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]

Origin of Mass ?of Hadrons

of Us

One of the Interesting problems of QCD=

Origin of Mass = quark condensate

Spontaneous Chiral Symmetry Breaking

☆ 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 ?

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)

☆ 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)

☆ 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 …

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

◎ 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-

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

2. Di-lepton spectrum from the dropping ρ

in the vector manifestation(Effect of the violation of

vector dominance)

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

◎ 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

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).

☆ 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

☆ 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)

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.

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

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

☆ T – dependence of ρ and A1 meson massesA simple assumption for the parameters of the GHLS Lagrangian

no-dropping dropping

◎ 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

◎ V-A mixing → small near Tc

◎ T-dependence of the mixing effect

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

☆ 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

☆ 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 !

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]

◎ 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

☆ 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)]

☆ 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

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 :

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

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)

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

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

◎ 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’’, …

☆ 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

☆ 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

☆ 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

☆ 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

◎ 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

☆ 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)

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

◎ 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

◎ 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