チャーム・ボトムのエキゾチックハドロン物理の最近の発展

安井 繁宏∈ 佐々木グループ ⊂ WPI-SKCM2 ⊂ 広島大学

瀬戸内サマーインスティチュートSSI2023@広島大学, 20-22 Sep. 2023

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内容
1. イントロダクション
1.1 ハドロンの基本的性質
1.2 なぜ重いハドロンを研究するのか？
2. 重いクォークのスピン対称性と有効理論
2.1 スピン対称性とハドロンスペクトロスコピー
2.2 重いクォークの有効理論
2.3 重いハドロンの有効理論
3. 重いエキゾチックハドロン -ハドロン相互作用の観点から3.1 なぜエキゾチックハドロンが面白いのか？
3.2 チャームメソン：X, Y, Zc
3.3 ボトムメソン：Zb
3.4 チャームペンタクォーク：Pc, Pcs
3.5 ダブルチャームメソン：Tcc
3.6 フルチャームメソン：Xcc
3.7 反応論ー重イオン衝突によるエキゾチックハドロン生成ー

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内容
1. イントロダクション
1.1 ハドロンの基本的性質
1.2 なぜ重いハドロンを研究するのか？
2. 重いクォークのスピン対称性と有効理論
2.1 スピン対称性とハドロンスペクトロスコピー
2.2 重いクォークの有効理論
2.3 重いハドロンの有効理論
3. 重いエキゾチックハドロン -ハドロン相互作用の観点から3.1 なぜエキゾチックハドロンが面白いのか？
3.2 チャームメソン：X, Y, Zc
3.3 ボトムメソン：Zb
3.4 チャームペンタクォーク：Pc, Pcs
3.5 ダブルチャームメソン：Tcc
3.6 フルチャームメソン：Xcc
3.7 反応論ー重イオン衝突によるエキゾチックハドロン生成ー

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2. Theoretical framework for heavy hadrons

Heavy-lightmeson42. Theoretical framework for heavy hadronsD*DB*
Bmesonmesonmesonmeson

D*DB*
Bmesonmesonmesonmeson52. Theoretical framework for heavy hadrons
JP=0JP=1-

JP=0JP=1-

D
*
DB
*
B

meson

meson

meson

meson

D*
DB
*
B

meson

meson

meson

meson
6

2. Theoretical framework for heavy hadrons
JP=0JP=1-

JP=0JP=1-

D
*
DB
*
B

meson (qc)meson (qc)meson (qb)meson (qb)

D
*
DB
*
B

meson (qc)meson (qc)meson (qb)meson (qb)7

2. Theoretical framework for heavy hadrons
Heavy quark effective theory (HQET)
Heavy Quark Physics
A. V. Manohar and M. W. Wise

Cambridge University Press 2000

8

2. Theoretical framework for heavy hadrons
Heavy quark effective theory (HQET)

9

For charm ed baryons the addition of the c quark to the light quarks extends the ﬂavor symmetry
to SU (4) f . D ue to the large mass of the c quark, this symm etry is much more strongly broken
than the SU (3) f of the three light quarks. N evertheless, the SU (4) f representation is still usefulfor bookkeeping purposes. W ith the additive charm quantum number C the baryons are classiﬁ ed
in a 3-dimensional representation with the three coordinates ( I z , Y , C ). F igure 15.5 shows theSU (4) f weight diagram s.

2. Theoretical framework for heavy hadrons
SU(4) flavor symmetry？？？

F igure 15.5: SU (4) f multiplets of ground state baryons m ade of u, d, s, and c quarks. (a) T hespin 12 20-plet extends the charmless SU (3) f octet to C = 1,2; (b) the spin 32 20-plet extends theSU (3) f decuplet to C = 1, 2, 3.
W ith four quarks the 64 possible conﬁ gurations decom pose into

H. Georgi,Particle
Physics(15.29)(1999, 2019)4 ¢ 4 ¢ 4Lie= 4̄A Algebras
ü 20S ü 20M S üin20MA,(for a review on SU(N ) symmetries see e.g. [69]). T he subscripts S and A refer to the sym metry
and antisymm etry properties of the ﬂavor wave functions. T he ﬂavor symmetric 20S multiplet,
associated with spin-32 baryons, contains the charmless SU (3) f decuplet at the bottom level. T he20M S and 20M A multiplets correspond to the m ixed symmetric and mixed antisymm etric ﬂavour
wave functions of the spin-12 baryons, with the charmless octet baryons at the bottom level. T here
are two dsc and two usc spin-12 states, labeled …c0 , …cÕ0 and …c+ , …cÕ+ . T his is because one of theqq pairs can have spin 1 (sym metric) or spin 0 (antisymmetric), giving both the total spin j = 12with the third quark (see also F ig. 15.6 below).
11th A ugust, 2022

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2. Theoretical framework for heavy hadrons

・Heavy mass of charm hadron
mc=1.3 GeV, mb=4.7 GeV >> ΛQCD=a few 100MeV・Heavy Quark Symmetry (HQS)
Hadron spin = light quark spin x heavy quark spin11

2. Theoretical framework for heavy hadrons
Heavy quark effective theory
Separating QCD Lagrangian to heavy quark part and light quark partWe focus on the heavy part.

gluons + light quarks

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2. Theoretical framework for heavy hadrons
Heavy quark effective theory① On-mass-shell part and off-mass-shell partyThe heavy quarkis at rest!

on-mass-shell condition

heavy quark(mass mQ)x

on-mass off-massshellshelloff-mass-shell condition② Effective field heavy quark (positive-energy and negative-energy)
Q:

Qbar:

Projection operator

We consider only positive-energy partin the effective theory.
rest frame

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2. Theoretical framework for heavy hadrons
Heavy quark effective theory

3 steps to get HQ effective Lagrangian① Expressing Lheavy in terms of Qv (heavy quark) and Qv (anti heavy quark).
no mass term(mQ was absorbed in Qv)② Eliminating Qv (anti heavy quark) by using equation of motion.

③ We get HQ effective Lagrangian!

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2. Theoretical framework for heavy hadrons
Heavy quark effective theory
HQ effective Lagrangian (Nh heavy flavors)

3 Key Points

1. Velocity rearrangement
2. Radiative corrections
3. Heavy quark spin symmetry (mQ→∞)15

2. Theoretical framework for heavy hadrons
Heavy quark effective theory
HQ effective Lagrangian (𝑁ℎ heavy flavors)

specific 𝒗𝝁1. Velocity rearrangement① LO: Superposition of various vμtotal② Change of vμ and kμ③ Condition of v2 = 1yy’heavy quark(mass mQ)

xx’④ Transformation of Qv
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2. Theoretical framework for heavy hadrons
Heavy quark effective theory
HQ effective Lagrangian (Nh heavy flavors)

specific 𝒗𝝁1. Velocity rearrangement① LO: Superposition of various vμ⑤ Transformation of Lagrangiantotal② Change of vμ and kμ③ Condition of v2 = 1④ Transformation of Qv⑥ Invariance of HQ effective Lagrangian: invariant for changing vμWe can choose any v-frame.

172. Theoretical framework for heavy hadrons
Heavy quark effective theory
HQ effective Lagrangian (Nh heavy flavors)

2. Radiative corrections
Tree level:

Leading logarithmic approximation:

Figure: M. Neubert, Subnucl. Ser. 34, 98 (1997)(arXiv:9610266 [hep-ph])18

2. Theoretical framework for heavy hadrons
Heavy quark effective theory
HQ effective Lagrangian (Nh heavy flavors)

3. Heavy quark spin symmetry (mQ→∞)① Leading order of 1/mQ

Where is γμ matrix?

② Heavy quark spin (HQS) transformation
Lagrangian L0 is invariant for HQS:
③ HQS is a conserved quantity.

Qor

spin updown

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2. Theoretical framework for heavy hadrons
Heavy quark symmetry

Heavy Quark Symmetry(HQS)

Q

Correspondence

massorspin upHeavy Hadron
Mass Spectrum

Energydown

decay width20

2. Theoretical framework for heavy hadrons
Heavy quark symmetry“Qqbar“ meson
spin

Qjq
S =1/2

HQS (S=1/2)
conserved

Spin J
conserved

Light spin jconserved!

→ Mass degeneracy for J=j±1/2
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2. Theoretical framework for heavy hadrons
Heavy quark symmetry“Qqq” baryon
spin

Qjq

qS =1/2

HQS (S=1/2)
conserved

Spin J
conserved

Light spin jconserved!

→ Mass degeneracy for J=j±1/2
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2. Theoretical framework for heavy hadrons
Heavy quark symmetry

ρ770 MeV

630 MeV

J=1J=0(j=1/2)→ HQS doublet @large mK*
890 MeV390 MeVP(*)=(P,P*)

D*2010 MeV
140 MeV1870 MeV

DB*5325 MeV
45 MeV

QQq

5280 MeV

BQq

500 MeVK

140 MeVπ23

2. Theoretical framework for heavy hadrons
Heavy quark symmetry→ HQS doublet

J = 3/2
J = 1/2

ΣQ(*)=(ΣQ,ΣQ*)

(j=1)

Δ
1230 MeVΣ*1385 MeV

290 MeV

195 MeVΣ940 MeVN

1190 MeV
1115 MeVΛJ = 1/2
(j=0)

Σ c*

2520 MeV
65 MeV
2455 MeV

Σc2286 MeV

Λc@large mQ

Σb*5830 MeV

Σb

20 MeV

5810 MeV
Qqq5619 MeVΛb→ HQS singlet @large m(ΛQ)

Qqq

Qqq

Q24

2. Theoretical framework for heavy hadrons
Heavy quark symmetry“Qbarq” meson + “qqq” baryonj

HQS (S=1/2)
conserved

spin

Qqq

S =1/2π,ρ,ω,...

qq

Spin J
conserved

Light spin jconserved!

→ Mass degeneracy for J=j±1/2
Example: Single-heavy hadronic molecules25

2. Theoretical framework for heavy hadrons
Heavy quark symmetry
Heavy hadron mass spectrum (example)

HQS doublet

light spin j’total spin J’=j’±1/2j’=j+L
S =1/2

HQS doublet

masslight spin j
total spin J=j±1/2

HQS singlet
Deacy width:

N. Isgur and M. B. Wise, Phys. Rev. Lett. 66, 1130 (1991)

Cf. NLO of 1/mQ: S. Yasui, Phys. Rev. D91, 014031 (2015)26

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory
Heavy Quark Effective Theory

Heavy Hadron Effective Theory
G. Burdman, J.F. Donoghue, Phys. Lett. B280, 287 (1992)
M.B. Wise, Phys. rev. D45, 2188 (1992)
T.-M Yan et al., Phys. Rev. D46, 1148 (1992)

Qheavy quark

qqqq

Heavy Hadron
gluon

π,ρ,ω,...

qqq

Cf. “Phenomenology of heavy meson chiral Lagrangians” (review)
R. Casalbuoni et al., Phys. Rep. 281, 45 (1997)

q27

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory

Let us construct “heavy hadron effective theory”

based on the heavy quark effective theory!
HHET/HQET
HQS

Qq28

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory

Let us construct “heavy hadron effective theory”

based on the heavy quark effective theory!
HHET/HQET
HQS

Qq

Heavy-light meson: Qqbar:
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2. Theoretical framework for heavy hadrons
Heavy hadron effective theory
Heavy-light meson: Qqbar:① Constructing effective fieldat rest frame
spin upspin downQ:qbar:
4 × 4 dim. Dirac matrix form

A. F. Falk, H. Georgi, B. Grinstein, andM. B. Wise, Nucl. Phys. B343, 1 (1990)
A. F. Falk, Nucl. Phys. B378, 79 (1992)→ See next pagefor Lorentz covariancerest framev-frame

spin 0:spin 1:polarization vector:30

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory
SU(2) spin doublet representation of particle and antiparticle (review)① SU(2) operator② Special property③ Fermion (2) and antifermion (2*)④ Unitary transformation

2≠2* ???

⑤ Transformation of 2* representation

2:= 2*:

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2. Theoretical framework for heavy hadrons
Heavy hadron effective theory① Constructing effective field (cont’d)spin 0:(pseudoscalar)spin 1:(vector)

Symmetries of effective fields: U(Nf)V: SU(2) HQS“Superposed” field of spin 0 and 132

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory① Constructing effective field (cont’d)Note: normalization of the state

Original state :HHET state :Q. How many particles do exist in volume 𝑉？

Not 1 even for“one-particle”
state!

2𝐸𝑝 (2𝑣 0 ) particles per unit volume← Important(Don't forget this...)This FACT is important for constructing the inter-particle potential.

332. Theoretical framework for heavy hadrons
Heavy hadron effective theory② Constructing effective Lagrangian (leading order of mQ→∞)chiral covariant derivative:

Non-linear chiral transformation

invariant under HQS
and chiral symmetryπNon-linear rep. of π field:

Vector current:

Axial-vector current:(even # of π)(odd # of π)

Example of vertex structure (axial-vector coupling)

D*timeDπ
D

D*ππ
D*

We will see details later.

D*34

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory③ Including NLO of 1/M
Lorentz boost: vμ→Vμ

Cf. “velocity-rearrangement” O(1/mQ) in HQETyy’

heavy hadron(mass M)xx’
Lorentz boost: vμ→Vμ up to O(1/M)

New effective field including O(1/M)

M. E. Luke, A. V. Manohar, Phys. Lett. B286, 348 (1992)
N. Kitazawa, T. Kurimoto, Phys. Lett. B323, 65 (1994) 35
S. Yasui, K. Sudoh, Phys. Rev. C88,015201(2013)

2. Theoretical framework for heavy hadrons
Heavy hadron effective theory③ Including NLO of 1/M (cont’d)“covariant” transformation

Lorentz boost:✓ construction of interaction term
invariant under velocity rearrangement

Possible form of current:: Dirac matrices(S,V,T,A,P)

Example of axial-vector current
LO:LO: unknown coupling

NLO:

NLO: unknown couplings← HQS breaking term (g2)36

2. Theoretical framework for heavy hadrons
Experiment ⇄ TheoryHadron Spectroscopy
Production✔️Massmass

✔️
Decay
mass

HQ
S

2 0 1 7 /0 2 /2 7 1 5 : 5 6
2 0 1 7 /0 2 /2 7 1 5 : 5 62001
177 /0/0 22 /2/2 77 115
5 :: 556
6237

Do you have questions?

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