SU(2)CS and SU(2NF) hidden symmetries L. Ya. Glozman Institut f ur - - PowerPoint PPT Presentation

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

SU(2)CS and SU(2NF) hidden symmetries

• L. Ya. Glozman

Institut f¨ ur Physik, FB Theoretische Physik, Universit¨ at Graz in collaboration with M. Denissenya, C. B. Lang, M. Pak

29th April 2017

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

1

Key questions

2

J = 0 mesons

3

J = 1 mesons

4

Left-right mixing. The chiralspin SU(2)CS and SU(4) groups.

5

J=2 mesons

6

J = 1/2 baryons

7

SU(2)CS and SU(4) symmetries of confinement in QCD

8

Conclusions

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

What is origin of hadron mass? Are chiral symmetry breaking and confinement uniquely connected? Is it possible to separate confinement and chiral symmetry breaking physics? What physics is responsible for confinement and for chiral symmetry breaking? What is the underlying systematics that drives a genesis of hadrons and hadron spectra?

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

Low mode truncation

Banks-Casher: < ¯ qq >= −πρ(0). What we do: S = SFull −

k

• i=1

1 λi |λiλi|. What one expects for J = 0 correlators or states, if they survive:

(1/2,1/2) (1/2,1/2)

a b

SU(2) * SU(2)

L R

SU(2) * SU(2)

L R

π σ η a U(1) U(1)

A A

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

M.Denissenya, L.Ya.G., C.B.Lang, PRD 89(2014)077502; 91(2015)034505

We use JLQCD Nf = 2 overlap gauge configurations and quark propagators.

10-4 10-3 10-2 4 8 12 16 20 24 28

t σ a0 η π k=30

SU(2)L × SU(2)R × U(1)A is restored in correlators.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 6 8 10 12 14 t 1st 2nd

π(k=0)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 6 8 10 12 14 t 1st 2nd

π(k=10)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 6 8 10 12 14 t 1st 2nd

π(k=60)

The ground states do not survive truncation: Without the near-zero modes π, σ, a0, η do not exist. These states are not a direct consequence of confinement.

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

M.Denissenya, L.Ya.G., C.B.Lang, PRD 89(2014)077502; 91(2015)034505

What one expects for J = 1 mesons: (0, 0) (1/2, 1/2)a (1/2, 1/2)b (1, 0) ⊕ (0, 1)

f1(0, 1++) !(0, 1−−) b1(1, 1+−) !(0, 1−−) ρ(1, 1−−) ρ(1, 1−−) h1(0, 1+−) a1(1, 1++) Ψ(τ a ⊗ γk)Ψ Ψ(τ a ⊗ γ5γk)Ψ Ψ(τ a ⊗ γ0γk)Ψ Ψ(τ a ⊗ γ5γ0γk)Ψ Ψ(1F ⊗ γ5γk)Ψ Ψ(1F ⊗ γ5γ0γk)Ψ Ψ(1F ⊗ γ0γk)Ψ Ψ(1F ⊗ γk)Ψ SU(2)A

SU(2)A SU(2)A U(1)A U(1)A

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

M.Denissenya, L.Ya.G., C.B.Lang, PRD 89(2014)077502; 91(2015)034505 Isovector J = 1 states.

10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 t

ρ(k=10)

1st 2nd 3rd 4th 5th 6th 7th 0.5 1 1.5 2 2 4 6 8 10 12 14 16 t 1st 2nd 3rd 4th 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 t

a1(k=10)

1st 2nd 3rd 4th 5th 0.5 1 1.5 2 2 4 6 8 10 12 14 16 t 1st 2nd 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 t

b1(k=10)

1st 2nd 3rd 4th 5th 6th 0.5 1 1.5 2 2 4 6 8 10 12 14 16 t 1st 2nd

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

M.Denissenya, L.Ya.G., C.B.Lang, PRD 89(2014)077502; 91(2015)034505

J = 1

500 1000 1500 2000 0 8 40 65 93 125 180 0 2 6 10 14 20 30 mass, MeV σ, MeV k

ρ ω h1 b1 a1 ρ′ f1 ω′ b1′ ρ′′ ω′′ a1′

We clearly see a larger degeneracy than the SU(2)L × SU(2)R × U(1)A symmetry of the QCD Lagrangian. What does it mean !?

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G., EPJA 51(2015)27

(i) (0,0): |(0, 0); ±; J = 1 √ 2 |¯ RR ± ¯ LLJ. (ii) (1/2, 1/2)a and (1/2, 1/2)b: |(1/2, 1/2)a; +; I = 0; J = 1 √ 2 |¯ RL + ¯ LRJ, |(1/2, 1/2)a; −; I = 1; J = 1 √ 2 |¯ RτL − ¯ LτRJ, |(1/2, 1/2)b; −; I = 0; J = 1 √ 2 |¯ RL − ¯ LRJ, |(1/2, 1/2)b; +; I = 1; J = 1 √ 2 |¯ RτL + ¯ LτRJ. (iii) (0,1)⊕(1,0): |(0, 1) + (1, 0); ±; J = 1 √ 2 |¯ RτR ± ¯ LτLJ,

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G., EPJA 51(2015)27

Consider rotations in an imaginary 3-dim space of doublets constructed from the Weyl spinors U = uL uR

• D =

dL dR

• U → U′ = ei ε·σ

2 U ,

D → D′ = ei ε·σ

2 D ,

where σ are the standard Pauli matrices: [σi, σj] = 2iǫijk σk . We refer to this imaginary three-dimensional space as the chiralspin space and denote this symmetry group as SU(2)cs A group that contains at the same time SU(2)L × SU(2)R and SU(2)CS is SU(4) with the fundamental vector Ψ =     uL uR dL dR    

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G., M. Pak, PRD 92(2015)016001

Instead of the states constructed with Weyl spinors we can consider the left- and right-handed Dirac bispinors and bilinear operators. Then the SU(2)cs chiralspin rotations are generated through Σ = {γ0, iγ5γ0, −γ5} , [Σi, Σj] = 2iǫijk Σk . The SU(4) group contains at the same time SU(2)L × SU(2)R and SU(2)CS ⊃ U(1)A with the fundamental vector Ψ =     uL uR dL dR     and has the following set of generators: {(τ a ⊗ 1D), (1F ⊗ Σi), (τ a ⊗ Σi)}

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G., M. Pak, PRD 92(2015)016001

(0, 0) (1/2, 1/2)a (1/2, 1/2)b (1, 0) ⊕ (0, 1)

f1(0, 1++) Ψ(1F ⊗ γ5γk)Ψ b1(1, 1+−) Ψ(τ a ⊗ γ5γ0γk)Ψ ρ(1, 1−−) Ψ(τ a ⊗ γ0γk)Ψ ρ(1, 1−−) Ψ(τ a ⊗ γk)Ψ !(0, 1−−) Ψ(1F ⊗ γk)Ψ !(0, 1−−) Ψ(1F ⊗ γ0γk)Ψ h1(0, 1+−) Ψ(1F ⊗ γ5γ0γk)Ψ a1(1, 1++) Ψ(τ a ⊗ γ5γk)Ψ

SU(2)CS

SU(2)CS SU(4)

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

• M. Denissenya, L.Ya.G, M.Pak, PRD 91(2015)114512

J=2 mesons.

(0, 0)

(1/2, 1/2)a (1/2, 1/2)b (1, 0) ⊕ (0, 1) !2(0, 2−−) π2(1, 2−+) a2(1, 2++) a2(1, 2++) f2(0, 2++) f2(0, 2++) η2(0, 2−+) ρ2(1, 2−−) SU(2)CS SU(2)CS

SU(4)

10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16

log λn(t) t a2 - n=1 a2 - n=2 ρ2 - n=1 π2 - n=1 ρ2 - n=2 π2 - n=2 a2 - n=3 a2 - n=4 T2 : k=0

10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16

log λn(t) t a2 - n=1 a2 - n=2 ρ2 - n=1 π2 - n=1 ρ2 - n=2 π2 - n=2 a2 - n=3 a2 - n=4 T2 : k=30

900 1200 1500 1800 2100 0 8 40 65 105 125 180 0 2 6 10 16 20 30 mass, MeV σ, MeV k a2 : T2, state 1 a2 : T2, state 2 ρ2 : T2, state 1 π2 : T2, state 1

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

J = 1/2 baryons: M. Denissenya, L.Ya.G, M.Pak, arXiv:1508.01413

(1/2, 0) + (0, 1/2) : ON± = εabcP±ua ubTCγ5dc ← 20 of SU(4) (1, 1/2) + (1/2, 1) : ON± = iεabcP±ua ubTCγ5γ0dc ← 20 of SU(4) (1, 1/2) + (1/2, 1) : O∆± = iεabcP±γiγ5ua ubTCγiuc ← 20′ of SU(4)

uT Cγ5γ0d uT Cd uT Cγ5d uT Cγ0d uT Cγ0u dT Cγ0d LT L ± RT R LT R ± RT L 0+ 0−

SU(2)A U(1)A SU(2)CS SU(2)CS

1 2, 1 2

• (0, 0)

I = 0 I = 0 I = 0 I = 1

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

J = 1/2 baryons: M. Denissenya, L.Ya.G, M.Pak, arXiv:1508.01413

10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16

log λn(t) t N+ - n=1 N+ - n=2 N− - n=1 N− - n=2

∆+ - n=1 ∆− - n=1

J=1/2 k=0 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 log λn(t) t

N+ - n=1 N+ - n=2 N− - n=1 N− - n=2

∆+ - n=1 ∆− - n=1

J=1/2 k=30 500 1000 1500 2000 2500 3000 3500 0 8 40 65 105 125 180 0 2 6 10 16 20 30

J=1/2

mass, MeV σ, MeV k

N+, n=1 N−, n=1 N+, n=2 N−, n=2 N+, n=3 N−, n=3 N+, n=4 N−, n=4 ∆+, n=1 ∆−, n=1

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G. EPJA 51(2015)27

What does this symmetry mean ? A unitary transformation (L.Ya.G., A. Nefediev, PRD 76 (2007) 096004): |(0, 1) + (1, 0); 1 1−− =

• 2

3 |1; 3S1 +

• 1

3 |1; 3D1, |(1/2, 1/2)b; 1 1−− =

• 1

3 |1; 3S1 −

• 2

3 |1; 3D1, |(0, 0); 0 1−− =

• 2

3 |0; 3S1 +

• 1

3 |0; 3D1, |(1/2, 1/2)a; 0 1−− =

• 1

3 |0; 3S1 −

• 2

3 |0; 3D1, |(0, 1) + (1, 0); 1 1++ = |1; 3P1, |(0, 0); 0 1++ = |0; 3P1, |(1/2, 1/2)a; 1 1+− = |1; 1P1, |(1/2, 1/2)b; 0 1+− = |0; 1P1. (1)

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G. EPJA 51(2015)27

We can invert this unitary transformation and obtain a chiral decomposition of vectors |0; 3S1, |1; 3S1, |0; 3D1, |1; 3D1, |0; 1P1, |1; 1P1, |0; 3P1, |1; 3P1. A degeneracy of all these states requires that there are no spin-spin, spin-orbit and tensor forces in the system. These forces are mediated by the magnetic field. We conclude: After reduction of the near-zero modes there are no magnetic interactions and only electric interactions are left in the system. Dynamical string?

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G., M. Pak, PRD 92(2015)016001 SU(2)CS and SU(4) are hidden in the QCD Lagrangian. The interaction part of the Lagrangian consists of electric and magnetic interactions: ΨγµDµΨ = Ψγ0D0Ψ − Ψγ · DΨ . We apply SU(2)CS and SU(4) transformations on the interaction part of the QCD Lagrangian, Ψ

′γ0D0Ψ′ = Ψγ0D0Ψ.

The γ0-part is invariant (singlet) under these transformations. It encodes the electric (Coulombic) Qφ interactions in QCD. The spatial (magnetic) part, ∼ j · A, is not invariant - it is a triplet under SU(2)CS and a 15-plet under SU(4) .

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

L.Ya.G., M. Pak, PRD 92(2015)016001

QCD Hamiltonian in Coulomb gauge (Christ and Lee): HQCD = HE + HB +

• d3xΨ†(x)[−iα · ∇ + βm]Ψ(x) + HT + HC,

HT = −g

• d3x Ψ†(x)α · A(x) Ψ(x) ,

HC = g 2 2

• d3x d3y J−1 ρa(x)F ab(x, y) J ρb(y) .

The Coulombic HC part is a SU(2)CS- and SU(4)- singlet. It is a confining part

• f the QCD Hamiltonian. This part generates a SU(4)-symmetric spectrum.

The transverse (magnetic) part HT is not SU(2)CS- and SU(4)-symmetric and therefore its expectation value vanishes in the SU(4)-symmetric hadron wave function. The quasi-zero modes are due to the magnetic part of QCD. The magnetic interactions break SU(4) and SU(2)CS symmetries of confinement explicitly and SU(2)L × SU(2)R × U(1)A - dynamically. Instanton fluctuations?

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

(Near) zero modes of Euclidean QCD and SU(2)CS-SU(4) breaking.

The Euclidean SU(2)CS generators: Σ = {γ4, iγ5γ4, −γ5} . (2) The SU(2)CS generators do not commute with the Dirac operator. These symmetries are missing in the Lagrangian. The intrinsic dynamical reason: the zero modes of the Dirac operator γµDµΨ0(x) = 0. (3) The zero mode is chiral, L or R, depending on the topological charge Q = 0. Atiyah-Singer: Q = nL − nR . At Q = 0, there is an asymmetry between L and R. Conclusion: The zero modes break explicitly SU(2)CS and SU(2NF).

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries

Outline Key questions J = 0 mesons J = 1 mesons Left-right mixing. The chiralspin SU(2)CS and SU(4) groups. J=2 mesons J = 1/2 baryons SU(2)CS and SU(4) symmetries of confinement in QCD Conclusions

Conclusions Observed on the lattice SU(4) symmetry of hadrons upon elimination of the near-zero modes is a symmetry of confinement in QCD that is due to color-electric charge-charge interaction. The magnetic interactions in QCD are responsible for generation

• f the quasi-zero modes. They break explicitly the SU(4)

symmetry of confinement and dynamically the SU(2)L × SU(2)R × U(1)A symmetry. Instantons? The hadron spectra observed in real world can be viewed as a result of splitting of the primary energy levels of the dynamical QCD string with the SU(4) symmetry by means of dynamics associated with the quasi-zero modes.

• L. Ya. Glozman

SU(2)CS and SU(2NF) hidden symmetries