strong and electroweak matter 2004. helsinki, 16-19 june. angel gómez nicola universidad...
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Strong and Electroweak Matter 2004. Helsinki, 16-19 June.
Angel Gómez Nicola
Universidad Complutense Madrid
Motivation
T>0 ChPT pion electromagnetic form factors
Thermal and poles
Motivation
After QGP hadronization and SB, the description of the meson gas must rely on Chiral Perturbation Theory
(model independent, chiral power counting p,T << 1 GeV )
Only NGB mesons (and photons) involved.
2 loops are O(p2) , divergences absorbed in 4 and so on.
...42 2422
6
Derivative and mass expansion
nonlinear -model
S.Weinberg, ‘79J.Gasser&H.Leutwyler ’84,’85
point towards Chiral Symmetry Restoration:
J.Gasser&H.Leutwyler ‘87
P.Gerber&H.Leutwyler ‘89
A.Bochkarev&J.Kapusta ‘96
A.Dobado, J.R.Peláez ’99 ’01.
0 50 100 150 200TemperatureHMeVL
0.2
0.4
0.6
0.8
1
<q q >€€€€€€€€€€€€€€€€€€€€€€€€€€€€€<qq >
o
SUH3LSUH2LSUH3Lpions
+ free K,h
< ss >€€€€€€€€€€€€€€€€€€€€€€€€€€€€< ss >
o
One loop ChPT
T
T=0
T
T=0
),...(),(, TfTqqP
)();(Im
)()();(Re
);(
6
62
22
pOTpg
pOTmTpg
Tpgmp
J.L.Goity&H.Leutwyler, ‘89
A.Schenk, ‘93
R.Pisarski&M.Tytgat, ‘96
D.Toublan, ‘97
J.M.Martinez Resco&M.A.Valle, ‘98
Pion dispersion law:
Nonequilibrium ChPT: f (t), amplification via parametric resonance.
AGN ‘01
However, ChPT alone cannot reproduce the light resonances (,, ...)
Needed to explain observed phenomena in RHIC.
K.Kajantie et al ’96C.Gale, J.Kapusta ’87 ‘91G.Q.Li,C.M.Ko,G.E.Brown ‘95H.J.Schulze, D.Blaschke ‘96,’03V.L.Eletsky et al ‘01
Enhancement consistent with a dropping
Mand a significant broadening in the hadron gas at freeze-out.
CHIRAL SYMMETRY BREAKING
UNITARITY+
Inverse Amplitude Method
““Thermal” polesThermal” poles
Dynamically generated (no explicit resonance fields)
OUR APPROACH
AGN, F.J.Llanes-Estrada, J.R.Peláez PLB550, 55 (2002), hep-ph/0405273A.Dobado, AGN, F.J.Llanes-Estrada, J.R.Peláez,
PRC66, 055201 (2002)
scattering amplitude and form factors in T > 0 SU(2) ChPT
Motivation
T>0 ChPT pion electromagnetic form factors
Thermal and poles
T>0 ChPT pion electromagnetic form factors
Pion form factors enter directly in the dilepton rate:
In the central region the dominant channel is pion annihilation:
e+
e-
e+
e-
~ + ...
)(),(),()()(2)2(2)2(
)( 21)4(
21*
21212
32
3
13
13
44qppppVppVEnEn
E
pd
E
pdqL
xqdd
dNBB
)()()0(0),( 2121 ppJppV EM
1
1)(
/
TEB eEn (thermal equilibrium)
At T>0 a more general structure is allowed:
),,(),,(),(
),,(),(
0000021
000210
kSSGkSkSSFkppV
kSSFkppV
sjsjj
t
k = p1 - p2
S = p1 + p2
|)|,(|||)|,(|)|,( 0)1(2
0)1(
0)1(
0 SSGSSSFSSFS sst
)1(
)1(, ,
1
s
st
GG
FF
s
st
ChPT to O(p4)
00 J
(At T = 0, Ft (S2)= Fs(S2), Gs = 0)
Related by gauge invariance to dispersion law in hot matter
T0 limit (J.Gasser&H.Leutwyler 1984).
Gauge invariance condition.
Thermal perturbative unitarity in the c.o.m. frame (see later)
T>0 ChPT calculation to O(p4):
2 one loop 4 tree level(including renormalization)
0
2
02
0
),0(6
charge Effective )0,0(
S
t
TT
tT
Sd
SSdF
Qr
SSFQ
2
0
20 fm 45.0 , 1 rQ
Model independent !
Confirms Dominguez et al ’94 (QCD sum rules)
The pion electromagnetic charge radius at T>0
22
3
3
2/32
)2(3)(
3
4)(
mknkd
Tn
rTV
B
T
(rough) deconfinement estimate:
1)()( cc TnTV
MeV 200cT
Charge screening
2/3
0
2
3
4)0( rV
MeV 265cT
Kapusta
H.A.Weldon ’92
Enhancement Absorption
Thermal perturbative unitarity:
Likewise, for the thermal amplitude:
220
)0(00
)1( )()();(Im SaSTSa IJTIJ
)()()();(Im 0)0(2
0)0(
1100)1( SFSaSTSF T )2( mE
Consider c.om. frame ( , back to back dileptons)21 pp
2thermal phase space:
(1+nB)2 nB2
)2/(214
1)( 020
2
0 SnS
mS BT
0
)0,()0,( 0)1(
0)1(
SFSF st
I=J=1 scattering partial wave
1 to lowest order
Motivation
T>0 ChPT pion electromagnetic form factors
Thermal and polesThermal and poles
Excellent T=0 data description up to 1 GeV energies and resonance generationas s poles in the complex amplitude.
T.N.Truong, ‘88A.Dobado, M.J.Herrero,T.N.Truong, ‘90
A.Dobado&J.R.Peláez, ’93,’97J.A.Oller, E.Oset, J.R.Peláez, ’99
A.Dobado, M.J.Herrero, E.Ruiz Morales ‘00AGN&J.R.Peláez ‘02
Unitarization: The Inverse Amplitude Method
Exact unitarity at T>0
11Im IAMIAMT
IAM FaF
)(Im Im12
TIAMIJ
IAMIJT
IAMIJ aaa
+ ChPT matching at low energies
)1()0(
)1(1
aaa
FFIAM
IAM
Valid to O(nB) (only 2 thermal states, dilute gas).
);()(
)();(
0)1(2
0)0(
2 20
)0(
0 TSaSa
SaTSa IAM
IJ
);();(Re)(
);(Re1);( 011
0)1(
1120
)0(11
0)1(
0 TSaTSaSa
TSFTSF IAMIAM
1 2 3 40.3, 5.6, 3.4, 4.3l l l l
SU(2) 4 constants from T=0 fit of phase shifts:
300 400 500 600 700 800 900 1000
ReH"#######spoleL=MHMeVL- 250
- 200
- 150
- 100
- 50
0mIH"####### s elopL=-G2HVeML T=100 MeV
T=200 MeV
T=25 MeV
T=100 MeV
T=125 MeVPole evolution withTemperature :
r pole
s pole
(770)
Thermal and poles
I=J=0 I=J=1
2nB (M/2) 0.3
Consistent with Chiral Symmetry Restoration::
M Mm (m(T) much softer)
first by phase space but decreases as Mm suppresses 2decay. (similar results to T.Hatsuda, T.Kunihiro et al, ’98,’00)
*
Small M change at low T (VMD*). Further decrease consistent with phenomenological estimates and observed behaviour (STAR )
* M.Dey, V.L.Eletsky&B.L.Ioffe, 1990
Significant broadening as required by dilepton data.
The unitarized form factor
Peak reduction and spreading around M compatible with dilepton spectrum (nB contributions alone overestimate data) and other calculations including explicitly resonances under VMD assumption (C.Song and V.Koch, ’96)
m= 139.6 MeV f= 92.4 MeV
6 18l
(T=0 formfactor fit)
Chiral Perturbation Theory provides model-independent predictions for meson gas properties.
In one-loop ChPT, we have calculated scattering amplitudes and the two independent form factors, checking gauge invariance and thermal unitarity. The electromagnetic pion radius grows for T>100 MeV, favouring a deconfinement temperature Tc~200 MeV.
Imposing unitarity in SU(2) allows to describe the thermal and poles in the amplitudes and form factors. Our results show a clear increase of (T) and a slow M (T) reduction consistently with theoretical and experimental analysis, including dilepton data. (T) and M (T) behave according to Chiral Symmetry Restoration.
Angular dependence, plasma expansion, +- e+e , baryon density, hadronic photon spectrum, ...