Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2013/repino.pdf Äàòà èçìåíåíèÿ: Sun Jun 23 14:12:16 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 20:05:49 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 5

An eective theory for QCD with an axial chemical potential
Xumeu Planells
With: A. A. Andrianov

,

, V. A. Andrianov



and D. Espriu





University of Barcelona, Spain Saint-Petersburg State University, Russia

Based on

A. A. Andrianov, V. A. Andrianov, D. Espriu & X. Planells, Phys. Lett. B 710 (2012) 230. A. A. Andrianov, D. Espriu & X. Planells, Eur. Phys. J. C (2013) 73:2294.
XXI International Workshop on High Energy Physics and Quantum Field Theory Repino, June 24th, 2013

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Study of LPB in HIC

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Outline

Motivation of local parity breaking (LPB) Axial baryon charge and axial chemical potential Eective scalar/pseudosalar meson theory with µ
5

Vector Meson Dominance (VMD) approach to LPB Conclusions

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Study of LPB in HIC

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Motivation of local parity breaking
Parity is one interactions. in a nite vo spontaneous of the well established global symmetries of strong Yet there are reasons to believe that it may be broken lume since no fundamental principle forbids parity breaking for µ = 0.

P - and CP -odd condensates = "pion" condensates

Topological uctuations

A. Vilenkin, Phys. Rev. D22, 3080 (1980); A.B. Migdal, Zh. Eksp. Teor. Fiz. 61 (1971); T. D. Lee and G. C. Wick, Phys. Rev. D 9, 2291 (1974); A. A. Andrianov & D. Espriu, Phys. Lett. B 663 (2008) 450; A. A. Andrianov, V. A. Andrianov & D. Espriu, Phys. Lett. B 678 (2009) 416 D. Kharzeev, R. D. Pisarski & M. H. G. Tytgat, Phys. Rev. Lett. 81, 512 (1998); K. Buckley, T. Fugleberg, & A. Zhitnitsky, Phys. Rev. Lett. 84 (2000) 4814; D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008); A. A. Andrianov, V. A. Andrianov, D. Espriu & X. Planells, Phys. Lett. B 710 (2012) 230; A. A. Andrianov, D. Espriu & X. Planells, Eur. Phys. J. C (2013) 73:2294

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Motivation of local parity breaking
Local large uctuations in the topological charge probably exist in a hot environment. For peripheral Heavy Ion Collisions (HIC) they lead to the Chiral Magnetic Eect (CME): Large B large E charge separation. For central collisions (and light quarks) they correspond to an ephemeral phase with axial chemical potential µ5 = 0.

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Axial baryon charge and axial chemical potential
Topological charge T5 may arise in a nite volume due to quantum uctuations in a hot medium due to sphaleron transitions [Manton, McLerran, Rubakov, Shaposhnikov]
T5 =

8

1

2

vol.

d 3 x jkl Tr G j k G l - i G j G k G

2 3

l

and survive for a sizeable lifetime in a heavy-ion reball. One can control the value of T5 introducing into the QCD Lagrangian a topological chemical potential µ via Ltop = µ T5 , where
T5 = T5 (tf ) - T5 (0) =

8

1

tf 2 0

dt

vol.

d 3 x Tr G µ Gµ .

The PCAC (broken by gluon anomaly) predicts the induced axial charge to be conserved during reball (in the chiral limit):
d q Q5 - 2Nf T5 dt

0,

q Q5 =

vol.

¯ d 3 x q 0 5 q = NL - NR
5

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Study of LPB in HIC


Axial baryon charge and axial chemical potential
The characteristic left-right oscillation time is governed by inverse quark masses. For u , d quarks 1/mq 1/5 MeV-1 40 fm reball and the left-right quark mixing can be neglected. For s quark 1/ms 1/150 MeV-1 1 fm reball and s Q5 0 due to left-right oscillations. For u , d quarks QCD with a background topological charge leads to q the generation of an axial chemical potential µ5 , conjugate to Q5 1 1 q T5 Q5 µ5 µ, 2Nf 2Nf
q Ltop = µ T5 Lq = µ5 Q5

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Eective scalar/pseudosalar meson theory with µ

5

In the scalar sector the spurion technique can be used by taking µ as the time component of some external axial-vector eld
D = D - i {Iq µ5 0 , ·} = D - 2i Iq µ5 0 .

5

Note the breaking of Lorentz symmetry. Two new processes are likely to appear inside the reball: the decays , that are strictly forbidden in QCD on parity grounds.
In a medium where parity is broken: are these processes relevant within the reball? Can these particles reach thermal equilibrium?!

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Generalized

Eective scalar/pseudosalar meson theory with µ

mo del

5

Eective Lagrangian:
L = Tr Dµ HD µ H - 1

1 4



+ Tr M (H + H ) +

b

2

M

2

2

Tr HH



2 where

+

d1

2

Tr (HH )2 -

2

4

Tr HH



2

+ (detH + detH )

c

Tr M (HH H + H HH ) +
= exp i

d2

2

2

Tr M (H + H ) Tr HH
= b b .
ii



H = ,

2f

,

= a a ,

The v.e.v. of the neutral scalars are dened as vi = i = u , d , s , and satisfy the following gap equations:
M 2 vi - 21 vi3 - 2 v 2 vi + c
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where

vu vd vs = 0. vi

Study of LPB in HIC

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Generalized

Eective scalar/pseudosalar meson theory with µ

mo del

5

For further purposes we need the non-strange meson sector and s
+ + 0 + 0 2 0 vu + + a0 2a0 0 q - 0 = 2 - q - 0 0 , = 2a0 vd + - a0 0 0 0 vs 0 0 2s q s =

cos sin - sin cos



For µ5 = 0, we assume vu = vd = vs = v0 f 92 MeV. The coupling constants (in units of MeV) are tted to phenomenology assuming isospin symmetry via 2 minimization (MINUIT):
b = -3510100/m, M 2 = 1255600, c = 1252.2, 1 = 67.007,
2 = 9.3126, d1 = -1051.7/m, d2 = 523.21/m,

where m mq = (mu + md )/2 and m/ms
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Study of LPB in HIC


New eigenstates of strong interactions with LPB (isotriplet)

Eective scalar/pseudosalar meson theory with µ

5

We present a simple case of mixing due to LPB in the isotriplet sector with and a0 . The kinetic and mixing terms in the Lagrangian are given by 1 1 22 1 2 1 L = ( a0 )2 + ( )2 - m1 a0 - m2 2 - 4µ5 a0 , 2 2 2 2 where
2 2 m1 = - 2[M 2 - 2(31 + 2 )vq - 2 vs2 2 m2 =

2m
vq

- cvs + 2(3d1 + 2d2 )mvq + 2d2 ms vs + 2µ2 ] 5
2 b + (d1 + 2d2 )vq + d2 vs2

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New eigenstates of strong interactions with LPB (isotriplet)

Eective scalar/pseudosalar meson theory with µ

5

After diagonalization in the momentum representation, the new (momentum-dependent) eigenstates are dened and ~0 . ~ a

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New eigenstates of strong interactions with LPB (isotriplet)

Eective scalar/pseudosalar meson theory with µ

5

c For high energies k0 , |k | > m1 m2 /(4µ5 ) k , in-medium goes ~ ~ tachyonic. Nevertheless, energies are always positive (no vacuum instabilities). ~0 mass shows an enhancement, but µ5 has to be a understood as a perturbatively small parameter.

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New eigenstates of strong interactions with LPB (isosinglet)

Eective scalar/pseudosalar meson theory with µ

5

In the isosinglet sector, we show the mixing of , and . The kinetic and mixing terms in the Lagrangian are given by 12 1 22 1 22 1 L = [( )2 + ( q )2 + ( s )2 ] - m3 2 - m4 q - m5 s 2 2 2 2 - 4µ5 q - 2 2cvq q s , where
2 2 m3 = - 2(M 2 - 6(1 + 2 )vq - 2 vs2 + cvs 2 m4 = 2 m5

2m
vq

+ 6(d1 + 2d2 )mvq + 2d2 ms vs + 2µ2 ), 5
2 b + (d1 + 2d2 )vq + d2 vs2 + 2cvs ,

=

2ms
vs

[b +

2 2d2 vq

+ (d1 +

d2 )vs2

2 cvq ]+ . vs

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New eigenstates of strong interactions with LPB (isosinglet)

Eective scalar/pseudosalar meson theory with µ
After diagonalization, the new eigenstates are , and . ~~ ~

5

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New eigenstates of strong interactions with LPB (isosinglet)

Eective scalar/pseudosalar meson theory with µ

5

c Again, for high energies k0 , |k | > k with ~ m3 c 22 2 k 4µ5 m5 m4 m5 - 8c 2 vq , in-medium goes tachyonic. ~ ~

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Decay widths

Eective scalar/pseudosalar meson theory with µ

5

The cubic couplings used to calculate the widths , , ~~~ ~~ from the Lagrangian are given by
2 Laa = 2[(3d1 + 2d2 )m - 2(31 + 2 )vq ] a0 ,

L



2 2 La = 2 a0 [ q vq - (b + (3d1 + 2d2 )vq + d2 vs2 )mq ], v
q

1 2 = 2 ( )2 vq - (b + 3(d1 + 2d2 )vq + d2 vs2 )m 2 , v
q

La = -

4µ

vq

5

a0 ,

Laa = -

2µ

vq

5

q a0 , 2

L



= 0.

After diagonalization, one replaces the initial {q , s , } to { , , } and { , a0 } to { , ~0 }. ~~~ ~a The widths are rstly computed at the rest frame of the decaying particle and then with a boost.
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Decay widths (at rest)

Eective scalar/pseudosalar meson theory with µ

5

exhibits a smooth behaviour with 60 MeV mean free ~ ~ path 3 fm Lreball 5 Â 10 fm. Possible thermalization! Down to µ5 100 MeV, width decreases and becomes stable. ~ The bumps seem to reect the tachyonic nature of the decaying . ~ width grows up to the GeV scale (violation of unitarity). ~

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Decay widths (moving

Eective scalar/pseudosalar meson theory with µ
) ~

5

Decay widths in the isosinglet case show strong dependences on the 3-momentum (nothing to do with Lorentz time dilatation).

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Vector Meson Dominance approach to LPB
If we assume that the vector mesons appear as part of a covariant derivative, no mixing term can be generated. However, such a mixing is not forbidden. This coupling is very model dependent. Vector mesons can be introduced in the Vector Meson Dominance framework with no mixing of states with dierent parities. The only LPB eect will be the Chern-Simons term
L
µ

Tr [µ V V ] ,

where µ µ5 µ0 . Vector mesons exhibit the following dispersion relation:
2 2 mV , - mV µ5 |k |,

where = 0, ±1 is the meson polarization. Note the breaking of Lorentz symmetry again. Massive vector mesons split into three polarizations with masses 2 2 2 mV ,- < mV ,L < mV ,+ .
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Vector Meson Dominance approach to LPB
Manifestation of LPB in heavy ion collisions

Polarization splitting in e + e - decay for LPB µ5 = 290 MeV compared with µ5 = 0 (shaded region). Note the polarization asymmetry aside the peak.
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Conclusions
LPB not forbidden by any physical principle in QCD at nite temperature/density. Topological uctuations transmit their inuence to hadronic physics via an axial chemical potential. LPB leads to unexpected modications of the in-medium properties of scalar and vector mesons. The new eigenstate seems to be in thermal equilibrium with ~ the pion gas in the HIC reball.

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Thank you for your attention!

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Backup I
MINUIT input in MeV
exp m exp v0 = 92 ± 5, exp exp m = 137 ± 5, ma = 980 ± 50, exp exp = 600 ± 120, m = 548 ± 50, m = 958 ± 100,

MINUIT output versus experimental values in MeV

exp = 60 ± 30, a

exp = 600 ± 120.

exp

-18 + arctan 2 36.7 , while MINUIT 35.46 . Xumeu Planells Study of LPB in HIC

Magnitude MINUIT Exp. value v0 92.00 92 m 137.84 137 ma 980.00 980 m 599.99 600 m 497.78 548 m 968.20 958 a 60.00 60 600.00 600

Error -3.52 â 10-7 6.10 â 10-3 -1.26 â 10-6 -1.66 â 10-5 -9.16 â 10-2 1.06 â 10-2 2.04 â 10-5 6.81 â 10-6

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Backup II
Vacuum: for non-vanishing isosinglet µ5 we impose our solutions to be vu = vd = vq = vs .

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Backup III
c µ5 -dependence of the tachyon critical energy for isotriplet k and ~ c isosinglet case k . ~

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