7 The
4He total photo-absorption cross section with two- plus three-nucleon
interactions from chiral effective field theory
Sofia Quaglioni, Petr Navrátil
Lawrence Livermore National Laboratory, L-414, P.O. Box 808, Livermore, CA 94551, USA
Abstract
The total photo-absorption cross section of 4He is evaluated microscopically using two- (NN) and three-nucleon (NNN) interactions
based upon chiral effective field theory (χEFT). The calculation is performed using the Lorentz integral transform method along
with the ab initio no-core shell model approach. An important feature of the present study is the consistency of the NN and NNN
interactions and also, through the Siegert theorem, of the two- and three-body current operators. This is due to the application
of the χEFT framework. The inclusion of the NNN interaction produces a suppression of the low-energy peak and enhancement
of the high-energy tail of the cross section. We compare to calculations obtained using other interactions and to representative
experiments. The rather confused experimental situation in the giant resonance region prevents discrimination among different
interaction models.
Key words:
PACS: 25.20.Dc, 21.30.-x, 21.60.Cs, 27.10.+h
Interactions among nucleons are governed by quantum
chromodynamics (QCD). In the low-energy regime rele-
vant to nuclear structure and reactions, this theory is non-
perturbative, and, therefore, hard to solve. Thus, theory
has been forced to resort to models for the interaction,
which have limited physical basis. New theoretical develop-
ments, however, allow us to connect QCD with low-energy
nuclear physics. Chiral effective field theory (χEFT) [1,2]
provides a promising bridge to the underlying theory, QCD.
Beginning with the pionic or the nucleon-pion system [3]
one works consistently with systems of increasing number
of nucleons [4]. One makes use of spontaneous breaking of
chiral symmetry to systematically expand the strong in-
teraction in terms of a generic small momentum and takes
the explicit breaking of chiral symmetry into account by
expanding in the pion mass. Nuclear interactions are non-
perturbative, because diagrams with purely nucleonic in-
termediate states are enhanced [1,2]. Therefore, the chiral
perturbation expansion is performed for the potential. The
χEFT predicts, along with the nucleon-nucleon (NN) inter-
action at the leading order, a three-nucleon (NNN) interac-
tion at the next-to-next-to-leading order or N2LO [2,5,6],
and even a four-nucleon (NNNN) interaction at the fourth
Email addresses: quaglioni1@llnl.gov (Sofia Quaglioni),
navratil1@llnl.gov (Petr Navrátil).
order (N3LO) [7]. The details of QCD dynamics are con-
tained in parameters, low-energy constants (LECs), not
fixed by the symmetry, but can be constrained by exper-
iment. At present, high-quality NN potentials have been
determined at N3LO [8]. A crucial feature of χEFT is the
consistency between the NN, NNN and NNNN parts. As a
consequence, at N2LO and N3LO, except for two parame-
ters assigned to two NNN diagrams, the potential is fully
constrained by the parameters defining the NN interaction.
The full interaction up to N2LO was first applied to the
analysis of nd scattering [6] and later the N3LO NN poten-
tial was combinedwith the availableNNN atN2LO to study
the 7Li structure [9]. In a recent work [10] the NN potential
at N3LO of Ref. [8] and the NNN interaction at N2LO [5,6]
have been applied to the calculation of various properties
of s- and mid-p-shell nuclei, using the ab initio no-core shell
model (NCSM) [11,12], up to now the only approach able to
handle the chiral NN+NNN potentials for systems beyond
A = 4. In that study, a preferred choice of the two NNN
LECs was found and the fundamental importance of the
chiral NNN interaction was demonstrated for reproducing
the structure of light nuclei. In the present work, we ap-
ply for the first time the same χEFT interactions to the ab
inito calculation of reaction observables involving the con-
tinuum of the four-nucleon system. In particular, we study
Preprint submitted to Elsevier 30 October 2018
http://arxiv.org/abs/0704.1336v1
the 4He total photo-absorption cross section.
Experimental measurements of the α particle photo-
disintegration suffer from a recurrent history of large
discrepancies in the near-threshold region, where the
4He(γ, p)3H and the 4He(γ, n)3He break-up channels dom-
inate the total photo-absorption cross section (we refer the
reader to the reviews of available data in Refs. [13,14,15]).
The latest examples date back to the past two years [15,16].
Of particular controversy is the height of the cross section
at the peak, alternatively found to be either pronounced
or suppressed with differences up to a factor of 2 between
different experimental data. With the exception of [17],
early evaluations of the 4He photo-disintegration [18,19,14]
showed better agreement with the high-peaked experi-
ments, and, ultimately, with those of Ref. [16]. The in-
ability of these calculations to reproduce a suppressed
cross section at low energy was often imputed to the semi-
realistic nature of the Hamiltonian and, in particular, to
the absence of the NNN force. The introduction of NNN in-
teractions leads, indeed, to a reduction of the peak height,
as it was recently shown in a calculation of the photo-
absorption cross section with the Argonne V18 (AV18)
NN potential augmented by the Urbana IX (UIX) NNN
force [20]. A damping of the peak was also found using the
correlated AV18 potential constructed within the unitary
correlation operator method (UCOM) [21]. In both cases,
however, the suppression is not sufficient to reach the low-
lying data, and in particular those of Ref. [15]. The latter
calculations represent a substantial step forward in the
study of the 4He photo-disintegration. However, they still
present a residual degree of arbitrariness in the choice of
the NNN force to complement AV18 in the first case, or
in the choice of the unitary transformation leading to the
non-local phase-equivalent interaction in the second case.
We note that the Illinois potential models have been found
to be more realistic NNN partners of AV18 in the repro-
duction of the structure of light p-shell nuclei [22]. From
a fundamental point of view, it is therefore important to
calculate the 4He photo-absorption cross section in the
framework of χEFT theory, where NN and NNN potentials
are derived in a consistent way and their relative strengths
is well established by the order in the chiral expansion.
When the wavelength of the incident radiation is much
larger than the spatial extension of the system under con-
sideration, the nuclear photo-absorption process can be de-
scribed in good approximation by the cross section
σγ(ω) = 4π
ωR(ω) , (1)
where ω is the incident photon energy and the inclusive
response function
R(ω) =
〈Ψf | D̂ |Ψ0〉
δ(Ef − E0 − ω) (2)
is the sum of all the transitions from the ground state
|Ψ0〉 to the various allowed final states |Ψf 〉 induced by the
dipole operator:
riY10(r̂i) . (3)
In the above equations ground- and final-state energies are
denoted by E0 and Ef , respectively, whereas τ
i and r i =
rir̂i represent the isospin third component and center of
mass frame coordinate of the ith nucleon. This form of
the transition operator includes the leading effects of the
meson-exchange currents through the Siegert’s theorem.
Additional contributions to the cross section (due to retar-
dation, higher electric multiples, magnetic multiples) not
considered by this approximation are found to be negligi-
ble in the A = 2 [23] and A = 3 [24] nuclei, in particular
for ω . 40 MeV. A similar behavior can be expected from
a system of small dimensions like the 4He.
Denoting by Ĥ the full Hamiltonian of the system,
p i − p j
V NNij +
i<j<k
V NNNijk , (4)
wherem is the nucleon mass, V NNij is the sum of N
3LO NN
and Coulomb interactions, and V NNNij is the N
2LO NNN
force, we i) solve the many-body Schrödinger equation for
the ground state |Ψ0〉, ii) obtain the response (2) by eval-
uation [25,26] and subsequent inversion [27] of an integral
transform with a Lorentzian kernel of finite width σI ∼
10−20 MeV (z = E0 + σR + iσI),
L(σR, σI) =−
〈ψ0|D̂
z − Ĥ
D̂|ψ0〉
(ω − σR)2 + σ
dω , (6)
and iii) calculate the photo-absorption cross section in the
long wave-length approximation using Eq. (1). Following
these steps, a fully microscopic result for the 4He photo-
absorption cross section can be reached through the use
of efficient expansions over localized many-body states. In-
deed, in the technique summarized by Eqs. (5-6) and known
as Lorentz integral transform (LIT) method [28], the con-
tinuum problem is mapped onto a bound-state-like prob-
The present calculations are performed in the framework
of the ab initio NCSM approach [11]. This method looks
for the eigenvectors of Ĥ in the form of expansions over a
complete set of harmonic oscillator (HO) basis states up
to a maximum excitation of Nmax~Ω above the minimum
energy configuration, where Ω is the HO parameter. The
convergence to the exact results with increasing Nmax is
accelerated by the use of an effective interaction derived,
in this case, from the adopted NN and NNN χEFT po-
tentials at the three-body cluster level [12]. The reliabil-
ity of the NCSM approach combined with the LIT method
was validated by comparing to the results obtained with
the effective-interaction hyper-spherical harmonics (EIHH)
technique [29] in a recent benchmark calculation [30]. A
complete description of the NCSM approachwas presented,
e.g., in Refs. [11,12]. Here, we emphasize some of the as-
pects involved in a calculation of the effective interaction
at the three-body cluster level in presence of a NNN poten-
tial. We use the Jacobi coordinate HO basis antisymmen-
trized according to the method described in Ref. [31]. The
NCSM calculation proceeds as follows. First, we diagonalize
the Hamiltonian with and without the NNN interaction in
a three-nucleon basis for all relevant three-body channels.
Second, we use the three-body solutions from the first step
to derive three-body effective interactions with and without
the NNN interaction. By subtracting the two effective in-
teractions we isolate the NN and NNN contributions. This
is needed due to a different scaling with particle number of
the two- and the three-body interactions. The 4He effective
interaction is then obtained by adding the two contribu-
tions with the appropriate scaling factors [12]. Note that
our effective interaction is model-space dependent. Conse-
quently, we need both the effective interaction for the 4He
ground state (JπT = 0+0), and the one for the 1−1 states,
entering the LIT calculation. Indeed, due to the change of
parity, the model-space size changes (Nmax → Nmax + 1).
With the effective interactions replacing the interactions in
the Hamiltonian (4), the four-nucleon calculations proceed
as described in the text following Eq. (4).
We start our discussion presenting the results obtained
for the ground state of the α particle using two different
values of the HO parameter, namely ~Ω = 22 and 28 MeV.
This choice for the HO frequencies is driven by our final
goal of evaluating the 4He photo-absorption cross section
and providing an estimate for its theoretical uncertainty.
Indeed, in the particular case of the 4He nucleus, frequen-
cies in the range 12 ≤ ~Ω ≤ 28MeV allow to achieve a good
description of both ground state and complex energy con-
tinuum, as required in a calculation of response functions
with the LIT method [30].
For all of the three observables examined in Fig. 1 the
χEFT NN and NN+NNN interactions lead to very sim-
ilar and smooth convergence patterns. In particular, an
accurate convergence is reached starting from Nmax = 18,
as we find independence from both model space and fre-
quency. Although χEFT forces are known to present a
relatively soft core, the use of effective interactions for both
the NN and NNN forces is the essential key to this remark-
able result. The summary of the extrapolated ground-sate
properties is presented in Table 1. The present results
for ground-state energy and point-proton radius with the
N3LONN interaction are consistent with a previous NCSM
evaluation (E0 = −25.36(4) MeV, 〈r
2 = 1.515(10) fm)
obtained using a two-body effective interaction in a model
space up to Nmax = 18 [35] and with that obtained by
the hyper-spherical harmonic variational calculation of
Ref. [36] (E0 = −25.38 MeV, 〈r
2 = 1.516 fm ) and
by the Faddeev-Yakubovsky method [37] (E0 = −25.37
MeV). Finally, with the present choice for the LECs [10]
the calculated binding-energy with inclusion of the NNN
force is within few hundred KeV of experiment. This leaves
room for additional effects expected from the inclusion
-31.0
-30.0
-29.0
-28.0
-27.0
-26.0
-25.0
0 2 4 6 8 10 12 14 16 18 20 22
PSfrag replacements
NN+NNN
h̄Ω = 22 MeV
h̄Ω = 28 MeV
Fig. 1. (Color online) The 4He ground-state energy E0 [panel a)],
point-proton root-mean-square radius 〈r2p〉
2 [panel b)] and total
dipole strength 〈Ψ0|D̂
†D̂|Ψ0〉 [panel c)] obtained with the χEFT
NN and NN+NNN interactions. Convergence pattern with respect
to the model space truncation Nmax for ~Ω = 22 and ~Ω = 28 MeV.
of the here missing N3LO NNN (not yet available) and
NNNN interaction terms [38].
At the ground-state level, the inclusion of the NNN force
affects mostly the energy, providing 3.21 MeV additional
binding, while only a weak suppression of about 3.8% is
found for the point-proton radius. That the total dipole
strength follows the same pattern as the radius and is re-
duced of 7.9% is not so surprising considering the approxi-
mate relation between them [39]:
〈Ψ0|D̂
†D̂|Ψ0〉 ≃
3(A− 1)
〈r2p〉 . (7)
The latter expression, which is exact for the deuteron and
the triton and for ground-state wave functions symmetric
under exchange of the spatial coordinates of any pair of nu-
cleons, represents a quite reasonable approximation for the
α-particle and is found to be within 9% off our calculations
with both the NN and NN+NNN χEFT potentials. As we
Table 1
Calculated 4He ground-state energy E0, point-proton root-mean-
square radius 〈r2p〉
2 , and total dipole strength 〈Ψ0|D̂
†D̂|Ψ0〉 ob-
tained using the χEFT NN and NN+NNN interactions compared
to experiment. The experimental value of the point-proton radius is
deduced from the measured alpha-particle charge radius, 〈r2c〉
1.673(1) fm [32], proton charge radius, 〈R2p〉
2 = 0.895(18) fm [33],
and neutron mean-square-charge radius, 〈R2n〉 = −0.120(5) fm
2 [34].
E0 [MeV] 〈r
2 [fm] 〈Ψ0|D̂
†D̂|Ψ0〉 [fm
NN -25.39(1) 1.515(2) 0.943(1)
NN+NNN -28.60(3) 1.458(2) 0.868(1)
Expt. -28.296 1.455(7) -
10 20 30 40 50 60
100 150 200
18/19
16/17
14/15
12/13
0 40 80 120 160 200
PSfrag replacements
σR [MeV]
σI = 20 MeV
Nmax = 18/19
h̄Ω = 22 MeV
h̄Ω = 22 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
NN+NNN
NN+NNN
Fig. 2. (Color online) The LIT of the 4He dipole response as a func-
tion of σR at σI = 20 MeV. Convergence pattern of the NN+NNN
calculation with respect to the model-space truncation Nmax for
~Ω = 28 MeV (upper panel), and frequency dependence of the best
(Nmax = 18/19) results with and without inclusion of the NNN force
(lower panel).
will see later, this also implies rather weak NNN effects on
the 4He photo-absorption cross section at low energy.
We turn now to the second part of our calculation, for
which the ground state is an input. The actual evaluation
of Eq. (5) is performed by applying the Lanczos algorithm
to the Hamiltonian of the system, using as starting vector
|ϕ0〉 = 〈Ψ0|D̂
†D̂|Ψ0〉
2 D̂|Ψ0〉 [26,30]. Indeed, the LIT can
be written as a continued fraction of the elements of the
resulting tridiagonal matrix, the so-called Lanczos coeffi-
cients an and bn:
L(σ) =
〈Ψ0|D̂
†D̂|Ψ0〉
(z − a0)−
(z−a1)−
(z−a2)−
. (8)
Due to the selection rules induced by the dipole opera-
tor (3), for a given truncationNmax in the 0
+0 model space
used to expand the ground state, a complete calculation
of Eq. (8) requires an expansion of |ϕ0〉 over a 1
−1 space
up to Nmax + 1. This is the oriigin of the even/odd no-
tation for Nmax introduced to describe the convergence of
the LIT in Fig. 2. The LITs obtained using the NN and
NN+NNN χEFT interactions show, once again, conver-
gence patterns very similar to each other. As an example,
in the upper panel of Fig. (2) we show the model-space de-
pendence of the LIT including the NNN force at ~Ω = 28
MeV. Thanks to the use of three-body effective interaction
for both the NN and NNN terms of the potential, a sta-
ble position and height of the peak in the low-σR region
and satisfactory quenching of the oscillations in the tail are
found for Nmax = 18/19. In this regard, our approach dif-
20 25 30 35
18/19
16/17
14/15
40 60 80 100 120
20 40 60 80 100 120
PSfrag replacements
ω [MeV]
Nmax = 18/19
h̄Ω = 22 MeV
h̄Ω = 22 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
NN+NNN
NN+NNN
Fig. 3. (Color online) The 4He photo-absorption cross section as
a function of the excitation energy ω. Convergence pattern of the
NN+NNN calculation with respect to the model-space truncation
Nmax for ~Ω = 28 MeV (upper panel), and frequency dependence
of the best (Nmax = 18/19) results with and without inclusion of
the NNN force (lower panel).
fers from the one of Ref. [20], where the effective interaction
(at the two-body cluster level) is constructed only for the
NN potential, while the NNN force is taken into account
as bare interaction. The bottom panel of Fig. 2 indicates
that for Nmax = 18/19 we find also a fairly good agree-
ment between the ~Ω = 22 and ~Ω = 28 MeV calculations,
in particular below σR = 60 MeV, where for both NN and
NN+NNN interactions the two curves are within 0.5% of
each other. At higher σR the ~Ω = 22 MeV results present
a weak oscillation (less than 5% in the range 60 ≤ σR ≤
140 MeV) around the ~Ω = 28 MeV curves, and the dis-
crepancy between the two frequencies becomes larger be-
yond σR = 140 MeV, where the absolute value of the LIT
is small. As we will see later, this small discrepancy will
be propagated to the cross section by the inversion proce-
dure [27], giving rise to the uncertainty of our calculations.
As for the NNN effects at the level of the LIT, the shift
of about 3 MeV in the position of the peak is due to the
different ground-state energies for the NN and NN+NNN
potentials. In addition one can notice a quenching of about
13% of the peak height.
In analogy with Fig. 2, Fig. 3 shows the convergence be-
havior of our results for the cross section. Starting from
Nmax = 14/15 the calculated LIT’s are accurate enough to
find stable inversions for the response function, and hence
deriving the corresponding results for the cross section. The
curves obtained for the NN+NNN interaction at the HO
frequency value of ~Ω = 28 MeV are shown in the upper
panel: the model space dependence is weak and the differ-
ence between Nmax = 16/17 and 18/19 never exceeds 5%
in the range from threshold to ω = 120 MeV. A somewhat
larger discrepancy (less than 7%) is found by comparing
the best results (Nmax = 18/19) for ~Ω = 22 MeV and 28
MeV. As with the LIT, the first oscillates slightly around
the second, particularly in the tail of the cross section. We
will use this discrepancy as an estimate for the theoretical
uncertainty of our calculations. Note that both the NN and
NN+NNN calculated cross sections are translated to the
experimental threshold for the 4He photo-disintegration,
Eth = 19.8 MeV (ω → ω+∆Eth, with ∆Eth being the dif-
ference of the calculated and experimental thresholds). The
same procedure will be applied later in the comparisonwith
experimental data and different potential models. Under
this arrangement, the position of the peak is not affected
by the inclusion of the NNN force, while the relative differ-
ence between the NN and NN+NNN cross sections varies
almost linearly from −10% at threshold to about +25% at
ω = 120 MeV. In particular, the peak height undergoes a
9% suppression and the two curves cross around ω = 40
MeV. In view of the inverse-energy-weighted integral of the
cross-section (1),
σγ(ω)
dω = 4π2
〈Ψ0|D̂
†D̂|Ψ0〉 , (9)
the mildness of the NNN force effects in the peak region
is a consequence of the small reduction found for the total
dipole strength. Considering in addition the approximate
relation (7), we can infer a weak sensitivity of the cross
section at low energy with respect to variations of the LECs
in the NNN force, for which we have embraced the preferred
choice suggested in Ref. [10].
We compare to experimental data in the region ω < 40
MeV, where corrections to the unretarded dipole approxi-
mation are expected to be largely negligible and the relative
uncertainty of our calculations is minimal. The data sets
from Nilsson et al. [16] and Shima et al. [15] are chosen here
as the latest examples of controversial experiments charac-
terizing the 4He photo-effect since the 50’s (see reviews of
available data in Refs. [13,14] and [15]). Note that in the
upper panel of Fig. 4, we estimate the total cross section
from the 4He(γ, n) measurements of Ref. [16] by assuming
σγ(ω) ≃ 2σγ,n(ω). The latter assumption, which relies on
the similarity of the (γ, p) and (γ, n) cross sections, pro-
vides a sufficiently safe estimate of the total cross section
below the three-body break-up threshold (ω = 26.1 MeV).
At higher energies it represents a lower experimental bound
for the total cross section, as in the energy range considered
here the data of Nillsson et al. do not contain the contribu-
tions of the 4He(γ, np)d and four-body break-up channels.
Shima et al. [15] provide total photo-disintegration data
obtained by simultaneous measurements of all the open
channels. Finally, we show also an indirect determination
of the photo-absorption cross section deduced from elas-
tic photon-scattering on 4He by Wells et al. [40]. We find
an overall good agreement with the photo-disintegration
data from bremsstrahlung photons [16], which are consis-
tent with the indirect measurements of Ref. [40], while we
20 25 30 35
20 25 30 35
PSfrag replacements
4He+γ →X
ω [MeV]
AV18+UIX
NN+NNN
χEFT NN
χEFT NN+NNN
Nilsson (‘07)
Shima (‘05)
Wells (‘92)
Fig. 4. (Color online) The 4He photo-absorption cross section as a
function of the excitation energy ω. Present NCSM results obtained
using the χEFT NN and NN+NNN interactions compared to: (upper
panel) the 4He(γ, n) data of Nilsson et al. [16] multiplied by a factor
of 2, the total cross section measurements of Shima et al. [15], the
total photo-absorption at the peak derived from Compton scattering
via dispersion relations from Wells et al. [40]; (lower panel) the EIHH
predictions for AV18, AV18+UIX [20] and UCOM [21]. The widths of
the χEFT NN and χEFT NN+NNN curves reflect the uncertainties
in the calculations (see text).
reach only the last of the experimental points of Ref. [15].
The lower panel of Fig. 4 compares our present results
with the prediction for the 4He photo-absorption cross sec-
tion obtained in the framework of the EIHH approach [29]
using the AV18, AV18+UIX [20] and UCOM [21] interac-
tions. Interestingly, both the results with AV18 and χEFT
NN interactions and those with AV18+UIX and χEFT
NN+NNN forces show similar peak heights (∼ 3.2 mb and
∼ 3.0mb respectively), but different peak positions (partic-
ularly for the first case) with an overall better agreement of
the second set of curves. In this regard we notice that the α-
particle ground-state properties obtained with AV18+UIX
and the χEFT NN+NNN are very close to each other and
to experiment. On the contrary, already at the ground-state
level the two NN interactions are less alike as the 4He with
the AV18 potential is more than 1 MeV less bound than
with the N3LO NN potential, while they still yield to the
same point-proton radius. A somewhat larger discrepancy
is found close to threshold between the cross sections ob-
tained with the χEFT NN+NNN and UCOM interactions.
Beyond ω = 80 MeV, in the range not shown in the Fig-
ure, the χEFT NN+NNN force leads to larger cross section
values than AV18+UIX or UCOM, which yield to similar
results in the region 45 ≤ ω ≤ 100 MeV. Keeping in mind
that at such high energies the cross section is small and the
uncertainty in our calculation larger, this effect can be re-
lated in part to differences in the details and interplay of
tensor and spin-orbit forces in the considered interaction
models. At the same time corrections to the unretarded
dipole operator play here a more important role.
In conclusion we summarize our work. We have calcu-
lated the total photo-absorption cross section of 4He us-
ing the potentials of χEFT at the orders presently avail-
able, the NN at N3LO and the NNN at N2LO. The micro-
scopic treatment of the continuum problem was achieved
by means of the LIT method, applied within the NCSM
approach. Accurate convergence in the NCSM expansions
is reached thanks to the use of three-body effective interac-
tions. Our result shows a peak around ω = 27.8 MeV, with
a cross section of 3 mb. The NNN force induces a reduction
of the peak and an enhancement of the tail of the cross sec-
tion. The fairly mild NNN effects are far from explaining
the low-lying experimental data of Ref. [15] while moder-
ately improve the agreement of the calculated cross section
with the measurements of Nilsson et. al. [16]. In view of the
overall good agreement between the χEFT NN+NNN and
AV18+UIX calculations, the photo-absorption cross sec-
tion at low energy appears to be more sensitive to change
in the α-particle size, than to the details of the spin-orbit
component of the NNN interaction. In this regard, a more
substantial role of the NNN force can be expected in the
photo-disintegration of p-shell nuclei, for which differences
in the spin-orbit strength have crucial effects on the spec-
trum [41,10]. Finally, the rather contained width of the the-
oretical band embracing the χEFT NN+NNN, AV18+UIX
and UCOM results within 15 MeV from threshold is re-
markable compared to the large discrepancies still present
among the different experimental data. Hence the urgency
for further experimental activity to help clarify the situa-
tion.
Acknowledgments We would like to thank Winfried Lei-
demann for supplying us with the computer code for the
inversion of the LIT. We are also thankful to Giuseppina
Orlandini and Sonia Bacca for useful discussions, and to Ian
Thompson for critical reading of the manuscript. This work
was performed under the auspices of the U. S. Department
of Energy by the University of California, Lawrence Liv-
ermore National Laboratory under contract No. W-7405-
Eng-48. Support from the LDRD contract No. 04–ERD–
058 and from U.S. DOE/SC/NP (Work Proposal Number
SCW0498) is acknowledged.
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	References

  The total photo-absorption cross section of $^4$He is evaluated
microscopically using two- (NN) and three-nucleon (NNN) interactions based upon
chiral effective field theory ($\chi$EFT). The calculation is performed using
the Lorentz integral transform method along with the {\em ab initio} no-core
shell model approach. An important feature of the present study is the
consistency of the NN and NNN interactions and also, through the Siegert
theorem, of the two- and three-body current operators. This is due to the
application of the $\chi$EFT framework. The inclusion of the NNN interaction
produces a suppression of the low-energy peak and enhancement of the
high-energy tail of the cross section. We compare to calculations obtained
using other interactions and to representative experiments. The rather confused
experimental situation in the giant resonance region prevents discrimination
among different interaction models.

7 The
4He total photo-absorption cross section with two- plus three-nucleon
interactions from chiral effective field theory
Sofia Quaglioni, Petr Navrátil
Lawrence Livermore National Laboratory, L-414, P.O. Box 808, Livermore, CA 94551, USA
Abstract
The total photo-absorption cross section of 4He is evaluated microscopically using two- (NN) and three-nucleon (NNN) interactions
based upon chiral effective field theory (χEFT). The calculation is performed using the Lorentz integral transform method along
with the ab initio no-core shell model approach. An important feature of the present study is the consistency of the NN and NNN
interactions and also, through the Siegert theorem, of the two- and three-body current operators. This is due to the application
of the χEFT framework. The inclusion of the NNN interaction produces a suppression of the low-energy peak and enhancement
of the high-energy tail of the cross section. We compare to calculations obtained using other interactions and to representative
experiments. The rather confused experimental situation in the giant resonance region prevents discrimination among different
interaction models.
Key words:
PACS: 25.20.Dc, 21.30.-x, 21.60.Cs, 27.10.+h
Interactions among nucleons are governed by quantum
chromodynamics (QCD). In the low-energy regime rele-
vant to nuclear structure and reactions, this theory is non-
perturbative, and, therefore, hard to solve. Thus, theory
has been forced to resort to models for the interaction,
which have limited physical basis. New theoretical develop-
ments, however, allow us to connect QCD with low-energy
nuclear physics. Chiral effective field theory (χEFT) [1,2]
provides a promising bridge to the underlying theory, QCD.
Beginning with the pionic or the nucleon-pion system [3]
one works consistently with systems of increasing number
of nucleons [4]. One makes use of spontaneous breaking of
chiral symmetry to systematically expand the strong in-
teraction in terms of a generic small momentum and takes
the explicit breaking of chiral symmetry into account by
expanding in the pion mass. Nuclear interactions are non-
perturbative, because diagrams with purely nucleonic in-
termediate states are enhanced [1,2]. Therefore, the chiral
perturbation expansion is performed for the potential. The
χEFT predicts, along with the nucleon-nucleon (NN) inter-
action at the leading order, a three-nucleon (NNN) interac-
tion at the next-to-next-to-leading order or N2LO [2,5,6],
and even a four-nucleon (NNNN) interaction at the fourth
Email addresses: quaglioni1@llnl.gov (Sofia Quaglioni),
navratil1@llnl.gov (Petr Navrátil).
order (N3LO) [7]. The details of QCD dynamics are con-
tained in parameters, low-energy constants (LECs), not
fixed by the symmetry, but can be constrained by exper-
iment. At present, high-quality NN potentials have been
determined at N3LO [8]. A crucial feature of χEFT is the
consistency between the NN, NNN and NNNN parts. As a
consequence, at N2LO and N3LO, except for two parame-
ters assigned to two NNN diagrams, the potential is fully
constrained by the parameters defining the NN interaction.
The full interaction up to N2LO was first applied to the
analysis of nd scattering [6] and later the N3LO NN poten-
tial was combinedwith the availableNNN atN2LO to study
the 7Li structure [9]. In a recent work [10] the NN potential
at N3LO of Ref. [8] and the NNN interaction at N2LO [5,6]
have been applied to the calculation of various properties
of s- and mid-p-shell nuclei, using the ab initio no-core shell
model (NCSM) [11,12], up to now the only approach able to
handle the chiral NN+NNN potentials for systems beyond
A = 4. In that study, a preferred choice of the two NNN
LECs was found and the fundamental importance of the
chiral NNN interaction was demonstrated for reproducing
the structure of light nuclei. In the present work, we ap-
ply for the first time the same χEFT interactions to the ab
inito calculation of reaction observables involving the con-
tinuum of the four-nucleon system. In particular, we study
Preprint submitted to Elsevier 30 October 2018
http://arxiv.org/abs/0704.1336v1
the 4He total photo-absorption cross section.
Experimental measurements of the α particle photo-
disintegration suffer from a recurrent history of large
discrepancies in the near-threshold region, where the
4He(γ, p)3H and the 4He(γ, n)3He break-up channels dom-
inate the total photo-absorption cross section (we refer the
reader to the reviews of available data in Refs. [13,14,15]).
The latest examples date back to the past two years [15,16].
Of particular controversy is the height of the cross section
at the peak, alternatively found to be either pronounced
or suppressed with differences up to a factor of 2 between
different experimental data. With the exception of [17],
early evaluations of the 4He photo-disintegration [18,19,14]
showed better agreement with the high-peaked experi-
ments, and, ultimately, with those of Ref. [16]. The in-
ability of these calculations to reproduce a suppressed
cross section at low energy was often imputed to the semi-
realistic nature of the Hamiltonian and, in particular, to
the absence of the NNN force. The introduction of NNN in-
teractions leads, indeed, to a reduction of the peak height,
as it was recently shown in a calculation of the photo-
absorption cross section with the Argonne V18 (AV18)
NN potential augmented by the Urbana IX (UIX) NNN
force [20]. A damping of the peak was also found using the
correlated AV18 potential constructed within the unitary
correlation operator method (UCOM) [21]. In both cases,
however, the suppression is not sufficient to reach the low-
lying data, and in particular those of Ref. [15]. The latter
calculations represent a substantial step forward in the
study of the 4He photo-disintegration. However, they still
present a residual degree of arbitrariness in the choice of
the NNN force to complement AV18 in the first case, or
in the choice of the unitary transformation leading to the
non-local phase-equivalent interaction in the second case.
We note that the Illinois potential models have been found
to be more realistic NNN partners of AV18 in the repro-
duction of the structure of light p-shell nuclei [22]. From
a fundamental point of view, it is therefore important to
calculate the 4He photo-absorption cross section in the
framework of χEFT theory, where NN and NNN potentials
are derived in a consistent way and their relative strengths
is well established by the order in the chiral expansion.
When the wavelength of the incident radiation is much
larger than the spatial extension of the system under con-
sideration, the nuclear photo-absorption process can be de-
scribed in good approximation by the cross section
σγ(ω) = 4π
ωR(ω) , (1)
where ω is the incident photon energy and the inclusive
response function
R(ω) =
〈Ψf | D̂ |Ψ0〉
δ(Ef − E0 − ω) (2)
is the sum of all the transitions from the ground state
|Ψ0〉 to the various allowed final states |Ψf 〉 induced by the
dipole operator:
riY10(r̂i) . (3)
In the above equations ground- and final-state energies are
denoted by E0 and Ef , respectively, whereas τ
i and r i =
rir̂i represent the isospin third component and center of
mass frame coordinate of the ith nucleon. This form of
the transition operator includes the leading effects of the
meson-exchange currents through the Siegert’s theorem.
Additional contributions to the cross section (due to retar-
dation, higher electric multiples, magnetic multiples) not
considered by this approximation are found to be negligi-
ble in the A = 2 [23] and A = 3 [24] nuclei, in particular
for ω . 40 MeV. A similar behavior can be expected from
a system of small dimensions like the 4He.
Denoting by Ĥ the full Hamiltonian of the system,
p i − p j
V NNij +
i<j<k
V NNNijk , (4)
wherem is the nucleon mass, V NNij is the sum of N
3LO NN
and Coulomb interactions, and V NNNij is the N
2LO NNN
force, we i) solve the many-body Schrödinger equation for
the ground state |Ψ0〉, ii) obtain the response (2) by eval-
uation [25,26] and subsequent inversion [27] of an integral
transform with a Lorentzian kernel of finite width σI ∼
10−20 MeV (z = E0 + σR + iσI),
L(σR, σI) =−
〈ψ0|D̂
z − Ĥ
D̂|ψ0〉
(ω − σR)2 + σ
dω , (6)
and iii) calculate the photo-absorption cross section in the
long wave-length approximation using Eq. (1). Following
these steps, a fully microscopic result for the 4He photo-
absorption cross section can be reached through the use
of efficient expansions over localized many-body states. In-
deed, in the technique summarized by Eqs. (5-6) and known
as Lorentz integral transform (LIT) method [28], the con-
tinuum problem is mapped onto a bound-state-like prob-
The present calculations are performed in the framework
of the ab initio NCSM approach [11]. This method looks
for the eigenvectors of Ĥ in the form of expansions over a
complete set of harmonic oscillator (HO) basis states up
to a maximum excitation of Nmax~Ω above the minimum
energy configuration, where Ω is the HO parameter. The
convergence to the exact results with increasing Nmax is
accelerated by the use of an effective interaction derived,
in this case, from the adopted NN and NNN χEFT po-
tentials at the three-body cluster level [12]. The reliabil-
ity of the NCSM approach combined with the LIT method
was validated by comparing to the results obtained with
the effective-interaction hyper-spherical harmonics (EIHH)
technique [29] in a recent benchmark calculation [30]. A
complete description of the NCSM approachwas presented,
e.g., in Refs. [11,12]. Here, we emphasize some of the as-
pects involved in a calculation of the effective interaction
at the three-body cluster level in presence of a NNN poten-
tial. We use the Jacobi coordinate HO basis antisymmen-
trized according to the method described in Ref. [31]. The
NCSM calculation proceeds as follows. First, we diagonalize
the Hamiltonian with and without the NNN interaction in
a three-nucleon basis for all relevant three-body channels.
Second, we use the three-body solutions from the first step
to derive three-body effective interactions with and without
the NNN interaction. By subtracting the two effective in-
teractions we isolate the NN and NNN contributions. This
is needed due to a different scaling with particle number of
the two- and the three-body interactions. The 4He effective
interaction is then obtained by adding the two contribu-
tions with the appropriate scaling factors [12]. Note that
our effective interaction is model-space dependent. Conse-
quently, we need both the effective interaction for the 4He
ground state (JπT = 0+0), and the one for the 1−1 states,
entering the LIT calculation. Indeed, due to the change of
parity, the model-space size changes (Nmax → Nmax + 1).
With the effective interactions replacing the interactions in
the Hamiltonian (4), the four-nucleon calculations proceed
as described in the text following Eq. (4).
We start our discussion presenting the results obtained
for the ground state of the α particle using two different
values of the HO parameter, namely ~Ω = 22 and 28 MeV.
This choice for the HO frequencies is driven by our final
goal of evaluating the 4He photo-absorption cross section
and providing an estimate for its theoretical uncertainty.
Indeed, in the particular case of the 4He nucleus, frequen-
cies in the range 12 ≤ ~Ω ≤ 28MeV allow to achieve a good
description of both ground state and complex energy con-
tinuum, as required in a calculation of response functions
with the LIT method [30].
For all of the three observables examined in Fig. 1 the
χEFT NN and NN+NNN interactions lead to very sim-
ilar and smooth convergence patterns. In particular, an
accurate convergence is reached starting from Nmax = 18,
as we find independence from both model space and fre-
quency. Although χEFT forces are known to present a
relatively soft core, the use of effective interactions for both
the NN and NNN forces is the essential key to this remark-
able result. The summary of the extrapolated ground-sate
properties is presented in Table 1. The present results
for ground-state energy and point-proton radius with the
N3LONN interaction are consistent with a previous NCSM
evaluation (E0 = −25.36(4) MeV, 〈r
2 = 1.515(10) fm)
obtained using a two-body effective interaction in a model
space up to Nmax = 18 [35] and with that obtained by
the hyper-spherical harmonic variational calculation of
Ref. [36] (E0 = −25.38 MeV, 〈r
2 = 1.516 fm ) and
by the Faddeev-Yakubovsky method [37] (E0 = −25.37
MeV). Finally, with the present choice for the LECs [10]
the calculated binding-energy with inclusion of the NNN
force is within few hundred KeV of experiment. This leaves
room for additional effects expected from the inclusion
-31.0
-30.0
-29.0
-28.0
-27.0
-26.0
-25.0
0 2 4 6 8 10 12 14 16 18 20 22
PSfrag replacements
NN+NNN
h̄Ω = 22 MeV
h̄Ω = 28 MeV
Fig. 1. (Color online) The 4He ground-state energy E0 [panel a)],
point-proton root-mean-square radius 〈r2p〉
2 [panel b)] and total
dipole strength 〈Ψ0|D̂
†D̂|Ψ0〉 [panel c)] obtained with the χEFT
NN and NN+NNN interactions. Convergence pattern with respect
to the model space truncation Nmax for ~Ω = 22 and ~Ω = 28 MeV.
of the here missing N3LO NNN (not yet available) and
NNNN interaction terms [38].
At the ground-state level, the inclusion of the NNN force
affects mostly the energy, providing 3.21 MeV additional
binding, while only a weak suppression of about 3.8% is
found for the point-proton radius. That the total dipole
strength follows the same pattern as the radius and is re-
duced of 7.9% is not so surprising considering the approxi-
mate relation between them [39]:
〈Ψ0|D̂
†D̂|Ψ0〉 ≃
3(A− 1)
〈r2p〉 . (7)
The latter expression, which is exact for the deuteron and
the triton and for ground-state wave functions symmetric
under exchange of the spatial coordinates of any pair of nu-
cleons, represents a quite reasonable approximation for the
α-particle and is found to be within 9% off our calculations
with both the NN and NN+NNN χEFT potentials. As we
Table 1
Calculated 4He ground-state energy E0, point-proton root-mean-
square radius 〈r2p〉
2 , and total dipole strength 〈Ψ0|D̂
†D̂|Ψ0〉 ob-
tained using the χEFT NN and NN+NNN interactions compared
to experiment. The experimental value of the point-proton radius is
deduced from the measured alpha-particle charge radius, 〈r2c〉
1.673(1) fm [32], proton charge radius, 〈R2p〉
2 = 0.895(18) fm [33],
and neutron mean-square-charge radius, 〈R2n〉 = −0.120(5) fm
2 [34].
E0 [MeV] 〈r
2 [fm] 〈Ψ0|D̂
†D̂|Ψ0〉 [fm
NN -25.39(1) 1.515(2) 0.943(1)
NN+NNN -28.60(3) 1.458(2) 0.868(1)
Expt. -28.296 1.455(7) -
10 20 30 40 50 60
100 150 200
18/19
16/17
14/15
12/13
0 40 80 120 160 200
PSfrag replacements
σR [MeV]
σI = 20 MeV
Nmax = 18/19
h̄Ω = 22 MeV
h̄Ω = 22 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
NN+NNN
NN+NNN
Fig. 2. (Color online) The LIT of the 4He dipole response as a func-
tion of σR at σI = 20 MeV. Convergence pattern of the NN+NNN
calculation with respect to the model-space truncation Nmax for
~Ω = 28 MeV (upper panel), and frequency dependence of the best
(Nmax = 18/19) results with and without inclusion of the NNN force
(lower panel).
will see later, this also implies rather weak NNN effects on
the 4He photo-absorption cross section at low energy.
We turn now to the second part of our calculation, for
which the ground state is an input. The actual evaluation
of Eq. (5) is performed by applying the Lanczos algorithm
to the Hamiltonian of the system, using as starting vector
|ϕ0〉 = 〈Ψ0|D̂
†D̂|Ψ0〉
2 D̂|Ψ0〉 [26,30]. Indeed, the LIT can
be written as a continued fraction of the elements of the
resulting tridiagonal matrix, the so-called Lanczos coeffi-
cients an and bn:
L(σ) =
〈Ψ0|D̂
†D̂|Ψ0〉
(z − a0)−
(z−a1)−
(z−a2)−
. (8)
Due to the selection rules induced by the dipole opera-
tor (3), for a given truncationNmax in the 0
+0 model space
used to expand the ground state, a complete calculation
of Eq. (8) requires an expansion of |ϕ0〉 over a 1
−1 space
up to Nmax + 1. This is the oriigin of the even/odd no-
tation for Nmax introduced to describe the convergence of
the LIT in Fig. 2. The LITs obtained using the NN and
NN+NNN χEFT interactions show, once again, conver-
gence patterns very similar to each other. As an example,
in the upper panel of Fig. (2) we show the model-space de-
pendence of the LIT including the NNN force at ~Ω = 28
MeV. Thanks to the use of three-body effective interaction
for both the NN and NNN terms of the potential, a sta-
ble position and height of the peak in the low-σR region
and satisfactory quenching of the oscillations in the tail are
found for Nmax = 18/19. In this regard, our approach dif-
20 25 30 35
18/19
16/17
14/15
40 60 80 100 120
20 40 60 80 100 120
PSfrag replacements
ω [MeV]
Nmax = 18/19
h̄Ω = 22 MeV
h̄Ω = 22 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
h̄Ω = 28 MeV
NN+NNN
NN+NNN
Fig. 3. (Color online) The 4He photo-absorption cross section as
a function of the excitation energy ω. Convergence pattern of the
NN+NNN calculation with respect to the model-space truncation
Nmax for ~Ω = 28 MeV (upper panel), and frequency dependence
of the best (Nmax = 18/19) results with and without inclusion of
the NNN force (lower panel).
fers from the one of Ref. [20], where the effective interaction
(at the two-body cluster level) is constructed only for the
NN potential, while the NNN force is taken into account
as bare interaction. The bottom panel of Fig. 2 indicates
that for Nmax = 18/19 we find also a fairly good agree-
ment between the ~Ω = 22 and ~Ω = 28 MeV calculations,
in particular below σR = 60 MeV, where for both NN and
NN+NNN interactions the two curves are within 0.5% of
each other. At higher σR the ~Ω = 22 MeV results present
a weak oscillation (less than 5% in the range 60 ≤ σR ≤
140 MeV) around the ~Ω = 28 MeV curves, and the dis-
crepancy between the two frequencies becomes larger be-
yond σR = 140 MeV, where the absolute value of the LIT
is small. As we will see later, this small discrepancy will
be propagated to the cross section by the inversion proce-
dure [27], giving rise to the uncertainty of our calculations.
As for the NNN effects at the level of the LIT, the shift
of about 3 MeV in the position of the peak is due to the
different ground-state energies for the NN and NN+NNN
potentials. In addition one can notice a quenching of about
13% of the peak height.
In analogy with Fig. 2, Fig. 3 shows the convergence be-
havior of our results for the cross section. Starting from
Nmax = 14/15 the calculated LIT’s are accurate enough to
find stable inversions for the response function, and hence
deriving the corresponding results for the cross section. The
curves obtained for the NN+NNN interaction at the HO
frequency value of ~Ω = 28 MeV are shown in the upper
panel: the model space dependence is weak and the differ-
ence between Nmax = 16/17 and 18/19 never exceeds 5%
in the range from threshold to ω = 120 MeV. A somewhat
larger discrepancy (less than 7%) is found by comparing
the best results (Nmax = 18/19) for ~Ω = 22 MeV and 28
MeV. As with the LIT, the first oscillates slightly around
the second, particularly in the tail of the cross section. We
will use this discrepancy as an estimate for the theoretical
uncertainty of our calculations. Note that both the NN and
NN+NNN calculated cross sections are translated to the
experimental threshold for the 4He photo-disintegration,
Eth = 19.8 MeV (ω → ω+∆Eth, with ∆Eth being the dif-
ference of the calculated and experimental thresholds). The
same procedure will be applied later in the comparisonwith
experimental data and different potential models. Under
this arrangement, the position of the peak is not affected
by the inclusion of the NNN force, while the relative differ-
ence between the NN and NN+NNN cross sections varies
almost linearly from −10% at threshold to about +25% at
ω = 120 MeV. In particular, the peak height undergoes a
9% suppression and the two curves cross around ω = 40
MeV. In view of the inverse-energy-weighted integral of the
cross-section (1),
σγ(ω)
dω = 4π2
〈Ψ0|D̂
†D̂|Ψ0〉 , (9)
the mildness of the NNN force effects in the peak region
is a consequence of the small reduction found for the total
dipole strength. Considering in addition the approximate
relation (7), we can infer a weak sensitivity of the cross
section at low energy with respect to variations of the LECs
in the NNN force, for which we have embraced the preferred
choice suggested in Ref. [10].
We compare to experimental data in the region ω < 40
MeV, where corrections to the unretarded dipole approxi-
mation are expected to be largely negligible and the relative
uncertainty of our calculations is minimal. The data sets
from Nilsson et al. [16] and Shima et al. [15] are chosen here
as the latest examples of controversial experiments charac-
terizing the 4He photo-effect since the 50’s (see reviews of
available data in Refs. [13,14] and [15]). Note that in the
upper panel of Fig. 4, we estimate the total cross section
from the 4He(γ, n) measurements of Ref. [16] by assuming
σγ(ω) ≃ 2σγ,n(ω). The latter assumption, which relies on
the similarity of the (γ, p) and (γ, n) cross sections, pro-
vides a sufficiently safe estimate of the total cross section
below the three-body break-up threshold (ω = 26.1 MeV).
At higher energies it represents a lower experimental bound
for the total cross section, as in the energy range considered
here the data of Nillsson et al. do not contain the contribu-
tions of the 4He(γ, np)d and four-body break-up channels.
Shima et al. [15] provide total photo-disintegration data
obtained by simultaneous measurements of all the open
channels. Finally, we show also an indirect determination
of the photo-absorption cross section deduced from elas-
tic photon-scattering on 4He by Wells et al. [40]. We find
an overall good agreement with the photo-disintegration
data from bremsstrahlung photons [16], which are consis-
tent with the indirect measurements of Ref. [40], while we
20 25 30 35
20 25 30 35
PSfrag replacements
4He+γ →X
ω [MeV]
AV18+UIX
NN+NNN
χEFT NN
χEFT NN+NNN
Nilsson (‘07)
Shima (‘05)
Wells (‘92)
Fig. 4. (Color online) The 4He photo-absorption cross section as a
function of the excitation energy ω. Present NCSM results obtained
using the χEFT NN and NN+NNN interactions compared to: (upper
panel) the 4He(γ, n) data of Nilsson et al. [16] multiplied by a factor
of 2, the total cross section measurements of Shima et al. [15], the
total photo-absorption at the peak derived from Compton scattering
via dispersion relations from Wells et al. [40]; (lower panel) the EIHH
predictions for AV18, AV18+UIX [20] and UCOM [21]. The widths of
the χEFT NN and χEFT NN+NNN curves reflect the uncertainties
in the calculations (see text).
reach only the last of the experimental points of Ref. [15].
The lower panel of Fig. 4 compares our present results
with the prediction for the 4He photo-absorption cross sec-
tion obtained in the framework of the EIHH approach [29]
using the AV18, AV18+UIX [20] and UCOM [21] interac-
tions. Interestingly, both the results with AV18 and χEFT
NN interactions and those with AV18+UIX and χEFT
NN+NNN forces show similar peak heights (∼ 3.2 mb and
∼ 3.0mb respectively), but different peak positions (partic-
ularly for the first case) with an overall better agreement of
the second set of curves. In this regard we notice that the α-
particle ground-state properties obtained with AV18+UIX
and the χEFT NN+NNN are very close to each other and
to experiment. On the contrary, already at the ground-state
level the two NN interactions are less alike as the 4He with
the AV18 potential is more than 1 MeV less bound than
with the N3LO NN potential, while they still yield to the
same point-proton radius. A somewhat larger discrepancy
is found close to threshold between the cross sections ob-
tained with the χEFT NN+NNN and UCOM interactions.
Beyond ω = 80 MeV, in the range not shown in the Fig-
ure, the χEFT NN+NNN force leads to larger cross section
values than AV18+UIX or UCOM, which yield to similar
results in the region 45 ≤ ω ≤ 100 MeV. Keeping in mind
that at such high energies the cross section is small and the
uncertainty in our calculation larger, this effect can be re-
lated in part to differences in the details and interplay of
tensor and spin-orbit forces in the considered interaction
models. At the same time corrections to the unretarded
dipole operator play here a more important role.
In conclusion we summarize our work. We have calcu-
lated the total photo-absorption cross section of 4He us-
ing the potentials of χEFT at the orders presently avail-
able, the NN at N3LO and the NNN at N2LO. The micro-
scopic treatment of the continuum problem was achieved
by means of the LIT method, applied within the NCSM
approach. Accurate convergence in the NCSM expansions
is reached thanks to the use of three-body effective interac-
tions. Our result shows a peak around ω = 27.8 MeV, with
a cross section of 3 mb. The NNN force induces a reduction
of the peak and an enhancement of the tail of the cross sec-
tion. The fairly mild NNN effects are far from explaining
the low-lying experimental data of Ref. [15] while moder-
ately improve the agreement of the calculated cross section
with the measurements of Nilsson et. al. [16]. In view of the
overall good agreement between the χEFT NN+NNN and
AV18+UIX calculations, the photo-absorption cross sec-
tion at low energy appears to be more sensitive to change
in the α-particle size, than to the details of the spin-orbit
component of the NNN interaction. In this regard, a more
substantial role of the NNN force can be expected in the
photo-disintegration of p-shell nuclei, for which differences
in the spin-orbit strength have crucial effects on the spec-
trum [41,10]. Finally, the rather contained width of the the-
oretical band embracing the χEFT NN+NNN, AV18+UIX
and UCOM results within 15 MeV from threshold is re-
markable compared to the large discrepancies still present
among the different experimental data. Hence the urgency
for further experimental activity to help clarify the situa-
tion.
Acknowledgments We would like to thank Winfried Lei-
demann for supplying us with the computer code for the
inversion of the LIT. We are also thankful to Giuseppina
Orlandini and Sonia Bacca for useful discussions, and to Ian
Thompson for critical reading of the manuscript. This work
was performed under the auspices of the U. S. Department
of Energy by the University of California, Lawrence Liv-
ermore National Laboratory under contract No. W-7405-
Eng-48. Support from the LDRD contract No. 04–ERD–
058 and from U.S. DOE/SC/NP (Work Proposal Number
SCW0498) is acknowledged.
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