Nuclear forces from chiral effective field theory
Nuclear Forces from Chiral Effective Field
Theory∗
R.Machleidt
Department of Physics, University of Idaho, Moscow, Idaho, U.S.A.
November 20, 2018
Abstract
In this lecture series, I present the recent progress in our under-
standing of nuclear forces in terms of chiral effective field theory.
Contents
1 Introduction and Historical Perspective 3
2 QCD and the Nuclear Force 5
3 Effective Field Theory for Low-Energy QCD 5
3.1 Symmetries of Low-Energy QCD . . . . . . . . . . . . . . . . 6
3.1.1 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . 6
3.1.2 Explicit Symmetry Breaking . . . . . . . . . . . . . . 9
3.1.3 Spontaneous Symmetry Breaking . . . . . . . . . . . . 9
3.2 Chiral Effective Lagrangians Involving Pions . . . . . . . . . 10
3.3 Nucleon Contact Lagrangians . . . . . . . . . . . . . . . . . . 12
4 Nuclear Forces from EFT: Overview 13
4.1 Chiral Perturbation Theory and Power Counting . . . . . . . 14
4.2 The Hierarchy of Nuclear Forces . . . . . . . . . . . . . . . . 14
∗Lecture series presented at the DAE-BRNS Workshop on Physics and Astrophysics
of Hadrons and Hadronic Matter, Visva Bharati University, Santiniketan, West Bengal,
India, November 2006.
5 Two-Nucleon Forces 16
5.1 Pion-Exchange Contributions in ChPT . . . . . . . . . . . . 16
5.1.1 Zeroth Order (LO) . . . . . . . . . . . . . . . . . . . 17
5.1.2 Second Order (NLO) . . . . . . . . . . . . . . . . . . 17
5.1.3 Third Order (NNLO) . . . . . . . . . . . . . . . . . . 19
5.1.4 Fourth Order (N3LO) . . . . . . . . . . . . . . . . . . 20
5.1.5 Iterated One-Pion-Exchange . . . . . . . . . . . . . . 20
5.2 NN Scattering in Peripheral Partial Waves Using the Pertur-
bative Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 NN Contact Potentials . . . . . . . . . . . . . . . . . . . . . 28
5.3.1 Zeroth Order . . . . . . . . . . . . . . . . . . . . . . . 29
5.3.2 Second Order . . . . . . . . . . . . . . . . . . . . . . . 30
5.3.3 Fourth Order . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Constructing a Chiral NN Potential . . . . . . . . . . . . . . 31
5.4.1 Conceptual Questions . . . . . . . . . . . . . . . . . . 31
5.4.2 What Order? . . . . . . . . . . . . . . . . . . . . . . . 33
5.4.3 Charge-Dependence . . . . . . . . . . . . . . . . . . . 34
5.4.4 A Quantitative NN Potential at N3LO . . . . . . . . 36
6 Many-Nucleon Forces 39
6.1 Three-Nucleon Forces . . . . . . . . . . . . . . . . . . . . . . 40
6.2 Four-Nucleon Forces . . . . . . . . . . . . . . . . . . . . . . . 42
7 Conclusions 42
A Fourth Order Two-Pion Exchange Contributions 44
A.1 One-loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . 44
A.1.1 c2i contributions. . . . . . . . . . . . . . . . . . . . . . 44
A.1.2 ci/MN contributions. . . . . . . . . . . . . . . . . . . . 44
A.1.3 1/M2N corrections. . . . . . . . . . . . . . . . . . . . . 45
A.2 Two-loop contributions. . . . . . . . . . . . . . . . . . . . . . 46
B Partial Wave Decomposition of the Fourth Order Contact
Potential 48
1 Introduction and Historical Perspective
The theory of nuclear forces has a long history (cf. Table 1). Based upon
the seminal idea by Yukawa [1], first field-theoretic attempts to derive the
nucleon-nucleon (NN) interaction focused on pion-exchange. While the one-
pion exchange turned out to be very useful in explaining NN scattering data
and the properties of the deuteron [2], multi-pion exchange was beset with
serious ambiguities [3, 4]. Thus, the “pion theories” of the 1950s are gen-
erally judged as failures—for reasons we understand today: pion dynamics
is constrained by chiral symmetry, a crucial point that was unknown in the
1950s.
Historically, the experimental discovery of heavy mesons [5] in the early
1960s saved the situation. The one-boson-exchange (OBE) model [6, 7]
emerged which is still the most economical and quantitative phenomenol-
ogy for describing the nuclear force [8, 9]. The weak point of this model,
however, is the scalar-isoscalar “sigma” or “epsilon” boson, for which the
empirical evidence remains controversial. Since this boson is associated with
the correlated (or resonant) exchange of two pions, a vast theoretical effort
that occupied more than a decade was launched to derive the 2π-exchange
contribution to the nuclear force, which creates the intermediate range at-
traction. For this, dispersion theory as well as field theory were invoked
producing the Stony Brook [10], Paris [11, 12], and Bonn [7, 13] potentials.
The nuclear force problem appeared to be solved; however, with the
discovery of quantum chromodynamics (QCD), all “meson theories” were
relegated to models and the attempts to derive the nuclear force started all
over again.
The problem with a derivation from QCD is that this theory is non-
perturbative in the low-energy regime characteristic of nuclear physics, which
makes direct solutions impossible. Therefore, during the first round of new
attempts, QCD-inspired quark models [14] became popular. These models
are able to reproduce qualitatively and, in some cases, semi-quantitatively
the gross features of the nuclear force [15, 16]. However, on a critical note,
it has been pointed out that these quark-based approaches are nothing but
another set of models and, thus, do not represent any fundamental progress.
Equally well, one may then stay with the simpler and much more quantita-
tive meson models.
A major breakthrough occurred when the concept of an effective field
theory (EFT) was introduced and applied to low-energy QCD. As outlined
by Weinberg in a seminal paper [17], one has to write down the most general
Lagrangian consistent with the assumed symmetry principles, particularly
Table 1: Seven Decades of Struggle: The Theory of Nuclear Forces
1935 Yukawa: Meson Theory
The “Pion Theories”
1950’s One-Pion Exchange: o.k.
Multi-Pion Exchange: disaster
Many pions ≡ multi-pion resonances:
1960’s σ, ρ, ω, ...
The One-Boson-Exchange Model: success
Refined meson models, including
1970’s sophisticated 2π exchange contributions
(Stony Brook, Paris, Bonn)
Nuclear physicists discover
1980’s QCD
Quark Cluster Models
Nuclear physicists discover EFT
1990’s Weinberg, van Kolck
and beyond Back to Pion Theory!
But, constrained by Chiral Symmetry: success
the (broken) chiral symmetry of QCD. At low energy, the effective degrees
of freedom are pions and nucleons rather than quarks and gluons; heavy
mesons and nucleon resonances are “integrated out”. So, the circle of his-
tory is closing and we are back to Yukawa’s meson theory, except that we
have learned to add one important refinement to the theory: broken chiral
symmetry is a crucial constraint that generates and controls the dynamics
and establishes a clear connection with the underlying theory, QCD.
Following the first initiative by Weinberg [18], pioneering work was per-
formed by Ordóñez, Ray, and van Kolck [19, 20] who constructed a NN
potential in coordinate space based upon chiral perturbation theory at next-
to-next-to-leading order. The results were encouraging and many researchers
became attracted to the new field [21, 22, 23, 24, 25, 26, 27]. As a conse-
quence, nuclear EFT has developed into one of the most popular branches
of modern nuclear physics [28, 29].
It is the purpose of these lectures to describe in some detail the recent
progress in our understanding of nuclear forces in terms of nuclear EFT.
2 QCD and the Nuclear Force
Quantum chromodynamics (QCD) is the theory of strong interactions. It
deals with quarks, gluons and their interactions and is part of the Standard
Model of Particle Physics. QCD is a non-Abelian gauge field theory with
color SU(3) the underlying gauge group. The non-Abelian nature of the
theory has dramatic consequences. While the interaction between colored
objects is weak at short distances or high momentum transfer (“asymptotic
freedom”); it is strong at long distances ( >∼ 1 fm) or low energies, leading
to the confinement of quarks into colorless objects, the hadrons. Conse-
quently, QCD allows for a perturbative analysis at large energies, whereas it
is highly non-perturbative in the low-energy regime. Nuclear physics resides
at low energies and the force between nucleons is a residual QCD interac-
tion. Therefore, in terms of quarks and gluons, the nuclear force is a very
complicated problem.
3 Effective Field Theory for Low-Energy QCD
The way out of the dilemma of how to derive the nuclear force from QCD
is provided by the effective field theory (EFT) concept. First, one needs to
identify the relevant degrees of freedom. For the ground state and the low-
energy excitation spectrum of an atomic nucleus as well as for conventional
nuclear reactions, quarks and gluons are ineffective degrees of freedom, while
nucleons and pions are the appropriate ones. Second; to make sure that this
EFT is not just another phenomenology, the EFT must observe all relevant
symmetries of the underlying theory. This requirement is based upon a ‘folk
theorem’ by Weinberg [17]:
If one writes down the most general possible Lagrangian, in-
cluding all terms consistent with assumed symmetry principles,
and then calculates matrix elements with this Lagrangian to any
given order of perturbation theory, the result will simply be the
most general possible S-matrix consistent with analyticity, per-
turbative unitarity, cluster decomposition, and the assumed sym-
metry principles.
Thus, the EFT program consists of the following steps:
1. Identify the degrees of freedom relevant at the resolution scale of (low-
energy) nuclear physics: nucleons and pions.
2. Identify the relevant symmetries of low-energy QCD and investigate if
and how they are broken.
3. Construct the most general Lagrangian consistent with those symme-
tries and the symmetry breaking.
4. Design an organizational scheme that can distinguish between more
and less important contributions: a low-momentum expansion.
5. Guided by the expansion, calculate Feynman diagrams to the the de-
sired accuracy for the problem under consideration.
We will now elaborate on these steps, one by one.
3.1 Symmetries of Low-Energy QCD
In this section, we will give a brief introduction into (low-energy) QCD, its
symmetries and symmetry breaking. A more detailed introduction can be
found in the excellent lecture series by Scherer and Schindler [30].
3.1.1 Chiral Symmetry
The QCD Lagrangian reads
LQCD = q̄(iγµDµ −M)q −
Gµν,aGµνa (1)
with the gauge-covariant derivative
Dµ = ∂µ + ig
Aµ,a (2)
and the gluon field strength tensor
Gµν,a = ∂µAν,a − ∂νAµ,a − gfabcAµ,bAν,c . (3)
In the above, q denotes the quark fields and M the quark mass matrix.
Further, g is the strong coupling constant and Aµ,a are the gluon fields.
The λa are the Gell-Mann matrices and the fabc the structure constants
of the SU(3)color Lie algebra (a, b, c = 1, . . . , 8); summation over repeated
indices is always implied. The gluon-gluon term in the last equation arises
from the non-Abelian nature of the gauge theory and is the reason for the
peculiar features of the color force.
On a typical hadronic scale, i.e., on a scale of low-mass hadrons which are
not Goldstone bosons, e.g., mρ = 0.78 GeV ≈ 1 GeV; the masses of the up
(u), down (d), and—to a certain extend—strange (s) quarks are small [31]:
mu = 2± 1 MeV (4)
md = 5± 2 MeV (5)
ms = 95± 25 MeV (6)
It is therefore of interest to discuss the QCD Lagrangian in the limit of
vanishing quark masses:
L0QCD = q̄iγ
µDµq −
Gµν,aGµνa . (7)
Defining right- and left-handed quark fields,
qR = PRq , qL = PLq , (8)
(1 + γ5) , PL =
(1− γ5) , (9)
we can rewrite the Lagrangian as follows:
L0QCD = q̄Riγ
µDµqR + q̄LiγµDµqL −
Gµν,aGµνa . (10)
Restricting ourselves now to up and down quarks, we see that L0QCD is
invariant under the global unitary transformations
7−→ exp
−iΘRi
7−→ exp
−iΘLi
, (12)
where τi (i = 1, 2, 3) are the generators of SU(2)flavor, the usual Pauli spin
matrices. The right- and left-handed components of massless quarks do not
mix. This is SU(2)R × SU(2)R symmetry, also known as chiral symmetry.
Noether’s Theorem implies the existence of six conserved currents; three
right-handed currents
i = q̄Rγ
qR with ∂µR
i = 0 (13)
and three left-handed currents
i = q̄Lγ
qL with ∂µL
i = 0 . (14)
It is useful to consider the following linear combinations; namely, three vec-
tor currents
i = R
i + L
i = q̄γ
q with ∂µV
i = 0 (15)
and three axial-vector currents
i = R
i − L
i = q̄γ
q with ∂µA
i = 0 , (16)
which got their names from the fact that they transform as vectors and
axial-vectors, respectively. Thus, the chiral SU(2)L × SU(2)R symmetry is
equivalent to SU(2)V ×SU(2)A, where the vector and axial-vector transfor-
mations are given respectively by
7−→ exp
−iΘVi
7−→ exp
−iΘAi γ5
. (18)
Obviously, the vector transformations are isospin rotations and, therefore,
invariance under vector transformations can be identified with isospin sym-
metry.
There are the six conserved charges,
QVi =
d3x V 0i =
d3x q†(t, ~x)
q(t, ~x) with
= 0 (19)
QAi =
d3x A0i =
d3x q†(t, ~x)γ5
q(t, ~x) with
= 0 , (20)
which are also generators of SU(2)V × SU(2)A.
3.1.2 Explicit Symmetry Breaking
The mass term −q̄Mq in the QCD Lagrangian Eq. (1) breaks chiral sym-
metry explicitly. To better see this, let’s rewrite M,
(mu +md)
(mu −md)
(mu +md) I +
(mu −md) τ3 . (23)
The first term in the last equation in invariant under SU(2)V (isospin sym-
metry) and the second term vanishes for mu = md. Thus, isospin is an exact
symmetry if mu = md. However, both terms in Eq. (23) break SU(2)A.
Since the up and down quark masses are small as compared to the typical
hadronic mass scale of ≈ 1 GeV [cf. Eqs. (4) and (5)], the explicit chiral
symmetry breaking due to non-vanishing quark masses is very small.
3.1.3 Spontaneous Symmetry Breaking
A (continuous) symmetry is said to be spontaneously broken if a symmetry
of the Lagrangian is not realized in the ground state of the system. There is
evidence that the chiral symmetry of the QCD Lagrangian is spontaneously
broken—for dynamical reasons of nonperturbative origin which are not fully
understood at this time. The most plausible evidence comes from the hadron
spectrum. From chiral symmetry, one would naively expect the existence
of degenerate hadron multiplets of opposite parity, i.e., for any hadron of
positive parity one would expect a degenerate hadron state of negative parity
and vice versa. However, these “parity doublets” are not observed in nature.
For example, take the ρ-meson, a vector meson with negative parity (1−)
and mass 776 MeV. There does exist a 1+ meson, the a1, but it has a mass
of 1230 MeV and, thus, cannot be perceived as degenerate with the ρ. On
the other hand, the ρ meson comes in three charge states (equivalent to
three isospin states), the ρ± and the ρ0 with masses that differ by at most
a few MeV. In summary, in the QCD ground state (the hadron spectrum)
SU(2)V (isospin symmetry) is well observed, while SU(2)A (axial symmetry)
is broken. Or, in other words, SU(2)V ×SU(2)A is broken down to SU(2)V .
A spontaneously broken global symmetry implies the existence of (mass-
less) Goldstone bosons with the quantum numbers of the broken generators.
The broken generators are the QAi of Eq. (20) which are pseudoscalar. The
Goldstone bosons are identified with the isospin triplet of the (pseudoscalar)
pions, which explains why pions are so light. The pion masses are not ex-
actly zero because the up and down quark masses are not exactly zero either
(explicit symmetry breaking). Thus, pions are a truly remarkable species:
they reflect spontaneous as well as explicit symmetry breaking.
3.2 Chiral Effective Lagrangians Involving Pions
The next step in our EFT program is to build the most general Lagrangian
consistent with the (broken) symmetries discussed above. An elegant formal-
ism for the construction of such Lagrangians was developed by Callan, Cole-
man, Wess, and Zumino (CCWZ) [32] who worked out the group-theoretical
foundations of non-linear realizations of chiral symmetry. The Lagrangians
given below are built upon the CCWZ formalism.
As discussed, the relevant degrees of freedom are pions (Goldstone bosons)
and nucleons. Since the interactions of Goldstone bosons must vanish at zero
momentum transfer and in the chiral limit (m→ 0), the low-energy expan-
sion of the Lagrangian is arranged in powers of derivatives and pion masses.
This is chiral perturbation theory (ChPT).
The Lagrangian consists of one part that deals with the interaction
among pions, Lππ, and another one that describes the interaction between
pions and the nucleon, LπN :
Leff = Lππ + LπN (24)
Lππ = L(2)ππ + L
ππ + . . . (25)
LπN = L
πN + L
πN + L
πN + . . . , (26)
where the superscript refers to the number of derivatives or pion mass inser-
tions (chiral dimension) and the ellipsis stands for terms of higher dimension.
The leading order (LO) ππ Lagrangian is given by [33]
L(2)ππ =
∂µU∂µU
† +m2π(U + U
and the LO relativistic πN Lagrangian reads [34]
L(1)πN = Ψ̄
iγµDµ −MN +
γµγ5uµ
Ψ (28)
Dµ = ∂µ + Γµ (29)
(ξ†∂µξ + ξ∂µξ
τ · (π × ∂µπ) + . . . (30)
uµ = i(ξ
†∂µξ − ξ∂µξ†) = −
τ · ∂µπ + . . . (31)
U = ξ2 = 1 +
τ · π −
(τ · π)3 +
8α− 1
π4 + . . . (32)
In Eq. (28) the chirally covariant derivative Dµ is applied which introduces
the “gauge term” Γµ (also known as chiral connection), a vector current that
leads to a coupling of pions with the nucleon. Besides this, the Lagrangian
includes a coupling term which involves the axial vector uµ. The SU(2)
matrix U = ξ2 collects the Goldstone pion fields.
In the above equations, MN denotes the nucleon mass, gA the axial-
vector coupling constant, and fπ the pion decay constant. Numerical values
will be given later.
The coefficient α that appears in Eq. (32) is arbitrary. Therefore, dia-
grams with chiral vertices that involve three or four pions must always be
grouped together such that the α-dependence drops out (cf. Fig. 4, below).
We apply the heavy baryon (HB) formulation of chiral perturbation the-
ory [35] in which the relativistic πN Lagrangian is subjected to an expansion
in terms of powers of 1/MN (kind of a nonrelativistic expansion), the lowest
order of which is
L̂(1)πN = N̄
iD0 −
~σ · ~u
i∂0 −
τ · (π × ∂0π)−
τ · (~σ · ~∇)π
N + . . . (33)
In the relativistic formulation, the nucleon is represented by a four-component
Dirac spinor field, Ψ, while in the HB version, the nucleon, N , is a Pauli
spinor; in addition, all nucleon fields include Pauli spinors describing the
isospin of the nucleon.
At dimension two, the relativistic πN Lagrangian reads
L(2)πN =
ciΨ̄O
i Ψ . (34)
The various operators O(2)i are given in Ref. [36]. The fundamental rule by
which this Lagrangian—as well as all the other ones—are assembled is that
they must contain all terms consistent with chiral symmetry and Lorentz
invariance (apart from other trivial symmetries) at a given chiral dimension
(here: order two). The parameters ci are known as low-energy constants
(LECs) and are determined empirically from fits to πN data.
The HB projected πN Lagrangian at order two is most conveniently
broken up into two pieces,
L̂(2)πN = L̂
πN, fix + L̂
πN, ct , (35)
L̂(2)πN, fix = N̄
~D · ~D + i
{~σ · ~D, u0}
N (36)
L̂(2)πN, ct = N̄
2 c1m
π (U + U
u20 + c3 uµu
~σ · (~u× ~u)
N . (37)
Note that L̂(2)πN, fix is created entirely from the HB expansion of the relativistic
L(1)πN and thus has no free parameters (“fixed”), while L̂
πN, ct is dominated
by the new πN contact terms proportional to the ci parameters, besides
some small 1/MN corrections.
At dimension three, the relativistic πN Lagrangian can be formally writ-
ten as
L(3)πN =
diΨ̄O
i Ψ , (38)
with the operators, O(3)i , listed in Refs. [36, 37]; not all 23 terms are of
interest here. The new LECs that occur at this order are the di. Similar
to the order two case, the HB projected Lagrangian at order three can be
broken into two pieces,
L̂(3)πN = L̂
πN, fix + L̂
πN, ct , (39)
with L̂(3)πN, fix and L̂
πN, ct given in Refs. [36, 37].
3.3 Nucleon Contact Lagrangians
Nucleon contact interactions consist of four nucleon fields (four nucleon legs)
and no meson fields. Such terms are needed to renormalize loop integrals,
to make results reasonably independent of regulators, and to parametrize
the unresolved short-distance contributions to the nuclear force. For more
about contact terms, see Sec. 5.3.
Because of parity, nucleon contact interactions come only in even num-
bers of derivatives, thus,
LNN = L
NN + L
NN + L
NN + . . . (40)
The lowest order (or leading order) NN Lagrangian has no derivatives
and reads [18]
L(0)NN = −
CSN̄NN̄N −
CT N̄~σNN̄~σN , (41)
where N is the heavy baryon nucleon field. CS and CT are unknown con-
stants which are determined by a fit to the NN data. The second order NN
Lagrangian is given by [19]
L(2)NN = −C
1[(N̄ ~∇N)
2 + (~∇NN)2]− C ′2(N̄ ~∇N) · (~∇NN)
−C ′3N̄N [N̄ ~∇
2N + ~∇2NN ]
−iC ′4[N̄ ~∇N · (~∇N × ~σN) + (~∇N)N · (N̄~σ × ~∇N)]
−iC ′5N̄N(~∇N · ~σ × ~∇N)− iC
6(N̄~σN) · (~∇N × ~∇N)
−(C ′7δikδjl + C
8δilδkj + C
9δijδkl)
×[N̄σk∂iNN̄σl∂jN + ∂iNσkN∂jNσlN ]
−(C ′10δikδjl + C
11δilδkj + C
12δijδkl)N̄σk∂iN∂jNσlN
C ′13(δikδjl + δilδkj)
+C ′14δijδkl)[∂iNσk∂jN + ∂jNσk∂iN ]N̄σlN . (42)
Similar to CS and CT , the C ′i are unknown constants which are fixed in a
fit to the NN data. Obviously, these contact Lagrangians blow up quite a
bit with increasing order, which why we do not give L(4)NN explicitly here.
4 Nuclear Forces from EFT: Overview
In the beginning of Sec. 3, we spelled out the steps we have to take to
accomplish our EFT program for the derivation of nuclear forces. So far,
we discussed steps one to three. What is left are steps four (low-momentum
expansion) and five (Feynman diagrams). In this section, we will say more
about the expansion we are using and give an overview of the Feynman
diagrams that arise order by order.
4.1 Chiral Perturbation Theory and Power Counting
In ChPT, we analyze contributions in terms of powers of small momenta
over the large scale: (Q/Λχ)ν , where Q stands for a momentum (nucleon
three-momentum or pion four-momentum) or a pion mass and Λχ ≈ 1 GeV
is the chiral symmetry breaking scale (hadronic scale). Determining the
power ν at which a given diagram contributes has become known as power
counting. For a non-iterative contribution involving A nucleons, the power
ν is given by
ν = −2 + 2A− 2C + 2L+
∆i , (43)
∆i ≡ di +
− 2 , (44)
where C denotes the number of separately connected pieces and L the num-
ber of loops in the diagram; di is the number of derivatives or pion-mass
insertions and ni the number of nucleon fields involved in vertex i; the sum
runs over all vertices contained in the diagram under consideration. Note
that for an irreducible NN diagram (A = 2), the above formula reduces to
ν = 2L+
∆i (45)
The power ν is bounded from below; e.g., for A = 2, ν ≥ 0. This fact is
crucial for the power expansion to be of any use.
4.2 The Hierarchy of Nuclear Forces
Chiral perturbation theory and power counting imply that nuclear forces
emerge as a hierarchy ruled by the power ν, Fig. 1.
The NN amplitude is determined by two classes of contributions: con-
tact terms and pion-exchange diagrams. There are two contacts of order
Q0 [O(Q0)] represented by the four-nucleon graph with a small-dot vertex
shown in the first row of Fig. 1. The corresponding graph in the second row,
four nucleon legs and a solid square, represents the seven contact terms of
O(Q2). Finally, at O(Q4), we have 15 contact contributions represented by
a four-nucleon graph with a solid diamond.
Now, turning to the pion contributions: At leading order [LO, O(Q0),
ν = 0], there is only the well-known static one-pion exchange (1PE), second
diagram in the first row of Fig. 1. Two-pion exchange (2PE) starts at next-
to-leading order (NLO, ν = 2) and all diagrams of this leading-order two-
pion exchange are shown. Further 2PE contributions occur in any higher
+... +... +...
2N Force 3N Force 4N Force
N LO3
Figure 1: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons
and dashed lines pions. Further explanations are given in the text.
order. Of this sub-leading 2PE, we show only two representative diagrams at
next-to-next-to-leading order (NNLO) and three diagrams at next-to-next-
to-next-to-leading order (N3LO).
Finally, there is also three-pion exchange, which shows up for the first
time at N3LO (two loops; one representative 3π diagram is included in
Fig. 1). At this order, the 3π contribution is negligible [38].
One important advantage of ChPT is that it makes specific predictions
also for many-body forces. For a given order of ChPT, two-nucleon forces
(2NF), three-nucleon forces (3NF), . . . are generated on the same footing (cf.
Fig. 1). At LO, there are no 3NF, and at NLO, all 3NF terms cancel [18,
39]. However, at NNLO and higher orders, well-defined, nonvanishing 3NF
occur [39, 40]. Since 3NF show up for the first time at NNLO, they are
weak. Four-nucleon forces (4NF) occur first at N3LO and, therefore, they
are even weaker.
5 Two-Nucleon Forces
In this section, we will elaborate in detail on the two-nucleon force contri-
butions of which we have given a rough overview in the previous section.
5.1 Pion-Exchange Contributions in ChPT
The effective pion Lagrangians presented in Sec. 3.2 are the crucial ingredi-
ents for the evaluation of the pion-exchange contributions to the NN inter-
action. We will derive these contributions now order by order.
We will state our results in terms of contributions to the momentum-
space NN amplitude in the center-of-mass system (CMS), which takes the
general form
V (~p ′, ~p) = VC + τ 1 · τ 2WC
+ [VS + τ 1 · τ 2WS ] ~σ1 · ~σ2
+ [VLS + τ 1 · τ 2WLS ]
−i~S · (~q × ~k)
+ [VT + τ 1 · τ 2WT ] ~σ1 · ~q ~σ2 · ~q
+ [VσL + τ 1 · τ 2WσL ] ~σ1 · (~q × ~k ) ~σ2 · (~q × ~k ) , (46)
where ~p ′ and ~p denote the final and initial nucleon momenta in the CMS,
respectively; moreover,
~q ≡ ~p ′ − ~p is the momentum transfer,
~k ≡ 1
(~p ′ + ~p) the average momentum,
~S ≡ 1
(~σ1 + ~σ2) the total spin,
and ~σ1,2 and τ 1,2 are the spin and isospin operators, respectively, of nucleon
1 and 2. For on-energy-shell scattering, Vα and Wα (α = C, S, LS, T, σL)
can be expressed as functions of q and k (with q ≡ |~q| and k ≡ |~k|), only.
Our formalism is similar to the one used by the Munich group [22, 41, 42]
except for two differences: all our momentum space amplitudes differ by an
over-all factor of (−1) and our spin-orbit potentials, VLS and WLS , differ by
an additional factor of (−2). Our conventions are more in tune with what
is commonly used in nuclear physics.
In all expressions given below, we will state only the nonpolynomial con-
tributions to the NN amplitude. Note, however, that dimensional regular-
ization typically generates also polynomial terms. These polynomials are
absorbed by the contact interactions to be discussed in a later section and,
therefore, they are of no interest here.
5.1.1 Zeroth Order (LO)
At order zero [ν = 0, O(Q0), lowest order, leading order, LO], there is only
the well-known static one-pion exchange, second diagram in the first row of
Fig. 1 which is given by:
V1π(~p
′, ~p) = −
τ 1 · τ 2
~σ1 · ~q ~σ2 · ~q
q2 +m2π
. (48)
At first order [ν = 1, O(Q)], there are no pion-exchange contributions
(and also no contact terms).
5.1.2 Second Order (NLO)
Non-vanishing higher-order graphs start at second order (ν = 2, next-to-
leading order, NLO). The most efficient way to evaluate these loop diagrams
is to use covariant perturbation theory and dimensional regularization. This
is the method applied by the Munich group [22, 41, 42]. One starts with
the relativistic versions of the πN Lagrangians (cf. Sec. 3.2) and sets up
four-dimensional (covariant) loop integrals. Relativistic vertices and nucleon
propagators are then expanded in powers of 1/MN . The divergences that
occur in conjunction with the four-dimensional loop integrals are treated by
means of dimensional regularization, a prescription which is consistent with
chiral symmetry and power counting. The results derived in this way are
the same obtained when starting right away with the HB versions of the
πN Lagrangians. However, as it turns out, the method used by the Munich
group is more efficient in dealing with the rather tedious calculations.
Two-pion exchange occurs first at second order, also know as leading-
order 2π exchange. The graphs are shown in the first row of Fig. 2. Since
a loop creates already ν = 2, the vertices involved at this order can only be
from the leading/lowest order Lagrangian L̂(1)πN , Eq. (33), i. e., they carry
only one derivative. These vertices are denoted by small dots in Fig. 2.
Concerning the box diagram, we should note that we include only the non-
iterative part of this diagram which is obtained by subtracting the iter-
ated 1PE contribution Eq. (65) or Eq. (66), below, but using M2N/Ep ≈
(N LO)2
(NLO)
Figure 2: Two-pion exchange contributions to the NN interaction at order two
and three in small momenta. Solid lines represent nucleons and dashed lines pions.
Small dots denote vertices from the leading order πN Lagrangian L̂(1)πN , Eq. (33).
Large solid dots are vertices proportional to the LECs ci from the second order
Lagrangian L̂(2)πN, ct, Eq. (37). Symbols with an open circles are relativistic 1/MN
corrections contained in the second order Lagrangian L̂(2)πN , Eqs. (35). Only a few
representative examples of 1/MN corrections are shown and not all.
M2N/Ep′′ ≈ MN at this order (NLO). Summarizing all contributions from
irreducible two-pion exchange at second order, one obtains [22]:
WC = −
384π2f4π
4m2π(5g
A − 4g
A − 1) + q
2(23g4A − 10g
A − 1)
48g4Am
, (49)
VT = −
VS = −
3g4AL(q)
64π2f4π
, (50)
where
L(q) ≡
w + q
4m2π + q2 . (52)
5.1.3 Third Order (NNLO)
The two-pion exchange diagrams of order three (ν = 3, next-to-next-to-
leading order, NNLO) are very similar to the ones of order two, except that
they contain one insertion from L̂(2)πN , Eq. (35). The resulting contributions
are typically either proportional to one of the low-energy constants ci or
they contain a factor 1/MN . Notice that relativistic 1/MN corrections can
occur for vertices and nucleon propagators. In Fig. 2, we show in row 2
the diagrams with vertices proportional to ci (large solid dot), Eq. (37),
and in row 3 and 4 a few representative graphs with a 1/MN correction
(symbols with an open circle). The number of 1/MN correction graphs
is large and not all are shown in the figure. Again, the box diagram is
corrected for a contribution from the iterated 1PE. If the iterative 2PE of
Eq. (65) is used, the expansion of the factor M2N/Ep = MN − p
2/2MN + . . .
is applied and the term proportional to (−p2/2MN ) is subtracted from the
third order box diagram contribution. Then, one obtains for the full third
order contribution [22]:
16πf4π
16MNw2
2m2π(2c1 − c3)− q
× w̃2A(q)
, (53)
128πMNf4π
3g2Am
4m2π + 2q
2 − g2A(4m
π + 3q
w̃2A(q)
, (54)
VT = −
9g4Aw̃
2A(q)
512πMNf4π
, (55)
WT = −
g2AA(q)
32πf4π
(10m2π + 3q
, (56)
VLS =
3g4Aw̃
2A(q)
32πMNf4π
, (57)
WLS =
g2A(1− g
32πMNf4π
w2A(q) , (58)
A(q) ≡
arctan
2m2π + q2 . (60)
As discussed in Sec. 5.1.5, below, we prefer the iterative 2PE defined in
Eq. (66), which leads to a different NNLO term for the iterative 2PE. This
changes the 1/MN terms in the above potentials. The changes are obtained
by adding to Eqs. (53)-(56) the following terms:
VC = −
256πf4πMN
2 + ω̃4A(q)) (61)
128πf4πMN
2 + ω̃4A(q)) (62)
VT = −
512πf4πMN
(mπ + ω
2A(q)) (63)
WT = −
WS = −
256πf4πMN
(mπ + ω
2A(q)) (64)
5.1.4 Fourth Order (N3LO)
This order, which may also be denoted by next-to-next-to-next-to-leading
order (N3LO), is very involved. Three-pion exchange (3PE) occurs for the
first time at this order. The 3PE contribution at N3LO has been calculated
by the Munich group and found to be negligible [38]. Therefore, we will
ignore it.
The 2PE contributions at N3LO can be subdivided into two groups, one-
loop graphs, Fig. 3, and two-loop diagrams, Fig. 4. Since these contributions
are very complicated, we have moved them to Appendix A.
5.1.5 Iterated One-Pion-Exchange
Besides all the irreducible 2PE contributions presented above, there is also
the reducible 2PE which is generated from iterated 1PE. This “iterative
2PE” is the only 2PE contribution which produces an imaginary part. Thus,
one wishes to formulate this contribution such that relativistic elastic uni-
tarity is satisfied. There are several ways to achieve this.
Kaiser et al. [22] define the iterative 2PE contribution as follows,
(KBW)
2π,it (~p
′, ~p) =
d3p′′
(2π)3
V1π(~p ′, ~p ′′)V1π(~p ′′, ~p)
p2 − p′′2 + i�
Q4 (N LO)3
Figure 3: One-loop 2π-exchange contributions to the NN interaction at order
four. Basic notation as in Fig. 2. Symbols with a large solid dot and an open circle
denote 1/MN corrections of vertices proportional to ci. Symbols with two open
circles mark relativistic 1/M2N corrections. Both corrections are part of the third
order Lagrangian L̂(3)πN , Eq. (39). Representative examples for all types of one-loop
graphs that occur at this order are shown.
with V1π given in Eq. (48).
Since we adopt the relativistic scheme developed by Blankenbecler and
Sugar [43] (BbS) (see beginning of Sec. 5.4), we prefer the following for-
mulation which is consistent with the BbS approach (and, of course, with
relativistic elastic unitarity):
2π,it (~p
′, ~p) =
d3p′′
(2π)3
V1π(~p ′, ~p ′′)V1π(~p ′′, ~p)
p2 − p′′2 + i�
. (66)
The iterative 2PE contribution has to be subtracted from the covariant
box diagram, order by order. For this, the expansion M2N/Ep = MN −
p2/2MN + . . . is applied in Eq. (65) and M2N/Ep′′ = MN −p
′′2/2MN + . . . in
Eq. (66). At NLO, both choices for the iterative 2PE collapse to the same,
� � � �
� � � �
� � �
(N LO)3
Figure 4: Two-loop 2π-exchange contributions at order four. Basic notation as in
Fig. 2. The oval stands for all one-loop πN graphs some of which are shown in the
lower part of the figure. The solid square represents vertices proportional to the
LECs di which are introduced by the third order Lagrangian L
πN , Eq. (38). More
explanations are given in the text.
while at NNLO there are obvious differences.
5.2 NN Scattering in Peripheral Partial Waves Using the
Perturbative Amplitude
After the tedious mathematics of the previous section, it is time for more
tangible affairs. The obvious question to address now is: How does the
derived NN amplitude compare to empirical information? Since our deriva-
tion includes only one- and two-pion exchanges, we are dealing here with
the long- and intermediate-range part of the NN interaction. This part of
the nuclear force is probed in the peripheral partial waves of NN scattering.
Thus, in this section, we will calculate the phase shifts that result from the
NN amplitudes presented in the previous section and compare them to the
empirical phase shifts as well as to the predictions from conventional meson
theory. Besides the irreducible two-pion exchanges derived above, we must
also include 1PE and iterated 1PE.
In this section [44], which is restricted to just peripheral waves, we
will always consider neutron-proton (np) scattering and take the charge-
dependence of 1PE due to pion-mass splitting into account, since it is ap-
preciable. With the definition
V1π(mπ) ≡ −
~σ1 · ~q ~σ2 · ~q
q2 +m2π
, (67)
the charge-dependent 1PE for np scattering is
1π (~p
′, ~p) = −Vπ(mπ0) + (−1)
I+1 2Vπ(mπ±) , (68)
where I denotes the isospin of the two-nucleon system. We use mπ0 =
134.9766 MeV, mπ± = 139.5702 MeV [31], and
2MpMn
Mp +Mn
= 938.9182 MeV . (69)
Also in the iterative 2PE, we apply the charge-dependent 1PE, i.e., in
Eq. (66) we replace V1π with V
The perturbative relativistic T-matrix for np scattering in peripheral
waves is
T (~p ′, ~p) = V (np)1π (~p
′, ~p) + V (EM,np)2π,it (~p
′, ~p) + V2π,irr(~p
′, ~p) , (70)
where V2π,irr refers to any or all of the irreducible 2PE contributions pre-
sented in Sec. 5.1, depending on the order at which the calculation is con-
ducted. In the calculation of the irreducible 2PE, we use the average pion
mass mπ = 138.039 MeV and, thus, neglect the charge-dependence due to
pion-mass splitting. The charge-dependence that emerges from irreducible
2π exchange was investigated in Ref. [45] and found to be negligible for
partial waves with L ≥ 3.
For the T -matrix given in Eq. (70), we calculate phase shifts for partial
waves with L ≥ 3 and Tlab ≤ 300 MeV. At order four in small momenta,
partial waves with L ≥ 3 do not receive any contributions from contact inter-
actions and, thus, the non-polynomial pion contributions uniquely predict
the F and higher partial waves. We use fπ = 92.4 MeV [31] and gA = 1.29.
Via the Goldberger-Treiman relation, gA = gπNN fπ/MN , our value for gA
is consistent with g2πNN/4π = 13.63± 0.20 which is obtained from πN and
NN analysis [46, 47].
The LECs used in this calculation are shown in Table 2, column “NN
periph. Fig. 5”. Note that many determinations of the LECs, ci and d̄i, can
be found in the literature. The most reliable way to determine the LECs
Table 2: Low-energy constants, LECs, used for a NN potential at N3LO,
Sec. 5.4.4, and in the calculation of the peripheral NN phase shifts shown
in Fig. 5 (column “NN periph. Fig. 5”). The ci belong to the dimension-
two πN Lagrangian, Eq. (37), and are in units of GeV−1, while the d̄i are
associated with the dimension-three Lagrangian, Eq. (38), and are in units of
GeV−2. The column “πN empirical” shows determinations from πN data.
LEC NN potential NN periph. πN
at N3LO Fig. 5 empirical
c1 –0.81 –0.81 −0.81± 0.15a
c2 2.80 3.28 3.28± 0.23b
c3 –3.20 –3.40 −4.69± 1.34a
c4 5.40 3.40 3.40± 0.04a
d̄1 + d̄2 3.06 3.06 3.06± 0.21b
d̄3 –3.27 –3.27 −3.27± 0.73b
d̄5 0.45 0.45 0.45± 0.42b
d̄14 − d̄15 –5.65 –5.65 −5.65± 0.41b
aTable 1, Fit 1 of Ref. [48].
bTable 2, Fit 1 of Ref. [37].
from empirical πN information is to extract them from the πN amplitude
inside the Mandelstam triangle (unphysical region) which can be constructed
with the help of dispersion relations from empirical πN data. This method
was used by Büttiker and Meißner [48]. Unfortunately, the values for c2 and
all d̄i parameters obtained in Ref. [48] carry uncertainties, so large that the
values cannot provide any guidance. Therefore, in Table 2, only c1, c3, and
c4 are from Ref. [48], while the other LECs are taken from Ref. [37] where
the πN amplitude in the physical region was considered. To establish a link
between πN and NN , we apply the values from the above determinations
in our calculations of the NN peripheral phase shifts. In general, we use
the mean values; the only exception is c3, where we choose a value that is,
in terms of magnitude, about one standard deviation below the one from
Ref. [48]. With the exception of c3, phase shift predictions do not depend
sensitively on variations of the LECs within the quoted uncertainties.
In Fig. 5, we show the phase-shift predictions for neutron-proton scat-
tering in F waves for laboratory kinetic energies below 300 MeV (for G and
H waves, see Ref. [26]). The orders displayed are defined as follows:
• Leading order (LO) is just 1PE, Eq. (68).
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
Figure 5: F -wave phase shifts of neutron-proton scattering for laboratory kinetic
energies below 300 MeV. We show the predictions from chiral pion exchange at lead-
ing order (LO), next-to-leading order (NLO), next-to-next-to-leading order (N2LO),
and next-to-next-to-next-to-leading order (N3LO). The solid dots and open circles
are the results from the Nijmegen multi-energy np phase shift analysis [49] and the
VPI single-energy np analysis SM99 [50], respectively.
• Next-to-leading order (NLO) is 1PE, Eq. (68), plus iterated 1PE,
Eq. (66), plus the contributions of Sec. 5.1.2 (order two), Eqs. (49)
and (50).
• Next-to-next-to-leading order (denoted by N2LO in the figures) con-
sists of NLO plus the contributions of Sec. 5.1.3 (order three), Eqs. (53)-
(58) and (61)-(64).
• Next-to-next-to-next-to-leading order (denoted by N3LO in the fig-
ures) consists of N2LO plus the contributions of Sec. 5.1.4 (order four),
Eqs. (99)-(112) and (115)-(124).
It is clearly seen in Fig. 5 that the leading order 2π exchange (NLO) is
a rather small contribution, insufficient to explain the empirical facts. In
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
Figure 6: F -wave phase shifts of neutron-proton scattering for laboratory kinetic
energies below 300 MeV. We show the results from one-pion-exchange (OPE),
and one- plus two-pion exchange as predicted by ChPT at next-to-next-to-next-
to-leading order (N3LO) and by the Bonn Full Model [13] (Bonn). Note that the
“Bonn” curve does not include the repulsive ω and πρ exchanges of the full model,
since this figure serves the purpose to compare just predictions by different mod-
els/theories for the π + 2π contribution to the NN interaction. Empirical phase
shifts (solid dots and open circles) as in Fig. 5.
contrast, the next order (N2LO) is very large, several times NLO. This is
due to the ππNN contact interactions proportional to the LECs ci that are
introduced by the second order Lagrangian L(2)πN , Eq. (34). These contacts
are supposed to simulate the contributions from intermediate ∆-isobars and
correlated 2π exchange which are known to be large (see, e. g., Ref. [13]).
At N3LO a clearly identifiable trend towards convergence emerges. Ob-
viously, 1F3 and 3F4 appear fully converged. However, in 3F2 and 3F3, N3LO
differs noticeably from NNLO, but the difference is much smaller than the
one between NNLO and NLO. This is what we perceive as a trend towards
convergence.
In Fig. 6, we conduct a comparison between the predictions from chi-
ral one- and two-pion exchange at N3LO and the corresponding predictions
from conventional meson theory (curve ‘Bonn’). As representative for con-
ventional meson theory, we choose the Bonn meson-exchange model for the
NN interaction [13], since it contains a comprehensive and thoughtfully con-
structed model for 2π exchange. This 2π model includes box and crossed
box diagrams with NN , N∆, and ∆∆ intermediate states as well as di-
rect ππ interaction in S- and P -waves (of the ππ system) consistent with
empirical information from πN and ππ scattering. Besides this the Bonn
model also includes (repulsive) ω-meson exchange and irreducible diagrams
of π and ρ exchange (which are also repulsive). However, note that in the
phase shift predictions displayed in Fig. 6, the “Bonn” curve includes only
the 1π and 2π contributions from the Bonn model; the short-range contri-
butions are left out since the purpose of the figure is to compare different
models/theories for π + 2π. In all waves shown we see, in general, good
agreement between N3LO and Bonn. In 3F2 and 3F3 above 150 MeV and
in 3F4 above 250 MeV the chiral model at N3LO is more attractive than
the Bonn 2π model. Note, however, that the Bonn model is relativistic and,
thus, includes relativistic corrections up to infinite orders. Thus, one may
speculate that higher orders in ChPT may create some repulsion, moving
the Bonn and the chiral predictions even closer together [51].
The 2π exchange contribution to the NN interaction can also be de-
rived from empirical πN and ππ input using dispersion theory, which is
based upon unitarity, causality (analyticity), and crossing symmetry. The
amplitude NN̄ → ππ is constructed from πN → πN and πN → ππN
data using crossing properties and analytic continuation; this amplitude is
then ‘squared’ to yield the NN̄ amplitude which is related to NN by cross-
ing symmetry [52]. The Paris group [11, 12] pursued this path and calcu-
lated NN phase shifts in peripheral partial waves. Naively, the dispersion-
theoretic approach is the ideal one, since it is based exclusively on empirical
information. Unfortunately, in practice, quite a few uncertainties enter into
the approach. First, there are ambiguities in the analytic continuation and,
second, the dispersion integrals have to be cut off at a certain momentum
to ensure reasonable results. In Ref. [13], a thorough comparison was con-
ducted between the predictions by the Bonn model and the Paris approach
and it was demonstrated that the Bonn predictions always lie comfortably
within the range of uncertainty of the dispersion-theoretic results. There-
fore, there is no need to perform a separate comparison of our chiral N3LO
predictions with dispersion theory, since it would not add anything that we
cannot conclude from Fig. 6.
Finally, we need to compare the predictions with the empirical phase
shifts. In F waves the N3LO predictions above 200 MeV are, in general,
too attractive. Note, however, that this is also true for the predictions by
the Bonn π + 2π model. In the full Bonn model, besides π + 2π, (repul-
sive) ω and πρ exchanges are included which move the predictions right
on top of the data. The exchange of a ω meson or combined πρ exchange
are 3π exchanges. Three-pion exchange occurs first at chiral order four.
It has be investigated by Kaiser [38] and found to be negligible, at this
order. However, 3π exchange at order five appears to be sizable [53] and
may have impact on F waves. Besides this, there is the usual short-range
phenomenology. In ChPT, this short-range interaction is parametrized in
terms of four-nucleon contact terms (since heavy mesons do not have a place
in that theory). Contact terms of order four (N3LO) do not contribute to
F -waves, but order six does. In summary, the remaining small discrepan-
cies between the N3LO predictions and the empirical phase shifts may be
straightened out in fifth or sixth order of ChPT.
5.3 NN Contact Potentials
In conventional meson theory, the short-range nuclear force is described by
the exchange of heavy mesons, notably the ω(782). The qualitative short-
distance behavior of the NN potential is obtained by Fourier transform of
the propagator of a heavy meson,
ei~q·~r
m2ω + ~q2
e−mωr
. (71)
ChPT is an expansion in small momenta Q, too small to resolve struc-
tures like a ρ(770) or ω(782) meson, because Q � Λχ ≈ mρ,ω. But the
latter relation allows us to expand the propagator of a heavy meson into a
power series,
m2ω +Q2
−+ . . .
, (72)
where the ω is representative for any heavy meson of interest. The above
expansion suggests that it should be possible to describe the short distance
part of the nuclear force simply in terms of powers of Q/mω, which fits in
well with our over-all power scheme since Q/mω ≈ Q/Λχ.
A second purpose of contact terms is renormalization. Dimensional reg-
ularization of the loop integrals of pion-exchanges (cf. Sec. 5.1) typically
generates polynomial terms with coefficients that are, in part, infinite or
scale dependent. Contact terms pick up infinities and remove scale depen-
dence.
The partial-wave decomposition of a power Qν has an interesting prop-
erty. First note that Q can only be either the momentum transfer between
the two interacting nucleons q or the average momentum k [cf. Eq. (47) for
their definitions]. In any case, for even ν,
Qν = f ν
(cos θ) , (73)
where fm stands for a polynomial of degree m and θ is the CMS scattering
angle. The partial-wave decomposition of Qν for a state of orbital-angular
momentum L involves the integral
QνPL(cos θ)d cos θ =
(cos θ)PL(cos θ)d cos θ , (74)
where PL is a Legendre polynomial. Due to the orthogonality of the PL,
L = 0 for L >
. (75)
Consequently, contact terms of order zero contribute only in S-waves, while
order-two terms contribute up to P -waves, order-four terms up to D-waves,
etc..
We will now present, one by one, the various orders of NN contact
terms together with their partial-wave decomposition [54]. Note that, due
to parity, only even powers of Q are allowed.
5.3.1 Zeroth Order
The contact potential at order zero reads:
V (0)(~p′, ~p) = CS + CT ~σ1 · ~σ2 (76)
Partial wave decomposition yields:
V (0)(1S0) = C̃1S0 = 4π (CS − 3CT )
V (0)(3S1) = C̃3S1 = 4π (CS + CT ) (77)
5.3.2 Second Order
The contact potential contribution of order two is given by:
V (2)(~p′, ~p) = C1q
2 + C2k
2 + C4k
~σ1 · ~σ2
−i~S · (~q × ~k)
+ C6(~σ1 · ~q) (~σ2 · ~q)
+ C7(~σ1 · ~k) (~σ2 · ~k) (78)
Second order partial wave contributions:
S0) = C1S0(p
C2 − 3C3 −
C4 − C6 −
P0) = C3P0 pp
C5 + 2C6 −
P1) = C1P1 pp
C2 + 2C3 −
P1) = C3P1 pp
S1) = C3S1(p
C2 + C3 +
S1 −3 D1) = C3S1−3D1p
P2) = C3P2 pp
5.3.3 Fourth Order
The contact potential contribution of order four reads:
V (4)(~p′, ~p) = D1q
4 +D2k
4 +D3q
2k2 +D4(~q × ~k)2
4 +D6k
4 +D7q
2k2 +D8(~q × ~k)2
~σ1 · ~σ2
2 +D10k
−i~S · (~q × ~k)
2 +D12k
(~σ1 · ~q) (~σ2 · ~q)
2 +D14k
(~σ1 · ~k) (~σ2 · ~k)
+ D15
~σ1 · (~q × ~k) ~σ2 · (~q × ~k)
The rather lengthy partial-wave expressions of this order have been relegated
to Appendix B.
5.4 Constructing a Chiral NN Potential
5.4.1 Conceptual Questions
The two-nucleon system is non-perturbative as evidenced by the presence
of a shallow bound state (the deuteron) and large scattering lengths. Wein-
berg [18] showed that the strong enhancement of the scattering amplitude
arises from purely nucleonic intermediate states. He therefore suggested to
use perturbation theory to calculate the NN potential and to apply this
potential in a scattering equation to obtain the NN amplitude. We adopt
this prescription.
Since the irreducible diagrams that make up the potential are calculated
using covariant perturbation theory (cf. Sec. 5.1), it is consistent to start
from the covariant Bethe-Salpeter (BS) equation [55] describing two-nucleon
scattering. In operator notation, the BS equation reads
T = V + V G T (81)
with T the invariant amplitude for the two-nucleon scattering process, V the
sum of all connected two-particle irreducible diagrams, and G the relativistic
two-nucleon propagator. The BS equation is equivalent to a set of two
equations
T = V + V g T (82)
V = V + V (G − g)V (83)
= V + V1π (G − g)V1π + . . . , (84)
where g is a covariant three-dimensional propagator which preserves rela-
tivistic elastic unitarity. We choose the propagator g proposed by Blanken-
becler and Sugar (BbS) [43] (for more details on relativistic three-dimensional
reductions of the BS equation, see Ref. [7]). The ellipsis in Eq. (84) stands
for terms of irreducible 3π and higher pion exchanges which we neglect.
Note that when we speak of covariance in conjunction with (heavy baryon)
ChPT, we are not referring to manifest covariance. Relativity and relativis-
tic off-shell effects are accounted for in terms of a Q/MN expansion up to
the given order. Thus, Eq. (84) is evaluated in the following way,
V ≈ V(on-shell) + V1π G V1π − V1π g V1π , (85)
where the pion-exchange content of V(on-shell) is V1π+V ′2π with V1π the on-
shell 1PE given in Eq. (48) and V ′2π the irreducible 2π exchanges calculated
in Sec. 5.1, but without the box. V1π denotes the relativistic (off-shell) 1PE.
Notice that the term (V1π G V1π−V1π g V1π) represents what has been called
“the (irreducible part of the) box diagram contribution” in Sec. 5.1 where
it was evaluated at various orders.
The full chiral NN potential V is given by irreducible pion exchanges
Vπ and contact terms Vct,
V = Vπ + Vct (86)
Vπ = V1π + V2π + . . . , (87)
where the ellipsis denotes irreducible 3π and higher pion exchanges which
are omitted. Two-pion exchange contributions appear in various orders
V2π = V
2π + V
2π + V
2π + . . . (88)
as calculated in Sec. 5.1. Contact terms come in even orders,
Vct = V
ct + V
ct + V
ct + . . . (89)
and were presented in Sec. 5.3. The potential V is calculated at a given order.
For example, the potential at NNLO includes 2PE up to V (3)2π and contacts
up to V (2)ct . At N
3LO, contributions up to V (4)2π and V
ct are included.
The potential V satisfies the relativistic BbS equation, Eq. (82). Defining
V̂ (~p ′, ~p) ≡
(2π)3
V (~p ′, ~p)
T̂ (~p ′, ~p) ≡
(2π)3
T (~p ′, ~p)
with Ep ≡
M2N + ~p
2 (the factor 1/(2π)3 is added for convenience), the
BbS equation collapses into the usual, nonrelativistic Lippmann-Schwinger
(LS) equation,
T̂ (~p ′, ~p) = V̂ (~p ′, ~p) +
d3p′′ V̂ (~p ′, ~p ′′)
p2 − p′′2 + i�
T̂ (~p ′′, ~p) . (92)
Since V̂ satisfies Eq. (92), it can be used like a usual nonrelativistic
potential, and T̂ is the conventional nonrelativistic T-matrix.
Iteration of V̂ in the LS equation requires cutting V̂ off for high momenta
to avoid infinities, This is consistent with the fact that ChPT is a low-
momentum expansion which is valid only for momenta Q � Λχ ≈ 1 GeV.
Thus, we multiply V̂ with a regulator function
V̂ (~p ′, ~p) 7−→ V̂ (~p ′, ~p) e−(p
′/Λ)2n e−(p/Λ)
≈ V̂ (~p ′, ~p)
+ . . .
with the ‘cutoff parameter’ Λ around 0.5 GeV. Equation (94) provides an
indication of the fact that the exponential cutoff does not necessarily affect
the given order at which the calculation is conducted. For sufficiently large
n, the regulator introduces contributions that are beyond the given order.
Assuming a good rate of convergence of the chiral expansion, such orders are
small as compared to the given order and, thus, do not affect the accuracy at
the given order. In our calculations we use, of course, the full exponential,
Eq. (93), and not the expansion. On a similar note, we also do not expand
the square-root factors in Eqs. (90-91) because they are kinematical factors
which guarantee relativistic elastic unitarity.
5.4.2 What Order?
Since in nuclear EFT we are dealing with a perturbative expansion, at some
point, we have to raise the question, to what order of ChPT we have to go
to obtain the precision we need. To discuss this issue on firm grounds, we
show in Table 3 the χ2/datum for the fit of the world np data below 290
MeV for a family of np potentials at NLO and NNLO. The NLO potentials
produce the very large χ2/datum between 67 and 105, and the NNLO are
between 12 and 27. The rate of improvement from one order to the other
is very encouraging, but the quality of the reproduction of the np data at
NLO and NNLO is obviously insufficient for reliable predictions.
Table 3: χ2/datum for the reproduction of the 1999 np database [56] by
families of np potentials at NLO and NNLO constructed by the Juelich
group [57].
Tlab bin # of np — Juelich np potentials —
(MeV) data NLO NNLO
0–100 1058 4–5 1.4–1.9
100–190 501 77–121 12–32
190–290 843 140–220 25–69
0–290 2402 67–105 12–27
Based upon these facts, it has been pointed out in 2002 by Entem and
Machleidt [25, 26] that one has to proceed to N3LO. Consequently, the first
N3LO potential was published in 2003 [27].
At N3LO, there are 24 contact terms (24 parameters) which contribute to
the partial waves with L ≤ 2 (cf. Sec. 5.3). In Table 4, column ‘Q4/N3LO’,
we show how these terms/parameters are distributed over the various partial
waves. For comparison, we also show the number of parameters used in
the Nijmegen partial wave analysis (PWA93) [49] and in the high-precision
CD-Bonn potential [9]. The table reveals that, for S and P waves, the
number of parameters used in high-precision phenomenology and in EFT at
N3LO are about the same. Thus, the EFT approach provides retroactively
a justification for what the phenomenologists of the 1990’s were doing. At
NLO and NNLO, the number of parameters is substantially smaller than for
PWA93 and CD-Bonn, which explains why these orders are insufficient for
a quantitative potential. This fact is also clearly reflected in Fig. 7 where
phase shifts are shown for potentials constructed at NLO, NNLO, and N3LO.
5.4.3 Charge-Dependence
For an accurate fit of the low-energy pp and np data, charge-dependence
is important. We include charge-dependence up to next-to-leading order of
the isospin-violation scheme (NLØ, in the notation of Ref. [58]). Thus, we
include the pion mass difference in 1PE and the Coulomb potential in pp
scattering, which takes care of the LØ contributions. At order NLØ, we
have the pion mass difference in 2PE at NLO, πγ exchange [59], and two
charge-dependent contact interactions of order Q0 which make possible an
accurate fit of the three different 1S0 scattering lengths, app, ann, and anp.
Table 4: Number of parameters needed for fitting the np data in phase-shift
analysis and by a high-precision NN potential versus the number of NN
contact terms of EFT based potentials at different orders.
Nijmegen CD-Bonn — Contact Potentials —
partial-wave high-precision Q0 Q2 Q4
analysis [49] potential [9] LO NLO/NNLO N3LO
1S0 3 4 1 2 4
3S1 3 4 1 2 4
3S1-3D1 2 2 0 1 3
1P1 3 3 0 1 2
3P0 3 2 0 1 2
3P1 2 2 0 1 2
3P2 3 3 0 1 2
3P2-3F2 2 1 0 0 1
1D2 2 3 0 0 1
3D1 2 1 0 0 1
3D2 2 2 0 0 1
3D3 1 2 0 0 1
3D3-3G3 1 0 0 0 0
1F3 1 1 0 0 0
3F2 1 2 0 0 0
3F3 1 2 0 0 0
3F4 2 1 0 0 0
3F4-3H4 0 0 0 0 0
1G4 1 0 0 0 0
3G3 0 1 0 0 0
3G4 0 1 0 0 0
3G5 0 1 0 0 0
Total 35 38 2 9 24
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
Figure 7: Phase parameters for np scattering as calculated from NN potentials at
different orders of ChPT. The dotted line is NLO [57], the dashed NNLO [57], and
the solid N3LO [27]. Partial waves with total angular momentum J ≤ 2 are dis-
played. Solid dots represent the Nijmegen multienergy np phase shift analysis [49]
and open circles are the GWU/VPI single-energy np analysis SM99 [50].
5.4.4 A Quantitative NN Potential at N3LO
NN Scattering. The fitting procedure starts with the peripheral partial
waves because they depend on fewer parameters. Partial waves with L ≥ 3
are exclusively determined by 1PE and 2PE because the N3LO contacts
contribute to L ≤ 2 only. 1PE and 2PE at N3LO depend on the axial-
vector coupling constant, gA (we use gA = 1.29), the pion decay constant,
fπ = 92.4 MeV, and eight low-energy constants (LECs) that appear in the
dimension-two and dimension-three πN Lagrangians, Eqs. (37) and (38).
In the fitting process, we varied three of them, namely, c2, c3, and c4. We
found that the other LECs are not very effective in the NN system and,
therefore, we kept them at the values determined from πN (cf. Table 2).
The most influential constant is c3, which has to be chosen on the low side
(slightly more than one standard deviation below its πN determination) for
an optimal fit of the NN data. As compared to a calculation that strictly
uses the πN values for c2 and c4, our choices for these two LECs lower
Table 5: χ2/datum for the reproduction of the 1999 np database [56]
by various np potentials. (Numbers in parentheses are the values of cutoff
parameters in units of MeV used in the regulators of the chiral potentials.)
Tlab bin # of np Idaho Juelich Argonne
(MeV) data N3LO [27] N3LO [60] V18 [61]
(500–600) (600/700–450/500)
0–100 1058 1.0–1.1 1.0–1.1 0.95
100–190 501 1.1–1.2 1.3–1.8 1.10
190–290 843 1.2–1.4 2.8–20.0 1.11
0–290 2402 1.1–1.3 1.7–7.9 1.04
Table 6: χ2/datum for the reproduction of the 1999 pp database [56] by
various pp potentials. Notation as in Fig. 5.
Tlab bin # of np Idaho Juelich Argonne
(MeV) data N3LO [27] N3LO [60] V18 [61]
(500–600) (600/700–450/500)
0–100 795 1.0–1.7 1.0–3.8 1.0
100–190 411 1.5–1.9 3.5–11.6 1.3
190–290 851 1.9–2.7 4.3–44.4 1.8
0–290 2057 1.5–2.1 2.9–22.3 1.4
the 3F2 and 1F3 phase shifts bringing them into closer agreement with the
phase shift analysis. The other F waves and the higher partial waves are
essentially unaffected by our variations of c2 and c4. Overall, the fit of all
J ≥ 3 waves is very good.
We turn now to the lower partial waves. Here, the most important fit
parameters are the ones associated with the 24 contact terms that contribute
to the partial waves with L ≤ 2. In addition, we have two charge-dependent
contacts which are used to fit the three different 1S0 scattering lengths, app,
ann, and anp.
In the optimization procedure, we fit first phase shifts, and then we
refine the fit by minimizing the χ2 obtained from a direct comparison with
the data. The χ2/datum for the fit of the np data below 290 MeV is shown
in Table 5, and the corresponding one for pp is given in Table 6. These tables
show that at N3LO a χ2/datum comparable to the high-precision Argonne
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
0 100 200 300
Lab. Energy (MeV)
Figure 8: Neutron-proton phase parameters as described by two potentials at
N3LO. The solid curve is calculated from the Idaho N3LO potential [27] while the
dashed curve is from the Juelich [60] one. Solid dots and open circles as in Fig. 7.
V18 [61] potential can, indeed, be achieved. The “Idaho” N3LO potential [27]
produces a χ2/datum = 1.1 for the world np data below 290 MeV which
compares well with the χ2/datum = 1.04 by the Argonne potential. In 2005,
also the Juelich group produced several N3LO NN potentials [60], the best
of which fits the np data with a χ2/datum = 1.7 and the worse with a
χ2/datum = 7.9 (see Table 5). While 7.9 is clearly unacceptable for any
meaningful application, a χ2/datum of 1.7 is reasonable, although it does
not meet the precision standard that few-nucleon physicists established in
the 1990’s.
Turning to pp, the χ2 for pp data are typically larger than for np be-
cause of the higher precision of pp data. Thus, the Argonne V18 produces a
χ2/datum = 1.4 for the world pp data below 290 MeV and the best Idaho
N3LO pp potential obtains 1.5. The fit by the best Juelich N3LO pp poten-
tial results in a χ2/datum = 2.9 which, again, is not quite consistent with
the precision standards established in the 1990’s. The worst Juelich N3LO
pp potential produces a χ2/datum of 22.3 and is incompatible with reliable
predictions.
Table 7: Deuteron properties as predicted by various NN potentials are com-
pared to empirical information. (Deuteron binding energy Bd, asymptotic S state
AS , asymptotic D/S state η, deuteron radius rd, quadrupole moment Q, D-state
probability PD; the calculated rd and Q are without meson-exchange current con-
tributions and relativistic corrections.)
Idaho Juelich
N3LO [27] N3LO [60] CD-Bonn[9] AV18[61] Empiricala
(500) (550/600)
Bd (MeV) 2.224575 2.218279 2.224575 2.224575 2.224575(9)
AS (fm−1/2) 0.8843 0.8820 0.8846 0.8850 0.8846(9)
η 0.0256 0.0254 0.0256 0.0250 0.0256(4)
rd (fm) 1.975 1.977 1.966 1.967 1.97535(85)
Q (fm2) 0.275 0.266 0.270 0.270 0.2859(3)
PD (%) 4.51 3.28 4.85 5.76
aSee Table XVIII of Ref. [9] for references; the empirical value for rd is from Ref. [62].
Phase shifts of np scattering from the best Idaho (solid line) and Juelich
(dashed line) N3LO np potentials are shown in Figure 8. The phase shifts
confirm what the corresponding χ2 have already revealed.
The Deuteron. The reproduction of the deuteron parameters is shown
in Table 7. We present results for two N3LO potentials, namely, Idaho [27]
and Juelich [60]. Remarkable are the predictions by the chiral potentials for
the deuteron radius which are in good agreement with the latest empirical
value obtained by the isotope-shift method [62]. All NN potentials of the
past (Table 7 includes two representative examples, namely, CD-Bonn [9]
and AV18 [61]) fail to reproduce this very precise new value for the deuteron
radius.
In Fig. 9, we display the deuteron wave functions derived from the N3LO
potentials and compare them with wave functions based upon conventional
NN potentials from the recent past. Characteristic differences are notice-
able; in particular, the chiral wave functions are shifted towards larger r
which explains the larger deuteron radius.
6 Many-Nucleon Forces
As noted before, an important advantage of the EFT approach to nuclear
forces is that it creates two- and many-nucleon forces on an equal footing.
0 1 2 3 4 5 6
r (fm)
Figure 9: Deuteron wave functions: the family of larger curves are S-waves,
the smaller ones D-waves. The thick lines represent the wave functions
derived from chiral NN potentials at order N3LO (thick solid: Idaho [27],
thick dashed: Juelich [60]). The thin dashed, dash-dotted, and dotted lines
refer to the wave functions of the CD-Bonn [9], Nijm-I [8], and AV18 [61]
potentials, respectively.
6.1 Three-Nucleon Forces
The first non-vanishing 3NF terms occur at NNLO and are shown in Fig. 10
(cf. also Fig. 1, row ‘Q3/NNLO’, column ‘3N Force’). There are three di-
agrams: the 2PE, 1PE, and 3N-contact interactions [39, 40]. The 2PE
3N-potential is given by
V 3NF2PE =
i 6=j 6=k
(~σi · ~qi)(~σj · ~qj)
(q2i +m
ijk τ
j (95)
with ~qi ≡ ~pi′ − ~pi, where ~pi and ~pi′ are the initial and final momenta of
nucleon i, respectively, and
ijk = δ
4c1m2π
~qi · ~qj
�αβγ τ
k ~σk · [~qi × ~qj ] . (96)
The vertex involved in this 3NF term is the two-derivative ππNN vertex
(solid square in Fig. 10) which we encountered already in the 2PE contribu-
tion to the NN potential at NNLO. Thus, there are no new parameters and
1 2 3
Figure 10: The three-nucleon force at NNLO (from Ref. [40]).
the contribution is fixed by the LECs used in NN . The 1PE contribution is
V 3NF1PE = D
i 6=j 6=k
~σj · ~qj
q2j +m
(τ i · τ j)(~σi · ~qj) (97)
and, finally, the 3N contact term reads
V 3NFct = E
j 6=k
τ j · τ k . (98)
The last two 3NF terms involve two new vertices (that do not appear in the
2N problem), namely, the πNNNN vertex with parameter D and a 6N ver-
tex with parameters E. To pin them down, one needs two observables that
involve at least three nucleons. In Ref. [40], the triton binding energy and
the nd doublet scattering length 2and were used. Alternatively, one may also
choose the binding energies of 3H and 4He [63]. Once D and E are fixed, the
results for other 3N, 4N, . . . observables are predictions. In Refs. [64, 63],
the first calculations of the structure of light nuclei (6Li and 7Li) were re-
ported. Recently, the structure of nuclei with A = 10−13 nucleons has been
calculated using the ab initio no-core shell model and applying chiral two
and three-nucleon forces [65]. The results are very encouraging. Concerning
the famous ‘Ay puzzle’, the above 3NF terms yield some improvement of
the predicted nd Ay, however, the problem is not solved [40].
Note that the 3NF expressions given in Eqs. (95)-(98) above are the ones
that occur at NNLO, and all calculations to date have included only those.
Since we have to proceed to N3LO for sufficient accuracy of the 2NF, then
consistency requires that we also consider the 3NF at N3LO. The 3NF at
N3LO is very involved as can be seen from Fig. 11, but it does not depend
on any new parameters. It is presently under construction [66]. So, for the
moment, we can only hope that the Ay puzzle may be solved by a complete
calculation at N3LO.
Figure 11: Three-nucleon force contributions at N3LO (from Ref. [66]).
6.2 Four-Nucleon Forces
In ChPT, four-nucleon forces (4NF) appear for the first time at N3LO
(ν = 4). Thus, N3LO is the leading order for 4NF. Assuming a good rate of
convergence, a contribution of order (Q/Λχ)4 is expected to be rather small.
Thus, ChPT predicts 4NF to be essentially insignificant, consistent with ex-
perience. Still, nothing is fully proven in physics unless we have performed
explicit calculations. Very recently, the first such calculation has been per-
formed: The chiral 4NF, Fig. 12, has been applied in a calculation of the
4He binding energy and found to contribute a few 100 keV [68]. It should
be noted that this preliminary calculation involves many approximations,
but it certainly provides the right order of magnitude of the result, which is
indeed very small as compared to the full 4He binding energy of 28.3 MeV.
7 Conclusions
The theory of nuclear forces has made great progress since the turn of the
millennium. Nucleon-nucleon potentials have been developed that are based
on proper theory (EFT for low-energy QCD) and are of high-precision, at
the same time. Moreover, the theory generates two- and many-body forces
on an equal footing and provides a theoretical explanation for the empirically
known fact that 2NF � 3NF � 4NF . . . .
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
(m) (n) (o) (p)
Figure 12: The four-nucleon force at N3LO (from Ref. [67]).
At N3LO [26, 27], the accuracy can be achieved that is necessary and
sufficient for reliable microscopic nuclear structure predictions. First cal-
culations applying the N3LO NN potential [27] in the conventional shell
model [69, 70], the ab initio no-core shell model [71, 72, 73], the coupled
cluster formalism [74, 75, 76, 77, 78], and the unitary-model-operator ap-
proach [79] have produced promising results.
The 3NF at NNLO is known [39, 40] and has been applied in few-nucleon
reactions [40, 80, 81] as well as the structure of light nuclei [64, 63, 65]. How-
ever, the famous ‘Ay puzzle’ of nucleon-deuteron scattering is not resolved
by the 3NF at NNLO. Thus, one important outstanding issue is the 3NF at
N3LO, which is under construction [66].
Another open question that needs to be settled is whether Weinberg
power counting, which is applied in all current NN potentials, is consistent.
This controversial issue is presently being debated in the literature [82, 83].
Acknowledgements
It is a pleasure to thank the organizers of this workshop, particularly,
Ananda Santra, for their warm hospitality. I gratefully acknowledge numer-
ous discussions with my collaborator D. R. Entem. This work was supported
in part by the U.S. National Science Foundation under Grant No. PHY-
0099444.
A Fourth Order Two-Pion Exchange Contributions
The fourth order 2PE contributions consist of two classes: the one-loop
(Fig. 3) and the two-loop diagrams (Fig. 4).
A.1 One-loop diagrams
This large pool of diagrams can be analyzed in a systematic way by intro-
ducing the following well-defined subdivisions.
A.1.1 c2i contributions.
The only contribution of this kind comes from the football diagram with
both vertices proportional to ci (first row of Fig. 3). One obtains [41]:
3L(q)
16π2f4π
w2 + c3w̃
2 − 4c1m2π
, (99)
WT = −
2L(q)
96π2f4π
. (100)
A.1.2 ci/MN contributions.
This class consists of diagrams with one vertex proportional to ci and one
1/MN correction. A few graphs that are representative for this class are
shown in the second row of Fig. 3. Symbols with a large solid dot and an
open circle denote 1/MN corrections of vertices proportional to ci. They are
part of L̂(3)πN , Eq. (39). The result for this group of diagrams is [41]:
VC = −
g2A L(q)
32π2MNf4π
(c2 − 6c3)q4 + 4(6c1 + c2 − 3c3)q2m2π
+6(c2 − 2c3)m4π + 24(2c1 + c3)m
, (101)
WC = −
2L(q)
192π2MNf4π
g2A(8m
π + 5q
2) + w2
, (102)
WT = −
c4L(q)
192π2MNf4π
g2A(16m
π + 7q
2)− w2
, (103)
VLS =
8π2MNf4π
w2L(q) , (104)
WLS = −
c4L(q)
48π2MNf4π
g2A(8m
π + 5q
2) + w2
. (105)
A.1.3 1/M2N corrections.
These are relativistic 1/M2N corrections of the leading order 2π exchange
diagrams. Typical examples for this large class are shown in row 3–6 of
Fig. 3. This time, there is no correction from the iterated 1PE, Eq. (65) or
Eq. (66), since the expansion of the factor M2N/Ep does not create a term
proportional to 1/M2N . The total result for this class is [42],
VC = −
32π2M2Nf
2m8πw
−4 + 8m6πw
−2 − q4 − 2m4π
(106)
WC = −
768π2M2Nf
q4 + 3m2πq
2 + 3m4π − 6m
−k2(8m2π + 5q
+ 4g4A
k2(20m2π + 7q
2 − 16m4πw
−2) + 16m8πw
+12m6πw
−2 − 4m4πq
2w−2 − 5q4 − 6m2πq
2 − 6m4π
− 4k2w2
16g4Am
, (107)
VT = −
g4A L(q)
32π2M2Nf
q2 +m4πw
, (108)
WT = −
1536π2M2Nf
7m2π +
q2 + 4m4πw
− 32g2A
m2π +
, (109)
VLS =
g4A L(q)
4π2M2Nf
q2 +m4πw
, (110)
WLS =
256π2M2Nf
16g2A
m2π +
4m4πw
q2 − 9m2π
, (111)
VσL =
g4A L(q)
32π2M2Nf
. (112)
A.2 Two-loop contributions.
The two-loop contributions are quite intricate. In Fig. 4, we attempt a
graphical representation of this class. The gray disk stands for all one-loop
πN graphs which are shown in some detail in the lower part of the figure.
Not all of the numerous graphs are displayed. Some of the missing ones
are obtained by permutation of the vertices along the nucleon line, others
by inverting initial and final states. Vertices denoted by a small dot are
from the leading order πN Lagrangian L̂(1)πN , Eq. (33), except for the four-
pion vertices which are from L(2)ππ , Eq. (27). The solid square represents
vertices proportional to the LECs di which are introduced by the third
order Lagrangian L(3)πN , Eq. (38). The di vertices occur actually in one-
loop NN diagrams, but we list them among the two-loop NN contributions
because they are needed to absorb divergences generated by one-loop πN
graphs. Using techniques from dispersion theory, Kaiser [41] calculated the
imaginary parts of the NN amplitudes, Im Vα(iµ) and Im Wα(iµ), which
result from analytic continuation to time-like momentum transfer q = iµ−0+
with µ ≥ 2mπ. From this, the momentum-space amplitudes Vα(q) and
Wα(q) are obtained via the subtracted dispersion relations:
VC,S(q) = −
ImVC,S(iµ)
µ5(µ2 + q2)
, (113)
VT (q) =
ImVT (iµ)
µ3(µ2 + q2)
, (114)
and similarly for WC,S,T .
In most cases, the dispersion integrals can be solved analytically and the
following expressions are obtained [26]:
VC(q) =
3g4Aw̃
2A(q)
1024π2f6π
(m2π + 2q
2mπ + w̃
2A(q)
+ 4g2Amπw̃
(115)
WC(q) = W
C (q) +W
C (q) , (116)
C (q) =
18432π4f6π
192π2f2πw
2g2Aw̃
(g2A − 1)w
6g2Aw̃
2 − (g2A − 1)w
384π2f2π
w̃2(d̄1 + d̄2) + 4m
+L(q)
4m2π(1 + 2g
A) + q
2(1 + 5g2A)
(5 + 13g2A) + 8m
π(1 + 2g
(117)
C (q) = −
ImW (b)C (iµ)
µ5(µ2 + q2)
, (118)
where
ImW (b)C (iµ) = −
3µ(8πf2π)3
g2A(2m
π − µ
2) + 2(g2A − 1)κ
− 3κ2x2 + 6κx
m2π + κ2x2 ln
m2π + κ2x2
µ2 − 2κ2x2 − 2m2π
m2π + κ2x2
; (119)
VT (q) = V
T (q) + V
T (q)
VS(q) = −
S (q) + V
S (q)
, (120)
T (q) = −
S (q) = −
2L(q)
32π2f4π
(d̄14 − d̄15) (121)
T (q) = −
S (q) =
ImV (b)T (iµ)
µ3(µ2 + q2)
, (122)
where
ImV (b)T (iµ) = −
2g6Aκ
µ(8πf2π)3
dx(1− x2)
m2π + κ2x2
 ; (123)
WT (q) = −
WS(q) =
2A(q)
2048π2f6π
w2A(q) + 2mπ(1 + 2g
, (124)
where κ ≡
µ2/4−m2π.
Note that the analytic solutions hold modulo polynomials. We have
checked the importance of those contributions where we could not find an
analytic solution and where, therefore, the integrations have to be performed
numerically. It turns out that the combined effect on NN phase shifts from
C , V
T , and V
S is smaller than 0.1 deg in F and G waves and smaller
than 0.01 deg in H waves, at Tlab = 300 MeV (and less at lower energies).
This renders these contributions negligible. Therefore, we omit W (b)C , V
and V (b)S in the construction of chiral NN potentials at order N
In Eqs. (117) and (121), we use the scale-independent LECs, d̄i, which
are obtained by combining the scale-dependent ones, dri (λ), with the chiral
logarithm, ln(mπ/λ), or equivalently d̄i = dri (mπ). The scale-dependent
LECs, dri (λ), are a consequence of renormalization. For more details about
this issue, see Ref. [37].
B Partial Wave Decomposition of the Fourth Or-
der Contact Potential
The contact potential contribution of order four, Eq. (80), decomposes into
partial-waves as follows.
V (4)(1S0) = D̂1S0(p
′4 + p4) +D1S0p
V (4)(3P0) = D3P0(p
′3p+ p′p3)
V (4)(1P1) = D1P1(p
′3p+ p′p3)
V (4)(3P1) = D3P1(p
′3p+ p′p3)
V (4)(3S1) = D̂3S1(p
′4 + p4) +D3S1p
V (4)(3D1) = D3D1p
V (4)(3S1 −3 D1) = D̂3S1−3D1p
4 +D3S1−3D1p
V (4)(1D2) = D1D2p
V (4)(3D2) = D3D2p
V (4)(3P2) = D3P2(p
′3p+ p′p3)
V (4)(3P2 −3 F2) = D3P2−3F2p
V (4)(3D3) = D3D3p
′2p2 (125)
The coefficients in the above expressions are given by:
D̂1S0 = D1 +
D3 − 3D5 −
D7 −D11 −
D12 −
D1S0 =
D4 − 10D5 −
D7 − 2D8 −
D12 −
D13 −
D14 −
D3P0 = −
D10 +
D11 +
D12 −
D1P1 = −
D2 + 4D5 −
D11 −
D3P1 = −
D10 − 2D11 −
D12 +
D̂3S1 = D1 +
D3 +D5 +
D11 +
D12 +
D3S1 =
D12 +
D13 +
D14 +
D3D1 =
D10 −
D11 +
D12 +
D13 −
D14 −
D̂3S1−3D1 = −
D11 −
D12 −
D13 −
D3S1−3D1 = −
D11 +
D12 +
D13 −
D14 +
D1D2 =
D12 +
D13 −
D14 +
D3D2 =
D10 +
D11 −
D12 −
D13 +
D14 +
D3P2 = −
D10 −
D11 +
D13 +
D3P2−3F2 =
D11 −
D12 +
D13 −
D3D3 =
D10 −
D15 (126)
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	Introduction and Historical Perspective
	QCD and the Nuclear Force
	Effective Field Theory for Low-Energy QCD 
	Symmetries of Low-Energy QCD
	Chiral Symmetry
	Explicit Symmetry Breaking
	Spontaneous Symmetry Breaking
	Chiral Effective Lagrangians Involving Pions  
	Nucleon Contact Lagrangians
	Nuclear Forces from EFT: Overview
	Chiral Perturbation Theory and Power Counting
	The Hierarchy of Nuclear Forces
	Two-Nucleon Forces
	Pion-Exchange Contributions in ChPT 
	Zeroth Order (LO) 
	Second Order (NLO) 
	Third Order (NNLO) 
	Fourth Order (N3LO) 
	Iterated One-Pion-Exchange 
	NN Scattering in Peripheral Partial Waves Using the Perturbative Amplitude 
	NN Contact Potentials 
	Zeroth Order
	Second Order
	Fourth Order
	Constructing a Chiral NN Potential 
	Conceptual Questions
	What Order?
	Charge-Dependence
	A Quantitative NN Potential at N3LO 
	Many-Nucleon Forces
	Three-Nucleon Forces
	Four-Nucleon Forces
	Conclusions
	Fourth Order Two-Pion Exchange Contributions
	One-loop diagrams
	ci2 contributions.
	ci/MN contributions.
	1/MN2 corrections.
	Two-loop contributions.
	Partial Wave Decomposition of the Fourth Order Contact Potential
