Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2013/minnesota2013.pdf Äàòà èçìåíåíèÿ: Sat Jun 22 14:06:21 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 20:14:43 2014 Êîäèðîâêà:

Chiral symmetry breaking, confinement and the mass generation of hadrons
L. Ya. Glozman ¨ Institut fur Physik, FB Theoretische Physik, Universitat Graz ¨

¨ with M. Denissenya, C.B. Lang, M. Schrock

L.Ya. Glozman


Contents of the Talk
Key questions to QCD Chiral symmetry and origin of hadron mass The quark condensate and the Dirac operator Extraction of the physical states on the lattice Hadrons after unbreaking of the chiral symmetry Chiral parity doublets Effect of the low-lying modes Conclusions

L.Ya.Glozman ­ p.0/18


Key questions to QCD
Key question to QCD: How is the hadron mass generated in the light quark sector?
· How impor tant is the chiral symmetr y breaking for the hadron mass? · Are confinement and chiral symmetr y breaking directly interrelated? · Is there parity doubling and does chiral symmetr y get effectively restored in high-lying hadrons? · Is there some other symmetr y? · How is the angular momentum of hadrons connected to the chiral symmetry breaking?
L.Ya.Glozman


What is the hadron mass origin in QCD?

Gell-Mann - Levy sigma model, Nambu - Jona-Lasinio mechanism, many "Bag-like" and microscopical models to QCD, SVZ sum rules: Chiral symmetry breaking in a vacuum is the source of the hadron mass in the light quark sector. A typical implication: In a dense medium upon smooth chiral restoration the hadron (,... ) mass should drop off (the Brown-Rho scaling). Is it true? Is chiral symmetry breaking in QCD and confinement are uniquely interconnected? (A key question for the QCD phase diagram).
L.Ya.Glozman


The quark condensate and the Dirac operator
Banks-Casher : A density of the lowest quasi-zero eigenmodes of the Dirac operator represents the quark condensate of the vacuum: < 0 | q q | 0 > = - ( 0 ) . ¯ Sequence of limits: V ; mq 0 . The lattice volume is finite and the spectrum is descrete. We remove an increasing number of the lowest Dirac modes from the valence quark propagators and study the effects of the remaining chiral symmetr y breaking on the masses of hadrons. S (k ) = S -
i k

µ

-1

|vi >< vi |5 ,

S - standard quark propagator in a given gauge configuration; µi are the real eigenvalues of the Hermitian D5 = 5 D Dirac operator ; |vi > - eigenvectors; k number of the removed lowest eigenmodes.
L.Ya.Glozman


Extraction of the physical states on the lattice
Assume we have hadrons (states) with energies n = 1, 2, 3, ... with fixed quantum numbers. C ( t ) i j = O i ( t ) O ( 0) =
n j

a

(n ) (n ) - E e i aj

(n)

t

(1 )

where a
(n ) i

= 0| O i | n .

The generalized eigenvalue problem: C ( t) i j u
(n ) j

=

(n )

( t, t0 ) C ( t0 ) i j u

(n ) j

.

(2 )

Each eigenvalue and eigenvector corresponds to a given state. If a basis Oi is complete enough, one extracts energies and "wave functions" of all states. C ( t) i j u C ( t)
(n ) j (n ) k j uj

=

a a

(n ) i (n ) k

.

(3 )
L.Ya.Glozman


Extraction of the physical states on the lattice
E.g., we want to study I = 1, 1 Then a basis of inter polators:
--

states = (770) and its excitations.

O

V

= q (x ) i q (x ); ¯

O

T

= q (x ) 0i q (x ); ¯

O = q ( x ) i q ( x ) ; ... ¯ plus inter polators with a Gaussian smearing of the quark fields in spatial directions in the source and sink. Some lattice details: · Unquenched QCD with 2 dynamical flavors. · L = 2.4 fm; a = 0.144 fm · m = 322 MeV (mu,d 15 MeV) · Chirally improved fermions We subtract the low-lying chiral modes from the valence quarks.
L.Ya.Glozman


( I = 1 , 1

--

) with 12 eigenmodes subtracted

The correlators n (t) exp (-En t) for all eigenstates (left) and the effective mass plot En (t) = log(n (t)/n (t + 1)) for the two lowest states (right).

Eigenvectors corresponding to the ground state (left) and 1st excited state (right)
L.Ya.Glozman


b1 ( I = 1 , 1

+-

) states

The correlators n (t) exp (-En t) for all eigenstates with 2 eigenmodes subtracted and the effective mass plot En (t) = log(n (t)/n (t + 1)) for the lowest state.

The same with 128 eigenmodes subtracted. The quality of the exponential decay essentially improves with increasing the number of removed eigenmodes for ALL hadrons. By unbreaking the chiral symmetr y we remove from the hadron its pion cloud and subtract all higher Fock components like N , , ,...

L.Ya.Glozman


What do meson degeneracies and splittings tell us?

The S U (2)L â S U (2)R â Ci (chiral-parity) multiplets for J = 1 mesons: (0 , 0 )
1 (2, 1) 2

:
a b

(0 , 1

--

) )

f1 ( 0 , 1 (1, 1

++

)

: : :

h 1 (0 , 1 (0 , 1

+-

--

) )

(1, 1) 22

--

) )

b 1 (1 , 1 (1, 1

+-

(0 , 1 ) + (1 , 0 )

a 1 (1 , 1

++

--

) group.

The h1 , , and b1 states would form an irreducible multiplet of the S U (2)L â S U (2)R â U (1)

A

L.Ya.Glozman


What do meson degeneracies and splittings tell us?
· Chiral symmetr y is restored but confinement is still there ! · Hadrons get their large chirally symmetric mass! · The S U (2)L â S U (2)R gets restored while the U (1)A is still broken! · The U (1)A explicit breaking comes not (not only) from the low-lying modes as the S U (2)L â S U (2)R !

· - degeneracy indicates higher symmetr y that includes S U (2)L â S U (2)
R

as a subgroup. What is this symmetry!? Is this symmetry
L.Ya.Glozman

related with the symmetry of the high-lying mesons?


Low and high lying meson spectra.
M

2

. .

1

. . . . .

.. ... . . . . . . .. . . .. .. . .. . . ..
...

.
f a ah b f a f a h b f a f ah 00 00 1 1 1 111 2 2 22 2 2 33 3 333 4 4 44 4 4 5 55 5

The high-lying mesons are from pp annihilation at LEAR (Anisovich, Bugg, ¯ Sarantsev,...). Missing parity par tners for highest spin states at each band. They ALL require higher par tial wave in pp that is strongly (10-100 times) suppressed in ¯ pp near threshold. Cannot be seen in pp? ¯ ¯ Large symmetr y: N = n + J plus chiral symmetr y. An alternative: N = n + L without chiral symmetr y. (Afonin, Shifman-Vainshtein, Klempt-Zaitsev,...).L is a conser ved quantum number ?! Naive string picture with quarks at the ends is intrinsically inconsistent. L.Ya.Glozman


Baryons

Three possible S U (2)L â S U (2)R â Ci (chiral-parity) multiplets for any spin (1 / 2 , 0 ) + (0 , 1 / 2 ); (3 / 2 , 0 ) + (0 , 3 / 2 ); (1 / 2 , 1 ) + (1 , 1 / 2 ) Our inter polators have J = 1/2 for N and J = 3/2 for , i.e. we cannot see (1/2, 1) + (1, 1/2) quar tets. · Chiral symmetr y is restored (all bar yons are in doublets), while confinement is still there. · Bar yons have large CHIRALLY SYMMETRIC mass. · Two J = 1/2 N doublets get degenerate - clear sign for a higher symmetr y. No this higher symmetr y for 's.
L.Ya.Glozman


Chiral parity doublet
A free I = 1/2 chiral doublet B in the (0, 1/2) + (1/2, 0) representation: B B
+

B=



-

.

(4 )

The axial rotation mixes the positive and negative parity components:
a A a B ex p i 2

1

B.

(5 )

A chiral-invariant Lagrangian ¯ ¯ L 0 = i B µ µ B - m0 B B · A nonzero chiral-invariant (!) mass m0 . A A A A · g+ = g- = 0, while the off-diagonal axial charge , |g+- | = |g-+ | = 1. · Pion decouples: GB± B± = 0. B. W. Lee, 1972: "We dismiss this model as physically uninteresting"
L.Ya.Glozman

(6 )


Low and high lying baryon spectra.
3000
[MeV]

** 2500

2000

*

*

**

**

** ** **

1500

1000

1 2

3 2

5 2

7 2

9 2

11 2

13 2

Low-lying spectrum: spontaneous breaking of chiral symmetr y is impor tant for physics. High-lying spectrum: parity doubling is suggestive of EFFECTIVE chiral symmetr y restoration.
Recent and most complete analysis on highly excited nucleons (elastic N and photoproduction data from Bonn and JLAB) repor ts evidence for some of the missing states and the parity doubling patterns L.Ya.Glozman look now better than before.


The role of the low-lying modes for (I = 1, 1
Eigenvalues (interpolators 1,8) 1 Normalized eigenvalues full trunc128 trunc64 trunc32 meff (t), MeV 2000 1600 1200 800 400 0 0 2 4 6 t 8 10 12 1 2 3 4 5 t 6 7 Effective mass curves (interpolators 1,8) 0.1

--

)

full trunc128 trunc64 trunc32

0.01

0.001

0.0001

8

9

10

The correlators n (t) exp (-En t) from the full quark propagators and from the lowest modes (32; 64; 128) (left) and the respective effective mass plots En (t) = log(n (t)/n (t + 1)) (right).

Nomalized Eigenvector Components

1 0.5 0 -0.5 -1 2 3 4 5 t 6 7

Nomalized Eigenvector Components

Eigenvectors (full) 1 8

Eigenvectors (trunc-64) 1 0.5 0 -0.5 -1 2 3 4 5 t 6 7 8 1 8

8

Eigenvectors corresponding to the full quark propagators (left) and from their lowest L.Ya.Glozman 64 modes (right).


The role of the low-lying modes for a1 (I = 1, 1
Eigenvalues (interpolators 1,4) 1 Normalized eigenvalues full trunc128 trunc64 trunc32 meff (t), MeV 2500 2000 1500 1000 500 0 1 2 3 4 5 t 6 7 8 9 10 1 2 3 4 t 5 Effective mass curves (interpolators 1,4) full trunc128 trunc64 trunc32 0.1

++

)

0.01

0.001

0.0001

6

7

The correlators n (t) exp (-En t) from the full quark propagators and from the lowest modes (32; 64; 128) (left) and the respective effective mass plots En (t) = log(n (t)/n (t + 1)) (right).

Eigenvectors (full) Nomalized Eigenvector Components 1 0.5 0 -0.5 -1 1 2 3 4 t 5 6 7 8 Nomalized Eigenvector Components 1 4 1 0.5 0 -0.5 -1 1 2 3

Eigenvectors (trunc-64) 1 4

4 t

5

6

7

8

Eigenvectors corresponding to the full quark propagators (left) and from their lowest L.Ya.Glozman 64 modes (right).


The role of the low-lying modes for b1 (I = 1, 1
Diagonal Correlators (interpolator 6) 1 full trunc64

+-

)

0.1 C(t)

0.01

0.001

0.0001 1 2 3 4 5 t 6 7 8 9 10

The correlators n (t) exp (-En t) from the full quark propagators and from the lowest 64 modes.

In contrast to other quantum numbers there is no b1 state from the low-lying modes. The low-lying modes do not provide confinement. L.Ya.Glozman


The role of the low-lying modes for N
Eigenvalues (interpolators 1,3,16) 1 Normalized eigenvalues full trunc128 trunc64 trunc32 meff (t), MeV 2500 2000 1500 1000 500 0 0 2 4 6 t 8 10 12 1 2 3 4 5 t 6 7 8 9 10 Effective mass curves (interpolators 1,3,16) full trunc128 trunc64 trunc32

0.1

0.01

0.001

0.0001

The correlators n (t) exp (-En t) from the full quark propagators and from the lowest modes (32; 64; 128) (left) and the respective effective mass plots En (t) = log(n (t)/n (t + 1)) (right).

Eigenvectors (full) Nomalized Eigenvector Components 1 0.5 0 -0.5 -1 2 3 4 5 t 6 7 8 9 Nomalized Eigenvector Components 1 3 16 1 0.5 0 -0.5 -1 2 3 4

Eigenvectors (trunc-128) 1 3 16

5 t

6

7

8

9

Eigenvectors corresponding to the full quark propagators (left) and from their lowest L.Ya.Glozman 128 modes (right).


Conclusions
· Removal of the low-lying modes from valence quarks restores chiral symmetr y and signals from all hadrons sur vive (except for a pion). The quality of the signals from the hadrons after removal of the quark condensate become much better than with the untruncated quark propagators. Most probably this is related to the fact that we ar tificially remove the pion cloud of the hadrons. · Chiral symmetr y is ar tificially restored but confinement is there. · There is a large chirally symmetric mass in this regime. · All hadrons in this regime fall into different representations of the chiral group. We obser ve a higher degree of degeneracy than simply S U (2)L â S U (2)R , i.e. there is a higher symmetr y in this regime that includes chiral group as a subgroup. · Removal of the low-lying modes from valence quarks does NOT restore the U (1) symmetr y. · The lowest-lying modes saturate pions and provide 2/3 of mass for N and . · The lowest-lying modes do not contain confinement.
L.Ya.Glozman

A