effects of vector – axial-vector mixing to dilepton spectrum in hot and/or dense matter masayasu...
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Effects of vector – axial-vector mixing to dilepton spectrum
in hot and/or dense matter
Masayasu Harada (Nagoya Univ.)
@ Heavy Ion Meeting 2010-12 at Yonsei (December 11, 2010)
based on M.H. and C.Sasaki, PRD74, 114006 (2006) M.H. and C.Sasaki, Phys. Rev. C 80, 054912 (2009) M.H., C.Sasaki and W.Weise, Phys. Rev. D 78, 114003 (2008)
see also M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010) M.H. and M.Rho, arXiv:1010.1971 [hep-ph]
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Origin of Mass ?of Hadrons
of Us
One of the Interesting problems of QCD=
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Origin of Mass = quark condensate
Spontaneous Chiral Symmetry Breaking
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☆ QCD under extreme conditions
・ Hot and/or Dense QCD
◎ Chiral symmetry restoration
Tcritical ~ 170 – 200 MeV
rcritical ~ a few times of normal nuclear matter density
Change of Hadron masses ?
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Masses of mesons become light due to chiral restoration
☆ Dropping mass of hadrons
◎ Brown-Rho scaling G.E.Brown and M.Rho, PRL 66, 2720 (1991)
for T → Tcritical and/or ρ → ρcritical
◎ NJL model T.Hatsuda and T.Kunihiro, PLB185, 304 (1987)
◎ QCD sum rule :T.Hatsuda and S.H.Lee, PRC46, R34 (1992)
T.Hatsuda, Quark Matter 91 [NPA544, 27 (1992)]
◎ Vector Manifestation M.H. and K.Yamawaki, PRL86, 757 (2001)M.H. and C.Sasaki, PLB537, 280 (2002)M.H., Y.Kim and M.Rho, PRD66, 016003 (2002)
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☆ Di-lepton data consistent with dropping vector meson mass ◎ KEK-PS/E325 experiment
mr = m0 (1 - /0) for = 0.09
mf = m0 (1 - /0) for = 0.03
K.Ozawa et al., PRL86, 5019 (2001)M.Naruki et al., PRL96, 092301 (2006)R.Muto et al., PRL98, 042501 (2007)F.Sakuma et al., PRL98, 152302 (2007)
☆ Di-lepton data consistent with NO dropping vector meson mass
Analysis : H.v.Hees and R.Rapp, NPA806, 339 (2008)
All PT
Analysis : J.Ruppert, C.Gale, T.Renk, P.Lichard and J.I.Kapusta, PRL100, 162301 (2008)
◎ NA60 ◎ CLASR. Nasseripour et al. PRL99, 262302 (2007).M.H.Wood et al. PRC78, 015201 (2008)
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☆ Quark Structure and Chiral representation◎ coupling to currents and densities
(S. Weinberg, 69’)longitudinal components
Note: ρ and A1 ( π and σ ) are chiral partners ?
These are partners …
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Chiral Restoration
linear sigma modelvector manifestation
Hybrid Scenario (π 、 ρ, A1, σ degenerate) : mσ = mA1 = mρ = mπ
π 、 σ degenerate: mσ → mπ
ρ, A1 degenerate: mA1 → mρ
π, ρ degenerate: mρ → mπ
σ, A1 degenerate : mσ → mA1
Either of 3 is expected to happen → dropping mass
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◎ Signal of chiral symmetry restoration axial-vector current (A1couples) = vector current (ρ couples)
These must agree with each other
e-
e+vector mesons(ρ etc.)
e
νaxial-vector mesons
Impossible experimentally
But, in medium, this might be seen through the vector – axial-vector mixing !
How these mixing effects are seen in the vector spectral function ?ρ etc etc
e+
e-
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Outline
1. Introduction
2. Di-lepton spectrum from the dropping ρ
in the vector manifestation
(Effect of the violation of r meson dominance)
3. Effect of Vector-Axial-vector mixing
in hot matter
4. Effect of V-A mixing in dense baryonic matter
5. Summary
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2. Di-lepton spectrum from the dropping ρ
in the vector manifestation(Effect of the violation of
vector dominance)
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Chiral Restoration
linear sigma modelvector manifestation
Hybrid Scenario (π 、 ρ, A1, σ degenerate) : mσ = mA1 = mρ = mπ
π 、 σ degenerate: mσ → mπ
ρ, A1 degenerate: mA1 → mρ
π, ρ degenerate: mρ → mπ
σ, A1 degenerate : mσ → mA1
only p and r are taken into account : Dropping r mass
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◎ View of the VM in Hot Matter
◎ Assumptions
・ Relevant d.o.f until near Tc-ε ・・・ only π and ρ
・ Other mesons (A1, σ, ...) ・・・ still heavy
・ Partial chiral restoration already at Tc-ε
☆ Formulation of the VM
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M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985)M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988)
H.Georgi, PRL 63, 1917 (1989); NPB 331, 311 (1990): M.H. and K.Yamawaki, PLB297, 151 (1992); M.Tanabashi, PLB 316, 534 (1993): M.H. and K.Yamawaki, Physics Reports 381, 1 (2003)
Systematic low-energy expansion including dynamical r
◎ Hidden Local Symmetry ・・・ EFT for r and p based on chiral symmetry of QCD
r = gauge boson of the HLS massive through the Higgs mechanism
loop expansion ⇔ derivative expansion
☆ Formulation of the VM
◎ VM is protected by the VM fixed point
・ stable against the quantum correctionsM.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001)
・ stable against the thermal corrections
・ stable against the density corrections
M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002)
M.H., Y. Kim and M. Rho, Phys. Rev. D 66, 016003 (2002).
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☆ Key 1 : Strong Violation of r meson dominance in the VM
e+
e-
◎ r meson dominance at T = 0
a = 2 vector dominance⇒a/21 – a/2
long standing problem not clearly explained in QCD !
◎ r dominance at T > 0 ?e+
e-
◎ a = 2 kept fixed in several analyses (No T-dependence on a)
0 → 1 1 → 1/2
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☆ Key 2 : Dropping ρ occurs only for T > Tf ~ 0.7 Tc
G.E.Brown, C.H.Lee and M.Rho, PRC74, 024906 (2006); NPA747, 530 (2005)
(H.v.Hees and R.Rapp, hep-ph/0604269)
+ Suppression by the violation of VD
When the r meson coming from the region T < Tf gives dominant contribution to the dilepton spectrum, the dropping r scenario may not be excluded by NA60 data
cf : “Hadronic freedom” leads to the small interactions for T > Tf, so that the r from T > Tf is highly suppressed. G.E.Brown, MH, J.W.Holts, M.Rho, C.Sasaki, PTP 121, 1209 (2009)
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T dependence of r with dropping mass for T > Tf = 0.7 Tc
Tf/Tc
ρ mass mρ → 0
VM
VM with VD T = 0.75 Tc
VM ~ VM with VD /1.8
VM ~ VM with VD / 2
T = 0.8 Tc ◎violation of “Vector meson dominance” → Large suppression near Tc
We need to include other hadrons such as A1 to make a comparison with experiment.
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3. Effect of Vector-Axialvector mixing in hot matter
MH, C.Sasaki and W.Weise, Phys. Rev. D 78, 114003 (2008)
Using an Effective field theory including p, r, A1 based on the generalized hidden local symmetry
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Hybrid Scenario (π 、 ρ, A1, σ degenerate) : mσ = mA1 = mρ = mπ
Chiral Restoration
linear sigma modelvector manifestation
π 、 σ degenerate: mσ → mπ
ρ, A1 degenerate: mA1 → mρ
π, ρ degenerate: mρ → mπ
σ, A1 degenerate: mσ → mA1
ρ mass is invariant → dropping A1 & σ masses
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☆ T – dependence of ρ and A1 meson massesA simple assumption for the parameters of the GHLS Lagrangian
no-dropping dropping
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◎ Vector spectral function at T/Tc = 0.8
e-
e+A1
π +ρ
e-
e+A1
π
ρ
Effects of pion mass ・ Enhancement around s1/2 = ma – mπ ・ Cusp structure around s1/2 = ma + mπ
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◎ V-A mixing → small near Tc
◎ T-dependence of the mixing effect
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Chiral Restoration
linear sigma modelvector manifestation
Hybrid Scenario (π 、 ρ, A1, σ degenerate) : mσ = mA1 = mρ = mπ
π 、 σ degenerate: mσ → mπ
ρ, A1 degenerate: mA1 → mρ
π, ρ degenerate: mρ → mπ
σ, A1 degenerate : mσ → mA1
Either of 3 is expected to happen → dropping mass
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☆ Dropping A1 with dropping ρ
T/Tc = 0.8
Effect of A1 meson
s1/2 = 2 mρ
Effect of dropping A1
・ Spectrum around the ρ pole is suppressed ・ Enhancement around A1-π threshold ・ Cusp structure around s1/2= 2mρ
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☆ Vanishing V-A mixing (ga1rp = 0) at Tc ?
In quark level
A1
r
pLeft chirality
L
L
R
We need to flip chirality once in a1-r-p coupling
ga1rp < q∝ bar q > → 0 for T → Tc
Vector – axial-vector mixing vanishes at T = Tc !
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4. Vector – Axial-vector mixingin dense baryonic matter
based on M.H. and C.Sasaki, Phys. Rev. C 80, 054912 (2009)see also M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010) M.H. and M.Rho, arXiv:1010.1971 [hep-ph]
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◎ V-A mixing from the current algebra analysis in the low density region B.Krippa, PLB427 (1998)
・ This is obtained at loop level in a field theoretic sense.・ Is there more direct V-A mixing at Lagrangian level ? such as £ ~ Vm Am ? ・・・ impossible in hot matter due to parity and charge conjugation invariance ・・・ possible in dense baryonic matter since charge conjugation is violated but be careful since parity is not violated
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☆ A possible V-A mixing term violates charge conjugation but conserves parity
generates a mixing between transverse r and A1
ex : for pm = (p0, 0, 0, p) no mixing between V0,3 and A0,3 (longitudinal modes) mixing between V1 and A2, V2 and A1 (transverse modes)
◎ Dispersion relations for transverse r and A1
+ sign ・・・ transverse A1 [p0 = ma1 at rest (p = 0)] - sign ・・・ transverse r [p0 = mr at rest (p = 0)]
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☆ Determination of mixing strength C ◎ An estimation from w dominance
e-
e+A1
w
ρ
+ Cw ~ 0.1 GeV × (nB / n0)
n0 : normal nuclear matter density
◎ An estimation in a holographic QCD (AdS/QCD) model・ Infinite tower of vector mesons (w, w’, w”, …) in AdS/QCD models・ These effects of infinite w mesons can generate V-A mixing
・ This summation was done in an AdS/QCD modelS.K.Domokos, J.A.Harvey, PRL99 (2007)
a very rough estimation
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Can infinite tower of w mesons contribute ?
This is related to a long-standing problem of QCD not clearly understood: Why does the r/w meson dominance work well ?
a1r, r’
pK
K
tn
K*a1
p
K
Ktn
wr, r’, r’’
pK
K
t
n
r, r’, r’’p
K
Ktn
K*
Example 1: vector form factor in t -> KpKbarnt at CLEO
CLEO, PRL92, 232001 (2004)
r : 1/(1+l+d) = 1.27r‘ : l/(1+l+d) = - 0.40r‘’ : d/(1+l+d) = 0.13
CLEO results :
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r meson dominance ⇒ ;
Example 2: p EM form factor• In an effective field theory for p and r based on the Hidden Local Symmetry p EM form factor is parameterized as
•In Sakai-Sugimoto model (AdS/QCD model), infinite tower of r mesons do contribute
k=1 : r mesonk=2 : r’ mesonk=3 : r” meson…
T.Sakai, S.Sugimoto, PTP113, PTP114
= 1.31 + (-0.35) + (0.05) + (-0.01) + …r’r r’’ r’’’
r a1(1260) r’(1450) PDG 776 1230 1465SS 776(input) 1190 1607
Note: AdS/QCD model is for large Nc limit
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Example 2: p EM form factor
rmeson dominancec2/dof = 226/53=4.3
;
SS model : c2/dof = 147/53=2.8best fit in the HLS : c2/dof=81/51=1.6
Exp data :NA7], NPB277, 168 (1996)J-lab F(pi), PRL86, 1713(2001)J-lab F(pi), PRC75, 055205 (2007)J-lab F(pi)-2, PRL97, 192001 (2006)
Infinite tower works well as the r meson dominance !
MH, S.Matsuzaki, K.Yamawaki, arXiv:1007.4715cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003)
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Example 3: wp transition form factor
1
• best fit in the HLS : c2/dof=24/30=0.8
• Sakai-Sugimoto model : c2/dof=45/31=1.5
Exp data: CMD-2, PLB613, 29 (2005) NA60, PLB677, 260 (2009)
MH, S.Matsuzaki, K.Yamawaki, arXiv:1007.4715cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003)
r meson dominance : c2/dof=124/31=4.0
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Example 4: Proton EM form factor M.H. and M.Rho, arXiv:1010.1971 [hep-ph]
rmeson dominance : c2/dof=187
• best fit in the HLS : c2/dof=1.5
a = 4.55 ; z = 0.55
Violation of r/w meson dominance may indicate existence of the contributions from the higher resonances.Contribution from heavier vector mesons actually exists in several physical processes even in the low-energy region
• Sakai-Sugimoto: Hong-Rho-Yi-Yee model : c2/dof=20.2
a = 3.01 ; z = -0.042
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◎ An estimation in a holographic QCD (AdS/QCD) model・ Infinite tower of vector mesons in AdS/QCD models
w, w’, w”, …・ These infinite w mesons can generate V-A mixing
・ This summation was done in an AdS/QCD model
In the following, I take C = 0.1 - 1 GeV.Note that e.g. C = 0.5 GeV corresponds to
C = 0.1 × (nB/n0) at nB = 5 n0
C = 0.5 × (nB/n0) at nB = n0
・・・
This may be too big, but we can expect some contributions from heavier w’, w’’, …
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☆ Dispersion relations
r meson A1 meson
・ C = 0.5 GeV : small changes for r and A1 mesons・ C = 1 GeV : small change for A1 meson substantial change in r meson
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☆ Vector spectral function for C = 1 GeV
note : Gr = 0 for √s < 2 mp ◎ low 3-momentum (pbar = 0.3 GeV)
・ longitudinal mode : ordinary r peak ・ transverse mode : an enhancement for √s < mr and no clear r peak a gentle peak corresponding to A1 meson ・ spin average (Im GL + 2 Im GT)/3 : 2 peaks corresponding to r and A1
◎ high 3-momentum (pbar = 0.6 GeV) ・ longitudinal mode : ordinary r peak ・ transverse mode : 2 small bumps and a gentle A1 peak ・ spin averaged : 2 peaks for r and A1 ; Broadening of r peak
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☆ Di-lepton spectrum at T = 0.1 GeV with C = 1 GeV
2 mp
・ A large enhancement in low √s region → result in a strong spectral broadening ・・・ might be observed in with low-momentum binning at J-PARC, GSI/FAIR and RHIC low-energy running
note : Gr = 0 for √s < 2 mp
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☆ Effects of V-A mixing for w and f mesons・ Assumption of nonet structure → common mixing strength C for r-A1, w-f1(1285) and f-f1(1420)
・ Vector current correlator
note : we used the following meson widths
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◎ f meson spectral functionspin averaged, integrated over 0 < p < 1 GeV
・ C = 1 GeV : suppression of f peak (broadening)・ C = 0.3 GeV : suppression for √s > mf enhancement for √s < mf
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☆ Integrated rate with r, w and f mesons for C = 0.3, 0.5, 1 GeV
・ An enhancement for √s < mr , mw (reduced for decreasing C)・ An enhancement for √s < mf from f-f1(1420) mixing → a broadening of f width
(at T = 0.1GeV)
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5. Summary ◎ chiral symmetry restoration in hot and/or dense medium
→ mass change of either ρ, A1 (or both) can be expected
◎ Dilepton spectrum from the dropping r in the VM
・ In the VM, strong violation of the vector (ρ) meson dominance is expected
→ large suppression of dilepton spectrum
◎ Effect of axial-vector (A1) meson
・ In the standard scenario, dropping A1 is expected.
Through the V-A mixing, we may see the dropping A1.
Note : The mixing becomes small associate with the chiral restoration.
ga1rp ~ Fp2 → 0 at T = Tc
→ Observation of dropping A1 may be difficult
・ In case of dropping A1 with dropping ρ,
effect of A1 suppress the di-lepton spectrum cusp structure around s1/2= 2mr
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◎ Effect of V-A mixing (violating charge conjugation) in dense baryonic matter
for r-A1, w-f1(1285) and f-f1(1420)
→ ・ substantial modification of rho meson dispersion relation ・ broadening of vector spectral function ・・・ might be observed at J-PARC and GSI/FAIR
◎ Large C ? :
・ If C = 0.1GeV, then this mixing will be irrelevant.・ If C > 0.3GeV, then this mixing will be important.・ We need more analysis for fixing C.
Note : Cw = 0.3 GeV at nB = 3 n0 . → This V-A mixing becomes relevant for nB > 3 n0
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◎ Including dropping mass (especially dropping r) ?
・ This mixing will not vanish at the restoration. (cf: V-A mixing in hot matter becomes small near Tc.)
☆ Future work
・ Dropping r may cause a vector meson condensation at high density.
ex: mr*/mr = ( 1 – 0.1 nB/n0 ) suggested by KEK-E325 exp. C = 0.3 (nB/n0 ) [GeV] just as an example
Vector meson condensation !
vacuum r longitudinal r
transverse r
p0
p
nB/n0 = 2 nB/n0 = 3 nB/n0 = 3.1
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