Yukawa's Pion, Low-Energy QCD and Nuclear Chiral Dynamics
arXiv:0704.1992v1  [nucl-th]  16 Apr 2007
Yukawa’s Pion, Low-Energy QCD and Nuclear Chiral Dynamics
Wolfram Weise
Physik-Department, Technische Universität München, D-85747 Garching, Germany
A survey is given of the evolution from Yukawa’s early work, via the understanding of
the pion as a Nambu-Goldstone boson of spontaneously broken chiral symmetry in QCD, to
modern developments in the theory of the nucleus based on the chiral effective field theory
representing QCD in its low-energy limit.
§1. The Beginnings
One of the most remarkable documentations in modern science history is the
series of articles which form the first volumes of Supplement of the Progress of
Theoretical Physics. These Collected Papers on Meson Theory1) celebrated the
twenty years’ anniversary of Yukawa’s pioneering work2) which laid the foundations
for our understanding of the strong force between nucleons. In those early days of
nuclear physics very little was known about this force. Heisenberg’s ”Platzwechsel”
interaction between proton and neutron gave a first intuitive picture, and some
unsuccessful attempts had been made to connect Fermi’s theory of beta decay with
nuclear interactions.
Yukawa’s 1935 article introduced the conceptual framework for a systematic
approach to the nucleon-nucleon interaction. His then postulated ”U field” did not
yet have the correct quantum numbers of what later became the pion∗), but it had
already some of its principal features. It provided the basic mechanism of charge
exchange between proton and neutron. From the form of the potential, e−µr/r,
and from the estimated range of the nuclear force, the mass of the U particle was
predicted to be about 200 times that of the electron. This U particle was erroneously
identified with the muon discovered in 1937 almost simultaneously by Anderson and
by Nishina and their collaborators. The ”real” π meson was discovered a decade
later3) in cosmic rays and then produced for the first time4) at the Berkeley cyclotron.
Its pseudoscalar nature was soon established. In 1949 the Nobel prize was awarded
to Yukawa.
In the footsteps of Yukawa’s original work, the decade thereafter saw an impres-
sively coherent effort by the next generation of Japanese theorists. A cornerstone of
these developments was the visionary conceptual design by Taketani et al.5) of the
inward-bound hierarchy of scales governing the nucleon-nucleon interaction, sketched
in Fig.1. The long distance region I is determined by one-pion exchange. It contin-
ues inward to the intermediate distance region II dominated by two-pion exchange.
The basic idea was to construct the NN potential in regions I and II by explicit
calculation of π and 2π exchange processes, whereas the detailed behaviour of the
∗) I am grateful to Professor H. Miyazawa for instructive communications on this point.
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1992v1
2 W. Weise
interaction in the short distance region III remains unresolved at the low-energy
scales characteristic of nuclear physics. This short distance part is given a suitably
parametrized form. The parameters are fixed by comparison with scattering data.
1 2 3
IIIIII
r [µ−1]
one-pion exchange 
two-pion 
exchange 
short 
distance   
one-pion
exchange
two-pion
exchange
Fig. 1. Hierarchy of scales governing the nucleon-nucleon potential (adapted from Taketani6)).
The distance r is given in units of the pion Compton wavelength µ−1 ≃ 1.4 fm.
Taketani’s picture, although not at all universally accepted at the time by the
international community of theorists, turned out to be immensely useful. Today
this strategy is the one conducted by modern effective field theory approaches. It is
amazing how far this program had already been developed by Japanese theorists in
the late fifties. One example is the pioneering calculation of the two-pion exchange
potential7) using dispersion relation techniques and early knowledge8) of the resonant
pion-nucleon amplitude which anticipated the ∆ isobar models of later decades.
§2. The Pion in the context of Low-Energy QCD
Today’s theory of the strong interaction is Quantum Chromodynamics (QCD).
There exist two limiting situations in which QCD is accessible with ”controlled” ap-
proximations. At momentum scales exceeding several GeV (corresponding to short
distances, r < 0.1 fm), QCD is a theory of weakly interacting quarks and gluons. At
low momentum scales considerably smaller than 1 GeV (corresponding to long dis-
tances, r > 1 fm), QCD is governed by color confinement and a non-trivial vacuum:
the ground state of QCD hosts strong condensates of quark-antiquark pairs and
gluons. Confinement implies the dynamical breaking of a global symmetry which is
exact in the limit of massless quarks: chiral symmetry. Spontaneous chiral symmetry
breaking in turn implies the existence of pseudoscalar Nambu-Goldstone bosons. For
two quark flavours (Nf = 2) with (almost) massless u and d quarks, these Goldstone
bosons are identified with the isospin triplet of pions. At low energy, Goldstone
bosons interact weakly with one another or with any massive particles. Low-energy
QCD is thus realized as an effective field theory in which these Goldstone bosons are
the active, light degrees of freedom.
2.1. Chiral Symmetry and the NJL Model
How does Yukawa’s pion figure in the frame of QCD? Historically, the founda-
tions for understanding the pion as a Nambu-Goldstone boson9), 10) of spontaneously
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 3
broken chiral symmetry were initiated in the 1960’s, culminating in the PCAC and
current algebra approaches11) of the pre-QCD era. A most inspiring work from this
early period is the one by Nambu and Jona-Lasinio10)(NJL). Just as the BCS model
provided an understanding of the basic mechanism behind superconductivity, the
NJL model helped clarifying the dynamics that drives spontaneous chiral symmetry
breaking and the formation of pions as Goldstone bosons.
Consider as a starting point the conserved color current of quarks, Jaµ = ψ̄γµt
where ta (a = 1, ..., 8) are the generators of the SU(Nc = 3) gauge group and ψ are
the quark fields with 4NcNf components representing their spin, color and flavor
degrees freedom. This current couples to the gluon fields. Any two quarks interact
through multiple exchanges of gluons. Assume that the distance over which color
propagates is restricted to a short correlation length lc. Then the interaction expe-
rienced by low-momentum quarks can be schematically viewed as a local coupling
between their color currents:
Lint = −Gc Jaµ(x)Jµa(x) , (2.1)
where Gc ∼ ḡ2 l2c is an effective coupling strength of dimension length2 which encodes
the QCD coupling, averaged over the relevant distance scales, in combination with
the squared correlation length, l2c .
Now adopt the local interaction (2.1) and write the following model Lagrangian
for the quark fields ψ(x):
L = ψ̄(x)(iγµ∂µ −m0)ψ(x) + Lint(ψ̄, ψ) . (2.2)
In essence, by ”integrating out” gluon degrees of freedom and absorbing them in
the four-fermion interaction Lint, the local SU(Nc) gauge symmetry of QCD is now
replaced by a global SU(Nc) symmetry. Confinement is obviously lost, but all other
symmetries of QCD are maintained. The mass matrix m0 incorporates small ”bare”
quark masses. In the limit m0 → 0, the Lagrangian (2.2) has a chiral symmetry of
right- and left-handed quarks, SU(Nf )R×SU(Nf )L, that it shares with the original
QCD Lagrangian for Nf massless quark flavors.
A Fierz transform of the color current-current interaction (2.1) produces a set
of exchange terms acting in quark-antiquark channels. For the Nf = 2 case:
Lint →
(ψ̄ψ)2 + (ψ̄ iγ5 τ ψ)
+ ... , (2.3)
with the isospin SU(2) Pauli matrices τ = (τ1, τ2, τ3). For brevity we have not
shown a series of terms with combinations of vector and axial vector currents, both
in color singlet and color octet channels. The constant G is proportional to the color
coupling strength Gc. The ratio of these two constants is uniquely determined by
Nc and Nf .
The steps just outlined can be viewed as a contemporary way of introducing
the time-honored NJL model10) . This model has been further developed and ap-
plied12), 13) to a variety of problems in hadron physics. The virtue of this schematic
model is its simplicity in illustrating the basic mechanism of spontaneous chiral sym-
metry breaking, as follows. In the mean-field (Hartree) approximation the equation
4 W. Weise
of motion derived from the Lagrangian (2.2) leads to a gap equation
M = m0 −G〈ψ̄ψ〉 , (2.4)
which links the dynamical generation of a constituent quark mass M to the appear-
ance of the chiral quark condensate
〈ψ̄ψ〉 = −Tr lim
x→ 0+
〈T ψ(0)ψ̄(x)〉 = −2iNfNc
(2π)4
M θ(Λ2 − ~p 2)
p2 −M2 + iε
. (2.5)
This condensate plays the role of an order parameter of spontaneous chiral symme-
try breaking. Starting from m0 = 0 a non-zero quark mass develops dynamically,
together with a non-vanishing chiral condensate, once G exceeds a critical value of
order Gcrit ∼ 10 GeV−2. The procedure requires a momentum cutoff Λ ≃ 2M be-
yond which the interaction is ”turned off”. Note that the strong non-perturbative
interactions, by polarizing the vacuum and turning it into a condensate of quark-
antiquark pairs, transmute an initially pointlike quark with its small bare mass m0
into a massive quasiparticle with a size of order (2M)−1.
2.2. The Pseudoscalar Meson Spectrum
The NJL model demonstrates lucidly the appearance of chiral Nambu-Goldstone
bosons. Solving Bethe-Salpeter equations in the color singlet quark-antiquark chan-
nels generates the lightest mesons as quark-antiquark excitations of the correlated
QCD ground state with its condensate structure. Several such calculations have been
performed in the past with Nf = 3 quark flavors
12)–14) . Such a model has an un-
wanted U(3)R×U(3)L symmetry to start with, but the axial U(1)A anomaly reduces
this symmetry to SU(3)R × SU(3)L × U(1)V . In QCD, instantons are considered
responsible for U(1)A breaking. In the NJL model, these instanton driven interac-
tions are incorporated in the form of a flavor determinant15) det[ψ̄i(1± γ5)ψj ]. This
interaction involves all three flavors u, d, s simultaneously in a genuine three-body
term.
Fig. 2. Symmetry breaking pattern in the pseudoscalar meson nonet calculated in the three-flavor
NJL model14) .
The symmetry breaking pattern resulting from such a calculation is apparent
in the pseudoscalar meson spectrum of Fig.2. Starting from massless u, d and s
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 5
quarks, the pseudoscalar octet emerges as a set of massless Goldstone bosons of
spontaneously broken SU(3) × SU(3), while U(1)A breaking drives the singlet η0
away from the Goldstone boson sector. Finite quark masses shift the Jπ = 0− nonet
into its empirically observed position, including η-η′ mixing.
The very special nature of the pion as a Nambu-Goldstone boson is manifest in
a famous relation16) derived from current algebra and PCAC:
m2π f
π = −
(mu +md)〈ψ̄ψ〉+O(m2u,d). (2.6)
It involves the pion decay constant, fπ ≃ 0.09 GeV, defined by the matrix element
which connects the pion with the QCD vacuum via the axial vector current, A
ψ̄γµγ5
〈0|Aµi (x = 0)|πi(p)〉 = ip
µfπ . (2.7)
Just like the chiral condensate, the pion decay constant is a measure of spontaneous
chiral symmetry breaking expressed in terms of a characteristic scale 4πfπ ∼ 1 GeV.
The non-zero pion mass, mπ ∼ 0.14 GeV ≪ 4πfπ, reflects the explicit symmetry
breaking by the small quark masses, with m2π ∼ mq. One should note that the
quark masses mu,d and the condensate 〈ψ̄ψ〉 are both scale dependent quantities.
Only their product is scale independent, i.e. invariant under the renormalization
group. At a renormalization scale of about 1 GeV, a typical average quark mass
(mu +md) ≃ 7 MeV implies |〈ψ̄ψ〉| ≃ (0.3 GeV)3.
2.3. Scales and Symmetry Breaking Patterns
The quark masses are the only parameters that set primary scales in QCD. Their
classification into sectors of ”light” and ”heavy” quarks determines very different
physics phenomena. While the heavy quarks (i.e. the t, b and - within limits - the c
quarks) offer a natural ”small parameter” in terms of their reciprocal masses, such
that non-relativistic approximations (expansions of observables in powers of m−1t,b,c)
tend to work increasingly well with increasing quark mass, the sector of the light
quarks (i.e. the u, d quarks and - to some extent - the s quark) is governed by
quite different principles and rules. Evidently, the quark masses themselves are now
”small parameters”, to be compared with a characteristic ”large” scale of dynamical
origin. This large scale is visible as a characteristic mass gap of about 1 GeV which
separates the QCD vacuum from almost all of its excitations, with the exception of
the pseudoscalar meson octet of pions, kaons and the eta meson. This mass gap is
in turn comparable to 4πfπ, the scale associated with spontaneous chiral symmetry
breaking in QCD.
In this context it is also instructive to have a look at the spectroscopic pattern
of pseudoscalar and vector mesons, starting from heavy-light quark-antiquark pairs
in 1S0 and
3S1 states and following those states downward in the mass of the quark.
This is illustrated in Fig.3 where we show the masses of mesons composed of a b,
c, s or u quark with an anti-d-quark attached. Bare quark masses are subtracted
from the meson masses in this plot in order to directly demonstrate the evolution
from perturbative hyperfine splitting in the heavy systems to the non-perturbative
mass gap in the light ones. In the B̄ and B̄∗ mesons, the d̄ quark is tightly bound
6 W. Weise
Fig. 3. Evolution of the splitting between spin singlet (lower) and triplet (upper) quark-antiquark
states (the pseudoscalar (Jπ = 0−) and vector (Jπ = 1−) mesons) with varying mass of one
of the quarks. The bare quark masses are subtracted from the physical meson masses for
convenience of demonstration.
to the heavy b quark at small average distance, within the range where perturbative
QCD is applicable. The spin-spin interaction is well approximated by perturbative
one-gluon exchange, resulting in a small hyperfine splitting. Moving downward in
mass to the D and D∗ systems, with the b quark replaced by a c quark, the hyperfine
splitting increases but remains perturbative in magnitude. As this pattern evolves
further into the light-quark sector, it undergoes a qualitative change via the large
mass difference of K̄ and K̄∗ to the non-perturbative mass gap in the π− ρ system,
reflecting the Goldstone boson nature of the pion.
2.4. Chiral Effective Field Theory
Low-energy QCD is the physics of systems of light quarks at energy and momen-
tum scales smaller than the 1 GeV mass gap observed in the hadron spectrum. This
1 GeV scale set by 4πfπ offers a natural separation between ”light” and ”heavy”
(or, correspondingly, ”fast” and ”slow”) degrees of freedom. The basic idea of an
effective field theory is to introduce the active light particles as collective degrees of
freedom, while the heavy particles are frozen and treated as (almost) static sources.
The dynamics is described by an effective Lagrangian which incorporates all rele-
vant symmetries of the underlying fundamental theory. In QCD, confinement and
spontaneous chiral symmetry breaking implies that the ”fast” degrees of freedom are
the Nambu-Goldstone bosons. With Yukawa’s pion in mind, we restrict ourselves to
Nf = 2.
We first briefly summarize the steps17), 18) required in the pure meson sector
(baryon number B = 0) and later for the pion-nucleon sector (B = 1). A chiral field
is introduced as
U(x) = ei τi πi(x)/fπ ∈ SU(2) , (2.8)
with the Goldstone pion fields πi(x) normalized by the pion decay constant fπ taken
in the chiral limit (mπ → 0). The QCD Lagrangian is replaced by an effective
Lagrangian which involves U(x) and its derivatives:
LQCD → Leff (U, ∂µU, ...). (2.9)
Goldstone bosons interact only when they carry non-zero momentum, so the low-
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 7
energy expansion of (2.9) is an ordering in powers of ∂µU . Lorentz invariance permits
only even numbers of derivatives. One writes
Leff = L(2) + L(4) + ... , (2.10)
omitting an irrelevant constant. The leading term (the non-linear sigma model)
involves two derivatives:
L(2) = f
Tr[∂µU
†∂µU ]. (2.11)
At fourth order, the terms permitted by symmetries are (apart from an extra con-
tribution from the QCD anomaly, not included here):
L(4) =
(Tr[∂µU
†∂µU ])2 +
Tr[∂µU
†∂νU ]Tr[∂
µU †∂νU ], (2.12)
and so forth. The constants l1, l2 (following canonical notations
18)) must be deter-
mined by experiment.
The symmetry breaking mass term is small, so that it can be handled pertur-
batively, together with the power series in momentum. The leading contribution
introduces a term linear in the quark mass matrix m = diag(mu,md):
L(2) = f
Tr[∂µU
†∂µU ] +
B0 Tr[m(U + U
†)], (2.13)
with B0 = −〈ψ̄ψ〉/2fπ The fourth order term L(4) also receives symmetry breaking
contributions with additional constants li.
To the extent that the effective Lagrangian includes all terms dictated by sym-
metries of QCD, the chiral effective field theory is the low-energy equivalent19), 20)
of the original QCD Lagrangian. Given the effective Lagrangian, the framework
for systematic perturbative calculations of the S-matrix involving Goldstone bosons,
Chiral Perturbation Theory (ChPT), is then defined by the following rules:
Collect all Feynman diagrams generated by Leff . Classify all terms according to
powers of a small quantity Q which stands generically for three-momentum or energy
of the Goldstone boson, or for the pion mass mπ. The small expansion parameter is
Q/4πfπ. Loops are subject to dimensional regularization and renormalization.
2.5. Pion-Pion Scattering
When using the Gell-Mann, Oakes, Renner (GOR) relation (2.6) to leading
order in the quark mass, it is tacitly assumed that the chiral condensate is large in
magnitude and plays the role of an order parameter for spontaneous chiral symmetry
breaking. This basic scenario needs to be confirmed. It has in fact been tested by
a detailed quantitative analysis of pion-pion scattering, the process most accurately
and extensively studied using ChPT.
Consider s-wave ππ scattering at very low energy. The scattering lengths in the
isospin I = 0, 2 channels, calculated to leading chiral order, are21)
32πf2π
, a2 = −
16πf2π
, (2.14)
8 W. Weise
showing that the ππ interaction properly vanishes in the chiral limit, mπ → 0. The
next-to-leading order introduces one-loop iterations of the leading L(2) part of the
effective Lagrangian as well as pieces generated by L(4). At that level enters the
renormalized constant l̄3 which also determines the correction to the leading-order
GOR relation:
m2π =
32π2f2
π + O(
π) , (2
where
π = −
mu +md
〈ψ̄ψ〉 (2.16)
involves the quark mass in leading order. Here f is the pion decay constant in the
chiral limit. An accurate determination of the I = 0 s-wave ππ scattering length
therefore provides a constraint for l̄3 which in turn sets a limit for the next-to-leading
order correction to the GOR relation.
Such an investigation has been performed22) using low-energy ππ phase shifts
extracted from the detailed final state analysis of the K → ππ + eν decay. The
result, when translated into a statement about the non-leading term entering (2.15),
implies that the difference between m2π and the leading GOR expression (2
.16) is
less than 5 percent. Hence the “strong condensate” scenario of spontaneous chiral
symmetry breaking in QCD appears to be confirmed∗).
2.6. The Pion in Lattice QCD
The leading-order relationship m2π ∼ mq is also observed23) in lattice QCD
up to surprisingly large quark masses. A detailed recent analysis24) is shown in
Fig.4. Within statistical errors, the data for squared pion mass versus quark mass
lie on a straight. The lattice results are remarkably compatible with one-loop chiral
perturbation theory up to mπ . 0.5 GeV.
Fig. 4. Lattice QCD simulation results24) for the squared pion mass m2π as function of the quark
mass m in units of the lattice spacing a. Pion masses converted to physical units are attached
to the lattice data points. A linear fit (dashed) is shown in comparison with the next-to-leading
order ChPT result (solid curve).
∗) One should note, however, that this conclusion is drawn at the level of QCD with only Nf = 2
flavours. Additional corrections may still arise when strange quarks are taken into account.
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 9
2.7. Pion-Nucleon Effective Lagrangian
The prominent role played by the pion as a Goldstone boson of spontaneously
broken chiral symmetry has its impact on the low-energy structure and dynamics
of nucleons as well.25) When probing the nucleon with long-wavelength electroweak
and strong fields, a substantial part of the response comes from the pion cloud, the
“soft” surface of the nucleon. The calculational framework for this, baryon chiral
perturbation theory26), 27) has been applied quite successfully to a variety of low-
energy processes (such as low-energy pion-nucleon scattering, threshold pion photo-
and electroproduction and Compton scattering on the nucleon).
Consider now the sector with baryon number B = 1 and the physics of the pion-
nucleon system. The nucleon is represented by an isospin-1/2 doublet, Dirac spinor
field ΨN (x) = (p, n)
T of proton and neutron. The free field Lagrangian
LN0 = Ψ̄N (iγµ∂µ −M0)ΨN (2.17)
includes the nucleon mass in the chiral limit, M0. One should note that the nucleon,
unlike the pion, has a large mass of the same order as the chiral symmetry breaking
scale 4πfπ, which survives in the limit of vanishing bare quark masses, mu,d → 0.
The previous pure meson Lagrangian Leff is now replaced by Leff (U, ∂µU,ΨN , ...)
which also includes the nucleon field. The additional term involving the nucleon, de-
noted by LNeff , is expanded again in powers of derivatives (external momenta) of the
Goldstone boson field and of the quark masses:
LNeff = L
πN + L
πN ... (2
In the leading term, L(1)πN there is a replacement of ∂µ by a chiral covariant deriva-
tive which introduces vector current couplings between the pions and the nucleon.
Secondly, there is an axial vector coupling. This structure of the πN effective La-
grangian is again dictated by chiral symmetry. We have
L(1)πN = Ψ̄N [iγµ(∂
µ − iVµ) + γµγ5 Aµ −M0]ΨN , (2.19)
with vector and axial vector quantities involving the Goldstone boson (pion) fields
in the form ξ =
Vµ = i
(ξ†∂µξ + ξ∂µξ†) = − 1
εabcτa πb ∂
µπc + ... , (2.20)
(ξ†∂µξ − ξ∂µξ†) = −
µπa + ... , (2.21)
where the last steps result when expanding Vµ and Aµ to leading order in the pion
fields. So far, the only parameters that enter are the nucleon mass,M0, and the pion
decay constant, fπ, both taken in the chiral limit.
The nucleon has its own intrinsic structure which leads to a modification of the
axial vector coupling term in (2.19). The analysis of neutron beta decay reveals that
the γµγ5 term is to be multiplied by the axial vector coupling constant gA, with the
empirical value gA ≃ 1.27.
10 W. Weise
At next-to-leading order (L(2)πN ), the symmetry breaking quark mass term enters.
It has the effect of shifting the nucleon mass from its value in the chiral limit to the
physical one:
MN =M0 + σN . (2.22)
The sigma term
σN = mq
= 〈N |mq(ūu+ d̄d)|N〉 (2.23)
measures the contribution of the non-vanishing quark mass, mq =
(mu+md), to the
nucleon mass MN . Its empirical value is in the range σN ≃ (45 − 55) MeV and has
been deduced37) by a sophisticated extrapolation of low-energy pion-nucleon data
using dispersion relation techniques. Up to this point, the πN effective Lagrangian,
expanded to second order in the pion field, has the form
LNeff = Ψ̄N (iγµ∂µ −MN )ΨN −
Ψ̄Nγµγ5 τ ΨN · ∂µπ (2.24)
Ψ̄Nγµ τ ΨN · π × ∂µπ +
Ψ̄NΨN π
2 + ... ,
where we have not shown a series of additional terms of order (∂µπ)2 included in
the complete L(2)πN . These terms come with further low-energy constants encoding
physics at smaller distances and higher energies. These constants need to be fitted
to experimental data, e.g. from pion-nucleon scattering.
The “effectiveness” of such an effective field theory relies on the proper identifi-
cation of the active low-energy degrees of freedom. Pion-nucleon scattering is known
to be dominated by the p-wave ∆(1232) resonance with spin and isospin 3/2. The
excitation energy of this resonance, given by the mass difference δM =M∆−MN , is
not large, just about twice the pion mass. If the physics of the ∆(1232) is absorbed
in low-energy constants of an effective theory that works with pions and nucleons
only (as commonly done in heavy-baryon ChPT), the limits of applicabilty of such
a theory is clearly narrowed down to an energy-momentum range small compared
to δM . The B = 1 chiral effective Lagrangian is therefore often extended28) by
incorporating the ∆ isobar as an explicit degree of freedom.
§3. Chiral Thermodynamics and Goldstone Bosons in Matter
Before turning to chiral dynamics in nuclear many-body systems, it is instructive
to make a brief digression and touch upon more general issues of chiral symmetry at
finite temperature and non-zero baryon density.
3.1. The Chiral Order Parameter
As outlined in the previous sections, the QCD ground state (the vacuum) is
characterized by the presence of the strong chiral condensate 〈ψ̄ψ〉. The light hadrons
are quasiparticle excitations of this condensed ground state, with Yukawa’s pion
playing a very special role as Nambu-Goldstone boson of spontaneously broken chiral
symmetry. A key question33) is then the following: how do the basic quantities and
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 11
scales associated with this symmetry breaking pattern (the chiral condensate, the
pion mass and decay constant) evolve with changing thermodynamical conditions
(temperature, baryon density)?
Assume a homogenous hadronic medium in a volume V at temperature T and
consider the pressure
P (T, V, µ) =
lnZ = T
ln Tr exp
d3x (H − µρ)
. (3.1)
Here µ denotes the chemical potential, ρ the baryon density. The Hamiltonian
density H of QCD is expressed in terms of the relevant degrees of freedom in the
hadronic phase, derived from the chiral effective Lagrangian Leff . The Nf = 2
Hamiltonian has a mass term, δH = ψ̄mψ = mu ūu+md d̄d, so that H = H0 + δH,
with H0 representing the massless limit. Now take the derivative of the pressure with
respect to the quark mass and use the GOR relation (2.6) to derive the condensate
〈ψ̄ψ〉T,ρ at finite T and density ρ = ∂P/∂µ, or rather its ratio with the condensate
at T = µ = 0,
〈ψ̄ψ〉T,ρ
〈ψ̄ψ〉0
= 1 +
dP (T, µ)
f2π dm
. (3.2)
The T dependence of this condensate, at zero chemical potential, is shown in compar-
ison with two-flavor lattice QCD results in Fig.5. Its behaviour reflects a continuous
crossover transition at a critical temperature Tc ∼ 0.2 GeV which turns into a sec-
ond order phase transition in the chiral limit of massless quarks. Above Tc chiral
symmetry is restored and the pion stops being realized as a Nambu-Goldstone mode.
The chiral condensate therefore has the features of an order parameter. However, it
is not directly observable. A related measurable quantity is the pion decay constant.
Its temperature and density dependence is in fact an indicator of tendencies towards
chiral symmetry restoration, in the following sense.
The GOR relation (2.6) continues to hold in matter at finite temperature T <
Tc and density ρ, when reduced to a statement about the time component, A
ψ†γ5(τa/2)ψ, of the axial current. We can introduce the in-medium pion decay
constant, f∗π(T, ρ) through the thermal matrix element 〈 |A0|π〉T,ρ , the in-medium
analogue of eq.(2.7). One finds
f∗π(T, ρ)
2m∗π(T, ρ)
2 = −
mu +md
〈ψ̄ψ〉T,ρ + ... (3.3)
to leading order in the quark mass. The in-medium pion mass m∗π (more precisely:
the average of the π+ and π− masses) is protected by the pion’s Goldstone boson
nature and not much affected by the thermal environment. The “melting” of the
condensate by heat or compression translates primarily to the in-medium change of
the pion decay constant.
The leading behaviour32), 33) of the pion condensate and, consequently, of the
pion decay constant, with increasing temperature and density is:
f∗π(T, ρ)
〈ψ̄ψ〉T,ρ
〈ψ̄ψ〉0
= 1− T
8 f2π
m2π f
ρ+ ... . (3.4)
12 W. Weise
Fig. 5. Temperature dependence of the chi-
ral condensate at zero chemical potential.
The curve results from a calculation30)
based on an extended NJL model with
inclusion of Polyakov loop dynamics (the
PNJL model). The data points areNf = 2
lattice QCD results taken from ref.31) .
Fig. 6. Pion decay constant as function of
temperature T and baryon density ρ. Cal-
culation29) based on the PNJL model.
Normal nuclear matter density ρ0 = 0.16
fm−3 is indicated for orientation.
A typical result for the in-medium behavior of the pion decay constant is displayed
in Fig.6. It should be noted that the dropping of the condensate’s magnitude with
density is significantly more pronounced than its temperature dependence. The
decreasing “chiral gap” 4πf∗π(T, ρ) with changing thermodynamic conditions should
thus imply observable changes in the low-energy dynamics of pions in dense matter.
3.2. Low-Energy Pion-Nucleus Interactions
Goldstone’s theorem implies that low-momentum pions interact weakly. This
is generally true also for low-momentum pions interacting with nuclear many-body
systems. As a starting point, consider homogeneous nuclear matter at zero temper-
ature with proton density ρp and neutron density ρn. A pion wave in matter has its
energy ω and momentum ~q related by the dispersion equation
ω2 − ~q 2 −m2π −Π(ω, ~q ; ρp, ρn) = 0 . (3.5)
The polarization function, or pion self-energy Π, summarizes all interactions of the
pion with the medium. At low densities,
Π(±)(ω, ~q ; ρp, ρn) = −T+(ω, ~q ) ρ± T−(ω, ~q ) δρ , (3.6)
in terms of the isospin-even (T+) and isospin-odd (T−) pion-nucleon forward scat-
tering amplitudes, with ρ = ρp + ρn and δρ = ρp − ρn. We have now specified the
self-energies Π(±) for a π+ or π−, respectively.
Applications to finite systems, in particular for low-energy pion-nucleus interac-
tions34), 35) relevant to pionic atoms, commonly make use of an energy-independent
effective potential. Such an equivalent potential is constructed36) by expanding the
polarization function for ω −mπ ≪ mπ and |~q |2 ≪ m2π around the physical thresh-
old, ω = mπ and |~q | = 0. By comparison with the Klein-Gordon equation for the
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 13
pion wave function φ(~r ) in coordinate space,
ω2 −m2π + ~∇2 − 2mπU(~r )
φ(~r ) = 0 , (3.7)
the (energy-independent) potential U(~r ) is identified as follows:
2mπU(~r ) =
)−1 [
Π(mπ, 0) − ~∇
∂~q 2
, (3.8)
with all derivatives taken at the threshold point. The wave function renormalization
factor (1 − ∂Π/∂ω2)−1 encodes the energy dependence of the polarization function
Π(ω, ~q ) in the equivalent energy-independent potential (3.8). This potential is ex-
pressed in terms of local density distributions ρp,n(~r ) for protons and neutrons, and
the standard prescription ~q 2f(ρ) → −~∇f(ρ(~r ))~∇ is used for the ~q 2- dependent
parts. In practical calculations of pionic atoms, the Coulomb potential Vc is in-
troduced by replacing ω → ω − Vc(~r ), and corrections of higher order beyond the
leading terms (3.6), resulting from double scattering and absorption, are added.
3.3. Deeply Bound States of Pionic Atoms
Accurate data on 1s states of a negatively charged pion bound to Pb and Sn
isotopes41), 42) have set new standards and constraints for the detailed analysis of
s-wave pion interactions with nuclei. Such deeply bound pionic states owe their
existence, with relatively long lifetimes, to a subtle balance between the attractive
Coulomb force and the repulsive strong π−-nucleus interaction in the bulk of the
nucleus. As a consequence, the 1s wave function of the bound pion is pushed toward
the edge of the nuclear surface. Its overlap with the nuclear density distribution is
small, so that the standard π−pn→ nn absorption mechanism is strongly suppressed.
The topic of low-energy, s-wave pion-nucleus interactions has a long34), 35) history.
Inspired by the measurements of deeply bound pionic atoms it has recently been
re-investigated38) from the point of view of the distinct energy dependence of the
pion-nuclear polarization operator in calculations based on systematic in-medium
chiral perturbation theory39), 40) .
Consider a negatively charged pion interacting with nuclear matter and recall the
π− self-energy from Eq.(3.6). In the long-wavelength limit (~q → 0), chiral symmetry
(the Tomozawa-Weinberg low-energy theorem) implies T−(ω) = ω/(2f2π) + O(ω3).
Together with the observed approximate vanishing of the isospin-even threshold am-
plitude T+(ω = mπ), it is clear that 1s states of pions bound to heavy, neutron rich
nuclei are a sensitive source of information for in-medium chiral dynamics. Terms
of non-leading order in density (double scattering (Pauli) corrections of order ρ4/3,
absorption effects of order ρ2 etc.) are important and systematically incorporated.
Absorption effects and corresponding dispersive corrections appear at the three-loop
level and through short-distance dynamics parametrized by πNN contact terms, not
explicitly calculable within the effective low-energy theory. The imaginary parts as-
sociated with these terms are well constrained by the systematics of observed widths
of pionic atom levels throughout the periodic table.
With these ingredients the Klein-Gordon equation for the nuclear pion field has
been solved with the explicitly energy dependent pion self-energy just described. As
14 W. Weise
an example we show predictions38) for binding energies and widths for pionic 1s states
bound to a series of Sn isotopes. These calculations include a careful assessment of
uncertainties in neutron distributions. Results are shown in Fig.7 in comparison
with experimental data.41)
Fig. 7. Binding energies (upper pannel) and widths (lower pannel) of pionic 1s states in Sn iso-
topes. The curves show predictions38) based on the explicitly energy dependent pionic s-wave
polarization operator calculated in two-loop in-medium chiral perturbation theory. Upper and
lower curves give an impression of uncertainties related to the πN sigma term. Data from ref.41)
The question has been raised43), 44) whether one can actually ”observe” finger-
prints of (partial) chiral symmetry restoration in the high-precision data of deeply
bound pionic atoms. Pionic atom calculations are usually done with energy inde-
pendent phenomenological optical potentials instead of explicitly energy dependent
pionic polarisation functions. The connection is provided by Eq.(3.8). Consider
a zero momentum π− in low density matter. Its energy dependent leading-order
polarisation operator is Π(ω) = −T+(ω) (ρp + ρn) + T−(ω) (ρn − ρp) , and the in-
medium dispersion equation at ~q = 0 is ω2−m2π −Π(ω) = 0 . The chiral low-energy
expansion of the off-shell amplitude T+(ω) at ~q = 0 implies leading terms of the
form T+(ω) = (σN − β ω2)/f2π , with β ≃ σN/m2π required to yield the empirical
T+(ω = mπ) ≃ 0. Using Eq.(3.8) one finds for the effective (energy-independent)
s-wave potential
ρn − ρp
4 f2π
1− σN ρ
m2π f
ρn − ρp
4 f∗2π (ρ)
, (3.9)
with the replacement fπ → f∗π(ρ) of the pion decay constant representing the in-
medium wave function renormalization. The expression (3.9) is just the one proposed
previously in ref.43) based on the relation (3.4) between the in-medium changes of the
chiral condensate and the pion decay constant associated with the time component
of the axial current. The explicitly energy dependent chiral dynamics represented
by Π(ω) ”knows” about these renormalization effects. Their translation into an
equivalent, energy-independent potential implies fπ → f∗π(ρ) as given in eq. (3.9).
This heuristic reasoning has recently been underlined by a more profound derivation
in ref.45) .
The analysis of the deeply bound pionic atom data41) along these lines comes to
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 15
the conclusion that, when extrapolated to nuclear matter density ρ0 = 0.16 fm
f∗π(ρ0) ≃ 0.8 fπ , (3.10)
which is compatible with the theoretical prediction
f∗π(ρ0)
2m2π f
ρ0 , (3.11)
assuming σN ≃ 50 MeV. It is quite remarkable that an optical potential fit to recent
precision measurements of π+ and π− differential cross sections at the lowest possible
energy (Tπ = 21.5 MeV) on a variety of nuclei reaches a similar conclusion,
46) namely
f∗π(ρ0) ≃ 0.83 fπ , although within a different procedure. With the interpretation
(3.9), the tendency towards chiral restoration in a nuclear medium as suggested by
Eq.(3.4) appears to be - at least qualitatively - visible in low-energy pion-nucleus
interactions.
§4. Nuclear Chiral Dynamics
We now approach a basic question at the origin of modern nuclear physics:
is there a path from QCD via its low-energy representation, chiral effective field
theory, to the observed systematics of the nuclear chart? Or equivalently: how does
Yukawa’s pion and its realization as a Nambu-Goldstone boson figure in the nuclear
many-body problem?
Pionic degrees of freedom in nuclei have been in the focus right from the begin-
nings. The field of exchange currents in nuclei, with the pion as the prime agent, was
started already in the early fifties47) and investigated in great breadth in the seven-
ties. An instructive overview of these developments is given in volume II of ref.48)
The important role played by one-pion exchange and its strong tensor force in the
deuteron49) is a long known and well established fact. The description of radiative
np capture (n + p → d + γ) in terms of the magnetic pion exchange current50) was
a key to establishing the pion as an observable degree of freedom in the deuteron.
For heavier nuclei, on the other hand, the role of the pion is not so directly
evident. One-pion exchange does not contribute to the bulk nuclear (Hartree) mean
field when averaged over nucleon spins. Fock exchange terms involving one-pion
exchange are relatively small. The role of pions in binding the nucleus is manifest
primarily in the intermediate range attractive force generated by two-pion exchange
processes. For decades, nuclear mean field models preferred to replace the complexity
of such processes by a phenomenological scalar-isoscalar “sigma” field, although the
more detailed treatment of the two-pion exchange nucleon-nucleon interaction had
been known before (see e.g. ref.51) and volume I of ref.48)). Recent developments
return to these basics by introducing chiral effective field theory as a systematic
framework for the treatment of NN interactions and nuclear systems, following ref.52)
For an updated review see ref.53)
16 W. Weise
4.1. In-medium chiral perturbation theory and nuclear matter
In nuclear matter the relevant momentum scale is the Fermi momentum kF .
Around the empirical saturation point with k
F ≃ 0.26 GeV ∼ 2mπ, the Fermi
momentum and the pion mass are scales of comparable magnitude. This implies that
at the densities of interest in nuclear physics, ρ ∼ ρ0 = 2(k
3/3π2 ≃ 0.16 fm−3 ≃
0.45m3π , pions must be included as explicit degrees of freedom: their propagation in
matter is ”resolved” at the relevant momentum scales around the Fermi momentum.
At the same time, kF and mπ are small compared to the characteristic chiral
scale, 4πfπ ∼ 1 GeV. Consequently, methods of chiral perturbation theory are ex-
pected to be applicable to nuclear matter at least in a certain window around k
In that range, the energy density
E(kF ) =
E(kF )
ρ . (4.1)
should then be given as a convergent power series in the Fermi momentum. This is
the working hypothesis. More precisely, the energy per particle has an expansion
E(kF )
Fn(kF /mπ) knF . (4.2)
The expansion coefficients Fn are in general non-trivial functions of kF /mπ, the
dimensionless ratio of the two relevant scales. These functions must obviously not
be further expanded. Apart from kF and mπ, a third relevant “small” scale is the
mass difference δM =M∆ −MN ≃ 0.3 GeV between the ∆(1232) and the nucleon.
The strong spin-isospin transition from the nucleon to the ∆ isobar is therefore to be
included as an additional important ingredient in nuclear many-body calculations,
so that the Fn become functions of both kF /mπ and mπ/δM .
In-medium chiral perturbation theory is the framework for treating pion ex-
change processes in the presence of a filled Fermi sea of nucleons. The chiral pion-
nucleon effective Lagrangian, with its low-energy constants constrained by pion-
nucleon scattering observables in vacuum, is used to construct the hierarchy of NN
interaction terms as illustrated in Fig.8. One- and two-pion exchange processes (as
well as those involving low-energy particle-hole excitations) are treated explicitly.
They govern the long-range interactions at distance scales d > 1/kF fm relevant to
the nuclear many-body problem, whereas short-range mechanisms, with t-channel
spectral functions involving masses far beyond those of two pions, are not resolved
in detail at nuclear Fermi momentum scales and can be subsumed in contact interac-
tions and derivatives thereof. This “separation of scales” argument makes strategies
of chiral effective field theory work even for nuclear problems, with the “small” scales
(kF ,mπ, δM) distinct from the “large” ones (4πfπ,MN ). In essence, this is the mod-
ern realization of Taketani’s programme mentioned in the beginning. Closely related
renormalization group considerations have motivated the construction of a universal
low-momentum NN interaction V(low k)54) from phase shift equivalent NN potentials
such that the ambiguities associated with unresolved short-distance parts disappear.
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 17
Fig. 8. NN amplitude in chiral effective field
theory: (upper:) one-pion exchange, two-
pion exchange (including ∆ isobar inter-
mediate states) and (lower:) contact terms
representing short-distance dynamics.
Fig. 9. Energy density from in-medium chi-
ral perturbation theory at three-loop or-
der. Dashed lines show pions. Each
(solid) nucleon line means insertion of the
in-medium propagator (4.4). Two- and
three-body terms involving contact inter-
actions are also shown.
The two-pion exchange interaction has as its most prominent pieces the second
order tensor force and intermediate ∆(1232) states which reflect the strong spin-
isospin polarizability of the nucleon. The latter produces a Van der Waals - like NN
interaction. At long and intermediate distances it behaves as55)
V2π(r) ∼ −
e−2mπr
P (mπr) , (4.3)
where P is a polynomial in mπr. In the chiral limit (mπ → 0), this V2π approaches
the characteristic r−6 dependence of a non-relativistic Van der Waals potential.
The two-pion exchange force is the major source of intermediate range attraction
that binds nuclei. This is, of course, not a new observation. For example, the
important role of the second-order tensor force from iterated pion exchange had
been emphasised long ago,56) as well as the close connection of the nuclear force to
the strong spin-isospin polarizability of the nucleon.57) The new element that has
entered the discussion more recently is the systematics provided by chiral effective
field theory in dealing with these phenomena.
With these ingredients, in-medium ChPT calculations of nuclear and neutron
matter have been performed58), 59) up to three-loop order in the energy density, as
illustrated in Fig.9. Each nucleon line in these diagrams stands for the in-medium
propagator
(γ · p+MN )
p2 −M2N + iε
− 2πδ(p2 −M2N )θ(p0)θ(kF − |~p |)
. (4.4)
The regularization of some divergent loops introduces a scale which is balanced by
counter terms (contact interactions) so that the result is independent of this reg-
ularization scale.60) A limited, small number of constants in these contact terms
18 W. Weise
must be adjusted to empirical information such as the equilibrium density of nuclear
matter. Stabilization and saturation of nuclear matter at equilibrium is achieved
in a non-trivial and model-independent way: the Pauli principle acting on nucleon
intermediate states in two-pion exchange processes produces a repulsive term pro-
portional to ρ4/3 in the energy per particle. This partial Pauli blocking counteracts
the leading attraction from the term linear in ρ. Three-body forces arise necessarily
and naturally in this approach. Their contributions is not large at normal nuclear
matter density, indicating a convergent hierarchy of terms in powers of the Fermi
momentum as long as the baryon density does not exceed about twice the density of
equilibrium nuclear matter.
Binding and saturation of nuclear matter can thus be seen, in this approach, as
a combination of phenomena and effects which relate to the names Yukawa, Van der
Waals and Pauli. It does then perhaps not come as a surprise that the resulting nu-
clear matter equation of state, see Fig.10, is reminscent of a Van der Waals equation
of state. The nuclear liquid turns into a gas at a critical temperature Tc ≃ 15 MeV,
quite close to the commonly accepted empirical range Tc ∼ 16− 18 MeV.
Fig. 10. The nuclear matter equation of state: pressure versus baryon density calculated in three-
loop in-medium chiral perturbation theory.59) Shown are isothermes with temperatures indi-
cated.
4.2. Finite nuclei: density functional strategies
A description of finite nuclei over a broad range, from 16O to the very heavy
ones, is successfully achieved using a (relativistic) universal energy density functional
guided by the nuclear matter results. The energy as a functional of density is written
E[ρ] = Ekin +
E(0)(ρ) + Eexc(ρ)
+ Ecoul , (4.5)
where Ekin and Ecoul are the kinetic and Coulomb energy contributions. The ba-
sic idea61), 62) is to construct Eexc from the in-medium chiral perturbation theory
calculations discussed previously, representing the pionic fluctuations built on the
non-perturbative QCD vacuum in the presence of baryons. Binding and saturation,
Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 19
in nuclear matter as well as in finite nuclei, is driven primarily by two-pion exchange
mechanisms in combination with the Pauli principle included in Eexc.
At the same time the QCD vacuum is populated by strong condensates. The
E(0) part of the energy density incorporates the leading changes of these condensates
at finite baryon chemical potential (or density). As discussed in ref.61), 62) and refer-
ences therein, QCD sum rules at non-zero baryon density suggest that these density
dependent changes of condensates generate strong scalar and vector mean fields with
opposite signs: at nuclear bulk densities, several hundred MeV of scalar attraction
are compensated by an almost equal amount of vector repulsion, such that the net
effect of the condensate mean fields almost vanishes and is hardly visible in infinite,
homogeneous nuclear matter. However, In finite nuclei, the coherent effect of the
strong scalar and vector mean fields produces the large spin-orbit splitting observed
empirically.
Calculations along these lines, using the chiral pion-nuclear dynamics framework
and constraints from the symmetry breaking pattern of low-energy, have been per-
formed in refs61), 62) throughout the nuclear chart. The results for nuclear binding
energies and radii are comparable in accuracy with those of the best phenomenolog-
ical relativistic mean field models available. Examples are shown in Figs.11, 12 and
Fig. 11. Deviations (in %) of calculated binding energies (upper pannel) and r.m.s. charge radii
(lower pannel) from measured values for a series of nuclei from A = 16 to A = 210. For details
of the computations see ref.62)
Of particular interest is the systematics through chains of isotopes of deformed
nuclei, increasing the number of neutrons by one unit in each step and changing the
deformation pattern along the way. The results are sensitive to the detailed isospin
dependence of the nuclear interaction. Chiral pion dynamics and its prediction for
the isospin structure of the two-pion exchange NN force in the nuclear medium
appears to account successfully for the observed properties of such isotopic chains.
§5. Concluding Remarks
Yukawa’s original U-field which then became the pion is still, more than seventy
years after its first release, a generic starting point for our understanding of nuclear
20 W. Weise
Fig. 12. Charge form factor of 48Ca. Calcu-
lated curve62) in comparison with experi-
mental data.
Fig. 13. Deviations (in %) of calculated from
measured binding energies for a series of
isotopic chains from Nd to Pt. For details
of the computations see ref.62)
systems and interactions. Its property as a Nambu-Goldstone boson of spontaneously
broken chiral symmetry is at the origin of a successful effective field theory which
represents QCD in its low energy limit. Indications so far are promising that this
framework, constrained by the symmetry breaking pattern of low-energy QCD, can
serve as foundation for a modern theory of the nucleus. Generating the nucleon-
nucleon interaction itself directly from QCD is still a major challenge. Recent lattice
QCD results,63) although still taken at pion masses large compared to the physical
one, point to very interesting developments in the near future.
Acknowledgements
It is a great pleasure to thank Professor Taichiro Kugo and his colleagues for
their hospitalty in Kyoto and for arranging a most inspiring Symposium. Avraham
Gal’s careful reading of the manuscript is gratefully acknowledged.
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