Symmetries of Mesons after
Unbreaking of Chiral Symmetry
Mikhail Denissenya
University of Graz
Schladming Winter School 2015
Introduction
QCD Lagrangian
1 a aμ ν
μ
̄
ℒ= Fμ ν F + ∑ ψ (i( γμ D )−mf ) ψ
4
f =u , d , s ,...
a
a
a
F μ ν =∂μ A ν −∂ ν A μ + g f
abc
b
c
Aμ A ν.
mu, md << ms,...
and
mu, md << MHADRON
A toy lagrangian for two massless flavours possesses
U(2)L x U(2)R = SU(2)L x SU(2)R x UA(1) x UV(1)
Quantum fluctuations break
global symmetry
SU(2)L x SU(2)R spontaneously by formation of the quark condensate 〈 ψ ψ〉
UA(1) by anomaly
simplified picture of SCSB
quark
M(p)
<ψψ>≠0
vacuum
π
σ
ρ
vacuum
a1
N
...
...
<ψψ>≠0
simplified picture of SCSB
quark
M(p)
<ψψ>≠0
vacuum
π
σ
ρ
a1
N
...
...
<ψψ>≠0
vacuum
How would the hadron spectrum look like
if we artificially remove the quark condensate?
???
vacuum
<ψψ>=0
What can we learn from resulting spectrum about
the physics of confinement?
Banks-Casher relation and SCSB
Low-lying eigenmodes of the Dirac operator encode the effects of
SCSB
〈 ψ ψ〉=〈Tr (S ( x , y))〉=〈 ∑
A
k
1
〉
λk
N
λ
12x32x16
3
〈 ψ ψ〉=−π ρ(0)
k
ρ(0)≠0 ⇔ S χ SB
λ=0
L≈1.9 fm
L
ρ(0) is the spectral denisity of eigenmodes of the Dirac operator
On the lattice : removal of the chiral symmetry breaking dynamics
can be achieved by exclusion of the low-lowing eigenmodes.
Quark Propagators
Define reduced quark propagators
Nλ
1
S k (x , y)= ∑ λ un ( x)un † ( y)=S 100−k + S stoch
n=k+1 n
- k is the number of excluded low-lying modes
Lattice setup
dynamical nf = 2 overlap fermions,
Ns x NT =163 x 32 lattice size
a = 0.12 fm, L = 1.9 fm
100 conf , M = 289(2) MeV, QT = 0.
S.Aoki et al (2008)
Nr = 1, Nd = Nc x ND x NT /2 - noise dilution scheme
Meson Propagators
G(t , x)=〈 Ο (t , x) Ο † (0)〉
for isospin I=0 mesons:
G(t , x)=C + D
for isospin I=1 mesons:
G(t , x)=C
Here we consider only J = 0, 1 non-exotic mesons
Symmetries among J=0 mesons
Restoration of both SUL(2) x SUR(2) and UA(1) symmetries requires
C π = C a & & D σ = D η=0
0
J=0 Meson Correlators
J=0 Meson Correlators
J=0 Meson Correlators
J=0 Meson Correlators
J=0 Meson Correlators
J=0 Meson Correlators
J=0 Meson Correlators
SU(2)L x SU(2)R x U(1)A symmetry is restored
J=0 Meson Correlators
SU(2)L x SU(2)R x U(1)A symmetry is restored
What about the existence of bound states?
J=0: π states
−M t
−M ' t
π +B e
π +...
C π (t )= A e
J=0: π states
−M t
−M ' t
π +B e
π +...
C π (t )= A e
J=0: π states
−M t
−M ' t
π +B e
π +...
C π (t )= A e
We don't observe the pion ground state → no pseudo-Goldstone boson
k > 20: η, σ, a0, are identical to the π states pattern
J=1 Mesons
k - number of excluded low-lying eigenmodes
J=1 Mesons
k - number of excluded low-lying eigenmodes
Evolution of J=1 meson masses
ρ
ω
b1
h1
a1
f1
ρ'
ω'
Evolution of J=1 meson masses
ρ
ω
b1
h1
a1
f1
ρ'
ω'
E1
E0
Evolution of J=1 meson masses
ρ
ω
b1
h1
a1
f1
ρ'
ω'
E1
E0
J=1 meson mass splittings vanish
|1; 3S1⟩ |0; 3S1⟩ |1; 1P1⟩ |0; 1P1⟩ |1; 3P1⟩ |0; 3P1⟩ |1; 3D1⟩ |0; 3D1⟩
Evolution of J=1 meson masses
ρ
ω
b1
h1
a1
f1
ρ'
ω'
E1
E0
J=1 meson mass splittings vanish
|1; 3S1⟩ |0; 3S1⟩ |1; 1P1⟩ |0; 1P1⟩ |1; 3P1⟩ |0; 3P1⟩ |1; 3D1⟩ |0; 3D1⟩
Energy of the system does not depend on the spin orientation
of quarks → QCDr dynamical string
En = (n+1) ћω, n = 0, 1, 2 ...
E0 = ћω = (960 ± 20) MeV
Symmetries among J=1 Mesons
J = 1 mesons follow a rich set of multiplets of the
parity-chiral group SU (2)L×SU (2)R ×C i
Degeneracy of all J=1 mesons is a hint for a higher symmetry
Symmetries among J=1 Mesons
J = 1 mesons follow a rich set of multiplets of the
parity-chiral group SU (2)L×SU (2)R ×C i
Degeneracy of all J=1 mesons is a hint for a higher symmetry
SU (4)⊃SU (2)L×SU (2)R ×U (1) A
talk of Markus Pak
Summary
Upon unbreaking of chiral symmetry
Disconnected contributions in the isoscalar channels vanish
Full SU(2)L x SU(2)R x UA(1) symmetry gets restored
In the chirally restored regime
J=0 meson ground states are not found in the spectrum
Non-exotic J=1 mesons become degenerate and can be
interpreted as the QCDr dynamical strings
M.Denissenya, L. Ya. Glozman, and C. B. Lang
Phys. Rev. D 91, 034505 (2015)
Phys. Rev. D 89, 077502 (2014)
J=2 Mesons and Baryons are under investigation (in collaboration with Markus Pak)
Thank You for Your Attention!