# Chiral perturbation theory

Post-publication activity

Curator: Heinrich Leutwyler


## Strong interaction at low energies

In the framework of the Standard Model, the strong interaction is described by Quantum Chromodynamics (QCD). The reason why an effective field theory is needed to analyze the low energy properties of this theory is that, in the case of QCD, the standard method of analysis (expansion in powers of the coupling constant) only works at high energies, where asymptotic freedom ensures that the interaction can be treated as a perturbation. At low energies, the relation between the degrees of freedom present in the Lagrangian of QCD (gluons, quarks) and those visible in the spectrum of physical states (mesons, baryons) cannot be analyzed in terms of an expansion in powers of the coupling constant. Many models that resemble QCD in one respect or the other have been proposed to understand this relation: constituent quarks, Nambu-Jona-Lasinio-model, linear $\sigma$ model, hidden local symmetry, holography, Anti-de-Sitter-space/Conformal Field Theory and many others. Some of these may be viewed as simplified versions of QCD that do catch some of the salient features of the theory at the semi-quantitative level, but none provides a basis for a coherent approximation scheme that would allow us, in principle, to solve QCD. Nonperturbative methods are required to analyze the low energy end of the spectrum. There are two methods that do not rely on an expansion in powers of the QCD coupling constant: effective field theory and simulation on a lattice.

At low energies, the most important property of the system is the energy gap, i.e. the difference between the energies of ground state and first excited state. In quantum field theory, the ground state is the vacuum, while the first excited state contains a single particle at rest, the lightest particle occurring in the spectrum of the theory. Since the lightest particle in QCD is the pion, the energy gap is given by $M_\pi c^2$. At low energies, the characteristic properties of QCD all derive from the fact that the pion is remarkably light, so that the energy gap is small.

## Symmetry

Already in 1960, more than a decade before the discovery of QCD, Nambu found out why the pion is so light: the strong interaction has a hidden, approximate symmetry (Nambu, 1960). At the time when Nambu proposed the idea, the strong interaction was not understood at all and the occurrence of approximate symmetries looked mysterious, but with the discovery of QCD, the mystery disappeared: in this theory, approximate symmetries do occur naturally, because the fermions come in several flavours. The interaction with the gluons is the same for all flavours - within QCD, the only difference between a $u$- and a $d$-quark, for instance, is that $m_u$ differs from $m_d$. If the masses of the two lightest quarks were the same, QCD would have an exact isospin symmetry: invariance under rotations in the internal space spanned by the two lightest flavours: $$\tag{1} q=\left\{ \hspace{0.1cm} \begin{array}{c}u\\d\end{array} \hspace{0.1cm}\right\}, \hspace{0.5cm} q'=V\cdot q, \hspace{0.5cm}V\in \mathrm{SU(2)\hspace{1cm}isospin \,\,rotation}$$ Heisenberg had introduced a symmetry group with this structure into nuclear physics, already in 1932 (Heisenberg, 1932). In the meantime, the evidence for the strong interaction to be approximately invariant under isospin rotations is overwhelming. In particular, the mesons and baryons do occur in nearly degenerate isospin multiplets. Disregarding the electromagnetic interaction, the splitting within these multiplets is due entirely to the difference between $m_u$ and $m_d$. For isospin to represent an approximate symmetry, the difference $m_u-m_d$ must be small and vice versa: if the difference is small, then QCD is approximately symmetric under the group SU(2) of isospin rotations.

Small compared to what? QCD has an intrinsic scale, $\Lambda_{QCD}$, which is independent of the quark masses and carries the dimension of an energy. Expressing the quark masses in energy units, the condition for isospin to be an approximate symmetry can be written as $|m_u − m_d|$ ≪ $\Lambda_{QCD}$. Note, however, that the mass of the proton, the mass of the ρ-meson or the pion decay constant can also serve as a reference scale: in the limit where the quark masses are sent to zero, all of these are proportional to $\Lambda_{QCD}$. The coefficient of proportionality is a number of order 1 and is calculable, at least in principle. These examples make it clear that the above inequality is compelling only in an algebraic sense. Numerically, the mass of the ρ-meson appears to be a more adequate measure of the scale of QCD than $\Lambda_{QCD}$.

Nambu's symmetry is connected with the fact that not only the difference between $m_u$ and $m_d$, but the masses themselves happen to be very small compared to the scale of QCD. In the theoretical limit where the two lightest quarks are both taken massless, QCD acquires an additional symmetry. The left- and right-handed components of the quark fields,

$q_\indR=\frac{1}{2}(1+\gamma_5)q\,,\hspace{2cm}q_\indL=\frac{1}{2}(1-\gamma_5)q\,,$

can then be subject to independent isospin rotations, $$\tag{2} q_\indR'=V_\indR\cdot q_\indR , \hspace{0.2cm} q_\indL'=V_\indL\cdot q_\indL ,\hspace{0.2cm}V_\indR,\,V_\indL\in \mathrm{SU(2)\hspace{1cm}chiral \,rotation}$$ The Lagrangian of QCD with two massless flavours remains invariant under this operation, a property referred to as chiral symmetry. For massless quarks, both the vector currents and the axial-vector currents are conserved. The group structure is of the form SU(2)$\times$SU(2). The isospin subgroup consists of those elements with $V_\indR=V_\indL$ and is generated by the charges of the vector currents.

## Spontaneous symmetry breakdown

Chiral symmetry is spontaneously broken or hidden: the Lagrangian of QCD with two massless flavours is symmetric under chiral rotations, but the state of lowest energy, the vacuum, is not. That this can happen was first observed in solid state physics, where the phenomenon of spontaneous magnetization was encountered long ago: although the Heisenberg model of a ferromagnet does not single out a direction for the magnetic moments, the configuration where all of these are aligned is lower in energy than configurations for which the different directions are equally likely. The magnetization chooses some direction and thereby breaks the symmetry with respect to rotations. Nambu pointed out that the spontaneous breakdown of a symmetry also occurs in particle physics and that this is the reason why the approximate SU(2)$\times$SU(2) symmetry is hidden: the ground state is approximately symmetric only under the subgroup SU(2) of isospin rotations.

The quark condensate

$\lvac \ubar u \rvac =\lvac \ubar_\indR u_\indL+\ubar_\indL u_\indR\rvac$

plays a role analogous to the magnetization and may be viewed as a quantitative measure for spontaneous symmetry breaking. If the ground state were invariant under a chiral rotation of the left-handed quark fields, the matrix element $\lvac \ubar_\indR u_\indL\rvac$ would have to vanish, because the operator $u_\indL$ is transformed into a linear combination of $u_\indL$ and $d_\indL$, while $\ubar_\indR$ stays put. Indeed, lattice calculations beautifully demonstrate that the quark condensate is different from zero - chiral symmetry is indeed spontaneously broken. Since the ground state is invariant under isospin rotations, the $u$- and $d$-quark condensates are the same. The numerical values, found when extrapolating lattice data to $m_u,m_d\rightarrow 0$, cluster around $\lvac \ubar u \rvac =\lvac \dbar d \rvac\simeq -(250\,\mathrm{MeV})^3$ [$\overline{\mathrm{MS}}$ scheme, running scale 2 GeV, all numerical values are given in units of energy; expressed in these units, $\hbar=c=1$].

## Nambu-Goldstone bosons

The spontaneous breakdown of an exact, continuous symmetry implies that the spectrum of the theory contains massless particles, referred to as Nambu-Goldstone bosons. In the case of a magnet, spin waves (magnons) play this role, while in the case of QCD, the spontaneous breakdown from SU(2)$\times$SU(2) to SU(2) is accompanied by the occurrence of three Nambu-Goldstone bosons with the quantum numbers of the pion: spin $0$, negative parity, isospin $1$. In the limit $m_u,m_d\rightarrow0$, which is referred to as the chiral limit, the symmetry becomes exact, the pion is strictly massless. In reality, we are dealing with an approximate symmetry: the pion is not massless, but it is significantly lighter than all other particles in the spectrum of QCD. In order to simplify the following discussion, I disregard isospin breaking and consider the theoretical world where $m_u=m_d$ and where the electromagnetic interaction is turned off, $e=0$. In this world, all three pions have the same mass.

At low energies, the structure of QCD is dominated by the Nambu-Goldstone bosons. In particular, pion exchange generates poles in matrix elements of the axial-vector current $A_i^\mu=\qbar\,\gamma^\mu\gamma_5\tfrac{1}{2}\!\tau_i\, q\,$ (the $2\times 2$ matrices $\tau_1,\tau_2,\tau_3$ occurring here stand for the Pauli matrices; they act in the space of the two lightest flavours). The two-point function formed with this current, for instance, contains such a pole: $$\tag{3} i\!\!\int\hspace{-1.5mm} \dx\, e^{i\, p\cdot x}\,\lvac T\,A^\mu_i(x)A^\nu_k(0) \rvac= \frac{F_\pi^2\,\delta_{ik}}{M_\pi^2-p^2}\;p^\mu p^\nu+\ldots$$ In this case, the residue of the pole is determined by the axial-vector current matrix element relevant for the decay $\pi^+\rightarrow\mu^+\nu_\mu$. Lorentz invariance determines the form of this matrix element up to a constant, which is denoted by $F_\pi$, $$\tag{4}\lvac A_i^\mu(x)|\pi_k(p)\rangle= i\, \delta_{ik}\,p^\mu\, F_\pi\,e^{-i\,p\cdot x}\,,$$ and is referred to as the pion decay constant. Numerically, $F_\pi\simeq 92$ MeV.

## Pion pole dominance

The hidden approximate symmetry not only requires the occurrence of approximately massless particles, but also governs their low energy properties. In particular, the symmetry implies that, at low energies, the Nambu-Goldstone bosons interact only weakly among themselves. Quite a few low energy properties of the strong interaction were derived from the fact that the pion pole contributions dominate the amplitudes at low momenta and low energies, starting, in 1958, with the Goldberger-Treiman relation (Goldberger and Treiman, 1958). An important step in this development was current algebra, a framework introduced by Gell-Mann in 1962 (Gell-Mann, 1962). Using current algebra and pion pole dominance, Weinberg showed in 1966 (Weinberg, 1966) that chiral symmetry fully determines the interaction among pions of low energy, in terms of the pion decay constant: at leading order in the expansion in powers of the pion momenta and the pion mass, the amplitude of the elastic collision $\pi+\pi\rightarrow \pi+\pi$ can be expressed in terms of the function $$\tag{5} A(s,t,u)=\frac{s-M_\pi^2}{F_\pi^2}+\ldots\,,$$ where $s,t,u$ are the Mandelstam variables of the reaction (the explicit expression for the scattering amplitude is given in equation (35), section $\pi\pi$ scattering). The formula illustrates the statement that, at low energies, the pions only interact weakly: if the relative velocity of the incoming particles tends to zero, the square of the center of mass energy, $s$, approaches $4M_\pi^2$, so that $A(s,t,u)$ tends to $3M_\pi^2/F_\pi^2$ and hence indeed disappears in the chiral limit - at zero energy, Nambu-Goldstone bosons behave like free particles.

In 1968, Gell-Mann, Oakes and Renner (Gell-Mann, Oakes, and Renner, 1968) then showed that the square of the mass of the Nambu-Goldstone bosons grows in proportion to $m_u+m_d$: $$\tag{6} M_{\pi}^2=(m_u+m_d)B + O(m^2)\,,\hspace{0.8cm} B=\left \vert \hspace{0.05cm}\frac{\lvac \ubar u \rvac }{F_\pi^2}\hspace{0.05cm}\right \vert_{\hspace{0.1cm}m_u,m_d\rightarrow 0}\,.$$ The constant $B$ is proportional to the quark condensate, which as discussed above measures the strength of spontaneous symmetry breaking. The GMOR relation (6) thus states that the square of the pion mass is proportional to the product of a sum of quark masses and the quark condensate. While the former explicitly break the chiral symmetry of the QCD Lagrangian, the latter measures the strength of spontaneous symmetry breaking.

## Effective field theory

Weinberg noted that the results derived from current algebra and pion pole dominance can be obtained in a much simpler way, from a field theory that involves pion fields as dynamical variables instead of quarks and gluons (Weinberg, 1979). The reason is that at low energies, the amplitudes are dominated by the pole terms arising from pion exchange: while the behaviour of QCD in the ultraviolet is governed by the quarks and gluons, the pions dominate the scenery in the infrared. The contributions generated by pion exchange may be viewed as tree graphs of an effective field theory that contains three pion fields, $\vec{\pi}(x)=\{\pi_1(x),\pi_2(x),\pi_3(x)\}$. The internal lines occurring in these graphs stand for pion pole terms $\propto (M_\pi^2-p^2)^{-1}$. In the language of the effective theory, the lines represent pion propagators.

The vertices can be expanded in powers of the momenta and the series can be ordered by counting powers of momenta. In the language of the effective theory, the vertices represent interaction terms in the Lagrangian. Momentum factors in the vertices translate into derivatives of the pion field. For pions on the mass shell, $p^2=M_\pi^2$, the momentum is of the same order as the pion mass. According to equation (6), $M_\pi^2$ is proportional to the sum of the quark masses. It is therefore convenient to treat the symmetry breaking parameters $m_u,m_d$ as quantities of $O(p^2)$. In this bookkeeping, the leading term in the low energy expansion of the two-point function in equation (3), for instance, represents a contribution of $O(1)$, while the expansion of the $\pi\pi$ scattering amplitude in (5) or of the square of the pion mass in (6) both start at $O(p^2)$. The simultaneous expansion in powers of momenta and quark masses is referred to as the chiral expansion and the resulting effective field theory is called Chiral Perturbation Theory ($\chi\hspace{-0.1em}$PT). In the following, I outline the main features of this framework, focusing on the foundations of the method and illustrating it with one or two examples. For a more thorough discussion of the work done in $\chi\hspace{-0.1em}$PT, I refer to the literature quoted in section Further reading, at the end of the present article.

In view of the derivative couplings occurring in the Lagrangian, $\chi\hspace{-0.1em}$PT is not renormalizable in the standard sense: it cannot be characterized by a finite number of coupling constants. Also, the tree graphs of a field theory cannot be the full story: for the S-matrix to be unitary, loop graphs cannot simply be discarded. The ordering in powers of momenta and quark masses is crucial for $\chi\hspace{-0.1em}$PT to represent a coherent framework (Weinberg, 1979): to a given order in the chiral expansion, only a finite number of graphs contributes. In particular, the leading contribution to the S-matrix is given by the tree graphs of the effective theory. At next-to-leading order (NLO), graphs containing one loop contribute, those with two loops enter at NNLO, etc. Moreover, the divergences occurring in the loop graphs can be absorbed in the bare coupling constants of the effective Lagrangian, so that the effective theory does represent a renormalizable framework, order by order in the chiral expansion.

## Foundations of Chiral Perturbation Theory

The original demonstration of the fact that the low energy structure of the strong interaction can be analyzed in terms of an effective pion field theory relied on heuristic arguments, in particular on what Weinberg (Weinberg, 2009) refers to as a folk theorem: "if one writes down the most general possible Lagrangian, including all terms consistent with the assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition, and the assumed symmetry properties."

In the meantime, the method has been put on firm footing. The basic elements of the framework introduced in (Gasser and Leutwyler, 1984), (Gasser and Leutwyler, 1985) are the Green functions formed with the local operators of the underlying theory. The symmetry properties, which play a central role in the low energy analysis, manifest themselves through the Ward-Takahashi identities, which express current conservation in terms of the Green functions. As shown in (Leutwyler, 1994b), Chiral Perturbation Theory does yield the general solution of these identities, order by order in the chiral expansion. From a mathematical point of view, the general solution involves an infinite string of free parameters (at a given order of the expansion, only a finite number of these are relevant). In the framework of the effective theory, the free parameters are the coupling constants of the effective Lagrangian, which are also referred to as low energy constants or LECs. I do not discuss the proof here, but add two remarks concerning the above folk theorem:

(i) The first concerns the occurrence of anomalies. As is well known, not all of the symmetries of a classical field theory survive quantization: as a consequence of the short distance singularities of the currents, some of the WT identities may contain anomalies.As the low energy analysis set up in (Gasser and Leutwyler, 1984), (Gasser and Leutwyler, 1985), (Leutwyler, 1994b) is based on these identities, it accounts for the anomalies ab initio. In the effective theory, their presence does not show up at leading order, but at NLO, the anomalies do in general affect the form of the effective Lagrangian, through a Wess-Zumino-Witten term. Apart from that specific contribution, which is fully determined by group geometry, the effective Lagrangian does have the assumed symmetry properties.

(ii) Lorentz invariance plays a crucial role in this context. For nonrelativistic systems, the folk theorem in general fails (Leutwyler, 1994a). The Heisenberg model of a ferromagnet, for instance, is invariant under rotations of the spin of the particles, but the effective Lagrangian is invariant only up to a total derivative. The phenomenon is related to a topological invariant, the Brouwer degree, which enters the effective Lagrangian already at leading order (Leutwyler, 1994a), (Bär, Imboden, and Wiese, 2004). The structure of this term is similar to the WZW term, but the phenomenon does not originate in the short distance behaviour of the currents: the WT identities obeyed by the Green functions of a ferromagnet are anomaly free.

## Green functions and external fields

The external field method is a very efficient tool in the analysis of the low energy structure. It is based on the observation that the Green functions formed with the local operators of the theory describe the response of the system to the perturbations generated if the Lagrangian of the theory is supplemented with terms that are proportional to these operators - the proportionality coefficients are referred to as external fields. In the case of QCD, the local operators of interest in the present context are the vector- and axial-vector currents as well as the scalar and pseudoscalar densities, so that the perturbation takes the form $$\tag{7} {\cal L}= {\cal L}_\QCD^0+{\cal L}_{\mathrm{\it ext}}\,,\hspace{0.5cm} {\cal L}_{\mathrm{\it ext}}= \qbar\,\gamma^\mu(v_\mu+a_\mu \gamma_5)q-\qbar (s-i\,\gamma_5\,p )q\,.$$ The external fields $v_\mu(x), a_\mu(x),s(x),p(x)$ represent colour neutral 2$\times$2 matrices, which act on the flavour of the quarks: $$\tag{8} v_\mu=\frac{1}{2} v_\mu^i\,\tau_i\,,\hspace{0.3cm}a_{\mu}=\frac{1}{2}a_{\mu}^i\,\tau_i\,,\hspace{0.3cm} s= s^0{\bf 1}+s^i \tau_i\,,\hspace{0.3cm}p= p^0{\bf 1}+p^i\tau_i\,.$$ The mass term of the two lightest quarks represents a linear combination of the scalar currents $\ubar u$, $\dbar d$. It is convenient to absorb the corresponding part of the quark mass matrix $$\tag{9}{\cal M}=\left\{\begin{array}{cc}m_u&0\\0&m_d\end{array}\right\}$$ in the scalar external field, $s={\cal M}+s'$. The unperturbed Lagrangian ${\cal L}_\QCD^0$ occurring on the right hand side of (7) then describes QCD with two massless flavours, while the quark fields $s,c,b,t$ are equipped with their physical masses. The external fields $v_\mu(x)$ and $a_\mu(x)$ enter together with the derivative of the quark field. In the chiral expansion, they count as quantities of $O(p)$, while $s(x)$ and $p(x)$ count as terms of order $O(p^2)$, like $m_u$ and $m_d$.

If the external fields are taken to be different from zero only in a finite region of space-time, then one may consider two different asymptotic states: the state $|0\,\mathrm{in},\hspace{-0.5mm}f\rangle$, where the system was in the state of lowest energy in the remote past, and the state $|0\,\mathrm{out},\hspace{-0.5mm}f\rangle$, where it will wind up in the ground state in the remote future - as indicated, these states depend on the external fields, which I collectively denote by $f=\{v,a,s,p\}$. The probability amplitude for the system to end up in the ground state if it started there, $\langle 0\,\mathrm{out},\hspace{-0.5mm}f|0\,\mathrm{in},\hspace{-0.5mm}f\rangle$, plays a central role in the external field method: it implicitly contains all of the Green functions of the theory. Indeed, the probability amplitude may be expressed in terms of the unperturbed vacuum $\!\rvac$ as \begin{equation*} \langle 0\,\mathrm{out},\hspace{-0.5mm}f|0\,\mathrm{in},\hspace{-0.5mm}f\rangle = \lvac T\exp i\!\!\int\hspace{-1.5mm}\dx\, {\cal L}_{\mathrm{\it ext}}\rvac\,. \end{equation*} The formula shows that the expansion of the probability amplitude in powers of the external fields yields the Green functions formed with the operators collected in ${\cal L}_{\mathrm{\it ext}}$: $\lvac \qbar q\rvac$, $\lvac T\,V^\mu_i(x)V^\nu_k(y) \rvac$, $\lvac T\,A^\mu_i(x)A^\nu_k(y) \rvac$, $\ldots$ More precisely, since $m_u$ and $m_d$ are included in the external scalar field, the expansion gives the Green functions in the chiral limit. To instead extract those for QCD with the physical values of $m_u,m_d$, the generating functional must be expanded around the point $s={\cal M}$ rather than around $s=0$.

## Generating functional, gauge invariance

It is convenient to remove the disconnected parts of the Green functions. A well-known formula of statistical mechanics shows that this can be done by writing the probability amplitude in the form an exponential: $$\tag{10}\exp i \,S\{v,a,s,p\} \equiv \lvac T\exp i\!\!\int\hspace{-1.5mm}\dx\, {\cal L}_{\mathrm{\it ext}}\rvac\,.$$ The quantity $S\{v,a,s,p\}$ is referred to as the generating functional. It collects the connected parts of the Green functions formed with the operators $V^\mu_i$, $A^\mu_i$, $\qbar\, q$, $\qbar\,\tau_i\hspace{0.05cm}q$, $\qbar\, i\gamma_5\, q$ and $\qbar\, i\gamma_5\tau_i\hspace{0.05cm}q$.

The efficiency of the external field method is illustrated by the fact that, in this framework, the WT identities take a remarkably simple form: they hold if and only if the generating functional is invariant under a gauge transformation of the external fields (Bell 1967). The right- and left-handed fields, $$\tag{11}r_\mu \equiv v_\mu+a_\mu\,,\hspace{1cm} l_\mu \equiv v_\mu-a_\mu\,,$$ transform with $V_\indR$ and $V_\indL$, respectively, $$\tag{12} r_\mu' = V_\indR r_\mu V_\indR^\dagger+i\, V_\indR \partial_\mu V_\indR^\dagger\,,\hspace{0.5cm} l_\mu'= V_\indL l_\mu V_\indL^\dagger+i \,V_\indL \partial_\mu V_\indL^\dagger\,,$$ while the transformation rule for the scalar and pseudoscalar fields reads $$\tag{13} s'+\,i\,p'= V_\indR(s+i\,p)V_\indL^\dagger\,.$$ The external fields thus promote the global chiral symmetry of the Lagrangian of QCD to a local symmetry: the SU(2)-matrices $V_\indR$, $V_\indL$ need not be constant, but may depend on space and time.

It is important here that the WT identities for the Green functions built with the currents $V_\mu^i$, $A_\mu^i$ do not contain anomalies. When extending the framework to include isoscalar currents, for instance, anomalies do occur. In particular, the electromagnetic current $j^\mu(x)$ does contain an isoscalar component. The WT identities for some of the Green functions containing this current, such as $\lvac TA^\mu_i(z) j^\rho(x) j^\sigma(y)\rvac$, do contain anomalies. In fact, these anomalies play a crucial role in our understanding of the decay $\pi^0\rightarrow\gamma\gamma$: at leading order of the chiral expansion, the above three-point-function can explicitly be calculated from the anomalies contained therein. Since the resulting expression exclusively involves the constant $F_\pi$, the decay rate can be predicted - the agreement with the observed lifetime of the neutral pion provides one of the basic tests of the theory.

Anomalies complicate life, but do not change the structure of the effective theory in an essential way. In the following, I stick to the anomaly free situation specified above, where the set of all Ward-Takahashi identities are obeyed if and only if the generating functional is gauge invariant.

## Effective Lagrangian at leading order

Consider first the generic situation, where the Lagrangian is invariant under the Lie group $G$ and the symmetry breaks down spontaneously, the ground state being invariant only under the subgroup $H\subset G$. In geometric terms, the variables of the effective theory needed to analyze the low energy structure of such a system represent the coordinates of the coset space $C=G/H$. In particular, the number of fields needed is given by the dimension of $C$, that is by dim[$G$] - dim[$H$]. In the case of QCD with two massless quarks, $G=\mathrm{SU(2)}\times\mathrm{SU(2)}$ and $H=\mathrm{SU(2)}$, so that $C=\mathrm{SU(2)}$: the variables of the corresponding effective theory represent the coordinates of SU(2).

Any unitary, unimodular 2$\times$2 matrix $U$ can be decomposed as $$\tag{14}U= \U^0\,{\bf 1}+i\hspace{0.1cm} \vec{\U}\cdot\vec{\tau}\co$$ where the four-vector $\U=\{\U^0,\U^1,\U^2,\U^3\}$ is real and of unit length. The components of the matrix $U$ thus represent linear combinations of those of the vector $\U$. The Nambu-Goldstone bosons of QCD with two massless flavours may be viewed as living on the group SU(2) or, equivalently, on a unit sphere embedded in four-dimensional Euclidean space, $C=S^3$.

A model of quantum field theory that does live on $S^3$ is well-known: the nonlinear $\sigma$ model. The dynamical variables are spinless fields that may be collected in a four-vector of unit length, $\U(x)=\{\U^0(x),\U^1(x),\U^2(x),\U^3(x)\}$. The Lagrangian reads $$\tag{15}{\cal L}_\sigma=\frac{1}{2}F^2 \partial^\mu \U \partial_\mu \U^T\, ,\hspace{0.5cm}\mathrm{with}\hspace{0.2cm}\U \,\U^T=1.$$ Both ${\cal L}_\sigma$ and the Lagrangian ${\cal L}_\QCD^0$ in equation (7) have a global SU(2)$\times$SU(2) symmetry. Indeed, ${\cal L}_\sigma$ represents the leading term in the chiral expansion of the effective Lagrangian of massless QCD. In the underlying theory, the symmetry group generates chiral rotations of the quark fields - at the level of the effective theory, it generates rotations of the vector $\U$. The action of SU(2)$\times$SU(2) on this vector takes a very simple form in matrix notation, where it can be represented as $$\tag{16} U'= V_\indR\,U\,V_\indL^\dagger\,.$$

The expression (15) represents the leading effective Lagrangian only in the absence of external fields. Turning them on, ${\cal L}_{QCD}^0\rightarrow {\cal L}_{QCD}^0+{\cal L}_{ext}$, the effective Lagrangian picks up additional contributions:

(i) The partial derivatives are replaced by covariant ones. In matrix notation, these take the form $$\tag{17}D_\mu\hspace{0.05cm} U=\partial_\mu\hspace{0.05cm} U -i\,r_\mu\,U+i\,U\,l_\mu\, .$$ (ii) The scalar and pseudoscalar external fields generate further terms. The transformation property (13) implies that the two vectors $$\tag{18}\chi\equiv 2\hspace{0.02cm}B\hspace{0.05cm}\{s^0,p^1,p^2,p^3\}\hspace{0.3cm}\mathrm{and}\hspace{0.3cm} \tilde{\chi}\equiv 2\hspace{0.02cm}B\hspace{0.05cm}\{p^0,-s^1,-s^2,-s^3\}$$ both transform in the same way as the vector $\U$, so that the expressions $\chi \,\U^T$ and $\tilde{\chi}\,\U^T$ are both invariant. The first is even under space reflections, the second is odd. The fact that the electric dipole moment of the neutron is too small to be seen with the presently available methods implies that the strong interaction conserves CP to a very high degree of accuracy. In QCD, this property implies that the theory is separately invariant under charge conjugation and under parity.

Claim: The most general gauge invariant expression consistent with the symmetries of QCD that can be formed within the effective theory at $O(p^2)$ reads $$\tag{19} {\cal L}_\eff^{(2)}=\frac{1}{2} F^2 D^\mu \U D_\mu\, \U^T +F^2\chi\, \U^T \fs$$ Proof: To verify that this indeed represents the most general expression, one may first observe that functions which depend on the vector $\U$ alone can only be invariant under chiral rotations if they are constant over the manifold on which the effective field lives. Hence terms that do not contain derivatives or external fields are without interest. At $O(p)$, Lorentz invariance does not allow an invariant and, at $O(p^2)$ at most two derivatives can occur. Since terms of the form $\partial_\mu f^\mu$ can be added without changing the action, one may use integration by parts to remove second derivatives. Gauge invariance then implies that the first derivatives of the effective field can only occur together with the external fields, in the combination $D_\mu U=D_\mu\, \U^0\,{\bf 1}+i\hspace{0.1cm} D_\mu\,\vec{\U}\cdot\vec{\tau}$. Hence these derivatives can only enter via the product $D_\mu\, \U^A D_\nu\, \U^B$, which contains a single invariant under chiral rotations. Lorentz invariance then selects the combination occurring in the first term of (19). Finally, since the external scalar and pseudoscalar fields count as quantities of $O(p^2)$, they can only enter linearly - as discussed above, the second term in (19) is the only invariant permitted by invariance under chiral rotation and parity.

This shows that, at leading order of the chiral expansion, the effective field theory of QCD with two massless quarks contains two low energy constants: $F,B$. The second one specifies the normalization of the field $\chi(x)$, while the external fields contained in $\tilde{\chi}(x)$ do not show up at leading order. The notation is explained in section Discussion: while $F$ represents the value of the pion decay constant $F_\pi$ in the limit $m_u,m_d\rightarrow 0$, the constant $B$ coincides with the one occurring in the Gell-Mann-Oakes-Renner relation (6).

As mentioned earlier, only the tree graphs of the effective theory contribute at leading order. These represent the corresponding classical field theory, where $\U(x)$ sits at the extremum of the classical action and obeys the classical equations of motion. The classical solution relevant for the time-ordered Green functions is the one that contains only positive (negative) frequencies for $t\rightarrow+\infty (-\infty)$.

For $v_\mu=a_\mu=p=0$ and $s={\cal M}$, the extremum occurs at $\bar{\U}= \{1,0,0,0\}$ and remains there if the quark masses are sent to zero, so that the symmetry becomes exact. The ground state thus spontaneously breaks the symmetry to the subgroup of those elements that leave the ground state invariant. In matrix notation, the ground state is characterized by $\bar{U}={\bf 1}$. According to equation (16), the ground state is thus invariant under those elements with $V_\indR=V_\indL$. As discussed above, these form the isospin subgroup. The perturbation generated by the external fields modifies the solution of the equations of motion, but - as long as the perturbation is small - the classical solution $\U(x)$ remains in the vicinity of the vector $\bar{\U}$.

The effective theory also yields a representation for the generating functional. At leading order, only the tree graphs contribute: the chiral expansion of the generating functional starts with the classical effective action, $$\tag{20}S\{v,a,s,p\}= \underset{\U}{\mathrm{extremum}} \hspace{0.1cm}\int\hspace{-1.5mm}\dx\,{\cal L}^{(2)}_\eff+\ldots$$ The formula states that the Green functions of QCD with two massless quarks can explicitly be calculated to leading order of the chiral expansion by evaluating the classical action of the nonlinear $\sigma$ model - the two theories could barely be more different, but the leading low energy behaviour of the Green functions is the same. The reason is that (i) the low energy properties are determined by the local symmetries of the theory and (ii) the nonlinear $\sigma$ model has the same local symmetries as QCD with two massless quarks: the effective Lagrangian (19) is gauge invariant under the simultaneous transformation of the external and effective fields specified in equations (12), (13) and (16).

## Discussion

The evaluation of the contribution to the classical action that is bilinear in the external field $a_\mu$ immediately leads to the expression (3) for the two-point function of the axial-vector current, except that the quantities $F_\pi$ and $M_\pi^2$ occurring in that expression are replaced by $F$ and $(m_u+m_d)B$, respectively. The calculation thus confirms that the square of the pion mass grows in proportion to the sum of the two lightest quark masses. Moreover, it shows that the two LECs occurring in the leading order effective Lagrangian represent the values of the pion decay constant and of the Gell-Mann-Oakes-Renner constant at leading order of the expansion in powers of $m_u,m_d$: $$\tag{21} F= F_\pi\hspace{0.03cm}\vert_{\hspace{0.1cm}m_u,m_d\rightarrow 0} \,,\hspace{0.8cm}B= M_{\pi}^2/(m_u+m_d)\hspace{0.03cm}\vert_{\hspace{0.1cm}m_u,m_d\rightarrow 0} \,.$$

The explicit expression (19) shows that, in ${\cal L}^{(2)}_\eff$, only the isoscalar component $s^0$ of the scalar external field enters. Since the quark mass difference $m_u-m_d$ is contained in the component $s^3$ of the external scalar field, it does not show up at all: at leading order only the mean of the masses of the two lightest quarks matters, $$\tag{22} m_{ud}=\frac{1}{2}(m_u+m_d)\fs$$ This implies that, in QCD, isospin breaking effects can manifest themselves only at higher orders of the chiral expansion - at leading order, chiral symmetry protects not only the pion mass, but the entire generating functional from isospin breaking. Note that the statement only holds in the limited framework considered here: currents formed with $u$- and $d$-quarks and pion matrix elements. Neither the mass of the kaon, nor the mass of proton and neutron are protected from first order isospin breaking.

The hidden symmetry requires the Nambu-Goldstone bosons to interact among themselves. To work out the form of this interaction, it is convenient to identify the pion field with $\vec{\pi}\equiv F\{\U^1,\U^2,\U^3\}$ and to solve the constraint $\U\, \U^T=1$ for $\U^0$. Setting $s={\cal M}$, switching the other external fields off and expanding in powers of $\vec{\pi}$, the Lagrangian turns into a string of terms involving 0, 2, 4, $\ldots$ pion fields: \begin{align} \tag{23} {\cal L}_\eff^{(2)} &= F^2M^2+ \frac{1}{2}\partial^\mu\vec{\pi}\cdot\partial_\mu\vec{\pi} -\frac{1}{2}M^2\,\vec{\pi}^{\,2}\\ &\quad+\frac{1}{2}F^{-2}(\vec{\pi}\cdot\partial^\mu\vec{\pi})(\vec{\pi}\cdot\partial_\mu\vec{\pi})-\frac{1}{8}F^{-2} M^2(\vec{\pi}^{\,2})^2+O(\vec{\pi}^{\,6})\,,\nonumber \end{align} with $M^2\equiv 2 B m_{ud}$. The kinetic part confirms that the pions propagate with mass $M$. Those terms in equation (23) that involve four pion fields describe elastic $\pi\pi$ scattering. The evaluation of the corresponding tree graphs immediately leads to the expression quoted in equation (5), which is inversely proportional to the square of the pion decay constant. The calculation also shows that the interaction is of derivative type. In the chiral limit, the scattering amplitudes vanish at zero momentum and grow with the square of the momenta, irrespective of the number of particles involved. The quark masses, which break the symmetry, generate an interaction $\propto m_{ud}$, which is momentum independent and hence survives at zero momentum.

Above, the vector $\vec{\pi}$ was identified with $F\hspace{0.05cm}\vec{\U}$. Any other set of three coordinates that parametrize the coset space $C$ may be used for this purpose, canonical coordinates, for instance, defined by $U=\exp i\hspace{0.05cm}\vec{\pi}\cdot\vec{\tau}/F$. The choice of the parametrization is irrelevant, because it does not affect physical quantities - the S-matrix, as well as the Green functions formed with the local operators of QCD are reparametrization invariant. In early $\chi\hspace{-0.1em}$PT calculations, the situation was much less transparent; the freedom in the choice of the coordinates was a plague, because the form of the interaction among the pions and the pion propagator do depend on the choice of the coordinates.

Side remark: Investigating the Green functions of the field $\vec{\pi}(x)$ amounts to be working with the generating functional obtained by adding a perturbation of the form $\vec{f}(x)\cdot\vec{\pi}(x)$ to the effective Lagrangian, where $\vec{f}(x)$ is an external field. Since the vector $\vec{\pi}(x)$ transforms in a complicated, nonlinear manner under the action of the symmetry group, the term $\vec{f}(x)\cdot\vec{\pi}(x)$ ruins the symmetry properties of the effective Lagrangian - it is extremely tedious to control the effect of such a perturbation beyond tree level. The problem arises because there is no counter part to $\vec{\pi}(x)$ in QCD: the Green functions of $\vec{\pi}(x)$ are devoid of physical significance.

## Higher orders

The leading order effective Lagrangian in equation (19) is of $O(p^2)$. At $O(p^4)$, the general solution of the Ward identities contains 7 low energy constants $(\ell_1,\ldots,\ell_7)$ and 3 contact terms ($h_1,h_2,h_3)$ (Gasser and Leutwyler, 1984), (Gasser and Leutwyler, 1985): \begin{align} \tag{24}{\cal L}^{(4)}_\eff &= \ell_1\,(D^\mu \U D_\mu\, \U^T)^2+\ell_2\,(D^\mu \U D^\nu \U^T)(D_\mu\, \U D_\nu\, \U^T)+ \ell_3\,(\chi\, \U^T)^2 \\ &\quad+ \ell_4\,(D^\mu \chi D_\mu\, \U^T)+\ell_5\,(\U F^{\mu\nu}F_{\mu\nu}\,\U^T)+ \ell_6\,(D^\mu \U F_{\mu\nu}D^\nu \U^T )\no &\quad+ \ell_7\,(\tilde{\chi}\,\U^T)^2+ h_1\,\chi \chi^T+h_2\,\mbox{tr}\,F_{\mu\nu}F^{\mu\nu}+h_3 \,\tilde{\chi}\tilde{\chi}^T\,. \nonumber \end{align} The contact terms do not contain the pion field and hence generate contributions to the Green functions that are proportional to $\delta$-functions or derivatives thereof - they vanish unless all arguments coincide. Contact terms are needed to absorb the divergences occurring in the one loop graphs. Such terms also show up in the underlying theory: a term proportional to the square of the electromagnetic field strength, for instance, is needed to renormalize the QCD contributions to the electric charge, that is, to the coupling constant that specifies the strength of the electromagnetic interaction. Since the values of the contact terms depend on the convention used in the definition of the relevant Green functions, they cannot occur in observable quantities, so that their numerical values are not of physical interest.

In contrast to the leading order effective Lagrangian, the contribution of $O(p^4)$ in equation (24) does contain the quark mass difference $m_d-m_u$, via the field $\tilde{\chi}$: if $s(x)$ is replaced by the quark mass matrix and $p(x)$ is turned off, this field reduces to $B\,\{0,0,0,m_d-m_u\}$. In particular, the term $\ell_7\,(\tilde{\chi}\,U^T)^2$ does generate a difference between the masses of the charged and neutral pions, proportional to $\ell_7\,(m_d-m_u)^2$: isospin breaking does manifest itself at first nonleading order of the chiral expansion.

## Mass of the pion to one loop

In order to illustrate how the effective theory works, I explicitly discuss one example, evaluating the pion mass at next-to-leading order of the chiral perturbation series. The Gell-Mann-Oakes-Renner formula (6) only yields the leading term in the expansion of $M_\pi^2$ in powers of $m_u,m_d$. In the framework of the effective theory, the contribution of NLO can be worked out in the standard way, known from perturbative quantum field theory. Unavoidably, this calculation makes use of the corresponding methods. If the reader is not familiar with this language, I recommend to jump to the result of the calculation, which is given in equation (33).

Figure 1: Self-energy graphs for the pion mass at NLO

For simplicity, I again consider the isospin limit, that is, set $m_u= m_d$ and denote the common value by $m_{ud}$. The pion mass can be read off, for instance, from the pole position in the Fourier transform of a two-point function, such as $\lvac TA^i_\mu(x)A^k_\nu(y)\rvac$. Two self-energy graphs contribute at NLO: the tree graph in Figure 1a , which contains a vertex from ${\cal L}^{(4)}_\eff$ proportional to the LEC $\ell_3$ and the tadpole graph in Figure 1b, which exclusively contains propagators and vertices from ${\cal L}^{(2)}_\eff$. The tadpole graph is similar to the one arising in the perturbation series of the $\lambda\phi^4$ interaction, but the second line in equation (23) shows that the effective theory contains two types of interaction vertices with four pion legs: in addition to the term proportional to $(\vec{\pi}^2)\hspace{-0.2mm}^2$, there is a vertex containing derivatives of the pion field. Evaluating these graphs, one finds that the pole is located at \begin{align} \tag{25}M_\pi^2=M^2+\frac{2\ell_3 M^4}{F^2}+\frac{M^2}{2F^2}\frac{1}{i}\Delta_M(0)+O(M^6) \end{align} (recall that $M^2$ differs from the mean mass of the two lightest quarks only by a proportionality constant, $M^2\equiv 2B\,m_{ud}$). The first two terms are the tree graph contributions from ${\cal L}^{(2)}_\eff$ and ${\cal L}^{(4)}_\eff$, respectively, while the third represents the tadpole graph in Figure 1b, which involves the propagator of a particle of mass $M$, \begin{align} \tag{26}\Delta_M(x)=\frac{1}{(2\pi)^{d}}\!\int\hspace{-0.8mm} d^{\hspace{0.1mm}d}\hspace{-0.4mm}p\, \frac{e^{-ip\cdot x}}{M^2-p^2-i\epsilon}\fs \end{align} In the present context, only the value of the propagator at the origin is relevant. In dimensional regularization, the explicit expression reads: \begin{align} \tag{27}\Delta_M(0)=i\,(4\pi)^{-\frac{d}{2}}\Gamma(1-\frac{d}{2})M^{d-2}\fs \end{align}

## Renormalization

In four dimensions, $\Delta_M(x)$ explodes at short distances, in proportion to $1/x^2$, so that $\Delta_M(0)$ is infinite. In dimensional regularization, the divergence manifests itself through a pole at $d=4$: $$\tag{28}\Gamma(1-\frac{d}{2})=\frac{2}{d-4}+\ldots$$ It is convenient to isolate the divergent part in terms of the quantity $$\tag{29}\lambda(\mu)=\frac{1}{2}(4\pi)^{-\frac{d}{2}}\Gamma(1-\frac{d}{2})\mu^{d-2}\co$$ which is independent of $M$, but instead contains the running scale $\mu$. In the vicinity of $d=4$, $\Delta_M(0)$ can be written as $$\tag{30}\Delta_M(0)=2M^2\left\{\lambda(\mu)+\frac{1}{32\pi^2}\ln\frac{M^2}{\mu^2}+O(d-4)\right\}\fs$$ Since the divergent part of $\Delta_M(0)$ is proportional to $M^2$, it can be absorbed in a renormalization of the low energy constant: only the combination $$\tag{31} \ell_3^r(\mu)= \ell_3+\frac{1}{2}\lambda(\mu)$$ counts. The renormalized low energy constant, $\ell_3^r$, remains finite as $d\rightarrow 4$, but logarithmically depends on the running scale: $$\tag{32}\ell^r_3(\mu)=\frac{1}{64\pi^2}\ln \frac{\mu^2}{\Lambda_3^2}\fs$$ The quantity $\Lambda_3$ is the renormalization group invariant scale of the low energy constant $\ell_3$. In this notation, the r.h.s. of equation (25) simplifies to $$\tag{33} M_\pi^2=M^2\left\{1+\frac{M^2}{32\pi^2 F^2}\ln\frac{M^2}{\Lambda_3^2}+O(M^4)\right\}\fs$$

## Values of the low energy constants

The result (33) shows that the expansion of $M_\pi^2$ in powers of $m_{ud}$ is not a simple Taylor series, but contains a term proportional to $m_{ud}^2\ln m_{ud}$. The phenomenon is typical of the one-loop formulae of $\chi\hspace{-0.1em}$PT - contributions of this type are referred to as chiral logarithms. The coefficient of the logarithm is determined by the ratio $B^2/F^2$, that is, by the two low energy constants that specify the effective Lagrangian at leading order. The scale $\Lambda_3$ of the logarithm, on the other hand, is not determined by these two constants, but by $\ell_3$, that is by one of the LECs occurring in ${\cal L}^{(4)}_\eff$.

In view of the remarkable progress achieved with simulations of QCD on a lattice, the dependence of the pion mass on the quark masses can now be calculated from first principles. The data indeed show evidence for the presence of the logarithmic term. The numerical results obtained in two recent determinations read $\bar{\ell}_3\equiv\ln(\Lambda_3^2/M_\pi^2) = 3.16\pm0.31$ (Borsanyi et al., 2012) and $2.91\pm 0.24$ (RBC/UKQCD, 2012). These results imply $\Lambda_3=0.63\pm 0.06\,\mbox{GeV}$, indicating that this scale is known to about 10%. For a thorough discussion of the lattice results which appeared before the end of November 2013, I refer to (Aoki et al., 2013).

The representation (33) implies that for small values of $m_{ud}$, the Gell-Mann-Oakes-Renner formula slightly overestimates the pion mass, but the NLO correction is small: the above result for $\Lambda_3$ implies that, at the physical value of $m_{ud}$, $M_\pi$ is smaller than $M$ by about 1%. The effect grows if the quark mass is taken larger (as usually the case in lattice simulations), but since the logarithm vanishes at $M=\Lambda_3$, the effect reaches a maximum and then decreases. Since the perturbative calculation discussed above treats $M$ as a small quantity of $O(p)$, while $\ell_3^r$ and hence $\Lambda_3$ represent terms of $O(1)$, quark masses for which $M$ and $\Lambda_3$ are of the same size are too large for the formula (33) to apply. In the region of validity of this formula, the second term on the r.h.s. does not exceed 6% of the first. The lattice data indicate that the first term in the expansion of $M_\pi^2$, the Gell-Mann-Oakes-Renner formula, represents a decent approximation out to values of $m_{ud}$ that are 10 times as large as in nature. This observation is consistent with $\chi\hspace{-0.1em}$PT, in particular also with the fact that the calculated NLO corrections are small, but it reaches far beyond the region where it is legitimate to drop NNLO contributions.

While the low energy constant $\ell_3$ controls the chiral expansion of $M_\pi^2$ at NLO, $\ell_4$ determines the analogous contribution in the chiral expansion of the pion decay constant. The renormalization formulae analogous to (31), (32) read $\ell_4^r(\mu)=\ell_4-2\lambda(\mu)=-\ln (\mu^2/\Lambda_4^2)/(16\pi^2)$. The expansion of $F_\pi$ in powers of $m_{ud}$ looks very similar to (33): $$\tag{34}F_\pi=F\left\{1-\frac{M^2}{16\pi^2F^2}\ln\frac{M^2}{\Lambda_4^2}+O(M^4)\right\}\fs$$ The formula states that the pion decay constant also contains a chiral logarithm; the sign is opposite to the one in $M_\pi^2$ and the coefficient is twice as large. The scale $\Lambda_4$ can be estimated on the basis of a dispersive calculation of the scalar form factor, with the result $\bar{\ell}_4\equiv\ln\Lambda_4^2/M_\pi^2= 4.4\pm 0.2$ (Colangelo, Gasser, and Leutwyler, 2001), which amounts to $\Lambda_4=1.22\pm0.12$ GeV, about twice as large as $\Lambda_3$. Accordingly, the effect from the term of NLO in (34) is larger than in (33): including an estimate of the effects of NNLO, the ratio of the physical decay constant to its value in the chiral limit is predicted to be $F_\pi/F=1.0719\pm 0.0052$ (Colangelo and Dürr, 2004).

Several lattice collaborations have calculated the quark mass dependence of the pion decay constant from first principles - for a detailed discussion, see (Aoki et al., 2013). The result for $\bar{\ell}_4$ obtained in two recent calculations reads $\bar{\ell}_4 = 4.03 \pm 0.16$ (Borsanyi et al., 2012) and $\bar{\ell}_4 = 3.99 \pm 0.18$ (RBC/UKQCD, 2012), respectively. The first reference also quotes a result for the ratio of the physical decay constant to its value in the limit $m_u,m_d\rightarrow 0$, $F_\pi/F=1.0627\pm 0.0028$, to be compared with the theoretical prediction quoted above. Note that the two determinations of $F_\pi/F$ are entirely independent, as they not only rely on an analysis of different physical quantities, but also on qualitatively different sources of information. Despite the slight tension, the fact that the results are consistent with one another corroborates the validity of the effective theory based on SU(2)$\times$SU(2).

All of the LECs occurring in ${\cal L}^{(4)}_\eff$ are now known to an accuracy that allows significant statements about quantities of physical interest. Partly, the results are based on experiment, partly on work done on the lattice. For some quantities, the chiral perturbation series has even been evaluated to NNLO, but for the LECs of ${\cal L}^{(6)}_\eff$, which enter these results, only crude estimates are available. In some cases, however, such as those couplings that enter the extension of the formulae (33) and (34) to the next order of the chiral expansion, the lattice results should soon provide the information needed to determine them reliably. It will be most interesting to compare the results with the available estimates and to check the theoretical picture used to obtain those.

## $\pi\pi$ scattering

Chiral symmetry also governs the interaction among the Nambu-Goldstone bosons. For an account of the history of $\pi\pi$ scattering, which starts with Yukawa's hypothesis that pion-exchange is responsible for the nuclear forces and ends with a discussion of recent work concerning the final state interaction in the decays $K^{\pm}\rightarrow \pi^+\pi^-e^{\pm}\nu_e$ and $K^{\pm}\rightarrow \pi^0\pi^0\pi^{\pm}$, I refer to (Gasser, 2009). In the following, I briefly discuss the implications of chiral symmetry for elastic $\pi\pi$ scattering and again disregard the isospin breaking effects due to $m_u\neq m_d$.

Lorentz invariance and crossing symmetry imply that the scattering amplitude of the process $\pi^a\pi^b\rightarrow \pi^c\pi^d$ only depends on the Mandelstam variables $s,t,u$. The dependence on the flavour index of the pions is of the form $$\tag{35} T^{abcd}(s,t,u)=\delta^{ab}\delta^{cd}A(s,t,u)+\delta^{ac}\delta^{bd}A(t,u,s)+\delta^{ad}\delta^{bc}A(u,s,t)\fs$$

### Tree approximation

The leading term of the chiral perturbation series for the $\pi\pi$ scattering amplitude is given by the tree graphs of the Lagrangian in equation (19) - the relevant vertices are those in the second line of equation (23). One readily verifies that the result is of the above form and that the explicit expression for $A(s,t,u)$ is indeed given by equation (5). As mentioned in section Pion pole dominance, Weinberg derived this result from current algebra, at a time when the effective Lagrangian method was yet to be invented. According to equation (5), the leading order expression for $A(s,t,u)$ only depends on $s$; this implies that only the S- and P-waves are different from zero at leading order of the chiral expansion. Denoting the amplitude of the partial wave with angular momentum $\ell$ and isospin $I$ by $t^I_\ell(s)$, the equations (35) and (5) amount to the following explicit expressions $$\tag{36} t_0^0(s)\Lo\frac{2s-M^2}{32\pi F^2}\co\quad t_1^1(s)\Lo\frac{s-4M^2}{96\pi F^2}\co \quad t_0^2(s)\Lo-\frac{s-2M^2}{32\pi F^2}\co$$ where the symbol $\Lo$ indicates that the equation is valid only at leading order of the chiral perturbation series. The three partial waves listed in (36) are the only waves that receive a contribution already at leading order. In particular, since two-pion-states with isospin $I=0,2$ are even with respect to an interchange of the flavour indices, Bose statistics implies that they must also be even under an interchange of the momenta and hence carry even angular momentum, $\ell = 0,2,\ldots$ Accordingly, S-waves ($\ell=0$) cannot carry $I=1$, nor can there be a P-wave ($\ell=1$) with $I=0$ or 2.

At threshold, $s=4M_\pi^2$, the partial waves reduce to the scattering lengths. The canonical definition of the isoscalar S-wave scattering length, for instance, reads $a^0_{0\,\mathrm{\it can}}\equiv- (\hbar/M_\pi c)\times t^0_0(4M_\pi^2)$. It converts the dimensionless partial wave amplitude into a length. In the context of the low energy analysis, this convention is not convenient, however, because the scale factor used in the conversion is inversely proportional to $M_\pi$ and hence diverges in the chiral limit: while the partial wave $t^0_0(s)$ is $O(p^2)$, the quantity $a^0_{0\,\mathrm{\it can}}$, which measures its value at threshold, is $O(p)$. For this reason, it is customary to instead identify the scattering length with the dimensionless quantity $a_0^0\equiv t^0_0(4M_\pi^2)$, which differs from $a^0_{0\,\mathrm{\it can}}$ not only in normalization, but also in sign. In this notation, the above formulae for the partial wave amplitudes lead to the following parameter free prediction: $$\tag{37}a_0^0\Lo\frac{7M^2}{32\pi F^2}\co\quad a_0^2\Lo-\frac{M^2}{16\pi F^2}\fs$$ The result shows that the scattering lengths vanish in the chiral limit: Nambu-Goldstone bosons of zero momentum do not interact. This can also be seen directly, in the expression (19) for the effective Lagrangian: in the chiral limit, $M\rightarrow 0$, only vertices with derivative coupling survive.

In the elastic region, $4M_\pi^2\leq s\leq16M_\pi^2$, unitarity imposes the condition $|1+2 i\rho(s)\, t(s)|=1$, with $\rho(s)\equiv\sqrt{1-4M_\pi^2/s}$. The leading contribution to the partial waves is real. The condition requires the presence of an imaginary part, whose chiral expansion starts with $\mathrm{Im}\,t(s)=\rho(s)\, \mathrm{Re}\,t(s)^2+O(p^6)$. The full representation of the scattering amplitude to NLO was worked out in (Gasser and Leutwyler, 1984), (Gasser and Leutwyler, 1985). It involves four of the seven low energy constants in ${\cal L}^{(4)}_\eff$: $\ell_1,\ell_2$, which show up in the energy dependence of the partial waves, and $\ell_3,\ell_4$, which concern the dependence on the mean quark mass $m_{ud}$ and were discussed above.

Eliminating the quantities $M,F$ in favour of $M_\pi,F_\pi$, the first two terms in the chiral expansion of the S-wave scattering lengths take the form $$a_0^0\NLo \frac{7M_\pi^2}{32\pi F_\pi^2}\left\{1+\frac{9M_\pi^2}{32\pi^2 F_\pi^2}\ln\frac{\lambda_{a^0_0}^2}{M_\pi^2}\right\}\!,\qquad a_0^2\NLo -\frac{M_\pi^2}{16\pi F_\pi^2}\left\{1-\frac{M_\pi^2}{32\pi^2 F_\pi^2}\ln\frac{\lambda_{a^2_0}^2}{M_\pi^2}\right\}\!.\nonumber$$ As indicated by the symbol $\NLo$, the expressions are valid modulo contributions of next-to-next-to-leading order. The scales $\lambda_{a^0_0}$ and $\lambda_{a^2_0}$ can be expressed in terms of those of the LECs $\ell_1,\ldots,\ell_4$. While the coefficient of the chiral logarithm in $a_0^2$ is small, of the same size as the one in equation (33) for $M_\pi^2$, the chiral logarithm in $a_0^0$ is 9 times larger. Indeed, in $a_0^0$, the NLO corrections are unusually large, while $a_0^2$ barely receives any such corrections. This is illustrated in Figure 2, where the three black dots indicate the $\chi\hspace{-0.1em}$PT predictions for $a_0^0$ and $a_0^2$ at LO (Weinberg, 1966), NLO (Gasser and Leutwyler, 1984), (Gasser and Leutwyler, 1985) and NNLO (Bijnens et al., 1996), (Bijnens et al., 1997), respectively.

 Figure 2: S-wave scattering lengths: $\chi\hspace{-0.1em}$PT. Figure 3: S-wave scattering lengths: experiment.

Figure 2 shows that the corrections increase the prediction for $a_0^0$ quite substantially, while $a_0^2$ nearly stays put. The reason is understood very well: the leading order approximation for $t^0_0(s)$ in equation (36) shows that, in the channel with $I=0$, the interaction is attractive; the amplitude rapidly grows with the energy and exceeds the unitarity limit $\mathrm{Re}\,t^0_0(s)\leq \frac{1}{2}\,\rho(s)^{-1}$ when the energy reaches about 0.5 GeV. In the channel with $I=2$, this does not happen: the interaction is repulsive and the amplitude remains small, also above threshold.

The ellipse in Figure 2 indicates the result obtained on the basis of the Roy equations (Roy, 1971). While the chiral perturbation series of the scattering amplitude is of use only in the vicinity of threshold, these equations are rigorously valid for $s<68M_\pi^2$, that is, up to a c.m. energy of 1.15 GeV. In this framework, the S-wave scattering lengths play a crucial role, because they represent the two subtraction constants that appear in the Roy equations and dominate the behaviour of their solutions at low energies. The net result of the analysis in (Colangelo, Gasser, and Leutwyler, 2001) is that $a_0^0,a_0^2$ as well as $\bar{\ell}_1,\bar{\ell}_2$ can be expressed in terms of $F_\pi,M_\pi, \bar{\ell}_3, \bar{\ell}_4$, up to small uncertainties associated with the contributions of NNLO and with the dispersion integrals that enter the calculation. The narrow band in Figure 2 is obtained by (i) treating $\bar{\ell}_3$ as a free parameter and (ii) using the value of $\bar{\ell}_4$ that follows from the dispersive analysis of the scalar pion form factor in (Colangelo, Gasser, and Leutwyler, 2001). The ellipse results if the value of $\bar{\ell}_3$ is not left open, but fixed with the crude estimate $\bar{\ell}_3=2.9\pm 2.4$, which is derived from the SU(3) mass formulae in (Gasser and Leutwyler, 1984), (Gasser and Leutwyler, 1985).

### Experimental results

In the meantime, the remarkably sharp theoretical predictions for the scattering lengths have been tested in a series of beautiful low energy precision experiments concerning the decays $K\rightarrow\ell\nu\pi\pi$ (Pislak et al., 2001), (Batley et al., 2010), $K\rightarrow \pi\pi\pi$ (Batley et al., 2009) and pionic atoms (Adeva et al., 2011). The results are shown in Figure 3. For a thorough comparison of theory and experiment I refer to (Gasser, 2009), (Colangelo, 2009). The precision of the data on the difference $a_0^0-a_0^2$ is comparable to the accuracy of the prediction and confirms it within errors - provided the isospin breaking effects are properly accounted for (at the experimental precision reached, these are by no means negligible (Gasser, 2009)). The data do not pin down the two individual scattering lengths to the same accuracy, but in combination with the fact that the dispersive evaluation of the scalar form factor leads to the sharp correlation between $a_0^0$ and $a_0^2$ (the narrow band mentioned above), the experimental results imply $a_0^0= 0.2198 (46)_{stat} (16)_{syst} (64)_{th}$, $a^2_0=-0.0445(11)_{stat} (4)_{syst}(8)_{th}$, thus confirming the predictions to a remarkable degree of accuracy. Alternatively, as pointed out by Stern and collaborators (Descotes-Genon, Fuchs, Girlanda, and Stern, 2002), the two scattering lengths may be disentangled by invoking the comparatively crude experimental results extracted from the reaction $\pi N\rightarrow \pi\pi N$. In the plot, the recent update of this analysis by García-Martín et al. (García-Martín et al., 2011) is shown as a green ellipse. Combining the $\pi N$ results with the data of (Batley et al., 2010), these authors obtain $a_0^0=0.220(8)$, $a_0^2=-0.042(4)$, also consistent with the predictions, albeit somewhat less precise. As pointed out by Stern and collaborators (Descotes-Genon, Fuchs, Girlanda, and Stern, 2002), chiral symmetry by itself does not ensure that the quark condensate is the leading order parameter of the spontaneously broken symmetry - operators of higher dimension might play an equally import role. In order to investigate this possibility in a systematic manner, these authors proposed a generalization of $\chi\hspace{-0.1em}$PT. The experimental results now close this chapter: as far as pion physics is concerned, order parameters of higher dimension do not play a significant role.

### Lattice results

Figure 4: Lattice results for the S-wave scattering lengths. While the horizontal bands indicate lattice determinations of $a_0^2$, the coloured ellipses represent the regions allowed by the lattice results for the low energy constants $\bar{\ell}_3,\bar{\ell}_4$ which dominate the uncertainties in the theoretical prediction.

The recent progress made on the lattice now allows a calculation of $a_0^2$ from first principles, via the volume dependence of the energy levels of the system in a box of finite size (Beane et al., 2008), (Feng, Jansen, and Renner, 2010), (Beane et al., 2012), (Yagi et al., 2011). The results of these calculations are indicated by the horizontal bands in Figure 4. Also, as discussed in section Values of the low energy constants, the quantities $\bar{\ell}_3,\bar{\ell}_4$, which dominate the uncertainties in the theoretical predictions for $a_0^0$ and $a_0^2$, can now be determined on the lattice, from the dependence of $M_\pi$ and $F_\pi$ on the quark masses. The coloured ellipses indicate the results for $a_0^0$, $a_0^2$ obtained in this way. The plot depicts a rather fertile acre of potatoes, but it also shows that there is some tension among some of the lattice results, indicating that not all of the systematic errors are yet under control.

## Extensions

### Expansion in the mass of the strange quark, SU(3)$\times$SU(3)

In the preceding discussion, the strange quark was treated on the same footing as the heavy quarks: $m_s,m_c,\ldots$ were all held fixed at the physical values. The strange quark can instead be treated like $u$ and $d$, expanding the Green functions of QCD not only in $m_u,m_d$, but also in $m_s$. At leading order of this expansion, all three light quarks are then massless, so that the symmetry under chiral rotations in equation (2) is extended to $V_\indR,V_\indL\in\mbox{SU(3)}$: QCD with 3 massless flavours is invariant under $G=\mbox{SU(3)}\times\mbox{SU(3)}$. The quark condensate breaks this symmetry down to the subgroup $H=\mbox{SU(3)}$ and the Nambu-Goldstone bosons now live on the coset space $G/H=\mbox{SU(3)}$.

The structure of the effective theory essentially remains the same. The basic variable is a space-time-dependent unitary, unimodular $3\times3$ matrix $U(x)\in \mbox{SU(3)}$, rather than a 2$\times$2 matrix. The main difference to the framework discussed previously is that the manifold on which the Nambu-Goldstone bosons live is not any more a sphere: the components of the matrix field $U(x)$ cannot be written as linear combinations of those of a vector field of unit length, but as before, one may parametrize the group elements with canonical coordinates: the effective field may be represented as $U(x)=\exp\{ i \pi (x)/F_0\}$ where $\pi(x)$ is a hermitian, traceless $3\times 3$ matrix field. Using the Gell-Mann matrices $\lambda_1,\ldots\lambda_8$, as a basis, the decomposition $\pi(x)=\pi^1(x)\,\lambda_1+\ldots+\pi^8(x)\,\lambda_8$ reduces the matrix $U(x)$ to a set of eight pseudoscalar fields: $\pi^1(x),\pi^2(x)$ represent charged pions, $\pi^3(x)$ describes the $\pi^0$, the components $\pi^4(x),\ldots,\pi^7(x)$ correspond to $K^+,K^0,\bar{K}^0,K^-$ and $\pi^8(x)$ is the effective field associated with the $\eta$.

The leading term in the effective chiral Lagrangian reads: $$\tag{38}{\cal L}^{(2)}_\eff=\frac{1}{4}F_0^2\langle D^\mu UD_\mu U^\dagger+\chi U^\dagger+U\chi^\dagger\rangle\co\quad \chi=2B_0(s+ip)\co$$ where $\langle\hspace{-0.3mm} A\rangle$ stands for the trace of the matrix $A$. While the SU(2)$\times$SU(2) low energy constants $F,B$ depend on $m_s$, their SU(3)$\times$SU(3) analogues $F_0,B_0$ are independent thereof - they represent the leading terms in the expansion of $F,B$ in powers of $m_s$: $F=F_0+O(m_s)$, $B=B_0+O(m_s)$. The quantities $F/F_0-1$ and $B/B_0-1$ measure the violation of the Okubo-Iizuka-Zweig rule: in the large $N_c$ limit, $F$ tends to $F_0$ and $B$ tends to $B_0$.

The mass of the strange quark is much larger than $m_{ud}$. In the recent review of the lattice results given in (Aoki et al., 2013), the ratio is quoted as $m_s/m_{ud}=27.46\pm 0.44$. Accordingly, when expanding the quantities of interest not only in $m_u,m_d$, but also in $m_s$, the expansion will converge less rapidly and the higher order contributions will be more important. In fact, one may question whether the physical value of $m_s$ is small enough for a framework that treats $m_s$ as a perturbation to be meaningful. The theoretical line of reasoning which indicates that the extension from SU(2)$\times$SU(2) to SU(3)$\times$SU(3) must make sense runs as follows. If the masses $m_u$, $m_d$ and $m_s$ are set equal, QCD acquires an exact flavour symmetry, which generalizes the invariance under isospin rotations in equation (1) to the group SU(3) of rotations in the space of all three light flavours. In fact, Gell-Mann (Gell-Mann, 1962) and Ne'eman (Ne'eman, 1961) had pointed out that nature does have an approximate symmetry of this structure, even before quarks had been discovered (the occurrence of such a symmetry was the first hint at the existence of quarks). The QCD Lagrangian contains two parameters that control the breaking of this symmetry: in addition to the mass difference $m_d-m_u$, which we encountered before and which measures the breaking of isospin symmetry, the Lagrangian contains the isospin invariant difference $m_s-m_{ud}$. If both of these mass differences vanish, the symmetry is exact, if they are small, then SU(3) is an approximate symmetry. Conversely, for this to be the case, it must be legitimate to treat the mass differences $m_u-m_d$ and $m_s-m_{ud}$ as perturbations and to expand the various quantities of interest in powers of these. Now comes the salient point: since $m_s$ is large compared to $m_{ud}$, the difference $m_s-m_{ud}$ can be small only if $m_s$ itself is small. In other words, nature has an approximate SU(3) symmetry, because the masses of the three lightest quarks are small. It so happens that $m_u,m_d,m_s$ are very different, but since they are small compared to the scale of QCD, this only produces modest symmetry breaking.

The mass pattern of the Nambu-Goldstone bosons offers a check. With the effective Lagrangian in equation (38), it is straightforward to calculate their masses to leading order of the expansion in powers of $m_u,m_d,m_s$. For $m_u=m_d$, the result reads $$\tag{39} M_{\pi}^2\Lo 2B_0\, m_{ud}\co\quad M_K^2\Lo B_0(m_{ud}+m_s)\co\quad M_\eta^2\Lo \frac{2}{3} B_0(2m_s+m_{ud})\fs$$ Note that, in contrast to SU(2)$\times$SU(2), where the leading order effective Lagrangian hides isospin breaking, the extended effective Lagrangian does break SU(3), already at leading order. Eliminating $m_{ud}$ and $m_s$ in (39), one obtains the famous Gell-Mann-Okubo formula, $M_\eta^2\Lo\frac{1}{3}(4M_K^2-M_\pi^2)$, which predicts $M_\eta$ in terms of the kaon and pion masses (Gell-Mann, 1962), (Okubo, 1962). The numerical result for $M_\eta$ differs from the physical mass by less than 4%. The agreement does not check whether the meson masses themselves are well approximated by the above LO formulae, but it does show that the ratio of mass differences, $(M_K^2-M_\pi^2):(M_\eta^2-M_\pi^2)$ is remarkably close to the value 3:4 predicted at leading order.

The kaon and pion decay constants also offer a measure for the size of SU(3) symmetry breaking. Since the operation $d\leftrightarrow s$ takes a $\pi^+$ into a $K^+$, it converts the matrix element $\lvac \dbar\gamma_\mu \gamma_5 u|\pi^+\rangle\propto F_\pi$ into $\lvac \sbar\gamma_\mu \gamma_5 u|K^+\rangle\propto F_K$. The leading order effective Lagrangian in (38) yields $F_\pi\Lo F_0, F_K\Lo F_0$. In this case, the symmetry breaking thus starts showing up only at NLO. Intuitively, $F_\pi$ and $F_K$ represent the quark wave functions at the origin. Since the $s$-quark is heavier than the $d$-quark, the kaon wave function is more narrow than the pion wave function, so that the value at the origin is larger, $F_K>F_\pi$. For the ratio, lattice determinations have now reached a precision of half a percent: $F_K/F_\pi =1.194\pm 0.004$ (Aoki et al., 2013). The result shows that, in the decay constants, the breaking of SU(3) symmetry is of order 20%.

For recent reviews of the work done in $\chi\hspace{-0.1em}$PT, I refer to the literature quoted in section Further reading. The formulae for the masses and decay constants have been worked out to NNLO quite some time ago. The quality of the lattice data should soon permit a reliable determination of the low energy constants, up to and including NNLO. Thereby, it will become possible to identify the domains in the space of the quark masses, where it is meaningful to truncate the chiral perturbation series at LO, NLO and NNLO, respectively. Much work yet remains to be done here, also concerning the size of those LECs that vanish in the large $N_c$ limit, to obtain reliable values for the ratios $F/F_0, B/B_0$, etc.

### $\chi\hspace{-0.1em}$PT on the lattice

The pioneering work of Symanzik (Symanzik, 1975), (Symanzik, 1983) has shown that the artifacts generated by the lattice regularization can be investigated in a systematic way, using effective field theory methods. In the limit where the lattice spacing $a$ tends to zero, the lattice artifacts must disappear. Understanding the dependence on $a$ is of central importance for the extrapolation of the lattice results to the continuum, which is needed to establish contact with physics. There is a general formalism, known as the Symanzik improvement programme, which removes the leading lattice artifacts in a controlled manner and thus ensures that quantities of physical interest converge more rapidly when the continuum limit is approached. The improvement programme is implemented by adding higher-dimensional operators to the lattice action. The coefficients of these terms must be tuned appropriately in order to cancel the leading lattice artifacts. The improvement depends on the particular version chosen for the discretization when formulating the theory on a lattice.

Symanzik's framework can be combined with $\chi\hspace{-0.1em}$PT. In order to account for the distortions generated by the regularization, additional terms need to be added to the effective chiral Lagrangian, proportional to powers of the lattice spacing $a$. In the formulation of QCD on a lattice, Minkowski space-time is replaced by a four-dimensional Euclidean space. Since the lattice regularization breaks Euclidean invariance, the additional terms are not in general invariant under Euclidean rotations and special care must be taken for the improved action to respect invariance under chiral rotations, at least as far as the leading lattice artifacts are concerned. Since the improvement depends on the particular form chosen for the discretization, different versions of Chiral Perturbation Theory are needed to analyze the dependence of the results on the quark masses and on the lattice spacing: Q$\chi\hspace{-0.1em}$PT, S$\chi\hspace{-0.1em}$PT, W$\chi\hspace{-0.1em}$PT, tm$\chi\hspace{-0.1em}$PT, PQ$\chi\hspace{-0.1em}$PT, MA$\chi\hspace{-0.1em}$PT, $\ldots$ The prefixes stand for quenched, staggered, Wilson, twisted-mass, partially quenched, mixed action, $\ldots$ For a review of these developments, I refer to (Aoki et al., 2013), (Sharpe, 2005), (Bijnens, Danielsson and Lähde, 2007), (Golterman,2009).

### Baryons, nuclear forces

The method of effective field theory can also be used to analyze the low energy properties of the baryons. In fact, as mentioned in section Pion pole dominance, the earliest low energy theorem related to the chiral symmetry of the strong interaction is the Goldberger-Treiman relation (Goldberger and Treiman, 1958), which predicts the strength of the pion-nucleon interaction, $g_{\pi N}$, in terms of the nucleon matrix element of the axial-vector current, $g_A$: $$g_{\pi N}= \frac{g_A\,M_N}{F_\pi}\{1+O(m_u,m_d,e^2)\}\fs$$ In principle, the analysis of the low energy structure in the sector with baryon number $N_B=1$ proceeds in the same manner as the one in the sector with $N_B=0$, discussed above. The resulting structure, however, is quite different, because the state of lowest energy with $N_B=1$ is not the vacuum, but the nucleon, a state that carries momentum, spin and isospin. For $m_u= m_d$ and $e=0$, the ground state is a degenerate isodoublet. In reality, the neutron is slightly heavier than the proton, $M_n>M_p$, because $m_d$ happens to be larger than $m_u$ (the contribution from the electromagnetic interaction is of opposite sign, but too small to invert the order). Two fields are needed to describe the degrees of freedom active at low energies: the nucleon field $\psi(x)$ and the pion field $\vec{\pi}(x)$. The corresponding effective field theory is referred to as Baryon $\chi\hspace{-0.1em}$PT. In the framework of the present short overview, I cannot discuss this theory any further, but refer the reader to recent reviews of the subject: (Bernard, 2008), (Scherer, 2010).

Effective field theory methods can be applied also to the interaction between nucleons. In particular, the properties of their bound states - nuclear physics - can be investigated on this basis. Nuclear physics is a mature field - sophisticated representations of the $NN$-potential, for instance, are available that offer an excellent account of a host of experimental data. The main aim of a reanalysis of these representations in the framework of an effective field theory is to anchor nuclear physics in the Standard Model. Many questions call for an answer here, such as: Why are the binding energies of the nuclei so much smaller than typical hadronic scales ? Is it possible to derive the properties of the simplest nuclear bound state, the deuteron, from QCD ? In particular, what happens to the deuteron if $m_u, m_d$ and $e$ are turned off ? Is it possible to understand from first principles why the mean binding energy per nucleon is around 8 MeV ? etc. The pioneering work in this field is due to Weinberg (Weinberg, 1990). For a review of the progress achieved in recent years, I refer to (Epelbaum and Meißner, 2012).

### Thermodynamics of QCD, finite size effects, $\epsilon$-regime

The properties of the strong interaction in thermal equilibrium are of physical interest in various contexts: big bang, interior of stars, heavy ion collisions, etc. As shown in (Gasser and Leutwyler, 1987a)(Gasser and Leutwyler, 1987b), chiral symmetry yields significant information about the partition function, provided the temperature is low enough for the energy density to be dominated by the lightest particles, the Nambu-Goldstone bosons. As an example, I quote the formula $$\tag{40} \langle\hspace{0.3mm} \ubar u \rangle_T=\lvac \ubar u \rvac\left\{1-\frac{T^2}{8F^2}-\frac{T^4}{384F^4}+O(T^6)\right\}\co$$ which concerns the temperature dependence of the $u$-quark condensate for $m_u=m_d=0$ (the quantity $T$ stands for the temperature in units of energy; expressed in these units, the Boltzmann constant $k$ is equal to 1). The equation is based on a $\chi\hspace{-0.1em}$PT calculation to two loops and amounts to a parameter free prediction for the expansion of the condensate in powers of $T$, valid up to and including NNLO. Similar formulae can be given for energy density and pressure, also for the situation encountered in nature, where $m_u$ and $m_d$ are different from zero. The contributions of NNNLO have been evaluated as well. Moreover, as a crude estimate for the contributions from states with $M>0.5\,\mbox{GeV}$, a gas of noninteracting resonances has been considered. In view of the fact that the number of levels observed up to a given mass rapidly rises with the mass, the contributions from the resonance gas rapidly grow with the temperature. These calculations indicate that $\chi\hspace{-0.1em}$PT yields reliable results only below a temperature of about 130 MeV.

Chiral Perturbation Theory also allows one to study the effects occurring if the system is enclosed in a box of finite size, as it is done in lattice simulations. In the absence of a mass gap, the box affects some of the properties of the system, even in the limit when the box is sent to infinity, because the spontaneous breakdown of a symmetry can occur only at infinite volume. Consider the quark condensate at temperature $T$ in a cubic box of side $L$ and set $m_u=m_d=0$, so that SU(2)$\times$SU(2) represents an exact symmetry. Since symmetries cannot break down spontaneously at finite volume, the quark condensate $\langle\hspace{0.3mm} \ubar u\rangle_{T,L}$ must vanish. It is essential here that $u$ and $d$ are taken strictly massless - a tiny mass suffices to trigger the spontaneous breakdown. In fact, $\chi\hspace{-0.1em}$PT allows one to calculate the manner in which $\langle\hspace{0.3mm} \ubar u\rangle_{T,L}$ disappears if the quark masses are sent to zero (Gasser and Leutwyler, 1987a), (Gasser and Leutwyler, 1987b).

The parameter that controls the behaviour of the partition function in the region where $m_u$ and $m_d$ are very small is the quantity $x=F^2M^2 L^3/T$, where $F$ is the pion decay constant in the chiral limit and $M^2\equiv (m_u+m_d)B$ represents the leading term in the expansion of $M_\pi^2$ in powers of $m_u$ and $m_d$. The standard analysis within $\chi\hspace{-0.1em}$PT is based on an expansion in powers of $T, 1/L$ and $m_u,m_d$. The first two variables are booked as terms of $O(p)$, while the quark masses are treated as quantities of $O(p^2)$. In this so-called $p$-regime, $x$ represents a large term of $O(1/p^2)$. At finite volume and finite temperature, however, $x$ becomes smaller than 1 if the quark masses are taken small enough. The region where this happens is referred to as the $\epsilon$-regime, because a different counting of powers is needed to obtain a coherent expansion: $T=O(\epsilon)$, $1/L=O(\epsilon)$ and $m_u,m_d=O(\epsilon^4)$. At leading order in this expansion, the partition function can be written down explicitly: $$\tag{41} {\cal Z} = A\, Z_2(x)\{1+O(\epsilon^2)\}\,,\quad Z_2(x)\equiv\sum_{n=0}^\infty \frac{x^{2n}}{2^{2n}\, n! \,(n+1)!}\co$$ with $x=F^2B(m_u+m_d)L^3/T$. The function $Z_2(x)$ is related to the Bessel functions and $A$ is an irrelevant normalization constant. Since the $u$-quark condensate can be expressed in terms of the logarithmic derivative of the partition function with respect to $m_u$, it can also be given explicitly: $$\tag{42} \langle\hspace{0.3mm} \ubar u \rangle_{T,L}= \frac{Z_2'(x)}{Z_2(x)}\,\lvac \ubar u \rvac \{1+O(\epsilon^2)\}\fs$$ At small values of $x$, the factor $Z_2'(x)/Z_2(x)$ is proportional to $x$. Hence the condensate disappears if the quark masses are sent to zero, so that chiral symmetry is indeed restored.

The properties of the partition function are being studied on the lattice, not only in the $\epsilon$-regime but also at high temperatures, where disorder wins over energy, such that the quark condensate melts. Under these conditions, matter takes the form of a plasma of quarks and gluons and chiral symmetry is also restored, but for a very different reason. The behaviour of matter in the presence of a chemical potential that enforces a nonzero expectation value for the number density of baryons is of considerable interest as well. This represents a currently very active field of research, but I cannot discuss it in a meaningful way in the framework of the present short overview.

## Conclusions

Today, the fact that the strong interaction is described by QCD is beyond reasonable doubt. This theory indeed has an approximate chiral SU(2)$\times$SU(2) symmetry, as conjectured by Nambu. The hypothesis that this symmetry breaks down spontaneously to SU(2) and that the pions represent the ensuing Goldstone bosons is also confirmed, not only by experiment, which has put the theoretical predictions that follow from it to stringent test, but also by numerical simulation of the theory on a lattice. Consequently, the mass gap of the strong interaction is now understood perfectly well. In particular, the lattice results provide a direct test of the Gell-Mann-Oakes-Renner relation and demonstrate that $M_\pi^2$ is indeed dominated by the contribution from the quark condensate. This also shows that the quark condensate is the leading order parameter of the hidden symmetry. Figure 4 illustrates the fact that the expansion in powers of $m_u$, $m_d$ yields a very accurate low energy representation of QCD - low energy pion physics has become a precision laboratory. Chiral Perturbation Theory is a very useful tool in this laboratory.

The situation with the extension to SU(3)$\times$SU(3) is different. Although the only coherent way I know to understand the fact that QCD possesses an approximate SU(3) symmetry is that the mass of the strange quark can be treated as a perturbation, the predictive power of the effective theory built on this hypothesis is weaker than in the case of SU(2)$\times$SU(2). The reason is that $m_s$ is large compared to $m_u$ or $m_d$, so that the perturbation series converges more slowly and the corrections to be expected from the neglected higher orders of the expansion are more important. The chiral perturbation series has been worked out to NNLO for quite a few quantities of physical interest, but as concerns the determination of the LECs appearing in these formulae, our knowledge still leaves much to be desired. The main problem here is posed by those low energy constants that determine the dependence on the quark masses. Since the corresponding sum rules involve scalar intermediate states, resonance saturation does not work for these - while vector meson dominance provides useful approximations, scalar meson dominance does not.

The range of energies and momenta where the truncated chiral perturbation series provides an accurate representation of the physical quantities of interest is limited. This range may be extended by means of dispersive methods. A considerable amount of work has been done in this direction, analyzing form factors and scattering amplitudes by means of dispersion relations. In the case of the $\pi\pi$ scattering amplitude, for instance, $\chi\hspace{-0.1em}$PT is needed only to calculate the two subtraction constants. The energy dependence of the partial waves then follows from the dispersive representations obeyed by these. The method also allows one to calculate the partial wave amplitudes off the real axis and to determine the position of the poles on the second sheet which reveal the occurrence of resonances. Even if the poles occur rather far away from the real axis, as in the case of the $\sigma$ or the $\kappa$, for instance, dispersive methods are much preferable to analytic continuation 'by hand'.

Finally, I draw the readers attention to the fact that in the present review, many applications of $\chi\hspace{-0.1em}$PT were not discussed at all: effective theory of the e.m.interaction, electroweak symmetry breaking, magnons, phonons, $\ldots$ The example of the effective Lagrangian relevant for the low energy structure of the strong interaction may serve as an introduction to a method that can be used whenever the system under study undergoes spontaneous symmetry breakdown.

## Acknowledgment

I thank Hans Bijnens, Gilberto Colangelo, Gerhard Ecker, Jürg Gasser, Ulf Meißner and Stefan Scherer for very useful comments and am indebted to Riccardo Guida and Prasanna Venkatesh for their kind help with the presentation.

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