Electromagnetic polarizabilities and the excited
states of the nucleon
Martin Schumacher
mschuma3@gwdg.de
Zweites Physikalisches Institut der Universität Göttingen, Friedrich-Hund-Platz 1
D-37077 Göttingen, Germany
Abstract
The electromagnetic polarizabilities of the nucleon are shown to be essentially composed
of the nonresonant αp(E0+) = +3.2, αn(E0+) = +4.1, the t-channel α
p,n = −βtp,n = +7.6
and the resonant βp,n(P33(1232)) = +8.3 contributions (in units of 10
−4fm3). The remaining
deviations from the experimental data ∆αp = 1.2 ± 0.6, ∆βp = 1.2 ∓ 0.6, ∆αn = 0.8 ± 1.7
and ∆βn = 2.0 ∓ 1.8 are contributed by a larger number of resonant and nonresonant
processes with cancellations between the contributions. This result confirms that dominant
contributions to the electric and magnetic polarizabilities may be represented in terms of two-
photon coupling to the σ meson having the predicted mass mσ = 666 MeV and two-photon
width Γγγ = 2.6 keV.
PACS. 11.55.Fv Dispersion relations – 11.55.Hx Sum rules – 13.60.Fz Elastic and Compton
scattering – 14.70.Bh Photons
1 Introduction
Recently, it has been shown [1] that the t-channel components αt and βt of the electric (α) and
magnetic (β) polarizabilities of the nucleon can be understood as a property of the constituent
quarks. The constituent quarks couple to π and σ fields and, mediated by these fields, they
couple to two photons. The coupling of two photons with perpendicular linear polarization
to the π0 meson provides the main contribution, γtπ, to the backward-angle spin-polarizability
γπ. Similarly, the coupling of two photons with parallel linear polarization provides the main
contribution, (α− β)t, to the difference (α−β) of the electric and the magnetic polarizabilities.
The quantitative prediction (α − β)tp,n = 15.2 (in units of 10−4fm3) makes use of the fact that
the mass of the particle of the σ field is predicted by the quark-level Nambu–Jona-Lasinio (NJL)
model to be mσ = 666 MeV and its two-photon width to be Γγγ = 2.6 keV [1].
The foregoing paragraph describes the résumé of a long and partial controversial history
of research. The scalar-isoscalar t-channel was introduced [2] in analogy to the pseudoscalar
t-channel [3]. But differing from the π0-pole contribution [3] to the scattering amplitude, the
meaning and importance of the scalar-isoscalar t-channel [2] was less well known, mainly because
the σ meson was not considered as a normal particle. One important step forward was the
formulation of the BEFT sum rule [4], relating the s-channel part of the difference of the electric
and magnetic polarizabilities, (α − β)s, to the multipole content of the total photoabsorption
cross section using a fixed-θ dispersion relation at θ = π, and by relating the t-channel part
(α− β)t to a dispersion relation for t with the imaginary part of the amplitude taken from the
reactions γγ → ππ and NN̄ → ππ via a unitarity relation. Furthermore, the scalar-isoscalar
phase δ00(t) was taken from the reaction ππ → ππ. One of the first evaluations of the BEFT
sum-rule showed that for pointlike uncorrelated pions the large value of (α − β)t = +17.51 is
obtained [5] which in other calculations has been reduced by very different factors (see [6] for an
overview) when the ππ correlation and the pion internal structure is taken into account. The
largest reduction amounting to a factor of 2 has been obtained in the latest of this early series of
http://arxiv.org/abs/0704.0200v1
calculations [7]. This unsatisfactory situation has recently been clarified by showing [1, 8] that
the arithmetic average of the most recent calculations of Drechsel et al. [9] and Levchuk et al.
(see [6]), (α − β)tp,n = 15.3 ± 1.3, leads to a very good agreement with the experimental result
and with a parameter-free calculation based on the quark-level NJL model or dynamical linear
σ model (LσM) [1,10,11], leading to (α− β)tp,n = 15.2.
After the size and the dynamics of the t-channel contribution to the electromagnetic polar-
izabilities has been well understood, it appears of interest to get a similar understanding for the
s-channel contribution. Especially, the question has to be answered what the individual contri-
butions of the resonant excited states of the nucleon to the electric and magnetic polarizabilities
are and how the contributions of the “pion cloud” to the electric and magnetic polarizabilities
may be specified. To the author’s knowledge such an investigation has not been carried out
before.
2 Electromagnetic polarizabilities obtained from the forward-
angle sum-rule for (α + β) and the backward-angle sum-rule
for (α− β)
The appropriate tool for the present investigation is to simultaneously apply the forward-angle
sum-rule for (α + β) and the backward-angle sum-rule for (α − β). This leads to the following
relations
α = αs + αt, (1)
[A(ω)σ(ω,E1,M2, · · · ) +B(ω)σ(ω,M1, E2, · · · )] dω
, (2)
5αe gπNN
12π2 m2σ fπ
= 7.6, (3)
β = βs + βt, (4)
[A(ω)σ(ω,M1, E2, · · · ) +B(ω)σ(ω,E1,M2, · · · )] dω
, (5)
βt = − 5αe gπNN
12π2 m2σ fπ
= −7.6, (6)
ω0 = mπ +
, (7)
A(ω) =
, (8)
B(ω) =
. (9)
In (1) – (9) ω is the photon energy in the lab-system, mπ the pion mass and m the nucleon
mass. The quantities αs, βs are the s-channel electric and magnetic polarizabilities, and αt, βt
the t-channel electric and magnetic polarizabilities, respectively. The multipole content of the
photoabsorption cross section enters through
σ(ω,E1,M2, · · · ) = σ(ω,E1) + σ(ω,M2) + · · · , (10)
σ(ω,M1, E2, · · · ) = σ(ω,M1) + σ(ω,E2) + · · · , (11)
i.e. through the sums of cross sections with change and without change of parity during the
electromagnetic transition, respectively1. The multipoles belonging to parity-change are favored
for the electric polarizability αs whereas the multipoles belonging to parity-nonchange are fa-
vored for the magnetic polarizability βs. The coefficients A(ω) and B(ω) in Eqs. (2), (5), (8)
and (9) multiplying the cross sections of the parity-favored and parity-nonfavored multipoles,
respectively, are A ∼ +1.07 and B ∼ −0.07 at the pion photoproduction threshold. They in-
crease with photon energy, as expected for relativistic correction factors. Using A(ω) and B(ω)
it is easy to prove that (α + β) ≡ (α + β)s is given by the Baldin or Baldin-Lapidus (BL) [14]
sum rule, whereas (α− β)s is given by the s-channel part of the BEFT [4] sum rule.
For the t-channel parts, αt and βt, we use the predictions obtained from the σ-meson pole
representation2 with properties as predicted by the quark-level Nambu–Jona-Lasinio model [1,
8]. The quantities entering into this prediction are αe = e
2/4π = 1/137.04, the pion-nucleon
coupling constant, gπNN = 13.169±0.057, the pion decay constant, fπ = (92.42±0.26) MeV, and
the σ-meson mass, mσ = 666.0 MeV [1, 10, 11]. For convenience we summarize the arguments
leading to the relations (3) and (6). The flavor wave-functions of the π0 and the σ meson are
given by
|π0〉 =
(−uū+ dd̄), |σ〉 =
(uū+ dd̄). (12)
This leads to the decay matrix elements
M(σ → γγ) = −5
M(π0 → γγ) = 5
. (13)
Using the NJL model or the dynamical LσM with dimensional regularization we arrive at [1,11]
mclσ =
4πf clπ√
, (14)
where mclσ and f
π = 89.8 MeV are the σ meson mass and the π decay constant in the chiral
limit (cl) and Nc = 3 the number of colors. Then the mass of the σ meson is given by
(mclσ )
2 +m2π = 666 MeV. (15)
Inserting this into
(α− β)t = gσNNM(σ → γγ)
2πm2σ
and using fσNN = fπNN and (α+ β)
t = 0 we arrive at (3) and (6).
1It should be noted that this separation into cross sections for separate multipoles is possible in the presently
used fixed-θ dispersion theory applied at θ = 0 and θ = π, whereas in the corresponding formulas based on fixed-t
dispersion theory [12] terms containing mixed products of CGLN [13] amplitudes occur (for a discussion see [6]).
2This σ-meson pole in the complex t-plane of the Compton scattering amplitude A(s, t) is not the same, but
has relations with the σ-meson pole introduced to parameterize the ππ scattering amplitude. These relations
have been discussed in detail in [1,6].
3 Components of electromagnetic polarizabilities from analyses
of total photoabsorption and meson photoproduction data
In the following we use different photoabsorption data to get information on partial contributions
to αs and βs. Analyses of total photoabsorption cross sections have been carried out in [15].
These analyses give a very good overview over the resonant and nonresonant contributions to
the electromagnetic polarizabilities. Further information is taken from the PDG2006 [16], the
GWSES [17] and the Mainz [18,19] analyses of meson photoproduction data.
3.1 Components of electromagnetic polarizabilities from analysis of the total
photoabsorption cross-section of the proton
In the following we wish to study the contributions of nucleon resonances and nonresonant
excited states to the s-channel electromagnetic polarizabilities. Only the resonances P33(1232),
P11(1440), D13(1520), S11(1535) and F15(1680) have to be taken into account. The contributions
of the resonances S11(1650), D15(1675) and higher lying resonances are negligible. For this
analysis we use the Walker [15,20] parameterization of nucleon resonances
I = Ir
W 2r ΓΓγ
(W 2 −W 2r )2 +W 2r Γ2
, (17)
Γ = Γr
)2l+1(
q2r +X
q2 +X2
, (18)
Γγ = Γr
k2r +X
k2 +X2
. (19)
s = 2ωm+m2, ω = photon energy in the lab system, (20)
W 2 = s (21)
k = |k| =
, |k| = photon momentum in the c.m. system, (22)
q = |q| =
E2π −m2π; Eπ =
s−m2 +m2π
, |q| = π momentum in the c.m. system, (23)
jγ , multipole angular momentum of the photon, (24)
l, single π angular momentum. (25)
The damping constants X are X = 160 MeV for the P33(1232) resonance and X = 350 MeV
else.
For the proton, parameters are given in [15] for the relevant resonant states, leading to the
results given in lines 3 – 5 of Table 1. The sum αp+βp of nonresonant contributions in line 7 of
Table 1 is in agreement with the corresponding number calculated from the nonresonant cross
section given in [15] if the nonresonant cross section data are extrapolated to about 3.5 GeV.
This shows that with the predicted t-channel contributions given in line 6 there is consistency
between the experimental electromagnetic polarizabilities and the predictions.
3.2 Components of electromagnetic polarizabilities from analyses of meson
photoproduction for the proton and the neutron
From isospin considerations it has been derived [21] that the amplitudes for meson photopro-
duction are composed of A(1/2) and A(3/2), referring to final states of definite isospin (1
Table 1: Partial contributions to the electromagnetic polarizabilities based on the analysis of
the total photoabsorption cross section [15]. The t-channel parts in line 6 are the predictions
based on the σ-meson pole representation (see section 2). Line 7 contains the differences between
the numbers in line 2 and the sums of numbers given in lines 3–6. The experimental data are
normalized to (α+ β)p = 13.9 ± 0.3 (see [6]).
1 αp βp
2 experiment 12.0 ± 0.6 1.9 ∓ 0.6
3 P33(1232) M1, E2 −1.1 +8.3
4 P11(1440) M1 −0.1 +0.3
5 D13(1520) E1,M2 +1.2 −0.3
6 S11(1535) E1 +0.1 −0.0
5 F15(1680) E2,M3 −0.1 +0.4
6 t-channel +7.6 −7.6
7 nonresonant +4.4 +0.8
Furthermore, there is an amplitude A(0) which may be related to “recoil” effects [21]. This latter
amplitude makes a contribution to I = 1/2 only. Therefore, the amplitudes
(1/2) = A(0) +
A(1/2), nA
(1/2) = A(0) −
A(1/2) (26)
may be introduced. Furthermore, with
A(+) =
(A(1/2) + 2A(3/2)), A(−) =
(A(1/2) −A(3/2)), (27)
the physical amplitudes may be expressed by the isospin combinations (see e.g. [19, 22])
A(γp → nπ+) =
2(A(−) +A(0)) =
(1/2) − 1
A(3/2)), (28)
A(γp → pπ0) = A(+) +A(0) =p A(1/2) +
A(3/2), (29)
A(γn → pπ−) = −
2(A(−) −A(0)) =
(1/2) +
A(3/2)), (30)
A(γn → nπ0) = A(+) −A(0) = −nA(1/2) +
A(3/2). (31)
The relation for the cross section of 1π photoproduction is given by
σ1π = 2π
(l + 1)2
(l + 2)(|El+|2 + |M(l+1)−|2) + l(|Ml+|2 + |E(l+1)−|2)
, (32)
∆σ1π = 2π
(l + 1)2(−1)l
(l + 2)(|El+|2 − |M(l+1)−|2) + l(|Ml+|2 − |E(l+1)−|2)
,(33)
∆σ1π = σ1π(E1,M2, · · · )− σ1π(M1, E2, · · · ). (34)
The peak cross section Ir introduced in (17) is given by
Ir = 2π
2J + 1
2J0 + 1
, (35)
where J and J0 are the spins of the excited state and the ground state, respectively, Γγ the
photon width and Γ the total width of the resonance. The photon width Γγ may be expressed
through the resonance couplings A1/2 and A3/2 by the relation [16]
(2J + 1)MR
|A1/2|2 + |A3/2|2
, (36)
where MN and MR are the nucleon and resonant masses. Combining (35) and (36) we arrive
|A1/2|2 + |A3/2|2
. (37)
Using (37) the quantity Ir can be calculated from the resonance couplings A1/2 and A3/2 given
by the PDG [16], by GWSES [17] and Mainz [19]. The results obtained for the electromagnetic
polarizabilities obtained from the data given in [19] are given in lines 3 – 7 of Table 2.
Table 2: Partial contributions to the electromagnetic polarizabilities. The resonant contributions
in lines 3–7 are obtained from the analysis of Drechsel et al. [19]. The t-channel parts in line
8 are the predictions based on the σ-meson pole representation (see section 2). The predicted
contribution due to the E0+ amplitude in line 9 is based on the analyses given in [17–19]. Line
10 contains the differences between the numbers in line 2 and the sums of numbers given in lines
3–9. The experimental data are normalized to (α+ β)p = 13.9 ± 0.3 and (α + β)n = 15.2 ± 0.5
(see [6]).
1 αp βp αn βn
2 experiment 12.0 ± 0.6 1.9∓ 0.6 12.5 ± 1.7 2.7 ∓ 1.8
3 P33(1232) M
(3/2)
1+ , E
(3/2)
1+ −1.1 +8.3 −1.1 +8.3
4 P11(1440) p,nM
(1/2)
1− −0.0 +0.2 −0.0 +0.1
5 D13(1520) p,nE
(1/2)
2− , p,nM
(1/2)
2− +0.6 −0.2 +0.5 −0.1
6 S11(1535) p,nE
(1/2)
0+ +0.1 −0.0 +0.1 −0.0
7 F15(1680) p,nE
(1/2)
3− , p,nM
(1/2)
3− −0.1 +0.3 −0.0 +0.0
8 t-channel +7.6 −7.6 +7.6 −7.6
9 E0+ (empirical) +3.2 −0.3 +4.1 −0.4
10 background +1.7 +1.2 +1.3 +2.4
The main contributions to the nonresonant parts of the electromagnetic polarizabilities are
expected from the E0+ amplitude which has to be taken from analyses of meson photoproduction
data. Multipole analyses of pion photoproduction based on fixed-t dispersion relations and
unitarity are given by Hanstein et al. [18] in a convenient form. Cross sections separated into
resonant and nonresonant parts are provided for the reactions γp → π+n and γn → π−p up to
energies of 500 MeV and extrapolations of the nonresonant parts are straightforward using the
data contained in [19] and [17]. In principle there is a problem in disentangling resonant and
nonresonant contributions because of interference effects. The interference of the amplitudes
(1/2)
0+ with the S11(1535) and S11(1650) resonances, however, does not lead to problems in
determining the nonresonant E0+ contributions because of the smallness of the resonant parts.
The results for the electromagnetic polarizabilities obtained from these empirical E0+ data are
contained in line 9 of Table 2.
3It should be noted that the quantity Ir of (37) contains the branching correction Γ/Γπ as required.
Up to this point the electromagnetic polarizabilities find an explanation in the numbers given
in lines 3 – 9 of Table 2, with the exception of the small contributions given in line 10 which
deserve a further investigation. These non-E0+ parts of the nonresonant contributions are partly
due to the M
(3/2)
1− , p,nM
(1/2)
1+ and p,nE
(1/2)
1+ amplitudes which interfere with the corresponding
resonant amplitudes p,nM
(1/2)
1− (P11(1440)), M
(3/2)
1+ (P33(1232)) and E
(3/2)
1+ (P33(1232)), respec-
tively [19]. Only the nonresonant parts of the M1− and M1+ amplitudes are expected to be to
some extent important in comparison with dominant E0+ amplitude. Therefore we restrict the
present discussion to the M1− and M1+ amplitudes. Using the data given in [19] we arrive at
the estimates
αnonres.p (M1−) = −0.0, βnonres.p (M1−) = +0.2, αnonres.n (M1−) = −0.1, βnonres.p (M1−) = +0.4,
αnonres.p (M1+) = −0.0, βnonres.p (M1+) = +0.3, αnonres.n (M1+) = −0.1, βnonres.p (M1+) = +0.6.
The conclusion we have to draw from this is that it is not possible to relate the numbers given in
line 10 of Table 2 to known photoproduction processes, unless the two-pion channels are taken
into account (see e.g. [23]). The ππN final states can be characterized either as quasi two-
body states such as π∆ and ρN , or as a ππN component in which both pions are in S waves.
Furthermore, in the Regge regime above ≈ 2000 MeV also f2(1270), a2(1320) and Pomeron
t-channel exchanges play a role. The π∆ contribution has been analyzed in terms of a ∆ Kroll-
Ruderman term and a ∆ pion-pole term [24]. Using data from this analysis [24] we arrive at
(αp,n + βp,n) ≈ 1.0 for this partial ππ channel. The nonresonant cross section above ≈ 2000
MeV makes a contribution of about (αp,n + βp,n) ≈ 0.7.
4 Discussion
4.1 Discussion of the s-channel contribution
For a long time there have been attempts to understand the electromagnetic polarizabilities
predominantly in terms of properties of the “pion cloud” of the nucleon. Among these attempts
CHPT in its original relativistic form [25] is among the most prominent ones. It has been
shown by L’vov [26] that the results obtained for the electromagnetic polarizabilities through
the evaluation of chiral loops [25] can be reproduced via dispersion theory when the Born
approximation of the electric-dipole CGLN amplitude E0+ is taken into account. The results
obtained in this way are shown in lines 2 and 3 of Table 3.
Table 3: Predictions for the “meson cloud” contribution to the electromagnetic polarizabilities
in different approaches.
1 method αp βp αn βn reference
2 CHPT +7.4 −2.0 +10.1 −1.2 Bernard [25]
3 piona) Born +7.3 −1.8 +9.8 −0.9 L’vov [26]
4 E0+ Born +7.5 −1.4 +9.9 −1.8 present
a) The use of fixed-t dispersion theory requires the consideration of interference terms of the
E0+ amplitude with other amplitudes.
It is of interest to use also the present approach based on forward and backward dispersion
relations for studies of this type. For this purpose use may be made of the Born approximation
(see [22] p. 286, [27] p. 35) given in the form
EBorn0+ (γN → π±N) = ±
(−)Born
0+ ± E
(0)Born
, (38)
(−)Born
1− v2
1 + v
, (39)
with v = |q|/
q2 +m2π being the velocity of the pion in the c.m. system. The expression given
in (39) corresponds to the static approximation discussed in detail in [22, 27]. Because of the
relation
σE0+(γn → π−p)
σE0+(γp → π+n)
≃ 1.3 (40)
(see [22] p. 276) the recoil terms E
(0)Born
0+ may be replaced by multiplying E
(−)Born
0+ with (1 +
)−1/2 and (1+mπ
)+1/2 in order to get the results for the proton and neutron, respectively. The
relation given in (40) is well justified at threshold but its approximate validity extends to higher
energies [18,19,27]. The pseudovector coupling constant f in (39) is given by f = gπNN (mπ/2m)
with gπNN = 13.169 ± 0.057. There is a remarkable agreement between the numbers given in
Table 3 but these numbers are larger by a factor ∼ 2.4 than the corresponding numbers in line
9 of Table 2. Two reasons for the deviation of the empirical E0+ amplitude from the Born
approximation have been discussed in [19]. The first reason is that the pseudovector (PV)
coupling is not valid at high photon energies but has to be replaced by some average of the PV
and pseudoscalar the (PS) coupling. The second reason are ρ and ω meson t-channel exchanges
which are not taken into account in the Born approximation.
In Table 2 (see also Table 1) we see that the different resonant contributions to the electric
polarizabilities cancel each other, so that the electric polarizabilities are mainly due to the
t-channel part αtp,n (∼60%) given in line 8 and a smaller nonresonant part α(E0+) (∼30%)
given in line 9. For the magnetic polarizabilities there is an almost complete cancellation of the
P11(1440), D13(1520) and F15(1680) contributions, so that the main remaining contributions are
due to the P33(1232) resonance, canceled to a large extent by the t-channel contribution β
t. The
nonresonant background given in line 10 of Table 2 amounts to about 10% of the experimental
electric polarizabilities and to about 70% of the experimental magnetic polarizabilities. This
means that precise predictions of these contributions are highly desirable, especially for the
magnetic polarizabilities. Unfortunately, the non-E0+ parts of the nonresonant photoabsorption
cross sections are dominated by two-pion channels where the information on the multipole
content is scarce.
4.2 Discussion of the t-channel contribution
In [1] it has been shown that there are two independent, but apparently equivalent and comple-
mentary options to calculate the scalar-isoscalar t-channel contribution to the electromagnetic
polarizabilities of the nucleon.
Option 1 makes use of the properties of the σ-meson as predicted by the quark-level NJL
model and in this respect is of course model dependent. The quark-level NJL model predicts
a definite σ-mesons mass, viz. mσ = 666 MeV, through a parameter-free relation of mσ to
the pion decay constant fπ. The result (α − β)t = 15.2 is in an excellent agreement with the
experimental result. The agreement between a prediction and an experimental result cannot be
used as an argument for the validity of the prediction without further support. This support is
provided by dispersion theory applied to the measured properties of the σ meson as showing up
in particle reactions with two pions in the intermediate state (Option 2).
Option 2 first takes into consideration that the σ meson has been observed in many data
analyses [16] as a pole on the second sheet of the isoscalar S wave of ππ scattering. This pole
describes part of the resonant structure of the σ meson without being a complete description.
This latter property of the pole follows from the fact that the 90◦ crossing of the scalar-isoscalar
phase δ00(s) is located at much higher energies than predicted by the structure of the pole. The
analyses of Colangelo et al. [28] and Caprini et al. [29] led to
s(pole) = (470± 30) − i(295 ± 20) MeV
s(δS = 90
◦) = (844± 13) MeV [28], (41)
Mσ = 441
−8 MeV, Γσ = 544
−25 MeV [29]. (42)
The numbers contained in (41) and (42) are extremely valuable in characterizing the properties
of the σ meson as a real particle but they can only qualitatively be compared with the mass
mσ = 666 MeV of the virtual σ meson, because in the latter case there is no open decay channel.
This means that there is no contradiction between the existence of the broad mass distribution
for the real σ meson and a precisely determined mass of the virtual σ meson. Furthermore, the
numbers contained in (41) and (42) are of no direct relevance for the prediction of (α − β)t.
First of all it certainly would lead only to a qualitative estimate for (α − β)t if the parameters
of the σ-meson pole in (41) and (42) would be used instead of mσ = 666 MeV. Furthermore,
such an insufficient attempt is not necessary because the BEFT [4] sum rule provides a precise
relation between (α − β)t and the properties of the real σ meson. In the BEFT sum rule the
imaginary part of the t-channel Compton scattering amplitude is given by an unitarity relation
where the two reaction γγ → σ → ππ and NN̄ → σ → ππ are exploited. In these reactions
the resonant structure of the σ meson enters via the experimentally determined scalar-isoscalar
phase δ00(s) which is considerably different from the corresponding quantity predicted by the
poles shown in (41) and (42). The real part of the t-channel Compton scattering amplitude is
obtained via a dispersion relation. The present status of the evaluation of the BEFT sum rule
(α−β)tnp = 15.3±1.3 is in good agreement with the experimental result as well as the prediction
based on the quark-level NJL model.
5 Conclusion
The good agreement of the result based on the BEFT sum rule with the experimental result
as well as the prediction based on the quark level NJL model may be understood as a strong
argument that the two predictions of (α−β)t are equivalent. This implies that in addition to the
poles in (41) and (42) also the mass mσ = 666 MeV of the virtual σ meson is an experimentally
verified property of the σ meson.
Acknowledgment
The author is indebted to Deutsche Forschungsgemeinschaft for the support of this work through
the projects SCHU222 and 436RUS113/510. He thanks M.I. Levchuk, A.I. L’vov and A.I. Mil-
stein for a long term cooperation which contributed to the motivation for the present investiga-
tion.
References
[1] M. Schumacher, Eur. Phys. J. A 30, 413 (2006); DOI 10.1140/epja/i2006-10103-0
[hep-ph/0609040].
[2] A.C. Hearn, E. Leader, Phys. Rev. 126, 789 (1962); R. Köberle, Phys. Rev. 166, 1558
(1968).
[3] E.E. Low, Phys. Rev. 120, 582 (1960) (and reference therein); M. Jacob, J. Mathews, Phys.
Rev. 117, 854 (1960).
[4] J. Bernabeu, T.E.O. Ericson, C. Ferro Fontan, Phys. Lett. 49 B, 381 (1974); J. Bernabeu,
B. Tarrach, Phys. Lett 69 B, 484 (1977).
[5] I. Guiasu, E.E. Radescu, Phys. Rev. D 14, 1335 (1976); Phys. Lett. 62 B, 193 (1976).
[6] M. Schumacher, Prog. Part. Nucl. Phys. 55, 567 (2005) [hep-ph/0501167].
[7] B.R. Holstein, A.M. Nathan, Phys. Rev. D 49, 6101 (1994).
[8] M.I. Levchuk, A.I. L’vov, A.I. Milstein, M. Schumacher, Proceedings of the Workshop
NSTAR2005, 12–15 October 2005, Tallahassee, Florida, edited by S. Capstick, V. Crede,
P. Eugenio (World Scientific 2006) 389 [hep-ph/0511193].
[9] D. Drechsel et al., Phys. Rep. 378, 99 (2003); Phys. Rev. C 61, 015204 (1999).
[10] T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 221 (1994).
[11] R. Delbourgo, M. Scadron, Mod. Phys. Lett. A 10, 251 (1995) [hep-ph/9910242]; Int. J.
Mod. Phys. A 13, 657 (1998) [hep-ph/9807504].
[12] A.I. L’vov, V.A. Petrun’kin, M. Schumacher, Phys. Rev. C 55, 359 (1997).
[13] G.F. Chew, M.L. Goldberger, F.E. Low, Y. Nambu, Phys. Rev. 106, 1345 (1957).
[14] A.M. Baldin, Nucl. Phys. 18, 310 (1960); L.I. Lapidus, Zh. Eksp. Teor. Fiz. 43, 1358 (1962)
[Sov. Phys. JETP 16, 964 (1963)].
[15] T.A. Armstrong et al., Phys. Rev. D 5, 1640 (1972); Nucl. Phys. B 41, 445 (1972).
[16] W.-M. Yao et al., (Particle Data Group) J. Phys. G 33, 1 (2006) [URL: http://pdg.lbl.gov].
[17] R.A. Arndt, et al. Phys. Rev. C 66, 055213 (2002).
[18] O. Hanstein, D. Drechsel, L. Tiator, Nucl. Phys. A 632, 561 (1998).
[19] D. Drechsel, O. Hanstein, S.S. Kamalov, L. Tiator, Nucl. Phys. A 645, 145 (1999).
[20] R.L. Walker, Phys. Rev. 182, 1729 (1969).
[21] K.M. Watson, Phys. Rev. 95, 228 (1954).
[22] T. Ericson, W. Weise, Pions and Nuclei, International Series of Monographs on Physics 74,
Oxford Science Publications (1988).
[23] D. Drechsel, L. Tiator, J. Phys. G: Nucl. Part. Phys. 18, 449 (1992).
[24] J.A. Gómez Tejedor, E. Oset, Nucl. Phys. A 571, 667 (1994); 600, 413 (1996).
http://arxiv.org/abs/hep-ph/0609040
http://arxiv.org/abs/hep-ph/0501167
http://arxiv.org/abs/hep-ph/0511193
http://arxiv.org/abs/hep-ph/9910242
http://arxiv.org/abs/hep-ph/9807504
http://pdg.lbl.gov
[25] V. Bernard, N. Kaiser, U.-G. Meissner, Phys. Rev. Lett. 67, 1515 (1991); Nucl. Phys. B
373, 346 (1992).
[26] A.I. L’vov, Phys. Lett. B 304, 29 (1993).
[27] A. Donnachie, in: High Energy Physics, Edited by E.H.S. Burhop V, 1 Academic Press
(1972)
[28] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603, 125 (2001).
[29] I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. 96, 132001 (2006).
	Introduction
	Electromagnetic polarizabilities obtained from the forward-angle sum-rule for (+) and the backward-angle sum-rule for (-)
	Components of electromagnetic polarizabilities from analyses of total photoabsorption and meson photoproduction data
	Components of electromagnetic polarizabilities from analysis of the total photoabsorption cross-section of the proton
	Components of electromagnetic polarizabilities from analyses of meson photoproduction for the proton and the neutron
	Discussion
	Discussion of the s-channel contribution
	Discussion of the t-channel contribution
	Conclusion

  The electromagnetic polarizabilities of the nucleon are shown to be
essentially composed of the nonresonant $\alpha_p(E_{0+})=+3.2$,
$\alpha_n(E_{0+})=+4.1$,the $t$-channel $\alpha^t_{p,n}=-\beta^t_{p,n}=+7.6$
and the resonant $\beta_{p,n}(P_{33}(1232))=+8.3$ contributions (in units of
$10^{-4}$fm$^3$. The remaining deviations from the experimental data
$\Delta\alpha_p=1.2\pm 0.6$, $\Delta\beta_p=1.2\mp 0.6$, \Delta\alpha_n=0.8\pm
1.7$ and $\Delta\beta_n=2.0\mp 1.8$ are contributed by a larger number of
resonant and nonresonant processes with cancellations between the
contributions. This result confirms that dominant contributions to the electric
and magnetic polarizabilities may be represented in terms of two-photon
couplings to the $\sigma$-meson having the predicted mass $m_\sigma=666$ MeV
and two-photon width $\Gamma_{\gamma\gamma}=2.6$ keV.

Introduction
Recently, it has been shown [1] that the t-channel components αt and βt of the electric (α) and
magnetic (β) polarizabilities of the nucleon can be understood as a property of the constituent
quarks. The constituent quarks couple to π and σ fields and, mediated by these fields, they
couple to two photons. The coupling of two photons with perpendicular linear polarization
to the π0 meson provides the main contribution, γtπ, to the backward-angle spin-polarizability
γπ. Similarly, the coupling of two photons with parallel linear polarization provides the main
contribution, (α− β)t, to the difference (α−β) of the electric and the magnetic polarizabilities.
The quantitative prediction (α − β)tp,n = 15.2 (in units of 10−4fm3) makes use of the fact that
the mass of the particle of the σ field is predicted by the quark-level Nambu–Jona-Lasinio (NJL)
model to be mσ = 666 MeV and its two-photon width to be Γγγ = 2.6 keV [1].
The foregoing paragraph describes the résumé of a long and partial controversial history
of research. The scalar-isoscalar t-channel was introduced [2] in analogy to the pseudoscalar
t-channel [3]. But differing from the π0-pole contribution [3] to the scattering amplitude, the
meaning and importance of the scalar-isoscalar t-channel [2] was less well known, mainly because
the σ meson was not considered as a normal particle. One important step forward was the
formulation of the BEFT sum rule [4], relating the s-channel part of the difference of the electric
and magnetic polarizabilities, (α − β)s, to the multipole content of the total photoabsorption
cross section using a fixed-θ dispersion relation at θ = π, and by relating the t-channel part
(α− β)t to a dispersion relation for t with the imaginary part of the amplitude taken from the
reactions γγ → ππ and NN̄ → ππ via a unitarity relation. Furthermore, the scalar-isoscalar
phase δ00(t) was taken from the reaction ππ → ππ. One of the first evaluations of the BEFT
sum-rule showed that for pointlike uncorrelated pions the large value of (α − β)t = +17.51 is
obtained [5] which in other calculations has been reduced by very different factors (see [6] for an
overview) when the ππ correlation and the pion internal structure is taken into account. The
largest reduction amounting to a factor of 2 has been obtained in the latest of this early series of
http://arxiv.org/abs/0704.0200v1
calculations [7]. This unsatisfactory situation has recently been clarified by showing [1, 8] that
the arithmetic average of the most recent calculations of Drechsel et al. [9] and Levchuk et al.
(see [6]), (α − β)tp,n = 15.3 ± 1.3, leads to a very good agreement with the experimental result
and with a parameter-free calculation based on the quark-level NJL model or dynamical linear
σ model (LσM) [1,10,11], leading to (α− β)tp,n = 15.2.
After the size and the dynamics of the t-channel contribution to the electromagnetic polar-
izabilities has been well understood, it appears of interest to get a similar understanding for the
s-channel contribution. Especially, the question has to be answered what the individual contri-
butions of the resonant excited states of the nucleon to the electric and magnetic polarizabilities
are and how the contributions of the “pion cloud” to the electric and magnetic polarizabilities
may be specified. To the author’s knowledge such an investigation has not been carried out
before.
2 Electromagnetic polarizabilities obtained from the forward-
angle sum-rule for (α + β) and the backward-angle sum-rule
for (α− β)
The appropriate tool for the present investigation is to simultaneously apply the forward-angle
sum-rule for (α + β) and the backward-angle sum-rule for (α − β). This leads to the following
relations
α = αs + αt, (1)
[A(ω)σ(ω,E1,M2, · · · ) +B(ω)σ(ω,M1, E2, · · · )] dω
, (2)
5αe gπNN
12π2 m2σ fπ
= 7.6, (3)
β = βs + βt, (4)
[A(ω)σ(ω,M1, E2, · · · ) +B(ω)σ(ω,E1,M2, · · · )] dω
, (5)
βt = − 5αe gπNN
12π2 m2σ fπ
= −7.6, (6)
ω0 = mπ +
, (7)
A(ω) =
, (8)
B(ω) =
. (9)
In (1) – (9) ω is the photon energy in the lab-system, mπ the pion mass and m the nucleon
mass. The quantities αs, βs are the s-channel electric and magnetic polarizabilities, and αt, βt
the t-channel electric and magnetic polarizabilities, respectively. The multipole content of the
photoabsorption cross section enters through
σ(ω,E1,M2, · · · ) = σ(ω,E1) + σ(ω,M2) + · · · , (10)
σ(ω,M1, E2, · · · ) = σ(ω,M1) + σ(ω,E2) + · · · , (11)
i.e. through the sums of cross sections with change and without change of parity during the
electromagnetic transition, respectively1. The multipoles belonging to parity-change are favored
for the electric polarizability αs whereas the multipoles belonging to parity-nonchange are fa-
vored for the magnetic polarizability βs. The coefficients A(ω) and B(ω) in Eqs. (2), (5), (8)
and (9) multiplying the cross sections of the parity-favored and parity-nonfavored multipoles,
respectively, are A ∼ +1.07 and B ∼ −0.07 at the pion photoproduction threshold. They in-
crease with photon energy, as expected for relativistic correction factors. Using A(ω) and B(ω)
it is easy to prove that (α + β) ≡ (α + β)s is given by the Baldin or Baldin-Lapidus (BL) [14]
sum rule, whereas (α− β)s is given by the s-channel part of the BEFT [4] sum rule.
For the t-channel parts, αt and βt, we use the predictions obtained from the σ-meson pole
representation2 with properties as predicted by the quark-level Nambu–Jona-Lasinio model [1,
8]. The quantities entering into this prediction are αe = e
2/4π = 1/137.04, the pion-nucleon
coupling constant, gπNN = 13.169±0.057, the pion decay constant, fπ = (92.42±0.26) MeV, and
the σ-meson mass, mσ = 666.0 MeV [1, 10, 11]. For convenience we summarize the arguments
leading to the relations (3) and (6). The flavor wave-functions of the π0 and the σ meson are
given by
|π0〉 =
(−uū+ dd̄), |σ〉 =
(uū+ dd̄). (12)
This leads to the decay matrix elements
M(σ → γγ) = −5
M(π0 → γγ) = 5
. (13)
Using the NJL model or the dynamical LσM with dimensional regularization we arrive at [1,11]
mclσ =
4πf clπ√
, (14)
where mclσ and f
π = 89.8 MeV are the σ meson mass and the π decay constant in the chiral
limit (cl) and Nc = 3 the number of colors. Then the mass of the σ meson is given by
(mclσ )
2 +m2π = 666 MeV. (15)
Inserting this into
(α− β)t = gσNNM(σ → γγ)
2πm2σ
and using fσNN = fπNN and (α+ β)
t = 0 we arrive at (3) and (6).
1It should be noted that this separation into cross sections for separate multipoles is possible in the presently
used fixed-θ dispersion theory applied at θ = 0 and θ = π, whereas in the corresponding formulas based on fixed-t
dispersion theory [12] terms containing mixed products of CGLN [13] amplitudes occur (for a discussion see [6]).
2This σ-meson pole in the complex t-plane of the Compton scattering amplitude A(s, t) is not the same, but
has relations with the σ-meson pole introduced to parameterize the ππ scattering amplitude. These relations
have been discussed in detail in [1,6].
3 Components of electromagnetic polarizabilities from analyses
of total photoabsorption and meson photoproduction data
In the following we use different photoabsorption data to get information on partial contributions
to αs and βs. Analyses of total photoabsorption cross sections have been carried out in [15].
These analyses give a very good overview over the resonant and nonresonant contributions to
the electromagnetic polarizabilities. Further information is taken from the PDG2006 [16], the
GWSES [17] and the Mainz [18,19] analyses of meson photoproduction data.
3.1 Components of electromagnetic polarizabilities from analysis of the total
photoabsorption cross-section of the proton
In the following we wish to study the contributions of nucleon resonances and nonresonant
excited states to the s-channel electromagnetic polarizabilities. Only the resonances P33(1232),
P11(1440), D13(1520), S11(1535) and F15(1680) have to be taken into account. The contributions
of the resonances S11(1650), D15(1675) and higher lying resonances are negligible. For this
analysis we use the Walker [15,20] parameterization of nucleon resonances
I = Ir
W 2r ΓΓγ
(W 2 −W 2r )2 +W 2r Γ2
, (17)
Γ = Γr
)2l+1(
q2r +X
q2 +X2
, (18)
Γγ = Γr
k2r +X
k2 +X2
. (19)
s = 2ωm+m2, ω = photon energy in the lab system, (20)
W 2 = s (21)
k = |k| =
, |k| = photon momentum in the c.m. system, (22)
q = |q| =
E2π −m2π; Eπ =
s−m2 +m2π
, |q| = π momentum in the c.m. system, (23)
jγ , multipole angular momentum of the photon, (24)
l, single π angular momentum. (25)
The damping constants X are X = 160 MeV for the P33(1232) resonance and X = 350 MeV
else.
For the proton, parameters are given in [15] for the relevant resonant states, leading to the
results given in lines 3 – 5 of Table 1. The sum αp+βp of nonresonant contributions in line 7 of
Table 1 is in agreement with the corresponding number calculated from the nonresonant cross
section given in [15] if the nonresonant cross section data are extrapolated to about 3.5 GeV.
This shows that with the predicted t-channel contributions given in line 6 there is consistency
between the experimental electromagnetic polarizabilities and the predictions.
3.2 Components of electromagnetic polarizabilities from analyses of meson
photoproduction for the proton and the neutron
From isospin considerations it has been derived [21] that the amplitudes for meson photopro-
duction are composed of A(1/2) and A(3/2), referring to final states of definite isospin (1
Table 1: Partial contributions to the electromagnetic polarizabilities based on the analysis of
the total photoabsorption cross section [15]. The t-channel parts in line 6 are the predictions
based on the σ-meson pole representation (see section 2). Line 7 contains the differences between
the numbers in line 2 and the sums of numbers given in lines 3–6. The experimental data are
normalized to (α+ β)p = 13.9 ± 0.3 (see [6]).
1 αp βp
2 experiment 12.0 ± 0.6 1.9 ∓ 0.6
3 P33(1232) M1, E2 −1.1 +8.3
4 P11(1440) M1 −0.1 +0.3
5 D13(1520) E1,M2 +1.2 −0.3
6 S11(1535) E1 +0.1 −0.0
5 F15(1680) E2,M3 −0.1 +0.4
6 t-channel +7.6 −7.6
7 nonresonant +4.4 +0.8
Furthermore, there is an amplitude A(0) which may be related to “recoil” effects [21]. This latter
amplitude makes a contribution to I = 1/2 only. Therefore, the amplitudes
(1/2) = A(0) +
A(1/2), nA
(1/2) = A(0) −
A(1/2) (26)
may be introduced. Furthermore, with
A(+) =
(A(1/2) + 2A(3/2)), A(−) =
(A(1/2) −A(3/2)), (27)
the physical amplitudes may be expressed by the isospin combinations (see e.g. [19, 22])
A(γp → nπ+) =
2(A(−) +A(0)) =
(1/2) − 1
A(3/2)), (28)
A(γp → pπ0) = A(+) +A(0) =p A(1/2) +
A(3/2), (29)
A(γn → pπ−) = −
2(A(−) −A(0)) =
(1/2) +
A(3/2)), (30)
A(γn → nπ0) = A(+) −A(0) = −nA(1/2) +
A(3/2). (31)
The relation for the cross section of 1π photoproduction is given by
σ1π = 2π
(l + 1)2
(l + 2)(|El+|2 + |M(l+1)−|2) + l(|Ml+|2 + |E(l+1)−|2)
, (32)
∆σ1π = 2π
(l + 1)2(−1)l
(l + 2)(|El+|2 − |M(l+1)−|2) + l(|Ml+|2 − |E(l+1)−|2)
,(33)
∆σ1π = σ1π(E1,M2, · · · )− σ1π(M1, E2, · · · ). (34)
The peak cross section Ir introduced in (17) is given by
Ir = 2π
2J + 1
2J0 + 1
, (35)
where J and J0 are the spins of the excited state and the ground state, respectively, Γγ the
photon width and Γ the total width of the resonance. The photon width Γγ may be expressed
through the resonance couplings A1/2 and A3/2 by the relation [16]
(2J + 1)MR
|A1/2|2 + |A3/2|2
, (36)
where MN and MR are the nucleon and resonant masses. Combining (35) and (36) we arrive
|A1/2|2 + |A3/2|2
. (37)
Using (37) the quantity Ir can be calculated from the resonance couplings A1/2 and A3/2 given
by the PDG [16], by GWSES [17] and Mainz [19]. The results obtained for the electromagnetic
polarizabilities obtained from the data given in [19] are given in lines 3 – 7 of Table 2.
Table 2: Partial contributions to the electromagnetic polarizabilities. The resonant contributions
in lines 3–7 are obtained from the analysis of Drechsel et al. [19]. The t-channel parts in line
8 are the predictions based on the σ-meson pole representation (see section 2). The predicted
contribution due to the E0+ amplitude in line 9 is based on the analyses given in [17–19]. Line
10 contains the differences between the numbers in line 2 and the sums of numbers given in lines
3–9. The experimental data are normalized to (α+ β)p = 13.9 ± 0.3 and (α + β)n = 15.2 ± 0.5
(see [6]).
1 αp βp αn βn
2 experiment 12.0 ± 0.6 1.9∓ 0.6 12.5 ± 1.7 2.7 ∓ 1.8
3 P33(1232) M
(3/2)
1+ , E
(3/2)
1+ −1.1 +8.3 −1.1 +8.3
4 P11(1440) p,nM
(1/2)
1− −0.0 +0.2 −0.0 +0.1
5 D13(1520) p,nE
(1/2)
2− , p,nM
(1/2)
2− +0.6 −0.2 +0.5 −0.1
6 S11(1535) p,nE
(1/2)
0+ +0.1 −0.0 +0.1 −0.0
7 F15(1680) p,nE
(1/2)
3− , p,nM
(1/2)
3− −0.1 +0.3 −0.0 +0.0
8 t-channel +7.6 −7.6 +7.6 −7.6
9 E0+ (empirical) +3.2 −0.3 +4.1 −0.4
10 background +1.7 +1.2 +1.3 +2.4
The main contributions to the nonresonant parts of the electromagnetic polarizabilities are
expected from the E0+ amplitude which has to be taken from analyses of meson photoproduction
data. Multipole analyses of pion photoproduction based on fixed-t dispersion relations and
unitarity are given by Hanstein et al. [18] in a convenient form. Cross sections separated into
resonant and nonresonant parts are provided for the reactions γp → π+n and γn → π−p up to
energies of 500 MeV and extrapolations of the nonresonant parts are straightforward using the
data contained in [19] and [17]. In principle there is a problem in disentangling resonant and
nonresonant contributions because of interference effects. The interference of the amplitudes
(1/2)
0+ with the S11(1535) and S11(1650) resonances, however, does not lead to problems in
determining the nonresonant E0+ contributions because of the smallness of the resonant parts.
The results for the electromagnetic polarizabilities obtained from these empirical E0+ data are
contained in line 9 of Table 2.
3It should be noted that the quantity Ir of (37) contains the branching correction Γ/Γπ as required.
Up to this point the electromagnetic polarizabilities find an explanation in the numbers given
in lines 3 – 9 of Table 2, with the exception of the small contributions given in line 10 which
deserve a further investigation. These non-E0+ parts of the nonresonant contributions are partly
due to the M
(3/2)
1− , p,nM
(1/2)
1+ and p,nE
(1/2)
1+ amplitudes which interfere with the corresponding
resonant amplitudes p,nM
(1/2)
1− (P11(1440)), M
(3/2)
1+ (P33(1232)) and E
(3/2)
1+ (P33(1232)), respec-
tively [19]. Only the nonresonant parts of the M1− and M1+ amplitudes are expected to be to
some extent important in comparison with dominant E0+ amplitude. Therefore we restrict the
present discussion to the M1− and M1+ amplitudes. Using the data given in [19] we arrive at
the estimates
αnonres.p (M1−) = −0.0, βnonres.p (M1−) = +0.2, αnonres.n (M1−) = −0.1, βnonres.p (M1−) = +0.4,
αnonres.p (M1+) = −0.0, βnonres.p (M1+) = +0.3, αnonres.n (M1+) = −0.1, βnonres.p (M1+) = +0.6.
The conclusion we have to draw from this is that it is not possible to relate the numbers given in
line 10 of Table 2 to known photoproduction processes, unless the two-pion channels are taken
into account (see e.g. [23]). The ππN final states can be characterized either as quasi two-
body states such as π∆ and ρN , or as a ππN component in which both pions are in S waves.
Furthermore, in the Regge regime above ≈ 2000 MeV also f2(1270), a2(1320) and Pomeron
t-channel exchanges play a role. The π∆ contribution has been analyzed in terms of a ∆ Kroll-
Ruderman term and a ∆ pion-pole term [24]. Using data from this analysis [24] we arrive at
(αp,n + βp,n) ≈ 1.0 for this partial ππ channel. The nonresonant cross section above ≈ 2000
MeV makes a contribution of about (αp,n + βp,n) ≈ 0.7.
4 Discussion
4.1 Discussion of the s-channel contribution
For a long time there have been attempts to understand the electromagnetic polarizabilities
predominantly in terms of properties of the “pion cloud” of the nucleon. Among these attempts
CHPT in its original relativistic form [25] is among the most prominent ones. It has been
shown by L’vov [26] that the results obtained for the electromagnetic polarizabilities through
the evaluation of chiral loops [25] can be reproduced via dispersion theory when the Born
approximation of the electric-dipole CGLN amplitude E0+ is taken into account. The results
obtained in this way are shown in lines 2 and 3 of Table 3.
Table 3: Predictions for the “meson cloud” contribution to the electromagnetic polarizabilities
in different approaches.
1 method αp βp αn βn reference
2 CHPT +7.4 −2.0 +10.1 −1.2 Bernard [25]
3 piona) Born +7.3 −1.8 +9.8 −0.9 L’vov [26]
4 E0+ Born +7.5 −1.4 +9.9 −1.8 present
a) The use of fixed-t dispersion theory requires the consideration of interference terms of the
E0+ amplitude with other amplitudes.
It is of interest to use also the present approach based on forward and backward dispersion
relations for studies of this type. For this purpose use may be made of the Born approximation
(see [22] p. 286, [27] p. 35) given in the form
EBorn0+ (γN → π±N) = ±
(−)Born
0+ ± E
(0)Born
, (38)
(−)Born
1− v2
1 + v
, (39)
with v = |q|/
q2 +m2π being the velocity of the pion in the c.m. system. The expression given
in (39) corresponds to the static approximation discussed in detail in [22, 27]. Because of the
relation
σE0+(γn → π−p)
σE0+(γp → π+n)
≃ 1.3 (40)
(see [22] p. 276) the recoil terms E
(0)Born
0+ may be replaced by multiplying E
(−)Born
0+ with (1 +
)−1/2 and (1+mπ
)+1/2 in order to get the results for the proton and neutron, respectively. The
relation given in (40) is well justified at threshold but its approximate validity extends to higher
energies [18,19,27]. The pseudovector coupling constant f in (39) is given by f = gπNN (mπ/2m)
with gπNN = 13.169 ± 0.057. There is a remarkable agreement between the numbers given in
Table 3 but these numbers are larger by a factor ∼ 2.4 than the corresponding numbers in line
9 of Table 2. Two reasons for the deviation of the empirical E0+ amplitude from the Born
approximation have been discussed in [19]. The first reason is that the pseudovector (PV)
coupling is not valid at high photon energies but has to be replaced by some average of the PV
and pseudoscalar the (PS) coupling. The second reason are ρ and ω meson t-channel exchanges
which are not taken into account in the Born approximation.
In Table 2 (see also Table 1) we see that the different resonant contributions to the electric
polarizabilities cancel each other, so that the electric polarizabilities are mainly due to the
t-channel part αtp,n (∼60%) given in line 8 and a smaller nonresonant part α(E0+) (∼30%)
given in line 9. For the magnetic polarizabilities there is an almost complete cancellation of the
P11(1440), D13(1520) and F15(1680) contributions, so that the main remaining contributions are
due to the P33(1232) resonance, canceled to a large extent by the t-channel contribution β
t. The
nonresonant background given in line 10 of Table 2 amounts to about 10% of the experimental
electric polarizabilities and to about 70% of the experimental magnetic polarizabilities. This
means that precise predictions of these contributions are highly desirable, especially for the
magnetic polarizabilities. Unfortunately, the non-E0+ parts of the nonresonant photoabsorption
cross sections are dominated by two-pion channels where the information on the multipole
content is scarce.
4.2 Discussion of the t-channel contribution
In [1] it has been shown that there are two independent, but apparently equivalent and comple-
mentary options to calculate the scalar-isoscalar t-channel contribution to the electromagnetic
polarizabilities of the nucleon.
Option 1 makes use of the properties of the σ-meson as predicted by the quark-level NJL
model and in this respect is of course model dependent. The quark-level NJL model predicts
a definite σ-mesons mass, viz. mσ = 666 MeV, through a parameter-free relation of mσ to
the pion decay constant fπ. The result (α − β)t = 15.2 is in an excellent agreement with the
experimental result. The agreement between a prediction and an experimental result cannot be
used as an argument for the validity of the prediction without further support. This support is
provided by dispersion theory applied to the measured properties of the σ meson as showing up
in particle reactions with two pions in the intermediate state (Option 2).
Option 2 first takes into consideration that the σ meson has been observed in many data
analyses [16] as a pole on the second sheet of the isoscalar S wave of ππ scattering. This pole
describes part of the resonant structure of the σ meson without being a complete description.
This latter property of the pole follows from the fact that the 90◦ crossing of the scalar-isoscalar
phase δ00(s) is located at much higher energies than predicted by the structure of the pole. The
analyses of Colangelo et al. [28] and Caprini et al. [29] led to
s(pole) = (470± 30) − i(295 ± 20) MeV
s(δS = 90
◦) = (844± 13) MeV [28], (41)
Mσ = 441
−8 MeV, Γσ = 544
−25 MeV [29]. (42)
The numbers contained in (41) and (42) are extremely valuable in characterizing the properties
of the σ meson as a real particle but they can only qualitatively be compared with the mass
mσ = 666 MeV of the virtual σ meson, because in the latter case there is no open decay channel.
This means that there is no contradiction between the existence of the broad mass distribution
for the real σ meson and a precisely determined mass of the virtual σ meson. Furthermore, the
numbers contained in (41) and (42) are of no direct relevance for the prediction of (α − β)t.
First of all it certainly would lead only to a qualitative estimate for (α − β)t if the parameters
of the σ-meson pole in (41) and (42) would be used instead of mσ = 666 MeV. Furthermore,
such an insufficient attempt is not necessary because the BEFT [4] sum rule provides a precise
relation between (α − β)t and the properties of the real σ meson. In the BEFT sum rule the
imaginary part of the t-channel Compton scattering amplitude is given by an unitarity relation
where the two reaction γγ → σ → ππ and NN̄ → σ → ππ are exploited. In these reactions
the resonant structure of the σ meson enters via the experimentally determined scalar-isoscalar
phase δ00(s) which is considerably different from the corresponding quantity predicted by the
poles shown in (41) and (42). The real part of the t-channel Compton scattering amplitude is
obtained via a dispersion relation. The present status of the evaluation of the BEFT sum rule
(α−β)tnp = 15.3±1.3 is in good agreement with the experimental result as well as the prediction
based on the quark-level NJL model.
5 Conclusion
The good agreement of the result based on the BEFT sum rule with the experimental result
as well as the prediction based on the quark level NJL model may be understood as a strong
argument that the two predictions of (α−β)t are equivalent. This implies that in addition to the
poles in (41) and (42) also the mass mσ = 666 MeV of the virtual σ meson is an experimentally
verified property of the σ meson.
Acknowledgment
The author is indebted to Deutsche Forschungsgemeinschaft for the support of this work through
the projects SCHU222 and 436RUS113/510. He thanks M.I. Levchuk, A.I. L’vov and A.I. Mil-
stein for a long term cooperation which contributed to the motivation for the present investiga-
tion.
References
[1] M. Schumacher, Eur. Phys. J. A 30, 413 (2006); DOI 10.1140/epja/i2006-10103-0
[hep-ph/0609040].
[2] A.C. Hearn, E. Leader, Phys. Rev. 126, 789 (1962); R. Köberle, Phys. Rev. 166, 1558
(1968).
[3] E.E. Low, Phys. Rev. 120, 582 (1960) (and reference therein); M. Jacob, J. Mathews, Phys.
Rev. 117, 854 (1960).
[4] J. Bernabeu, T.E.O. Ericson, C. Ferro Fontan, Phys. Lett. 49 B, 381 (1974); J. Bernabeu,
B. Tarrach, Phys. Lett 69 B, 484 (1977).
[5] I. Guiasu, E.E. Radescu, Phys. Rev. D 14, 1335 (1976); Phys. Lett. 62 B, 193 (1976).
[6] M. Schumacher, Prog. Part. Nucl. Phys. 55, 567 (2005) [hep-ph/0501167].
[7] B.R. Holstein, A.M. Nathan, Phys. Rev. D 49, 6101 (1994).
[8] M.I. Levchuk, A.I. L’vov, A.I. Milstein, M. Schumacher, Proceedings of the Workshop
NSTAR2005, 12–15 October 2005, Tallahassee, Florida, edited by S. Capstick, V. Crede,
P. Eugenio (World Scientific 2006) 389 [hep-ph/0511193].
[9] D. Drechsel et al., Phys. Rep. 378, 99 (2003); Phys. Rev. C 61, 015204 (1999).
[10] T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 221 (1994).
[11] R. Delbourgo, M. Scadron, Mod. Phys. Lett. A 10, 251 (1995) [hep-ph/9910242]; Int. J.
Mod. Phys. A 13, 657 (1998) [hep-ph/9807504].
[12] A.I. L’vov, V.A. Petrun’kin, M. Schumacher, Phys. Rev. C 55, 359 (1997).
[13] G.F. Chew, M.L. Goldberger, F.E. Low, Y. Nambu, Phys. Rev. 106, 1345 (1957).
[14] A.M. Baldin, Nucl. Phys. 18, 310 (1960); L.I. Lapidus, Zh. Eksp. Teor. Fiz. 43, 1358 (1962)
[Sov. Phys. JETP 16, 964 (1963)].
[15] T.A. Armstrong et al., Phys. Rev. D 5, 1640 (1972); Nucl. Phys. B 41, 445 (1972).
[16] W.-M. Yao et al., (Particle Data Group) J. Phys. G 33, 1 (2006) [URL: http://pdg.lbl.gov].
[17] R.A. Arndt, et al. Phys. Rev. C 66, 055213 (2002).
[18] O. Hanstein, D. Drechsel, L. Tiator, Nucl. Phys. A 632, 561 (1998).
[19] D. Drechsel, O. Hanstein, S.S. Kamalov, L. Tiator, Nucl. Phys. A 645, 145 (1999).
[20] R.L. Walker, Phys. Rev. 182, 1729 (1969).
[21] K.M. Watson, Phys. Rev. 95, 228 (1954).
[22] T. Ericson, W. Weise, Pions and Nuclei, International Series of Monographs on Physics 74,
Oxford Science Publications (1988).
[23] D. Drechsel, L. Tiator, J. Phys. G: Nucl. Part. Phys. 18, 449 (1992).
[24] J.A. Gómez Tejedor, E. Oset, Nucl. Phys. A 571, 667 (1994); 600, 413 (1996).
http://arxiv.org/abs/hep-ph/0609040
http://arxiv.org/abs/hep-ph/0501167
http://arxiv.org/abs/hep-ph/0511193
http://arxiv.org/abs/hep-ph/9910242
http://arxiv.org/abs/hep-ph/9807504
http://pdg.lbl.gov
[25] V. Bernard, N. Kaiser, U.-G. Meissner, Phys. Rev. Lett. 67, 1515 (1991); Nucl. Phys. B
373, 346 (1992).
[26] A.I. L’vov, Phys. Lett. B 304, 29 (1993).
[27] A. Donnachie, in: High Energy Physics, Edited by E.H.S. Burhop V, 1 Academic Press
(1972)
[28] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603, 125 (2001).
[29] I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. 96, 132001 (2006).
	Introduction
	Electromagnetic polarizabilities obtained from the forward-angle sum-rule for (+) and the backward-angle sum-rule for (-)
	Components of electromagnetic polarizabilities from analyses of total photoabsorption and meson photoproduction data
	Components of electromagnetic polarizabilities from analysis of the total photoabsorption cross-section of the proton
	Components of electromagnetic polarizabilities from analyses of meson photoproduction for the proton and the neutron
	Discussion
	Discussion of the s-channel contribution
	Discussion of the t-channel contribution
	Conclusion
