Two-pion exchange three-nucleon potential: O(q^4) chiral expansion
arXiv:0704.0711v1  [nucl-th]  5 Apr 2007
Two-pion exchange three-nucleon potential:
O(q4) chiral expansion
S. Ishikawa1 and M. R. Robilotta2
1Department of Physics, Science Research Center,
Hosei University, 2-17-1 Fujimi, Chiyoda, Tokyo 102-8160, Japan
2Instituto de F́ısica, Universidade de São Paulo,
C.P. 66318, 05315-970, São Paulo, SP, Brazil
(Dated: November 4, 2018)
Abstract
We present the expansion of the two-pion exchange three-nucleon potential (TPE-3NP) to chiral
order q4, which corresponds to a subset of all possibilities at this order and is based on the πN
amplitude at O(q3). Results encompass both numerical corrections to strength coefficients of
previous O(q3) terms and new structures in the profile functions. The former are typically smaller
than 10% whereas the latter arise from either loop functions or non-local gradients acting on the
wave function. The influence of the new TPE-3NP over static and scattering three-body observables
has been assessed and found to be small, as expected from perturbative corrections.
PACS numbers: 13.75.Cs, 21.30.Fe, 13.75.Gx, 12.39.Fe
http://arxiv.org/abs/0704.0711v1
I. INTRODUCTION
The research programme for nuclear forces, outlined more than fifty years ago by Taketani,
Nakamura, and Sasaki [1], treats pions and nucleons as basic degrees of freedom. This insight
proved to be very fruitful. On the one hand, it implies the interconnection of all nuclear
processes, both among themselves and with a class of free reactions. On the other, it
determines a close relationship between the number of pions involved in a given interaction
and its range. As a consequence, the outer components of nuclear forces are dominated by
just a few basic subamplitudes, describing either single (N → πN) or multipion (ππ → ππ,
πN → πN , πN → ππN , ...) interactions.
Nevertheless, it took a long time for a theoretical tool to be available which allows the
precise treatment of these amplitudes. Nowadays, owing to the development of chiral per-
turbation theory (ChPT) in association with effective lagrangians [2, 3], the roles of pions
and nucleons in nuclear forces can be described consistently. The rationale for this approach
is that the quarks u and d, which have small masses, dominate low-energy interactions.
One then works with a two-flavor version of QCD and treats their masses as perturbations
in a chiral symmetric lagrangian. The systematic inclusion of quark mass contributions is
performed by means of chiral perturbation theory, which incorporates low-energy features
of QCD into the nuclear force problem. In performing perturbative expansions, one uses a
typical scale q, set by either pion four-momenta or nucleon three-momenta, such that q ≪ 1
Nuclear forces are dominated by two-body (NN) interactions and leading contributions
are due to the one-pion exchange potential (OPEP), which begins [4] at O(q0). The two-
pion exchange potential (TPEP) begins at O(q2) and, at present, there are two independent
expansions up to O(q4) in the literature, based on either heavy baryon [5] or covariant [6, 7]
ChPT. The TPEP is closely related with the off-shell πN amplitude and, at this order,
two-loop diagrams involving intermediate ππ scattering already begin to contribute.
In proper three-nucleon (3N) interactions, the leading term is due to the process known
as TPE-3NP, in which the pion emitted by a nucleon is scattered before being absorbed
by another one. It has been available since long [8–10], involves only tree-level interactions
and has the longest possible range. This contribution begins at O(q3) and consistency with
available NN forces demands the extension of the chiral series for the 3NP up to O(q4).
However, the implementation of this programme is not straightforward, since it requires
the evaluation of a rather large number of diagrams. With the purpose of exploring the
magnitude of O(q4) effects, in this work we concentrate on the particular subset of processes
which still belong to the TPE-3NP class. Our presentation is divided as follows. In section
II we display the general relationship between the TPE-3NP and the πN amplitude, in order
to discuss how it affects chiral power counting in the former. The πN amplitude relevant for
the O(q4) potential is derived in section III and used to construct the three-body interaction
in section IV. We concentrate on numerical changes induced into both potential parameters
and observables in sections V and VI, whereas conclusions are presented in section VII.
There are also four appendices, dealing with kinematics, πN subthreshold coefficients, loop
integrals and non-local terms.
II. GENERAL FORMULATION
Potentials to be used into non-relativistic equations can be derived from field the-
ory by means of the T -matrix. In the case of three-nucleon potentials, one starts
from the non-relativistic transition matrix describing the process N(p1) N(p2) N(p3) →
N(p′1) N(p
2) N(p
3), which includes both kernels and their iterations. The former corre-
spond to proper interactions, represented by diagrams which cannot be split into two pieces
by cutting positive-energy nucleon lines only, whereas the latter are automatically gener-
ated by the dynamical equation. Therefore, just the kernels, denoted collectively by t̄3, are
included into the potential.
The transformation of a T -matrix into a potential depends on both the dynamical equa-
tion adopted and conventions associated with off-shell effects. The latter were discussed
in a comprehensive paper by Friar [11]. Here we use the kinematical variables defined in
Appendix A and relate t̄3 to the momentum space potential operator Ŵ by writing [12]
〈p′1,p′2,p′3 |Ŵ |p1,p2,p3〉 = −(2π)3 δ3(P ′−P ) t̄3(p′1,p′2,p′3,p1,p2,p3) . (1)
In configuration space, internal dynamics is described by the function
W (r′,ρ′; r,ρ) = −
(2π)3
(2π)3
(2π)3
(2π)3
×ei[Qr·(r′−r)+Qρ·(ρ′−ρ)+qr ·(r′+r)/2+qρ·(ρ′+ρ)/2] t̄3(Qr,Qρ, qr, qρ) , (2)
which is to be used in a non-local version of the Schrödinger equation:
ρ′ − ǫ
ψ(r′,ρ′) = −
dr dρ W (r′,ρ′; r,ρ) ψ(r,ρ) . (3)
Non-local effects are associated with the variablesQr andQρ. When these effects are not too
strong, they can be represented by gradients acting on the wave function and the potential
W is rewritten as
W (r′,ρ′; r,ρ) = δ3(r′−r) δ3(ρ′−ρ)
V (r,ρ) . (4)
The two-pion exchange three-nucleon potential is represented in Fig. 1a. It is closely related
with the πN scattering amplitude, which is O(q) for free pions and becomes O(q2) within
the three-nucleon system. As a consequence, the TPE-3NP begins at O(q3) and, at this
order, it also receives contributions from interactions (c) and (d), which have shorter range.
The extension of the chiral series to O(q4) requires both the inclusion of single loop effects
into processes that already contribute at O(q3) and the evaluation of many new amplitudes,
especially those associated with diagram (b).
(a) (b) (c) (d)
FIG. 1: (Color online) Classes of three-nucleon forces, where full and dashed lines represent nucleons
and pions respectively; diagram (a) corresponds to the TPE-3NP.
In this paper we concentrate on the particular set of processes which belong to the TPE-
3NP class, represented by the T -matrix Tππ and evaluated using the kinematical conditions
given in Fig. 2. The coupling of a pion to nucleon i = (1, 2) is derived from the usual lowest
order pseudo-vector lagrangian L(1) and the Dirac equation yields the equivalent forms for
the vertex
(gA/2fπ) [τ ū (p
′−p) γ5 u](i) = (mgA/fπ) [τ ū γ5 u](i) , (5)
where gA, fπ and m represent, respectively, the axial nucleon decay, the pion decay and the
nucleon mass.
k´, b
   k , a
FIG. 2: (Color online) Two-pion exchange three-nucleon potential.
The amplitude for the intermediate process πa(k)N(p) → πb(k′)N(p′) has the isospin
structure
Tba = δab T
+ + iǫbacτc T
− (6)
and Fig. 2 yields
Tππ =−
[ū γ5 u]
[ū γ5 u]
(2) 1
k2−µ2
′2−µ2
(1) ·τ (2) T+ − i τ (1)×τ (2) ·τ (3) T−
, (7)
µ being the pion mass. Results in Appendix A show that [ū γ5 u]
(i) → O(q), whereas pion
propagators are O(q−2). As a consequence, in the O(q4) expansion of the potential one
needs Tππ to O(q) and T± to O(q3). For on-shell nucleons, the sub amplitudes T± can be
written as
T± = ū(p′)
D± − i
σµν(p
′−p)µKν B±
u(p) , (8)
with K = (k′+k)/2. The dynamical content of the πN interaction is carried by the functions
D± and B± and their main properties were reviewed by Höhler [13]. The chiral structure
of these sub amplitudes was discussed by Becher and Leutwyler [14, 15] a few years ago,
in the framework of covariant perturbation theory, and here we employ their results. As
far as power counting is concerned, in Appendix A one finds [ū(p′) u(p)](3) → O(q0) and
ū(p′) σµν(p
′−p)µKν u(p)](3) → O(q2), indicating that one needs the expansions of D±
and B± up to O(q3) and O(q) respectively.
At low and intermediate energies, the πN amplitude is given by a nucleon pole superim-
posed to a smooth background. One then distinguishes the pseudovector (PV) Born term
from a remainder (R) and writes
T± = T±pv + T
R . (9)
The former contribution depends on just two observables, namely the nucleon mass m and
the πN coupling constant g, as prescribed by the Ward-Takahashi identity [16]. The calcu-
lation of these quantities in chiral perturbation theory may involve loops and other coupling
constants but, at the end, results must be organized so as to reproduce the physical values
of both m and g in T±pv [17]. For this reason, one uses the constant g, instead of (gA/fπ),
since the former is indeed the observable determined by the residue of the nucleon pole
[13, 15, 18]. The pv Born sub amplitudes are given by
D+pv =
k′ ·k
k′ ·k
, (10)
B+pv = −g2
, (11)
D−pv =
k ·k′
k ·k′
, (12)
B−pv = −g2
, (13)
where s and u are the usual πN Mandelstam variables. In the case of free pions, their chiral
orders are respectively [D+pv, B
pv, D
pv, B
pv] → O[q2, q−1 , q, q0], but important changes do
occur when the pions become off-shell.
The amplitudes T±R receive contributions from both tree interactions and loops. The
former can be read directly from the basic lagrangians and correspond to polynomials in
t = (k′−k)2 and ν = (p′+p)·(k′+k)/4m, with coefficients given by renormalized LECs [15].
The latter are more complex and depend on Feynman integrals. In the description of πN
amplitudes below threshold, one approximates both types of contributions by polynomials
and writes [13, 19]
2mtn , (14)
where XR stands for D
R/ν, D
R/ν or B
R . The subthreshold coefficients xmn have the
status of observables, since they can be obtained by means of dispersion relations applied
to scattering data. As such, they constitute an important source of information about the
values of the LECs to be used in effective lagrangians.
The isospin odd subthreshold coefficients include leading order terms, which implement
the predictions made by Weinberg [20] and Tomozawa [21] for πN scattering lengths, given
D−WT =
2f 2π
, B−WT =
2f 2π
. (15)
For free pions, one has [D−WT , B
WT ] → O[q, q0], but these orders of magnitude also change
when pions become virtual.
Quite generally, the ranges of nuclear interactions are determined by t-channel exchanges.
At O(q3), the TPE-3NP involves only single-pion exchanges among different nucleons and
has the longest possible range. Another t-channel structure becomes apparent at O(q4),
associated with the pion cloud of the nucleon, which gives rise to both scalar and vector
form factors [18]. These effects extend well beyond 1 fm [22, 23] and a limitation of the
power series given by Eq. (14) is that they cannot accommodate these ranges, since Fourier
transforms of polynomials yield only δ-functions and its derivatives. In the description of the
πN amplitude produced by Becher and Leutwyler [15], one learns that the only sources of
medium range (mr) effects are their diagrams k and l, which contain two pions propagating
in the t-channel. In our derivation of the TPE-3NP, the loop content of these diagrams is not
approximated by power series and, for free pions, the non-pole subamplitudes are written as
D+R = D
mr(t) +
d̄+00 + d
2 + d̄+01t
d+20ν
4 + d+11ν
2t + d̄+02t
, (16)
B+R = B
mr(t) +
b+00ν
, (17)
D−R = D
mr(t) +
ν/(2f 2π)
d̄−00ν + d
3 + d̄−01νt
, (18)
B−R = B
mr(t) +
1/(2f 2π) + b̄
b−10ν
2 + b̄−01t
, (19)
where the labels (n) outside the brackets indicate the presence of O(qn) leading terms
and mr denotes terms associated with the nucleon pion cloud. The bar symbol over some
coefficients indicates that they do not include both Weinberg-Tomozawa and medium range
contributions, which are accounted for explicitly. The functions D±R and B
R depend on the
parameters fπ, gA, µ, m and on the LECs ci and d̄i, which appear into higher order terms
of the effective lagrangian. The subthreshold coefficients are the door through which LECs
enter our calculation and their explicit forms are given in Appendix B.
The dynamical content of the O(q3) πN amplitude is shown in Fig. 3. The first two
diagrams correspond to PV Born amplitudes, whereas the third one represents the Weinberg-
=    + +
FIG. 3: (Color online) Representation of the πN amplitude used in the construction of the TPE-
Tomozawa contact interaction, all of them with physical masses and coupling constants. The
fourth graph summarizes the terms within square brackets in Eqs. (16-19) and depends on
the LECs. Finally, the last two diagrams describe medium range effects owing to the nucleon
pion cloud, associated with scalar and vector form factors. This decomposition of the πN
amplitude has also been used in our derivation of the two-pion exchange components of the
NN interaction [6, 7] and hence the present calculation is consistent with those results.
III. INTERMEDIATE πN AMPLITUDE
The combination of Figs. 2 and 3 gives rise to the TPE-3NP, associated with the six
diagrams shown in Fig. 4. In the sequence, we discuss their individual contributions to the
subamplitudes D± and B±. We are interested only in the longest possible component of
the potential and numerators of expressions are systematically simplified by using k2 → µ2
and k
′2 → µ2. In configuration space, this corresponds to keeping only those terms which
contain two Yukawa functions and neglecting interactions associated with Figs. 1 (c) and 1
• diagrams (a) and (b): The crosses in the nucleon propagators of Figs. 4 (a) and 4 (b)
indicate that they do not include forward propagating components, so as to avoid double
counting when the potential is used in the dynamical equation. The covariant evaluation of
these contributions is based on Eqs. (10-13). Denoting by p̄ the momenta of the propagating
 + +=
 +  + +
(a) (b) (c)
(d) (e) (f)
FIG. 4: (Color online) Structure of the O(q4) two-pion exchange three-nucleon potential
nucleons, the factors 1/(s−m2) and 1/(u−m2) are decomposed as
(p̄0)2−Ē2
2Ē(p̄0−Ē)
2Ē(p̄0+Ē)
, (20)
with Ē =
m2+p̄2. The first term represents forward propagating nucleons, associated with
the iteration of the OPEP, whereas the second one gives rise to connected contributions.
Discarding the former and using the results of Appendix A, one has
1/({su} −m2)→ −1/
4m2 +
3q2r+q
ρ/3+16Q
ρ/3± 10qr ·Qρ/
3∓ 2qρ ·Qr/
. (21)
After appropriate truncation, one obtains
D+ab = −
(2µ2−t)→ O(q2) , (22)
B+ab → O(q
2) , (23)
D−ab = −
ν → O(q2) , (24)
B−ab → O(q
2) , (25)
where we have used the fact that, in the case of virtual pions, ν → O(q2).
• diagrams (c) and (d): These contributions are purely polynomial, can be read directly
from Eqs. (16-19), and are given by
D+cd = −
16 πf 4π
(2µ2−t)→ O(q2) , (26)
B+cd → O(q
2) , (27)
D−cd =
2f 2π
ν → O(q2) , (28)
B−cd =
2f 2π
2 c4m
8 πf 4π
→ O(q0) . (29)
• diagrams (e) and (f): The medium range components of the intermediate πN amplitude
D+e =
64π2f 4π
(2t−µ2)
(1−t/2µ2) Πt − 2π
→ O(q3) , (30)
D+ef → O(q
4) , (31)
B−e =
g2Amµ
16π2f 4π
(1−t/4µ2) Πt − π
→ O(q) , (32)
where Πt is the dimensionless Feynman integral
µ2 F (a)
← M = 2µ/a , F (a) =
tan−1
µ (1−a2/2)
. (33)
The amplitude D−ef , proportional to ν, is O(q3) for free pions and here becomes O(q4). Thus,
in fact, diagram (f) does not contribute to the TPE-3NP at O(q4).
• full results: The Golberger-Treiman relation g/m = gA/fπ is valid up to O(q2) and can
be used in diagrams (a) and (b). One then has
σ(2µ2)
(2µ2−t)
+ c3 +
g2A(1+g
16πf 2π
128π2f 2π
(1−2t/µ2) Πt
, (34)
where
σ(t = 2µ2) = −4 c1 µ2 −
3g2Aµ
32πf 2π
is the value of the scalar form factor at the Cheng-Dashen point [14]. The remaining ampli-
tudes read
B+ → O(q2) , (36)
1−g2A
2f 2π
ν , (37)
1 + 4 c4m
2f 2π
A(1+2g
16 πf 4π
g2Amµ
16 π2f 4π
(1−t/4µ2) Πt . (38)
The subamplitudes D± and B± begin at O(q2) and one needs just the leading terms in the
spinor matrix elements of Eq. (8), which is rewritten as
T+ = 2mD+ , (39)
T− = 2mD− + iσ(3) ·k′×kB− , (40)
with D+ → O(q2)+O(q3), D− → O(q2), and B− → O(q0)+O(q).
• O(q3) reduction: In order to compare our amplitudes with previous O(q3) results, one
notes that, in case corrections are dropped, one would have
(2µ2−t)
, (41)
2f 2π
2 c4m
. (42)
These expressions agree with those derived directly from a chiral lagrangian [24], except
for the terms within square brackets in both D+ and B−. The former corresponds to a
Born contribution whereas the latter is due to diagram (c) in Fig. 4, associated with the
Weinberg-Tomozawa term.
IV. TWO-PION EXCHANGE POTENTIAL
The expansion of the TPE-3NP up to O(q4) requires only leading terms in vertices and
propagators. In order to derive the non-relativistic potential in momentum space, one divides
Eq. (7) by the relativistic normalization factor
2m for each external nucleon leg
and writes1
t̄3 =
4f 2π
′2+µ2
(1) ·k σ(2) ·k′
(1) ·τ (2)D+ − i τ (1) × τ (2) ·τ (3)
(3) ·k′×k B−
. (43)
The configuration space potential has the form
V3(r,ρ) = τ
(1) ·τ (2) V +3 (r,ρ) + τ (1) × τ (2) ·τ (3) V −3 (r,ρ) + cyclic permutations, (44)
1 One notes that this expression is identical with Eq. (33) of Ref. [10] divided by 8m3.
V +3 (r,ρ) = C
(1) ·x̂31 σ(2) ·x̂23 U1(x31)U1(x23)
+ C+2
(1/9)σ(1) ·σ(2) [U(x31)−U2(x31)] [U(x23)−U2(x23)]
+ (1/3)σ(1) ·x̂23 σ(2) ·x̂23 [U(x31)−U2(x31)] U2(x23)
+ (1/3)σ(1) ·x̂31 σ(2) ·x̂31 U2(x31) [U(x23)−U2(x23)]
+ σ(1) ·x̂31 σ(2) ·x̂23 x̂31 ·x̂23 U2(x31)U2(x23)
+ C+3 σ
(1) ·∇I31 σ(2) ·∇I23 ∇I31 ·∇I23
I0 − 2 I1
, (45)
V −3 (r,ρ) = C
(1/9)σ(1)×σ(2) ·σ(3) [U(x31)−U2(x31)] [U(x23)−U2(x23)]
+ (1/3)σ(3)×σ(1) ·x̂23 σ(2) ·x̂23 [U(x31)−U2(x31)] U(x23)
+ (1/3)σ(1) ·x̂31 σ(2)×σ(3) ·x̂31 U2(x31) [U(x23)−U2(x23)]
+ σ(1) ·x̂31 σ(2) ·x̂23 σ(3) ·x̂31×x̂23 U2(x31)U2(x23)
+ C−2
(1) ·
31 −i∇
(2) ·x̂23 [U(x31)−U2(x31)] U1(x23)
+ σ(1) ·x̂31 σ(2) ·
31 −i∇
U1(x31) [U(x23)−U2(x23)]
+ 3σ(1) ·x̂31 σ(2) ·x̂23
31 −i∇
· [x̂31 U2(x31)U1(x23) + x̂23 U1(x31)U2(x23)]
+ C−3 σ
(1) ·∇I31 σ(2) ·∇I23 σ(3) ·∇I31×∇I23
I0 − I1/4
. (46)
The profile functions are written in terms of the dimensionless variables xij = µ rij and read
U(x) =
, (47)
U1(x) = −
, (48)
U2(x) =
, (49)
In = −
(2π)3
(2π)3
ei(k·r31+k
·r23)
′2+µ2
Πt(t) . (50)
The last function involves the loop integral given in Eq. (33) and is discussed further in
Appendix C. The gradients ∇Iij act on the functions I
n, whereas the ∇
ij act only on the
wave function and give rise to non-local interactions, as discussed in Appendix D.
The strength coefficients are the following combinations of the basic coupling constants
C+1 =
64 π2f 4π
σ(2µ2) , (51)
C+2 =
32 π2f 4π m
+mc3 +
g2A(1+g
16πf 2π
, (52)
C+3 =
4096 π3f 6π
, (53)
C−1 =
256 π2f 4π m
1 + 4mc4 −
g2A(1+2g
8πf 2π
, (54)
C−2 =
g2A(g
A−1)µ6
768 π2f 4π m
, (55)
C−3 = −
2048 π3f 6π
. (56)
V. STRENGTH COEFFICIENTS
The strength constants of the potential involve a blend of four well determined param-
eters, namely m = 938.28 MeV, µ = 139.57 MeV, gA = 1.267 and fπ = 92.4 MeV, with
the scalar form factor at the Cheng-Dashen point and the LECs c3 and c4, which are less
precise. As far as σ(2µ2) is concerned, we rely on the results [25] σ(2µ2)−σ(0) = 15.2± 0.4
MeV, σ(0) = 45±8 MeV, and adopt the central value σ(2µ2) = 60 MeV. The values quoted
for the LECs in the literature vary considerably, depending on the empirical input employed
and the chiral order one is working at. A sample of values is given in Table I.
Our work is based on the O(q3) expansion of the intermediate πN amplitude and, for
the sake of consistency, one must use LECs extracted at the same order. The kinematical
conditions of the three-body interaction are such that the variable ν is O(q2), an order of
magnitude smaller than the threshold value, ν = µ. This makes information encompassed in
the subthreshold coefficients better suited to this problem and we use results from Appendix
TABLE I: Some values of the LECs c3 and c4; m is the nucleon mass.
Reference Chiral order πN input mc3 mc4
[26] 3 amplitude at ν = 0, t = 0 −5.00± 1.43 3.62 ± 0.04
[26] 3 amplitude at ν = 0, t = 2µ2/3 −5.01± 1.01 3.62 ± 0.04
[27] 3 scattering amplitude −5.69± 0.04 3.03 ± 0.16
[15] 4 subthreshold coefficients -3.4 2.0
[15] 4 scattering lengths -4.2 2.3
tree 2 subthreshold coefficients -3.6 2.0
this work 3 subthreshold coefficients -4.9 3.3
B in order to write
mc3 = −mf 2π d+01 −
g4A mµ
16 π f 2π
− 77 g
768 π f 2π
, (57)
mc4 =
f 2π b
g2A(1+g
16 π f 2π
. (58)
Adopting the values for the subthreshold coefficients given by Höhler [13], namely d+01 =
1.14 ± 0.02µ−3 and b−00 = 10.36 ± 0.10µ−2, one finds the figures shown in the last row of
Table I. These, in turn, produce the strength coefficients displayed in Table II. For the sake
of comparison, we also quote values employed in our earlier calculation [10] and in two TM’
versions [28] of the Tucson-Melbourne potential [8].
TABLE II: Strength coefficients in MeV.
reference C+1 C
this work 0.794 -2.118 0.034 0.691 0.014 -0.067
Brazil [10] 0.92 -1.99 - 0.67 - -
TM’(93) [28] 0.60 -2.05 - 0.58 - -
TM’(99) [28] 0.91 -2.26 - 0.61 - -
Changes in these parameters represent theoretical progress achieved over more than two
decades and it is worth investigating their origins in some detail. With this purpose in
mind, we compare present results with those of our previous O(q3) calculation [10]. At the
chiral order one is working here, new qualitative effects begin to show up, associated with
both loops and non-local interactions. They are represented by terms proportional to the
coefficients C+3 , C
2 and C
3 in Eqs. (45) and (46).
The πN coupling is now described by g2Aµ
2/f 2π = 3.66 whereas, previously, the factor
g2µ2/m2 = 3.97 was used. From a conceptual point of view, the latter should be preferred,
since g is indeed the proper coupling observable. In chiral perturbation theory, the difference
between both forms is ascribed to the parameter ∆GT = −2d18µ2/g, which describes the
Goldberger-Treiman discrepancy [15]. As this is a O(q2) effect, both forms of the coupling
become equivalent in the present calculation. On the other hand, the empirical value of g is
subject to larger uncertainties and the form based on gA is more precise. Our present choice
accounts for a decrease of 8% in all parameters.
The relations C+1 ↔ Cs, C+2 ↔ Cp and C−1 ↔ −C ′p allow one to compare Eqs. (45)
and (46) with Eq. (67) of Ref. [10]. One notes that the latter contains an unfortunate
misprint in the sign of the term proportional to C ′p, as pointed out in Ref. [29]. In the
earlier calculation, the coefficient Cs was based on a parameter [30] ασ = 1.05µ
−1, which
corresponds to σ(2µ2) = 64 MeV. The results of Table II show that the values of C+2 and
C−1 are rather close to those of Cp and −C ′p. This can be understood by rewriting Eqs. (52)
and (54) in terms of the subthreshold coefficient d+01 and b
00 as follows
C+2 = −
32 π2f 4π m
mf 2π d
29g2Amµ
768πf 2π
, (59)
C−1 =
128 π2f 4π m
f 2π b
g2Amµ
16πf 2π
. (60)
Numerically, this amounts to C+2 = −(1.845 + 0.110 + [0.163]) MeV and C−1 = (0.624 +
[0.067]) MeV. The second term in the former equation was overlooked in Ref. [10] and
should have been considered there. The square brackets2 correspond to next-to-leading
order contributions and yield corrections of about 8% and 11% to the leading terms in C+2
and C−1 , respectively.
3 As the model used in Ref. [10] was explicitly designed to reproduce
the subthreshold coefficients quoted by Höhler [13], it produces the very same contributions
as the first terms in Eqs. (59) and (60).
2 These factors can be traced back to loop diagrams in Fig. 3 and are dynamically related with the term
proportional to C±
, as we discuss in Appendix C.
3 When comparing the new coefficients with those in the second row of Table II, one should also take into
account the 8% effect due to the Goldberger-Treiman discrepancy.
VI. NUMERICAL RESULTS FOR THREE-NUCLEON SYSTEMS
In order to test the effects of the TPE-3NP at O(q4), in this section, we present some
numerical results of Faddeev calculations for three-nucleon bound and scattering states. The
calculations are based on a configuration space approach, in which we solve the Faddeev
integral equations [31–33],
Φ3 = Ξ12,3 +
E + iǫ−H0 − V12
× [V12 (Φ1 + Φ2) +W3 (Φ1 + Φ2 + Φ3)] ,
(and cyclic permutations), (61)
where Ξ12,3, which does not appear in the bound state problem, is an initial state wave
function for the scattering problem, H0 is a three-body kinetic operator in the center of
mass, V12 is a nucleon-nucleon (2NP) potential between nucleons 1 and 2, and W3 is the
3NP displayed in Fig. 2. Partial wave states of a 3N system, in which both NN and 3N
forces act, are restricted to those with total NN angular momenta j ≤ 6 for bound state
calculations, and j ≤ 3 for scattering state calculations. The total 3N angular momentum
(J) is truncated at J = 19/2, while 3NP is switched off for 3N states with J > 9/2 for
scattering calculations. These truncation procedures are confirmed to give converged results
for the purposes of the present work.
When just local terms are retained, t̄3 in Eq. (43) can be cast in the conventional form
[8–10]
t̄3 = −
4f 2π
F (k2)
2 + µ2
F (k′2)
′2 + µ2
(σ(1) · k)(σ(2) · k′)
(τ (1) · τ (2)){a+ b(k · k′)}
−(iτ (1) × τ (2) · τ (3))(iσ(3) · k′ × k)d
, (62)
where the coefficients, a, b, and d are related with our potential strength coefficients by
[C+1 , C
2 , C
1 ] =
(4π)2
[−aµ4, bµ6, −dµ6] . (63)
The values of the coefficients, a, b, and d for the TPE-3NP at O(q4) are shown in Table III,
as BR-O(q4). In this table, the values for the older version of the Brazil TPE-3NP, BR(83)
[10], and the potential up to O(q3) given by Eqs. (41-42), BR-O(q3), are shown as well.
TABLE III: Coefficients a, b, and d of the TPE-3NP.
3NP a µ b µ3 d µ3
BR-O(q4) -0.981 -2.617 -0.854
BR-O(q3) -0.736 -3.483 -1.204
BR(83) -1.05 -2.29 -0.768
In Eq. (62), the function F (k2) represents a πNN form factor. We apply a dipole form
factor with the cut off mass Λ,
Λ2−µ2
, which modifies the profile functions U(x), U1(x),
and U2(x) in Eqs. (47-49) as
U(x) =
Λ̄2 − 1
, (64)
U1(x) = −
+ Λ̄2
e−Λ̄x
Λ̄2 − 1
e−Λ̄x, (65)
U2(r) =
(Λ̄x)2
e−Λ̄x
−Λ̄(Λ̄
2 − 1)
e−Λ̄x, (66)
with Λ̄ = Λ/µ.
We choose the Argonne V18 model (AV18) [34] for a realistic NN potential, by which the
triton binding energy (B3) becomes 7.626 MeV, underbinding it by about 0.9 MeV compared
to the empirical value, 8.482 MeV. As it is well known, the introduction of the TPE-3NP
remedies this deficiency. The amount of attractive contribution depends on the cutoff mass
Λ, as shown in Fig. 5. The solid curve shows the dependence of B3 on Λ for the calculation
with the BR-O(q4) 3NP in addition to the AV18 2NP (AV18+BR-O(q4)). In the figure,
the empirical value and the AV18 result are displayed by the dashed and dotted horizontal
lines, respectively. Due to the strong attractive character of the 3NP, B3 is reproduced by
choosing a rather small value of Λ, namely 660 MeV. In the same figure, the Λ-dependence
of B3 for AV18+BR-O(q3) is displayed by a dashed curve and that for the AV18+BR(83) by
a dotted curve. From these curves we see that AV18+BR-O(q3) reproduces B3 for Λ = 620
MeV and AV18+BR(83) for Λ = 680 MeV. In other words, the BR-O(q4) 3NP is slightly
more attractive than the BR(83) 3NP and a large attractive effect occurs when one moves
from the TPE O(q4) 3NP to the O(q3) 3NP. This tendency is strongly correlated with the
magnitude of the coefficient b, as shown in Table III. This can be understood as a dominant
contribution to B3 from the component of the TPE-3NP associated with the coefficients b.
This dominance is shown in Table IV, where we tabulate calculated B3 for the AV18 plus
the BR-O(q4) 3NP and plus each term of the BR-O(q4) coming from the coefficients a, b,
and d.
550 600 650 700 750 800
L (MeV)
FIG. 5: (Color online) The triton binding energy B3 as functions of the cutoff mass Λ of the πNN
dipole form factor. The solid curve denotes the result for AV18+BR-O(q4), the dashed curve for
AV18+BR-O(q3), and the dotted curve for AV18+BR(83). The horizontal lines denote the AV18
result (dotted line) and the empirical value (dashed line).
In Fig. 6, we compare six calculated observables for proton-deuteron elastic scattering,
namely differential cross sections σ(θ), vector analyzing powers of the proton Ay(θ) and
of the deuteron iT11(θ), and tensor analyzing powers of the deuteron T20(θ), T21(θ), and
T22(θ), at incident proton energy E
N = 3.0 MeV, (or incident deuteron energy E
d = 6.0
MeV,) with experimental data of Ref. [35, 36]. In the figure, the solid curves designate
the AV18 calculations and the dashed curves the AV18+BR-O(q4) calculations, which are
almost indistinguishable from the AV18+BR-O(q3) and AV18+BR(83) calculations, once
TABLE IV: Triton binding energy for the AV18 2NP plus the BR-O(q4) 3NP for each term of the
BR-O(q4) 3NP with Λ = 660 MeV. ∆B3 means the difference of the calculated binding energy
from that of the AV18 calculation.
B3 (MeV) ∆B3 (MeV)
AV18+BR-O(q4) 8.492 0.866
AV18+BR-O(q4)-a 7.673 0.047
AV18+BR-O(q4)-b 8.241 0.615
AV18+BR-O(q4)-d 7.787 0.161
the cut off masses are chosen so that B3 is reproduced.
It is reminded that the TPE-3NF gives minor effects on the vector analyzing powers.
This happens because the exchange of pions gives essentially scalar and tensor components
of nuclear interaction in spin space, which are not so effective to the vector analyzing powers.
On the other hand, as is noticed in Refs. [37, 38], at ElabN = 3.0 MeV, the TPE-3NP gives a
wrong contribution to the tensor analyzing power T21(θ) around θ = 90
In Fig. 7, we compare calculations of observables in neutron-deuteron elastic scattering
at ElabN = 28.0 MeV with experimental data of proton-deuteron scattering Ref. [39]. At
this energy, discrepancies between the calculations and the experimental data in the vector
analyzing power iT11(θ) appear at θ ∼ 100◦, where iT11(θ) has a minimum, and at θ ∼ 140◦,
where iT11(θ) has a maximum, which are not compensated by the introduction of the TPE-
3NP. On the other hand, while the AV18 calculation almost reproduces the experimental
data of T21(θ) at θ ∼ 90◦, the introduction of the TPE-3NP gives a wrong effect, as in the
ElabN = 3 MeV case.
These results set the stage for the introduction of terms associated with the coefficients
C+3 , C
2 , and C
3 , Eqs. (44-45), which are new features of the O(q4) expansion of the TPE-
3NP. Terms proportional to C±3 , which include the rather complicated function I(r31, r23)
given in Appendix C, arise from a loop integral, Eq. (33). On the other hand, the term with
C−2 corresponds to a non-local potential and includes the gradient operator ∇
ij , which acts
on the wave function and arises from the kinematical variable ν. Both kinds of contributions
are not expressed in the conventional local form shown in Eq. (62), which involves only
the coefficients C+1 , C
2 , and C
1 , and the full evaluation of their effects would require an
0 60 120 180
-0.04
-0.03
-0.02
-0.01
0 60 120 180
0 60 120 180
-0.03
-0.02
-0.01
 (deg) 
 = 3.0 MeV   (E
 = 6.0 MeV)
 (deg) 
 (deg) 
 (deg) 
 (deg) 
 (deg) 
FIG. 6: (Color online) Proton-deuteron elastic scattering observables at ElabN = 3.0 MeV. Solid
curves are calculations for the AV18 potential, and dashed curves for the AV18+BR-O(q4). Ex-
perimental data are taken from Refs. [35, 36].
extensive rebuilding of large numerical codes. However, the coefficients of the new terms are
small, and in this exploratory paper we estimate their influence over observables as follows.
The function I(r31, r23) is approximated by Eq. (C11), which amounts to replacing Πt(t)
by a factor −π. Further, the kinematical factors in front of Πt(t) in Eqs. (34) and (38),
namely 1 − 2t/µ2 and 1 − t/4µ2, are approximately evaluated by putting t ≈ 2µ2, which
yields −3 and 1/2, respectively. By this procedure, the coefficients C+3 and C−3 are absorbed
into C+2 and C
1 , or in b and d respectively, and one has
∆C+2 = −3C+3 , ∆C−1 = C−3 /2. (67)
Numerically, this corresponds to ∆C+2 = −0.102 MeV ∼ 120C
2 and ∆C
1 = −0.034 MeV ∼
C−1 , or ∆b = −0.125(µ−3) and ∆d = 0.042(µ−3). The net change produced in the triton
binding energy is +0.026 MeV (+0.037 MeV from ∆C+2 and -0.011 MeV from ∆C
1 ), just
about 1/30 of the total increase in B3 due to the local terms of the BR-O(q4) TPE-3NP.
The non-local term proportional to C−2 is more involved and we restrict ourselves to a
rough assessment of its role. We replace the variable ν by a constant 〈ν〉 and assume, for
0 60 120 180
0 60 120 180
0 60 120 180
 (deg) 
 = 28.0 MeV   �E
 = 56.0 MeV�
 (deg) 
 (deg) 
 (deg) 
 (deg) 
 (deg) 
FIG. 7: (Color online) Nucleon-deuteron elastic scattering observables at ElabN = 28.0 MeV. Curves
are calculations for neutron-deuteron scattering. Solid curves denote calculations for the AV18
potential and dashed curves for the AV18+BR-O(q4). Experimental data are those for proton-
deuteron scattering taken from Ref. [39].
example, that 〈ν〉 = µ2
. This changes the C−2 term in Eq. (46) into the very simple form
V −3 (r,ρ) = C
1 (· · · ) + iC̃−2 σ(1) · x̂31σ(2) · x̂23U1(x31)U1(x23) + C−3 (· · · ) , (68)
C̃−2 = −
4f 2π
1− g2A
2f 2π
(4π)2
g2A(1− g2A)µ6
512π2f 4πm
= 0.021 MeV . (69)
Except for the isospin factor, this term is similar to that with C+1 (or a), which adds
about 0.05 MeV to the triton binding energy. Since the potential strength C̃−2 is about 3 %
of C+1 , its contribution to the binding energy may be estimated to be a tiny 0.001 MeV.
VII. CONCLUSIONS
In the framework of chiral perturbation theory, three-nucleon forces begin at O(q3), with
a long range component which is due to the exchanges of two pions and relatively simple.
At O(q4), on the other hand, a large number of different processes intervene and a full
description becomes rather complex. For this reason, here we concentrate on a subset of
O(q4) interactions, namely that which still involves the exchanges of just two pions. This
part of the 3NP is closely related with the πN amplitude, and the expansion of the former
up to O(q4) depends on the latter at O(q3).
Our expressions for the potential are given in Eqs. (44-56) and the new chiral layer of the
TPE-3NP considered in this work gives rise to both numerical corrections to strength coef-
ficients of already existing terms (C+1 , C
2 , C
1 ) and new structures in the profile functions.
Changes in numerical coefficients lay in the neighborhood of 10% and can be read in Tables
II and III. New structures, on the other hand, arise either from loop functions representing
form factors or the non-local terms associated with gradients acting on the wave function.
They correspond to the terms proportional to the parameters C+3 , C
2 and C
3 , which are
small and compatible with perturbative effects.
In order to insert our results into a broader picture, in Table V we show the orders at
which the various effects begin to appear, including the drift potential derived recently [40].
TABLE V: Chiral picture for two- and three-body forces.
beginning TWO-BODY TWO-BODY THREE-BODY
O(q0) OPEP: V −T , V
O(q2) OPEP: V −D TPEP: V
T , V
O(q3) TPEP: V −LS, V
T , V
SS ;V
C , V
LS TPEP: C
1 , C
O(q4) TPEP: V −D ;V
Q , V
D TPEP: C
3 , C
The influence of the new TPE-3NP over three-body observables has been assessed in both
static and scattering environments, adopting the Argonne V18 potential for the two-body
interaction. In order to reproduce the empirical triton binding energy, the O(q4) potential
requires a cutoff mass of 660 MeV. Comparing this with the value of 680 MeV for the 1983
Brazil TPE-3NP, one learns that the later version is more attractive.
In the study of proton-deuteron elastic scattering, we have calculated cross sections σ(θ),
vector analyzing powers Ay(θ) of the proton and iT11(θ) of the deuteron, and tensor analyzing
powers T20(θ), T21(θ), and T22(θ) of the deuteron, at energies of 3 and 28 MeV. Results are
displayed in Figs. 6 and 7, where it is possible to see that there is little sensitivity to
the changes induced in the strength parameters when one goes from O(q3) to O(q4). Old
problems, as the Ay(θ) puzzle, remain unsolved.
The present version of the TPE-3NP contains new structures, associated with loop inte-
grals an non-local operators. Their influence over observables has been estimated and found
to be at least one order of magnitude smaller than other three-body effects. A more detailed
study of this part of the force is being carried on.
APPENDIX A: KINEMATICS
The coordinate describing the position of nucleon i is ri and one uses the combinations
R = (r1+r2+r3)/3 , r = r2−r1 , ρ = (2 r3−r1−r2)/
3 , (A1)
which yield
r1 = R−
, r2 = R +
, r3 = R +
. (A2)
The momentum of nucleon i is pi and one defines
P = p1+p2+p3 , pr = (p2−p1)/2 , pρ = (2p3−p1−p2)/2
3 . (A3)
Initial momenta p and final momenta p′ are used in the combinations
Q = (P ′+P )/2 , q = (P ′−P ) , (A4)
Qr = (p
r+pr)/2 , qr = (p
r−pr) , (A5)
Qρ = (p
ρ+pρ)/2 , qρ = (p
ρ−pρ) . (A6)
In the CM, one has P = 0 and the three-momenta are given by
p1 = −(Qr−qr/2)− (Qρ−qρ/2)/
3 , p′1 = −(Qr+qr/2)− (Qρ+qρ/2)/
3 , (A7)
p2 = (Qr−qr/2)− (Qρ−qρ/2)/
3 , p′2 = (Qr+qr/2)− (Qρ+qρ/2)/
3 , (A8)
p3 = 2(Qρ−qρ/2)/
3 , p′3 = 2(Qρ+qρ/2)/
3 . (A9)
Energy conservation for on-shell particles yield the non-relativistic constraint
Qr ·qr +Qρ ·qρ = 0 . (A10)
The momenta of the exchanged pions are written as
k = p1 − p′1 , k′ = p′2 − p2 , (A11)
k0 = −(qr+qρ/
3)·(Qr+Qρ/
3)/m , k = qr+qρ/
3 , (A12)
′0 = (qr−qρ/
3)·(Qr−Qρ/
3)/m , k′ = qr−qρ/
3 , (A13)
and the Mandelstam variables for nucleon 3 read
s = (p3+k)
2 = m2 − (qr+qρ/
3) · (qr+2Qr−qρ/
3Qρ) +O(q4) , (A14)
u = (p3−k′)2 = m2 − (qr−qρ/
3) · (qr+2Qr+qρ/
3Qρ) +O(q4) , (A15)
ν = (s−u)/4m = −2 qr ·Qρ/
3 +O(q4) . (A16)
In the evaluation of the intermediate πN amplitude, one needs
[ū(p′) u(p)](3) ≃ 2m+O(q2) , (A17)
ū(p′) σµν(p
′−p)µKν u(p)](3) ≃ 2 iσ(3) ·qρ×qr/
3 +O(q4) . (A18)
The πN vertex for nucleon 1 is associated with
[ū(p′) γ5 u(p)]
(1) ≃ σ(1) ·(qr+qρ/
3) +O(q3) , (A19)
and results for nucleon 2 are obtained by making qr → −qr.
APPENDIX B: SUBTHRESHOLD COEFFICIENTS
The polynomial parts of the amplitudes T±R , Eqs. (30-35), are determined by the sub-
threshold coefficients of Ref. [15]. The terms relevant to the O(q3) expansion are written as
d+00 = −
2 (2c1 − c3) µ2
8 g4A µ
64 π f 4π
3 g2A µ
64 π f 4π
, (B1)
d+01 = −
48 g4A µ
768 π f 4π
77 g2A µ
768 π f 4π
, (B2)
d+02 =
193 g2A
15360 π f 4π µ
, (B3)
d−00 =
2 f 2π
+O(q2) , (B4)
b−00 =
2 f 2π
2 c4 m
g4A m µ
8 π f 4π
g2A m µ
8 π f 4π
, (B5)
b−01 =
g2A m
96 π f 4π µ
, (B6)
where the parameters ci and d̃i are the usual coupling constants of the chiral lagrangians
of order 2 and 3 respectively [41] and the tilde over the latter indicates that they were
renormalized [15]. Terms within square brackets labeled (mr) in these results are due to
the medium range diagrams shown in Fig. 3 and have been included explicitly into the
functions D±mr and B
mr. Terms bearing the (WT ) label were also explicitly considered in
Eqs. (15-19). The subthreshold coefficients are determined from πN scatterig data and a
set of experimental values is given in Ref. [13].
APPENDIX C: FUNCTIONS In
The functions In, describing loop contributions, are given by
In(r31, r23) = −
(2π)3
(2π)3
ei(k·r31+k
·r23)
′2+µ2
Πt(t) . (C1)
Using the definition Eq. (33) and the Jacobi variables Eq. (A1), one writes
In(r31, r23) =
I(r31, r23) , (C2)
I(r31, r23) = 128π
da tan−1
µ (1−a2/2)
L(a; r,ρ) (C3)
L(a; r,ρ) =
(2π)3
(2π)3
ei(Q·r−
3q·ρ/2)
a2q2+4µ2
[(Q−q)2+µ2]
[(Q+q)2+µ2]
. (C4)
The numerical evaluation of the function L is can be simplified by using alternative repre-
sentations.
• form 1: One uses the Feynman procedure for manipulating denominators, which yields
L(a; r,ρ) =
(2π)3
(2π)3
ei(Q·r−
3q·ρ/2)
a2q2+4µ2
[(Q2+q2/4+µ2)−(1−2b)q ·Q]2
(2π)3
ei[(1−2b)r−
3ρ]·q/2
a2q2+4µ2
e−Θ r
µ2+b(1−b) q2 . (C5)
Performing the angular integration over q, one has
L(a; r,ρ) =
16 π3
e−Θ r
Θ (a2q2+4µ2)
sin q [(1− 2b) r −
3ρ]/2
[(1− 2b) r −
3ρ]/2
. (C6)
• form 2: The Fourier transform
dx e−ik·x
allows one to write
L(a; r,ρ) =
e−µ|r31+z|
|r31+z|
e−µ|r23−z|
|r23−z|
e−2µ z/a
. (C8)
These results may be further simplified by means of approximations.
• heavy baryon approximation: In the limit m → ∞, corresponding to the heavy
baryon case, one uses F (a)→ 4π/a2 in Eq. (33) and Eqs. (C5) and (C7) yield, respectively,
I(r31, r23) ≃
tan−1
e−Θ r
sin q [(1− 2b) r −
3ρ]/2
[(1− 2b) r −
3ρ]/2
, (C9)
I(r31, r23) ≃
e−µ|r31+z|
|r31+z|
e−µ|r23−z|
|r23−z|
e−2µ z
2µ z2
. (C10)
• multipole approximation: The integrand in Eq. (C10) is peaked around z = 0 and a
multipole expansion of the Yukawa functions produces
I(r31, r23) ≃ U(x31) U(x23) + · · · . (C11)
The same result can also be obtained by using the expansion Πt(t) ∼ −π[1 + t/12µ2 +
t2/80µ4 + · · · ], valid for low t, directly into Eq. (C1).
APPENDIX D: NON-LOCAL TERM
In configuration space, the variable Qρ corresponds to a non-local operator, represented
by a gradient acting on the wave function. In order to make the dependence of t̄3 on Qρ
explicit, one writes
t̄3 = [Qρ]i Xi(qr, qρ) , (D1)
where X is a generic three-vector, and evaluates the matrix element
〈ψ |W |ψ〉 =−
]12 ∫
dr′ dρ′ dr dρ ψ∗(r′,ρ′) ψ(r,ρ)
dQr dQρ dqr dqρ
× ei[Qr·(r
′−r)+Q
·(ρ′−ρ)+q
·(r′+r)/2+q
·(ρ′+ρ)/2] t̄3(Qr,Qρ, qr, qρ)
dr dρ
∗(r,ρ)
ψ(r,ρ) + ψ∗(r,ρ)
∇ρ ψ(r,ρ)
dqr dqρ e
·ρ] Xi(qr, qρ) . (D2)
This yields the potential
V3(r,ρ) = −
(2π)6
dqr dqρ e
·ρ] Xi(qr, qρ) , (D3)
where the operator ∇
acts only on the wave function. An alternative form can be
obtained by integrating Eq. (D2) by parts, and one finds
V3(r,ρ) =−
(2π)6
dqr dqρ e
·ρ] X(qr, qρ)
− i∇wfρ
dqr dqρ e
·ρ] X(qr, qρ)
. (D4)
In the case of the three-body force, the only non-local contribution is associated with the
subamplitude D−, Eq. (37), which yields
Xi = −i τ (1) × τ (2) ·τ (3)
′2+µ2
(1) ·k σ(2) ·k′
g2A(g
A − 1)√
3 8f 4π m
(k′+k)i . (D5)
The action of ∇ρ on the second term of Eq. (D4) gives rise to an integrand proportional to
′2−k2), which has short range and does not contribute to the TPE-3NP. Therefore it is
neglected.
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