Exotic Hadrons in poledominated QCD sum rules based

Exotic Hadrons in pole-dominated QCD sum rules based on Phys. Rev. C 74, 045206 (2006) 古城　徹 (京大理)、林垣　新 (Gothe Univ. ) 、慈道　大介 （基研） 2007. 3. 1. ストレンジネスとエキゾティックス

相関関数の解析で用いる近似 even odd hadron (現象論) side QCD side （スペクトル関数の積分） Operator Product Expansion （OPE） q soft q ？ hard sum of local operators , … information of QCD vacuum simple parametrization

M に対する制限 知りたい低エネルギーの情報 Borel trans. OPE good OPE bad Borel window small large Borel window の範囲で, 物理量 mass & residue を unphysical な展開パラメータ M ( と物理的な Sth ) の関数として表現. , physical parameter M に依るべきではない！

Mass ( even & odd part ) We take t = 0. 9, 1. 1 for even & odd part, respectively to get the widest Borel window. even (t=0. 9) odd (t=1. 1) t = 0. 9 mass 1. 63~1. 64 Ge. V 2. 1~2. 3 Ge. V t = 1. 1 mass 1. 71~1. 73 Ge. V 2. 0~2. 2 Ge. V

Residue ( even & odd part ) even (t=0. 9) similar value with close t relative sign odd (t=1. 1) probably same state Positive parity

About KN scattering states KN ? OPE lattice: volume dep. (ex. Takahashi et al. , PRD 71, 114509 (2005) ) QSR: soft kaon theorem. (ex. S. H. Lee et al. , PLB 609, 252 (2005) ) Speculations: The response of the mass against variation of provides the some information of KN contaminations. K: Not strongly depend on N: The variation of . Ioffe’s formula from – 0. 21 Ge. V to －0. 25 Ge. V shows: KN threshold MK + MN → ~0. 18 Ge. V 　　　enhance Our observed state even: 0. 1 Ge. V reduce odd : 0. 05 Ge. V enhance Optimistically speaking, our observed state seems to be not the mimic of the KN scattering states. E

Our criterion for the Borel window Lower bound for M: ( OPE convergence ) 　　The highest dim. terms share less than 10% of whole OPE. Upper bound for M: ( Continuum suppression & Pole dominance ) 　　Pole contribution exceeds the continuum ones : usually used constraint A B OPE Sth S

Linear combination of the correlators (not operators) Set up: We prepare similar 5 quark operators mixing parameter Only difference is the chirality of the diquark operator When t ~ ± 1. 0, we can employ the concepts of Weinberg Sum Rule. similar high energy behavior (chiral sym. is good sym. in high energy) OPE ? E Cancellation of lower dimension terms OPE ? Eth Phys. Rev. Lett. 18, 507(1967) Eth E And high energy contaminations is expected to be canceled. If the low energy contributions remain enough, we can extract the information of low energy.

About mixing t OPE side phen. side (odd WSR) dim 1, 3, 5 terms cancel (even WSR) dim 0, 4 terms cancel Chiral even part: constructed from the chirality conserving terms Chiral odd part: constructed from the chirality violating terms sign for parity

spectrum approximated by OPE Our choice for t qualitative behavior of spectrum: even 2 Ge. V good cont. suppression ! _ _ PP + SS _ PP _ _ PP －SS _odd_ － SS contaminated 2 Ge. VPP still t ~ 1. 0 from 2. 5~ 4. 0 _ Ge. V _ PP _ + SS PP _ _ ~ still. PPlarge －SS !! 　_terms dim 15, 17, … dominant －SS good OPE convergence ! We can take small M 2 The choice t ~ 1. 0 provides the widest Borel window. (when we take reasonable value for Sth = 1. 9 ~ 2. 5 Ge. V) 0

OPE dependence of the mass even ( t = 0. 9 ) odd ( t = 1. 1 ) After dim. 12, the OPE dependence of the mass becomes small. For other t, the OPE dep. is quite large.

For further development It is important to clarify the meaning of factorization hypothesis more accurately to judge which physics are taken into account and are neglected in when we use the vacuum saturation. The meaning of the inclusion of complete set between the normal product of the operators is not so clear.

Appendix. Our criterions on diagram selection for OPE results Qualitative behavior of the spectral function ( approximated by OPE ) Our parameter set Our previous study for another interpolating field

How large dim. terms should be calculated ? soft hard dim 3 dim 5 + + …. Cutting loops generates the large factor． leads to the extremely slow OPE convergence． dim 12 After dim. 12, no additional large factor emerges． hard ( due to momentum conservation ) Asymptotic expansion becomes stable.

Our criterions on diagram selection for OPE 1, Loop corrections are neglected. 2, Strange quark mass is evaluated to O(ms). 3, The higher dimensional gluon condensates such as triple gluon condensates are neglected. Since they are expected to be smaller than the quark condensates entering the tree diagrams. ( This statement is usually used in the baryon sum rules. ) All other diagrams are calculated (within factorization hypothesis).

OPE Results ( up to dim. 15 )

Qualitative behavior of spectral function even part: _ _ SS PP + dim. 8 dim. 0 + 4 + dim. 8 + dim. 10 + dim. 6 + dim. 10

After the subtraction (even part) _ PP _ ― SS + dim. 8 dim. 0 + 4 + dim. 10 + dim. 6

Qualitative behavior of spectral function odd part: _ PP _ SS dim. 3 dim. 9 dim. 7 dim. 9 dim. 1 dim. 5 dim. 11 dim. 5 dim. 3 dim. 11

After the subtraction (odd part) _ PP ― _ SS + dim. 9 + dim. 7 + dim. 11 + dim. 5 + dim. 3

Our parameter set gives the error of the mass ~ 0. 08 – 0. 12 Ge. V gives the error of the mass ~ 0. 03 – 0. 05 Ge. V gives the error of the mass ~ 0. 01 – 0. 02 Ge. V

Our previous study using the SDO operator (T. K, master thesis. 2004) , , …. ( Sugiyama, Doi, Oka, Phys. Lett. B 581(2004)167 ) should be suppressed. ? OPE E KN [diquark]2 - antiquark type: well-physically motivated operator _ s spin color ud ud attractive spin, color, flavour anti-symmetric composite boson This op. does’nt have the nonrelativistic limit. Small coupling with KN ？

Pole ratio for SDO op. ( at higher dimension effect even = 2. 0 Ge. V ) M 2 small odd too small ! Too small pole ratio to contaminations ( In experience, ratio is hoped to exceed 50 % ) We failed to find the Borel window despite of inclusion of higher dim. terms of OPE (low energy correlations). Too small pole contribution? or too large high energy contaminations? To clarify this point, we need the additional treatments to suppress the high energy contaminations.

even odd 40 % sum rule’s artifact ! mass No stability mass 2. 3~2. 4 Ge. V 2. 6~2. 8 Ge. V

The idea of Weinberg sum rule S. Weinberg, Phys. Rev. Lett. 18, 507(1967) chiral partner continuum side cancellation in high energy region ( Chiral symmetry is good symmetry in high energy ) p. QCD ρ(~780 Me. V) OPE side E ρ(~1400 Me. V) E a 1 (~1200 Me. V) a 1 (~1600 Me. V) cancellation of leading OPE term

Application； ρ(~780 Me. V) p. QCD E E a 1 (~1200 Me. V) a 1 (~1600 Me. V) ρ(~1400 Me. V) simple parametrization λ m cancellation in high energy region λ’ m’ We can get the equations relating the physical parameters for ρ& A 1. solutions λ 2 = λ’ 2 　 m’ = 1. 6 ×m ~　1. 3 Ge. V