1. Mesons in medium and QCD sum rule with dim 6 operators
HyungJoo Kim
Yonsei Univ.
ASRC, JAEA, Japan
references
[1] Phys.Lett. B748 (2015)
[2] Phys.Rev. D93 (2016) no.1,016001
[3] Nucl.Phys. A968 (2017)
[4] Phys.Lett. B772 (2017)
collaborators
Su Houng Lee
Philipp Gubler
Kenji Morita

2. Out line QCD sum rule for Mesons in medium dim 6 gluon operators
Application to Charmonium at finite T

3. Mesons in medium Study, To understand, At finite T or 𝜌 vacuum
𝜌(𝑠)
𝜌(𝑠)
vacuum
At finite T or 𝜌 𝑠 𝑠
𝑚, Γ
𝑚′, Γ′
Study,
Modification of mesons in medium
ex) Mass shift, Broadening …
To understand,
- Non trivial structure of QCD vacuum
- Phase transition of quark matter
ex) Quarkonia – indicators of QGP in heavy-ion collision
Light mesons – probes of partial restoration of chiral symmetry

4. Mesons in medium - Charmonium Sequential dissociation
Sequential dissociation scenario
Lattice QCD and potential model expect that the charmonium ground state can survive up to higher T than excited states.
-𝐽/𝜓
- 𝜒 𝑐
-𝜓′
⇒ real dissociation T ? ⇒
detailed mechanism ?
⇔ deconfinement, QGP
ex)
L=0
L=1

5. QCD, non-perturbative methods
QCD at high E ⇒ asymptotic freedom ⇒ pert.
QCD at low E ⇒ confinement, non-pert, hard to describe hadrons directly from ℒ 𝑄𝐶𝐷
Various approaches to treat Non-perturbativity
Lattice QCD
NJL
ChPT
pNRQCD
Dyson-Schwinger
Ads/QCD correspondence
QCD sum rules …
hadrons

6. QCD sum rule, as a non-perturbative method
“QCD sum rule” by Shifman-Vainshtein-Zakharov in 1979
Hadrons are represented by quark currents.
Factorize short distance(asymptotic freedom) and long distance(confinement) using the OPE.
Wide applications in hadron phenomenology
Determination quark(u,d,s,c,b) masses.
Mass, Decay constant, … of mesons and baryons
successful for charmonium in vacuum.
well reproduce masses of the ground sate charmonium
electromagnetic decay width of 𝐽/Ψ
expect mass difference between 𝐽/Ψ and 𝜂 𝑐 before experiment … etc

7. QCD sum rule, Main Object
Correlation function ⟨𝑇{𝑗 𝑥 𝑗 0 }⟩
= Amplitude of the quark-pair creation and annihilation
Π 𝑞 2 ≪0
Highly virtual photon( 𝑞 2 ≪0) from hard scattering process
Quark-pair propagate at short distance
Asymptotic freedom, Almost free quark propagator
Π 𝑞 2 >0
Quark-pair propagate at long distance
Confinement, Quark-pair confined as vector mesons
Observed by dilepton decay ( √𝑞 2 = 𝑚 𝑉 )
𝑗 𝑥 : same quantum number with hadron.
ex) scalar meson : 𝑞 𝑞 , vector meson : 𝑞 𝛾 𝜇 𝑞 𝛾
𝑞 2
ex) 𝑗 𝜇 = 𝜓 𝛾 𝜇 𝜓
Π 𝑞 2 =∫𝑑𝑥 𝑒 𝑖𝑞𝑥 ⟨𝑇{ 𝑗 𝜇 𝑥 𝑗 𝜇 0 }⟩
𝑞, 𝑞 pair
𝑞 2 >0
(= 𝑚 𝑉 2 )

8. QCD sum rule, OPE part
Wilson Coefficient x Local operator
Wilson’s Operator Product Expansion At short distance ( Q 2 =− q 2 ≫1 ),
: Condensates
Local operators
Wilson coefficients = ⅹ
pert.
⇒ Non-perturbative corrections

9. QCD sum rule, Phenomenological part
By Kallen-Lehman representation, : spectral function at 𝑞 2 =𝑠>0, all hadrons can couple to j(x)| 0 Using Dispersion relation ( Q 2 =− q 2 ),
Modeling,
“1-pole + continuum”
s=Re[z]

10. QCD sum rule, final relation
Correlation function
From Π 𝑞 2 ≪0
From Π 𝑞 2 >0
quark mass, 𝛼 𝑠 , Condensates
Hadronic parameters 𝑚 𝑅 , 𝑓 𝑅
Approximate relation, but useful

11. QCD sum rule, Borel transformation
To pick out the ground resonance state, Borel Transformation 𝑩
× 𝑒 −𝑠𝜎
𝑄 2 →𝜎
√𝜎 : borel distance
larger σ probes longer distance scale

12. QCD sum rule, Borel transformation
× 𝑒 −𝑠𝜎
𝑚 𝑅 2 (𝜎) depends on 𝜎
limited σ window = Borel window
𝑚 𝑅 (𝜎)
𝑚 𝑒𝑥𝑝 σ

13. 𝐶 𝑑 𝑂 𝑑 0 → 𝐶 𝑑 𝑂 𝑑 𝑇,𝜌 QCD sum rule, in medium
For not extremely high 𝑇,𝜌
All medium effects are put into the condensates
Medium breaks Lorentz symmetry Non-scalar operators
ex) 𝐺 𝜇𝜈 𝑎 𝐺 𝜇𝜈 𝑎 𝐺 𝜇𝜈 𝑎 𝐺 𝜇𝜈 𝑎 , 𝐺 𝜇𝜎 𝑎 𝐺 𝜎𝜈 𝑎 ≠0
𝑂 𝑇,𝜌 are estimated by various approaches or Lattice QCD.
ex) Finite density
𝑂 𝜌 = 𝑂 0 +𝜌⋅ 𝑂 𝑁 : linear density approx.
Finite temperature
𝑂 𝑇 from Lattice QCD
𝐶 𝑑 𝑂 𝑑 0 → 𝐶 𝑑 𝑂 𝑑 𝑇,𝜌

14. QCD sum rule, charmonium at finite T w/ dim4
Phys.Rev.D82,054008, 2010.
K.Morita, S.H.Lee
Analysis by K.Morita et al.
𝑇 J/Ψ > 𝑇 χ c (without introducing width)
But sum rule breaks down slightly above 𝑇 𝑐 .
MEM analysis by P.Gubler et al.
Peak positions agree well with the experimental values
J/Ψ, 𝜂 𝑐 ~ 1.1 𝑇 𝑐 and χ c ~ 𝑇 𝑐
but ground , 1st excited melt almost simultaneously.
arXiv:
K.Araki, K.Suzuki, P.Gubler, M.Oka
Necessity of dimension 6 gluon operators ?

15. QCD sum rule, why dim 6 gluon operators?
pole becomes weaker, continuum becomes stronger in medium
→ larger 𝜎 to see the ground state better.
Charmonium size becomes large near Tc.
→ larger 𝜎 to probe larger distance scale.
OPE side,
OPE convergence becomes worse,
Higher dimensional operators become important.
In medium, worthwhile to include dim 6 operators.
𝑇,𝜌↑
× 𝑒 −𝑠𝜎
~ 1 𝑟 +𝜅𝑟
~ 1 𝑟

16. Dim 6 gluon operators, independent set
dim 6 gluon operators can be made with 𝐷 𝜇 and 𝐺 𝜇𝜈 .
2-scalar, 3-twist2, and 1-twist4 are independent.
(twist # : twist=symmetric+traceless, #=dimension-spin )
Renormalization of twist4
[ 𝐷 𝜇 ]=1, [ 𝐺 𝜇𝜈 ]=2

17. Dim 6 gluon operators, Wilson coefficients
Wilson coefficients of dim 6 non-scalar gluon operators
for Heavy S, P, and A currents (𝜓=𝑐,𝑏)
for Light S, P, V, and A currents (𝜓=𝑢,𝑑,𝑠)
Current structure
𝑗 𝑃 = 𝜓 𝑖 𝛾 5 𝜓
𝑗 𝑉 = 𝜓 𝛾 𝜇 𝜓
𝑗 𝑆 = 𝜓 𝜓
𝑗 𝐴 = 𝜓 ( 𝑞 𝜇 𝑞 𝜈 / 𝑞 2 − 𝑔 𝜇𝜈 ) 𝛾 𝜈 𝛾 5 𝜓
C d=6 (Q) of

18. Dim 6 gluon operators, Wilson coefficients
C G light ≠ lim m→0 C G heavy
= lim m→0 C G heavy − C Q→G heavy
C G light = lim m→0 C G heavy ?
C(Q) of Light quark(u,d,s)
C(Q) of heavy quark (c,b)
𝑞 𝑞
𝐺 𝜇𝜈 𝑎 𝐺 𝜇𝜈 𝑎
C Q→G heavy
ℎ ℎ =− 1 12 𝑚 ℎ 𝛼 𝑠 𝜋 𝐺 𝜇𝜈 𝑎 𝐺 𝜇𝜈 𝑎 +…
C G light

19. Dim 6 gluon operators, for heavy quark system
Heavy quark system is well described in the pure gauge theory,
For charmonium system,
only 5 gluon operators up to dimension 6.
dim 4
dim 6

20. Dim 6 gluon operators, T dependece
K.Morita, Phys.Rev. D79 (2009)
𝑂 𝑑=6 𝑇 =?
𝛼 𝑠 𝜋 𝐸 2 𝑇 and 𝛼 𝑠 𝜋 𝐵 2 𝑇 from Lattice QCD.
dim 4
dim 6
NPA 679 (2001) S.S.Kim, S.H.Lee
Phys.Rev. C (1998) P.Levai, U.W.Heinz

21. Dim 6 gluon operators, T dependence
Assumption
assume fields are isotropic and ignore angular correlations 𝑇 𝑇
dim 4
dim 6 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇

22. Application : charmonia at finite T
Using,
Π OPE = C I : pert
+ C G G 0 T + C G G 2 T : dim 4
+ C f G 3 f G 3 T + C G G 3 T + C G G 4 T : dim 6
Wilson coefficients of dim 6 gluon operators.
Estimated T dependence of dim 6 gluon condensates.
Investigate relative temperature behavior of charmonia near Tc.
(sequential dissociation)
Particle
Current structure
JPC
𝒎 𝒆𝒙𝒑 (GeV)
𝜂 𝑐
𝑗 𝑃 = 𝑐 𝑖 𝛾 5 𝑐
0 −+
2.98
𝐽/Ψ
𝑗 𝑉 = 𝑐 𝛾 𝜇 𝑐
1 −−
3.10
𝜒 𝑐0
𝑗 𝑆 = 𝑐 𝑐
0 ++
3.41
𝜒 𝑐1
𝑗 𝐴 = 𝑐 ( 𝑞 𝜇 𝑞 𝜈 / 𝑞 2 − 𝑔 𝜇𝜈 ) 𝛾 𝜈 𝛾 5 𝑐
1 ++
3.51

23. [GeV]
Result
𝜂 𝑐
𝐽/Ψ
𝜒 𝑐0
𝜒 𝑐1
[GeV-2]

24. Criterion to determine reliable 𝝈 window
OPE Convergence vs Ground Sate Dominance
At low 𝝈, continuum contribution is not suppressed sufficiently.
At large 𝝈, higher dimensional operator↑ ⇒Truncated OPE is failed
𝜎 𝑚𝑖𝑛 :
𝜎 𝑚𝑎𝑥 :

25. Result [GeV] [GeV-2] Psuedo Vector Scalar Axial 1.04 Tc 1.05 Tc
(narrow 𝜎)

26. Sign of 𝐆 𝟒 ? at 𝑇=1.05 𝑇 𝑐 G 4 ′ = G 4 x R 𝑠 0 ∆𝜎~0.7 ∆𝜎~0.8 - 𝐺 4
[GeV]
𝑠 0
∆𝜎~0.7
∆𝜎~0.8
- 𝐺 4
∆𝜎~0.3
∆𝜎~0.1
[GeV-2]
(-) sign seems to be better for G4.
Borel windows shrink faster for S and A.

27. Summary & Conclusions
In medium, dimension 6 operators become important.
We identify idep. dim 6 gluon operators.
We complete OPE for heavy S, P, V, and A currents up to dim 6.
We estimate T dependence of dimension 6 gluon condensate.
In our estimation, 𝐽/Ψ looks more stable than 𝜒 𝑐 , but not for 𝜂 𝑐 .
In view point of borel window, 𝐽/Ψ, 𝜂 𝑐 are more stable than 𝜒 𝑐 .
We need better estimation about T dep of dim 6 condensates.