複素Borel和則を用いた有限温度でのクォーコニウムの解析
荒木賢志（東工大）、鈴木渓（東工大）、Philipp Gubler （理研）、岡真（東工大）

クォーコニウム抑制
クォーコニウム抑制：QGP中でクォーコニウムが消失する現象

cRRRRcR



この消失温度が理論的興味の対象になっているR﹅

﹅

﹅﹅

3
近年では、励起状態が基底状態に比べてより強J. Matthew Durham (for the PHENIX Collaboration) / Nuclear Physics A 00 (2014) 1–4

く抑制されるという実験結果が報告されている

J/ψ
ψ’Figure 5. Relative modification of (2s) and J/ as a function ofFigure
4.
Thenuclear
modification
factorRforJ/and(2s)propertime spent in the nucleus, for PHENIX, NA50, and E866dA
A. Adare et al. (PHENIX Collaboration), PRL111 (2013)
202301
production at midrapidity.
[4, 12, 13]. A model which describes the fixed target data breaks
down for the PHENIX data [11].

基底状態だけでなく励起状態の消失温度も求めたい

両者に違いはあるのか（？）

方法
QCD和則
（スペクトル関数の積分値の情報）
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]

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P. Gubler and M. Oka, Prog. Theor. Phys. 124, 995 (2010).
M. Asakawa, T. Hatsuda, and Y. Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001).
K. Ohtani, P. Gubler, and M. Oka, Eur. Phys. J. A 47, 114 (2011).
K. Ohtani, P. Gubler, and M. Oka, Phys. Rev. D 87, 034027 (2013).
P. Gubler, K. Morita, and M. Oka, Phys. Rev. Lett. 107, 092003 (2011).
K. Suzuki, P. Gubler, K. Morita, and M. Oka, Nucl. Phys. A897 (2013).
B. L.Ioﬀe and K. N. Zyablyuk, Nucl. Phys A687, 437 (2001).
S. Borsanyi et al., Phys.Rev. D 88, 014513 (2013).
P. Colangelo and A. Khodjamirian, At the Frontier of Particle Physics/ Handbook of QCD 3, 1495
(2000).
[13] Particle Data Group, J. Beringer et al., Phys. Rev. D 86, 010001 (2012).
[14] D. B. Leinweber, Ann. Phys. 254, 328 (1997).
M.[15]Asakawa,T. Hatsuda,
Y. Suppl.
Nakahara,Prog.
Part. Nucl. Phys. 46, 459 (2001)
S. Bethke,
Nucl. Phys.andProc.
234, 229(2013).
[16] E. V. Shuryak, Rev. Mod. Phys. 65, 1 (1993).
[17] K. Kodaira, An Introduction to Calculus (Kaiseki Nyumon I) (Iwanami-Shoten, 2003).
[18] K.-J. Araki, K. Ohtani, P. Gubler, and M. Oka, PTEP 2014, 073B03 (2014), 1403.6299.

MEM(積分値の情報からスペクトル関数を決定する方法)	 .	

スペクトル関数

これの温度依存性を見る

B̂[−q2 ] (q ) ln −!

µ2= −k!(M ) ,1 "k1
1−s/(M 2 )B̂[−q2 ]=e.
s − q2(k − 1)! (M 2 )kThese all suggest that the Borel transformation of the analytically continued of OPE equals
that of the original OPE with a complex valued Borel mass. We can thus set

複素Borel和則B̂[|z|] Π(z) = GOPE (M 2 eiθ ),(9)

Borel
sum
whereGOPErules(M 2 ) is のパラメータを複素数に一般化できる
defined as GOPE (M 2 ) ≡ B̂[−q ] ΠOPE (q 2 ).
2Araki, K. Ohtani, P. Gubler, M. Oka, PTEP, 073B03(2014)
Finally, weK.J.
obtain# ∞1−s/M2
2GOPE (M2 ) =e
ρ(s)ds(Re[M] > 0)
2M 0(10)where M2 ≡ M 2 eiθ . This form of the complex plane QCD sum rule is nothing but thewell known real valued Borel sum rule, in which the 積分核の振る舞い
real Borel mass is replaced by its
パラメータ領域
complex analogue, M2 . Therefore, the complex Borel plane sum rules is found to be asimple generalization of the ordinary Borel sum rules to the complex Borel mass plane andof course includes the latter at θ = 0.

2.4.

複素Borel和則sProperties of the CBSR

Although the CBSR of Eq.(10) looks similar to its real counterpart, its content is quite
diﬀerent. Since Eq.(10) is complex valued, it simultaneously gives two sum rules which canbe obtained from its real and imaginary part. Specifically, we have⎧# ∞⎪⎨Re[GOPE (M2 )] =K R (M2 ; s)ρ(s)ds,
従来のBorel和則#0 ∞(11)
OPE
2I2⎪⎩Im[G(M )] =K (M ; s)ρ(s)ds,0

where K R (M2 ; s) and K I (M2 ; s) are defined as( 1)2
K R (M2 ; s) ≡ Re
e−s/Ms(J/Ψ)

解析結果
真空@2.97GeV

有限温度

spectral function
default model

0.08

0.06l(t)@3.71GeV

0.04

0.02

012J/ψ(3097)

34
t [GeV]5678

ψ’(3686)