Two-pion-exchange contributions to
the pp→ ppπ0 reaction
Y. Kim(a,b), T. Sato(c), F. Myhrer(a) and K. Kubodera(a)
(a) Department of Physics and Astronomy, University of South
Carolina, Columbia, South Carolina 29208, USA
(b) School of Physics, Korea Institute for Advanced Study, Seoul
130-012, Korea
(c) Department of Physics, Osaka University, Toyonaka, Osaka
560-0043, Japan
Abstract
Our previous study of the near-threshold pp → ppπ0 reaction based on
a hybrid nuclear effective field theory is further elaborated by examining the
momentum dependence of the relevant transition operators. We show that the
two-pion exchange diagrams give much larger contributions than the one-pion
exchange diagram, even though the former is of higher order in the Weinberg
counting scheme. The relation between our results and an alternative counting
scheme, the momentum counting scheme, is also discussed.
http://arxiv.org/abs/0704.1342v2
In the standard nuclear physics approach (SNPA), a nuclear reaction amplitude
is calculated with the use of the transition operator derived from a phenomenological
Lagrangian and nuclear wave functions generated by a high-precision phenomenolog-
ical NN potential. SNPA has been enormously successful in explaining a vast range
of nuclear phenomena. Meanwhile, a nuclear chiral perturbation approach based on
heavy-baryon chiral perturbation theory (HBχPT) is gaining ground as a powerful
tool for addressing issues that cannot be readily settled in SNPA. HBχPT is a low-
energy effective field theory of QCD, based on a systematic expansion in terms of the
expansion parameter ǫ ≡ Q/Λχ ≪ 1, where Q is a typical energy-momentum involved
in a process under study or the pion mass mπ, and the chiral scale Λχ ≃ 4πfπ ≃ 1
GeV. HBχPT has been applied with great success to low-energy processes includ-
ing e.g., pion-nucleon scattering and electroweak reactions on a nucleon and in few-
nucleon systems. Our present work is concerned with a HBχPT study of the near-
threshold pp→ppπ0 reaction. A motivation of this study may be stated in reference
to the generic NN→NNπ processes near threshold. Although HBχPT presupposes
the small size of its expansion parameter Q/Λχ, the pion-production reactions involve
somewhat large energy- and three-momentum transfers even at threshold. Therefore
the application of HBχPT to the NN→NNπ reactions may involve some delicate
aspects, but this also means that these processes may serve as a good test case for
probing the limit of applicability of HBχPT. Apart from this general issue to be
investigated, a specific aspect of the pp→ppπ0 reaction makes its study particularly
interesting. For most isospin channels, the NN→NNπ amplitude near threshold is
dominated by the pion rescattering diagram where the πN scattering vertex is given
by the Weinberg-Tomozawa term, which represents the lowest chiral order contri-
bution. However, a quantitatively reliable description of the NN→NNπ reactions
obviously requires detailed examinations of the corrections to this dominant ampli-
tude. Meanwhile, since the Weinberg-Tomozawa vertex does not contribute to the
pion-nucleon rescattering diagram for pp→ppπ0, this reaction is particularly sensi-
tive to higher chiral-order contributions and hence its study is expected to provide
valuable information to guide us in formulating a quantitative description of all the
NN→NNπ reactions (including the channels that involve a deuteron).
The first HBχPT-based study of the near-threshold pp→ppπ0 reaction was made
in Refs. [1, 2]. In HBχPT one naturally expects a small cross section for this reac-
tion since, for s-wave pion production, the pion-nucleon vertex in the impulse ap-
proximation (IA) diagram and the pion-rescattering vertex in the one-pion-exchange
rescattering (1π-Resc) diagram arise from the next-to-leading-order (NLO) chiral la-
grangian. A remarkable feature found in Refs. [1, 2] is that a drastic cancellation
between the IA and 1π-Resc amplitudes leads to the suppression of the pp→ppπ0
amplitude far beyond the above-mentioned naturally expected level. This destruc-
tive interference is in sharp contrast with the constructive interference reported in
SNPA-based calculations [3, 4]. It is to be recalled that the pp→ppπ0 cross section
obtained in Refs. [3, 4] was significantly smaller (by a factor of ∼5) than the experi-
mental value [5]. The drastic cancellation between the IA and 1π-Resc terms found
in the HBχPT calculations [1, 2] leads to even more pronounced disagreement be-
tween theory and experiment. In this connection it is worth noting that, according
to Lee and Riska [6], the heavy-meson (σ and ω) exchanges can strongly enhance
the pp→ppπ0 amplitude. It is also to be noted that σ-meson-exchange introduced
in many NN potentials is more properly described by correlated two-pion-exchange
(see e.g., Refs. [7, 8]), and that there have been substantial developments in deriving
a two-pion exchange NN potential using HBχPT, see e.g. [9]. These developments
were conducive to a HBχPT study of two-pion-exchange (TPE) contributions to the
pp→ppπ0 reaction [10, 11]. In the plane-wave approximation it was found [10] that
TPE contributions are indeed very large (as compared to the 1π-Resc amplitude), a
result that is in line with the finding in Ref.[6]. A subsequent DWBA calculation [11]
indicates that this feature remains essentially unchanged when the initial- and final-
state interactions are taken into account. More recent investigations [12, 13, 14, 15],
however, have raised a number of important issues that call for further investigations,
and the purpose of our present note is to address these issues.
In Ref. [10], to be referred to as DKMS, were derived all the transition operators
for pp→ppπ0 belonging to next-to-next-to-leading order (NNLO) in the Weinberg
counting, and these operators were categorized into Types I ∼ VII, according to the
patterns of the corresponding Feynman diagrams; see Figs. 2 - 5 in DKMS. Types I,
II, III and IV belong to diagrams of the two-pion exchange (TPE) type, while Types
V, VI and VII arise from diagrams of the vertex correction type. A notable feature
pointed out in DKMS is that the contributions of Types II ∼ IV are by far the largest,
and that they even exceed those of the 1π-Resc amplitude, which is formally of lower
chiral order. On the other hand, the possibility of strong cancellation among the TPE
diagrams was pointed out in Refs. [12, 13]. This motivates us to make here a further
study of the behavior of the TPE diagrams.1
A remark is in order here on a counting scheme to be used. At the NN→NNπ
threshold the nucleon three-momentum must change from the initial value p ∼√
mπmN to zero, entailing a rather large momentum transfer. To take this large
1For a brief report on this study, see Ref. [16].
momentum transfer into account, Cohen et al. [2] proposed a new counting scheme,
to be called the momentum counting scheme (MCS); see Ref. [13] for a detailed
review. In MCS the expansion parameter is ǫ̃ ≡ p/mN ≃ (mπ/mN)1/2, which is
larger than the usual HBχPT expansion parameter ǫ ≃ mπ/mN . A study based
on MCS [13] indicates that the 1π-Resc diagram for pp→ppπ0 is higher order in ǫ̃
(and hence less important) than a certain class of TPE diagrams, called “leading
order loop diagrams”, and that MCS is consistent with the estimates of the TPE and
other diagrams reported in DKMS. Furthermore, according to Hanhart and Kaiser
(HK) [12], the “leading parts” (see below) of these MCS “leading order” diagrams
exhibit exact cancellation among themselves;2 see also Lensky et al. [14]. Although
these studies are illuminating, we consider it important to examine the behavior of
the “sub-leading” parts (in MCS counting) of these TPE diagrams in order to see
whether they can be still as large as indicated by the phenomenological success of the
Lee-Riska heavy-meson exchange mechanism. In what follows we shall demonstrate
that this is indeed the case.
Analytic expressions for the pp→ppπ0 transition operators to NNLO in HBχPT
were given in DKMS. Although these expressions are valid for arbitrary kinematics,
we find it illuminating to concentrate here on their simplified forms obtained with
the use of fixed kinematics approximation (FKA), wherein the energies associated
with particle propagators are “frozen” at their threshold values. In FKA, the TPE
operator corresponding to each of the above-mentioned Types I ∼ IV can be written
~Σ · ~k
t(p, p′, x) (1)
where ~p (~p ′) is the relative three-momentum in the initial (final) pp state (~p1−~p2 = 2~p,
~p ′1 − ~p ′2 = 2~p ′), ~k ≡ ~p − ~p ′, x = p̂ · p̂′, and ~Σ = 12(~σ1 − ~σ2). The function t(p, p
′, x)
diverges as k → ∞, and it is useful to decompose t(p, p′, x) into terms that have
definite k-dependence as k → ∞. It turns out [18] that t(p, p′, x) can be expressed as
t(p, p′, x)
k→∞∼ t1
gA/(8f
|~k|+ t2
ln{|~k|2/Λ2}
+ t3 + δt(p, p
′, x), (2)
where t3 is asymptotically k-independent, and δt(p, p
′, x) is O(k−1). For each of
Types I ∼ IV, analytic expressions for ti’s (i = 1, 2, 3) can be extracted [18] from
the amplitudes T given in DKMS [10]. The first term with t1 in eq.(2) is the leading
2HK [12] pointed out that the sign of the contribution of Type II in Ref. [10] should be reversed;
we have confirmed the necessity of this correction.
part in MCS discussed by HK [12], whereas the remaining terms, which we refer to as
the “sub-leading” terms, were not considered by HK. The study of these sub-leading
terms is an important theme in what follows. Table 1 shows the value of t1 for Type
K (K= I ∼ IV) extracted from the results given in DKMS. The third row in Table 1
gives the ratio RK = TK/TResc, where TK is the plane-wave matrix element of T in
eq.(1) for Type K (K=I ∼ IV) normalized by TResc, the plane-wave matrix element of
the 1π-Resc diagram. The fourth row in Table 1 gives R ⋆K = T
K/TResc, where T
the plane-wave matrix element of T with the t1 term in eq.(2) subtracted. We can see
from the table that the most divergent t1 terms of the TPE diagrams add up to zero,
confirming the result of Ref. [12]. However, this does not necessarily mean that the
TPE diagrams are unimportant, because we still need to examine the contributions
of the “sub-leading” terms (the t2, t3 and δt terms) in eq.(2). Comparison of RK
and R ⋆K indicates that the subtraction of the t1 term reduces the magnitude of TK
drastically (except for Type I which has no t1 term), but the fact that |R ⋆K | is of the
order of unity (Types I, II and IV) or larger than 1 (Type III) suggests that the TPE
contributions can be quite important. The sum of the contributions of Types I ∼ IV
R ⋆K (=
RK) = −4.65 , (3)
which indicates that, at least in plane-wave approximation, the TPE contributions
are more important than the 1π-Resc contribution.
Table 1: For the four types of TPE diagrams, K= I, II, III and IV, the
second row gives the value of t1 defined in eq.(2), and the third row gives
the ratio RK = TK/TResc, where TK is the plane-wave matrix element of
T in eq.(1) for Type K, and TResc is the 1π-Resc amplitude. The last row
gives R ⋆K = T
K /TResc, where T
K is the plane-wave matrix element of T in
eq.(1) with the t1 term in eq.(2) subtracted.
Type of diagrams : K = I II III IV
(t1)K 0 1 1/2 −3/2
RK −.70 −6.54 −6.60 9.19
R ⋆K −.70 −0.82 −3.73 0.61
Next we investigate the behavior of the TPE diagrams as we go beyond the plane-
wave approximation by using distorted waves (DW) for the initial- and final-state
NN wave functions. For formal consistency we should use the NN potential derived
from HBχPT, but we adopt here a “hybrid EFT” approach and use phenomenological
potentials. A conceptual problem in adopting this hybrid approach is that, whereas
the TPE transition operators derived in HBχPT are valid only for a momentum range
sufficiently lower than Λχ∼1 GeV, a phenomenological NN potential can in principle
contain any momentum components.3 To stay close to the spirit of HBχPT, we
therefore introduce a Gaussian momentum regulator, exp(−p2/Λ2G), in the initial and
final distorted wave integrals, suppressing thereby the high momentum components
of the phenomenological NN potentials; this is similar to the MEEFT method used in
Ref.[19]. ΛG should be larger than the characteristic momentum scale of the pp→ppπ0
reaction, p ≃ √mNmπ ≃ 360 MeV/c, but it should not exceed the chiral scale Λχ;
in the present study we shall consider the range, 500 MeV< ΛG <1 GeV. As high-
precision phenomenological NN potentials, we consider the Bonn-B potential [20],
the CD-Bonn potential [21], and the Nijm93 potential of the Nijmegen group [22].
It is worth noting here that several groups [23, 24] have developed a systematic
approach to construct from a phenomenological potential an effective NN potential,
called Vlow−k, that resides within a model space which only contains momentum
components below a specified cutoff scale Λlow−k. In this work we will use Vlow−k as
derived by the Stony Brook group [24]. It is conceptually natural to use Vlow−k in
conjunction with transition operators derived from HBChPT [25]. A problem however
is Vlow−k [24], primarily meant for describing sub-pion-threshold phenomena, was
obtained with the use of a rather low cutoff, Λlow−k ∼ 2 fm−1. This cutoff is perhaps
too close to the characteristic momentum scale p ∼ 360 MeV/c for the pion production
reaction. It therefore seems worthwhile to “rederive” Vlow−k employing a momentum
cut-off higher than 2 fm−1 and use it in the present DWBA calculation. Below we
will use Vlow−k generated from the CD-Bonn potential for Λlow−k= 4 and 5 fm
−1. We
remark that, as is well known, Vlow−k’s generated from any realistic phenomenological
potentials lead to practically equivalent half-off-shell NN K-matrices and hence the
same NN wave function.
We evaluate the TPE contributions in DWBA for a typical case of Tlab = 281
MeV. Since the t1 terms in eq.(2) add up to zero, we drop the t1 terms in our cal-
culation.4 Thus, in eq.(1), we use t⋆(p, p′, x) instead of t(p, p′, x), where t⋆(p, p′, x) is
3 A pragmatic problem associated with this conceptual issue is that, in a momentum-space calcu-
lation of the matrix elements of the TPE operators sandwiched between distorted pp wave-functions
generated by a phenomenological NN potential, the convergence of momentum integrations is found
to be extremely slow [17, 18].
4 Removing the t1 term lessens the severity of the convergence problem in our momentum inte-
obtained from t(p, p′, x) by suppressing the t1 term. The partial-wave projected form
of t⋆(p, p′, x) in a DWBA calculation is written as:
J = −
p2dp p′ 2dp′
dx ψ1S0(p
′) t⋆(p, p′, x) (p− p′x)ψ3P0(p) (4)
Here ψα(p) is a distorted two-nucleon relative wave function in the α partial-wave
(1S0 for the initial state and
3P0 for the final state) given by
ψα(p) = cos(δα)
δ(p− pon)/p2 + P
Kα(p, pon)
(E − Ep)
, (5)
where δα is the phase-shift for the α partial wave, and Kα(p, pon) is the partial-
wave K-matrix pertaining to the asymptotic on-shell momentum pon. The plane-wave
approximation corresponds to the use of the wave functions of the generic form:
ψ(p) = δ(p− pon)/p2 . (6)
We show in Table 2 the values of J , eq.(4), for the TPE operators of Types I ∼ IV,
calculated at Tlab = 281 MeV, with the use of the Nijm93 potential of the Nijmegen
group [22]5 and Vlow−k. For the Nijm93 potential case, we present the results for five
different values of ΛG between 500 and 1000 MeV/c. For the Vlow−k case, the results
for two choices of Λlow−k are shown: Λlow−k = 4 fm
−1 and 5 fm−1. For comparison, the
values of J corresponding to plane-wave approximation are also shown (bottom row).
From Table 2 we learn the following: (1) The results for the Nijm93 potential with the
gaussian cutoff ΛG are stable against the variation of ΛG within a reasonable range
(500 - 1000 MeV/c); (2) There is semi-quantitative agreement between the results for
the Nijm93 potential and those for Vlow−k; (3) A semi-quantitative agreement is also
seen between the DWBA and PWBA calculations; (4) The feature found in the plane-
wave approximation that the contributions of the TPE diagrams are more important
than the 1π-Resc contribution remains unchanged in the DWBA calculation; the
summed contribution of the TPE operators is larger (in magnitude) than that of
1π-Resc by a factor of 2∼3.5.
gration mentioned in footnote 3.
5 We have checked the results obtained using the Bonn-B and CD-Bonn NN potentials are very
similar to those for the Nijm93 potential case, which we show here as a representative case.
Table 2: The values of J , eq.(4), corresponding to the TPE diagrams of
Types I ∼ IV, evaluated in a DWBA calculation for Tlab = 281 MeV. The
column labeled “Sum” gives the combined contributions of Types I ∼ IV,
and the last column gives the value of J for 1π-Resc. For the Nijm93
potential case, the results for five different choices of ΛG are shown. For the
case with Vlow−k, CD-4 (CD-5) represents Vlow−k generated from the CD-
Bonn potential with a momentum cut-off Λlow−k = 4 fm
−1 (5 fm−1). The
last row gives the results obtained in plane-wave approximation.
I II III IV Sum 1π−Resc
VNijm : ΛG = 500MeV/c −0.11 −0.12 −0.55 0.08 −0.70 0.18
VNijm : ΛG = 600MeV/c −0.12 −0.12 −0.57 0.07 −0.74 0.20
VNijm : ΛG = 700MeV/c −0.12 −0.11 −0.57 0.06 −0.74 0.21
VNijm : ΛG = 800MeV/c −0.12 −0.11 −0.55 0.04 −0.74 0.22
VNijm : ΛG = 1000MeV/c −0.12 −0.10 −0.52 0.03 −0.71 0.23
Vlow−k (CD−4) −0.12 −0.09 −0.46 0.03 −0.65 0.23
Vlow−k (CD−5) −0.09 −0.06 −0.30 −0.01 −0.46 0.22
Plane−wave −0.06 −0.07 −0.30 0.05 −0.37 0.080
We now discuss the above results in the context of MCS [13]. A subtlety in MCS is
that a loop diagram of a given order ν in ǫ̃ not only contains a contribution of order ν
(“leading part”) but, in principle, can also involve contributions of higher orders in ǫ̃
(“sub-leading part”) due to the non-analytic functions generated by the loop integral.
As mentioned, however, HK [12] considered only the leading part, which correspond
to the t1 term in eq.(2). According to MCS, for the reaction pp → ppπ0, the loop
diagrams corresponding to our Type II, III and IV diagrams belong to NLO in the
ǫ̃ parameter, whereas those corresponding to Type I and the 1π-Resc tree diagram
are next order in ǫ̃ (NNLO); see Table 11 in Ref. [13]. Meanwhile, as discussed
earlier, the sum of the “leading parts” of the NLO diagrams vanishes, and therefore,
in calculating J ’s in Table 2, we have dropped the t1 term contribution, retaining only
the “sub-leading” parts of these NLO diagrams. This means that all the entries in
Table 2 represent “sub-leading contributions” (NNLO) in MCS. If we look at Table 2
from this perspective, we note that the order-of-magnitude behavior of our numerical
results is in rough agreement with MCS, although Type IV tends to be rather visibly
smaller (in magnitude) than the others. However, it is striking that J for Type III
is significantly (if not by an order of magnitude) larger than the other sub-leading
contributions. (A similar feature was also seen in R⋆ in Table 1.) In view of the fact
that Type III arises from crossed-box TPE diagrams [10], there is a possibility that
the enhancement of the Type III diagrams may be related to the strong attractive
scalar NN potential that is known to arise from TPE crossed-box-diagrams [7, 8].
We have studied the “sub-leading” parts, which are of NNLO in the momentum
counting scheme (MCS) [13], of the TPE amplitudes for the pp→ppπ0 reaction in both
PWBA and DWBA calculations. We have shown in fixed kinematics approximation
(FKA) that, even though the leading parts of the TPE amplitudes cancel among
themselves [12, 14], the contributions of the sub-leading parts are quite significant.
They are in general comparable to the 1π-Resc amplitude, and the sub-leading part
of the Type III diagrams is even significantly larger than the 1π-Resc diagram. The
total contribution of the TPE diagrams is larger (in magnitude) than that of the
1π-Resc diagram by a factor of ∼5 (PWBA) or 2∼3 (DWBA). We have focused here
on the TPE loop diagrams but, to obtain theoretical cross section for pp→ppπ0 that
can be directly compared with the experimental value, we must consider the other
diagrams discussed in DKMS as well as the relevant counter terms. These will be
discussed in a forthcoming article [26] .
The authors are indebted to Christoph Hanhart and Anders G̊ardestig for useful
discussions. A helpful communication from Ulf Meissner is also grateful acknowl-
edged. This work is supported in part by the US National Science Foundation, Grant
No. PHY-0457014, and by the Japan Society for the Promotion of Science, Grant-in-
Aid for Scientific Research (C) No.15540275 and Grant-in-Aid for Scientific Research
on Priority Areas (MEXT), No. 18042003.
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  Our previous study of the near-threshold pp\to pp\pi^0 reaction based on a
hybrid nuclear effective field theory is further elaborated by examining the
momentum dependence of the relevant transition operators. We show that the
two-pion exchange diagrams give much larger contributions than the one-pion
exchange diagram, even though the former is of higher order in the Weinberg
counting scheme. The relation between our results and an alternative counting
scheme, the momentum counting scheme, is also discussed.

Two-pion-exchange contributions to
the pp→ ppπ0 reaction
Y. Kim(a,b), T. Sato(c), F. Myhrer(a) and K. Kubodera(a)
(a) Department of Physics and Astronomy, University of South
Carolina, Columbia, South Carolina 29208, USA
(b) School of Physics, Korea Institute for Advanced Study, Seoul
130-012, Korea
(c) Department of Physics, Osaka University, Toyonaka, Osaka
560-0043, Japan
Abstract
Our previous study of the near-threshold pp → ppπ0 reaction based on
a hybrid nuclear effective field theory is further elaborated by examining the
momentum dependence of the relevant transition operators. We show that the
two-pion exchange diagrams give much larger contributions than the one-pion
exchange diagram, even though the former is of higher order in the Weinberg
counting scheme. The relation between our results and an alternative counting
scheme, the momentum counting scheme, is also discussed.
http://arxiv.org/abs/0704.1342v2
In the standard nuclear physics approach (SNPA), a nuclear reaction amplitude
is calculated with the use of the transition operator derived from a phenomenological
Lagrangian and nuclear wave functions generated by a high-precision phenomenolog-
ical NN potential. SNPA has been enormously successful in explaining a vast range
of nuclear phenomena. Meanwhile, a nuclear chiral perturbation approach based on
heavy-baryon chiral perturbation theory (HBχPT) is gaining ground as a powerful
tool for addressing issues that cannot be readily settled in SNPA. HBχPT is a low-
energy effective field theory of QCD, based on a systematic expansion in terms of the
expansion parameter ǫ ≡ Q/Λχ ≪ 1, where Q is a typical energy-momentum involved
in a process under study or the pion mass mπ, and the chiral scale Λχ ≃ 4πfπ ≃ 1
GeV. HBχPT has been applied with great success to low-energy processes includ-
ing e.g., pion-nucleon scattering and electroweak reactions on a nucleon and in few-
nucleon systems. Our present work is concerned with a HBχPT study of the near-
threshold pp→ppπ0 reaction. A motivation of this study may be stated in reference
to the generic NN→NNπ processes near threshold. Although HBχPT presupposes
the small size of its expansion parameter Q/Λχ, the pion-production reactions involve
somewhat large energy- and three-momentum transfers even at threshold. Therefore
the application of HBχPT to the NN→NNπ reactions may involve some delicate
aspects, but this also means that these processes may serve as a good test case for
probing the limit of applicability of HBχPT. Apart from this general issue to be
investigated, a specific aspect of the pp→ppπ0 reaction makes its study particularly
interesting. For most isospin channels, the NN→NNπ amplitude near threshold is
dominated by the pion rescattering diagram where the πN scattering vertex is given
by the Weinberg-Tomozawa term, which represents the lowest chiral order contri-
bution. However, a quantitatively reliable description of the NN→NNπ reactions
obviously requires detailed examinations of the corrections to this dominant ampli-
tude. Meanwhile, since the Weinberg-Tomozawa vertex does not contribute to the
pion-nucleon rescattering diagram for pp→ppπ0, this reaction is particularly sensi-
tive to higher chiral-order contributions and hence its study is expected to provide
valuable information to guide us in formulating a quantitative description of all the
NN→NNπ reactions (including the channels that involve a deuteron).
The first HBχPT-based study of the near-threshold pp→ppπ0 reaction was made
in Refs. [1, 2]. In HBχPT one naturally expects a small cross section for this reac-
tion since, for s-wave pion production, the pion-nucleon vertex in the impulse ap-
proximation (IA) diagram and the pion-rescattering vertex in the one-pion-exchange
rescattering (1π-Resc) diagram arise from the next-to-leading-order (NLO) chiral la-
grangian. A remarkable feature found in Refs. [1, 2] is that a drastic cancellation
between the IA and 1π-Resc amplitudes leads to the suppression of the pp→ppπ0
amplitude far beyond the above-mentioned naturally expected level. This destruc-
tive interference is in sharp contrast with the constructive interference reported in
SNPA-based calculations [3, 4]. It is to be recalled that the pp→ppπ0 cross section
obtained in Refs. [3, 4] was significantly smaller (by a factor of ∼5) than the experi-
mental value [5]. The drastic cancellation between the IA and 1π-Resc terms found
in the HBχPT calculations [1, 2] leads to even more pronounced disagreement be-
tween theory and experiment. In this connection it is worth noting that, according
to Lee and Riska [6], the heavy-meson (σ and ω) exchanges can strongly enhance
the pp→ppπ0 amplitude. It is also to be noted that σ-meson-exchange introduced
in many NN potentials is more properly described by correlated two-pion-exchange
(see e.g., Refs. [7, 8]), and that there have been substantial developments in deriving
a two-pion exchange NN potential using HBχPT, see e.g. [9]. These developments
were conducive to a HBχPT study of two-pion-exchange (TPE) contributions to the
pp→ppπ0 reaction [10, 11]. In the plane-wave approximation it was found [10] that
TPE contributions are indeed very large (as compared to the 1π-Resc amplitude), a
result that is in line with the finding in Ref.[6]. A subsequent DWBA calculation [11]
indicates that this feature remains essentially unchanged when the initial- and final-
state interactions are taken into account. More recent investigations [12, 13, 14, 15],
however, have raised a number of important issues that call for further investigations,
and the purpose of our present note is to address these issues.
In Ref. [10], to be referred to as DKMS, were derived all the transition operators
for pp→ppπ0 belonging to next-to-next-to-leading order (NNLO) in the Weinberg
counting, and these operators were categorized into Types I ∼ VII, according to the
patterns of the corresponding Feynman diagrams; see Figs. 2 - 5 in DKMS. Types I,
II, III and IV belong to diagrams of the two-pion exchange (TPE) type, while Types
V, VI and VII arise from diagrams of the vertex correction type. A notable feature
pointed out in DKMS is that the contributions of Types II ∼ IV are by far the largest,
and that they even exceed those of the 1π-Resc amplitude, which is formally of lower
chiral order. On the other hand, the possibility of strong cancellation among the TPE
diagrams was pointed out in Refs. [12, 13]. This motivates us to make here a further
study of the behavior of the TPE diagrams.1
A remark is in order here on a counting scheme to be used. At the NN→NNπ
threshold the nucleon three-momentum must change from the initial value p ∼√
mπmN to zero, entailing a rather large momentum transfer. To take this large
1For a brief report on this study, see Ref. [16].
momentum transfer into account, Cohen et al. [2] proposed a new counting scheme,
to be called the momentum counting scheme (MCS); see Ref. [13] for a detailed
review. In MCS the expansion parameter is ǫ̃ ≡ p/mN ≃ (mπ/mN)1/2, which is
larger than the usual HBχPT expansion parameter ǫ ≃ mπ/mN . A study based
on MCS [13] indicates that the 1π-Resc diagram for pp→ppπ0 is higher order in ǫ̃
(and hence less important) than a certain class of TPE diagrams, called “leading
order loop diagrams”, and that MCS is consistent with the estimates of the TPE and
other diagrams reported in DKMS. Furthermore, according to Hanhart and Kaiser
(HK) [12], the “leading parts” (see below) of these MCS “leading order” diagrams
exhibit exact cancellation among themselves;2 see also Lensky et al. [14]. Although
these studies are illuminating, we consider it important to examine the behavior of
the “sub-leading” parts (in MCS counting) of these TPE diagrams in order to see
whether they can be still as large as indicated by the phenomenological success of the
Lee-Riska heavy-meson exchange mechanism. In what follows we shall demonstrate
that this is indeed the case.
Analytic expressions for the pp→ppπ0 transition operators to NNLO in HBχPT
were given in DKMS. Although these expressions are valid for arbitrary kinematics,
we find it illuminating to concentrate here on their simplified forms obtained with
the use of fixed kinematics approximation (FKA), wherein the energies associated
with particle propagators are “frozen” at their threshold values. In FKA, the TPE
operator corresponding to each of the above-mentioned Types I ∼ IV can be written
~Σ · ~k
t(p, p′, x) (1)
where ~p (~p ′) is the relative three-momentum in the initial (final) pp state (~p1−~p2 = 2~p,
~p ′1 − ~p ′2 = 2~p ′), ~k ≡ ~p − ~p ′, x = p̂ · p̂′, and ~Σ = 12(~σ1 − ~σ2). The function t(p, p
′, x)
diverges as k → ∞, and it is useful to decompose t(p, p′, x) into terms that have
definite k-dependence as k → ∞. It turns out [18] that t(p, p′, x) can be expressed as
t(p, p′, x)
k→∞∼ t1
gA/(8f
|~k|+ t2
ln{|~k|2/Λ2}
+ t3 + δt(p, p
′, x), (2)
where t3 is asymptotically k-independent, and δt(p, p
′, x) is O(k−1). For each of
Types I ∼ IV, analytic expressions for ti’s (i = 1, 2, 3) can be extracted [18] from
the amplitudes T given in DKMS [10]. The first term with t1 in eq.(2) is the leading
2HK [12] pointed out that the sign of the contribution of Type II in Ref. [10] should be reversed;
we have confirmed the necessity of this correction.
part in MCS discussed by HK [12], whereas the remaining terms, which we refer to as
the “sub-leading” terms, were not considered by HK. The study of these sub-leading
terms is an important theme in what follows. Table 1 shows the value of t1 for Type
K (K= I ∼ IV) extracted from the results given in DKMS. The third row in Table 1
gives the ratio RK = TK/TResc, where TK is the plane-wave matrix element of T in
eq.(1) for Type K (K=I ∼ IV) normalized by TResc, the plane-wave matrix element of
the 1π-Resc diagram. The fourth row in Table 1 gives R ⋆K = T
K/TResc, where T
the plane-wave matrix element of T with the t1 term in eq.(2) subtracted. We can see
from the table that the most divergent t1 terms of the TPE diagrams add up to zero,
confirming the result of Ref. [12]. However, this does not necessarily mean that the
TPE diagrams are unimportant, because we still need to examine the contributions
of the “sub-leading” terms (the t2, t3 and δt terms) in eq.(2). Comparison of RK
and R ⋆K indicates that the subtraction of the t1 term reduces the magnitude of TK
drastically (except for Type I which has no t1 term), but the fact that |R ⋆K | is of the
order of unity (Types I, II and IV) or larger than 1 (Type III) suggests that the TPE
contributions can be quite important. The sum of the contributions of Types I ∼ IV
R ⋆K (=
RK) = −4.65 , (3)
which indicates that, at least in plane-wave approximation, the TPE contributions
are more important than the 1π-Resc contribution.
Table 1: For the four types of TPE diagrams, K= I, II, III and IV, the
second row gives the value of t1 defined in eq.(2), and the third row gives
the ratio RK = TK/TResc, where TK is the plane-wave matrix element of
T in eq.(1) for Type K, and TResc is the 1π-Resc amplitude. The last row
gives R ⋆K = T
K /TResc, where T
K is the plane-wave matrix element of T in
eq.(1) with the t1 term in eq.(2) subtracted.
Type of diagrams : K = I II III IV
(t1)K 0 1 1/2 −3/2
RK −.70 −6.54 −6.60 9.19
R ⋆K −.70 −0.82 −3.73 0.61
Next we investigate the behavior of the TPE diagrams as we go beyond the plane-
wave approximation by using distorted waves (DW) for the initial- and final-state
NN wave functions. For formal consistency we should use the NN potential derived
from HBχPT, but we adopt here a “hybrid EFT” approach and use phenomenological
potentials. A conceptual problem in adopting this hybrid approach is that, whereas
the TPE transition operators derived in HBχPT are valid only for a momentum range
sufficiently lower than Λχ∼1 GeV, a phenomenological NN potential can in principle
contain any momentum components.3 To stay close to the spirit of HBχPT, we
therefore introduce a Gaussian momentum regulator, exp(−p2/Λ2G), in the initial and
final distorted wave integrals, suppressing thereby the high momentum components
of the phenomenological NN potentials; this is similar to the MEEFT method used in
Ref.[19]. ΛG should be larger than the characteristic momentum scale of the pp→ppπ0
reaction, p ≃ √mNmπ ≃ 360 MeV/c, but it should not exceed the chiral scale Λχ;
in the present study we shall consider the range, 500 MeV< ΛG <1 GeV. As high-
precision phenomenological NN potentials, we consider the Bonn-B potential [20],
the CD-Bonn potential [21], and the Nijm93 potential of the Nijmegen group [22].
It is worth noting here that several groups [23, 24] have developed a systematic
approach to construct from a phenomenological potential an effective NN potential,
called Vlow−k, that resides within a model space which only contains momentum
components below a specified cutoff scale Λlow−k. In this work we will use Vlow−k as
derived by the Stony Brook group [24]. It is conceptually natural to use Vlow−k in
conjunction with transition operators derived from HBChPT [25]. A problem however
is Vlow−k [24], primarily meant for describing sub-pion-threshold phenomena, was
obtained with the use of a rather low cutoff, Λlow−k ∼ 2 fm−1. This cutoff is perhaps
too close to the characteristic momentum scale p ∼ 360 MeV/c for the pion production
reaction. It therefore seems worthwhile to “rederive” Vlow−k employing a momentum
cut-off higher than 2 fm−1 and use it in the present DWBA calculation. Below we
will use Vlow−k generated from the CD-Bonn potential for Λlow−k= 4 and 5 fm
−1. We
remark that, as is well known, Vlow−k’s generated from any realistic phenomenological
potentials lead to practically equivalent half-off-shell NN K-matrices and hence the
same NN wave function.
We evaluate the TPE contributions in DWBA for a typical case of Tlab = 281
MeV. Since the t1 terms in eq.(2) add up to zero, we drop the t1 terms in our cal-
culation.4 Thus, in eq.(1), we use t⋆(p, p′, x) instead of t(p, p′, x), where t⋆(p, p′, x) is
3 A pragmatic problem associated with this conceptual issue is that, in a momentum-space calcu-
lation of the matrix elements of the TPE operators sandwiched between distorted pp wave-functions
generated by a phenomenological NN potential, the convergence of momentum integrations is found
to be extremely slow [17, 18].
4 Removing the t1 term lessens the severity of the convergence problem in our momentum inte-
obtained from t(p, p′, x) by suppressing the t1 term. The partial-wave projected form
of t⋆(p, p′, x) in a DWBA calculation is written as:
J = −
p2dp p′ 2dp′
dx ψ1S0(p
′) t⋆(p, p′, x) (p− p′x)ψ3P0(p) (4)
Here ψα(p) is a distorted two-nucleon relative wave function in the α partial-wave
(1S0 for the initial state and
3P0 for the final state) given by
ψα(p) = cos(δα)
δ(p− pon)/p2 + P
Kα(p, pon)
(E − Ep)
, (5)
where δα is the phase-shift for the α partial wave, and Kα(p, pon) is the partial-
wave K-matrix pertaining to the asymptotic on-shell momentum pon. The plane-wave
approximation corresponds to the use of the wave functions of the generic form:
ψ(p) = δ(p− pon)/p2 . (6)
We show in Table 2 the values of J , eq.(4), for the TPE operators of Types I ∼ IV,
calculated at Tlab = 281 MeV, with the use of the Nijm93 potential of the Nijmegen
group [22]5 and Vlow−k. For the Nijm93 potential case, we present the results for five
different values of ΛG between 500 and 1000 MeV/c. For the Vlow−k case, the results
for two choices of Λlow−k are shown: Λlow−k = 4 fm
−1 and 5 fm−1. For comparison, the
values of J corresponding to plane-wave approximation are also shown (bottom row).
From Table 2 we learn the following: (1) The results for the Nijm93 potential with the
gaussian cutoff ΛG are stable against the variation of ΛG within a reasonable range
(500 - 1000 MeV/c); (2) There is semi-quantitative agreement between the results for
the Nijm93 potential and those for Vlow−k; (3) A semi-quantitative agreement is also
seen between the DWBA and PWBA calculations; (4) The feature found in the plane-
wave approximation that the contributions of the TPE diagrams are more important
than the 1π-Resc contribution remains unchanged in the DWBA calculation; the
summed contribution of the TPE operators is larger (in magnitude) than that of
1π-Resc by a factor of 2∼3.5.
gration mentioned in footnote 3.
5 We have checked the results obtained using the Bonn-B and CD-Bonn NN potentials are very
similar to those for the Nijm93 potential case, which we show here as a representative case.
Table 2: The values of J , eq.(4), corresponding to the TPE diagrams of
Types I ∼ IV, evaluated in a DWBA calculation for Tlab = 281 MeV. The
column labeled “Sum” gives the combined contributions of Types I ∼ IV,
and the last column gives the value of J for 1π-Resc. For the Nijm93
potential case, the results for five different choices of ΛG are shown. For the
case with Vlow−k, CD-4 (CD-5) represents Vlow−k generated from the CD-
Bonn potential with a momentum cut-off Λlow−k = 4 fm
−1 (5 fm−1). The
last row gives the results obtained in plane-wave approximation.
I II III IV Sum 1π−Resc
VNijm : ΛG = 500MeV/c −0.11 −0.12 −0.55 0.08 −0.70 0.18
VNijm : ΛG = 600MeV/c −0.12 −0.12 −0.57 0.07 −0.74 0.20
VNijm : ΛG = 700MeV/c −0.12 −0.11 −0.57 0.06 −0.74 0.21
VNijm : ΛG = 800MeV/c −0.12 −0.11 −0.55 0.04 −0.74 0.22
VNijm : ΛG = 1000MeV/c −0.12 −0.10 −0.52 0.03 −0.71 0.23
Vlow−k (CD−4) −0.12 −0.09 −0.46 0.03 −0.65 0.23
Vlow−k (CD−5) −0.09 −0.06 −0.30 −0.01 −0.46 0.22
Plane−wave −0.06 −0.07 −0.30 0.05 −0.37 0.080
We now discuss the above results in the context of MCS [13]. A subtlety in MCS is
that a loop diagram of a given order ν in ǫ̃ not only contains a contribution of order ν
(“leading part”) but, in principle, can also involve contributions of higher orders in ǫ̃
(“sub-leading part”) due to the non-analytic functions generated by the loop integral.
As mentioned, however, HK [12] considered only the leading part, which correspond
to the t1 term in eq.(2). According to MCS, for the reaction pp → ppπ0, the loop
diagrams corresponding to our Type II, III and IV diagrams belong to NLO in the
ǫ̃ parameter, whereas those corresponding to Type I and the 1π-Resc tree diagram
are next order in ǫ̃ (NNLO); see Table 11 in Ref. [13]. Meanwhile, as discussed
earlier, the sum of the “leading parts” of the NLO diagrams vanishes, and therefore,
in calculating J ’s in Table 2, we have dropped the t1 term contribution, retaining only
the “sub-leading” parts of these NLO diagrams. This means that all the entries in
Table 2 represent “sub-leading contributions” (NNLO) in MCS. If we look at Table 2
from this perspective, we note that the order-of-magnitude behavior of our numerical
results is in rough agreement with MCS, although Type IV tends to be rather visibly
smaller (in magnitude) than the others. However, it is striking that J for Type III
is significantly (if not by an order of magnitude) larger than the other sub-leading
contributions. (A similar feature was also seen in R⋆ in Table 1.) In view of the fact
that Type III arises from crossed-box TPE diagrams [10], there is a possibility that
the enhancement of the Type III diagrams may be related to the strong attractive
scalar NN potential that is known to arise from TPE crossed-box-diagrams [7, 8].
We have studied the “sub-leading” parts, which are of NNLO in the momentum
counting scheme (MCS) [13], of the TPE amplitudes for the pp→ppπ0 reaction in both
PWBA and DWBA calculations. We have shown in fixed kinematics approximation
(FKA) that, even though the leading parts of the TPE amplitudes cancel among
themselves [12, 14], the contributions of the sub-leading parts are quite significant.
They are in general comparable to the 1π-Resc amplitude, and the sub-leading part
of the Type III diagrams is even significantly larger than the 1π-Resc diagram. The
total contribution of the TPE diagrams is larger (in magnitude) than that of the
1π-Resc diagram by a factor of ∼5 (PWBA) or 2∼3 (DWBA). We have focused here
on the TPE loop diagrams but, to obtain theoretical cross section for pp→ppπ0 that
can be directly compared with the experimental value, we must consider the other
diagrams discussed in DKMS as well as the relevant counter terms. These will be
discussed in a forthcoming article [26] .
The authors are indebted to Christoph Hanhart and Anders G̊ardestig for useful
discussions. A helpful communication from Ulf Meissner is also grateful acknowl-
edged. This work is supported in part by the US National Science Foundation, Grant
No. PHY-0457014, and by the Japan Society for the Promotion of Science, Grant-in-
Aid for Scientific Research (C) No.15540275 and Grant-in-Aid for Scientific Research
on Priority Areas (MEXT), No. 18042003.
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