:

$\delta(\Delta E_{2P\rightarrow 2S})=0.329(50)\,{\textrm{meV}}$ (19)

where the error is dominated by the electronic experiments. This result is in
very good agreement with the result of eq. (17) and explains the ”radius
puzzle”.

For the difference of the eigenvalues according to eq. (10) calculated with
the numerical solution of the wave equation eq. (9) we get $\Delta
E^{\,\textrm{point proton}}_{2P\rightarrow 2S}=205.156(1)\,\textrm{meV}$. If
we take the difference of the eigenvalues directly we get $\Delta
E^{\,\textrm{point proton}}_{2P\rightarrow 2S}=205.159(3)\,\textrm{meV}$.
Since we do not use the difference of the eigenvalues we have not insisted to
improve this error. Carroll et al. [21] get for this difference
$205.1706(5)\,\textrm{meV}$ from the numerical solution of the Dirac equation.
According to [28, 30] we have to add a relativistic correction of 0.0169 meV
to our non-relativistic result yielding $\Delta E^{\,\textrm{point
proton}}_{2P\rightarrow 2S}=205.172(1)\,\textrm{meV}$ in good agreement with
the relativistic result of [21].

##  V Conclusions

If the Lamb shift of a point charge is taken as the dynamical QED effect due
to the interaction of the muon with itself via the vacuum polarization, its
self-energy, and not as a shift caused by the ”Uehling potential” taken as an
time-independent external potential, one gets agreement for the radius
determined from the Lamb shift in muonic hydrogen and the combined electronic
experiments.

The finite size effect as given in eq. (18) is practically not modified by our
considerations. The other corrections used for eq. (18), in particular the
”double vacuum polarization” of 0.151 meV, are to sufficient approximation not
changed by the modified muon wave functions proposed here and we assume that
they are the same as used in the analysis of Antognini et al. [3]. If we
correct the Lamb shift of the point charge in eq. (18) with the difference of
eq. (17) we arrive at a new value for the rms radius of the proton derived
from the muon experiment of Antognini et al. [3]:

$r_{p}=0.87455(48)\,\textrm{fm}$ (20)

which includes the error of the correction in eq. (17). This value is in good
agreement with the best electron scattering rms radius 0.879(8) fm [7, 10, 11]
and that of CODATA-2010 0.8775(51) fm [5]. For a final result for the radius
derived from muonic hydrogen one had to redo the relativistic calculations
including the finite size effects as e.g. Carroll et al. [21] realizing the
considerations of this paper.

###### Acknowledgements.

The author is indebted to Joerg Friedrich and Volker Metag for a critical
reading of the manuscript and for encouragement, and to Michael Distler for
help with Fig. 1 and beyond.

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