100 years of subatomic physics || chiral symmetry in subatomic physics

May 9, 2013 10:7 World Scientific Review Volume - 9.75in x 6.5in chiral

Chapter 8

Chiral Symmetry in Subatomic Physics

ULF-G. MEIßNER

Helmholtz-Institut fur Strahlen- und Kernphysik

and Bethe Center for Theoretical Physics

Universitat Bonn, D-53115 Bonn, Germany

and

Institut fur Kernphysik, Institute for Advanced Simulation

and Julich Center for Hadron Physics

Forschungszentrum Julich, D-52425 Julich, Germany

These are some personal thoughts on the role of chiral symmetry in subatomic

physics.

1. Introduction

Symmetries play an important role in our understanding of subatomic physics.

Remarkably, the most important consequences are related to the violation of sym-

metries such as the breaking of CP invariance which is a necessary ingredient in the

generation of the observed matter–antimatter asymmetry in the universe. For the

strong interactions, chiral symmetry has always played a prominent role. That be-

came even more important when appropriate effective field theories were developed

that allowed one to systematically work out the strictures of the spontaneous and

explicit broken chiral symmetry of Quantum Chromodynamics (QCD), the gauge

theory that describes the strong interactions on a microscopic basis. In this con-

tribution, I will discuss the role of chiral symmetry in particle and nuclear physics

and, in particular, show how these seemingly separate fields are strongly linked as

QCD generates various forms of strongly interacting matter — the hadrons and

nuclei. Here, the pions which are the Goldstone bosons of the spontaneously bro-

ken chiral symmetry, play a special role as they have profound influences on the

structure and dynamics of hadrons and also the forces between nucleons. When

the pion was predicted by Yukawa in 1935 as the force carrier of the strong interac-

tions, it could hardly be foreseen that almost a century later a systematic theory of

hadron and nuclear interactions could be formulated in which the pion is one of the

main ingredients. Such line of thought started long before QCD as chiral symmetry

has played an important role in shaping our understanding of strong interaction

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200 Ulf-G. Meißner

physics. The heydays of current algebra, soft pion theorems and attempts to go

beyond in the 1960s led to a cornucopia of interesting predictions. However, at that

time many of these could hardly be tested and field theory was no longer consid-

ered the appropriate tool for a microscopic theory of the strong interactions. This

changed with the formulation of QCD and the experimental findings of the proton

substructure, leading to new directions in subatomic physics. However, with the

advent of effective field theories and their applications to QCD spear-headed by

Weinberg1 and Gasser and Leutwyler,2,3 a whole new world of precision physics at

low energies opened up — something that many believed could never be achieved

or at most be realized using numerical simulations only. In this contribution, I try

to convey the fascination of working in this challenging field. I will also argue that

the combination of chiral symmetry with other methods such as dispersion relations

or lattice simulations is one of the main directions to further elucidate the role of

chiral symmetry.

This contribution is organized as follows: Section 2 contains a very basic in-

troduction into chiral symmetry, followed by a short discussion of chiral symmetry

breaking and its consequences in Section 3. Chiral symmetry and its realization in

QCD is discussed in Section 4. This is followed by an introduction to the concepts

and foundations of chiral perturbation theory, see Section 5. Then, applications of

chiral perturbations theory and its various extensions are discussed in the following

sections, including pion–pion scattering (Section 6), the role of strange quarks and

pion–kaon scattering (Section 7), the pion cloud of the nucleon (Section 8), meson–

baryon three flavor chiral dynamics, in particular (anti)kaon–nucleon scattering, in

Section 9, and then the role of chiral symmetry in nuclear forces and atomic nuclei

in Section 10. Section 11 is devoted to a short discussion of chiral symmetry in

heavy hadron physics. Some final remarks are collected in Section 12.

2. Chiral Symmetry

In this section, I briefly introduce the concept of chiral symmetry. We consider a

theory of massless fermions, given by the Lagrangian

L = iψγµ∂µψ . (1)

Such a theory possesses a chiral symmetry. To see this, perform a left/right (L/R)-

decomposition of the spin-1/2 field

ψ =1

2(1− γ5)ψ +

1

2(1 + γ5)ψ = PLψ + PRψ = ψL + ψR , (2)

using the projection operators PL/R, that obey P 2

L = PL, P2

R = PR, PL · PR =

0, PL + PR = 1I. The ψL/R are helicity eigenstates

1

2hψL,R = ±

1

2ψL,R , h =

~σ · ~p

|~p |, (3)

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Chiral Symmetry in Subatomic Physics 201

where ~p denotes the fermion momentum and ~σ are the Pauli spin matrices. In terms

of the left- and right-handed fields, the Lagrangian takes the from

L = iψLγµ∂µψL + iψRγµ∂

µψR , (4)

which means that the L/R fields do not interact and, by use of Noether’s theorem,

one has conserved L/R currents. We note that a fermion mass term breaks chiral

symmetry, as a mass term mixes the left- and right-handed components, ψMψ =

ψRMψL+ψLMψR. Physically, this is easy to understand. While massless fermions

move with the speed of light, this is no longer the case for massive fermions. Thus,

for a massive fermion with a given handedness in a certain frame, one can always

find a boost such that the sign of ~σ ·~p changes. If the mass term is sufficiently small

(where “small” depends on other scales in the theory), one can treat this explicit

chiral symmetry breaking in perturbation theory and speaks of an approximate

chiral symmetry — more on that later.

3. Chiral Symmetry Breaking

In many fields of physics, broken symmetries play a special role. Of highest interest

is the phenomenon of spontaneous symmetry breaking, which means that the ground

state of a theory shares a lesser symmetry than the corresponding Lagrangian or

Hamiltonian. A key ingredient in this context is Goldstone’s theorem:4,5 To every

generator of a spontaneously broken symmetry corresponds a massless excitation

of the vacuum. This can be understood in a nut-shell (ignoring subtleties like the

normalization of states and alike — the argument also goes through in a more

rigorous formulation). Let H be some Hamiltonian that is invariant under some

charges Qi, i.e. [H, Qi] = 0, with i = 1, . . . , n. Assume further that m of these

charges (m ≤ n) do not annihilate the vacuum, that is Qj|0〉 6= 0 for j = 1, . . . ,m.

Define a single-particle state via |ψ〉 = Qj|0〉. This is an energy eigenstate with

eigenvalue zero, since H |ψ〉 = HQj|0〉 = QjH |0〉 = 0. Thus, |ψ〉 is a single-particle

state with E = ~p = 0, i.e. a massless excitation of the vacuum. These states are

the Goldstone bosons, collectively denoted as pions π(x) in what follows. Through

the corresponding symmetry current the Goldstone bosons couple directly to the

vacuum,

〈0|J0(0)|π〉 6= 0 . (5)

In fact, the non-vanishing of this matrix element is a necessary and sufficient con-

dition for spontaneous symmetry breaking.

Another important property of Goldstone bosons is the derivative nature of

their coupling to themselves or matter fields. Again, in a hand-waving fashion,

this can be understood easily. As above, one can repeat the operation of acting

with the non-conserved charge Qj on the vacuum state k times, thus generating a

state of k Goldstone bosons that is degenerate with the vacuum. Assume now that

the interactions between the Goldstone bosons is not vanishing at zero momentum.

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202 Ulf-G. Meißner

Then, the ground state ceases to be degenerate with the k Goldstone boson state,

thus the assumption must be incorrect. Of course, this argument can also be made

rigorous. In the following, the derivative nature of the pion couplings will play an

important role.

4. Chiral Symmetry in QCD

In this section, I give a short introduction to chiral symmetry in the context of

Chromodynamics. QCD is a non-Abelian SU(3)color gauge theory. Matter is com-

posed of quarks which come in Nf flavors, three of them being light (u, d, s) and the

other three heavy (c, b, t). Here, light and heavy refers to a typical hadronic scale

of about 1 GeV. We will come back to the special case of the strange quark later,

see Section 7. The color force is mediated by gauge bosons, the gluons, that come

in different types. In what follows, I will mostly consider light quarks (the heavy

quarks are to be considered as decoupled). The QCD Lagrangian reads

LQCD = −

1

2g2Tr(GµνG

µν) + q iγµDµ q − qM q = L

0

QCD− qM q , (6)

where we have absorbed the gauge coupling in the definition of the gluon field and

color indices are suppressed. Gµν is the gluon field strength tensor, which includes

the well-known gluon self-couplings. The three-component vector q collects the

quark fields, qT (x) = (u(s), d(x), s(x)). As far as the strong interactions are con-

cerned, the different quarks u, d, s have identical properties, except for their masses.

The quark masses are free parameters in QCD — the theory can be formulated for

any value of the quark masses. In fact, light quark QCD can be well approximated

by a fictitious world of massless quarks, denoted L0

QCDin Eq. (6). Remarkably,

this theory contains no adjustable parameter — the gauge coupling g merely sets

the scale for the renormalization group invariant scale ΛQCD. The Lagrangian of

massless QCD is invariant under separate unitary global transformations of the L/R

quark fields,

qI → VIqI , VI ∈ U(3) , I = L,R , (7)

leading to 32 = 9 conserved left- and 9 conserved right-handed currents by virtue

of Noether’s theorem. These can be expressed in terms of vector (V ∼ L+R) and

axial-vector (A ∼ L−R) currents

Vµ0(Aµ

0) = q γ

µ(γ5) q , Vaµ (A

aµ) = q γ

µ(γ5)λa

2q , (8)

Here, a = 1, . . . , 8, and the λa are Gell-Mann’s SU(3) flavor matrices. We remark

that the singlet axial current is anomalous, and thus not conserved. The actual

symmetry group of massless QCD is generated by the charges of the conserved

currents, it is G0 = SU(3)R×SU(3)L×U(1)V . The U(1)V subgroup ofG0 generates

conserved baryon number since the isosinglet vector current counts the number of

quarks minus antiquarks in a hadron. The remaining group SU(3)R × SU(3)L is

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often referred to as chiral SU(3). Note that one also considers the light u and d

quarks only (with the strange quark mass fixed at its physical value), in that case,

one speaks of chiral SU(2) and must replace the generators in Eq. (8) by the Pauli-

matrices. Let us mention that QCD is also invariant under the discrete symmetries

of parity (P ), charge conjugation (C) and time reversal (T ) (as long as we ignore

the tiny θ-term).

The chiral symmetry is a symmetry of the Lagrangian of QCD but not of the

ground state or the particle spectrum — to describe the strong interactions in

nature, it is crucial that chiral symmetry is spontaneously broken. This can be

most easily seen from the fact that hadrons do not appear in parity doublets. If

chiral symmetry were exact, from any hadron one could generate by virtue of an

axial transformation another state of exactly the same quantum numbers except of

opposite parity. The spontaneous symmetry breaking leads to the formation of a

quark condensate in the vacuum 〈0|qq|0〉 = 〈0|qLqR + qRqL|0〉, thus connecting the

left- with the right-handed quarks. In the absence of quark masses this expectation

value is flavor-independent: 〈0|uu|0〉 = 〈0|dd|0〉 = 〈0|qq|0〉. More precisely, the

vacuum is only invariant under the subgroup of vector rotations times the baryon

number current, H0 = SU(3)V × U(1)V . This is the generally accepted picture

that is supported by general arguments6 as well as lattice simulations of QCD. In

fact, the vacuum expectation value of the quark condensate is only one of the many

possible order parameters characterizing the spontaneous symmetry violation — all

operators that share the invariance properties of the vacuum (Lorentz invariance,

parity, invariance under SU(3)V transformations) qualify as order parameters. The

quark condensate nevertheless enjoys a special role, it can be shown to be related

to the density of small eigenvalues of the QCD Dirac operator (see Ref. 7 and more

recent discussions in Refs. 8 and 9),

limM→0

〈0|qq|0〉 = −π ρ(0) . (9)

For free fields, ρ(λ) ∼ λ3 near λ = 0. Only if the eigenvalues accumulate near

zero, one obtains a non-vanishing condensate. This scenario is indeed supported

by lattice simulations and many model studies involving topological objects like

instantons or monopoles.

In QCD, we have eight (three) Goldstone bosons for SU(3) (SU(2)) with spin

zero and negative parity — the latter property is a consequence that these Gold-

stone bosons are generated by applying the axial charges on the vacuum. The

dimensionful scale associated with the matrix element Eq. (5) is the pion decay

constant (in the chiral limit)

〈0|Aaµ(0)|π

b(p)〉 = iδabFpµ , (10)

which is a fundamental mass scale of low-energy QCD. In the world of massless

quarks, the value of F differs from the physical value by terms proportional to the

quark masses, to be introduced later, Fπ = F [1+O(M)]. The physical value of Fπ

is 92.2MeV, determined from pion decay, π → νµ.

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204 Ulf-G. Meißner

Of course, in QCD the quark masses are not exactly zero. The quark mass term

leads to the so-called explicit chiral symmetry breaking. Consequently, the vector

and axial-vector currents are no longer conserved (with the exception of the baryon

number current)

∂µVµa =

1

2iq [M, λa] q , ∂µA

µa =

1

2iq {M, λa} γ5 q . (11)

However, the consequences of the spontaneous symmetry violation can still be an-

alyzed systematically because the quark masses are small. QCD possesses what

is called an approximate chiral symmetry. In that case, the mass spectrum of the

unperturbed Hamiltonian and the one including the quark masses cannot be sig-

nificantly different. Stated differently, the effects of the explicit symmetry breaking

can be analyzed in perturbation theory. This perturbation generates the remark-

able mass gap of the theory — the pions (and, to a lesser extent, the kaons and the

eta) are much lighter than all other hadrons. To be more specific, consider chiral

SU(2). The second formula of Eq. (11) is nothing but a Ward-identity that relates

the axial current Aµ = dγµγ5u with the pseudoscalar density P = diγ5u,

∂µAµ = (mu +md)P . (12)

Taking on-shell pion matrix elements of this Ward-identity, one arrives at

M2

π = (mu +md)Gπ

Fπ, (13)

where the coupling Gπ is given by 〈0|P (0)|π(p)〉 = Gπ . This equation leads to

some intriguing consequences: In the chiral limit, the pion mass is exactly zero —

in accordance with Goldstone’s theorem. More precisely, the ratio Gπ/Fπ is a

constant in the chiral limit and the pion mass grows as√

mu +md as the quark

masses are turned on.

There is even further symmetry related to the quark mass term. It is observed

that hadrons appear in isospin multiplets, characterized by very tiny splittings of the

order of a few MeV. These are generated by the small quark mass differencemu−md

(small with respect to the typical hadronic mass scale of a few hundred MeV) and

also by electromagnetic effects of the same size (with the notable exception of the

charged to neutral pion mass difference that is almost entirely of electromagnetic

origin). This can be made more precise: For mu = md, QCD is invariant under

SU(2) isospin transformations:

q → q′ = Uq , q =

(

u

d

)

, U =

(

a∗ b∗

−b a

)

, |a|2 + |b|

2 = 1 . (14)

In this limit, up and down quarks cannot be disentangled as far as the strong

interactions are concerned. Rewriting of the QCD quark mass term allows one to

make the strong isospin violation explicit:

HSB

QCD= mu uu+md dd =

1

2(mu +md)(uu+ dd) +

1

2(mu −md)(uu− dd) , (15)

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where the first (second) term is an isoscalar (isovector). Extending these consider-

ations to SU(3), one arrives at the eightfold way of Gell-Mann and Ne’eman10 that

played a decisive role in our understanding of the quark structure of the hadrons.

The SU(3) flavor symmetry is also an approximate one, but the breaking is much

stronger than it is the case for isospin. From this, one can directly infer that the

quark mass difference ms −md must be much bigger than md −mu.

There is one further source of symmetry breaking, which is best understood in

terms of the path integral representation of QCD. The effective action contains an

integral over the quark fields that can be expressed in terms of the so-called fermion

determinant. Invariance of the theory under chiral transformations not only requires

the action to be left invariant, but also the fermion measure.11 Symbolically,∫

[dq][dq] · · · → |J |

∫

[dq′][dq′] · · · . (16)

If the Jacobian is not equal to one, |J | 6= 1, one encounters an anomaly. Of course,

such a statement has to be made more precise since the path integral requires reg-

ularization and renormalization, still it captures the essence of the chiral anomalies

of QCD. One can show in general that certain 3-, 4-, and 5-point functions with an

odd number of external axial-vector sources are anomalous. As particular exam-

ples we mention the famous triangle anomalies of Adler, Bell and Jackiw and the

divergence of the singlet axial current,

∂µ(qγµγ5q) = 2iqmγ5q +

Nf

8πG

aµνG

µν,a, (17)

that is related to the generation of the η′ mass. There are many interesting aspects

of anomalies in the context of QCD and chiral perturbation theory.12

5. The Essence of Chiral Perturbation Theory

As the pions are Goldstone bosons, their interactions are of derivative nature. This

allows to formulate an effective field theory (EFT) at low energies/momenta, as

derivatives can be translated into small momenta. Such an EFT is necessarily non-

renormalizable, as one can write down an infinite tower of terms with increasing

number of derivatives consistent with the underlying symmetries, in particular chiral

symmetry. Consequently, such an EFT can only be applied for momenta and masses

(setting the “soft” scale) that are small compared to masses of the particles not

considered (setting the “hard” scale). For the case at hand, the hard scale is of the

order of 1 GeV. I will now show that there is a hierarchy of terms that allows one

to make precise predictions with a quantifiable theoretical order. This scheme runs

under the name of power counting. To be precise, consider an effective Lagrangian

Leff =∑

d

L(d)

, (18)

where d is supposed to be bounded from below. For interacting Goldstone bosons,

d ≥ 2, and the pion propagator is D(q) = i/(q2 −M2

π), with Mπ the pion mass.

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206 Ulf-G. Meißner

Consider now an L-loop diagram with I internal lines and Vd vertices of order d.

The corresponding amplitude scales as follows

Amp ∝

∫

(d4q)L1

(q2)I

∏

d

(qd)Vd , (19)

where we only count powers of momenta. Now let Amp ∼ qν , therefore using

Eq. (19) gives ν = 4L− 2I +∑

d dVd. Topology relates the number of loops to the

number of internal lines and vertices as L = I−∑

d Vd+1, so that we can eliminate

I and arrive at the compact formula1

ν = 2 + 2L+∑

d

Vd(d− 2) . (20)

The consequences of this simple formula are far-reaching. To lowest order (LO), one

has to consider only graphs with d = 2 and L = 0, which are tree diagrams. Explicit

symmetry breaking is also included as the quark mass counts as two powers of q,

cf. Eq. (13). This LO contribution is nothing but the current algebra result, which

can also be obtained with different — though less elegant — methods. However,

Eq. (20) tells us how to systematically construct corrections to this. At next-to-

leading order (NLO), one has one loop graphs L = 1 build from the lowest order

interactions and also contact terms with d = 4, that is higher derivative terms that

are accompanied by parameters, the so-called low-energy constants (LECs), that

are not constrained by the symmetries. These LECs must be fitted to data or can

eventually be obtained from lattice simulations, that allow to vary the quark masses

and thus give much easier access to the operators that involve powers of quark mass

insertions or mixed terms involving quark masses and derivatives. Space forbids to

discuss this interesting field, I just refer to the recent compilation in Ref. 13. At

next-to-next-to-leading order (NNLO), one has to consider two-loop graphs with

d = 2 insertions, one-loop graphs with one d = 4 insertion and d = 6 contact terms.

Matter fields can also be included in this scheme. For stable particles like the

nucleon, this is pretty straightforward, the main difference to the pion case is the

appearance of operators with an odd number of derivatives. For unstable states,

the situation is more complicated, as one has to account for the scales related

to the decays. For example, in case of the ∆(1232)-resonance, one can set up a

consistent power counting if one considers the nucleon-delta mass difference as a

small parameter. Here, I will not further elaborate on these issues but rather refer

to some recent related works on the ∆ and vector mesons.14–17

Coming back to chiral perturbation theory in this pure setting (considering

pions and possibly nucleons), one finds in the literature statements that the whole

approach is nothing more than parameter-fitting. This is, of course, incorrect. The

chiral Ward identities of QCD, that are faithfully obeyed in chiral perturbation

theory,18,19 connect a tower of different processes involving various numbers of pions

and external sources so that fixing the low-energy constants through a number of

processes allows one to make quite a number of testable predictions. Furthermore,

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Fig. 1. The LECs ci (circles) in pion–nucleon scattering (left), the two-nucleon (NN) interaction

(center) and the three-nucleon (NNN) interaction (right).

as the order increases, the number of LECs also increases, but again for a specific

process this is not prolific. The prime example is elastic pion–pion scattering, which

features four LECs at one-loop order but only two new LECs appear at two loops —

all other local two-loop contributions to this reaction merely correspond to quark

mass renormalizations of operators existing already at one-loop. This is a more

general phenomenon as one can group the various operator structures in two classes:

The so-called dynamical operators refer to terms with derivatives on the hadronic

fields (e.g. powers of momenta) and are independent of the quark masses, whereas

the so-called symmetry-breakers come with certain powers of quark mass insertions

and thus vanish in the chiral limit. As stated before, lattice simulations that allow

one to vary the quark masses can be used efficiently to learn about this type of

operators. But back to the interconnections between various processes in terms

of the LECs. A particularly nice and timely example is related to the dimension-

two couplings ci in the chiral effective pion–nucleon Lagrangian, see Fig. 1 (for

precise definitions and further details, see the review Ref. 20). The corresponding

operators can, e.g. be fixed in a fit to pion–nucleon scattering data. Then the same

operators play not only an important role in the two-pion exchange contribution to

nucleon–nucleon scattering but also they give the longest range part of the three-

nucleon forces, that are an important ingredient in the description of atomic nuclei

and their properties. In fact, there have also been attempts to determine these

couplings directly from nucleon–nucleon scattering data, leading to values consistent

with the ones determined from pion–nucleon scattering. Furthermore, this clearly

established the role of pion-loop effects (see the middle graph in Fig. 1) in nucleon–

nucleon scattering beyond the long-established tree-level pion exchange, already

proposed by Yukawa in 1935. For details, see Ref. 21.

One important issue to be discussed is unitarity. From the power counting out-

lined above, it is obvious that imaginary parts of scattering amplitudes or form

factors are only generated at subleading orders, or, more precisely, the one-loop

graphs generate the leading contributions to these. In general, this does not cause

any problem, with the exception of the strong pion–pion final state interactions to

be discussed in more detail later. In fact, one can turn the argument around and

use analyticity and unitarity to calculate the leading loop corrections without ever

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208 Ulf-G. Meißner

working out a loop diagram — the most famous examples are Lehmann’s analysis

of pion–pion scattering in 197223 and Weinberg’s general analysis of the structure of

effective Lagrangians.1 A pedagogic introduction to the relation between unitarity

and CHPT can be found in Ref. 24. As first stressed by Truong, see Ref. 25 (and

references therein), unitarization of chiral scattering amplitudes can generate reso-

nances — however, this extension of CHPT to higher energies comes of course with

a price, as one resums certain classes of diagrams and thus cannot make the direct

connection to QCD Green functions easily (if at all). The issue of unitarization of

CHPT will be picked up again in Section 9.

6. The Long Road Towards Precision at Low Energies:

Pion–Pion Scattering

Elastic pion–pion scattering is the purest and most well-studied process which allows

one to understand how chiral perturbation theory (CHPT) can operate and how one

can increase its precision by connecting it with dispersion relations. In addition, it

also tells us how experiment had to come a long way to achieve a precise extraction

of the pertinent observables. For a lucid discussion of the history of pion–pion

scattering, I refer to Ref. 26. As I will show in this chapter, theory and experiment

have converged at a very high level of precision so that the ππ S-wave scattering

lengths constitute one of the finest tests of the Standard Model at low energies.

Let us start with current algebra (CA), the precursor of CHPT, which amounts

to the LO (tree level) prediction. To be specific, consider the S-wave isospin zero

scattering length a00. Weinberg’s famous CA prediction from 196627 was followed in

1983 by the ground-breaking one-loop work of Gasser and Leutwyler28 and in 1997

by the two-loop results of Bijnens et al.29 The central values of their predictions

read

a0,tree0

= 0.16 , a0,1-loop0

= 0.20 , a0,2-loop0

= 0.217 , (21)

which shows convergence but one might be worried about the large one-loop cor-

rections (about 25%), as the expansion parameter is ξ = (Mπ/4πFπ)2' 0.014.

However, the physics behind this is well understood: S-wave pions in an isospin

zero state suffer from large final state interactions (rescattering), that is also visi-

ble in other processes like γγ → π0π0 (Ref. 30) or in the scalar pion form factor

(Ref. 31). Because of this fact, it was early recognized that for such cases com-

bining CHPT with unitarity might be a way to sharpen the predictions. In fact,

this turned out to be the case in particular for elastic pion–pion scattering, where

a combination of the Roy equation machinery with chiral symmetry constraints led

to the remarkably precise prediction32

a0

0= 0.220± 0.005 , (22)

which is truly amazing, as one has an accuracy of about 2% for a strong interaction

observable in the non-perturbative regime of QCD. For pions in the S-wave in

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an isospin-two state, matters are very different, the precise prediction of Ref. 32,

a20= −0.0444± 0.0010, is not very different from the tree level prediction a2,tree

0=

−0.0457. This is due to the fact that in this case the pion–pion interaction is very

weak.

From the experimental side, it has been quite a feat to reach such a precision.

The most accurate determination of the pion–pion S-wave scattering lengths comes

from a combination of kaon decays, more precisely of Ke4(K±

→ π+π−e±ν) and

K0→ 3π0. InKe4 decays, the generated pion pair is sensitive to the phase difference

δ0 − δ1 in the threshold region, however, it is of utmost importance to include

isospin breaking effects33 to achieve the required precision. The second process

features the cusp due to the rescattering process π0π0→ π+π−

→ π0π0 in the

invariant mass distribution of the two-pion system in the final-state. Here, one had

to develop a non-relativistic EFT that does not employ a chiral expansion but rather

an expansion in terms of the scattering length and is only applicable around the

cusp.34 Using this sophisticated framework, the NA 48/2 collaboration at CERN

was able to extract the desired scattering lengths with good precision35

a0

0= 0.2210± 0.0047stat± 0.0040sys , a

2

0= −0.0429± 0.0044stat± 0.0028sys . (23)

These values are in fine agreement with the predictions from Ref. 32 and have a

comparable uncertainty. It is still remarkable that the prediction preceeded its

precise verification by a decade. See also Fig. 2 for the present situation for the

comparison of theory and experiment.

And how about lattice QCD, that in principle allows for ab initio calculations

in QCD? First, there are two different paths that allow one to calculate the S-wave

scattering lengths. The first and more direct one is to calculate the low-energy

phase shifts using Luscher’s finite volume approach that relates the energy shift of

an interacting two-particle system in a finite volume to the continuum phase shift.36

This is feasible for the isospin-two scattering length as due to the maximal isospin

in the two-pion system only so-called stretched diagrams contribute, and these

are necessarily connected (that is all valence quark lines run through the whole

Feynman graph). The situation is different for the isospin zero state, that features

also disconnected graphs (where some valence fermion lines are closed and connected

through gluon exchanges to the initial/final state hadrons). Disconnected diagrams

are very noisy in Monte Carlo simulations and thus very hard to compute with

small uncertainties. Therefore, direct computations exist at present only for a20—

and these agree quite well with the prediction (see the horizontal bands in Fig. 2).

The second and indirect method is to compute the LECs ¯3,¯4 (that parametrize

symmetry breaking beyond leading order) and inject these into the pertinent one-

loop formulas for the scattering lengths — this leads to the filled ellipses in Fig. 2

and the agreement with the chiral plus Roy equation prediction is quite good. Still,

the lattice practitioners have to perform the direct computation of a00before they

can claim success — something one has to remind them on a regular basis.

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210 Ulf-G. Meißner

0.16 0.18 0.2 0.22 0.24 0.26

a00

-0.06 -0.06

-0.05 -0.05

-0.04 -0.04

-0.03 -0.03

a20

universal bandtree, one loop, two loopsscalar radius CGL 2001

E865 Ke4 2010NA48 Ke4 2010NA48 K3π 2010DIRAC 2011Garcia-Martin et al. 2011

CERN-TOV 2006JLQCD 2008PACS-CS 2009MILC 2010ETM 2010RBC/UKQCD 2011NPLQCD 2011Scholz et al. 2011

NPLQCD 2008 Feng et al. 2010NPLQCD 2011Yagi et al. 2011

Fig. 2. Determination of the ππ S-wave scattering lengths a00and a2

0. The black circles are the

tree, one-loop and two-loop CHPT predictions from left to right. The improved prediction based

on the Roy equations is the open ellipse (CGL). The various experimental determinations are given

by the light hatched areas. Indirect/direct lattice determinations are given by ellipses/horizontal

bands. Figure courtesy of Heiri Leutwyler.

7. Chiral Symmetry and Strange Quarks

The strange quark enjoys a special role in chiral dynamics as its mass ms ∼ ΛQCD

is sizeably larger than the mass of its up and down cousins. This is reflected in

the fact that the expansion parameter ξs = M2

K/(4πFπ)2' 0.18 is considerably

larger than its two-flavor counterpart. In addition, on general grounds7,8,37 one can

show that the three-flavor condensate is smaller than its two-flavor counterpart, but

by how much is matter of debate. The pioneering sum rule study by Moussallam38

seems to indicate a large suppression of the three-flavor condensate, Σ(3) = Σ(2)[1−

0.54± 0.27], whereas recent lattice studies give more modest changes, e.g. Ref. 39

gives Σ(3) = Σ(2)[1 − 0.23 ± 0.39]. Here, Σ(2) = − limmu,md→0〈uu〉 and Σ(3)

correspondingly. Both of these observations point towards possible problems in the

chiral expansion when strange quarks are involved. There are indications from some

lattice simulations on pseudoscalar masses and decay constants40 and K`3-decays41

that the three-flavor chiral series does not converge as signaled, e.g. by very large

NNLO corrections. In fact, resummation techniques have been proposed to deal

with such a situation, see e.g. Ref. 42. However, this chapter is not closed, so I

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Table 1. The pion–kaon scattering S-wave lengths in appropriate powers of the in-

verse charged pion mass.

CA 1-loop 2-loop Roy–Steiner Lattice

a1/2

00.14 0.18 ± 0.03 0.220 0.224 ± 0.022 0.1725 ± 0.0017+0.0023

−0.0156

a3/20

−0.07 −0.05 ± 0.02 −0.047 −0.0448 ± 0.0077 −0.0574± 0.0016+0.0024−0.0058

will discuss the simplest scattering process involving strange quarks, namely elastic

kaon–pion scattering, that nicely exhibits some of these interesting features.

As in pion–pion scattering, the scattering amplitude for πK → πK boils down to

two numbers at threshold, the S-wave scattering lengths a1/20

and a3/20

, respectively,

as the kaon has isospin 1/2. These fundamental parameters of three-flavor QCD

have been analyzed utilizing current algebra,43 chiral perturbation theory, Roy–

Steiner equations and lattice QCD. The one-loop CHPT results have been given in

Ref. 44, the two-loop calculation in Ref. 45, the dispersive representation based on

Roy–Steiner equations in Ref. 46 and the pioneering lattice calculation in Ref. 47.

The corresponding results are collected in Table 1. A few remarks are in order:

First, for these two quantities, there is agreement of the two-loop prediction with

the dispersive result, although for the two-loop result no specific errors were given in

Ref. 45. However, from the figures given in that paper one can read off the possible

ranges, which are asymmetric around the central values, a1/20

= 0.17, . . . , 0.225

and a3/20

= −0.075, . . . ,−0.04. Second, the lattice determination is actually for

a(π+K+) = a3/20

and then the CHPT machinery for the isoscalar and isovector

scattering lengths, a+, a−, respectively, are used, with a1/20

= a+ + 2a− and a3/20

=

a+ − a−. As was pointed out in Ref. 48, the chiral analysis of the isoscalar and

isovector scattering lengths exhibits a remarkably different behavior: at one-loop

order, a− depends only on one single LEC, L5, which can be determined by the

ratio FK/Fπ, such that a− can be predicted with good accuracy, reminiscent of

the chiral expansion for the isovector pion–nucleon scattering length.49 On the

contrary, a combination of seven LECs enters the one-loop prediction for a+, some

of these are not well determined. Therefore, the uncertainty of the prediction for

a+ is quite sizeable. Third, while the dispersive prediction appears quite precise,

the situation is less clear than in the case of pion–pion scattering, as the dispersive

representation below the matching point does not agree well with the data for the

I = 3/2 S-wave and for the I = 1/2 P-wave phase.46 It would be very interesting

to have a direct lattice calculation of a1/20

to shed further light on these issues.

8. On the “Pion Cloud” of the Nucleon

Chiral symmetry has played an important role in our understanding of hadron

structure and dynamics. In the sixties, this was encoded in current algebra theorems

for nucleons, pions and photons and later in terms of models, like chiral bag models,

Skyrmions or chiral quark models. With the advent of chiral perturbation theory,

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212 Ulf-G. Meißner

it became evident that better founded predictions could be made and with the

ever increasing experimental possibilities, one was finally able to test many of these

predictions. Recent reviews on these exciting developments are Refs. 50 and 51.

Here, I will rather dwell on one particular topic, namely on the so-called “pion

cloud” of the nucleon. This is a much debated topic and regularly leads to heated

discussions — so let us start with a solid definition and then discuss a very recent

application, namely the proton radius puzzle.

Many models of the nucleon feature a compact core (including possibly quarks)

and a longer-ranged component, often called the pion cloud. When I was a graduate

student in the early 1980s, there were heated debates about the size of the inner

core — little bags versus cloudy bags. While one has realized that such debates

were essentially senseless, such type of picture for the nucleon structure remained

(at various levels of sophistication). Clearly, chiral symmetry requires the pion

coupling to the nucleon! In the framework of chiral perturbation theory (in its

pre-EFT formulation) it was realized early that the long-ranged pion cloud can

have profound effects on the structure of the nucleon, i.e. pion loops can lead to

contributions that diverge as 1/Mπ or lnMπ as the pion mass vanishes.52 Such

a behavior is easy to understand, the contributions of massive pions are Yukawa-

suppressed ∼ exp(−Mπr)/r. This suppression becomes Coulomb-like ∼ 1/r (alas of

infinite range) asMπ → 0 and thus divergent matrix elements can emerge. At finite

pion mass, these loop effects contribute with different strength to various nucleon

properties, but they are certainly the best representation of the pion cloud. In

fact, the first calculation of a pion cloud effect dates back to Frazer and Fulco22

long before current algebra — unitarity and analyticity do encode aspects of chiral

symmetry. I will come back to this pioneering calculation later.

As already stated, beyond lowest order, any observable calculated in CHPT re-

ceives contributions from tree and loop graphs. Naively, these loop diagrams qualify

as the natural candidate for a precise definition of the “pion cloud” of any given

hadron. The loop graphs not only generate the imaginary parts of the pertinent

observables but are also — in most cases — divergent, requiring regularization

and renormalization. The method of choice in CHPT is dimensional regularization

(DR), which introduces the scale λ. Varying this scale has no influence on any

observable O (renormalization group invariance),

d

dλO(λ) = 0 , (24)

but this also means that it makes little sense to assign a physical meaning to the sep-

arate contributions from the contact terms and the loops. Physics, however, dictates

the range of scales appropriate for the process under consideration — describing the

pion vector radius (at one loop) by chiral loops alone would necessitate a scale of

about 1/2 TeV (as stressed long ago by Leutwyler). In this case, the coupling of

the ρ-meson generates the strength of the corresponding one-loop counterterm that

gives most of the pion radius. In DR, all one-loop divergences are simple poles

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in 1/(d− 4), where d is the number of space-time dimensions. Consequently, these

divergences can be absorbed in the pertinent LECs that accompany the correspond-

ing local operators at that order in harmony with the underlying symmetries. For a

given LEC, Li this amounts to Li → Lren

i + βi L(λ), where L ∼ 1/(d− 4) and βi is

the corresponding β-function. The renormalized and finite Lren

i must be determined

by a fit to data (or calculated eventually using lattice QCD). Having determined

the values of the LECs from experiment, one is faced with the issue of trying to

understand these numbers. Not surprisingly, the higher mass states of QCD leave

their imprint in the LECs. Consider again the ρ-meson contribution to the vector

radius of the pion. Expanding the ρ-propagator in powers of t/M2

ρ , its first term is

a contact term of dimension four, with the corresponding finite LEC L9 given by

L9 = F 2

π/2M2

ρ ' 7.2 · 10−3, close to the empirical value L9 = 6.9 · 10−3 at λ =Mρ.

This so-called resonance saturation (pioneered in Refs. 53 and 54) holds more gen-

erally for most LECs at one loop and is frequently used in two-loop calculations

to estimate the O(p6) LECs. Let us now discuss the “pion cloud” of the nucleon

in the context of these considerations. Consider as an example the isovector Dirac

radius of the proton.55 The first loop contributions appear at third order in the

chiral expansion, leading to

〈r2〉V1=

(

0.61−(

0.47GeV2)

d(λ) + 0.47 logλ

1GeV

)

fm2, (25)

where d(λ) is a dimension three pion–nucleon LEC that parametrizes the “nucleon

core” contribution. Compared to the empirical value (rv1)2 = 0.585 fm2, we note

that several combinations of (λ, d(λ)) pairs can reproduce the empirical result, e.g.(

1 GeV,+0.06 GeV−2)

,(

0.943 GeV, 0.00 GeV−2)

,(

0.6 GeV,−0.46 GeV−2)

. (26)

An important observation to make is that even the sign of the “core” contribution

to the radius can change within a reasonable range typically used for the scale

λ. Physical intuition would tell us that the value for the coupling d should be

negative such that the nucleon core gives a positive contribution to the isovector

Dirac radius, but field theory tells us that for (quite reasonable) regularization

scales above λ = 943 MeV this need not be the case. In essence, only the sum of

the core and the cloud contribution constitutes a meaningful quantity that should

be discussed. This observation holds for any observable — not just for the isovector

Dirac radius discussed here.

Coming back to the seminal work of Frazer and Fulco — phrased in a more

modern language — they were reconstructing the isovector spectral function of the

nucleon form factors as a product of the pion vector form factor FVπ (t) and the

t-channel P-wave πN partial waves f1

±(t) as (more precisely, we give the results for

the imaginary parts of the Sachs form factors GE and GM , respectively)

Im GVE(t) =

q3

t

mN

√

t(FV

π )∗(t) · f1

+(t) , Im G

VM (t) =

q3

t√

2t(FV

π )∗(t) · f1

−(t) , (27)

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214 Ulf-G. Meißner

0 20 40t [Mπ

2]

0

0.02

0.04

0.06

spec

tral

func

tion

[1/M

π4 ]2ImGE/t

2

2ImGM/t2

Fig. 3. The two-pion spectral function based on modern data for the pion vector form factor.56

The spectral functions weighted by 1/t2 are shown for GE (solid line) and GM (dash-dotted line).

The previous results by Hohler et al.57 (without ρ-ω mixing) are shown for comparison by the

gray/green lines. The dot-dot-dashed (red) line indicates the ρ-meson contribution to Im GM with

a width Γρ = 150MeV.

with qt =√

t/4−M2π the pion momentum in the intermediate state. This repre-

sentation is exact up to t ' 50M2

π. The resulting spectral functions are exhibited

in Fig. 3. The contribution from the ρ-meson is shown by the red dot-dot-dashed

line — the aforementioned enhancement of the two-pion continuum on the left shoul-

der of the resonance is clearly visible. Upon integration, this contribution amounts

to about 50% of the isovector nucleon size, first stressed by Hohler and Pietarinen.58

Naturally, in the one-loop approximation, this mechanism is correctly recovered in

chiral perturbation theory, see e.g. Ref. 59 for a detailed discussion (there, it is

also shown that a similar effect does not appear in the isoscalar spectral functions).

It is remarkable that this so important and visible effect is often ignored in mod-

ern attempts to extract the nucleon size from electron–proton scattering data. This

brings me to the so-called “proton size puzzle.” Until 2010, the electric radius of the

proton was believed to be 0.8768(69) fm (CODATA value),60 from here on referred

to as the “large value”. I would like to stress, however, that the most sophisticated

dispersion theoretical analysis of the nucleon electromagnetic form factors, that in-

clude the two-pion continuum, always led to a small value, rpE ' 0.84 fm.61 In 2010,

the result of the Lamb shift measurement in muonic hydrogen, that are sensitive

to the proton radius, became available: rpE = 0.84184(67) fm.62 This “small value”

led to a flurry of papers questioning either the analysis of the experiment or our

understanding of the proton structure. The underlying theory of strong interac-

tions effects in muonic hydrogen was also scrutinized, see e.g. Refs. 63 and 64.

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The situation was further complicated by the high-precision measurement of

electron–proton scattering at the Mainz Microtron MAMI-C.65 The analysis of these

data led to rpE = 0.879(5)(stat.)(4)(syst.)(2)(model)(4)(group) fm, where the various

types of fits functions (polynomials and splines that do not represent the two-pion

continuum) were used, and depending on the class of fits functions, a model error

is defined and in the end, the results of the two groups of fit functions were av-

eraged, leading to the uncertainty labeled “group”. The Mainz value is in perfect

agreement with the CODATA one but differs by many standard deviations from the

muonic hydrogen result. We have recently reanalyzed the Mainz cross-section data

together with the world data on the neutron form factors. The spectral functions

of the underlying form factors contain besides isoscalar and isovector vector meson

poles the two-pion continuum (updated with new pion form factor data) as well as

representations of the KK66 and the πρ67 continua. For the proton electric and

magnetic radius we find68

rpE = 0.84+0.01

−0.01 fm , rpM = 0.86+0.02

−0.03 fm , (28)

where the uncertainties mostly stem from generous variations of the two-meson

continua. The proton charge radius is completely in agreement with muonic hy-

drogen result — which is entirely due to the inclusion of the two-pion contin-

uum. The magnetic radius is also consistent with earlier determinations, see

e.g. Ref. 69, but again in stark contrast to the analysis of the Mainz group,

rpM = 0.777(13)(stat.)(9)(syst.)(5)(model)(2)(group) fm.65

9. Three-Flavor Chiral Dynamics Reloaded

In the case of three-flavor chiral dynamics with baryons, it was realized early that

the fairly large expansion parameter MK/(4πFπ) ' 0.43 can lead to convergence

problems, see e.g. the pioneering work in Ref. 70. In addition, if one investi-

gates the most fundamental process involving strange quarks and baryons, namely

(anti)kaon–nucleon scattering, the situation is further complicated by the appear-

ance of subthreshold resonances in some channels. More precisely, we have to deal

with the famous Λ(1405) resonance in the isospin zero antikaon-proton interac-

tion, first investigated by Dalitz and Tuan.71 Such a resonance is, of course, not

amenable to a perturbative treatment. However, it was realized by the Munich

group72 that combining chiral symmetry with coupled-channel dynamics allows for

a dynamic generation of such a state (other groups have picked up this idea, see

e.g. Refs. 73–75). To be specific, one considers a Bethe–Salpeter (or Lippman–

Schwinger) equation for the scattering matrix (in a highly symbolic notation),

T = V + V GT , (29)

where V is the potential and G the meson–baryon propagator in the intermediate

state. Here, I have suppressed all channel-indices, so in fact T , V and G are matrices

in channel space. To leading order in the chiral expansion, the potential is given by

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216 Ulf-G. Meißner

the Weinberg-Tomozawa term and the s- and u-channel Born terms. However, the

meson–baryon loop function is divergent and requires regularization. In the early

days, a momentum cutoff was used, but that requires extreme fine-tuning and leads

to a large sensitivity of the results on the choice of the cutoff value. A better method

was proposed in Ref. 74, which is based on a dispersive representation of the loop

function using dimensional regularization with a subtraction constant taking the

role of the regulator. Then, the dependence on the regulator is only logarithmic.

Also, as the power counting is only performed on the level of the potential and

not the scattering matrix, it is absolutely mandatory to calculate the higher order

corrections in V and check a posteriori the convergence in T . Fortunately, this has

been done forK−p scattering,76–78 for S-wave pion–nucleon scattering79 and also for

photo-kaon processes.80 Simply performing calculations with the lowest order chiral

potential is meaningless! Also, in most calculations the on-shell approximation is

used, which turns the integral equations (29) into a set of algebraic equations, that

can be solved easily. However, it is not known how good this approximation really

is, although it seems to work quite well in many cases, see e.g. the early review

Ref. 81. Therefore, calculations avoiding this approximation are required. These are

technically demanding and presently do not incorporate proper crossing symmetry,

for some attempts see Refs. 82–84 and 79. More work in this direction is certainly

required.

Let me now consider the extraction of the (anti)kaon–nucleon scattering lengths.

The one-loop CHPT calculation has been performed by Kaiser.85 It shows that

CHPT converges quite well in the channels without resonances, e.g. for the KN

scattering length with isospin one, quite in contrast to the isospin zero channel,

that features the Λ(1405). There, coupled channel unitarization is required. The

antikaon–nucleon scattering lengths can be extracted from scattering data and also

from the level shift and width of kaonic hydrogen. There has been a long-standing

puzzle related to the discrepancy between the DEAR86 and the earlier and less

accurate KpX experiment at KEK.87 The DEAR data have been puzzling the com-

munity for a long time. As first pointed out in Ref. 88, the energy shift and width

of kaonic hydrogen measured by DEAR is incompatible with the predicted val-

ues taking the underlying KN scattering lengths from scattering data only. This

was resolved by the recent measurement of the energy level shift (ε1s) and width

(Γ1s) of the kaonic hydrogen ground-state by the SIDDHARTA collaboration,89

ε1s = −283±36 (stat)±6 (syst) eV , and Γ1s = 541±89 (stat)±22 (syst) eV. Two

groups have taken up the charge and shown that the SIDDHARTA data together

with the older scattering data indeed allow for a fairly precise determination of the

K−p scattering lengths, based on the chiral potential at NLO.90,91 I collect here the

results obtained in Ref. 91, noting that they are quite consistent with the earlier

results of Ref. 90. The values for the K−p scattering lengths are

a0 = −1.81+0.30−0.28 + i 0.92+0.29

−0.23 fm , a1 = +0.48+0.12−0.11 + i 0.87+0.26

−0.20 fm . (30)

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óóçç

ææ

ó ó

´

çç

ææ

-2 -1 0 10.2

0.4

0.6

0.8

1.0

1.2

1.4

Re a@fmD

Ima@

fmD

a0

a1

Fig. 4. Real and imaginary part of the isospin 0 and 1 (anti)kaon–nucleon scattering lengths.

The light shaded (green) areas correspond to the 1σ region of the approach from Ref. 91 around

the central value (full circles). The darker (blue) areas correspond to the 1σ region around central

value (empty circle) from Ref. 77. The cross and empty triangles denote older experimental values

from Refs. 92 and 93, respectively.

The comparison of these results with the earlier determination based on scattering

data only77 in Fig. 4 shows the clear improvement due to the kaonic hydrogen

data. The scattering length for the elastic K−p channel reads aK−p = −0.68+0.18−0.17+

i 0.90+0.13−0.13 fm. For comparison, taking the SIDDHARTA data only, one obtains

aK−p = −0.65+0.15−0.15 + i 0.81+0.18

−0.18 fm, while Ikeda et al. find aK−p = −0.70+0.13−0.13 +

i 0.89+0.16−0.16 fm. Therefore, these fundamental chiral SU(3) parameters can now

be considered to be determined with about an accuracy of ∼ 15%. The impact

of the measurement of kaonic hydrogen X-rays by the SIDDHARTA collaboration

on the allowed ranges for the kaon–deuteron scattering length in the framework

of non-relativistic EFT was analyzed in Ref. 94. Based on consistent input values

for K−p scattering, one predicts the kaon–deuteron scattering length to be AKd =

(−1.46 + i1.08) fm, with an estimated uncertainty of about 25% in both the real

and the imaginary part. This prediction could be further improved by including

corrections along the lines of Ref. 95. Most interestingly would, however, be a

precision measurement of the energy shift in kaonic deuterium.

10. Chiral Symmetry, Nuclear Forces and the “Level of Life”

As already discussed before, chiral symmetry also plays an important role in a

consistent and precise description of the forces between nucleons. While this was

known for a long time (see e.g. the recent Festschrift for Gerry Brown, one of the

pioneers in this field96), a truly systematic approach based on the chiral effective

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218 Ulf-G. Meißner

Lagrangian of QCD only became available through the groundbreaking work of

Weinberg.97 As realized by Weinberg, the power counting does not apply directly to

the S-matrix, but rather to the effective potential — these are all diagrams without

N -nucleon intermediate states. Such diagrams lead to pinch singularities in the

infinite nucleon mass limit (the so-called static limit), so that, e.g. the nucleon box

graph is enhanced as m/Q2, with m the nucleon mass and Q a small momentum.

The power counting formula for the graphs contributing with the νth power of Q

or a pion mass to the effective potential reads (considering only connected pieces):

ν = 2−N − 2L+∑

i

Vi

[

di +ni

2− 2

]

. (31)

Here, N is the number of in-coming and out-going nucleons, L the number of pion

loops, Vi counts the vertices of type i with di derivatives and/or pion mass insertions

and ni is the number of nucleons participating in this kind of vertex. Because of

chiral symmetry, the term in the square brackets is larger than or equal to zero and

thus the leading terms contributing, e.g., to the two-nucleon potential can easily

be identified. These are the time-honored one-pion exchange and two four-nucleon

contact interactions without derivatives. The so-constructed effective potential is

then iterated in the Schrodinger or Lippman–Schwinger equation, generating the

shallow nuclear bound states as well as scattering states. The resulting contribu-

tions at various orders to the 2N , the 3N and the 4N forces are depicted in Fig. 5.

Remarkably, by now the 2N , 3N and 4N force contributions have been worked out

to NNNLO, the last missing piece, namely the short-range and 1/mN -corrections

to the 3N forces, was only provided recently.98 This EFT approach shares a few

advantages over the very well developed and precise semi-phenomenological ap-

proaches, just to mention the consistent derivation of 2N , 3N and 4N forces as

well as electroweak current operators, the possibility to work out theoretical uncer-

tainties and to improve the precision by going to higher orders and, of course, the

direct connection to the spontaneously and explicitely broken chiral symmetry of

QCD. There has been a large body of work on testing and developing these forces

in few-nucleon systems, for comprehensive reviews see Refs. 99 and 100.

As one beautiful example of combining chiral perturbation theory calculations

and nuclear EFT, I want to discuss the recent extraction of the fundamental S-wave

pion–nucleon scattering lengths from the high-precision data on pionic hydrogen and

deuterium taken at PSI.101,102 To achieve the corresponding precision in theory, the

authors of Ref. 103 used chiral perturbation theory to calculate the π−d scattering

length with an accuracy of a few percent, including isospin-violating corrections both

in the two- and three-body sector. Here, two- and three-body refers to the photon

coupling to the two-nucleon and the two-nucleon plus one pion intermediate states,

in more conventional language the impulse approximation and the meson-exchange

current contributions, respectively. In particular, the isospin-breaking contributions

to the three-body part of aπ−d due to mass differences, isospin violation in the πN

scattering lengths, and virtual photons were studied. This last class of effects is

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2N LO

N LO3

NLO

LO

3N force 4N force2N force

Fig. 5. Contributions to the effective potential of the 2N , 3N and 4N forces based on Weinberg’s

power counting. Here, LO denotes leading order, NLO next-to-leading order and so on. Dimension

one, two and three pion–nucleon interactions are denoted by small circles, big circles and filled

boxes, respectively. In the 4N contact terms, the filled and open box denote two- and four-

derivative operators, respectively.

ostensibly infrared enhanced due to the smallness of the deuteron binding energy.

However, the authors of Ref. 103 showed that the leading virtual-photon effects that

might undergo such enhancement cancel, and hence the standard chiral perturbation

theory (Weinberg) counting provides a reliable estimate of isospin violation in aπ−d

due to virtual photons. This allowed to extract the isoscalar and isovector scattering

lengths to high precision, see also Fig. 6,

a+ = (7.6± 3.1) · 10−3

/Mπ , a− = (86.1± 0.9) · 10−3

/Mπ . (32)

Most remarkable is the fact that for the first time, the small isoscalar scatter-

ing length could be extracted with a definite sign, this was not possible based

on scattering data only, see e.g. Ref. 104. Also, one should note that the dom-

inant isovector scattering length could be determined with an uncertainty of 1%

only — this is truly amazing and demonstrates again the power of combining

EFT with chiral symmetry. In fact, the LO chiral perturbation theory result

of Weinberg already nicely captures the essence of these results, a+CA

= 0 and

a−CA

= (M2

π/8πF2

π)/(1 +Mπ/mp) = 79.5 · 10−3/Mπ, but only now one knows pre-

cisely how much these are affected by higher order corrections — the first attempt

to calculate these dates back almost two decades.105

Now I turn to the nuclear many-body problem. More precisely, this refers to

nuclei with atomic number A > 4. Nuclear lattice simulations combine the power

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220 Ulf-G. Meißner

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Fig. 6. Constraints in the a+-a− plane from the data on pionic hydrogen (level shift and width)

and pionic deuterium (level shift). For the precise relation between the quantity a+ and the

scattering length a+, see Ref. 103. Figure courtesy of Martin Hoferichter.

of EFT to generate few-nucleon forces with numerical methods to exactly solve the

non-relativistic A-body system, where in a nucleus A counts the number of neu-

trons plus protons. For a detailed review, I refer to Ref. 106 and here I give only

a very short account of this method. The basic idea is to introduce a smallest

length (the lattice spacing) in the spatial directions and in the temporal direction,

denoted a and at, respectively and then to discretize the finite space-time volume

L × L × L × Lt in integer numbers of a and at. A Wick rotation to Euclidean

space is naturally implied. Note that the lattice spacing entails an UV cutoff (a

maximal momentum), pmax = π/a. In typical simulations of atomic nuclei, one has

a ' 2 fm and thus pmax ' 300MeV. In contrast to lattice QCD, the continuum limit

a → 0 is not taken. This formulation allows to calculate the correlation function

Z(t) = 〈ψA| exp(−tH)|ψA〉, where t is the Euclidean time and |ψA〉 an A–nucleon

state. Using standard methods, one can derive any observable from the correlation

function, e.g. the ground-state energy is simply the infinite time limit of the loga-

rithmic derivative of Z(t) with respect to the time. Similarly, excited states can be

generated by starting with an ensemble of standing waves, generating a correlation

matrix Zji(t) = 〈ψjA| exp(−tH)|ψi

A〉, which upon projection onto internal quantum

numbers and diagonalization generates the ground and excited states — the larger

the initial state basis, the more excited states can be extracted. Another recently

developed method is based on position-space wave functions.107 In a first step, one

constructs from the general wave functions ψj(~n ) (j = 1, . . . , A) states with well-

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Table 2. The even-parity spectrum of 12C from nuclear lattice simulations. The ground state is

denoted as O+

1and the Hoyle state as O+

2. The NLO corrections include strong isospin breaking

as well as the Coulomb force. The NNLO corrections are generated by the leading three-nucleon

forces. The theoretical errors include both Monte Carlo statistical errors and uncertainties due to

extrapolation at large Euclidean time.

0+1

2+1

0+2

2+2

LO −96(2) MeV −94(2) MeV −88(2) MeV −84(2) MeV

NLO −77(3) MeV −72(3) MeV −71(3) MeV −66(3) MeV

NNLO −92(3) MeV −86(3) MeV −84(3) MeV −79(3) MeV

−80.7(4) MeV (Ref. 117)

Exp. −92.2 MeV −87.7 MeV −84.5 MeV −82.6 MeV (Ref. 118)

−81.1(3) MeV (Ref. 119)

defined momentum using all possible translations, L−3/2∑

~m ψj(~n+ ~m ) exp(i ~P · ~m).

A proper choice for the ψj allows one to prepare certain types of initial states, such

as shell-model wave functions,

ψj(~n ) = exp[−c~n2] , ψ′j(~n ) = nx exp[−c~n

2] , ψ′′j (~n ) = ny exp[−c~n

2] , . . . , (33)

or, for later use, alpha-cluster wave functions,

ψj(~n ) = exp[−c(~n− ~m)2] , ψ′j(~n ) = exp[−c(~n− ~m

′)2] , . . . . (34)

The possibility to construct all these different types of initial/final states is a reflec-

tion of the fact that in the underlying EFT all possible configurations to distribute

nucleons over all lattice sites are generated. This includes in particular the con-

figuration where four nucleons are located at one space-time point, so there is no

restriction like, e.g. in a no-core-shell model approach, in which one encounters se-

rious problems with the phenomenon of clustering, that is so prominent in nuclear

physics. It is also important to note that the nuclear forces have an approximate

spin–isospin SU(4) symmetry (Wigner symmetry)108 that is of fundamental impor-

tance in suppressing the malicious sign oscillations that plague any Monte Carlo

simulation of strongly interacting fermion systems at finite density. The relation

of the Wigner symmetry to the nuclear EFT formulation has been worked out in

Ref. 109.

As one application of this method, I want to discuss the so-called Hoyle state.

It plays a crucial role in the helium burning of stars heavier than our sun and in

the production of carbon and other elements necessary for life. This excited state

of 12C was postulated by Hoyle110 as a necessary ingredient for the fusion of three

α-particles to produce a sufficient amount of carbon at stellar temperatures. For

this reason, the Hoyle state plays also a very important role in the context of the

anthropic principle, although such considerations did not play any role when this

state was predicted.111 The Hoyle state was dubbed the “level of life” by Andrei

Linde.112 This excited state has been an enigma for nuclear structure theory since

decades, even the most successful Greens function MC methods based on realis-

tic two- and three-nucleon forces113 or the no-core-shell-model employing modern

(chiral or Vlow k) interactions114,115 have not been able to describe this state. The

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222 Ulf-G. Meißner

first ab initio calculation of the Hoyle state based on nuclear lattice simulations was

reported in Ref. 116. In the meantime, the calculation of the spectrum of 12C has

been considerably improved, using the aforementioned position-space initial and

final state wave functions. The predictions for the even-parity states in the 12C

spectrum are collected in Table 2. In all cases, the LO calculation is within 10%

of the experimental number, and the three-nucleon forces at NNLO are essential to

achieve agreement with experiment. The Hoyle state is clearly recovered and comes

out at almost the same energy as the 4He + 8Be threshold — thus allowing for

the resonant enhancement of carbon production that was first considered by Hoyle

half a century ago. Furthermore, one finds a second 2+ excited state that has been

much debated in the literature. Recent determinations of this level based on various

methods (alpha and proton scattering off 12C as well as an R-matrix analysis of the

β-decays of 12N and 12B) are in fair agreement with the calculation. The chiral

nuclear EFT will also allow one to investigate how the closeness of the Hoyle state

to the 4He + 8Be threshold depends on the fundamental parameters like the light

quark masses, thus allowing for a test of the anthropic principle. For a first attempt

within a alpha-cluster model, see Ref. 120.

11. Chiral Symmetry and Heavy Quarks

Naively, chiral symmetry has little to do with the physics of heavy quarks as the

sectors of light and heavy quarks are quite different in QCD. However, it was realized

in the early 1990s that chiral and heavy quark symmetry can indeed be intertwined

in decays and reactions of heavy hadrons. The implications of chiral symmetry

for heavy hadron physics was pioneered by Wise,121 Burdman and Donoghue122

and the group of Yan et al.123 This field has become ever more important since

then partly due to the many puzzling results that have emerged in charm quark

spectroscopy. The phenomenology of heavy meson chiral Lagrangians has been

reviewed in Ref. 124.

Arguably, the most interesting interplay between chiral and heavy quark dy-

namics are the decays of ψ′ into J/ψπ0 and J/ψη that were suggested to be a

reliable source for extracting the light quark mass ratio mu/md.125,126 See also the

review by Donoghue on “Light quark masses and chiral symmetry.”127 The decay

ψ′→ J/ψπ0 violates isospin symmetry. Both the up-down quark mass difference

and the electromagnetic (em) interaction can contribute to isospin breaking. How-

ever, it was shown that the em contribution to the decay ψ′→ J/ψπ0 is much

smaller than the effect of the quark mass difference.128,129 Based on the QCD mul-

tipole expansion and the axial anomaly, the relation between the quark mass ratio

1/R ≡ (md −mu)/(ms − m), where m = (md +mu)/2, and the ratio of the decay

widths of these two decays was worked out up to the next-to-leading order in the

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chiral expansion.130,131 One obtains

Γ(ψ′→ J/ψπ0)

Γ(ψ′→ J/ψη)

= 3

(

md −mu

md +mu

)2F 2

π

F 2η

M4

π

M4η

∣

∣

∣

∣

~qπ

~qη

∣

∣

∣

∣

3

, (35)

where Fπ(η) and Mπ(η) are the decay constant and mass of the pion (eta), respec-

tively, and ~qπ(η) is the pion (eta) momentum in the ψ′ rest frame. The most recent

measurement of the decay-width ratio,132 Rπ0/η = (3.88±0.23±0.05) allows one in

principle to extract the quark mass ratio R within a few percent precision, see e.g.

Ref. 133 for an earlier use of this approach. However, it was shown in Ref. 134 that

effects due to charmed meson triangle diagrams (dispersive effects) are much larger

than originally thought and that the sizeable uncertainty related to the calculation

of these effects makes a precise extraction of the light quark mass ratio from these

decays very difficult. Quite in contrast to the charmonium system, similar transi-

tions in the bottomnium spectrum that are sensitive to the quark masses are not

afflicted with such large heavy meson loop effects, as explained in detail in Ref. 135.

An intriguing implication of this suppression of the bottom meson loops in the tran-

sitions Υ(4S) → hbπ0(η) is that the decay amplitudes are dominated by the quark

mass differences, and hence it is possible to extract the light quark mass ratio from

the ratio of the branching fractions of the transitions Υ(4S) → hbπ0(η) with good

accuracy. The theoretical uncertainty on the extraction of the light quark mass

ratio mu/md has been estimated in Ref. 135 to be about 25%. Now it remains to

be seen how precisely the decays Υ(4S) → hbπ0 and Υ(4S) → hbη can be measured.

Finally, I only want to discuss a recent application of the interplay of chiral

symmetry, lattice QCD and charmonium physics.136 Charmonium states, the most

prominent one being the J/ψ, are bound states of a heavy quark and a heavy anti-

quark, so in general heavy quarkonia do not contain any valence light quarks. Thus,

one would naively expect that the light quark mass dependence of their properties

would be suppressed, so that one can use a simple formula linear in the light quark

masses to do the chiral extrapolation, as done in, e.g. Refs. 137 and 138 for the

mass splittings. The higher order corrections to the charmonium mass splittings

were already analyzed a long time ago in Ref. 139. While such a lowest order extrap-

olation might be sufficient for the low-lying states, a similar simple extrapolation

may not be reasonable for the higher excited states. In fact, dramatic and even

non-analytic dependences in the light quark masses can arise. Hence, for the ex-

cited states which are close to open-flavor thresholds, a formula taking into account

the non-analyticity should be utilized for the chiral extrapolation. Furthermore, for

the radiative transitions with strong coupled-channel effects, simulations at several

pion masses are necessary in order to extract the physical results. The effects of

light quarks in heavy quarkonium systems are due to quantum fluctuations of the

sea quarks. Sea quark and antiquark pairs are created and annihilated in the color

singlet heavy quarkonium. The low-energy fluctuations can be described in the

framework of chiral perturbation theory. The quarkonium states can be included

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224 Ulf-G. Meißner

as matter fields. Let me focus on the quark mass dependence of the quarkonium

mass. Two types of sea quark fluctuations are possible. Class (a) can be described

by closed light quark loops that are disconnected from the heavy quark–antiquark

systems. Such diagrams are suppressed according to the Okuba–Zweig–Iizuka rule.

Class (b) subsumes all diagrams in which the heavy quark (antiquark) and the vir-

tual sea antiquark (quark) can form a color singlet state, a heavy meson (antime-

son), i.e. a pair of virtual heavy meson and antimeson is created and annihilated

after a short propagation. Type (a) can be parametrized using an effective chiral

Lagrangian containing unknown low-energy constants. The resulting quark mass

dependence is analytic in the light quark masses up to chiral logarithms. Complex-

ity comes from fluctuations of type (b), which can lead to non-analyticity. This is

indeed the case for states that are close to open-flavor thresholds. For such a state,

the appropriate chiral extrapolation formula for its mass M takes the form

M(Mπ) =◦

M + dM2

π +√

e+ fM2π , (36)

where◦

M (the hadron mass in the chiral limit), d, e and f are parameters to be fit

to the lattice data. For states far away from any open-flavor threshold, e will be

much larger than fM2

π so that the square root can be expanded, and one may use

only the first two terms in the above equation up to O(M2

π). Explicit modeling of

various charmonium excitations indeed shows that the light quark mass dependence

is not always suppressed for heavy quarkonium systems.136 As an example, the

decay widths of the hindered M1 transitions between P-wave charmonia (see e.g.

Ref. 140) have a strong dependence on the pion mass. Lattice results at a pion

mass around 500 MeV can deviate by a factor of two from the actual values at the

physical pion mass. It would be interesting to investigate how such effects would

affect the recent lattice results for excited and exotic charmonium spectroscopy that

have been obtained for pion masses of about 400 MeV.141

12. Some Final Remarks

In this contribution, I have elucidated the role of chiral symmetry in hadronic and

nuclear physics, stressing the many connections between various seemingly disjunct

fields. Chiral perturbation theory as the effective field theory of the strong interac-

tions has matured over the years and methods have developed to either sharpen its

predictions or enlarge its range of validity. Also, the strange quark plays a special

role — chiral three-flavor dynamics is only one of the frontiers of research, where a

union of chiral symmetry and lattice simulations will play an even more important

role in the years to come. Furthermore, the fruitful interplay between chiral symme-

try and dispersion relations is expected to shed more light on scattering processes

and resonance excitation. In nuclear physics, three-nucleon forces are presently

at the forefront of structure research, where it is practically unthinkable to make

progress without invoking the powerful constraints from chiral symmetry. I would

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like to finish with one comment. Often it is said that chiral perturbation theory will

not be needed any more as the lattice starts to generate results at physical quark

masses. What a misconception! Time will show that chiral symmetry will continue

to be one major ingredients to understand the physics encoded in all these numbers

that will be generated from lattice QCD simulations in the years to come.

Acknowledgments

I would like to thank Ernest Henley for giving me the opportunity to contribute to

this volume. All my collaborators are thanked for sharing their insights into the

topics discussed here. I thank Veronique Bernard, Feng-Kun Guo and Dean Lee for

a careful reading of this manuscript. This work was supported in part by the DFG

(SFB/TR 16 and SFB/TR 110), the BMBF (grant 06BN9006), the HGF (grant

VH-VI-417) and the EU (FP7, HadronPhysics3).

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100 years of subatomic physics || chiral symmetry in subatomic physics

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