Realistic Few-Body Physics in the $dd\to \alpha\pi^0$ Reaction
ABSTRACT We use realistic two- and three-nucleon interactions in a hybrid chiral-perturbation-theory calculation of the charge-symmetry-breaking reaction $dd\to\alpha\pi^0$ to show that a cross section of the experimentally measured size can be obtained using LO and NNLO pion-production operators. This result supports the validity of our power counting scheme and demonstrates the necessity of using an accurate treatment of ISI and FSI.
arXiv:nucl-th/0602003v1 1 Feb 2006
Realistic Few-Body Physics in the dd → απ0
A. Nogga1, A. C. Fonseca2, A. G˚ ardestig3, C. Hanhart1, C. J. Horowitz4,
G. A. Miller5, J. A. Niskanen6, and U. van Kolck7
1Institut f¨ ur Kernphysik, Forschungszentrum J¨ ulich, J¨ ulich, Germany
2Centro de F´isica Nuclear, Universidade de Lisboa, 1649-003 Lisboa, Portugal
3Department of Physics and Astronomy, University of South Carolina, Columbia,
SC 29208, USA
4Department of Physics and Nuclear Theory Center, Indiana University,
Bloomington, IN 47405, USA
5Department of Physics, University of Washington, Seattle, WA 98195-1560, USA
6Department of Physical Sciences, University of Helsinki, Helsinki, Finland
7Department of Physics, University of Arizona, Tucson, AZ 85721, USA
We use realistic two- and three-nucleon interactions in a hybrid chiral-perturbation-
theory calculation of the charge-symmetry-breaking reaction dd → απ0to show that
a cross section of the experimentally measured size can be obtained using LO and
NNLO pion-production operators. This result supports the validity of our power
counting scheme and demonstrates the necessity of using an accurate treatment of
ISI and FSI.
1. For most purposes, hadronic isospin states can be considered as charge symmetric, i.e.,
invariant under a rotation by 180◦around the 2-axis in isospin space. Charge symmetry (CS) is
a subset of the general isospin symmetry, charge independence (CI), which requires invariance
under any rotation in isospin space. In quantum chromodynamics (QCD), CS implies that
dynamics are invariant under the exchange of the up and down quarks . However, since
the up and down quarks do have different masses (mu ?= md) [2,3], the QCD Lagrangian is
not charge symmetric. This symmetry violation is called charge symmetry breaking (CSB).
The different electromagnetic interactions of the up and down quarks break CI. Observing the
effects of CSB interactions therefore provides a probe of muand md.
Two exciting recent observations of CSB in experiments involving the production of neutral
pions stimulate our attention. Many years of effort led to the observation of CSB in np → dπ0
Preprint submitted to Elsevier Science9 February 2008
at TRIUMF. The CSB forward-backward asymmetry of the differential cross section was found
to be Afb= [17.2±8(stat)±5.5(sys)]×10−4. In addition, the final experiment at the IUCF
Cooler ring reported a very convincing dd → απ0signal near threshold (σ = 12.7 ± 2.2 pb at
Td = 228.5 MeV and 15.1 ± 3.1 pb at 231.8 MeV) . The dd → απ0reaction violates CS
since the deuterons and the α-particle are self-conjugate under the CS operator, with a positive
eigenvalue, while the neutral pion wave function changes sign.
The study of CSB π0production reactions presents an exciting new opportunity to learn about
the influence of quark masses in nuclear physics, and to use effective field theory (EFT) to
improve our understanding of how QCD works . This is because chiral symmetry of QCD
determines the form of pionic interactions. Electromagnetic CSB is typically of the same order
of magnitude as the strong one, and also can be handled using EFT.
The EFT for the Standard Model at momenta comparable to the pion mass, Q ∼ mπ, is chiral
perturbation theory (χPT) . This EFT has been extended [8,9,13,11,10,12] to momenta
relevant to pion production, Q ∼√mπM with M the nucleon mass. (For a review and further
references, see Ref. .)
EFT with the operators of Ref.  was used to correctly predict the sign of the forward-
backward asymmetry in np → dπ0. For the dd → απ0reaction, we surveyed various mecha-
nisms using initial-state plane-wave functions and simplified final-state wave functions . In
this simplified model, we found that the formally leading-order (LO) production mechanism
is suppressed through symmetries in the wave functions and studied other mechanisms. The
contributions from next-to-next-to-leading-order (NNLO) diagrams are too small to account
for the observed cross section — a cross section of only 0.9 pb was found. We also included
short-range pion emission, which contributes at N4LO through contact vertices whose strengths
are a priori unknown. We used resonance saturation, by means of CSB effects in Z–diagrams,
as motivated by a successful phenomenological model  of the charge-symmetry-conserving
(CSC) reaction pp → ppπ0. For the simplified wave functions, we then found a cross section of
the observed order of magnitude.
Our aim here is to take advantage of recent significant advances in four-body theory [16,17]
that allow us to include the effects of deuteron-deuteron interactions in the initial state, and
to use bound-state wave functions with realistic two- and three-nucleon interactions. The cal-
culations presented here are hybrid: the pion-production operators are constructed using EFT,
but the nuclear interactions used to obtain the wave functions are not. No calculation of this
kind can be considered to be completely well-founded unless the operators and wave functions
are constructed from the same convergent EFT. However, the interactions do include one-pion
exchange, so their long-range behavior is founded in EFT. Moreover, we employ several poten-
tials to gauge the sensitivity of the various production operators to the shorter-ranged parts of
The present study does not include all diagrams appearing at NNLO. A complete analysis
demands the inclusion of loop diagrams. Their evaluation requires a careful treatment of di-
vergences as pointed out in Ref. , and understood only recently . For technical reasons,
Fig. 1. Diagrams of np → dπ0; the solid circle indicates CSB.
photon exchange is so far only considered in the final-state wave function. In this letter, we
concentrate on the important effects of initial- and final-state interactions (ISI and FSI), and
anticipate that the use of the present incomplete set of operators should be sufficient to get
order-of-magnitude estimates and to demonstrate the technique.
2. We summarize the power counting of the previous study , which contains explicit ex-
pressions for the operators. At LO, there is only one contribution, represented by Fig. 1a:
pion rescattering in which the CSB occurs through the seagull pion-nucleon terms linked to
the nucleon-mass splitting. This contribution stems from the chiral transformation properties
of the quark operators that generate CSB, which are two: i) the up-down mass difference,
which breaks chiral symmetry as a component of a chiral four-vector; ii) electromagnetic quark
interactions, which break chiral symmetry as components of a chiral anti-symmetric rank-
two tensor. In lowest order, there exist two seagull operators involving a nucleon interacting
with two pions, one of which is neutral. Their strengths are determined by the quark-mass
and electromagnetic contributions to the nucleon mass splitting, δmN and¯δmN, respectively.
δmN is proportional to ε(mu+ md), where ε ≡ (mu− md)/(mu+ md) ≈ 1/3, while¯δmN is
the fine-structure constant α times a typical hadronic mass. The only existing constraint on
these two terms is δmN+¯δmN = Mn− Mp= 1.29 MeV and the model dependent estimate
¯δmN= −(0.76±0.3) MeV based on the Cottingham sum rule . Verifying the theory requires
that the two terms be constrained independently. The most natural reaction to study is πN
scattering. Ref.  predicted a significant difference between the π0p and π0n scattering lengths
that is not presently observable, as discussed in Ref. . Effects of these terms in the nuclear
potential  are relatively small or suffer from other unknowns as in πd scattering . This
leaves the investigation of CSB in the two reactions, np → dπ0and dd → απ0, as very promising
possibilities. For definiteness, in this paper we use the central value of the estimate from the
Cottingham sum rule. The leading diagram, Fig. 1a, is O[εm2
denotes the pion decay constant and Q ≈√mπM a typical momentum.
πMQ)], where fπ= 92.4 MeV
We refer to this contribution as “pion exchange”. For completeness, we distinguish the parts
proportional to δmNand¯δmNand denote the contributions by MPE= MPE,δmN+MPE,¯δmN.
There is no NLO contribution. At NNLO, suppressed by O(mπ/M), there exists a recoil cor-
rection of the LO term (labeled Mrec= Mrec,δmN+ Mrec,¯δmN). Its strength is also determined
by δmNand¯δmN. Therefore, its contribution allows us to estimate the size of NNLO contribu-
tions. The recoil correction to the πNN vertex is linear in the energy of the virtual pion. Such
operators were studied in Ref.  for the reactions NN → NNπ, and we use the prescription
provided there and applied in Ref. . Demonstrating the validity of this recipe for a four-body
environment deserves further study.
At the same order new parameters appear. In particular, a term arises in which a one-body CSB
operator (∝ β1+¯β3) is sandwiched between initial- and final-state wave functions, as illustrated
in, e.g., Fig. 1b. We refer to this as the one-body term (M1b). The terms β1= O(ǫm2
and¯β3= O(α/π) arise from, respectively, the quark-mass-difference and electromagnetic con-
tributions to the isospin-violating pion-nucleon coupling. Neither β1nor¯β3can be extracted
from experiment yet. To allow us to provide numerical results we estimate these terms by
modeling  β1by π-η mixing, see Fig. 1b,
β1= ¯ gη?π0|H|η?/m2
where ?π0|H|η? = −4200 MeV2is the π-η–mixing matrix element , and ¯ gη= gηNNfπ/M
the η-nucleon coupling constant. An early analysis  using one-boson-exchange potentials
in NN scattering gave g2
to η exchange and high-accuracy fits can be achieved  using g2
possibility of a vanishing coupling constant had been raised earlier. The detailed analysis of
NN total cross sections and p¯ p data using dispersion relations  found that g2
This is consistent with extractions from the nucleon pole in the amplitude πN → ηN that give
 0.5 > g2
small value g2
for the results shown below, but also examine the effects of using g2
are roughly consistent with the size expected using power counting arguments . We assume
the sign predicted by SU(3) symmetry, as in Ref. .
ηNN/4π = 3.86 (used in Ref. ), but the data show little sensitivity
ηNN/4π = 0. Indeed, the
ηNN/4π = 0.
ηNN/4π ≥ 0. Photoproduction reactions on a nucleon  (see their Fig. 2) yield the
ηNN/4π = 0.1. To be consistent with our earlier study , we use g2
ηNN/4π = 0.51
ηNN/4π = 0.10. Both values
The effects of electromagnetic interactions as well as strong CSB were included in computing
the α-particle wave functions, where the former effect is dominant. These interactions generate a
small isospin T = 1 component of the wave function that enables a non-zero contribution of CSC
production operators. To estimate the effects of the admixtures, we calculate the production
matrix element using the CSC counterpart of diagram Fig. 1b (referred to as MWF).
A number of other CSB mechanisms enter at N3LO or higher, including additional loop dia-
grams and short-range interactions. The lowest order where four-nucleon contact interactions
start to contribute is N4LO, that is, O(mπ/M) below NNLO. To estimate their strength,
Ref.  evaluated certain tree-level contributions as indicated by Fig. 1c, which represents
the exchange of heavy mesons (σ, ω, ρ) via a Z-graph mechanism, with π-η mixing generating
CSB at pion emission (Mσ, Mωand Mρ). Another Z-graph (labeled as Mρω) arises in which
the CSB occurs in the heavy-meson exchange via ρ-ω mixing along with strong pion emission
at the vertex. The Z-graphs are believed to be important because their inclusion leads to a
quantitative description of the total cross section for the reaction pp → ppπ0near threshold
. Our present results use the coupling constants and parameters of Ref. , see their Ta-
ble I. However, in the future it will be necessary to reassess the procedure in light of recent
developments concerning the treatment of divergences in EFT loop diagrams .
3. The various mechanisms generate pion-production kernels that are sandwiched between
final- and initial-state wave functions to provide a transition matrix element M. We restrict
our analysis to Td= 228.5 MeV, as the effects of a small change in energy are captured mainly
by the change in the phase-space factor. The cross section is related to the matrix elements by
σ = 4.303 pb
We present our new results in stages. First we introduce realistic bound-state wave functions,
while continuing to use the plane-wave approximation (PWA). The techniques to solve the four-
body problem have been presented by Nogga et al. . To be specific, we present results using
both the AV18  and CD-Bonn 2000  two-nucleon potentials combined with a properly
adjusted Tucson-Melbourne (TM99)  three-nucleon force. The combination guarantees that
the α-particle binding energy is reproduced with high accuracy. Additional calculations using
the Urbana-IX  three-nucleon potential resulted in essentially identical results and will be
presented elsewhere .
Table 1 summarizes our results for the transition amplitudes that add to M, labeled according
to the various mechanisms described above. The one-body term is predicted rather model
independently. Using these matrix elements for the one-body operator leads to a cross section
of 10 − 13 pb, which is accidentally in good agreement with the experiment. Compared to our
toy-model calculation , we find an increase of the cross section by a factor of 10, showing
that the high-momentum tail of the wave function is important, especially for the one-body
term. Using the smaller, but also realistic, coupling g2
cross section by a factor of 5. The one-body term is formally subleading. However, the toy-
model calculation showed that the pion-exchange term is suppressed due to the symmetry of
the α-particle wave function. This result persists for the realistic α-particle wave functions: the
amplitude does not vanish exactly, but remains smaller than the one-body term. This term is
quite sensitive to the chosen nuclear interaction, pointing to sensitivity to the short-range part
of the potential and to the small components of the α-particle wave function.
ηNN/4π = 0.10 would reduce the resulting
Since the LO term is suppressed and the one-body term is not well constrained by resonance
saturation, it is interesting to look at the pion-recoil term. Its parameters are better determined
than β1, since they are related to the nucleon mass difference and the Cottingham sum rule.
Our calculation may therefore give a trustful estimate of the size of the NNLO contributions.
The results are a rather model-independent amplitude of approximately 1/3 the size of the
one-body term and are in line with the power counting.
In contrast, we find that all the Z-graphs give unexpectedly large contributions, especially the
ρ-ω exchange operator. Also, the contributions add constructively, so that their sum tends to
overwhelm the one-body term. This model of resonance saturation thus gives results in vast
disagreement with the power counting.
CSB effects on the final-state wave functions (MWF) are smaller than pion recoil and insen-
sitive to the chosen nuclear interaction, indicating that these terms are well constrained using
Complex dd → απ0amplitudes at Td= 228.5 MeV in units of 10−4fm−2. PWA denotes the plane-wave
approximation. ISI results also include the initial-state interaction.
−1.51 + i 1.87
−0.28 + i 0.35
−1.79 + i 2.22
−0.81 + i 0.74
−0.15 + i 0.14
−0.96 + i 0.88
−2.51 + i 1.84
−0.56 + i 0.64
−0.53 + i 0.44
−0.33 + i 0.34
−1.32 + i 1.51
+0.51 − i 0.13
−0.76 + i 0.74
−0.14 + i 0.14
−0.90 + i 0.88
−0.63 + i 0.59
−0.12 + i 0.11
−0.75 + i 0.70
−1.94 + i 1.60
−0.32 + i 0.42
−0.35 + i 0.34
−0.18 + i 0.19
−0.84 + i 1.07
+0.41 − i 0.14
4. The next step is to present the effects of including the ISI. The correct treatment of this
involves the solution of the four-body scattering problem at center-of-mass energies greater
than 200 MeV. In spite of the tremendous progress achieved in recent years on the solution of
the four-nucleon problem [34,17], advances in obtaining exact solutions are limited to energies
below the four-particle breakup threshold. To understand our pion-production reaction it is
necessary to go beyond the distortions obtained through an effective optical-model potential
fitted to the elastic?dd scattering data. This is because important pion production occurs in
which the deuterons interact, break up, and then emit a pion. At very high energies the use
of Glauber approximation is justified, but in the threshold energy regime for pion production
the wave length associated with the relative d + d on-shell momentum is close to the size of
the deuteron. Therefore we obtain an approximate solution of the Yakubovsky  equation
for the four-nucleon scattering wave function that is made up of two terms: the first involves
the bound-state wave functions of the two deuterons times a plane wave describing the relative
motion between them; the second requires the breakup of one of the deuterons followed by the
three-body scattering of the N +d system into the three-particle continuum in the presence of
the remainder spectator nucleon.
Such an approximation is based on the lowest-order terms in the Neumann series expansion of
the four-particle Yakubovsky equation, leading to the following expression for the scattering
|Ψρ0? ≃ |φρ0? +
where ρ denotes one of the seven two-body partitions, four of (3)+1 type and three of (2)+(2)
type, and i is a pair interaction that is internal to ρ; j is both internal to ρ and ρ0. The
initial-state wave function component |φρ0
components of the target and projectile times a relative plane wave between their respective
center of mass. As usual, ρ0specifies the two-body entrance channel,¯δρρ0= 1 − δρρ0, G0is the
four-free-particle Green’s function and tithe t-matrix for pair i. If ρ0corresponds to a 2 + 2
initial state, then ρ can only be a (3)+1 two-body partition and Uρ
body Alt, Grassberger and Sandhas (AGS)  equation for the three-particles that make up
subsystem ρ. The first term in Eq. (3) corresponds to the initial-state wave function; the second
term requires the breakup of one of the bound pairs followed by the scattering of either one of
the particles from the remaining bound pair, leading to four free particles in the continuum.
The particle-pair scattering into the continuum takes place in the presence of the fourth one,
and therefore, by energy conservation, the total energy available is the total four-body center-
of-mass energy minus the relative kinetic energy of the fourth particle relative to the center of
mass of the other three. Thus the four-body scattering wave function we construct contains all
orders in the pair interaction but also three-particle correlations in first-order perturbation at
all possible energies that are consistent with four-particle energy conservation.
j? carries the appropriate bound-state wave function
ijis the solution of the three-
For four identical nucleons Eq. (3) may be written as
dd? = |φdd? +
√12(1 + P − P34P +˜P)(1 − P34)|ψ(12,3)4
where P = P12P23+ P13P23 and˜P = P13P24 are permutation operators whose appropriate
combination generates the 6 (12) components of the Yakubovsky wave function that are of
2 + 2 (1 + 3) type. The symmetrized d + d initial-state wave function  |φdd? is given by
√6(1 + P − P34P +˜P) ξd(12) ξd(34) Exp(12 − 34), (5)
where ξd(ij) is the deuteron wave function for the pair (ij), and Exp(12 − 34) represents the
relative plane wave between the two deuteron pairs. The second term in Eq. (4) mandates the
use of a specific choice of wave-function component for nucleons 1, 2, 3 and 4,
? = G0?12,3| U0(z)|(12)3?ξd(34),
where connectivity increases from left to right and z = E −4
momentum between nucleon 4 and the center of mass of (123). The breakup operator U0= tG0U
3k2+ i0, k being the relative
and the corresponding matrix element ?12,3| U0(z)|(12)3? represents the scattering of nucleon
3 from the bound state of (12) leading to three free nucleons in the continuum where nucleons
1 and 2 are last to interact through their respective t-matrix t. The operator U is the AGS
three-body scattering operator that satisfies the integral equation
U = PG−1
0 + PtG0U, (7)
from which one calculates Nd elastic scattering amplitudes. The permutation operator P is the
same as used in Eq. (4) and corresponds to the sum of the two cyclic permutations of particles
1, 2 and 3. We extract from Eq. (4) the3P0partial wave in the entrance channel to compute
the threshold cross section for dd → απ0.
Details regarding the numerical solution of the scattering problem and evaluation of the pion-
production matrix-element will be presented elsewhere . Here, in Table 1, we simply present
the results for the transition amplitudes, which acquire imaginary parts due to the presence of
the initial-state interaction. Generally, we observe a significant enhancement of all contributions
to the amplitude.
It is immediately apparent that the pion-exchange term, which is supposed to be LO, is now of
the size of the NNLO terms considered here, namely the pion-recoil and one-body operators.
It still shows a sizable model dependence, which could visibly influence our final result for the
cross section. A more consistent treatment of nuclear interactions and production operators
will be necessary in future.
The pion-recoil and one-body terms remain relatively model independent. Both, therefore,
can serve as an order-of-magnitude estimate of the cross section. Our results for these matrix
elements correspond to cross sections between 4.5 and 42 pb. Again, the one-body contribution
is larger than the pion-recoil term. Both would come close to each other for the smaller choices of
the strength of the one-body term. Our explicit calculation shows that the NNLO contribution
provides a strength consistent with the experiment. We stress that the strong enhancement due
to initial-state interactions and higher-momentum components of the α-particle wave functions
are necessary to find NNLO contributions of the required size.
The ISI also enhances short-range contributions from Z-graphs, but by far-smaller amounts.
We still find relative contributions much larger than expected from the power counting and,
again, all the contributions add up constructively. One possible explanation would be that there
is simply no convergent EFT for the reaction dd → απ0and the Z-graphs would need to be
included as done in this work. However, if this was true the value of the empirical cross section
would be the result of subtle cancellations amongst various terms from very different origins, a
very unlikely coincidence. The more likely interpretation is that the power counting works, but
the Z-graphs simply provide the wrong model to estimate the four-nucleon operators. If this
is the case we may even drop them all together from our investigations, since they are of high
5. This paper extends our earlier study of the reaction dd → απ0by using realistic wave func-
tions for the four-nucleon ISI and FSI. We also provide numerical estimates for some diagrams
that lead to a cross section of the right order of magnitude supporting the power counting given
the suppression of the LO. In addition, the present results allow us to identify a few issues that
deserve further study (in addition to the inclusion of all diagrams up to NNLO).
• Given the dramatic influence of initial-state interactions, it is of paramount importance that
new experimental constraints be obtained for the deuteron-deuteron interactions in the energy
region close to the pion-production threshold. Besides data on elastic dd scattering, also data
on other pion-production reactions with the same initial3P0 state are needed. The most
obvious examples are the CSC reactions dd →3H/HeNπ recently measured at COSY ,
which are an interesting alternative to elastic dd scattering because only a few partial waves
contribute in the entrance channel.
• Another important issue is to better understand the role of the Z-graphs used to estimate
the size of four-nucleon operators. As shown above, their size is much larger than indicated
by their N4LO power-counting order. Thus, it is necessary to reassess the procedure in light
of recent developments in EFT, especially concerning the treatment of divergences in loop
diagrams . Given the large number of experimental data, especially for pp → ppπ0, much
insight can be obtained.
• Finally, we found some dependence on the chosen nuclear interaction model. This is prob-
ably related to the presumed inconsistency between the chosen nuclear interaction and the
production operators. Such an inconsistency can only be resolved by applying nuclear inter-
actions based on chiral perturbation theory [7,38] that need to be extended to higher cutoffs
as outlined in Ref. .
The central result of this letter is that the inclusion of ISI and FSI enhances the contribution of
NNLO diagrams of the production operator enough to able to account for the measured cross
section. Together with the insight that the LO is suppressed, this supports the EFT approach to
this reaction. But it also demonstrates that a careful treatment of the nuclear effects is required
for a final analysis.
In view of a planned measurement of the reaction?dd → απ0at higher energies at COSY 
—where p waves will be relevant— and of the experimental determination of Afb, we will
have an increased database on cross sections, which will allow us to disentangle the various
contributions. Although much remains to be done before any precise statements about the
values of the parameters δmN,¯δmN can be made, we are now at threshold of understanding
how the light-quark masses makes a difference in nuclear physics.
We thank Andy Bacher, Allena Opper, and Ed Stephenson for useful discussions and encour-
agement, without which this work would not have been completed. This research was partially
funded by FCT grant POCTI/37280/FNU/2001 (ACF), NSF grant PHY-0457014 (AG), DOE
grants DE-FG02-97ER41014 (GAM) and DE-FG02-04ER41338 (UvK), and a Sloan Research
 G.A. Miller, B.M.K. Nefkens, and I.ˇSlaus, Phys. Rept. 194, 1 (1990).
 S. Weinberg, Trans. N.Y. Acad. Sci. 38, 185 (1977).
 H. Leutwyler, Phys. Lett. B 378, 313 (1996).
 A.K. Opper et al. Phys Rev. Lett. 91, 212302 (2003).
 E.J. Stephenson et al., Phys. Rev. Lett. 91, 142302 (2003).
 G.A. Miller, A.K. Opper, and E.J. Stephenson, Ann. Rev. Nucl. Part. Sci. (to appear).
 V. Bernard, N. Kaiser, and U.-G. Meißner, Int. J. Mod. Phys. E 4, 193 (1995); P.F. Bedaque and
U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52, 339 (2002).
 T.D. Cohen, J.L. Friar, G.A. Miller, and U. van Kolck, Phys. Rev. C 53, 2661 (1996); C. Hanhart,
U. van Kolck, and G.A. Miller, Phys. Rev. Lett. 85, 2905 (2000); C. Hanhart and N. Kaiser, Phys.
Rev. C 66, 054005 (2002).
 U. van Kolck, J.A. Niskanen, and G.A. Miller, Phys. Lett. B 493, 65 (2000).
 A. G˚ ardestig et al. [CSB theory collaboration], Phys. Rev. C 69, 044606 (2004).
 A. G˚ ardestig, D.R. Phillips, and C. Elster, Phys. Rev. C (to appear), arXiv:nucl-th/0511042.
 V. Lensky et al., arXiv:nucl-th/0511054.
 C. Hanhart, Phys. Rept. 397, 155 (2004).
 U. van Kolck, Ph.D. Dissertation, U. of Texas (1993); Few-Body Syst. Suppl. 9, 444 (1995);
 T.S.-H. Lee and D.O. Riska, Phys. Rev. Lett. 70, 2237 (1993); C.J. Horowitz, H.O. Meyer, and
D.K. Griegel, Phys. Rev. C 49, 1337 (1994); J.A. Niskanen, Phys. Rev. C 53, 526 (1996).
 A. Nogga, H. Kamada, and W. Gl¨ ockle, Phys. Rev. Lett. 85, 944 (2000); A. Nogga, H. Kamada,
W. Gl¨ ockle, and B.R. Barrett, Phys. Rev. C 65, 054003 (2002).
 A.C. Fonseca, Phys. Rev. Lett. 83, 4021 (1999).
 N. Fettes and U-.G. Meißner, Nucl. Phys. A 693, 693 (2001).
 J.A. Niskanen, Phys. Rev. C 65, 037001 (2002); J.L. Friar, U. van Kolck, G.L. Payne, and S.A.
Coon, Phys. Rev. C 68, 024003 (2003); E. Epelbaum, U.-G. Meißner, and J.E. Palomar, Phys.
Rev. C 71, 024001 (2005); J.L. Friar, G.L. Payne, and U. van Kolck, Phys. Rev. C 71, 024003
 U.-G. Meißner, U. Raha, and A. Rusetsky, arXiv:nucl-th/0512035.
 J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982).
 C. Hanhart et al., Phys. Rev. C 63, 044002 (2001); V. Malafaia and M.T. Pe˜ na, Phys. Rev. C 69,
 U. van Kolck, J.L. Friar, and T. Goldman, Phys. Lett. B 371, 169 (1996).
 S.A. Coon and M.D. Scadron, Phys. Rev. C 51, 2923 (1995).
 O. Dumbrajs et al., Nucl. Phys. B 216, 277 (1983).
 R. Machleidt, Phys. Rev. C 63, 024001 (2001).
 W. Grein and P. Kroll, Nucl. Phys. A 338, 332 (1980); Nucl. Phys. A 377, 505 (1982).
 S.R. Deans and J.W. Wooten, Phys. Rev. 185, 1797 (1969).
 L. Tiator et al., Nucl. Phys. A 580, 455 (1994).
 R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995).
 J.L. Friar, D. H¨ uber, and U. van Kolck, Phys. Rev. C 59, 53 (1999); S.A. Coon and H.K. Han,
Few-Body Syst. 30, 131 (2001).
 B.S. Pudliner et al., Phys. Rev. C 56, 1720 (1997).
 A. Nogga et al. [CSB theory collaboration], in preparation.
 F. Ciesielski, J. Carbonell, and C. Gignoux, Phys. Lett. B 447 199 (1999); M. Viviani et al., Phys.
Rev. Lett. 86, 3739 (2001); R. Lazauskas et al., Phys. Rev. C 71, 034004 (2005).
 O.A. Yakubovsky, Sov. J. Nucl. Phys. 5, 937 (1967).
 E.O. Alt, P. Grassberger, and W. Sandhas, Phys. Rev. C 1, 85 (1970); JINR No E4 - 6688 (1972).
 COSY Proposal #139: Near threshold π production in dd →3HeNπ and dd → tNπ.
 D.R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003); E. Epelbaum, W. Gl¨ ockle, and
U.-G. Meißner, Nucl. Phys. A 747, 362 (2005);
 A. Nogga, R.G.E. Timmermans, and U. van Kolck, Phys. Rev. C 72, 054006 (2005).
 H.H. Adam et al. [WASA-at-COSY Collaboration], arXiv:nucl-ex/0411038.