PHYSICAL REVIEW A 87, 052511 (2013)

Bound-state field-theory approach to proton-structure effects in muonic hydrogen

Peter J. Mohr*

National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8420, USA

J. Griffith† and J. Sapirstein‡

Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA

(Received 7 April 2013; published 17 May 2013)

A bound-state field-theory approach to muonic hydrogen is set up using a variant of the Furry representation

in which the lowest-order Hamiltonian describes a muon in the presence of a point Coulomb field, but the

origin of the binding field is taken to be three charged quarks in the proton, which are modeled as Dirac

particles that move freely within a spherical well. Bound-state field-theory techniques are used to evaluate oneand two-photon effects. Particular attention is paid to two-photon-exchange diagrams, which include the effect

of proton polarizability. In addition, the modification of the electromagnetic self energy of the proton by the

electric field of the muon is examined. Finally, the model is used to carry out a calculation of the static electric

polarizability of the proton.

DOI: 10.1103/PhysRevA.87.052511 PACS number(s): 31.30.jr, 12.39.Ba, 31.30.jd

I. INTRODUCTION

One of the simplest ways to model the proton is as three very

light quarks confined in a spherical well. Choosing the radius

of the well to be 1.2 fm leads to moderately good agreement

with experiment for its electromagnetic properties, such as

the charge radius, magnetic moment, and static electric and

magnetic polarizabilities. This model, a simplified version of

the MIT bag model [1], will be referred to in the following

as the static-well model. It allows an alternative approach

to the calculation of the electromagnetic properties of the

proton, generally treated with methods quite different in

character, that uses the methods of conventional bound-state

QED. The latter theory is characterized by wave functions

that satisfy the Dirac equation in an external field along

with electron propagators defined in terms of the same field.

When the external field is that of a point Coulomb source, a

modification of the interaction representation introduced by

Furry [2] allows a systematic Feynman diagram treatment

of radiative corrections. This approach can also be applied

to many-electron systems, and a Feynman diagram treatment

of electron-electron interactions is also possible. As will be

explained below, the present paper is patterned on a calculation

of these interactions in heliumlike ions involving two-photon

exchange [3].

The approach we will use in this paper was applied some

time ago [4] to the computation of the electromagnetic self

energy of the proton and neutron. In that work, both the

effect of exchange of a photon between quarks along with the

electromagnetic self energy of the quarks were evaluated and

found to sum to 0.53 MeV for the proton and −0.28 MeV for

the neutron for the case of nearly zero-mass quarks. The fact

that the proton is lighter than the neutron remains explained

by the fact that the down quark is heavier than the up quark,

but it is of note that the electromagnetic correction to the mass

*mohr@nist.gov

†

jgriff8@nd.edu

‡

jsapirst@nd.edu

splitting, −0.81 MeV, is the same order of magnitude as the

neutron-proton mass difference, 1.2 MeV.

The proton can be studied with electron-scattering experiments, which have a long history of providing information

about its properties, in particular the root-mean-square (rms)

radius, rp. The proton size has recently received considerable

attention because of unexpected results for the 2s1/2−2p3/2

transition energy of muonic hydrogen [5]. The issue of determining rp from scattering data can be problematic, as extrapolating the slope of the Dirac form factor to q2 = 0 involves a

number of assumptions [6]. An alternative approach is to determine the proton size by doing precise measurements of atomic

transitions that are sensitive to the effect of the size. The 2010

CODATA result [7] in fact uses this procedure with hydrogen and

deuterium, where the experiment and theory are so accurate

that the proton size can be inferred with an accuracy comparable to that available from scattering experiments as of 2010.

Because of its smaller size, muonic hydrogen has long

been recognized as a system whose spectrum could be used to

determine a much more accurate rms radius of the proton than

that obtained from hydrogen and deuterium, but the associated

experimental obstacles have only recently been overcome.

While indeed much more accurate, the result of Ref. [5] for

the proton size,

rp = 0.841 84(67) fm, (1)

is significantly smaller than the CODATA result,

rp = 0.8768(69) fm. (2)

This discrepancy is referred to as the muonic hydrogen puzzle.

One possible explanation of the puzzle involves the electromagnetic structure of the proton, and the largest theoretical

uncertainty comes from an effect called proton polarizability.

This is generally evaluated by relating the energy shift

to forward virtual photon-proton scattering. The amplitude

describing this scattering, T μν (ν,q2), can then be related to

proton form factors through dispersion relations. A recent

paper that covers all contributions to muonic hydrogen with

particular attention to proton polarizability is Ref. [8]; in

1050-2947/2013/87(5)/052511(13) ©2013 American Physical Society 052511-1
PETER J. MOHR, J. GRIFFITH, AND J. SAPIRSTEIN PHYSICAL REVIEW A 87, 052511 (2013)

the conclusion, we compare our results to results quoted in

that paper. Another recent summary is given in Ref. [9]. A

number of issues involving convergence of the dispersion

theory integrals and the need for experimental data complicate

that approach. The purpose of the present paper is to provide

an alternative analysis patterned after bound-state field-theory

calculations in atomic physics. This will be done by using

the static-well model of the proton together with standard

bound-state QED. As we will show, there is a natural way

of setting up a consistent QED calculation for hydrogen and

muonic hydrogen, with the proton treated as a bound state

of three quarks interacting with an electron or a muon, that

requires no scattering information for its predictions; rather it

depends only on the radius of the well.

Regarding the proton as three relativistic particles confined

to a small volume is closely analogous to treating lithiumlike

highly charged ions, where the electrons for large nuclear

charge Z are quite relativistic and the ion has a size of

1/Z Bohr radius. This problem has recently been addressed

with techniques similar to those used for heliumlike ions

mentioned above [3], and have been shown to provide an

accurate description of lithiumlike ions [10,11], the spectra

of which have been measured with high accuracy [12].

In these calculations almost all of the important physics

is described by Feynman diagrams with one or two photons.

The same turns out to hold for the present calculation, though

in this paper, while we will show all relevant diagrams, we

concentrate our attention on two effects dependent on proton

structure, the polarizability of the proton and the screening of

the proton electromagnetic self energy.

Our model of the proton is extremely simple, but there

are three reasons we have chosen it. The first is that proton

structure effects are generally very small, with even the largest,

the effect of its finite size, accounting for about 2 percent of the

transition energy in muonic hydrogen. Thus even a crude determination of a proton structure effect will have a small relative

theoretical error. The second is that while the proton polarizability correction has been evaluated with other methods, a

contribution we term the proton Lamb shift has not, and the

results presented here may stimulate more sophisticated calculations. The final reason is that mentioned above, to explore a

method of calculating the effect of proton structure on atomic

energy levels that does not require the use of dispersion theory.

We will in the following consider the effect of proton

structure on both electronic and muonic hydrogen. Because

our formalism does not include recoil, we will present results

in terms of the electron mass me and the muon mass mμ even

though reduced-mass effects on the latter are about 10 percent.

When we give a general formula, we refer to a lepton with

mass ml. The state of the lepton, in practice either 2s or 2p3/2,

will be denoted v, and the index m will be used for sums

over intermediate leptonic states: for the corresponding case

of quarks, we use g to denote a ground-state quark and reserve

n for sums over intermediate quark states.

The plan of the paper is as follows. We begin in Sec. II with

a quantum mechanical (QM) treatment of the shift in energy

levels arising from the perturbation of replacing the potential of

a point proton Coulomb field with that of a general distribution

of charge ρ(r). This perturbation theory is evaluated through

second order. In Sec. III we turn to a quantum-field-theoretic

approach to the problem in the context of the static-well model.

We do this by modifying the standard Furry representation [2]

through forcing the lowest-order Hamiltonian to be the same

as that used in that representation, but having the quarks in the

proton provide the Coulomb field instead of assuming a point

source. This requires the introduction of an additional term

in the interaction Hamiltonian we call the counterterm, the

effects of which however are quite simple to evaluate. We also

define the static-well model and briefly review the calculation

of the proton electromagnetic self energy. In Sec. IV we use

bound-state field theory to treat one-photon exchange, and

show that the results agree with the first-order QM energy.

We note here that in this paper we use the Coulomb gauge and

treat only Coulomb photons for all exchanged photons. For the

QM approach this corresponds to ignoring magnetic effects,

and in field theory to leaving out transverse-photon exchange.

In Sec. V we then turn to two-photon exchange diagrams,

which we break into two classes, one in which only one

photon attaches to the lepton, with the other being emitted and

reabsorbed in the proton, and a second in which each photon

is exchanged between the lepton and a quark. In Sec. VI A we

treat the first class, which has no QM analog, and present a

calculation of the contribution, which we call the proton Lamb

shift. In Sec. VI B we treat the second class, but again make a

breakup of the diagrams into firstly a part in which the proton is

left unchanged, and secondly a part where it is excited. (In our

model this means that the spectral representation for the quark

propagator is either saturated with the 1s state, or else that state

is excluded.) The first part will be shown to correspond exactly

to the second-order QM energy. The second part, the proton

polarizability, is then evaluated. The related calculation of the

proton’s static electric polarizability is carried out in Sec. VII

and it is shown that in the κ = 1 angular momentum channel a

complete cancellation between positive- and negative-energy

state terms occurs, leaving only contributions from the κ = −2

intermediate states. In the conclusion we compare our results

to the results of other calculations and describe directions for

future progress.

II. PERTURBATION THEORY OF FINITE-NUCLEAR-SIZE

EFFECTS

We consider a central potential for hydrogen or muonic

hydrogen coming from a finite charge distribution ρ(x),

normalized to unity. The corresponding static potential is

V (x) = −Zα dxρ(x

)

|x − x|

. (3)

While we have Z = 1, the following discussion can also be

applied to the case Z = 1. We start with a point-Coulomb

binding field, so this distribution leads to the perturbation

δV (x) = −Zα dxρ(x

) − δ(x)

|x − x| . (4)

The first-order correction,

E(1) =



dxφ†

v(x) δV (x) φv(x), (5)

is valid for either a relativistic or nonrelativistic calculation.

We first consider the nonrelativistic limit. Then in leading

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BOUND-STATE FIELD-THEORY APPROACH TO PROTON- ... PHYSICAL REVIEW A 87, 052511 (2013)

order the wave functions may be replaced by their value at the

origin and the first-order energy is

E(1)

0 = −Zα |φv(0)|

2



dx



dxρ(x

) − δ(x)

|x − x|

= −Zα |φv(0)|

2



dx

4π

 1

u2 − 16

x2 +···

× [ρ(x

) − δ(x)]

= 2πZα

3 |φv(0)|

2



dx x2

ρ(x)

= 2(Zα)

4

3n3 m3

l



dx x2

ρ(x), (6)

where in the last step we have assumed v to be an ns state, and

u is defined in the following.

The integral over dx has been carried out using a cutoff

procedure, which we now describe. We introduce a parameter

u, understood to ultimately be taken to zero, and work with

the basic identity



dx

1

|x2 − x|

e−u|x−x1|

|x − x1| = 4πu2

1 − e−u|x2−x1|

|x2 − x1| , (7)

which for small u has the expansion



dx

1

|x2 − x|

e−u|x−x1|

|x − x1|

= 4π

1

u − 12 |x2 − x1| + u6 |x2 − x1|

2

− u2

24 |x2 − x1|

3 +···

. (8)

By differentiating once or twice with respect to u, we have



dx e−u|x−x1|

|x2 − x| = 4π

 1

u2 − 16 |x2 − x1|

2 + O(u)



(9)

and



dx

e−u|x−x1|

|x2 − x|

|x − x1| = 4π

 2

u3 − 112|x2 − x1|

3 + O(u)



.

(10)

We have used Eq. (9) in the derivation of Eq. (6), and use

Eq. (10) to evaluate the correction coming from the variation

of the wave function to leading order. For S states, this arises

from

|φv(x)|

2 = |φv(0)|2 (1 − 2Zαml|x|+···) , (11)

which yields an additional contribution of

E(1)

1 = 2(Zα)

2

ml |φv(0)|

2



dx |x|



dxρ(x

) − δ(x)

|x − x|

= 2(Zα)

2

ml |φv(0)|

2



dx

4π

 2

u3 − 112 |x

|

3 +···

× [ρ(x

) − δ(x)]

= −2π(Zα)

2

3 ml |φv(0)|

2



dx |x|

3

ρ(x)

= −2(Zα)

5

3n3 m4

l



dx x3

ρ(x). (12)

This term cancels a corresponding term in second-order

perturbation theory, which is given by the standard form

E(2) =



dx2



dx1 φ†

v(x2) δV (x2)

× 

m=v

φm(x2) φ†

m(x1)

	v − 	m

δV (x1) φv(x1). (13)

The sum over terms involving m is −1 times the reduced

Green’s function. This expression is again valid for either a

relativistic or nonrelativistic calculation. The correction to the

potential is only non-zero inside the nucleus, which means that

the wave functions and reduced Green’s function are evaluated

for small arguments, because the Bohr radius for both the muon

and electron is large compared to the nuclear size. We again

take the nonrelativistic limit. Then the wave functions may be

evaluated at the origin and the reduced Green’s function may

be replaced by the nonrelativistic free Green’s function to give

E(2)

0 =− ml

2π

|φv(0)|

2



dx2



dx1 δV (x2)

1

|x2 − x1|

δV (x1)

=−(Zα)

2ml

2π

|φv(0)|

2



dx

2



dx

1



dx2



dx1

× ρ(x

2) − δ(x2)

|x2 − x

2|

1

|x2 − x1|

ρ(x

1) − δ(x1)

|x1 − x

1| . (14)

Introducing cutoffs allows us to carry out the integrals over the

unprimed variables with the formulas given above, resulting

in

E(2)

0 = (Zα)

2

ml |φv(0)|

2



dx

2



dx

1



dx1

× [ρ(x

2) − δ(x2)]|x2 − x1|

ρ(x

1) − δ(x1)

|x1 − x

1| , (15)

and

E(2)

0 = −π(Zα)

2ml |φv(0)|2

3

 

dx2



dx1 ρ(x2)|x2 − x1|

3

× ρ(x1) − 2



dx |x|

3 ρ(x)



= −(Zα)

5

3n3 m4

l

 

dx2



dx1 ρ(x2)|x2 − x1|

3

ρ(x1)

− 2



dx |x|

3 ρ(x)



. (16)

As alluded to above, the second term in the square brackets in

Eq. (16) is canceled by Eq. (12). The first term in the square

brackets is the third Zemach moment, which we denote r3Z.

An interesting feature about this term is that it too is canceled

by a term that arises when the nucleus is allowed to undergo

low-energy excitations (proton polarizability), though we will

not use this fact directly, and instead just evaluate the entire

effect. We turn now to a field-theoretic approach based on the

static-well model, and begin by introducing a formalism for

bound-state field theory.

III. FORMALISM

The formalism we use here is a simple extension of the

Furry representation [2], which we now briefly review. We

052511-3
PETER J. MOHR, J. GRIFFITH, AND J. SAPIRSTEIN PHYSICAL REVIEW A 87, 052511 (2013)

will use this representation both for leptons and quarks, and

begin by describing how it is used for the former. The full QED

Hamiltonian used for describing the scattering of free leptons

is H = H0 + HI , with ({h} = 1,{c} = 1,{e} = 1),

H0 =



dx ψ†(x) [α · p + βml] ψ(x) (17)

and

HI = qe



dx ψ(x)γμψ(x)Aμ(x), (18)

with qe = −e. We suppress normal ordering and the self-mass

counterterm for simplicity. The Furry representation is used

when H0 is replaced by

H ̃0 =



dx ψ†(x)



α · p + βml − Zα

|x|



ψ(x). (19)

This builds in a classical Coulomb field from an infinite mass

proton. Carrying out a unitary transformation to eliminate H ̃0

rather than H0 leads to the Furry representation in place of the

interaction representation. While the interaction Hamiltonian

HI of a lepton with photons keeps the same form, the

lowest-order spectrum now consists of hydrogenic bound and

scattering states, and the lepton Green’s function obeys the

relation



−iα · ∇x + βml − Zα

|x| − z



G(x, y; z) = δ(x − y), (20)

which has the spectral representation

G(x, y; z) = 

m

φm(x)φ†

m( y)

	m − z . (21)

In the present case we treat proton structure using the staticwell model to specify the wave functions and Green’s functions

of the constituent quarks. However, the influence of the proton

on a lepton in a bound state cannot be treated perturbatively, so

we need to build the binding of the lepton into the formalism

nonperturbatively. We do this by modifying the breakup of the

Hamiltonian given above to H = (H0 + HX) + (HI − HX). If

we choose

HX = − 

dx ψ†(x)

Zα

|x|

ψ(x), (22)

then this breakup is H = H ̃0 + (HI − HX). In the following

we refer to HX as the counterterm, though of course it

is to be distinguished from the electron mass counterterm.

We stress that we do not assume that a classical Coulomb

field is present. However, because we have added a new

term to the interaction Hamiltonian, H ̃0 is unchanged from

the usual Furry representation, and the same wave functions

and Green’s functions used in Feynman diagram calculations

in that representation can be used, although extra Feynman

diagrams involving HX need to be included.

The use of the Furry representation must be extended to

quarks in order to account for the proton’s Coulomb field. In

this case H0 and HI are almost identical to the free case, but

in H0 we assume the presence of a static well that confines

the quarks, the details of which are given below, so that the

up and down quark fields are expanded in terms of solutions

to the Dirac equation in this well. The proton then consists

of the usual two up quarks and one down quark, and the

sum of their charges leads to the Coulomb field felt by the

lepton.

Since the lowest-order problem describes the basic physics

of the atom, there has to be cancellation between diagrams in

which a Coulomb photon is exchanged between the lepton and

the quarks in the proton and diagrams with one counterterm.

In second order another cancellation between two-photon

exchange diagrams and diagrams involving one and two

counterterms must take place, and so on. Because the proton

is now modeled as a finite object, the cancellation will not be

complete, and we will identify the parts remaining after the

near cancellation as the subject of this paper, proton structure

effects. It is of course vital for this procedure to make sense that

the cancellation not only takes place, but that the perturbation

expansion converges. We will present results for the first and

second terms in the expansion below. In determining the order

of the expansion we note that the counterterm HX is of the

same order as two HI .

The static-well model has well-known solutions, which we

show partly to establish notation. We represent the solution

to the free Dirac equation in a spherically symmetric well,

centered at the same origin as used in Furry representation,

by

φq (r) =

 f1(r)χκμ(rˆ)

if2(r)χ−κμ(rˆ)

	

. (23)

Here χκμ(rˆ) is a spherical spinor and the radial wave functions

obey the equations

 ∂

∂r +

1 + κ

r



f1(r) − [m(r) + E] f2(r) = 0 (24)

 ∂

∂r +

1 − κ

r



f2(r) + [E − m(r)] f1(r) = 0. (25)

Confinement is enforced by choosing m(r) to be constant

inside the well and tending to infinity forr>R, which leads to

the MIT bag model boundary conditions [1]. The ground-state

solution for the case m(r) = 0 for r<R, which will be used

throughout this paper, is

f1(r) = N

r sin(wx)

f2(r) = N

r



cos(wx) − sin(wx)

wx 

, (26)

with x = r/R and

N = 1

√R




2w2

2w2 + cos(2w) − 1

. (27)

Here w = 2.042 787 for the ground state, which gives 	g =

335.9 MeV for R = 1.2 fm. When this wave function is used

for the up and down quarks the second and third moments,

052511-4
BOUND-STATE FIELD-THEORY APPROACH TO PROTON- ... PHYSICAL REVIEW A 87, 052511 (2013)

which can be calculated analytically, are

r2

 = R2 2w3 − 2w2 + 4w − 3

6w2(w − 1) = R2 0.531 392 (28)

and

r3

 = R3 2w2 − 2w + 3

8w(w − 1) = R3 0.426 041. (29)

While an analytic form can be derived for the third Zemach

moment, it is lengthy and involves the sine integral, so we give

only its numerical value,

r3

Z = R3 1.280 621. (30)

The wave functions of the up and down quarks are identical

in this zero-mass case, and we denote them as φg(x). This

approximation leads to the important simplification that the

charge density of the proton, even though it consists of three

quarks, can be written in terms of φg(x),

ρg(x) = φ†

g(x)φg(x). (31)

We will see that in our calculations of one- and two-Coulomb

photon exchange this density will enter in exactly the same

way as it does in Sec. II in parts of the calculation, thereby

reproducing the results of that section with the static-well

model charge density. Extra terms arising from the field-theory

approach will be identified with polarizability effects.

Were we to use different masses, the charge densities of the

up and down quarks would differ, and in that case we would

use

ρg(x) = 4

3 φ†

u(x)φu(x) − 1

3 φ†

d (x)φd (x). (32)

The 2s1/2 and 2p3/2 atomic wave functions for the lepton are

the standard Dirac-Coulomb solutions and will be denoted

as φv(x). As proton structure effects are strongly suppressed

for the 2p3/2 case, even though we will continue to use v in

formulas, in practice we will always assume v = 2s1/2.

The radius R is the only variable in this calculation, and we

will use different values to study the R dependence of what is

by far the largest proton structure correction, the finite-size correction from one-photon exchange. However, because all other

proton structure effects are much smaller, the value 1.2 fm

is understood to be used for those corrections.

For the calculation carried out here, which involves a quark

propagating in the well, we use the same kind of spectral

decomposition as given in Eq. (21),

Gq (x, y; z) = 

n

φn(x)φn

†

( y)

	n − z . (33)

The sum over m for the lepton and n for the quark Green’s

functions can be carried out using the method of finite basis sets

[13], which have been used extensively for atomic calculations,

with only minor modifications of the associated computer code

required for application to the quark Green’s function. This is

because the atomic calculations were set up in the same kind of

confining well as used here, but in that case only for the purpose

of discretizing the spectrum, with the well radius chosen to be

much larger than the atom or ion being considered.

We will need the explicit form for the spin-up and spindown proton wave function, with the former being

|p↑ =

	ijk

√72

[−2b†

id b†jab†ka + b†icb†jbb†ka + b†icb†jab†kb]|0.

(34)

In Eq. (34) a and b denote spin up and down states of an up

quark and c and d spin up and down states of a down quark;

	ijk is the Levi-Civita symbol, which makes the proton a color

singlet after the implicit sum over colors ijk is carried out.

Because we are taking the up and down quark masses equal

to zero, the associated wave function φc(x) can be replaced

by φa(x) and φd (x) by φb(x), which simplifies later formulas.

When the spin state of the quark is not important, we simply

use φg(x).

Energy shifts are calculated with the use of S-matrix

techniques, where we use Sucher’s generalization of the

Gell-Mann Low formula [14],

E = lim 	→0,λ→1

i	

2

∂

∂λ

lnv|T[e−iλHI (	)

]|v. (35)

Here HI (	) indicates that a factor e−	|t| is included in the time

integral over the Hamiltonian density in order to adiabatically

turn off the interaction at large positive and negative times. The

advantage of this formula is that the S matrix can be described

with standard Feynman diagram techniques, with the adiabatic

factors usually trivially leading to a factor 1/	 that cancels

the 	 in the numerator of the above formula, though when

we deal with two-photon diagrams, the formalism is needed

to cancel disconnected diagrams. Details of how this works

along with other technical issues can be found in Ref. [3].

That work described a calculation of two-photon exchange

diagrams contributing to energy shifts of excited states of

heliumlike ions, but the basic approach is almost identical.

The most important difference is that while in that work |v

describes two electrons, here it describes one electron or muon

and three quarks, and is given by

|v = b†

v |p. (36)

The diagrams that involve one photon are given in Figs. 1, 2,

and 3(a). Figure 1 is the standard one-loop Lamb shift, which

has been evaluated to spectroscopic accuracy. Figure 2 is

the electromagnetic self energy of the proton, which was

calculated using the Feynman gauge in Ref. [4]. The simplest

diagram to evaluate is Fig. 2(a), photon exchange between

pairs of quarks. It contributes −0.222 MeV to the proton

electromagnetic self energy, all of which comes from the vector

part of the photon exchange. More difficult to calculate is

the self-energy diagram of Fig. 2(b). This diagram in general

requires the inclusion of a self-mass counterterm, though not

for the zero-mass case. After this subtraction an ultravioletdivergent vertex term generally remains, but because the

quarks move freely within the well the ultraviolet-divergent

part of this term vanishes, and numerical evaluation yields

0.658 MeV (the results given in Ref. [4] referred to in the

introduction are based on the radius R = 1 fm, but because

they scale as 1/R a factor 1.2 must be inserted for comparison

to the present work). We note that vacuum polarization terms

do not contribute for an isolated proton, but may be of interest

for muonic hydrogen, an issue that will be discussed further in

052511-5
PETER J. MOHR, J. GRIFFITH, AND J. SAPIRSTEIN PHYSICAL REVIEW A 87, 052511 (2013)

FIG. 1. The two diagrams contributing to the Lamb shift in the hydrogenic bound state.

the conclusion. Finally, Figs. 3(a) and 3(b) describe exchange

of a photon between the lepton and the quarks in the proton

and the first-order effect of HX respectively, and we now turn

to their numerical evaluation.

IV. ONE-PHOTON EXCHANGE

We evaluate Fig. 3(a) in the Coulomb gauge, and consider

only Coulomb photon exchange. This leads to the energy shift

E1C = −α

dx d y

|x − y|

φv

†

(x)φv(x)φ†

g( y)φg( y). (37)

We have used our approximation of having the up and

down quark wave functions being equal, so the sum of the

contribution of the three quarks gives the single term φ†

gφg.

A direct evaluation of this diagram for a state with principal

quantum number n gives a result very close to −mlα2/n2,

with the difference attributable to relativistic effects and the

finite size of the proton built into our model. The associated

counterterm in Fig. 3(b) contributes

EX = α

dx

x φv

†

(x)φv(x). (38)

For the 2s1/2 state it has the value

EX = mlα2

γ

√8(1 + γ )

, (39)

where γ ≡ 

1 − (Zα)2.

Comparing the sum of E1C and EX to E(1) [Eq. (5)], one

sees that it is exactly reproduced by the field-theory expression.

However, because we now have a specific model for ρ(x), we

do not make the approximations made in the QM treatment

and instead numerically evaluate it. We find

E(1) = 2.278 68 × 10−11 a.u. [hydrogen]

(40)

E(1) = 2.006 26 × 10−4 a.u. [muonic hydrogen],

where all digits are significant. The root-mean-square charge

radius rp for R = 1.2 fm is

rp = 0.728 966 R = 0.874 760 fm. (41)

We emphasize that this is not meant to be a prediction

of the proton’s rms radius, it is simply the static-well result

when R = 1.2 fm. However, if we use this in the standard

nonrelativistic expression for the finite-size effect in hydrogen

given in Eq. (6) we get

E(1)

0 = 2.277 15 × 10−11 a.u. [hydrogen]

(42)

E(1)

0 = 2.013 00 × 10−4 a.u. [muonic hydrogen],

which differ from the exact result by 0.07 % and 0.34 %

respectively. Thus we see that the calculation reproduces the

bulk of the nonrelativistic expression for the effect of finite

nuclear size on 2s1/2 energy levels. In the following, while

the basic parameter of the static-well model is the well radius

R, we use Eq. (41) to replace it by 1.3718 rp in all formulas

dependent on R.

As we have a complete model of the charge distribution,

these small deviations can be attributed to higher moments and

relativistic effects. By carrying out the calculation for a range

of R around 1.2 fm, we find the fits

E(1)(r ̃p) = 

2.977 91 × 10−11 r ̃

2γ

p − 6.188 69 × 10−16 r ̃

3

p




a.u.

(43)

for hydrogen and

E(1)(r ̃p) = 

2.631 70 × 10−4 r ̃

2γ

p − 1.126 29 × 10−6

r ̃

3

p




a.u.

(44)

for muonic hydrogen, where r ̃p denotes rp in units of fermis

(femtometers).

We first note that the coefficients of the first and second

terms for hydrogen increase by close to a factor of (mμ/me)

3

and (mμ/me)

4 respectively for muonic hydrogen, consistent

FIG. 2. The two diagrams contributing to the Lamb shift in the proton bound state.

052511-6
BOUND-STATE FIELD-THEORY APPROACH TO PROTON- ... PHYSICAL REVIEW A 87, 052511 (2013)

FIG. 3. Order α contributions.

with the dependence on ml shown in Eqs. (6) and (12). The

coefficients agree at a level of the order of one tenth of

a percent. We originally attempted the fit with a quadratic

term in r ̃p instead of a term with the exponent 2γ , but were

forced to use the latter form for hydrogen to get a proper fit.

(The effect is less important for muonic hydrogen). In fact,

it is known that the actual dependence of leading finite-size

correction on r ̃p is not the nonrelativistic quadratic form,

but instead the relativistic form used above, as shown in

Ref. [15]. In perturbation theory, the leading effect of the

fractional power is a correction given by the Taylor expansion

of r ̃

2γ −2 p , which leads to a logarithmic term of relative

order (Zα)

2 ln r ̃p.

V. SCREENING OF THE PROTON SELF ENERGY

As mentioned in the introduction, the proton electromagnetic self energy has contributions from the Feynman diagrams

of Fig. 2. Before discussing how these diagrams are modified

when the lepton interacts with the proton, which can be

thought of as the Lamb shift of the proton, we mention another

Lamb shift related term. This other effect, while negligible for

hydrogen because it is of relative order (mr/MN )

2, makes a

small contribution for muonic hydrogen, and is not suppressed

at all for positronium, accounting for the self energy of the

positron in that system. It was first derived by Fulton and

Martin [16], and is given by

ESEN = 4α5

3πn3

m3

r

m2

N



ln mN

mrα2



δl0 − ln k0(n,l)



, (45)

where ln k0(n,l) is the Bethe logarithm. This recoil effect, which shifts the 2s1/2 energy in muonic hydrogen by

0.010 meV, is not included in our approach.

An isolated proton can of course emit and reabsorb a

photon, giving rise to the electromagnetic self energy of the

proton just mentioned. This contributes to the mass of the

proton, but when the proton is in a bound state an additional

shift arises, described in lowest order by the Feynman diagrams

in Figs. 4(a) and 4(b).We note in passing that these diagrams do

not have an analog in the QM treatment, as they involve internal

electromagnetic interactions in the proton. In this paper we

restrict our attention to the second of these diagrams, which

we refer to as exchange corrections to the EM self energy

of the proton and label as Eex(pLS). This is justified by

the behavior of the lowest-order proton electromagnetic self

energy, where the size of the exchange term and the quark self

energy are of the same magnitude. The photon propagators

are both taken to be Coulomb propagators, and it is straightforward to show that this set of diagrams gives the energy

shift

Eex(pLS) = −2α2 q ̃i q ̃ 2

j

16π2

dx d y d z dw

|x − y||w − z|

ψ†

v (w)ψv(w)

× 

n

1

	g − 	n

p|:ψ†

i (x)ψi(x)ψ†j ( y)

× φn( y)φ†

n(z)ψj (z):|p. (46)

In this equation i and j are understood to be summed over

the two flavors and appropriate sums over color indices are

implicit; q ̃i is the charge of the up or down quark in units of

e depending on whether i = 1 or 2 respectively; ψi, ψj , and

their adjoints are field operators, but φn and its adjoint are

wave functions, with the sum over n going over all allowed

values of angular momentum, κn and μn, and the positive- and

negative-energy states associated with κn. For two Coulomb

photons, one can show that only κn = −1 yields a nonzero

contribution. After taking the normal-ordered product of the

quark fields between the spin up wave function of the proton

and noting the integrations over angles are all elementary, one

FIG. 4. Order α2 corrections to proton Lamb shift.

052511-7
PETER J. MOHR, J. GRIFFITH, AND J. SAPIRSTEIN PHYSICAL REVIEW A 87, 052511 (2013)

is left with the sum

Eex(pLS) = 8

9

α2

[n]

1

	g − 	n

R

0

dxx2Rgg(x)

×

R

0

dy y2Rgn(y)

R

0

dzz2Rng(z)

×

∞

0

dww2Rvv(w)

1

max(x,y)

1

max(z,w)

,

(47)

where [n] denotes summation over all s states except the 1s

state, and

Rij (x) ≡ f1i(x)f1j (x) + f2i(x)f2j (x). (48)

We find, using finite basis sets to carry out the sum over [n],

that

Eex(pLS) = 0.5 × 10−3 meV. (49)

The smallness of this effect is related to the fact the integral

over z in Eq. (47) is

R

0

dzz2Rng(z)

1

max(z,w)

, (50)

which vanishes when w>R from the orthogonality of the

radial wave functions. This restricts the muon wave function

to lie within the nucleus, which gives a large suppression factor

that scales as (R/a0)

3.

This same suppression factor should make the diagram of

Fig. 4(a), which is more difficult to evaluate, numerically unimportant. However, an interesting application of our approach

would be the calculation of vacuum polarization. This was

shown in Ref. [4] to vanish for a free proton, but when bound

in an atom the arguments for its vanishing no longer apply, and

in fact this should correspond to the effect of hadronic vacuum

polarization, which is treated as a small effect, estimated in

Ref. [5] to be 0.011 meV. However, we are not aware of any

direct calculation of this term with zero-mass quarks, and will

discuss how such a calculation could be carried out in the

conclusions section.

Before turning to two-photon effects, we give more details

about the use of finite basis sets. As described in Ref. [13],

atomic finite basis set calculations are carried out in a well

much larger than the atom, but with the same boundary

conditions as used for the quarks. We continue to use an atomic

basis set appropriate for hydrogen, but add a second basis set

for the quarks, which is obtained by simple modifications of

the atomic code, involving changing the fermion mass to zero,

eliminating the potential, and changing from atomic units to

MeV-fm units. Atomic grids are created on an exponential grid

of the form r(i) = r0[eh(i−1) − 1]. We use the same grid for

both quark and lepton wave functions. This is done by choosing

parameters such that if, for example, a 1000-point grid were

used for the atom, the two hundredth point would be at r =

1.2 fm, so that the quark wave function would be put on a

200-point grid that matched the atomic grid, though of course

the quark wave function vanishes for i > 200. Several grids

were used to test numerical stability. As is also the case for

leptons, a complete set of positive and negative energy states

result for each possible value of κn, in this case N positive

energy s states and N negative energy s states, with a typical

value of N being 50. For leptons the effect of the negativeenergy states is generally very small, entering at the order of the

Lamb shift. However, for quarks they play a more important

role.

VI. TWO-PHOTON EXCHANGE

We now turn our attention to diagrams shown in Fig. 5.

In this section we will show that they in part reproduce the

second-order perturbation theory expression for E(2), Eq. (13)

in Sec. II, but have in addition extra terms we identify as

proton polarizability. To compare with individual diagrams, it

is useful to employ Eq. (4) to represent δV in terms of ρ − δ,

which yields four terms for Eq. (13)

E(2)

δδ = α2



dx d y φ†

v(x)

1

|x|



m=v

φm(x) φ†

m( y)

	v − 	m

1

| y|

φv( y),

(51)

E(2)

δρ = −α2



dx d y dw φ†

v(x)

1

|x|

× 

m=v

φm(x) φ†

m( y)

	v − 	m

ρ(w)

| y − w|

φv( y), (52)

E(2)

ρδ = −α2



dx d y d z φ†

v(x) ρ(z)

|x − z|

× 

m=v

φm(x) φ†

m( y)

	v − 	m

1

| y|

φv( y), (53)

and

E(2)

ρρ = α2



dx d y d z dw φ†

v(x) ρ(z)

|x − z|

× 

m=v

φm(x) φ†

m( y)

	v − 	m

ρ(w)

| y − w|

φv( y). (54)

The simplest diagram, Fig. 5(a), is easily shown to give

E(2)

δδ . The diagrams of Fig. 5(b) and its complex conjugate give

E(2)

δρ and E(2)ρδ if we identify ρ(x) = ρg(x) as in the treatment of

one-photon exchange. This leaves only E(2)

ρρ to be accounted

for. In general one contribution to it comes from Fig. 5(c),

given by

E2(ij ) = α2

q ̃i q ̃j



m

dx d y d z dw

|x − z|| y − w|

× φ†

v(x)φm(x)φ†

m( y)φv( y)

	v − 	m

×p|:ψ†

i (z)ψi(z)ψ†j (w)ψj (w) : |p, (55)

where the definition of q ̃i is the same as in Sec. V. The various

contributions to this term happen to sum to zero for the proton

in our model, but we note they would not were we considering

the neutron, or if we were taking the up and down quark wave

functions to be different.

The final two diagrams, Figs. 5(d) and 5(e), are referred to

as the ladder (L) and crossed ladder (XL), respectively. The

closed loop in these diagrams is associated with an integration

052511-8
BOUND-STATE FIELD-THEORY APPROACH TO PROTON- ... PHYSICAL REVIEW A 87, 052511 (2013)

FIG. 5. Order α2 contributions to polarizability.

over a virtual energy. Because in this paper we consider

only two-Coulomb photon exchange, the analysis of the loop

integral is considerably simpler than the case where the photon

propagators have energy dependence. That complication was

encountered in the full Feynman gauge analysis of ladder

and crossed ladder diagrams in excited states of heliumlike

ions [3], upon which the present calculation is patterned.

Issues involved in carrying out the full calculation will be

discussed in the conclusion. In this simpler case we can carry

out the integral over the timelike component of the loop

momentum with Cauchy’s theorem, which requires identifying

the poles coming from the propagators. We partition the lepton

and quark propagators into positive and negative energy parts,

which could lead to four contributions for each diagram, but

the position of the poles is such that only two contributions

fail to vanish. The surviving terms are

EL(++) = q ̃i

2

α2

dx d y d z dw

|x − z|| y − w|

× 

m+n+

φ†

v(x)φm(x)φ†

m( y)φv( y)

	v + 	g − 	m − 	n

×p|:ψ†

qi

(z)ψn(z)ψ†

n(w)ψqi(w):|p (56)

and

EL(−−) = −q ̃i

2

α2

dx d y d z dw

|x − z|| y − w|

× 

m−n−

φ†

v(x)φm(x)φ†

m( y)φv( y)

	v + 	g − 	m − 	n

×p|:ψ†

qi

(z)ψn(z)ψ†

n(w)ψqi(w):|p (57)

for the ladder diagram, and

EXL(+−) = q ̃i

2

α2

dx d y d z dw

|x − w|| y − z|

× 

m+n+

φ†

v(x)φm(x)φ†

m( y)φv( y)

	g − 	v + 	m − 	n

×p|:ψ†

qi

(z)ψn(z)ψ†

n(w)ψqi(w):|p (58)

and

EXL(−+) = −q ̃i

2

α2

dx d y d z dw

|x − w|| y − z|

× 

m−n−

φ†

v(x)φm(x)φ†

m( y)φv( y)

	g − 	v + 	m − 	n

×p|:ψ†

qi

(z)ψn(z)ψ†

n(w)ψqi(w):|p (59)

for the crossed ladder. The relative minus sign between the two

ladder and the two crossed ladder terms comes from closing

the contour in different ways, and we note that the Coulomb

propagators are different for the ladder and crossed ladder

diagrams. In the following we will give formulas only for

EL(++) and EXL(−+) as the other terms differ from these

only by an overall minus sign and a reversed role of positive

and negative energy states.

A. Elastic contribution

At this point we divide the sum over intermediate quark

states into a part with n = g, which leaves the proton

unchanged and corresponds to elastic scattering, and the

remaining part, corresponding to inelastic scattering, which

gives the polarizability contribution. We now show that the

052511-9
PETER J. MOHR, J. GRIFFITH, AND J. SAPIRSTEIN PHYSICAL REVIEW A 87, 052511 (2013)

TABLE I. Angular coefficient factors for the allowed values of κm, κn for l = 1 and l = 2. Factors of Al are for the ladder diagram; factors

of Bl are for the crossed ladder. “T” refers to the triplet atomic state, while “S” refers to the singlet.

l = 1 l = 2

κm κn A1(T) A1(S) B1(T) B1(S) κm κn A2(T) A2(S) B2(T) B2(S)

1 1 17/729 57/729 37/729 17/729 2 2 26/1125 66/1125 46/1125 26/1125

1 −2 64/729 24/729 44/729 64/729 2 −3 64/1125 24/1125 44/1125 64/1125

−2 1 64/729 24/729 44/729 64/729 −3 2 64/1125 24/1125 44/1125 64/1125

−2 −2 98/729 138/729 118/729 98/729 −3 −3 71/1125 111/1125 91/1125 71/1125

elastic term contains E(2)

ρρ . As 	n is a positive energy, only

EL(++) and EXL(−+) contribute. Because the quark

states are now all s states, the angular-momentum dependence

is particularly simple. The Coulomb propagators can be

simplified with the replacements

1

|x − z|

→ 1

r1a (60) 1

| y − w|

→ 1

r2a

with r1a = max(|x|,|z|), r2a = max(| y|,|w|). (For later use

we define r1b = min(|x|,|z|), r2b = min(| y|,|w|).) Integration

over angles then yields

EL(++) = α2



dV

1

r1ar2a

× 

m+

Rvm(x)Rmv(y)Rgg(z)Rgg(w)

	v − 	m

, (61)

with m being forced to be an s state. We have introduced the

shorthand



dV ≡

∞

0

dxx2

∞

0

dy y2

R

0

dzz2

R

0

dww2. (62)

Because Rgg(w)/4π = ρg(w), this reproduces the part of

E(2)

ρρ in which the electron propagator involves sums over

positive energy states. In order to include the part with negative

energy states the crossed ladder must be considered. The minus

sign mentioned above is needed for the energy denominators

to match, and the fact that ρ(z)ρ(w) is symmetric under

interchange of z and w is also needed. Thus part of the ladder

and crossed ladder diagrams together with the other diagrams

of Fig. 5 simply reproduce second-order perturbation theory

with a particular form for the charge distribution of the proton.

We evaluated these terms numerically as a test of the coding,

since the QM treatment yields a nonrelativistic (NR) limit for

comparison. The result for muonic hydrogen is

E(2)

2s = −3.948 × 10−7 a.u., (63)

which is within 4 % of the NR formula. We recall that formula

has a mixture of the Zemach term and a r3 term, where the

latter cancels part of the one-loop result.

B. Proton polarizability

We now turn to the evaluation of the remaining parts of the

ladder and crossed ladder. The quark propagator is treated in

the same manner as described in Sec. V. The feature of the basis

set mentioned in that section, whereby it automatically breaks

into a positive and a negative energy part, makes separation of

the various diagrams a simple matter. We begin with the case

where κm = κn = −1, but where we do not allow the intermediate quark to be in the ground state. The result for the ladder

is 0.0005 meV, and is almost completely canceled by the

crossed ladder, which contributes −0.0005 meV, making this

channel completely negligible.

In order to treat higher partial waves we must now specify

the spin state of the atom. Polarizability is important only for

the 2s1/2 state, which of course can be a triplet or singlet

state once the spin of the proton is considered. For oneCoulomb-photon exchange these states have the same energy,

as hyperfine splitting comes from transverse photon exchange,

which is not treated here. However, with two-photon exchange,

even when both are Coulomb photons, the triplet and singlet

energies differ except for the l = 0 partial wave discussed

above. We therefore present in the following formulas for the

energy levels of the 2s1/2 singlet and triplet.

While each Coulomb propagator has its own partial wave

expansion, leading in general to a double sum over l1 and l2, the

fact that the quarks are all in s states leads to the simplification

that these values are equal, l1 = l2 ≡ l. Another simplification

is that for a given l, only κm and κn values associated with

that value give a nonvanishing contribution. We define angular

factors Al(κm,κn) for the ladder and Bl(κm,κn) for the crossed

ladder diagrams, which are rational fractions resulting from

the integrations over angles and sums over magnetic quantum

numbers, tabulated in Table I for l = 1 and l = 2. Both

the triplet and singlet state results are given, though in this

paper we are only interested in fine structure. The case l = 1

corresponds to the dominant dipole transition. In terms of these

coefficients, the formula for the ladder is

EL(++) = α2

l



dV rl

1brl2b

rl+1

1a rl+12a



m+n+

Al(κm,κn)

× Rvm(x)Rmv(y)Rgn(z)Rng(w)

	v + 	g + 	n − 	m

(64)

and for the crossed ladder is

EX(+−) = α2

l



dV rl

3brl4b

rl+1

3a rl+14a



m+n−

Bl(κm,κn)

× Rvm(x)Rmv(y)Rgn(z)Rng(w)

	v − 	g + 	n − 	m

, (65)

where r3a = max(|x|,|w|),r3b = min(|x|,|w|) and r4a =

max(| y|,|z|),r4b = min(| y|,|z|).

052511-10
BOUND-STATE FIELD-THEORY APPROACH TO PROTON- ... PHYSICAL REVIEW A 87, 052511 (2013)

These can be evaluated with the techniques described

above with only simple changes, as the spline basis set

contains all values of κ needed and the radial integrals over

rl

b/rl+1 a are evaluated with techniques valid for arbitrary l.

The main numerical problem was ensuring that the typical

rapid convergence of splines in atomic physics carried over

to this problem. The calculation shows that the effect of the

l = 2 channel is very small, with almost the entire effect of

polarizability coming from the l = 1 dipole channel, which

shifts the 2s1/2 energy down by 0.026 meV. This leads to our

main polarizability result for the splitting,

E(2s1/2 − 2p3/2) = 0.026 meV, (66)

consistent with the results from dispersion relation analyses,

as will be discussed in the conclusions.

VII. STATIC POLARIZABILITIES

One of the basic electromagnetic properties of the proton is

its static polarizability, which has an electric and a magnetic

component. We concentrate on the static electric polarizability,

which is given by the particle data group (PDG) as

αp = 12.0(6) × 10−4 fm3

. (67)

FIG. 6. Diagram for static polarizability.

While this value is extracted from Compton scattering data

together with dispersion theory arguments [17], in the staticwell model the lowest-order expression comes from the

diagram of Fig. 6, where the line with a cross indicates a

constant electric field with magnitude Eext. The energy shift is

related to the static electric polarizability through

Eα = −1

2 αp E2

ext. (68)

We note two similar calculations using the closely related

MIT bag model [18,19] have been presented, but detailed

comparison with their results is not possible. A factor of 4π

has been absorbed into the external field.

The basic equation for the energy shift of a proton in the

presence of a constant electric field described by the potential

φ(r) = −Eextx3 is

Eα = q ̃i

2

E2

ext

n

1

	g − 	n



dx d yp| :ψ†

qi

(x)x3ψn(x)ψ†

n( y)y3ψqi( y): |p (69)

which reduces to

Eα = α

9

E2

ext

n



dx d y

7φ†

a(x)x3φn(x)φ†n( y)y3φa( y) + 2φ†

b(x)x3φn(x)φ†

n( y)y3φb( y)

	g − 	n

. (70)

For this case the orientation of the quark spin does not matter,

so the expression simplifies to

αp = −2



n

1

	g − 	n



dx d y φ†

g(x)x3φn(x)φ†n( y)y3φg( y),

(71)

where we have now isolated the factor −1

2E2

ext to give the

formula in terms of αp. We emphasize at this point that this is

a field-theoretic derivation, and that the sum over n is complete,

including negative-energy states.

Introducing the notation [see Eq. (48)]

rij ≡

R

0

dxx3 Rij (x), (72)

after integration over coordinate angles we find

αp = −2

9



n

κn = 1

rgnrng

	g − 	n

− 4

9



n

κn = −2

rgnrng

	g − 	n

. (73)

We calculated the two terms with basis set techniques. The

most striking result found was that the κ = 1 channel vanishes.

If one sums over only positive energy states one obtains a

nonzero result, but inclusion of the negative energy states leads

to an exact cancellation. This situation does not occur for the

κ = −2 channel, which in our model is solely responsible for

the proton static polarizability.

We also calculated αp using a form for the quark propagator

that does not rely on the spectral decomposition given in

Eq. (33). After rewriting Eq. (71) as

αp = 2



dx d y φ†

g(x)x3Gq (x, y; 	g)y3φg( y), (74)

we use the fact that the quark Green’s function satisfies the

differential equation

(−iα · ∇x + βmq − 	g)Gq (x, y; 	g) = δ3

(x − y) (75)

with appropriate boundary conditions. Without those conditions this is the same equation that a free fermion Green’s

function satisfies, and a simple modification of the well-known

partial-wave expansion of the latter propagator can be made

to solve for the quark propagator. We illustrate this for the

Green’s function of a massless quark with positive κ = l and

052511-11
PETER J. MOHR, J. GRIFFITH, AND J. SAPIRSTEIN PHYSICAL REVIEW A 87, 052511 (2013)

x>y, in which case

Gq (x, y; 	g)

= i	2

g



κμ  [hl(	gx) + Aljl(	gx)]χκμ(xˆ)

i[hl−1(	gx) + Aljl−1(	gx)]χ−κμ(xˆ)

	

×(jl(	gy)χ†

κμ( yˆ) −ijl−1(	gy)χ†

−κμ( yˆ)). (76)

The coefficients Al are determined by the boundary condition.

The fact that the propagator ranges over a finite volume

allows the admixture of the solution proportional to Al which

is forbidden for a free propagator because of the boundary

condition as x → ∞. When κ = 1, the integrand in Eq. (74)

includes the factor

(j1(	gy)χ†

1μ( yˆ) −ij0(	gy)χ†−1μ( yˆ))

 j0(	gy)χ−1μ( yˆ)

−ij1(	gy)χ1μ( yˆ)

	

,

(77)

which vanishes. An analogous factor on the left-hand-side

when x<y also vanishes. However, for κ = −2 there is no

such cancellation, and using

A1 = h2(	gR) − h1(	gR)

j2(	gR) − j1(	gR)

, (78)

we find the numerical result

αp = 25.4 × 10−4 fm3 . (79)

The spectral decomposition discussed previously gives the

same result, and we find that the sum is dominated by the first

p3/2 state, with higher-n positive-energy and negative-energy

states entering at under one tenth of a percent. This result is

a factor of two larger than the experimental value in Eq. (67).

However, we note that the relation of the PDG result quoted

to static polarizability determined from energy shifts involves

some subtleties, discussed in Ref. [17] and more recently in

Ref. [20].

VIII. CONCLUSION

In this paper we present an approach to the calculation of the

effect of the electromagnetic structure of the proton on energy

levels of muonic hydrogen that uses a simple bound-state

model for the proton, with only one free parameter, the radius

of the well, which we have chosen to be R = 1.2 fm. Once

the formalism is set up, standard techniques of quantum field

theory can be used to evaluate proton structure effects using

this parameter, in contrast to the standard approach, which

involves the analysis of forward photon-proton scattering. We

do not claim our approach is better, only that it introduces a

different way of looking at the problem.

We have considered only Coulomb photons in this paper.

This is because our primary concern was in setting up the

basic formalism, and testing it in the relatively tractable case

of Coulomb photon exchange. To continue, while one could

stay in the Coulomb gauge and introduce transverse photons, it

is simpler to change to the Feynman gauge. Because the quarks

all have the same energy, much of the work here is effectively

already in the Feynman gauge, with γμ ··· γ μ → γ0 ··· γ0.

Continuing the calculation to this more complete approach

will allow the treatment of the interesting case of hyperfine

splitting. The difficulties of proton structure are well known to

be exacerbated in this case, with even ground-state hydrogen

hfs uncertain at the sixth digit. Our approach should allow a

direct calculation of what is often referred to as dynamic proton

polarizability, as well as allowing a systematic treatment of

other spin-dependent effects that sometimes are difficult to

disentangle. These calculations are complicated by the fact

that the loop energy integral can no longer be evaluated with

Cauchy’s theorem. Instead one must carry out a Wick rotation

to the imaginary axis and carry out the integral numerically.

This rotation requires care because of the presence of poles

and cuts in the complex plane, which leads to a number of

extra terms.

We have shown that in our method a term, while extremely

small, that can be thought of as the Lamb shift of the proton

arises. The smallness of the effect had very much to do with

the fact that all three quarks are taken to be in the 1s state,

which limits the κ values allowed for the propagator. It is an

interesting open question as to how this effect would change

if corrections to the proton wave function involving non-s

states were present, as they presumably are because of gluon

exchange, but that is outside the scope of this paper.

We have shown that the one-photon diagrams together with

the elastic part of the two-photon diagrams give the following

contribution to the Lamb shift:

Efns(nonrecoil) = 

−7.161 22 r ̃

2γ

p + 0.046 70 r ̃

3

p




meV, (80)

which is the sum of Eqs. (44) and (63) with r ̃3

p factored

out of the latter value. This was calculated without the large

recoil corrections present in muonic hydrogen: multiplying the

appropriate factors of (mr/mμ)

3 and (mr/mμ)4 into the first

and second terms, our result becomes

Efns = 

−5.199 5 r ̃

2γ

p + 0.030 5 r ̃

3

p




meV. (81)

The polarizability of the proton was calculated. The basic

result of this paper is that despite the difference in approach,

a similarly small result is found. Specifically, again restoring

recoil corrections by multiplying our result by (mr/mμ)

4, we

have found the proton polarizability correction

Epol = 0.017 meV. (82)

While inclusion of transverse photons may lead to quantitative

changes, we consider it unlikely that a qualitative change will

result. We would expect this additional correction to be of

relative order α, unlike in the case for two-photon exchange

between two electrons bound by a high-Z nucleus where the

correction is of relative order α2. The difference arises from

the fact that the “small” components of the highly relativistic

quark wave functions are of the same order as the “large”

components, rather than being smaller by a factor of α.

We now compare our results with Ref. [8], where the

formula

ELS = 

206.0336(15) − 5.2275(10)r ̃

2

p


 meV + ETPE

(83)

corresponds to their Eq. (32), with the breakdown of ETPE

given in their Table I. Our quadratic term compares well when

a term −0.0275 r ̃2

p meV arising from radiative corrections

to finite size, not included in our work, is removed [9]. As

052511-12
BOUND-STATE FIELD-THEORY APPROACH TO PROTON- ... PHYSICAL REVIEW A 87, 052511 (2013)

TABLE II. Comparison of this work with previous calculations. See the text for an explanation of the entries.

Correction Ref. [21] Refs. [22,23] Ref. [24] This work Eq.

Esubt −0.005(2) meV −0.002 meV −0.002 meV

Einel 0.013(1) meV 0.014 meV 0.016 meV 0.017 meV (82)

Eel 0.030(1) meV 0.023 meV 0.023 meV 0.020 meV (81)

ETPE 0.037(2) meV 0.035 meV 0.037 meV 0.037 meV

shown in Table II, which gives contributions of the shift of the

2s1/2 level to the 2s1/2−2p3/2 transition energy, the breakup

of ETPE involves three terms, a subtraction term Esubt that

has no counterpart in our calculation, an inelastic contribution

Einel, which is the proton polarizability effect, and an elastic

term Eel corresponding to unexcited proton size effects.

The elastic term is not entirely a two-photon contribution in

our approach, because there is a partial cancellation between

the one- and two-photon diagrams, as discussed in Sec. II.

Here we identify the number in the table as that contribution

beyond the term quadratic in r ̃p from the sum of one- and

two-photon-exchange diagrams with unexcited proton states.

This is just the second (cubic in r ̃p) term in Eq. (81). The

agreement of the results of our calculation with previous

results is remarkably good in view of the wide difference in

the methods.

Perhaps the most interesting calculation left undone in this

framework is vacuum polarization. This of course dominates

the muonic hydrogen 2s1/2−2p3/2 splitting when an electron

is in the vacuum polarization loop, but in our framework we

could also put in the zero-mass quarks confined in the well into

the loop. For an isolated proton, symmetry arguments [4] show

that vacuum polarization vanishes, but once in an atom the

arguments no longer hold, and a finite effect should be present.

The standard approach is to introduce pion loops, in which

case the previously mentioned very small “hadronic vacuum

polarization” value of 0.011 meV results, but this approach

is quite different. Such calculations are, however, particularly

challenging because of the high degree of divergence present,

which always presents difficulties for bound-state methods.

We are investigating whether techniques that have proved

useful in studying vacuum polarization effects in atoms [25],

where careful grouping of angular momentum contributions

allows an accurate treatment of highly divergent terms, can be

extended to this case.

ACKNOWLEDGMENTS

The work of J.G. and J.S. was supported in part by NSF

Grant No. PHY1068065. We thank S.A. Blundell, K.T. Cheng,

and E. Tiesinga for useful discussions.

[1] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F.

Weisskopf, Phys. Rev. D 9, 3471 (1974).

[2] W. Furry, Phys. Rev. 81, 115 (1951).

[3] P. J. Mohr and J. Sapirstein, Phys. Rev. A 62, 052501 (2000).

[4] P. J. Mohr and J. Sapirstein, Phys. Rev. Lett. 54, 514 (1985).

[5] R. Pohl et al., Nature (London) 466, 213 (2010); A. Antognini

et al., Science 339, 6618 (2013).

[6] R. J. Hill and G. Paz, Phys. Rev. D 82, 113005 (2010).

[7] P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys. 84,

1527 (2012).

[8] R. Pohl, R. Gilman, G. A. Miller, Pohl, and K. Pachucki,

arXiv:1301.0905.

[9] A. Antognini, F. Kottmann, F. Biraben, P. Indelicato, F. Nez, and

R. Pohl, Ann. Phys. (NY) 331, 127 (2013).

[10] A. N. Artemyev, V. M. Shabaev, and V. A. Yerokhin, Phys. Rev.

A 52, 1884 (1995).

[11] J. Sapirstein and K. T. Cheng, Phys. Rev. A 83, 012504 (2011).

[12] P. Beiersdorfer, A. L. Osterheld, J. H. Scofield, J. R. Crespo

Lopez-Urrutia, and K. Widmann, Phys. Rev. Lett. 80, 3022

(1998).

[13] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A

37, 307 (1988).

[14] J. Sucher, Phys. Rev. 109, 1010 (1958); M. Gell-Mann and

F. Low, ibid. 84, 350 (1951).

[15] P. J. Mohr, At. Data Nucl. Data Tables 29, 453 (1983).

[16] T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954).

[17] A. I. L’vov, Int. Jour. Mod. Phys. A 8, 5267 (1993).

[18] A. Schafer, B. Muller, D. Vasak, and W. Greiner, Phys. Lett. B

143, 323 (1984).

[19] B. Hecking and G. F. Bertsch, Phys. Lett. B 99, 237 (1981).

[20] R. J. Hill, Gabriel Lee, Gil Paz, and Mikhail P. Solon, Phys. Rev.

D 87, 053017 (2013).

[21] C. E. Carlson and M. Vanderhaeghen, Phys. Rev. A 84, 020102

(2011).

[22] K. Pachucki, Phys. Rev. A 53, 2092 (1996).

[23] K. Pachucki, Phys. Rev. A 60, 3593 (1999).

[24] A. P. Martynenko, Yad. Fiz. 69, 1344 (2006) [Phys. At. Nucl.

69, 1309 (2006)].

[25] Gerhard Soff and Peter J. Mohr, Phys. Rev. A 38, 5066

(1988).

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