Mesons in medium and QCD sum rule with dim 6 operators HyungJoo Kim Yonsei Univ. 2018/1/17 @ ASRC, JAEA, Japan references [1] Phys.Lett. B748 (2015) 352-355 [2] Phys.Rev. D93 (2016) no.1,016001 [3] Nucl.Phys. A968 (2017) 254-274 [4] Phys.Lett. B772 (2017) 194-199 collaborators Su Houng Lee Philipp Gubler Kenji Morita

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

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

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

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

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

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 )

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

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]

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

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

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

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

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:1703.01438 K.Araki, K.Suzuki, P.Gubler, M.Oka Necessity of dimension 6 gluon operators ?

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

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

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

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

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

Dim 6 gluon operators, T dependece K.Morita, Phys.Rev. D79 (2009) 011501 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. C57 1879 (1998) P.Levai, U.W.Heinz

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

Application : charmonia at finite T Using, Π OPE = C I : pert + C G 0 G 0 T + C G 2 G 2 T : dim 4 + C f G 3 f G 3 T + C G 3 G 3 T + C G 4 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

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

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 :

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

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.

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.