1. Chiral Structure of Hadronic Currents
陈华星
北京航空航天大学
April 21, 武汉

2. Contents Motivations Flavor structure of tetraquark
Chiral structure of baryon
Chiral structure of tetraquark
Summary

3. 1. Motivation Conventional 𝑞 𝑞 mesons and 𝑞𝑞𝑞 baryons
QCD allows much richer hadron spectrum
Exotic hadrons:
glueballs ,
multiquark states ,
hybrids
molecular states

4. 1. Motivation Exotic in quantum numbers:
mesons : JPC=0--,0+-,1-+,2+-, etc.
baryons : S=+1 & B=1, I=5/2, etc.
Candidates: π1(1400), π1(1600) and π1(2000) with IGJPC =1-1-+
Zc(3900), Zc(4020) charged charmonium
Exotic in structure:
hadron molecule
tetraquark, pentaquark
Λ(1405), X(3872)

5. 2. Tetraquark Currents The flavor structure of meson is
SU(3)F 3⨂ 3 =1⨁8
The flavor structure of baryon is
SU(3)F 3⨂3⊗3=1⨁8⨁8⨁10

6. 2. Tetraquark Currents Tetraquark is complicated:
For each state, there may exist more than one currents.
We try to do a systematical study on tetraquark currents.
SU(3)F
Flavor structure

7. 2. Tetraquark Currents Meson Currents Scalar JP=0+ Vector JP=1-
Tensor JP=1+-
Axial-vector JP=1+
Pseudo-scalar JP=0-
A, B are the flavor indices; a is the color index. By adding δAB and λAB, we can obtain singlet and octet, respectively.

8. Diquark Currents

9. 2. Tetraquark Currents
We find that there are five independent currents for σ(600) JPC=0++ states:

10. Fierz Transformations

11. Chiral Transformation

12. 3. Chiral structure of Baryon
Chiral structure of quark:
𝑞= 𝑞 𝐿 + 𝑞 𝑅 = 1− 𝛾 5 2 𝑞+ 1+ 𝛾 5 2 𝑞 ,1 ⨁ 1,3
Chiral structure of meson
3,1 ⨁ 1,3 ⨂ 3 ,1 ⨁ 1, 3 = 1,1
⨁[ 8,1 ⨁ 1,8 ]
⨁[ 3 ,3 ⨁ 3, 3 ] 𝟏 𝟖
𝟏⨁𝟖
𝑞 𝛾 5 𝑞= 𝑞 𝐿 𝑞 𝑅 − 𝑞 𝑅 𝑞 𝐿 ∈[ 3 ,3 ⨁ 3, 3 ]
𝑞 𝛾 𝜇 𝑞= 𝑞 𝐿 𝛾 𝜇 𝑞 𝐿 + 𝑞 𝑅 𝛾 𝜇 𝑞 𝑅 ∈ 1,1 ,[ 8,1 ⨁ 1,8 ]

13. 3. Chiral structure of Baryon
3,1 ⨁ 1,3 3 = 1,1
⨁ 8,1 ⨁ 1,8
⨁[ 3 ,3 ⨁ 3, 3 ]
⨁[ 6,3 ⨁ 3,6 ]
⨁[ 10,1 ⨁ 1,10 ] 𝟏 𝟖
𝟏⨁𝟖
𝟖⨁10 𝟏𝟎

14. 3. Chiral structure of Baryon
We investigate chiral properties of local fields of baryons consisting of three quarks with flavor SU(3) symmetry. We construct explicitly independent local three-quark fields:
Λ= 𝜖 𝑎𝑏𝑐 𝜖 𝐴𝐵𝐶 ( 𝑞 𝐴 𝑎𝑇 𝐶 𝑞 𝐵 𝑏 ) 𝛾 5 𝑞 𝐶 𝑐 ,
𝑁 1 𝑁 = 𝜖 𝑎𝑏𝑐 𝜖 𝐴𝐵𝐷 𝜆 𝑁 𝐷𝐶 ( 𝑞 𝐴 𝑎𝑇 𝐶 𝑞 𝐵 𝑏 ) 𝛾 5 𝑞 𝐶 𝑐 ,
𝑁 2 𝑁 = 𝜖 𝑎𝑏𝑐 𝜖 𝐴𝐵𝐷 𝜆 𝑁 𝐷𝐶 ( 𝑞 𝐴 𝑎𝑇 𝐶 𝛾 5 𝑞 𝐵 𝑏 ) 𝑞 𝐶 𝑐 ,
𝑁 𝜇 𝑁 = 𝜖 𝑎𝑏𝑐 𝜖 𝐴𝐵𝐷 𝜆 𝑁 𝐷𝐶 ( 𝑞 𝐴 𝑎𝑇 𝐶 𝛾 𝜇 𝛾 5 𝑞 𝐵 𝑏 ) 𝛾 5 𝑞 𝐶 𝑐 ,
Δ 𝜇 𝑃 = 𝜖 𝑎𝑏𝑐 𝑆 𝑃 𝐴𝐵𝐶 ( 𝑞 𝐴 𝑎𝑇 𝐶 𝛾 𝜇 𝑞 𝐵 𝑏 ) 𝑞 𝐶 𝑐 ,
Δ 𝜇𝜈 𝑃 = 𝜖 𝑎𝑏𝑐 𝑆 𝑃 𝐴𝐵𝐶 ( 𝑞 𝐴 𝑎𝑇 𝐶 𝜎 𝜇𝜈 𝑞 𝐵 𝑏 ) 𝛾 5 𝑞 𝐶 𝑐 ,
where a,b,c are color indices, A,B,C are flavor indices, 𝑆 𝑃 𝐴𝐵𝐷 are totally symmetric tensors.

15. 3. Chiral structure of Baryon
We can perform chiral transformations, and verify:
𝑁 1 𝑁 + 𝑁 1 𝑁 ∈(8,1)⨁(1,8),
Λ, 𝑁 1 𝑁 − 𝑁 1 𝑁 ∈( 3 ,3)⨁(3, 3 ),
𝑁 𝜇 𝑁 , Δ 𝜇 𝑃 ∈(6,3)⨁(3,6),
Δ 𝜇𝜈 𝑃 ∈(10,1)⨁(1,10).
We can calculate their axial charges, such as:

16. 3. Chiral structure of Baryon
We construct all SUL(3)xSUR(3) chirally invariant non-derivative one-pseudoscalar-meson-baryon interactions, i.e., all chiral-singlet Lagrangians made by baryons and 3 ,3 ⨁ 3, 3 mesons:

17. 3. Chiral structure of Baryon
We construct all SUL(3)xSUR(3) chirally invariant non-derivative one-vector-meson-baryon interactions, i.e., all chiral-singlet Lagrangians made by baryons and 8,1 ⨁ 1,8 mesons:

18. 4. Chiral structure of Tetraquark
Chiral structure of quark:
𝑞= 𝑞 𝐿 + 𝑞 𝑅 = 1− 𝛾 5 2 𝑞+ 1+ 𝛾 5 2 𝑞 ,1 ⨁ 1,3
Chiral structure of tetraquark

20. 4. Chiral structure of Tetraquark
We systematically studied the chiral structure of local scalar and pesudoscalar tetraquark currents that belong to the “non-exotic” 3 ,3 ⨁ 3, 3 tetraquark chiral multiplets.
We find that they transform differently from 𝑞 𝑞 mesons under the 𝑈 𝐴 1 chiral transformations, but transform in the same way as 𝑞 𝑞 mesons under 𝑈 𝑉 1 , 𝑆𝑈 𝑉 3 and 𝑆𝑈 𝐴 3 chiral transformations.
The different 𝑈 𝐴 1 chiral transformation may be reasons for the 𝑈 𝐴 1 anomaly.

21. 4. Chiral structure of Tetraquark
We investigate the chiral structure of local vector and axial-vector tetraquark currents that belong to the “non-exotic” 8,1 ⨁ 1,8 tetraquark chiral multiplets. They transform in the same way as 𝑞 𝑞 mesons:
We find that there is a one to one correspondence among all the isovector vector and axial-vector local tetraquark currents of quantum numbers
𝐼 𝐺 𝐽 𝑃𝐶 = 1 − 1 − 𝜋 1 (1600)
𝐼 𝐺 𝐽 𝑃𝐶 = − − 𝜌(1570)
𝐼 𝐺 𝐽 𝑃𝐶 = 1 − 𝑎 1 (1640)
𝐼 𝐺 𝐽 𝑃𝐶 = − ? 𝑏 → 𝑓 0 𝜋,𝜌𝜌…

22. 5. Summary
We investigate the chiral structure of local baryon and non-local baryon fields, and study their chiral transformation properties.
We investigate the chiral structure of local scalar and pseudoscalar tetraquark currents, and study their chiral transformation properties.
We investigate the chiral structure of local vector and axial-vector tetraquark currents, and study their chiral transformation properties.

23. Thank you very much!

24. 4. Baryon masses
We assume that their masses originate from three different sources:
1. bare mass term
2. electromagnetic terms
3. spontaneous chiral symmetry breaking terms