Use r as a probe for the restoration of chiral symmetry (Pisarski, 1982)

1
Prime goal
Use r as a probe for the restoration of chiral
symmetry (Pisarski, 1982)
Principal difficulty properties of r in hot
and dense matter unknown (related to the
mechanism of mass generation) properties of
hot and dense medium unknown (general goal of
studying nuclear collisions) ? coupled
problem of two unknowns need to learn
on both
2
General question of QCD

• Origin of the masses of light hadrons?
• Expectation
• Mh10-20 MeV
• approximate chiral SU(nf)L SU(nf)R
  symmetry
• chiral doublets, degenerate in mass
• Observed
• MN1 GeV
• spontaneous chiral symmetry breaking ?
  ltqqgt ? 0
• M? 0.77 GeV ? Ma1 1.2 GeV

3
Several theoretical approaches including lattice
QCD still in development
Lattice QCD(for mB0 andquenched approx.) two
phase transitions at the same critical
temperature Tc
cL
-
cm
qq
L
1.0 T/Tc
1.0 T/Tc
deconfinement chiral
symmetry transition
restoration
hadron spectral functions on the lattice only now
under study
explicit connection between spectral properties
of hadrons (masses,widths) and the value of the
chiral condensate ltqqgt ?
4
High energy nuclear collisions

• Principal experimental approach
• measure lepton pairs (ee- or µµ-)
• - no final state interactions
• - continuous emission during the whole
  space-time
• - evolution of the collision system
• dominant component at low invariant masses
• thermal radiation, mediated by the vector mesons
  ?,(?,?)

Gtot MeV r (770)
150 (1.3fm/c) w(782)
8.6 (23fm/c)
f(1020) 4.4 (44fm/c)
in-medium radiation dominated by the ?

1. life time t 1.3 fm/c ltlt tcollision gt 10 fm/c
2. continuous regeneration by ??

5
CERES/NA45 at the CERN SPS

• Pioneering experiment,
• built 1989-1992
• results on p-Be/Au, S-Au
• and Pb-Au
• first measurement of
• strong excess radiation
• above meson decays
• vacuum-? excluded
• resolution and statistical
• accuracy insufficient to
• determine the in-medium
• spectral properties of the ?

6
Which processes populate the dimuon mass spectrum
below 1 GeV?
7
Measuring the collision centrality
The collision centrality can be measured via the
charged particle multiplicity as measured by the
pixel vertex telescope
Track multiplicity of charged tracks for
triggered dimuons for
opposite-sign pairs combinatorial background
signal pairs
4 multiplicity windows
Centrality bin multiplicity ltdNch/d?gt3.8
Peripheral 428 17
SemiPeripheral 2892 63
Semi-Central 92160 133
Central gt 160 193
8
Comparison of hadron decay cocktail to peripheral
data
No (significant) in-medium effects are expected
in peripheral collisions We can try to perform a
fit with the hadron cocktail
Very good description, also at low mass and pT
where acceptance is smallest
9
Isolate possible excess by subtracting cocktail
(without r) from the data
How to fit in the presence of an unknown source?
? Try to find excess above cocktail (if it
exists) without fit constraints

• ? and ? fix yields such as to get, after
  subtraction, a smooth underlying continuum
• ?
• (?) set upper limit, defined by saturating the
  measured yield in the mass region close to 0.2
  GeV (lower limit for excess).
• (?) use yield measured for pT gt 1.4 GeV/c

10
Evolution of the excess shape with centrality
The evolution of the excess with centrality can
be studied with precision with a rather fine
binning in multiplicity
data cocktail (all pT)

• Clear excess above the cocktail ?, centered at
  the nominal r pole and rising with centrality
• Excess even more pronounced at low pT

cocktail ?/? 1.2
11
Sensitivity of the difference procedure
Change yields of ?, ? and ? by 10 ? enormous
sensitivity, on the level of 1-2, to mistakes
in the particle yields.
The difference spectrum is robust to mistakes
even on the 10 level, since the consequences of
such mistakes are highly localized.
12
Systematics
The largest source of systematic error comes from
the subtraction of combinatorial and fake matches
background. In principle there are other
uncertainty sources as the form factors, but
these are negligible compared to the background.
Illustration of sensitivity ? to correct
subtraction of combinatorial background
and fake matches ? to variation of the ? yield
The systematic errors of continuum 0.4ltMlt0.6 and
0.8ltMlt1GeV are 25 (at most) in the most central
collisions
The structure in ? region looks rather robust
13
Dilepton Rate in a strongly interacting medium
dileptons produced by annihilation of thermally
excited particles ??- in hadronic phase
qq in QGP phase
at SPS energies ? ? - ???µµ- dominant
hadron basis
photon selfenergy
spectral function
Vector-Dominance Model
14
r meson in vacuum
Introduce r as gauge boson into free p r
Lagrangian ?
? is dressed with free pions
vacuum spectral function
(like ALEPH data V(t? 2pnt ))
15
Physics objective in heavy ion collisions
Goal Study properties of the r spectral
function Im Dr in a hot and dense medium
ProcedureSpectral function accessible through
rate equation, integrated over space-time and
momenta LimitationContinuously varying
values of temperature T and baryon density rB,
(some control via multiplicity dependences)
16
r spectral function in hot and dense hadronic
matter
Hadronic many-body approach Rapp/Wambach et
al., Weise et al.
hot matter
hot and baryon-rich matter
? is dressed with hot pions Prpp , baryons
Pr B (N,D ..) mesons Pr M (K,a1..)

• melts in hot and dense matter
• - pole position roughly unchanged - broadening
  mostly through baryon interactions

17
r spectral function in hot and dense hadronic
matter
Dropping mass scenario Brown/Rho et al.,
Hatsuda/Lee
explicit connection between hadron masses and
chiral condensate
universal scaling law

continuous evolution of pole mass with T and r
broadening at fixed T,r ignored
18
Final mass spectrum
integration of rate equation over
space-time and momenta required
continuous emission of thermal radiation during
life time of expanding fireball
example broadening scenario
19
How to compare data with predictions?
There are two possibilities, in principle

• correct data for acceptance in 3-dim. space
  M-pT-y and compare directly to predictions at the
  input
• 2) use the predictions in the form
• and generate Monte Carlo decays of the
  virtual photons g into mm- pairs, propagate
  these through the acceptance filter and compare
  results to uncorrected data at the output (done
  presently)
• The conclusions on the agreement or
  disagreement between data and predictions in
  principle should be independent of whether
  comparison is done at input or output (provided
  you understand the effect of your detector on
  data well)

20
Acceptance filtering of theoretical prediction in
NA60
Input (example) thermal radiation based on RW
spectral function
all pT
Output spectral shape much distorted relative
to input, but somehow reminiscent of the spectral
function underlying the input by chance?
21
Comparison to the main models that appeared in
the 90s
Rapp-Wambach hadronic model predicting strong
broadening/no mass shift Brown/Rho scaling
dropping mass due to dropping of chiral condensate
Predictions for In-In by Rapp et al (2003) for
dNch/d? 140, covering all scenarios
Theoretical yields normalized to data in mass
interval lt 0.9 GeV
After acceptance filtering, data and predictions
display spectral functions, averaged over
space-time and momenta
Only broadening of ? (RW) observed, no mass
shift (BR)
22
Brown-Rho vs Rapp-Wambach
Modification od BR by change of the fireball
parameters
Van Hees and Rapp, hep-ph/0604269
even switching out all temperature effects does
not lead to agreement between BR and the data
23
The role of baryons (Rapp-Hees)

• Without baryons
• Not enough broadening
• Lack of strength below the r peak

24
Semicentral collisions (Rapp-Hees)
Something is missing at high pT. What?
25
The vacuum r contribution
Ruppert-Renk
At high pT there is an important contribution
from the vacuum r r decays at kinetic
freeze-out We will see later why it is important
at high pT (does not dominate the yield
integrated in pT)
Rapp-Hees
26
Intermediate summary
In the last 2 years significative advance in
understanding the in-medium effects on the r
spectral function.

The main result is Hadronic many body
approaches predict a broadening of the r without
mass shift which is in fair agreement with
data Models predicting a decrease of the r mass
are ruled out data The main open question
is What is the connection to chiral symmetry?
27
The mass region above 1 GeV vector-axial vector
mixing
Above 1 GeV we can have contributions from 4p
processes. The spectral shape can be found for
instance from ee-?4p or studying (ALEPH) t?(2np)?
3p, 5p
2p, 4p, 6p
In addition, because of the pion heat bath, it
is possible also to have processes in which an
axial vector particle interacts with a pion, as
pa1?mm-. This effectively introduces a mixing
between vector and axial-vector states (at the
correlator level). This mixing depends on the
amount of chiral symmetry restoration
28
The mass region above 1 GeV models vs data
Ruppert / Renk, Phys.Rev.C (2005)
Rapp/Hees
Mass region above 1 GeV described dominantly in
terms of hadronic processes, 4 p
Mass region above 1 GeV described dominantly in
terms of partonic processes, dominated by qqbar
annihilation
? Hadron-parton duality
29
s(ee-?hadrons) in vacuum
e e-
p - p
r I 1
r
2p 4p ...
pp
e e-
h1 h2
r w f
KK
q q
_
qq

_
s sdual(1.5GeV)2 pQCD
continuum s lt sdual Vector-Meson
Dominance
30
Transverse momentum spectra
In a static fireball at temperature T the
differential particle momentum distribution is
Lorentz invariant phase space element
Assume a thermal Boltzmann shape
? transverse mass spectra (integrated over
rapidity)
mT scaling all particle spectra have the same T
slope
31
The fireball produced in a heavy ion collision
expands. The thermal energy is converted in
mechanical work and collective motion (flow)
develops.
Due to the very high particle density at the
center of the source, particles only get out to
the side of the source ? transverse direction
radial flow
energy of the particle in the local rest frame
of fluid element boosted to the observer rest
frame
32
The Cooper-Frye formula
The number of particles that cross a closed
surface can be written as
When we count the particles produced in a
collision we count for instance the number of
particles across a surface for a long time ?
three dimensional space or hypersurface in
space-time dSm
Cooper-Frye formula
33
At kinetic freeze-out the stable hadrons stream
free to the detector ? S can be taken as the last
scattering surface Only the contributions from
the narrow temperature window around kinetic
freeze-out must be considered. With some
mathematics one can show that
Integrated over f
Transverse flow-field
Once the mass is fixed (the particle is
specified), the function has only three
parameters vT, Tf and a normalization In
principle they can be extracted with a two
parameter fit to experimental distributions Evalu
ate c2 for fixed vT and Tf Create a c2 map as a
function of vT and Tf
34
Stable hadrons reflect the kinetic freeze-out
conditions. Using exp(-mT/T) gives a T dependent
on the momentum range ? T from exponential fit
(call Tslope) is not anymore the source
temperature Tf. At high pT the spectra are still
exponential with a common slope which reflects a
freeze-out temperature blue-shifted by the flow
transverse velocity vT
The pT spectra appear flattened at low pT and mT
scaling is broken. The T slope becomes mass
dependent (mT scaling is broken)
In principle allows to separate the thermal from
the collective motion
35

• Common flow velocity in p,K,p and their
  anti-particles is seen at SPS and AGS energies

NA49/SPS results Common flow velocity seen for
very wideparticle species (Nucl.Phys A 715
61) Pion and deuteron are taken out from fit
procedure (many pions come from resonance decays
- deuterons are most likely produced with
proton-neutron coalescence) However, spectra
described are very well described with the
thermal parameter extracted with other particles
Common flow velocities are seen also in RHIC
Au-Au data (PHENIX and STAR)
36
Other effects of transverse flow Peripheral
collisions shorter fireball lifetime ? less
time to develop flow (smaller vT) earlier
decoupling at higher Tf Central collisions
bigger fireball lifetime ? more time to develop
flow (larger vT) later decoupling at smaller
Tf Tf and vT are strongly anticorrelated
NA57 158 GeV Centrality classes 0 ? 40 to 53
most central 1 ? 23 to 40 most central 2 ? 11
to 23 most central 3 ? 4.5 to 11 most
central 4 ? 4.5 most central
37
f transverse momentum spectra
T slope extracted fitting
f pT spectra are corrected for acceptance after
background and side-window subtraction
38
T slope as a function of centrality
Fit with exp(-mT/Tslope) vs centrality increase
of Tslope (indication of radial flow)
NA60 (pT fit range 0-2.6 GeV)
NA49 (pT fit range 0-1.6 GeV) NA50 (pT fit range
1.2-2.6 GeV)
NA60 Preliminary
NA50 and NA49 differerences (f puzzle) Decay
channel (mm vs KK) pT fit range (high vs low)
The In-In measurement of NA60follows the NA49
systematics
39
T slope, fit range dependence
Visible T dependence on the fit range Low pT
Higher absolute values, steeper rise with
centrality agreement with NA49 High pT Lower
absolute values, flatter rise with centrality
tendency towards NA50 but still some quantitative
difference
40
Dimuon excess pT spectra
Strategy of acceptance correction
? reduce 3-dimensional acceptance correction
in M-pT-y to 2-dimensional correction in
M-pT, using measured y distribution as an
input ? use slices of ?m 0.1 GeV
and ?pT 0.2 GeV ? resum to three
extended mass windows 0.4ltMlt0.6
GeV 0.6ltMlt0.9 GeV
1.0ltMlt1.4 GeV
subtract charm from the data before acceptance
correction (based on IMR results we pospone
this discussion)
41
Dimuon excess pT spectra for three centrality bins
(spectra arbitrarily normalized)
hardly any centrality dependence ? integrate over
centrality Significant mass dependence
42
r-like region mT spectrum (vs f)
physics differences are better visible in mT-
than in pT
f mT spectrum nearly pure exponential Teff
nearly independent of fit range with some hint of
radial flow Excess spectra show an increase
(not flattening) at low mT What does it mean?
43
differential fits to pT spectra, assuming locally
1-parameter mT scaling and using gliding windows
of ?pT0.8 GeV ? local slope Teff
? slope Teff vs central pT of the 0.8 GeV moving
window
(very) soft slopes at at low pT hard slopes at
high pT
Why?
44
Consequences of continuum emission over the
fireball life-time
Important qualitative difference between f and
dimuon excess f spectral slope only
contributions from the narrow temperature region
around kinetic freeze-out important Excess
dimuons continuum emission during all the
fireball lifetime we see not only the emission
at freeze-out! No flow superposition from
contributions of static fireballs at different T,
weighted by the radiating volume In presence of
flow superposition from contributions at
different T with different blue-shifts of T ?
what we see in the spectra are effective pT
dependent slopes Flow affects cold source
emission, but not hot source emission ? Softer
slopes at low pTs correspond to early
emission with small flow Harder slopes at high
pTs corresponds to late emission with large flow
45
If this is correct, the higher the pT more
evident the role of the vacuum r produced at
kinetic freeze-out (but notice that it would
dominate the spectrum only at very large pTs)
If we look at the 0.4-0.6 mass region, where only
a contribution from the in-medium r is seen, then
we should see a lower slope This is qualitatively
true
46
Reasonably accounted for by a theoretical model
(Ruppert-Renk hep-ph/0612113)
Is that all good then? Unfortunately not. the
very low slopes at very low pT (lt0.5 GeV/c) shown
by the data are at the moment without explanation
47
The high mass continuum region (1ltmmmlt1.4 GeV)

• The slopes of the high mass continuum region are
  smaller than the ones of the low mass continuum
  region (mmmlt0.6 GeV)
• Can we reconcile this with an hadronic source as
  4p processes?
• Shouldnt 4p contribute all the way down to
  kinetic freeze-out?
• then effect of flow is important and T slope
  cannot be so low if 4p is the dominant source

This is a very much debated and very hot
point But not yet a final answer agreed by
everybody
48
The high mass continuum region (1ltmmmlt1.4 GeV)
49
Local slope vs mmm
50
The open charm and the intermediate mass region
In all the discussions so far we assumed to know
the charm yield and it was subtracted. But how do
we know it? This brings us to discuss in more
detail the intermediate mass region 1ltmmmlt2.5
GeV NA60 was mainly approved to assess the
nature of an excess seen in the intermediate
mass region above the expected yields of open
charm and Drell-Yan This excess is the same we
have been discussing above the f (interpreted
either as an hadronic source or as a partonic
source) Historically it was already noticed by
previous experiments (HELIOS, NA38/50) but none
of them was able to attribute it to a prompt
source or to enhanced charm Only NA60 has clearly
attributed it to a prompt source.
51
Disentangling the signal sources in the IMR
The dileptons from charm decay can be identified
by tagging their production point with respect
to the primary interaction vertex

• Identify the typical offset of
• D-meson decay (100 µm)

• Need a very good vertexing accuracy
• (20-30 µm, in the transverse plane)

• Obtained at SPS by NA60

• Expected for the PHENIX update
• and in ALICE (electrons)

• Distinguish thermal, prompt component
• from heavy quark decay component

52
Measuring the muon offset
OffsetsdX, dY between the vertex and the track
impact point in the transverse plane at
Zvertex Resolution depends on track momentum use
offset weighted by the covariance matrices of the
vertex and of the muon track
For dimuons
53
Is the excess enhanced charm?
Procedure Fix the prompt contribution to the
expected DY yield and see if the offset
distribution can be described with enhanced Charm
dN/d?
New alignment
Answer No, Charm cant fill the small offset
region ? more prompts are needed
54
How many prompts are needed?
Procedure Leave both contributions free and see
if we can describe the offset distribution for
1.2 lt Mµµ lt 2.7
dN/d?
New alignment
Answer The best fit requires 2.6 times more
prompts than the expected Drell-Yan yield
55
pT dependence of the excess (1.16ltmmmlt2.56 GeV)
High pT tail strongly depends on the correctness
of Drell-Yan description by Pythia
?Teff fits are performed in
0lt PT lt2.5 GeV/c
preliminary
56
Towards a unification of low and intermediate
dimuon mass regions evolution of excess Teff vs
mmm across the low and intermediate mass
preliminary
57
Comparison of Pb-Pb data to calculations of
thermal radiation
R.Rapp E.Shuryak, Phys.Lett B473 (2000) 13
Data also consistent with thermal radiation
Need direct measurement of open charm
contribution
58
NA50 IMR results on pT spectra
The thermal dimuons (Rapp-Shuryak) used to
describe NA50 data (mmmgt1.6 GeV) have T 172 MeV
L. Capelli phd thesis
59
backup
60
? and ? two body decays
The ?, f?mm decays can be described using VMD as
the r. At the tree level
e p1 e- p2
?, q
g
?-g vertex factor
Electromagnetic current of quark fields
Vector meson currents
SU(3) prediction for V-g couplings
61
? Dalitz decays
The ??gg can be derived from ??gg with q2?0
Effect of ? - ? mixing
62
Further contribution arising from ? coupling to r
and ?
g(k)
?
e(p1)
g(q2)
e-(p2)
? Dalitz form factor
63
Vector meson dominance
? Dalitz form factor
Previous data (Landsberg et al.) fitted with a
pole formula
Dilepton mass spectrum
ee-
mm-
64
? Dalitz decays
p0(k)
?
e(p1)
g(q2)
e-(p2)
Vector meson dominance
Anomaly in the form-factor VMD predicts a
(significantly) smaller value
65
Two body decays of pseudoscalar mesons (?, ?)
4-th order electromagnetic process
e(p1)
g
?,?
g
e-(p2)
Electron BR negligible wrt muon BR ? BR
negligible wrt ? BR
PDG value
66
Vector meson dominance
What does it mean? Hadron matter couples to a
qqbar pair which propagates as a vector meson
which then materializes as a photon All QCD
complexity, gluon self interactions and
confinement are incorporated in the physical
vector meson which forms the intermediate state
The hadron components of the photon vacuum
polarization come exclusively from the know
vector mesons this hypothesis works very well
close to the vector meson masses
67
p-r, and electromagnetic interactions in vacuum
Pions (pseudo-)scalar particles
The r vector meson ? a photon with a non zero
mass
self interactions are neglected
p-r and p-g interactions introduced via gauge
couplings
grp pion-r coupling constant
Direct g-r coupling
The r couples only to conserved currents, so that
68
It is interesting to consider the case
(universality of the couplings)
If we look at the electromagnetic field equations
we find
The r meson is the only hadronic source of the
electromagnetic field The hadronic part of the
electromagnetic current is then proportional to
the r meson field
69
The process r ? pp
At the tree-level the relevant part of the
lagrangian describing the r coupling to a pair of
pions is
This leads to the following vertex factor (and
Feynman rule)
rpp vertex factor
p - p- p p
r, k
decay amplitude
r polarization vector
The tree-level coupling constant g can be
determined from the empirical decay width.
Assuming Gr?pp 150 MeV we find
70
The r self-energy
The rm field describes a bare meson which we
can interpret as the qqbar component of the
physical r meson. The bare r is stable because
quarks are confined. The bare r propagator is
given by
Because of the coupling to pions (mainly) and to
the photon (much much less) the physical r meson
appears as a broad resonance.

In order to get a realistic description of
its properties (like the width) we must consider
the second order diagrams
r
r
r
r
r
r
r


1PI
71
r dressed propagator and renormalization
Without loss of generality we can write
where
r polarization scalar
The full (dressed) propagator comes from an
infinite sum of diagrams with self-energy
insertions
1PI
1PI
1PI
This infinite series can be easily summed ...
The r field is always coupled to conserved
currents (qmJm 0) and so the terms proportional
to qmqn can be neglected
72
The complete dressed r propagator is (beside a i
factor)
If the r meson were stable no final state could
contribute to the right hand side of the optical
theorem and so Pr(q2) would be real. In that case
the propagator has a pole determined by the
equation
which has a real solution mr. Therefore the pole
lies on the real q2 axis below the multi-particle
branch cut. mr is the physical (renormalized)
mass which can be measured. Its important to
realize that the parameter m0r that enters in the
Lagrangian is in general different (can be fixed
from experimental data).
Since the r can decay, mr will lie above the
multi-particle branch-cut ? Pr takes an imaginary
part
73
Lets expand (the real part of) Pr(q2) in power
series around m2r (the physical mass)
The physical mass is now defined by the condition
Close to the pole mass, the bare mass can be
replaced with the physical mass with
Now we require that the effect of the real part
of the self-energy is completely absorbed in the
shift from the bare mass m0r to the physical mass
mr, so that
74
The meaning of the imaginary part of Pr
r
r
scattering r?r

1PI
According to the optical theorem
general expression of the decay width
In this specific case the final state is
(dominantly) pp. Thus we come to the result
mass dependent width
75
The real part of Pr
RePr(q2) contains a divergent part. For instance,
the tadpole contribution is quadratically
divergent. Regularization cut-off or dispersion
relations
The constants are parameters of the effective
model
Needed to keep the photon massless
Fixes c1
76
pp elastic scattering and P wave phase shifts
m0r can be fixed from the comparison to the
measured pp- ? pp- elastic cross section
p- p - p p
p- p - p p
r
r-pp vertex
r-pp vertex
full (dressed) r propagator
The pp cross-section evaluated by VMD may be
related to the measured P wave phase shifts by
77
The mass shift induced by the r?pp self-energy is
small
? justification for considering only contribution
to g2 order.
78
The process r ?ll- (dilepton decay)
At the tree-level the r decay to a lepton pair is
described by the diagram
e p1 e- p2
r polarization vector
g
r, q
Fermion vertex
r-g vertex factor
In addition there is a diagram with pion loop at
the next to leading order
e p1 e- p2
g
r, q
79
The decay amplitude is given by the sum of the
two diagrams
e p1 e- p2
g
r, q
effective coupling constant 1/grg(q2)
The decay width is then
If we consider r?ee-, the lepton mass can be
neglected and
grg is renormalized by the loop diagram
80
Pion annihilation into dileptons (pp-?ll-)
81
The scattering amplitude can be arranged in the
following way
Amplitude of point-like coupling g-p
e e-
p - p
Correction to point-like scattering pion form
factor Fp(q2)
This is correct at the leading order in a and at
all orders of strong interactions.
82
The cross section calculation gives
r-? interference not included in present
calculation
83
Dilepton production rate in a pion gas at
temperature T
The number of p with momentum p1 in a volume d3x
is
statistical occupation factor
Let s the cross section. The volume swept by s
due to the relative motion is
The number of p- with momentum p2 in d3p2
interacting with a p in a unit time is
Dividing by d3x, the rate for unit 4-volume rate
is thus
84
We can express the relative velocity in covariant
way as
If the temperature is sufficiently high the
quantum effects are not very important and we can
assume that f(E) is a Boltzmann factor
inserting a delta function
the rate becomes
85
The phase space factor
Contains all Lorentz invariant factors and so
its Lorentz invariant. It can be easily
evaluated in the CMS with the result
Having found the phase space factor, the rate
becomes
86
Differential rate dR/dq4
Let R dN/d4x. Then the differential rate per
unit 4-volume and unit 4-momentum is
In case of pion annihilation into a lepton pair,
the use of the VMD cross-section gives
If q2gtgtm2l (strictly true only for electrons)
(from
)
87
Connection with the photon self-energy
The imaginary part of the r propagator is
This quantity is called the r spectral
function. Since g/grg1, we have
In this approximation the hadronic part of the
photon self-energy is simply
g
g
r
g
g
1PI
The dilepton rate probes the (hadronic) part of
the photon polarization tensor
88
Chiral transformations
Consider the system of the two light quarks up
and down. We can put them in a doublet
We can imagine this doublet as a state in an
isospin space (both u and d have isospin 1/2)
which can be described with the same formalism
used for spin 1/2 systems Lets neglect quark
masses (for up and down they are indeed small)
and interactions. The Lagrangian is simply
Treating u and d independently, the Lagrange
equations lead to the Dirac equation both for u
and d
89
In terms of the right- and left-handed fields
the Lagrangian can be rewritten in terms of left
and right-handed fields as
What happens if we perform a rotation in the
isospin space?
Consider indipendent rotations (SU(2)
transformations) on ?L and ?R (SU(2)L x SU(2)R)
These transformations are called chiral
transformations and leave the Lagrangian
invariant ? quark handedness is not changed.
1
90
There are two particular important cases 1.
corresponds to the
transformation on ?
Rotation in the isospin space
2. corresponds to the
transformation on ?
Axial transformation
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Breaking of chiral symmetry
Now we introduce a mass term so that quarks are
not anymore massless
The mass term can be rewritten as
Under a chiral transformation

• Thus
• If eV eA (isospin transformation) the mass term
  is invariant (the L and R field rotate together)
• If eV -eA (axial transformation) chiral
  symmetry is spoiled by the mass term

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Action of chiral transformations on particle
states
What happens if a chiral transformation acts on a
pion o a r meson? Lets associate a particle to
some Dirac bilinear as
with the correct particle
transformation properties with respect to Lorentz
boosts and parity.
We introduce the particles
s meson (scalar 0)
p meson (pseudoscalar 0-)
r meson (vector 1--)
a1 meson (axial 1-)
Consider the pion triplet and apply a vector
(isospin) transformation. It is easy to show
The pion vector is rotated in the isospin space
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The isospin invariance obviously means that all
the three pion states should have the same mass,
which is well satisfied (with just a small
electromagnetic violation). But what about
applying an axial transformation? In this case,
because of the g5 matrix
and
Since the anticommutator is
it follows that
The pion and the sigma are mixed ... rotated
among themselves. But if chiral symmetry is a
good symmetry, this must imply that the pion and
the sigma should have the same mass
In similar way an axial transformation on the s
field gives
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Now lets turn our attention to the vector and
axial vector mesons. If we apply an isospin
transformation on the r we get a similar result
as for the pion
we have again a rotation in isospin space and
isospin symmetry implies that all the 3 r states
should have the same mass. If we consider an
axial transformation on the other hand
Again mixing, with the axial state this
time. Thus, chiral symmetry implies that the r
and the a1 should have the same mass
However ... apart that its not clear what is the
s meson, for the r and the a1 we know their
masses very well
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So, apparently the axial symmetry doesnt seem to
be a good symmetry of QCD.
On the other hand, the quark masses (which break
the symmetry in the QCD Lagrangian) are quite low
and cannot be responsible for this large
difference. Moreover, there is other experimental
evidence (slightly more in the next slide) that
the axial symmetry is not so a bad symmetry.
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Spontaneous breakdown of chiral symmetry
There is a apparent contraddiction The fact that
we dont see 1 and 1- mesons with the same mass
suggest that chiral symmetry is broken On the
other hand, the weak pion decay is consistent
with PCAC. The Goldberger-Treiman relation seems
successful. The solution is chiral symmetry is
spontaneously broken
When does spontaneous breakdown of a symmetry
occur? The hamiltonian governing the system
possesses the symmetry However, the ground state
of the system (the state with lowest energy) is
not symmetric
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Imagine now that the QCD hamiltonian has a
similar form when the (x,y) coordinates are
replaced by the (s,p) fields. The space
rotations are replaced by the axial
transformation LA which rotates the p into the s
In quantum field theory the ground state is the
vacuum state. Since the ground state is shifted
away from the origin one of the fields must have
a non zero expectation value with respect to the
vacuum. Since the vacuum has quantum numbers 0,
it must be the expectation value of the s field
In the quark language so that
finite scalar (chiral) quark condenstate
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Excitations in the p direction small rotations
away from the ground state along the valley which
dont cost energy ? the pion mass must be
zero Excitations in the s direction radial
excitations that cost energy ? the s field must
be massive Consequently the spontaneous
breakdown of chiral symmetry removes the mass
degeneracy of the s and p fields. However, since
the interaction is still symmetric, pions are
massless consistent with PCAC
It can be shown (Weinberg) that the spontaneous
breakdown of chiral symmetry in the case of
vector mesons brings to
in fair agreement with the observed values of the
r and a1 masses.