nucl-th/9512029 20 Dec 1995 ph emen as mo dels suc ysics of the These tary Nucle essen lectures concepts ultrarelativistic hea h as the linear and nonlinear sigma mo del will b e discussed as w ar tial ideas Scienc In tro duction are of e of Division an c hiral Berkeley c hiral p erturbation attempt Decem symmet vy ion collisions will b e presen V L Abstract olk to awr CA to b er er enc Chiral ry a Ko c p edagogical e in Berkeley n theory h uclear USA Symm National ph Some applications in tro duction ysics et ry LBNLRep ort ted L Eectiv ab or in atory to e the to c hiral the el ell

Intro duction

Chiral symmetry is a symmetry of QCD in the limit of vanishing quark masses We know

however that the current quark masses are nite But compared with hadronic scales the

masses of the two lightest quarks up and down are very small so that chiral symmetry

may b e considered an approximate symmetry of the strong interactions

Long b efore QCD was known to b e the theory of strong interactions phenomenological

indications for the existence of chiral symmetry came from the study of the nuclear b eta

decay There one nds that the weak coupling constants for the vector and axialvector

hadroniccurrents C and C did not in case of C or only slightly in case

V A V

of C dier from those for the leptonic counterparts Consequently strong interaction

A

radiative corrections to the weak vector and axial vector charge are absent The same

is true for the more familiar case of the electric charge and there we know that it is its

conservation which protects it from radiative corrections Analogously we exp ect the

weak vector and axial vector charge or more generally currents to b e conserved due to

some symmetry of the strong interaction In case of the vector current the underlying

symmetry is the well known isospin symmetry of the strong interactions and thus the

hadronic vector current is identied with the isospin current The identication of the

axial current on the other hand is not so straightforward This is due to another very

imp ortant and interesting feature of the strong interaction namely that the symmetry

asso ciated with the conserved axial vector current is sp ontaneously broken By that one

means that while the Hamiltonian p ossesses the symmetry its ground state do es not An

imp ortant consequence of the sp ontaneous breakdown of a symmetry is the existence of

a massless mo de the so called Goldstoneb oson In our case the Goldstone b oson is the

pion If chiral symmetry were a p erfect symmetry of QCD the pion should b e massless

Since chiral symmetry is only approximate we exp ect the pion to have a nite but small

compared to all other hadrons mass This is indeed the case

The fact that the pion is a Goldstone b oson is of great practical imp ortance Low

energytemp erature hadronic pro cesses are dominated by pions and thus all observables

can b e expressed as an expansion in pion masses and momenta This is the basic idea of

chiral p erturbation theory which is very successful in describing threshold pion physics

At high temp eratures andor densities one exp ects to restore chiral symmetry By

that one means that unlike the ground state the state at high temp eraturedensity

p osses the same symmetry as the Hamiltonian the symmetry of the Hamiltonian of

course will not b e changed As a consequence of this so called chiral restoration we

exp ect the absence of any Goldstone mo des and thus the pions if still present should

b ecome as massive as all other hadrons To create a system of restored chiral symmetry

1

If of course chiral restoration and deconnement take place at the same temp erature as current

lattice gauge calculations suggest the concept of hadrons in the restored phase may b ecome meaningless

in the lab oratory is one of the ma jor goals of the ultrarelativistic heavy ion exp eriments

These lectures are intended to serve as an intro duction into the ideas of chiral symme

try in particular for exp erimentalists interested or working in this eld Thus emphasis

will b e put on the ideas and concepts rather than formalism Consequently most argu

ments presented will b e heuristic andor based on simple eective mo dels References will

b e provided for those seeking more rigorous derivations

In the rst section we will intro duce some basic concepts of quantum eld theory which

are necessary to discuss the eect of symmetries on the dynamics Then we will intro duce

the chiral symmetry transformations and derive some results such as the Goldb erger

Treiman relation In the second section we will present the linear sigma mo del as the most

simple eective chiral mo del Using this rather intuitive mo del we will discuss explicit

chiral symmetry breaking As an application we will consider pionnucleon scattering

The third section will b e devoted to the so called nonlinear sigma mo del which then

serves as a basis for the intro duction into chiral p erturbation theory In the last section

we will give some examples for chiral symmetry in the physics of hot and dense matter

Preparing these lectures I have b orrowed from many sources Those which I p ersonally

found most useful are listed at the end of this contribution This is certainly a p ersonal

choice as there are many other b o oks and articles on sub ject available If not stated

otherwise the conventions of Bjorken and Drell are used for metric gammamatrices

etc

Theory Primer

Basics of quantum eld theory

Quantum eld theory is usually written down in the Lagrangian formulation see eg

the b o ok of Bjorken and Drell Lets start out with what we know from classical

mechanics There one obtains the equations of motion from the Hamilton principle

R

where one requires that the variation of the action S dt Lq q t vanishes

d L L

S

dt dq q

Here S is called the action and L T V is the Lagrangefunction For example Newtons

equations of motion for a particle in a p otential V q derive from

L mq V q

V V

mq F mq

q q

If one go es over to a eld theory the co ordinates q are replaced by the elds x and

the velo cities q are replaced by the derivatives of the elds

q x

x

q x

x

and the Lagrangefunction is given by the spatial integral over the Lagrangian density L

or Lagrangian as we shall call it from now on

Z

L d x Lx x t

Z Z

S dt L d x Lx x t

Lorentz invariance implies that the action S and thus the Lagrangian L transform like

Lorentzscalars The equations of motion for the elds are again obtained by requiring

that the variation of the action S vanishes This variation is carried out by a variation of

the elds

with

Consequently

Z

S d x L L

Z

L L

L d x L

Z

L L

d x

where equ has b een used The derivatives of L with resp ect to the elds are so

called functional derivatives but for all practical purp oses they just work like normal

derivatives where L is considered a function of the elds Partial integration of the

second term of equ nally gives

Z

L L

S d x

which leads to the following equations of motion since the variation are arbitrary

L L

If we are dealing with more than one eld such as in case of pions where we have

three dierent charge states the equations of motion have the same form as in equ

only that the elds carry now an additional index lab eling the dierent elds under

consideration

L L

i i

As example let us consider the Lagrangian of a free b oson and fermion eld resp ectively

i free scalar b osons of mass m

L m

KG

L

m

L

i

Thus according to equ the equation of motion is

m r m

t

which is just the well known KleinGordon equation for a free b oson

ii free fermions of mass m

L i m

F D

By using the conjugate eld in the equation of motion

L

i m

L

we obtain the Dirac equation for

i m

whereas inserting for in leads to the conjugate Dirac equation

i

m i

Symmetries

One of the big advantages of the Lagrangian formulation is that symmetries of the La

grangian lead to conserved quantities currents In classical mechanics we know that

symmetries of the Lagrange function imply conserved quantities For example if the La

grange function is indep endent of space and time momentum and energy are conserved

resp ectively

Let us assume that L is symmetric under a transformation of the elds

meaning

L L

L L

L L

where we have expanded the rst term to leading order in Using equ and the

equation of motion we have

L L

L

so that

L

J

i

i

is a conserved current with J In the last equation we have included the indices

for p ossible dierent elds

i

As an example let us discuss the case of a unitary transformation on the elds such

as eg an isospin rotation among pions For obvious reasons unitary transformations

are the most common ones and the chiral symmetry transformations also b elong to this

class

a

i T

i i a j

ij

a

is a matrix usually called the generator where corresp onds to the rotation angle and T

ij

of the transformation isospin matrix in case of isospin rotations The index a indicates

that there might b e several generators asso ciated with the symmetry transformation in

case of isospin rotations we have three isospin matrices Equation corresp onds to

the expansion for small angles of the general transformation

a a

T i

e

where the vector on indicates the several comp onents of the eld such as

and From equ and equ we nd the following expression for the conserved

currents

L

a a

i J T

k

j k

j

a

where we have divided by the angle This current is often referred to as a No ether

current after E No ether who rst showed its existence

2

Note that some of the No ether currents are not conserved on the quantumlevel In other word not

every symmetry of the classical eld theory has a quantum analog If this is not the case one sp eaks of

anomalies For a discussion of anomalies see eg

Of course a conserved current leads to a conserved charge

Z

d

Q d xJ x Q

d t

Finally let us add a small symmetry breaking term to the Lagrangian

L L L

where L is symmetric with resp ect to a given symmetry transformation of the elds

and L breaks this symmetry Consequently the variation of the Lagrangian L do es not

vanish as b efore but is given by

L L

Following the steps ab ove we can easily convince ourselves that the variation of the

Lagrangian can still b e expressed as the divergence of a current which is given by equ

or in case of unitary transformations of the elds Thus we have

L L J

Since L the current J is not conserved Relation nicely shows how the

symmetry breaking term of the Lagrangian is related to the nonconservation of the

current It will also prove very useful when we later on intro duce the slight breaking of

chiral symmetry due to the nite quark masses

Example Massless fermions

As an example for the No ether current let us consider the Lagrangian of two avors of

massless fermions Since we will only discuss transformations on the fermions the results

will b e directly applicable to massless QCD

The Lagrangian is given by see eq

L i

j j

where the index j refers to the two dierent avors lets say up and down and is

the usual shorthand for

i Consider the following transformation

i

2

i e

V

where refers to the Pauli Isospin matrices and where we have switched to a iso

spinor notation for the fermions u d The conjugate eld transforms under

V

as follows

i

2

e i

and hence the Lagrangian is invariant under

V

i i i i i

i

Following equ the asso ciated conserved current is

a

a

V

and is often referred to as the vectorcurrent

ii Next consider the transformation

i

5

2

i e

A

i

5

2

i e

where we have made use of the anticommutation relations of the gamma matrices sp ecif

ically The Lagrangian transforms under as follows

A

i i i i i

i

where the second term vanishes b ecause anticommutes with Thus the Lagrangian

is also invariant under with the conserved axial vector current

A

a

A

In summary the Lagrangian of massless fermions and hence massless QCD is invariant

under b oth transformations and symmetry is what is meant by chiral symmetry

V A

3

Note that the ab ove Lagrangian is also invariant under the op erations expi and

expi The rst op eration is related to the conservation of the baryon numb er while the second

5

symmetry is broken on the quantum level This is referred to as the U axial anomaly which is real

breaking of the symmetry in contrast to the sp ontaneous breaking discussed b elow see eg

4

Often p eople talk ab out chiral symmetry but actually only refer to the axial transformation

A

This is due to its sp ecial role is plays since it is sp ontaneously broken in the ground state

The chiral symmetry is often referred to by its group structure as the SU SU

V A

symmetry

Now let us see what happ ens if we intro duce a mass term

L m

From the ab ove L is obviously invariant under the vector transformations but not

V

under

A

m i

A

Thus is not a go o d symmetry if the fermions quarks have a nite mass But as long

A

as the masses are small compared to the relevant scale of the theory one may treat as

A

an approximate symmetry in the sense that predictions based under the assumption of

the symmetry should b e reasonably close to the actual results

In case of QCD we know that the masses of the light quarks are ab out MeV

whereas the relevant energy scale given by MeV is considerably larger We

QC D

therefore exp ect that should b e an approximate symmetry and that the axial current

A

should b e approximately partially conserved This slight symmetry breaking due to the

quark masses is the basis of the so called Partial Conserved Axial Current hyp othesis

PCAC Furthermore as long as the symmetry breaking is small one would also exp ect

that its eect can b e describ ed in a p erturbative approach This is carried out in a

systematic fashion in the framework of chiral p erturbation theory

Chiral Symmetry and PCAC

Chiral transformation of mesons

In order to develop a b etter feeling for the meaning of the symmetry transformations

V

and let us nd out pions and rhomesons transform under these op erations To this

A

end let us consider combinations of quark elds which carry the quantum numb ers of

the mesons under consideration This should give us the correct transformation prop erties

pionlike state i sigmalike state

rholike state a like state a

5

A wheel which is slightly b ent and thus not p erfectly invariant under rotations can for most practical

purp oses still b e considered as b eing round as long as the b ending is small compared to the radius of

the wheel

i vector transformations see eqs

V

j j

i i

i i i i j i

i i

i j ij k k

where we have used the commutation relation b etween the matrices i

i j ij k k

In terms of pions this can b e written as

which is nothing else than an isospin rotation namely the isospin direction of the pion is

rotated by The same result one obtains for the meson

Consequently the vectortransformation can b e identied with the isospin rotations

V

and the conserved vector current with the isospin current which we know to b e conserved

in strong interactions

i axial transformations see eqs

A

j j

i i

i i i i i j

i

i i

where we have made use of the anticommutation relation of the matrices f g

i j ij

and of In terms of the mesons this reads

The pion and the sigmameson are obviously rotated into each other under the axial

transformations Similarly the rho rotates into the a

A

a

Ab ove we just have convinced ourselves that is a symmetry of the QCD Hamiltonian

A

Naively this would imply that states which can b e rotated into each other by this sym

metry op eration should have the same Eigenvalues ie the same masses This however

MeV We certainly do not is clearly not the case since m MeV and m

a

1

exp ect that the slight symmetry breaking due to the nite current quark masses is resp on

sible for this splitting This should lead to mass dierences which are small compared

to the masses themselves In case of the and a however the mass dierence is of the

same order as the mass of the The resolution to this problem will b e the sp ontaneous

breakdown of the axial symmetry Before we discuss what is meant by that let us rst

convince ourselves that the axial vector is conserved to a go o d approximation so that

the axial symmetry must b e present somehow

Pion decay and PCAC

Let us rst consider the weak decay of the pion In the simple Fermi theory the weak

interaction Hamiltonian is of the currentcurrent typ e where b oth currents are a sum

of axial and vector currents as we have dened them ab ove see eg Because of

parity the weak decay of the pion is controlled by the matrix element of the axial current

b etween the vacuum and the pion jA j This matrix element must b e prop ortional

to the pion momentum b ecause this is the only vector around

a b ab iq x

jA xj q if q e

and the prop ortionality constant f M eV is determined from exp eriment Let us

now take the divergence of equ

a b ab iq x ab iq x

j A xj q f q e f m e

To the extent that the pion mass is small compared to hadronic scales the axial current

is approximately conserved Or in other words the smallness of the pion mass is directly

related to the partial conservation of the axial current ie to the fact that the axial

transformation is an approximate symmetry of QCD In the literature the ab ove relation

is often referred to as the PCAC relation The ab ove relations also suggest

that the axial current carried by a pion is

f x A

or that the divergence of the axialvector current can b e identied with the pion eld up

to a constant Here x is the pion eld Sometimes this relation b etween pion eld

and axial current is also referred to as the PCAC relation

Goldb ergerTreiman relation

There is more evidence for the conservation of the axial current Let us consider the axial

current of a nucleon This is simply given by see equ

N

A g

a N N

where pr oton neutr on is now an isospinor representing proton and neutron The

N

factor g is due to the fact that the axial current of the nucleon is renormalized

a

p

6

The are several denitions of f around dep ending on whether factors of are present in equ

by as seen in the weak b eta decay of the neutron Since the nucleon has a large

mass M we do not exp ect that its axial current is conserved and indeed by using the

N

free Dirac equation for the nulceon one can show that

N

A ig M

a N N N

which vanishes only in case of vanishing nucleon mass We know however that the

nucleon interacts strongly with the pion Therefore let us assume that the total axial

current is the sum of the nucleon and the pion contribution Using the PCACrelation

and equ we have

A g f x

N a N

If we require that the total current is conserved A we obtain

M

N

g i

N N a

f

where we have used This is nothing else but a Klein Gordon equation for a massless

b oson pion coupled to the nucleon Hence requiring the conservation of the total axial

current immediately leads us to predict that the pion should b e massless This is exactly

what we also concluded from the weak pion decay If we now allow for a nite pion mass

which is equivalent to requiring that the divergence of the axial current is consistent with

the PCAC result then we arrive at the Klein Gordon equation for a pion coupled to

the nucleon

M

N

g i m

a N N

f

where the pionnucleon coupling constant is given by

M

N

g g

N N a

f

This is to b e compared with the value for the pionnucleon coupling as extracted eg from

pionnucleon scattering exp eriments

exp

g

N N

which is in remarkeble close agreement considering the fact that equ relates the

stronginteraction pionnucleon coupling g with quantities extracted from the weak

N N

interaction namely g and f Of course the reason why this works is that there is some

a

symmetry namely chiral symmetry at play which allows to connect semingly dierent

pieces of physics Equation is usually called the Goldb ergerTreiman relation

Sp ontaneous breakdown of chiral symmetry

There app ears to b e some contradiction On the one hand the meson mass sep ctrum do es

not reect the axialvector symmetry On the other hand the weak pion decay seems

to b e consistent with a partially conserved axialvector current Also the success of

the Goldb ergerTreiman relation indicates that the axialvector current is conserved and

hence that the axial transformation is a symmetry of the strong interactions

A

The solution to this puzzle is that the axialvector symmetry is spontaneously broken

What do es one mean by that One sp eaks of a sp ontaneously broken symmetry if a

symmetry of the Hamiltonian is not realized in the ground state

This is b est illustrated in a classical mechanics analog In g we have two rotation

ally invariant p otentials interactions In a the ground state is right in the middle

and the p otential plus ground state are still invariant under rotations In b on the

other hand the ground state is at a nite distance away from the center The p oint at

the center is a lo cal maximum of the p otential and thus unstable If we put a little ball in

the middle it will roll down somewhere and nd its groundstate some place in the valley

which represents the true minimum of the p otential By picking one p oint in this valley

ie picking the ground state the rotational symmetry is obviously broken Potential plus

groundstate are not symmetric anymore The symmetry has b een broken spontaneously

by cho osing a certain direction to b e the groundstate However eects of the symmetry

are still present Moving the ball around in the valley rotational excitations do es not

cost any energy whereas radial excitations do cost energy

Let us now use this mechanics analogy in order to understand what the sp ontaneous

breakdown of the axialvector symmetry of the strong interaction means Assume that

the eective QCDhamiltonian at zero temp erature has a form similar to that depicted in

g b where the xyco ordinates are replaced by elds The spacial rotations

are then the mechanics analog of the axialvector rotation which rotates into see

A

equ Since the ground state is not in the center but a some nite distance away

from it one of the elds will have a nite exp ectation value This can only b e the eld

b ecause it carries the quantum numb ers of the vacuum In the quark language this means

we exp ect to have a nite scalar quark condensate q q In this picture pionic

excitation corresp ond to small rotations away from the groundstate along the valley

which do not cost any energy Consequently the mass of the pion should b e zero In other

words due to the sp ontaneous breakdown of chiral symmetry we predict a vanishing pion

mass Excitations in the direction corresp ond to radial excitations and therefore are

massive

This scenario is in p erfect agreement with what we have found ab ove The sp onta

neous breakdown of the axialvector symmetry leads to dierent masses of the pion and

sigma However since the interaction itself is still symmetric pions b ecome massless

which is exactly what we nd from the PCAC relation provided that the axial current (a)

(y,π)

(x,σ)

(b) (y,π)

(x,σ)

Figure Eective p otentials a No sp ontaneous breaking of symmetry b Sp ontaneous

breaking of symmetry

is p erfectly conserved Thus the mesonic mass sp ectrum as well as the PCAC and the

Goldb ergerTreiman relation are consistent with a sp ontaneous breakdown of the axial

vector symmetry The pion app ears as a massless mo de Goldstone b oson as a result

A

of the symmetry of the interaction

Incidentally the assumption of a sp ontaneously broken axialvector symmetry also ex

p

plains the mass dierence b etween the and a meson and one predicts that m m

a

1

in go o d agreement with the measured masses The derivation of this result however is

to o involved to b e presented here and the interested reader is referred to the literature

One exp ects that at high temp eraturedensities the nite exp ectation value of the

scalar quark condensate melts away resulting in a system where chiral symmetry is not

sp ontaneously broken anymore In this as it is often called chirally restored phase

pionsigma as well as rhoa if they exist should b e degenerate and the pion lo oses its

identity as a Goldstone b oson ie it will b ecome massive The eective interaction in

this phase would then have a shap e similar to g a It is one of the ma jor goals of the

ultrarelativistic heavy ion program to create and identify a macroscopic sample of this

phase in the lab oratory

In the following section we will construct a chiral invariant Lagrangian the so called

Linearsigmamo del in order to see how the concept of sp ontaneous breakdown of chiral

symmetry is realized in the framework of a simple mo del We will also discuss how to

inco orp orate the eect of the nite quark masses leading to the explicit breaking of chiral

symmetry

7

If deconnement and chiral restoration o ccur at the same temp erature it may b ecome meaningless

to talk ab out mesons ab ove the critical temp erature

Linear sigmamo del

Chiral limit

In this section we will construct a simple chirally invariant mo del involving pions and nu

cleons the so called linear sigma mo del This mo del was rst intro duced by GellMann

and Levy in long b efore QCD was known to b e the theory of the strong interac

tion In order to construct such a mo del we have to write down a Lagrangian which is a

Lorentzscalar and which is invariant under the vector and axialvector transformations

and

V A

In the previous section we have shown that the pion transforms under and as

V A

V i i ij k j k A i i i

Similarly one can also show that the eld transforms like

V A i i

Since is simply an isospin rotation the squares of the elds are invariant under this

V

transformation

V

whereas under they transform like

A

A i i i i

However the combination is invariant under b oth transformations and

V A

V A

Since this combination is also a Lorentzscalar we can build a chirally invariant La

grangian around this structure

Pionnucleon interaction

The standard pion nucleon interaction involves a pseudoscalar combination of the

nucleon eld multiplied by the pion eld

i g

where from now on we denote the pionnucleon coupling constant simply by g

Under the chiral transformations this transforms exactly like b ecause the term

involving the nucleon has the same quantum numb ers as the pion Chiral invariance

requires that there must b e another term which transforms like in order to have

the invariant structure The simplest choice is a term of the form

g

so that the interaction term b etween nucleons and the mesons is

h i

L g

Nucleon mass term

We know that an explicit nucleon mass term breaks chiral invariance see section

The nucleon mass is also to o large to b e simply a result of the small explicit

chiral symmetry breaking as reected in the PCAC relation The simplest

way to give the nucleon a mass without breaking chiral symmetry is to exploit the

coupling of the nucleon to the eld which has the structure of a nucleon

mass term This however requires that the eld as a nite vacuum expectation

value

f

where choice of f is dictated by the Goldb ergerTreiman relation in

the limit of g A nite vacuum exp ectation value for the eld immediately

a

implies that chiral symmetry will b e sp ontaneously broken as discussed in the last

section In order for our mo del to generate such an exp ectation value we have to

intro duce a p otential for the sigma eld which has its minimum at f This

brings us to the next ingredient of our mo del

Pion sigma p otential

The p otential which generates the vacuum exp ectation value of the eld has to

b e a function of the invariant structure in order to b e chirally invariant The

simplest choice is

V V f

This p otential which is plotted in g see also g for a threedimensional

view indeed has its minimum at f for Due to its shap e it is often

referred to as the Mexican hat p otential

8

Actually one can allow for an explicit nucleon mass term if one also includes the chiral partner of



the nucleon which is b elieved to b e the N This is an interesting alternative approach which is

discussed in detail in ref V(σ, π=0)

fπ σ

Figure Potential of linear sigmamo del

Kinetic energy terms

Finally we have to add kinetic energy terms for the nucleons and the mesons which

have the form i and resp ectively Both are chirally invari

ant The rst term is just the Lagrangian of free mass less fermions which we have

shown to b e invariant The second term again has the invariant structure

Putting everything together the Lagrangian of the linear sigmamo del reads remem

b er that the p otential V enters with a minussign into the Lagrangian

L i g

LS

f

What are the prop erties of this mo del Let us start with the ground state As already

mentioned in the ground state the eld has a nite exp ectation value whereas the

pion has none b ecause of parity Furthermore the nucleon obtains its mass from its

interaction with the sigma eld But what are the masses of the and mesons There

are no explicit mass terms for the and elds in the Lagrangian but as with

the nucleon there could b e some coupling to the exp ectation value of the eld which

gives rise to mass terms From the structure of the p otential see gs and as well

as from our discussion of the sp ontaneous breakdown of chiral symmetry we exp ect the

pion to b e massless and the meson to b ecome massive In order to verify that let us

expand the p otential for small uctuations around the ground state

Actually it is these uctuations which are to b e b e identied with the

observed particles and meson Since a b osonic mass term is quadratic in the elds

see Lagrangian let us expand the p otential up to quadratic order in the uctuations

Expanding around a minimum the linear order vanishes and we have

V f O

where we have used that f Comparing with the Lagrangian of a free b oson we

identify the mass of the sigma to b e rememb er that L T V

m f

We nd no mass term for the pion in agreement with our exp ectation that the pion

should b e the massless Goldstone b oson of the sp ontaneously broken chiral symmetry

In summary the prop erties of the ground state of the linear sigmamo del are

f

M g g f

N

f m

m

Before we conclude this section let us calculate the conserved axial current and check

if the PCACrelation is satised in our mo del The innitesimal axial transformations of

the nucleon pion and sigma elds are given by see and

a

a

i

i i a ia

a a

Comparing with the general form for unitary transformations we nd that the gen

a

erator of the axial transformation T act on the elds in the following way

a

a a

T

a j aj

T i

a a

T i

Using the expression for the conserved current the conserved axial current is given

by

a

a a a

A

In order to check the PCACrelation we again expand the elds around the ground

state see eq

a

a a a a

f A

where we have used that f Since the PCACrelation involves the matrix element

j a

j only the last term of contributes The other terms would require either jA

nucleons or sigmamesons in the nal or initial state Thus as far as the PCAC relation

is concerned the axial current reduces to

a

A x f x

P C AC

in agreement with the PCACresults eq

Explicit breaking of chiral symmetry

So far we have assumed that the axialvector symmetry is a p erfect symmetry of the

strong interactions From our discussion in section we know however that the

small but nite current quark masses of the up and down quark break the axialvector

symmetry explicitly This explicit breaking of the symmetry should not b e confused with

the sp ontaneous breakdown we have discussed b efore In case of a sp ontaneous breaking

of a symmetry the Hamiltonian is still symmetric whereas in case of an explicit breaking

already the Hamiltonian is not symmetric

One may wonder if the whole concept of sp ontaneous symmetry breaking makes any

sense if already the Hamiltonian is not symmetric The answer to that again dep ends

on the scales involved If the explicit symmetry breaking is small ie if the quark masses

are small compared to to relevant energy scale of QCD as we b elieve they are then it

will b e sensible to apply the notion of a sp ontaneously broken symmetry

To illustrate that let us again utilize our little mechanics analogy which we have

develop ed in the previous section An explicit symmetry breaking would imply that

b oth p otentials of gure are not invariant under rotation This could for instance

b e achieved by slightly tilting them towards say the xdirection As a result also the

ground state of p otential a is away from the center x y But the dislo cation is

small compared to that due to the sp ontaneous breaking Furthermore as long as the

p otentials are tilted only slightly rotational excitation pions in p otential b are still

considerably softer than the radial ones sigmamesons So in this sense we exp ect the

eect due to the sp ontaneous breakdown of chiral symmetry to dominate the dynamics

as long as the explicit breaking is small In the linear sigmamo del the mass scale

generated by the sp ontaneous breakdown is the nucleon mass whereas that generated by

the explicit breakdown will b e the mass of the pion as we shall see Thus indeed the

explicit breaking is small and our picture develop ed under the assumption of p erfect

axialvector symmetry will survive the intro duction of the explicit breaking to a very

go o d approximation

After these remarks let us now intro duce a symmetry breaking term into the linear

sigmamo del In QCD we know that the symmetry is explicitly broken by a quark

massterm

L mqq

X S B

where the subscript X S B stands for explicit chiral symmetry breaking If we identify

as we have done b efore the scalar quarkeld combination q q with the eld this would

suggest the following symmetry breaking term in the sigmamo del

L

SB

where is the symmetry breaking parameter This term clearly is not invariant under the

axial transformation but preserves the vector symmetry Including this term the

A V

p otential V now has the form

V v

where we now have replaced f of eq by a general parameter v which in limit of

will go to f The eect of the symmetry breaking term is to tilt the p otential

slightly towards the p ositive direction and thus to break the symmetry see g

What are the consequences of this additional term First of all the minimum has

shifted slightly If we require that the value of the new minimum is still f in order to

preserve the Goldb ergerTreiman relation we nd for the parameter v to leading order

in

v f

f

V(σ, π=0)

σ

fπ

Figure Potential of linear sigmamo del with explicit symmetry breaking

Also the mass of the sigma is slightly changed

V

m f

f

0

But most imp ortantly the pion now acquires a nite mass

V

m

f

0

which xes the parameter

f m

Thus the square of the pion mass is directly prop ortional to the symmetry breaking

parameter as we would have exp ected it from our previous discussion

Due to our choice of f the nucleon mass is not changed which however do es

not mean that there is no contribution to the nucleon mass from the explicit symmetry

breaking If we split the nucleon mass into a contribution from the symmetric part of the

p otential v and one from the symmetry breaking term

M g g v

N

f

we nd that the contribution from the symmetry breaking which is often referred to as

the pionnucleon sigmaterm is given by

m

X S B

M g g f

N

N

f m

As we shall see b elow the pionnucleon sigmaterm can b e measured in pionnucleon

scattering exp eriments and its is currently b elieved to b e MeV

N

Since chiral symmetry is now explicitly broken the axialvector current is not con

served anymore The functional form of the axial current is the same however as in the

symmetric case eq b ecause the symmetry breaking term do es not involve any

derivatives see equ Its divergence is related to the variation of the symmetry

breaking term in the Lagrangian as shown at the end of section

a a

f m A

which leads directly to the PCAC relation Here denotes the variation of the

eld with resp ect to the axialvector transformation not the uctuation around the

A

a

ground state As in equ the angel has b een divided out

The main eect of the explicit chiral symmetry breaking was to give the pion a mass

But we can utilize the symmetry breaking further to derive some rather useful relations

b etween exp ectation values of the scalar quark op erator q q and measurable quantities like

f m and

N

When we intro duced the symmetry breaking term into our mo del we had required

that it has the same transformation prop erties under the chiral transformations as the

QCDsymmetry breaking term The overall strength of the symmetry breaking we then

adjusted to repro duce the ground state prop erties namely the pion mass Therefore it

seems reasonable to exp ect that that the vacuum exp ectation value of the symmetry

breaking terms in QCD and in the eective mo del are the same

j j j mq q j

9

This denition of the pionnucleon sigma term should b e taken with some care For a rigorous

denition see eg In the framework of the sigmamodel this denition however is correct to

leading order in

10

These derivations are merely heuristic but I feel they nicely demonstrate the physics which is going

on For a rigorous derivation see eg

If we insert for m f and use j j f we arrive at the so called GellMann

Oakes Renner GOR relation

m m

u d

m f juu ddj

where we have written out explicitly the average quark mass m and the quark op erator

q q The GOR relation is extremely useful since it relates the quark condensate with f

andor the pion mass with the currentquark mass

Similarly but less convincingly one can argue that the contribution to the nucleon

mass due to chiral symmetry breaking is the exp ectation value of the symmetry

N

breaking Hamiltonian H L b etween nucleon states This leads to the

X S B X S B

exact expression of the pionnucleon sigmaterm in terms of QCD variables

m m

u d

N juu ddjN

N

This relation will turn out to b e very helpful in order to estimate the change of the chiral

condensate in nuclear matter at nite density

Swave pionnucleon scattering

In order to see how chiral symmetry aects the dynamics let us as an example study

pionnucleon scattering in the sigmamo del Let us b egin by intro ducing some notation

q π π q'

p N N p'

The invariant scattering amplitude T q q is commonly decomp osed into a scalar and

a vector part see g for the notation of momenta

T q q As t B s t q q

where s t are the usual Mandelstam variables and q and q denote the incoming and

outgoing pion fourmomenta The relativistic scattering amplitude is related to the more

familiar scattering amplitude in the center of mass frame F q q by

M

N

p

F up sT up s

s

Here are Paulispinors for the nucleon representing spin and isospin and up s stand

for a relativistic spinors for a nucleon of momentum p

The scattering amplitude can b e decomp osed into isospineven and o dd comp onents

T T T

a b ab ab

where the indices a b refer to the isospin

In the discussion of pionnucleon scattering instead of st one usually uses the in

variant variables

s u

M

N

q q t q q

B

M M

N N

11

For details see eg the app endix of

12

Notice that the isospino dd amplitude is the negative of what in the literature is commonly called

the isovector amplitude whereas the isospineven amplitude is identical to the so called isoscalar one see

The spinaveraged nonspinip s s forward scattering p p amplitude which

will b e most relevant for the asp ects of chiral symmetry is usually denoted by D and is

given in terms of the ab ove variables by

X

D u p sT up s A B

s

Finally if one wants to extract eects due to explicit chiral symmetry breaking one b est

analyses the so called subtracted amplitude

g

B

D D D D

PV

M

N

B

Now let us calculate the pionnucleon scattering amplitude in the sigma mo del At

tree level the diagrams shown in g contribute to the amplitude The rst two

pro cesses represent the the simple absorption and reemission of the pion by the nucleon

Provided that there is a coupling b etween pion and nucleon one would have written

down these diagrams immediately without any knowledge of chiral symmetry The third

diagram c which involves the exchange of a sigmameson is a direct result of chiral

symmetry and as well shall see is crucial in order to give the correct value for the amplitude

q q' q q' q q' σ

p τa τ b p' p τb τa p' p p'

(a) (b) (c)

Figure Diagrams contributing to the pionnucleon scattering amplitude T

ab

In the following we will restrict ourselves to the forward scattering amplitudes ie

q q and p p Using standard Feynmanrules see eg the ab ove diagrams can

b e evaluated in a straightforward fashion For diagram a we obtain

a

u pT up

ab

p q m

g up up

a b

p q m

q

g u p

a b ab

s m

and the Dirac where we have used that

a b a b ab

equation p mup Obviously diagram a contributes only to the vector piece

of the amplitude B and the isospineven and o dd amplitudes are the same

g

B B

a a

s M

N

The contribution of the crossed or uchannel diagram b one obtains by replacing

s u

a b b a

q q

with the result

g

B B

b b

u M

N

Here isospineven and o dd amplitudes have the opp osite sign

It is instructive to calculate the scattering amplitude resulting from the rst two

diagrams only If we didnt know ab out chiral symmetry and hence the existence of

the exchange diagram this is what we would naively obtain At threshold q the

combined amplitudes are

g

B C

B

A

2

ab

m

M

N

2

M

N

m

B C

g B

A

2

ab

m

M

N

2

M

N

Using equations the resulting swave isospineven and isospino dd scattering

scattering length which is related to the scattering amplitude D at threshold by

D a

at threshold

m

M

N

would b e

m g

O a a b m

m

f M

N

M

N

m m

O a a b m

m

f M

N

M

N

where we have made of the Goldb ergerTreiman relation g f M This is to b e

N

compared with the exp erimental values of

a exp m a exp m

While we nd reasonable agreement for the isospino dd amplitude the isospin even

amplitude is o by two orders of magnitude A dierent choice of the pionnucleon

coupling g would not x the problem but just shift it from one amplitude to the other

Before we evaluate the remaining diagram c let us p oint out that in the chiral limit

ie m the isospino dd amplitude vanishes

In order to evaluate the exchange diagram we need to extract the pionsigma cou

pling from our Lagrangian This is done by expanding the p otential V up to third

p ower in the eld uctuations and The terms prop ortional to

then give the desired coupling

L f

The resulting amplitude is then given by

f

c

upT up g

ab

ab

t m

It only contributes to the scalar part of the amplitude A and only in the isospineven

see eqs we nd m m channel Using f

g t m m m g

A

c

f m t f t m

To leading order the contribution to the swave scattering lengths of diagram c is

m g

O c a

m

f M

N

M

N

a c

Thus to leading order the contribution of the exchange diagram c exactly cancels that

of the nucleonp ole diagrams a and b and the total isospineven scattering length

vanishes

m m

a O

M m

N

in much b etter agreement with exp eriment The cancelation b etween the large individual

contributions to the isospineven amplitude is a direct consequence of chiral symmetry

which required the exchange diagram In the chiral limit this cancelation is p erfect ie

the isospineven scattering amplitude vanishes identically b ecause the corrections m

are zero in this case

Furthermore since the third diagram c do es not contribute to the isospino dd am

plitude the go o d agreement found ab ove still holds In other words with the help of

chiral symmetry b oth amplitudes are repro duced well

Putting all terms together the isospineven amplitude D is given in terms of the

variables and

B

D A B

B

t m g g

f f t m

B

g g t m

B

f f t m

B

Here the rst term in the second line is the contribution form diagrams a and b and

2

m

the other two term are from diagram c At threshold where m and

B

M

N

t this reduces to

m m g

D

at threshold

f M m m

N

As already p ointed out to leading order m or in the chiral limit this amplitude

vanishes as a result chiral symmetry However the contribution next to leading order

involve also the mass of the meson which has not yet b een clearly identied m

in exp eriment In the sigmamo del this mass essentially is a free parameter since it

is directly prop ortional to the coupling Since gives the strength of the invariant

p otential V chiral symmetry considerations will not determine this parameter Thus

aside from the very imp ortant nding that the isospineven scattering length should b e

small the linear sigmamo del as no predictive p ower for the actual small value of the

scattering length

Notice that although D is the spin averaged forward t scattering amplitude

we can obviously study it an any value of t or equivalently and A kinematical

B

p oint of particular interest is the so called ChengDashen p oint given by

t m

B

13

In the framework of chiral p erturbation theory the value of the isospineven amplitude is essentially

regarded as an input to x the parameters of the expansion There are attempts to relate the value of

the scattering length to contributions from the Delta In this approach the problem is shifted to the

determination of an unknown oshell parameter app earing in the Deltapropagator

At this kinematical p oint the subtracted amplitude D is directly related with the

pionnucleon sigmaterm

N

N

D t m

f

In the sigmamo del we nd for the subtracted amplitude to leading order in the pion

mass

m g

N

D t m

f m f

where we have used the expression for the sigmaterm derived ab ove from the con

tribution of the explicit chiral symmetry breaking to the nucleon mass Notice although

the ChengDashen p oint is in an unphysical region it can nevertheless b e reached via dis

p ersion relation techniques and thus the sigmaterm can b e extracted from pionnucleon

scattering data For a detailed discussion see ref

Nonlinear sigmamo del

One of the disturbing features of the linear sigmamo del is the existence of the eld

b ecause it cannot really b e identied with any existing particle Furthermore at low

energies and temp eratures one would exp ect that excitations in the direction should

b e much smaller than pionic ones which in the chiral limit are massless see g

This is supp orted by our results for the pionnucleon scattering where in the nal result

the mass of the sigmameson only showed up in next to leading order corrections which

vanish in the chiral limit

Let us therefore remove the meson as a dynamical eld by sending its mass to

innity Formally this can b e achieved by assuming an innitely large coupling in the

linear sigmamo del As a consequence the mexican hat p otential gets innitely steep in

the sigmadirection see g This connes the dynamics to the circle dened by

the minimum of the p otential

f

V(σ, π=0) V(σ, π=0)

fπ σ fπ σ

This additional condition removes one degree of freedom which close to the ground

state where f is the sigma eld and we are left with pionic excitations only

Because of the ab ove constraint the dynamics is now restricted to rotation on the

so called chiral circle actually it is a sphere Therefore the elds can b e expressed in

terms of angles

x

f O x f cos

f

x

x O x f sin

f

q

Clearly which to leading order can b e identied with the pion eld Here

this ansatz fullls the constraint Equivalently one can chose a complex notation

for the elds as it is commonly done in the literature

(x)

x x

i

f

i sin i cos U x e

f f f

where U represents a unitary matrix The constraint is then equivalent to

tr U U

f

Since chiral symmetry or more precisely axialvector symmetry corresp onds to a symme

try with resp ect to rotation around the chiral circle all structures of the form

tr U U tr U U

are invariant Already at this p oint it b ecomes obvious that we eventually will need some

scheme which tells us which structures to include and which ones not This will lead us

to the ideas of chiral p erturbation theory in the following section

Let us continue by rewriting the Lagrangian of the linear sigmamo del in terms

of the new variables U or After a little algebra we nd that the kinetic energy term of

the mesons is given by

f

tr U U

Next we realize that nucleonmeson coupling term can b e written as

g i sin cos g f

f f

(x)

i

5

f

g f e

g f

where we have dened

(x)

i

5

2f

e

If we now redene the nucleon elds

W

f g

0 5

W

the interaction term can b e simply written as

g f g f M

W W N W W

where we have used the Goldb ergerTreiman relation In terms of the new elds

the entire interaction term as b een reduced to the nucleon mass term If we want

W

to identify the nucleons with the redened elds we also have to rewrite the nucleon

W

kinetic energy term in terms of those elds

i i

W W

Since is spacedep endent through the elds x the derivative also acts on giving

rise to additional terms After some straightforward algebra one nds

i i V A

W W W W

with

i h

V

h i

i

A

(x)

i

2f

e U

We do not need to transform the p otential of the linear sigmamo del V since

it vanishes on the chiral circle due to the constraint condition Putting everything

together the Lagrangian of the nonlinear sigmamo del which is often referred to as the

Weinb ergLagrangian reads in the ab ove variables

f

tr U U L i V A M

W N

were we have dropp ed the subscript from the nucleon elds Clearly this Lagrangian

dep ends nonlinearly on the elds It is instructive to expand the Lagrangian for small

uctuations f around the ground state This gives

L i M

W N

f f

where is now to b e identied with the pion Comparing with the linear sigmamo del the

eld has disapp eared and the coupling b etween nucleons and pions has b een changed to

a pseudovectorone involving the derivatives momenta of the pioneld In addition

an explicit isovector couplingterm has emerged From this Lagrangian it is immediately

clear that the swave pion nucleon scattering amplitudes vanishes in the chiral limit

b ecause all couplings involve the pion fourmomentum which at threshold is zero in case

of massless pions Thus the imp ortant cancelation b etween the nucleon p olediagrams

and the exchange diagram which we found in the linear sigmamo del has b een moved

into the derivative coupling of the pion through the ab ove transformations

On the level of the expanded Lagrangian the explicit breaking of chiral symme

try is intro duced by an explicit pion mass term Consequently corrections to the scattering

lengths due to the nucleon p ole diagrams should b e of the order of m since two derivative

couplings are involved However the coupling L which

2

f

m

contributes to rst order to the isospino dd amplitude should give rise to a term

2

f

in agreement with our previous ndings Not to o surprisingly one nds that the

ab ove Lagrangian gives exactly the same results for the scatteringlength as the linear

are absent b ecause we have assumed that the sigmamo del except that correction

2

m

mass of the meson is innite However the full Lagrangian would give rise to

many more terms if we expand to higher orders in the elds which then would lead to

lo ops etc How to control these corrections in a systematic fashion will b e the sub ject of

the following section where we discuss the ideas of chiral p erturbation theory

Basic ideas of Chiral Perturbation Theory

In the previous sections we were concerned with the most simple chiral Lagrangian in

order to see how chiral symmetry enters into the dynamics As we have already p ointed

out many more chirally invariant terms terms can b e included into the Lagrangian and

thus we need some scheme which tells what to include and what not This scheme is

provided by chiral p erturbation theory

Roughly sp eaking the essential idea of chiral p erturbation theory is to realize that at

low energies the dynamics should b e controlled by the lightest particles the pions and the

symmetries of QCD chiral symmetry Therefore smatrix elements ie scattering am

plitudes should b e expandable in a Taylorseries of the pionmomenta and masses which

is also consistent with chiral symmetry This scheme will b e valid until one encounters a

resonance such as the meson which corresp onds to a singularity of the smatrix Prac

tically sp eaking ab ove the resonance a BreitWigner distribution cannot b e expanded in

a Taylor series

It is not to o surprising that such a scheme works Imagine we did not know anything

ab out QED We still could go ahead and parameterize the say electronproton scattering

amplitude in p owers of the momentum transfer t In this case the Taylor co ecients

would b e related to the total charge the charge radius etc With this information we

could write down an eective protonelectron Lagrangian where the couplings are xed

by the ab ove Taylorco ecients namely the charge and the charge radius This eective

theory will of course repro duce the results of QED up to the order which has b een xed

by exp eriment It is in this sense the eective Lagrangian obtained in chiral p erturbation

theory should b e understo o d namely as a metho d of writing smatrix elements to a given

order in pionmomentummass And to the order considered the the eective Lagrangian

obtained with chiralp erturbation theory should b e equivalent with QCD

It should b e stressed that chiral p erturbation theory is not a p erturbation theory in

the usual sense ie it is not a p erturbation theory in the QCDcoupling constant In this

resp ect it is actually a nonp erturbative metho d since it takes already innitely many

order of the QCD coupling constant in order to generate a pion Instead as already

p ointed out Chiral p erturbation theory is an expansion of the smatrix elements in terms

of pionmomentamasses

From the ab ove arguments one could get the impression that chiral p erturbation

theory has no predictive p ower since it represents simply a p ower expansion of measured

scattering amplitudes Although this may true in some cases one could easily imagine

that one xes the eective Lagrangian from some exp eriments and then is able to calculate

other observables For example imagine that the eective pionnucleon interaction has

b een xed from pion nucleonscattering exp eriments This interaction can then b e used

to calculate eg the photopro duction of pions

To b e sp ecic let us discuss the case of pure pionic interaction ie without any

nucleons As p ointed out in the previous section chiral invariance requires that the

eective Lagrangian has to b e build from structures involving U U such as

tr U U tr U U tr U U tr U U

(x)

i

f

Furthermore each U e contains any p ower of the pioneld which may give rise

to lo ops etc To sp ecify which of the ab ove terms should b e included into the eective

Lagrangian and how much each term should b e expanded in terms of the pion eld one

has to count the p owers of pion momenta contributing to the desired pro cess scattering

amplitude

Consider a given Feynmandiagram contributing to the scattering amplitude It will

have a certain numb er L of lo ops a certain numb er V of vertices of typ e i involving d

i i

derivatives of the pion eld an a certain numb er of internal lines I The p ower D of the

p

pion momentum q this diagram will have at the end can b e determined as follows

R

each lo op involves an integral over the internal momenta d q q

each internal pion line corresp onds to a pion propagator and thus contributes as

2

q

d

i

each vertex V involving d derivatives of the pion eld contributes like q

i i

D

Consequently the total p ower of q q is given by

X

V d D L I

i i P

i

This can b e simplied by using the general relation b etween the numb ers of lo ops internal

lines and vertices of a given diagram

X

V L I

i p

i

to give

X

V d D L

i i

i

With this formula we can determine to which order of the Taylor expansion of the scat

tering amplitude a given diagram contributes

In order to see how this counting rule leads to an eective Lagrangian of a given

order we b est study the simple example of pionpion scattering Since U U do es not

Figure Leading order diagram for scattering

contribute to the dynamics the simplest contribution to the eective Lagrangian is given

by

f

L tr U U

where the subscript denotes the numb er of derivatives involved Since we are discussing

pionpion scattering we have to expand at least up to fourth order in the pion elds

h i

O L

f

where the second term contributes to the pionpion scattering amplitude Although this

term has two contributions for the purp oses of p ower counting the second term may b e

considered as one vertex function b ecause b oth contributions have the same numb er of

derivatives Thus to lowest order we have just one diagram which is shown in g

It has no lo ops L and the vertex function carries two derivatives of the pion eld

Using the ab ove counting rule the order of this diagram is D

We can easily convince ourselves that there are no more terms contributing to this

order Including terms into the Lagrangian with four derivatives of the pions eld such as

eg tr U U immediately leads to D Also expanding the ab ove Lagrangian

up to sixth order in the pion eld leads to D b ecause two of the pion elds have

to b e combined into a lo op since we are only considering a pro cess with four external

pions

Obviously the order of the eective Lagrangian dep ends on the pro cess under con

sideration Whereas a term involving six pion elds contributes to the order D to

pionpion scattering it would contribute to order D to a pro cess with three initial

and three nal pions Of course having realized that we are actually parameterizing

smatrix elements this is not such a surprise

As already mentioned to order D we have contributions from dierent sources

First of all form higher derivative terms in the Lagrangian and secondly from the ex

pansion to higher order in the pion elds giving rise to lo ops The b eauty of chiral

p erturbation theory is that the eects of lo ops can b e systematically b e absorb ed into

renormalized couplings and masses For details see eg

By now the astute reader will have asked himself How do I know that a momentum

is small or in other words what is the expansion scale There are several answers on the

market Georgi argues based on renormalization arguments that the scale should b e

f GeV whereas others argue that the mass of the lowest lying resonance

should give the scale since this is the energy where the entire game seizes to work

This seems to b e a reasonable argument and assuming that there is no meson of mass

MeV the mass of the meson should provide a reasonable b enchmark

So far we have worked in the chiral limit ie assuming that the pion mass vanishes

The explicit breaking of chiral symmetry is intro duced by terms of the form tr U U

and and the simplest symmetry breaking is

f m

L O tr U U m

X S B

which to leading order in the pionelds corresp onds to a pion massterm the constant

term do es not contribute to the dynamics Again one can have many symmetry breaking

term involving the ab ove structure such as

tr U U tr U U tr U U

so that an ordering scheme is necessary Therefore in the realistic case of explicit chiral

symmetry breaking the scattering amplitudes are not only expanded in terms of the pion

momenta but also in terms of the pion masses The countingrule is the same as given

ab ove where d now gives the numb er of derivatives and pion masses of a given

i

vertex of typ e i The total eective Lagrangian for pionpion scattering to order D

is then given by

h i

m

L m

f f

In principle the adjustable parameters of this Lagrangian are the pionmass and the

piondecay constant which have to b e xed to the exp erimental values

The resulting pionpion scattering length and volumes are then given by

m m

a a a

f f f m

where the subscript denotes the angular momentum and the sup erscript the isospin of

the amplitude As shown in table the leading order results agree reasonably well

with exp eriment and are improved by the next to leading order corrections Apparently

Exp eriment Lowest Order First Two Orders

a m

a m

a m

Table Pionpion scattering length

we do not nd p erfect agreement with exp eriment even for the swave scattering lengths

although already to leading order we haven taken into account terms quadratic in the

momenta so that higher orders in the pion momentum will not improve the situation

However rememb er that we not only expand in terms of the pion momenta but also as

a result of the explicit symmetry breaking in terms of the pion mass which in principle

can contribute to any order to the swave scattering length

As already p ointed out in the b eginning of this section chiral p erturbation theory or

more precisely the expansion in momenta breaks down once we get close to a resonance

This one easily understands by lo oking at the BreitWigner formula for the scattering

amplitude involving a resonance

f E

E E i

R

For energies which are small compared to the resonance energy E E this amplitude

R

may b e expanded in terms of a p ower series and the concept of chiral p erturbation theory

works well

E i

f E E E

R

E E

R R

However once we get close to the resonanceenergy we need to expand to higher and

higher order until at E E the p owerseries in E seizes to converge To b e sp ecic

R

we exp ect that in the the isovector pwave channel which is dominated by the meson

resonance the chiral p erturbation expansion should fail for energies E m

Finally let us include the nucleons into the chiral counting Naively one would think

that this should destroy the entire concept b ecause the nucleon has a large mass which is

of the order of the expansion scale However since at low energies the scattering amplitude

may also b e calculated in a nonrelativistic framework we do not exp ect the nucleon mass

2

p

which is small to enter directly but to leading order only via the kinetic energy

M

N

compared to that of the pion at the same momentum Therefore chiral p erturbation

theory should also work with nucleons present for details see The ab ove argument

can b e formalized by realizing that the nucleon only enters the amplitudes through the

nucleon propagator see eg the results of section At low momenta the nucleon

propagator contributing to diagram a of g can b e written as

p q M q M M

N N N

O

p q M M q q M

N N

N

where

M M

N N

M

N

pro jects on p ositive energy states Hence to leading order each nucleon propagator

to the p ower of pion momentum of the scattering amplitude This leads contributes like

q

to the following counting rule which now also includes the nucleons

X

D L V d E n

i i N i

i

Here the notation is as in equ and E denotes the numb er of external nucleon

N

lines and n the numb er of nucleon elds of vertex i which is typically n

i i

For the simple nucleonp ole diagram using pseudovector coupling we thus would have

n and D L E d n such that d

N

On top of the expansion in terms of pionmomenta and pion masses from equ

q

we therefore also have an expansion in the velo city of the nucleons v This is

M

N

carried out in a systematic fashion in the so called HeavyBaryon ChiralPerturbation

Theory as intro duced by Jenkins and Manohar This approach essentially corre

sp onds to a systematic nonrelativistic expansion for the nucleon wavefunction on the

basis that the nucleon baryon is heavy compared to the momenta involved We should

mention that the eect of the nucleon can also b e included in a fully covariant fashion

as discussed by Gasser et al

Including the nucleon gives rise to additional structures which explicitly break the

chiral symmetry such as

L a tr U U a

f

To leading order this is just a contribution to the nucleon mass which allows us to

identify the co ecient a with the sigmaterm

N

N

L tr U U

N N

f

The next to leading term in the ab ove expression is an attractive interaction b etween

pion and nucleon which contributes to the order D to the amplitude This term

by itself is quite large and would lead to a wrong prediction for the swave pionnucleon

amplitude However there are additional terms contributing to the same order which in

the heavyfermion expansion comes from the nucleonp ole diagrams The co ecients of

these terms then need to b e chosen such that the resulting scattering length acquire the

small value observed in exp eriment

Applications

In this last section we want to discuss a few applications of chiral symmetry relevant

for the physics of dense and hot matter First we briey address the issue of in medium

masses of pions and kaons Then we will discuss the temp erature and density dep endence

of the quark condensate We will conclude with some general remarks on the prop erties

of vector mesons in matter

Pion and kaon masses in dense matter

Changes of the pion mass in the nuclear medium should show up in the isoscalar pion

swave optical p otential To leading order in the density this is related to the swave

isoscalar scatteringlength a by

m

a U

M

N

where is the pion energy Since the swave isoscalar scattering length is small as a

result of chiral symmetry and slightly repulsive we predict a small increase of the pion

mass in the nuclear medium which at nuclear matter amounts to m M eV One

arrives at the same result by evaluating the eective Lagrangian as obtained from chiral

p erturbation theory at nite density This is not surprising since the swave

isoscalar amplitude is used to x the relevant couplings

In case of the kaons which can also b e understo o d as Goldstone b osons of an extended

SU SU chiral symmetry some interesting features o ccur Chiral p erturbation

theory predicts a repulsive swave scattering length for K nucleon scattering and a large

attractive one for K Using the ab ove relation for the optical p otential this

led to sp eculations ab out a p ossible swave kaon condensate in dense matter with

rather interesting implications for the structure and stability of neutron stars

Exp erimentally however one nds that the isoscalar swave scattering length for the

K is repulsive calling into question the results from chiral p erturbation theory The

resolution to this puzzle is the presence of the resonance just b elow the kaon

nucleon threshold This resonance which has not b een taken into account in the chiral

p erturbation analysis gives a large repulsive contribution to the scattering amplitude at

threshold Do es that mean that chiral p erturbation theory failed Yes and no Yes

b ecause as already p ointed out it is not able to generate any resonances and thus leads

to bad predictions in the neighb orho o d of the resonance No b ecause it predicts a

14

Lee et al have attempted to include the as an explicit state in a chiral p erturbation

theory analyses of the kaonnucleon scattering length While this approach may b e a reasonable thing to

do phenomenologically it app ears to b e b eyond the original philosophy of chiral p erturbation theory

strong attraction b etween the proton and the K which if iterated to innite order can

generate the resonance as a b ound state in the continuum in the continuum

b ecause the decays into

This situation is well known from nuclear physics The protonneutron scattering

length in the deuteron channel is repulsive although the protonneutron interaction is

attractive The reason is that in this channel a b ound state can b e formed the deuteron

which gives rise to a strong repulsive contribution to the scattering amplitude at threshold

To carry this analogy further we know that in nuclear matter the deuteron has dis

app eared essentially due to Pauliblo cking revealing the true attractive nature of the

nuclear interaction As a result we have an attractive mean eld p otential for the nucle

ons Similarly one can argue that the if it is a K proton b ound state

should eventually disapp ear resulting in an attractive swave optical p otential for the K

in nuclear matter Indeed an analysis of K atoms shows that the optical p otential

turns attractive already at rather low densities Extrap olated to nuclear mat

ter density the extracted optical p otential would b e as deep as MeV in reasonable

agreement with the predictions from chiral p erturbation theory

Change of the quarkcondensate in hot and dense matter

Temp erature dep endence

One of the applications of chiral p erturbation theory relevant to the physics of hot and

dense matter is the calculation of the temp erature dep endence of the quark condensate

Here we just want derive the leading order result A detailed discussion which includes

also higher order corrections can b e found in ref The basic idea is to realize that

the op erator of the quarkcondensate q q enters into the QCDLagrangian via the quark

mass term Thus we may write the QCDHamiltonian as

H H m q q

q

The quark condensate at nite temp erature is then given by the following statistical sum

P

H t

ijq q e ji

i

q q

P

T

H T

ije ji

i

Since H m q q this can b e written as

q

q q T ln Z m

T q

m

q

P

H T

where the partition function Z is given by Z ije ji

i

In chiral p erturbation theory we do not calculate the partition function of QCD but

rather that of the eective Lagrangian To make contact with the ab ove relations we

utilize the GellMann Oakes Renner relation To leading order in the pion mass the

derivative with resp ect to the quark mass therefore can b e written as

jq q j

m f m

q

Next to leading order contributions arise among others from the quarkmass dep endence

of the vacuum condensate

To leading order the partition function is simply given by that of a noninteracting

pion gas

Z

d p ln expE T ln Z ln Z ln Z ln Z

gas

where Z stands for the vacuum contribution which we of course cannot calculate in

chiral p erturbation theory since we are only concerned ab out uctuation around that

vacuum Thus the temp erature dep endence of the quark condensate in the chiral limit is

given by

jq q j

Z q q jq q j

gas T

f m

m

T

jq q j

f

Thus to leading order the quark condensate drops like T ie at low temp eratures

the change in the condensate is small

Corrections include the eect of pion interactions which in the chiral limit are pro

p ortional to the pion momentum and thus contribute to higher orders in the temp erature

Including contributions up to three lo ops one nds see eg

T T q q T

q T

c c ln O T c

q q f f f T

For N avors of massless quarks the co ecients are given in the chiral limit by

f

N N

f f

c N N c c

f

f

N N

f

f

The scale can b e xed from pion scattering data to b e MeV In

q q

g we show the temp erature dep endence of the quarkcondensate as predicted by 1.0

0.8 0 q> - 0.6 / - 0.4

0.0 0 50 100 150 200

T [MeV]

Figure Temp erature dep endence of the quark condensate from chiral p erturbation

theory chiral limit

the ab ove formula Currently lattice gauge calculations predict a critical temp erature

T MeV ab ove which the quark condensate has disapp eared At this temp erature

c

chiral p erturbation theory predicts only a drop of ab out which gets even smaller once

pion masses are included However we do not exp ect chiral p erturbation to work well

close to the critical temp erature The strength of this approach is at low temp eratures

The prediction that to leading order the condensate drops quadratic in the temp erature

is a direct consequence of chiral symmetry and can b e used to check chiral mo dels as well

as any other conjectures involving the change of the quarkcondensate such as eg the

change of hadron masses

Density dep endence

For low densities the density dep endence of the quark condensate can also b e determined

in a mo del indep endent way We exp ect that to leading order in density the change in

the quark condensate is simply given by the amount of quark condensate in a nucleon

multiplied by the nuclear density

q q q q N jq q jN higher orders in

15

Again we give a heuristic argument A rigorous derivation based on the HellmannFeynman theorem

can b e found eg in

All we need to know is the matrix element of q q b etween nucleon states This matrix

element however enters into the pionnucleon sigmaterm

q q

N

N jq q jN

N

m m f

q

2 2

f m

Thus where we also have made use of the GORrelation namely m

q

q q

0

we predict that the quark condensate drops linearly with density as compared to the

quadratic temp erature dep endence found ab ove

N

q q q q

f m

Corrections to higher order in density arrise among others from nuclear binding eects

These have b een estimated by Bro ckmann to b e at most of the order of for

denities up to twice nuclearmatter density Assuming a value for the sigma term of

MeV we nd that the condensate has dropp ed by ab out at nuclear matter

N

density

q q q q

Thus nite density is very ecient in reducing the quark condensate and we should

exp ect that any in medium mo dication due to a dropping quark condensate should

already b e observable at nuclear matter density The ab ove ndings also suggest that

chiral restoration ie the vanishing of the quarkcondensate is b est achieved in heavy

ion collisions at b ombarding energies which still lead to full stopping of the nuclei

Masses of vector mesons

Finally let us briey discuss what chiral symmetry tells us ab out the masses of vector

mesons in the medium Vector mesons such as the meson are of particular interest

b ecause they decay into dileptons Therefore p ossible changes of their masses in medium

are accessible to exp eriment

Using current algebra and PCAC Dey et al could show that at nite temp erature

the mass of the rhomeson do es not change to order T Instead to order T the vector

correlation function gets an admixture from the axialvector correlation function

C T C T C T

V V A

2

T

with The imaginary part of this vectorcorrelation function is directly related to

2

f

the dileptonpro duction crosssection As depicted in g the ab ove result therefore C C V T = 0 V O(T2)

mρ mρ m

MMa1

Figure Vectorsp ectral functions at T and to leading order in temp erature as given

by equ

predicts that to leading order in the temp erature the dilepton invariant mass sp ectrum

develops a p eak at the mass of the a meson in addition to that at the mass of the At

the same time the contribution at the p eak is reduced in comparison to the free case

Furthermore the p osition of the p eaks is not changed to this order in temp erature This

general result is also conrmed by calculations in chiral mo dels which have b een extended

to include vector mesons Notice that the ab ove nding also rules out that the

mass of the meson scales linearly with the quark condensate b ecause previously see

section we found that the quark condensate already drops to order T whereas the

mass of the do es not change to this order

Corrections to higher order in the temp erature however are not controlled by chiral

symmetry alone and therefore one nds mo del dep endencies Pisarski for instance

predicts in the framework of a linear sigma mo del with vector mesons that to order T

the mass of the decreases and that of the a increases Song on the other uses

a nonlinear mo del and nds the opp osite namely that the go es up and the a go es

down At the critical temp erature b oth again agree qualitatively in that the masses of a

and b ecome degenerate at a value which is roughly given by the average of the vacuum

masses GeV This agreement again is a result of chiral symmetry

At and ab ove the critical temp erature where chiral symmetry is not anymore sp on

taneously broken chiral symmetry demands that the vector and axial vector correlation

functions are the same One way to realize that is by the having the same masses for

the vector and axialvector a However this is not the only p ossibility As nicely

discussed in a pap er by Kapusta and Shuryak there are at least three qualitatively

dierent p ossibilities which are sketched in g

The masses of and a are the same In this case clearly the vector and axial

vector correlation functions are the same Note however that we cannot make any

statement ab out the value of the common mass It may b e zero as suggested by

some p eople it may b e somewhere in b etween the vacuum masses as the chiral C CV,A CV,A V,A

mρ = m mρ m a1 M a1 M M

(1) (2) (3)

Figure Several p ossibilities for the vector and axialvector sp ectral functions in the

chirally restored phase

mo dels seem to predict and it my b e even much larger than the mass of the a

We may have a complete mixing of the sp ectral functions Thus b oth the vector

and axialvector sp ectral functions have p eaks of equal strength at b oth the mass

of the and the mass of the a leading to two p eaks of equal strength in the

dilepton sp ectrum mo dulo Boltzmannfactors of course One example would b e

given by the low temp erature result with the mixing parameter T

c

2

T

Using the low temp erature result for would give a critical temp erature of

2

f

p

T f M eV which is surprisingly close to the value given by recent

c

lattice calculations

Both sp ectral functions could b e smeared over the entire mass range Due to thermal

broadening of the mesons and the onset of deconnement the structure of the

sp ectral function may b e washed out and it b ecomes meaningless to talk ab out

mesonic states

To summarize the only unique prediction derived from chiral symmetry current al

gebra ab out the temp erature dep endence of the mass is that it do es not change to

order T ie at low temp eratures Furthermore at and ab ove the critical temp erature

chiral symmetry requires that the vector and axialvector sp ectral functions are identical

which however do es not necessarily imply that b oth exhibit just one p eak lo cated at

the same p osition Corrections of the order T cannot b e obtained from chiral symmetry

alone

Finally let us p oint out that the ab ove ndings do not rule out scenarios which

relate the mass of the with the temp erature dep endence of the bagconstant or gluon

condensate such as prop osed by Pisarski and Brown and Rho This ideas

however involve concepts which go b eyond chiral symmetry such as the melting of the

gluon condensate Consequently in these scenarios a certain b ehavior of the mass of the

meson can only indirectly b e brought in connection with chiral restoration

Acknowledgments I would like to thank C Song for useful discussions concerning

the eects of chiral symmetry on the vector mesons This work was supp orted by the

Director Oce of Energy Research Oce of High Energy and Nuclear Physics Division

of the Department of Energy under Contract No DEACSF

App endix Useful references

This is a selection of references which the author found useful in preparing these lectures

It is by no means a complete representation of the available literature

DK Campb ell Chiral symmetry pions and nuclei

in Nuclear Physics with Heavy Ions and Mesons Volume Editors Balian Rho

and Ripka Les Houches XXX North Holland

Comment Very nice intro duction to chiral symmetry

H Georgi Weak Interactions and mo dern Particle Physics

Benjamin Cummings

Comment Go o d intro duction to the idea of chiral p ert theory sometimes a little

brief

JJ Sakurai Currents and Fields

Chicago Univ Press

Comment Nice little b o ok ab out current algebra etc These are lecture notes and

therefore rather explicit

De Alfaro Fubini Furlan and Rosseti Currents in hadron Physics

North Holland

Comment The current algebra bible Contains everything up to no chiral

p ert theory

U Meissner Recent developments in chiral p erturbation theory

Rep Prog Phys

Comment Go o d review ab out the technical asp ects of chiral p ert theory

H Leutwyler Principles of Chiral Perturbation Theory

Lectures from Hadrons workshop Gramado RS Brasil hepph

Comment Very nice review ab out the conceptual asp ects of chiral p erturbation

theory Somewhat complementary to that of Meissner

References

JD Bjorken and SD Drell Relativistic Quantum Fields Mc Graw Hill New York

ED Commins and PH Bucksbaum Weak interactions of leptons and quarks Cam

bridge University Press Cambridge UK

S Weinb erg Phys Rev Lett

JJ Sakurai Currents and mesons University of Chicago Press Chicago

M GellMann and M Levy Nuovo Cimento

C DeTar and T Kunihiro Phys Rev D

H Hohler Pionnucleon scattering ed H Schopper LandoltBornstein volume

Springer Berlin

V de Alfaro V Fubini S Furlan and G Rossetti Currents in hadron physics

NorthHolland Amsterdam

M GellMann RJ Oakes and B Renner Phys Rev

J Gasser and H Leutwyler Phys Rep

T Ericson and W Weise Pions and Nuclei Clarendon Press Oxford

V Bernard N Kaiser and UG Meissner Phys Lett B

S Weinb erg Physica A

H Leutwyler Principles of chiral p ertubation theory In Proceedings of Hadron

Workshop Gramado Brazil hepph World Scientic

JF Donoghue E Golowich and BR Holstein Dynamics of the Standard Model

Cambridge University Press Cambridge UK

H Georgi Weak interactions and modern particle theory BenjaminCummings

Menlo Park Ca

H Leutwyler Foundations and scop e of chiral p ertubation theory In Proceed

ings Workshop on Chiral Dynamics Theory and Experiment MIT Lecture

Notes in Physics hepph AM Bernstein and BR Holstein editors

Springer

S Weinb erg Phys Rev Lett

S Weinb erg Nucl Phys B

E Jenkins and A Manohar Nucl Phys B

J Gasser ME Sanio and A Svarc Nucl Phys B

V Thorsson and A Wirzba Nucl Phys A

GE Brown CH Lee M Rho and V Thorsson Nucl Phys A

DB Kaplan and AE Nelson Phys Lett B

AE Nelson and DB Kaplan Phys Lett B

GE Brown V Thorsson K Kub o dera and M Rho Phys Lett B

V Thorsson M Prakash and JM Lattimer Nucl Phys A

CH Lee GE Brown and M Rho Phys Lett

N Kaiser PB Siegel and W Weise nuclth

V Ko ch Phys Lett B

E Friedman A Gal and CJ Batty Phys Lett B

P Gerb er and H Leutwyler Nucl Phys B

EG Drukarev and EM Levin Nucl Phys A

TD Cohen RJ Furnstahl and DK Griegel Phys Rev C

R Bro ckmann and W Weise The chiral condensate in matter In Proceedings

of the Interanational Workshop on Hadrons in Nuclear Matter Hirschegg H

Feldmeier and W Norenb erg editor volume XXI I I page

M Dey VL Eletzky and BL Ioe Phys Lett B

C Song Phys Rev D

RD Pisarski Phys Rev D

JI Kapusta and EV Shuryak Phys Rev D

RD Pisarski Phys Lett B

GE Brown and M Rho Phys Rev Lett