Hadron Spectroscopy: Probing QCD Dynamics · Hadron Spectroscopy: Probing QCD Dynamics Estia...

Estia Eichten Hadron Physics Workshop@KEK, Tsukuba, JAPAN January 16, 20131

Hadron Spectroscopy:Probing QCD Dynamics

Estia Eichten (Fermilab)


• QCD Dynamics

• Heavy Quark-AntiQuark Systems

– Narrow States Below Threshold

• Color binding force

• Spin dependent interactions

– Transitions - Probing structure

• States Above Threshold

– New degrees of freedom

– Surprising hadronic transitions

• Hybrid spectrum

• Summary

2

Outline


• Matter fields interact with massless spin one gauge bosons. (The Gauge Principle)

– Global Symmetry ⇒ Gauged Local Symmetry

– QED: Charged particles ψ(matter fields) interact with the photon Aμ(gauge particle)

• QCD - SU(3) gauge interactions: • color octet gluons (g) and color triplet quarks (u,d,s,c,b,t)(L,R)

• Confinement -> physical states color singlets.

• Gluons are massless but unlike the photon they have charges . (Aμ -> ta A μa )

Hence gluons have self interactions. L = - (1/4g2) (Fμν) (Fμν) with Fμν = -i [∂μ + ig Aμ(x), ∂ν + ig Aν(x) ]

– Asymptotic freedom. The interactions become weaker at shorter distances (higher momentum). Perturbative QCD calculations are reliable.

3

_

ψ(x)-> exp{i eΛ(x)} ψ(x) ≡ G ψ(x) ; Aμ(x) -> GAμ(x)G-1 - (1/ie)[∂μ, G]G-1

L = ψ(i γ⋅D - M) ψ - (1/4g2) (Fμν) (Fμν)

with Dμ = [∂μ + ig Aμ(x)] and (Fμν) = -i[Dμ, Dν]

_ψ-> exp{iQ} ψ

L = ψ(i γ⋅∂ - M) ψ_

QCD Dynamics


– -

– Conversely the color coupling become stronger as the momentum decreases (distances increase) until at some scale (ΛQCD ⋍ 0.22 GeV) the dynamics is nonperturbative.

– Confinement - All physical states are color singlets.

– QCD has a rich spectrum of physical states: Color singlet states consisting of light quarks (q), gluons (G) and heavy quarks (Q).

4

QCD Dynamics

_


• Hadron spectroscopy from a modern viewpoint

– Parameters:

• g --> sets mass scale of dynamics (proton mass or lightest glueball mass)

• N (SU(N)) --> Large N limit - AdS/CFT correspondence

• quark masses: mu, md, ms; mc, mb, mt --> light quarks chiral symmetry; heavy quark symmetry

– Counting degress of freedom: heavy quark (Q), light quarks (q) and gluonic (G)

• simplest systems: QQ, QQQ

• gluonic excitations: QQG, QQGG, ....• one light quark: Qq, QQq• Glueballs: GG, GGG, ...• more than one light quark: qq, qqq, qqG, QQqq, ...• ....

– Lattice QCD allows the studies in different limits of the parameters

• pure QCD - glueball spectrum• no light quarks - only one heavy quark and gluons• different gauge groups and light quarks representations

5

QCD Dynamics

_

_

_

_

__

_

_


QCD Dynamics

• Glueball spectrum - LQCD

– low-lying states; 0++, 2++, 0-+, 1+-, ...

– BESIII looking in ψ’-> Ɣ + X

-> Ɣ + (π+ π- η’ )

– mixing with light quark states makes identification difficult.

• In order to understand QCD dynamics best to start with the simpliest hadrons.

6

[arXiv:1208.1858]


QCD with heavy quarks

• QCD interactions simplify for heavy quarks: mQ >> ΛQCD

• The typical momentum transfers to heavy quarks are small relative to mQ. The heavy quark remains near mass shell p0 ~ mQ. So one can redefine the field using

• Then Ψ is two component field familiar in the nonrelativistic theory.

Similiarly the antiquark is the projection (near the mass shell) of the charge conjugate field ψc. It is treated as independent field.

• One can eliminate the lower components of ψ using equations of motion to

rewrite the interactions of a heavy quark with QCD in a form that has a well-defined limit as mQ -> ∞

• This form is appropriate for both heavy-light (Qq) systems and quarkonium heavy quark-antiquark systems (QQ)

7

__

• QCD Hamiltonian for heavy quark with mQ ≫ ΛQCD (in Coulomb gauge)

• Heavy light systems (Qq) or (Qqq): HQET – Expansion parameter (ΛQCD/mQ)

– Leading behaviour independent of heavy quark mass and spin. (Heavy Quark Symmetry)

• Heavy Quark -Antiquark systems (QQ’): Nonrelativistic QCD (NRQCD)

– Expansion parameter (vQ=pQ/mQ) (v2cc ≃ 0.26 v2bb ≃0.12)

– Dynamics of heavy quarks independent of spin in leading order.



Kinetic Potential

relativistic corrections

_

_

_ _

• Nonrelativistic Quantum Mechanics: Born-Oppenheimer approximation

• Relativistic corrections treated perturbatively


Spectroscopic notation of atoms

�QQ(⌃r) =unl(r)

rYlm(�,⇥)

9


• Threshold: (QQ) -> (Qq) + (qQ) strong decays allowed. (Γ≈ 100 MeV)

– (cc) -> D + D and (bb) -> B + B– (bc) -> B + D

• Below threshold – Narrow states allow precise experimental probes of

the subtle nature of QCD. (Γ≲ 1 MeV)

– Consistency between (bb) and (cc) systems validates NRQCD approach. At LHCb the (bc) system can also be studied.

– NRQCD approach is a spectacular success• masses and spin splittings (pot -> LQCD)• direct decays (pQCD)• EM transitions (ME)• hadronic transitions (QCDME)

– Lattice QCD can provide nonperturbative elements

• Well described by potential models (~30 yrs)


Narrow States Below Threshold

0.2 0.4 0.6 0.8 1 1.2 1.4

R (fm)

-1.00

-0.50

0.00

0.50

1.00

1.50

E(R

) (G

eV

)

1S

!

4S

!

1P

!

3S

!

2S

!

"1S

"1P

"2S

"1D

"3S

"4S

"2P

"5S

!＜R2＞

cc bb

1D

!

CornellPotential Model

__

_

_

_

_ _ _

_

_

_ _ _

• Static Energy (Potential)

• At short distance– (a) dominate antiscreening term for gluons

(negative contribution β-function)

– (b) and (c) usual dielectric screening terms for gluons ( ) and nf light quarks ( )


Static Energy

+

+

+

+

Q

Q_ ↓k

(a)

(b)

(c)

1

-4 Nc Y

1/3 Nc Y

2/3 nf Y

X

Y ≣ αs(μ2)/4π ln(k2/μ2)

⇒[-β0]

• Lattice calculations of color binding force

• In good agreement with potential model expectations.

• Short distance behaviour shows running of αs (1/r):


Static EnergyB. Leder and F. Knechtli [Alpha Collaboration] [arXiv:1112.1245]

r0 ≃ 0.5 fm

Cornell

Richardson

Q <------------- R -------------> Q_


-0.8

-0.6

-0.4

-0.2

0.0

V1'(

r) [

GeV

2]

0.70.60.50.40.30.20.10.0

r [fm]

β = 6.0 β = 6.3

Fig. 8. Spin-orbit potential V !1(r) at ! = 6.0 and ! = 6.3. The dotted line is the fit

curve Eq. (3.10), applied to the data of ! = 6.0.

0.8

0.6

0.4

0.2

0.0

V2'(

r) [

GeV

2]

0.70.60.50.40.30.20.10.0

r [fm]

β = 6.0 β = 6.3

Fig. 9. Spin-orbit potential V !2(r) at ! = 6.0 and ! = 6.3. The dotted line is the fit

curve Eq. (3.8), applied to the data of ! = 6.0.

We start our discussion with the spin-orbit potentials V !1(r) and V !

2(r).For V !

1(r), we find that the potential is negative and almost constant at r !0.25 fm (see Fig. 8). This behavior clearly contradicts Eq. (3.9) and suggeststhe existence of the Lorentz-scalar piece in the interaction kernel in termsof the BS equation. Our data at ! = 6.3 suggest that one cannot exclude adeviation from a constant at small distances, an observation which was alsomade by Bali et al. [29,30]. For V !

2(r), we see a decreasing behavior with r (seeFig. 9), which is not restricted to the short range, but rather seems to have afinite tail up to intermediate distances.

Before discussing the functional form of V !1(r) and V !

2(r) quantitatively, we

20

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

V3(r

) [G

eV

3]

0.70.60.50.40.30.20.10.0

r [fm]

β = 6.0 β = 6.3

Fig. 12. Spin-spin (tensor) potential V3(r) at ! = 6.0 and ! = 6.3. The dotted lineis the fit curve Eq. (3.11), applied to the data of ! = 6.0.

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

V4(r

) [G

eV

3]

0.70.60.50.40.30.20.10.0

r [fm]

β = 6.0 β = 6.3

Fig. 13. Spin-spin potential V4(r) at ! = 6.0 and ! = 6.3. The dotted line is the fitcurve Eq. (3.12), applied to the data of ! = 6.0.

V3(r) (see Fig. 12) if the ansatz motivated by one-gluon-exchange in Eq. (3.9)is appropriate. The fit to this function yields the coe!cient c = 0.214(2) with!2

min/Ndf = 3.7. This value of !2min/Ndf is relatively large and the result for c

is 28 % smaller than the Coulombic coe!cient in V !0(r). A better fit can be

achieved using an ansatz in which the power of 1/r is left as a free parameter,i.e.

V3;fit(r) =3c!

rp. (3.11)

23

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

V3(r

) [G

eV

3]

0.70.60.50.40.30.20.10.0

r [fm]

β = 6.0 β = 6.3

Fig. 12. Spin-spin (tensor) potential V3(r) at ! = 6.0 and ! = 6.3. The dotted lineis the fit curve Eq. (3.11), applied to the data of ! = 6.0.

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

V4(r

) [G

eV

3]

0.70.60.50.40.30.20.10.0

r [fm]

β = 6.0 β = 6.3

Fig. 13. Spin-spin potential V4(r) at ! = 6.0 and ! = 6.3. The dotted line is the fitcurve Eq. (3.12), applied to the data of ! = 6.0.

V3(r) (see Fig. 12) if the ansatz motivated by one-gluon-exchange in Eq. (3.9)is appropriate. The fit to this function yields the coe!cient c = 0.214(2) with!2

min/Ndf = 3.7. This value of !2min/Ndf is relatively large and the result for c

is 28 % smaller than the Coulombic coe!cient in V !0(r). A better fit can be

achieved using an ansatz in which the power of 1/r is left as a free parameter,i.e.

V3;fit(r) =3c!

rp. (3.11)

23

Relativistic Corrections

• Heavy quark potential to order (1/m2)

• Lattice QCD calculations

m1=m2 (cc), (bb)m1⧧m2 (bc)

(perturbative)

Y. Koma, M. Koma and H. Wittig [Nucl. Phys. B769: 79 (2007)T. Kawanai and S. Sasaki [arXiv:1110.0888]

Spin-Orbit

Spin-SpinTensor

__

_


Mas

s (G

eV/c

2 )

4.0

3.5

3.0

Charmoniumfamily

2 M(D)

!c(3S)

!c(2S)

!c(1S) "M1M1

DD

DD

#+#$J/%%(4S) or hybrid%(2D)

%(3S)

%(13D1) %(2S) DD

DD*

####, !, #0

"M1M1"E1E1

#0

"E1E1

"E1E1hc(1P)

J/J/%

&c0(1P)&c1(1P)

&c2(1P)

&c1(2P) &c2(2P)X(3872)?

(',(,")J/%(J/% """"

DD,

Stephen Godfrey, Hanna Mahlke, Jonathan L. Rosner and E.E. [Rev. Mod. Phys. 80, 1161 (2008)]

Low-Lying Charmonium Spectrum

Narrow

Wide

Threshold for strong decays to DD mesons


3D2 state (ψc2)

• Predicted including the effects of coupling to decay channels

• Note 1D2 (2-+) should be at 3831 MeV - Not X(3872) !!!15

K.Lane, C. Quigg, EE PRD 69 094019 (2004) [hep-ph/0401210]

Moriond 2012 Talk - Vishal Bhardwaj (Belle)

Text


11P1 (hc) state

• CLEO observed the hadronic transition: 23D1 (ψ(4170)) -> π+π- 11P1 (hc)

• Unexpectedly large hadronic transition rate. (More later)

16

T.K. Pedlar et.al [CLEO Collaboration]PRL 107, 041803 (2011) [arXiv:1104.2025]


21S0 (ηc’) state

• BES III observed the M1 transition: 23S1 (ψ(3686)) -> γ 21S0 (ηc’)

• Transition rate in good agreeement with theory. Hyperfine splitting in 2S multiplet shows effects of coupling to decay channels.

17

Liangliang Wang [BESIII Collaboration] [arXiv:1110.2560]

K.Lane, C. Quigg, EE PRD 69 094019 (2004) [hep-ph/0401210]


Updated Charmonium Spectrum

• 13D2 observed - Only 2 narrow states remaining unobserved: 11D2 and 13D3

• New transitions: 23D1 -> π+π- 11P1; 13D2 -> Ɣ 13P1; 23S1 -> Ɣ 21S0

18

,2

Mas

s (G

eV/c

2 )

4.0

3.5

3.0

Charmoniumfamily

2 M(D)

!c(3S)

!c(2S)

!c(1S) "M1M1

DD

DD

#+#$J/%%(4S) or hybrid%(2D)

%(3S)

%(13D1) %(2S) DD

DD*

####, !, #0

"M1M1"E1E1

#0

"E1E1

"E1E1hc(1P)

J/J/%

&c0(1P)&c1(1P)

&c2(1P)

&c1(2P) &c2(2P)X(3872)?

(',(,")J/%(J/% """"

DD,

2--

2

ѱ2(1D)ππ

ΥE1ΥM1

2-+

23--

2

11D213D3

Observed transitions 12 Ɣ 8 ππ, π, η, ω


Stephen Godfrey, Hanna Mahlke, Jonathan L. Rosner and E.E. [Rev. Mod. Phys. 80, 1161 (2008)]

Low-Lying Bottomonium Spectrum

Narrow

Wide

Threshold for strong decays to BB mesons

!b(3S)

!b(2S)

!b(1S)

"(3S)

"(4S)

"(2S)

"(1S)

hb(2P)

hb(1P)

#b(2P)

#b(1P)

$(32D)

$(31D)

Mas

s (G

eV/c

2 )

2 M(B)

10.50

10.25

10.00

9.75

9.50

Bottomoniumfamily

%%%%

&

%%%%

'M1M1

'M1M1 'M1M1

'M1M1

'E1E1

!,%0%+%(,%0

%%%%

!,%0

+ S+ S)P)D ( ('E1E1)

ϒ

ϒ


33PJ (χ’’b)

• ATLAS discovers the 33PJ states of Upsilon family in ϒ(1S) + gamma and ϒ(2S) + gamma final states.

• Mass in good agreement with potential model expectations.

• Effects of coupling to decay channels not included in these theory predictions

20

ATLAS Collaboration [arXiv:1112.5154]


11P1b, 21P1b (hb, hb’) states

• Using the transition: Υ(5S) -> π+π- X

BELLE discovers the 11P1 (hb) and 21P1 (hb’) states

21

I. Adachi et al, Phys. Rev. Lett. 108 (2012) 032001. {BELLE Collaboration) {arXiv:1103.3419]

• Masses of hb(1P) and hb(2P) in agreement with theoretical expectations:ΔMHF = M(n1P1) - M(n3PJ) c.o.g

(n=1):

(n=2):


21S0(η’b)

• Using the transition chain: Υ(5S) -> π+π- hb(nP) -> π+π- Ɣ X (n=1,2)

• BELLE has observed the

21S0 (η’b ) and 11S0 (ηb ) states

• CLEO also reported evidence for the 21S0 state.

• Masses and widths are in good agreement with theoretical expectations.

• Hyperfine splittings as expected:

22

arXiv:1205.6351

m(ηb’) = (9999.0 ±3.5 +2.8−1.9) MeV/c2

and for the ηb

m(ηb) = ((9402.4 ± 1.5 ± 1.8) MeV/c2

Γ(ηb) = (10.8 +4.0−3.7+4.5−2.0)MeV

MHF(1S) = (57.9 ± 2.3) MeV/c2 and MHF(2S) = (24.3+4.0−4.5) MeV/c2


13DJb states in hadronic transitions

• Observed in hadronic transitions: Υ(5S) -> π+ π- + 3DJ

23


Updated Bottomonium Spectrum

Υ(10860)

Υ(11020)

!b(3S)

!b(2S)

!b(1S)

"(3S)

"(4S)

"(2S)

"(1S)

hb(2P)

hb(1P)

#b(2P)

#b(1P)

$(32D)

$(31D)

Mas

s (G

eV/c

2 )

2 M(B)

10.50

10.25

10.00

9.75

9.50

Bottomoniumfamily

%%%%

&

%%%%

'M1M1

'M1M1 'M1M1

'M1M1

'E1E1

!,%0%+%(,%0

%%%%

!,%0

+ S+ S)P)D ( ('E1E1)

3+-

(2,3,4)++ 3

1F

4-+

(3,4,5)-- 4

1Gχb(3P)

10.75

11.00

ππ ππ

19 narrow states still unobserved

2D

Observed transitions 25 Ɣ 18 ππ, π, η, ω

2-+

1D

hb(3P) BB Threshold

_


The BC Spectrum

• Only ground state Bc observed m(Bc) = 6277± 6 MeV 15 more very narrow states

• All excited states below strong decay threshold make transitions (γ, ππ, π0, η, ...) to the Bc.

• Unequal masses -> spin-splittings have different pattern: j-j versus LS.

• Study at hadron colliders (LHCb)

_BD


Transitions - Probing Structure

• EM transitions:

– Multipole expansion (E1, M1, ...)

– Excellent agreement between theory and experiment for EM transitions.

26

A

B

!

k

r

kQ Q

expansion kr/2

I. QUARKONIUM AND MULTIPOLE EXPANSIONS

LNRQCD = ⇤†�

iD0 +D2

2mQ

⇥⇤ +

cF

2mQ⇤†� · gB⇤ + o(

1

m2Q

)

+[⇤ ⇧ i�2⇥⇥, Aµ ⇧ �ATµ ]

where ⇤ is the Pauli spinor field that annihilates a heavy quark of mass m, flavor Q and

electrical charge eeQ, ⇥ is the corresponding one that creates a heavy antiquark.

HI = ieQ⇤†�

D · eA + eA ·D2mQ

⇥⇤ +

cF eQ

2mQ⇤†� · eB⇤ + ... (1)

II. RADIATIVE TRANSITIONS

For quarkonium states, Q1Q2, above the ground state but below threshold for strong

decay into a pair of heavy flavored mesons, electromagnetic transitions are often significant

decay modes. In fact, the first charmonium states not directly produced in e+e� collisions,

the ⇥Jc states, were discovered in photonic transitions of the ⇤⇤ resonance. Even today, such

transitions continue to be used to observe new quarkonium states [1].

A. E�ective Lagrangian

The theory of electromagnetic transitions between these quarkonium states is straightfor-

ward. Much of the terminology and techniques are familiar from the study EM transitions

in atomic and nuclear systems. The photon field Aµem couples to charged quarks through

the electromagnetic current:

jµ ⌅⇧

i=u,d,s

jiµ +

⇧

i=c,b,t

jiµ (2)

The heavy valence quarks (c, b, t) can described by the usual e�ective action:

LNRQCD = ⇤†⇤

iD0 +D2

2m+ cF g

� ·B2m

+ cD g[D·,E]

8m2+ icS g

� · [D⇤,E]

8m2+ . . .

⌅⇤ (3)

where the E and B fields are the chromoelectric and chomomagentic fields. Corrections to

the leading NR behaviour are determined by expansion in the quark and antiquark velocities.

For photon momentum small compared to the heavy quark masses, the form of the EM

2

Electric Magnetic


LNRQCD = ⇤†�

iD0 +D2

2mQ

⇥⇤ +

cF

2mQ⇤†� · gB⇤ + o(

1

m2Q

)

+[⇤ ⇧ i�2⇥⇥, Aµ ⇧ �ATµ ]



HI = ieQ⇤†�

D · eA + eA ·D2mQ

⇥⇤ +

cF eQ

2mQ⇤†� · eB⇤ + ... (1)

A(Rcm, r, t) = A(Rcm, t) + x ·�A(Rcm, t) + ... (2)

II. RADIATIVE TRANSITIONS

For quarkonium states, Q1Q2, above the ground state but below threshold for strong

decay into a pair of heavy flavored mesons, electromagnetic transitions are often significant

decay modes. In fact, the first charmonium states not directly produced in e+e� collisions,

the ⇥Jc states, were discovered in photonic transitions of the ⇤⇤ resonance. Even today, such

transitions continue to be used to observe new quarkonium states [1].

A. E�ective Lagrangian

The theory of electromagnetic transitions between these quarkonium states is straightfor-

ward. Much of the terminology and techniques are familiar from the study EM transitions

in atomic and nuclear systems. The photon field Aµem couples to charged quarks through

the electromagnetic current:

jµ ⌅⇧

i=u,d,s

jiµ +

⇧

i=c,b,t

jiµ (3)

The heavy valence quarks (c, b, t) can described by the usual e�ective action:

LNRQCD = ⇤†⇤

iD0 +D2

2m+ cF g

� ·B2m

+ cD g[D·,E]

8m2+ icS g

� · [D⇤,E]

8m2+ . . .

⌅⇤ (4)

2


LNRQCD = ⇤†�

iD0 +D2

2mQ

⇥⇤ +

cF

2mQ⇤†� · gB⇤ + o(

1

m2Q

)

+[⇤ ⌅ i�2⇥�, Aµ ⌅ �ATµ ]



HI = ieQ⇤†�

D · eA + eA ·D2mQ

⇥⇤ +

cF eQ

2mQ⇤†� · eB⇤ + · · ·

A(Rcm, r, t) = A(Rcm, t) + x · A(Rcm, t) + · · ·

1

mQ{p,A(Rcm, t) + · · ·} = r · E(Rcm, t) + · · ·

1

mQ{p,A(Rcm, t) + · · ·} =

where

i[H, r] =2p

mQ

and

i[H,A] =⇧

⇧tA = E

We have

eQ⇤†r · eE⇤ + · · ·

remembering

p ⇤ 1

r⇤ mQv

2

Electric

Magnetic


LNRQCD = ⇤†�

iD0 +trD2

2mQ

⇥⇤ +

cF

2mQ⇤†� · gB⇤ + o(

1

m2Q

)

+[⇤ ⌅ i�2⇥�, Aµ ⌅ �ATµ ]



A. photonic transitions

HI = ieQ⇤†�

D · eA + eA ·D2mQ

⇥⇤ +

cF eQ

2mQ⇤†� · eB⇤ + · · ·

A(Rcm, r, t) = A(Rcm, t) + x · A(Rcm, t) + · · ·

1

mQ{p,A(Rcm, t) + · · ·} = r · E(Rcm, t) + · · ·

1

mQ{p,A(Rcm, r, t)} = r · E(Rcm, t) + · · ·

where

i[H, r] =2p

mQ

and

i[H,A] =⇧

⇧tA = E

We have

eQ⇤†r · eE⇤ + · · ·

remembering

p ⇤ 1

r⇤ mQv

2

E1, E2, E3, ...M1, M2, M3, ...

Selection Rules e

ir·k2 =

�

n

1

n!

(ir · k)

2

n

• Biggest disagreements in hindered M1 or E1 transitions with large cancellations associated with nodes in the wavefunctions for radially excited states.

• No large enhancements in EM transitions rates for states above threshold observed.

• For more details see: (1) S. Godfrey, H. Mahlke, J. L. Rosner and E.E. [Rev.Mod. Phys. 80, 1161 (2008)](2) N. Brambilla et. al. [Eur.Phys.J.C71:1534,2011]


• QCD Multipole Expansion (QCDME). Analogous to the QED multipole expansion with gluons replacing photons.

– color singlet physical states means lowest order terms involve two gluon emission. So lowest multipoles E1 E1, E1 M1, E1 E2, ....

– factorize the heavy quark and light quark dynamics

– assume a model for the heavy quarkonium states Φi, Φf and a model for the intermediate states |KL> hybrid states.

– use chiral effective lagrangians to parameterize the light hadronic system.

27

10 May 15, 2010: Quarkonia Decays

Many authors contributed to the early development ofQCDME approach[101–103], but Yan[104] was the first topresent a gauge invariant formulation within QCD. Fora heavy QQ bound state, a dressed (constituent) quark(⌥(x, t)) is defined as

⌥(x, t) ⌅ U�1(x, t)⌥(x) (11)

where ⌥(x) is the usual quark field and U is defined as apath ordered exponential along a straight line path fromX ⌅ (x1 + x2)/2 (the c.o.m. coordinate of Q and Q) to x,

U(x, t) = P exp⌅igs

� x

XA(x⇥, t) · dx⇥

⇧(12)

For gluon fields the color indices have been suppressed.The dressed gluon field (A(x, t)) is defined by

Aµ(x, t) ⌅ U�1(x, t)Aµ(x)U(x, t)� i

gsU�1(x, t)�µU(x, t).

(13)Now we can make the QCD multipole expansion in pow-ers of (x�X) ·⌦ operating on the gluon field in exactanalogy with QED:

A0(x, t) = A0(X, t)� (x�X) ·E(X, t) + · · · ,

A(X, t) = �12(x�X)⇤B(X, t) + · · · , (14)

where E and B are color-electric and color-magnetic fields,respectively. The resulting Hamiltonian for a heavy QQsystem is then [104]

He�QCD = H(0)

QCD + H(1)QCD + H(2)

QCD, (15)

with H(0)QCD taken as the zeroth order Hamiltonian even

though it does not represent free fields but the sum of thekinetic and potential energies of the heavy quarks; and

H(1)QCD ⌅ QaAa

0(X, t), (16)

where Qa the color charge of QQ system (zero for colorsinglets); and finally

H(2)QCD ⌅ �da ·Ea(X, t)�ma ·Ba(X, t) + · · · , (17)

is treated perturbatively. dia = gE

⌃d3x⌥†(x�X)ita⌥

and mia = gM/2

⌃d3x⌥†⌅ijk(x�X)j⇥kta⌥ are the color-

electric dipole moment (E1) and the color-magnetic dipolemoment (M1) of the QQ system, respectively. Higher or-der terms (not shown) give rise to higher order electric(E2, E3, ...) and magnetic moments. (M2, ...)

Because H(2)QCD in Eq. 17 couples color singlet to octet

QQ states. The transitions between eigenstates |i� and |f�of H(0)

QCD is at least second order in H(2)QCD. The leading

order term is given by:�f⇤⇤H2

1

Ei �H(0)QCD + i�0 �H1

H2

⇤⇤i⇥

= (18)

⌥

KL

�f⇤⇤H2

⇤⇤KL⇥ 1Ei � EKL

�KL⇤⇤H2

⇤⇤i⇥,

where the sum KL is over a complete set of color octetQQ states |KL� with associated energy EKL. Finally con-nection is made to the physical hadronic transitions Eq.10 by assuming a factorization of the heavy quark inter-actions and the production of light hadrons. For examplethe leading order E1-E1 transition the amplitude is:

M(�i ⇧ �f + h) = (19)

124

⌥

KL

�f⇤⇤dia

m

⇤⇤KL⇥⌥⇤⇤KL⇤⇤dj

ma

⇤⇤i⇥

Ei � EKL

�h⇤⇤EaiEj

a

⇤⇤0⇥

The allowed light hadronic final state h is determined byquantum numbers of gluonic operator. The leading orderterm E1-E1 in Eq.19 has CP=++ and L = 0, 2 and hencecouples to 2⌃ and 2K in I = 0 states. Higher order terms(in powers of v) couple as follows: E1-M1 in O(v) with(CP=--) couples to �; E1-M1, E1-E2 in O(v) and M1-M1, E1-M2 in O(v2) with (CP=+-) couples to ⌃0 (isospinbreaking) and ⇧ (SU(3) breaking); and M1-M1, E1-E3, E2-E2 (CP=++) are higher order corrections to the E1-E1terms.

Applying this formulation to observed hadronic tran-sitions requires addition phenomenological assumptions.Following Kuang and Yan[104,108], the heavy QQ boundstates spectrum of H(0)

QCD is calculated by solving the SEwith a given potential model. The intermediate octet QQstates are modeled by the Buchmueller-Tye quark confin-ing string (QCS) model[109]. Then chiral symmetry rela-tions can be employed to parameterize the light hadronicmatrix element. The remaining unknown coe⇤cients inthe light hadron matrix elements are set by experimentor calculated using a duality argument between the phys-ical light hadron final state and associated two gluon finalstate. A detailed discussion of all these assumptions canbe found in the previous QWG review[110].

For the most common transitions h = ⌃1 + ⌃2 thee�ective chiral lagrangian form is [111]

g2E

6�⌃1⌃2

⇤⇤Eai Eaj

⇤⇤0⇥

=1

(2�1)(2�2)[C1⇤ijq

µ1 q2µ (20)

+ C2(q1kq2l + q1lq2k �23⇤ijq

µ1 q2µ)]

If the polarization of the heavy QQ initial and final statesis measured more information can be extracted form thesetransitions and a more general form of Eq. 21 is appropri-ate[112].

Important single light hadron transitions include the⇧, ⌃0 and � transitions. The general form the light hadronicfactor for the eta transition which is dominantly (E1-M2)is [117]

gegM

6�⇧⇤⇤Ea

i ⇤iBaj

⇤⇤0⇥

= i(2⌃)3/2C3qj (21)

The ⌃0 transitions and ⇧ transitions are related by thestructure of chiral symetry breaking[114]. Many more de-tails for these and other transitions within the contextof the Kuang-Yan model can be found in the review ofKuang[117].



⌥(x, t) ⌅ U�1(x, t)⌥(x) (11)



� x


⇧(12)





A0(x, t) = A0(X, t)� (x�X) ·E(X, t) + · · · ,

A(X, t) = �12(x�X)⇤B(X, t) + · · · , (14)


He�QCD = H(0)


QCD, (15)



H(1)QCD ⌅ QaAa

0(X, t), (16)





and mia = gM/2







1

Ei �H(0)QCD + i�0 �H1

H2

⇤⇤i⇥

= (18)

⌥

KL

�f⇤⇤H2


�KL⇤⇤H2

⇤⇤i⇥,


M(�i ⇧ �f + h) = (19)

124

⌥

KL

�f⇤⇤dia

m


ma

⇤⇤i⇥

Ei � EKL

�h⇤⇤EaiEj

a

⇤⇤0⇥





g2E

6�⌃1⌃2

⇤⇤Eai Eaj

⇤⇤0⇥

=1

(2�1)(2�2)[C1⇤ijq

µ1 q2µ (20)


µ1 q2µ)]



gegM

6�⇧⇤⇤Ea

i ⇤iBaj

⇤⇤0⇥

= i(2⌃)3/2C3qj (21)


zero for color singlet



⌥(x, t) ⌅ U�1(x, t)⌥(x) (11)



� x


⇧(12)





A0(x, t) = A0(X, t)� (x�X) ·E(X, t) + · · · ,

A(X, t) = �12(x�X)⇤B(X, t) + · · · , (14)


He�QCD = H(0)


QCD, (15)



H(1)QCD ⌅ QaAa

0(X, t), (16)





and mia = gM/2







1

Ei �H(0)QCD + i�0 �H1

H2

⇤⇤i⇥

= (18)

⌥

KL

�f⇤⇤H2


�KL⇤⇤H2

⇤⇤i⇥,


M(�i ⇧ �f + h) = (19)

124

⌥

KL

�f⇤⇤dia

m


ma

⇤⇤i⇥

Ei � EKL

�h⇤⇤EaiEj

a

⇤⇤0⇥





g2E

6�⌃1⌃2

⇤⇤Eai Eaj

⇤⇤0⇥

=1

(2�1)(2�2)[C1⇤ijq

µ1 q2µ (20)


µ1 q2µ)]



gegM

6�⇧⇤⇤Ea

i ⇤iBaj

⇤⇤0⇥

= i(2⌃)3/2C3qj (21)


E1 M1 ...10 May 15, 2010: Quarkonia Decays


⌥(x, t) ⌅ U�1(x, t)⌥(x) (11)



� x


⇧(12)





A0(x, t) = A0(X, t)� (x�X) ·E(X, t) + · · · ,

A(X, t) = �12(x�X)⇤B(X, t) + · · · , (14)


He�QCD = H(0)


QCD, (15)



H(1)QCD ⌅ QaAa

0(X, t), (16)





and mia = gM/2







1

Ei �H(0)QCD + i�0 �H1

H2

⇤⇤i⇥

= (18)

⌥

KL

�f⇤⇤H2


�KL⇤⇤H2

⇤⇤i⇥,


M(�i ⇧ �f + h) = (19)

124

⌥

KL

�f⇤⇤dia

m


ma

⇤⇤i⇥

Ei � EKL

�h⇤⇤EaiEj

a

⇤⇤0⇥





g2E

6�⌃1⌃2

⇤⇤Eai Eaj

⇤⇤0⇥

=1

(2�1)(2�2)[C1⇤ijq

µ1 q2µ (20)


µ1 q2µ)]



gegM

6�⇧⇤⇤Ea

i ⇤iBaj

⇤⇤0⇥

= i(2⌃)3/2C3qj (21)


+ higher order multipole terms.

g

gA

B

!

!

Model: Kuang & Yan [PR D24, 2874 (1981)]



• Two pion transitions: (E1-E1), ...

– nS -> mS transitions

• Higher order transitions (E1-M1); (M1-M1), (E1-M2), ... – Combine chiral symmetry and SU(3) symmetry breaking to relate

(π0,η,η’) transitions

28

D-waveS-wave

�(n3IS1⌅n3

F S1 � �) = |C1|2G|f 111nI0nF 0|2, (13)

where the phase-space factor G is [7]

G ⇥ 3

4

M�F

M�I

�3

⇤K

⇧

1� 4m2�

M2��

(M2�� 2m2

�)2 dM2��, (14)

d� ⇤ K

⇧

1� 4m2�

M2��

(M2�� 2m2

�)2 dM2��, (15)

with

K ⇥⌅

(MA + MB)2 �M2��

⌅(MA �MB)2 �M2

��

2MA, (16)

and

fLPIPFnI lInF lF

⇥⇥

K

�RF (r)rPF R�

KL(r)r2dr�

R�KL(r⇥)r⇥PIRI(r⇥)r⇥2dr⇥

MI � EKL, (17)

VI. NEW STATES

VII. SUMMARY

VIII. REFERENCES

[1] K. Gottfried, in Proc. 1977 International Symposium on Lepton and Photon Interactions at

High Energies, edited by F. Gutbrod, DESY, Hamburg, 1977, p. 667; Phys. Rev. Lett. 40

(1978) 598.

[2] G. Bhanot, W. Fischler and S. Rudas, Nucl. Phys. B 155 (1979) 208.

[3] M.E. Peskin, Nucl. Phys. B 156 (1979) 365; G. Bhanot and M.E. Peskin, ibid. 156 (1979)

391.

[4] M.B. Voloshin, Nucl. phys. B 154 (1979) 365; M.B. Voloshin and V.I. Zakharov, Phys. Rev.

Lett. 45 (1980) 688; V.A. Novikov and m.A. Shifman, Z. Phys. C 8 (1981) 43.

10

� = G |�EEAB C1|2

Phase Space Overlap - Vibrating String Model

M(ππ)



Known Hadronic Transitions

71

for these and other transitions within the context ofthe Kuang-Yan model can be found in the review ofKuang [521].

A summary of all experimentally observed hadronictransitions and their corresponding theoretical expecta-tions within the Kuang-Yan (KY) model is presented inTable 36. The experimental partial widths are deter-mined from the measured branching fractions and totalwidth of the initial state. If the total width is not well-measured, the theoretically-expected width is used, asindicated. The theory expectations are adjusted usingthe current experimental inputs to rescale the model pa-rameters |C1| and |C2| in Eq. (135) and |C3| in Eq. (136).

The multipole expansion works well for transitions ofheavy QQ states below threshold [81]. Within the spe-cific KY model a fairly good description of the rates forthe two-pion transitions is observed. The partial width�(⇥(3S) ⌅ ⇥(1S)⇥+⇥�) was predicted to be suppresseddue to cancellations between the various QCS interme-diate states [517], allowing nonleading terms, O(v2), tocontribute significantly. The non-S-wave behavior of them⇥+⇥� dependence in ⇥(3S) decays, also observed in the⇥(4S) ⌅ ⇥(2S)⇥+⇥� transitions, may well reflect thisinfluence of higher-order terms. Other possibilities arediscussed in Sect. 3.3.11. For single light-hadron transi-tions some puzzles remain. For example, the ratio

�(⇥(2S) ⌅ �⇥(1S))

�(⇤(2S) ⌅ �J/⇤(1S))(137)

is much smaller than expected from theory (seeSect. 3.3.6).

The situation is more complicated for above-threshold,strong open-flavor decays. The issues are manifest for⇥(5S) two-pion transitions to ⇥(nS) (n = 1, 2, 3). First,states above threshold do not have sizes that are smallcompared to the QCD scale (e.g.,

�⇧r2⌃�(5S) = 1.2 fm),

making the whole QCDME approach less reliable. Sec-ond, even within the KY model, the QCS intermediatestates are no longer far away from the initial-state mass.Thus the energy denominator, Ei � EKL in Eq. (134),can be small, leading to large enhancements in the tran-sition rates that are sensitive to the exact position of theintermediate states [528]. This is the reason for the largetheory widths seen in Table 36. Third, a number of newstates (see Sects. 2.3) that do not fit into the conventionalQQ spectra have been observed, implying additional de-grees of freedom appearing in the QCD spectrum beyondnaive-quark-model counting. Hence the physical quarko-nium states have open-flavor meson-pair contributionsand possible hybrid (QQg) or tetraquark contributions.The e⇤ect of such terms on hadronic transitions is notyet understood [531]. A possibly-related puzzle is thestrikingly-large ratio

R�[⇥(4S)] ⇥ �(⇥(4S) ⌅ ⇥(1S) �)

�(⇥(4S) ⌅ ⇥(1S)⇥+⇥�)⇤ 2.5 . (138)

This ratio is over a hundred times larger than one wouldexpect within the KY model, which is particularly sur-

TABLE 36: Partial widths for observed hadronic transitions.Experimental results are from PDG08 [18] unless otherwisenoted. Partial widths determined from known branching frac-tions and total widths. Quoted values assume total widthsof �tot(⇤b2(2P )) = 138 ± 19 keV [523], �tot(⇤b1(2P )) =96 ± 16 keV [523], �tot(⇥(13D2)) = 28.5 keV [524, 525] and�tot(⇥(5S)) = 43 ± 4 MeV [36]. Only the charged dipiontransitions are shown here, but the corresponding measured⇥0⇥0 rates, where they exist, are consistent with a parentstate of I = 0. Theoretical results are given using the Kuangand Yan (KY) model [517, 521, 526]. Current experimentalinputs were used to rescale the parameters in the theory par-tial rates. (|C1| = 10.2 ± 0.2 � 10�3, C2/C1 = 1.75 ± 0.14,C3/C1 = 0.78± 0.02 for the Cornell case)

Transition �partial (keV) �partial (keV)

(Experiment) (KY Model)

⌅(2S)

⇤ J/⌅ + ⇥+⇥� 102.3± 3.4 input (|C1|)⇤ J/⌅ + � 10.0± 0.4 input (C3/C1)⇤ J/⌅ + ⇥0 0.411± 0.030 [446] 0.64 [522]⇤ hc(1P ) + ⇥0 0.26± 0.05 [47] 0.12-0.40 [527]

⌅(3770)

⇤ J/⌅ + ⇥+⇥� 52.7± 7.9 input (C2/C1)⇤ J/⌅ + � 24± 11

⌅(3S)⇤ J/⌅ + ⇥+⇥� < 320 (90% CL)

⇥(2S)

⇤ ⇥(1S) + ⇥+⇥� 5.79± 0.49 8.7 [528]⇤ ⇥(1S) + � (6.7± 2.4)� 10�3 0.025 [521]

⇥(13D2)

⇤ ⇥(1S) + ⇥+⇥� 0.188± 0.046 [63] 0.07 [529]

⇤b1(2P )

⇤ ⇤b1(1P ) + ⇥+⇥� 0.83± 0.33 [523] 0.54 [530]⇤ ⇥(1S) + ⇧ 1.56± 0.46

⇤b2(2P )

⇤ ⇤b2(1P ) + ⇥+⇥� 0.83± 0.31 [523] 0.54 [530]⇤ ⇥(1S) + ⇧ 1.52± 0.49

⇥(3S)

⇤ ⇥(1S) + ⇥+⇥� 0.894± 0.084 1.85 [528]⇤ ⇥(1S) + � < 3.7� 10�3 0.012 [521]⇤ ⇥(2S) + ⇥+⇥� 0.498± 0.065 0.86 [528]

⇥(4S)

⇤ ⇥(1S) + ⇥+⇥� 1.64± 0.25 4.1 [528]⇤ ⇥(1S) + � 4.02± 0.54⇤ ⇥(2S) + ⇥+⇥� 1.76± 0.34 1.4 [528]

⇥(5S)

⇤ ⇥(1S) + ⇥+⇥� 228± 33⇤ ⇥(1S) +K+K� 26.2± 8.1⇤ ⇥(2S) + ⇥+⇥� 335± 64⇤ ⇥(3S) + ⇥+⇥� 206± 80

71

for these and other transitions within the context ofthe Kuang-Yan model can be found in the review ofKuang [521].

A summary of all experimentally observed hadronictransitions and their corresponding theoretical expecta-tions within the Kuang-Yan (KY) model is presented inTable 36. The experimental partial widths are deter-mined from the measured branching fractions and totalwidth of the initial state. If the total width is not well-measured, the theoretically-expected width is used, asindicated. The theory expectations are adjusted usingthe current experimental inputs to rescale the model pa-rameters |C1| and |C2| in Eq. (135) and |C3| in Eq. (136).

The multipole expansion works well for transitions ofheavy QQ states below threshold [81]. Within the spe-cific KY model a fairly good description of the rates forthe two-pion transitions is observed. The partial width�(⇥(3S) ⌅ ⇥(1S)⇥+⇥�) was predicted to be suppresseddue to cancellations between the various QCS interme-diate states [517], allowing nonleading terms, O(v2), tocontribute significantly. The non-S-wave behavior of them⇥+⇥� dependence in ⇥(3S) decays, also observed in the⇥(4S) ⌅ ⇥(2S)⇥+⇥� transitions, may well reflect thisinfluence of higher-order terms. Other possibilities arediscussed in Sect. 3.3.11. For single light-hadron transi-tions some puzzles remain. For example, the ratio

�(⇥(2S) ⌅ �⇥(1S))

�(⇤(2S) ⌅ �J/⇤(1S))(137)

is much smaller than expected from theory (seeSect. 3.3.6).

The situation is more complicated for above-threshold,strong open-flavor decays. The issues are manifest for⇥(5S) two-pion transitions to ⇥(nS) (n = 1, 2, 3). First,states above threshold do not have sizes that are smallcompared to the QCD scale (e.g.,

�⇧r2⌃�(5S) = 1.2 fm),

making the whole QCDME approach less reliable. Sec-ond, even within the KY model, the QCS intermediatestates are no longer far away from the initial-state mass.Thus the energy denominator, Ei � EKL in Eq. (134),can be small, leading to large enhancements in the tran-sition rates that are sensitive to the exact position of theintermediate states [528]. This is the reason for the largetheory widths seen in Table 36. Third, a number of newstates (see Sects. 2.3) that do not fit into the conventionalQQ spectra have been observed, implying additional de-grees of freedom appearing in the QCD spectrum beyondnaive-quark-model counting. Hence the physical quarko-nium states have open-flavor meson-pair contributionsand possible hybrid (QQg) or tetraquark contributions.The e⇤ect of such terms on hadronic transitions is notyet understood [531]. A possibly-related puzzle is thestrikingly-large ratio

R�[⇥(4S)] ⇥ �(⇥(4S) ⌅ ⇥(1S) �)

�(⇥(4S) ⌅ ⇥(1S)⇥+⇥�)⇤ 2.5 . (138)

This ratio is over a hundred times larger than one wouldexpect within the KY model, which is particularly sur-

TABLE 36: Partial widths for observed hadronic transitions.Experimental results are from PDG08 [18] unless otherwisenoted. Partial widths determined from known branching frac-tions and total widths. Quoted values assume total widthsof �tot(⇤b2(2P )) = 138 ± 19 keV [523], �tot(⇤b1(2P )) =96 ± 16 keV [523], �tot(⇥(13D2)) = 28.5 keV [524, 525] and�tot(⇥(5S)) = 43 ± 4 MeV [36]. Only the charged dipiontransitions are shown here, but the corresponding measured⇥0⇥0 rates, where they exist, are consistent with a parentstate of I = 0. Theoretical results are given using the Kuangand Yan (KY) model [517, 521, 526]. Current experimentalinputs were used to rescale the parameters in the theory par-tial rates. (|C1| = 10.2 ± 0.2 � 10�3, C2/C1 = 1.75 ± 0.14,C3/C1 = 0.78± 0.02 for the Cornell case)

Transition �partial (keV) �partial (keV)

(Experiment) (KY Model)

⌅(2S)

⇤ J/⌅ + ⇥+⇥� 102.3± 3.4 input (|C1|)⇤ J/⌅ + � 10.0± 0.4 input (C3/C1)⇤ J/⌅ + ⇥0 0.411± 0.030 [446] 0.64 [522]⇤ hc(1P ) + ⇥0 0.26± 0.05 [47] 0.12-0.40 [527]

⌅(3770)

⇤ J/⌅ + ⇥+⇥� 52.7± 7.9 input (C2/C1)⇤ J/⌅ + � 24± 11

⌅(3S)⇤ J/⌅ + ⇥+⇥� < 320 (90% CL)

⇥(2S)

⇤ ⇥(1S) + ⇥+⇥� 5.79± 0.49 8.7 [528]⇤ ⇥(1S) + � (6.7± 2.4)� 10�3 0.025 [521]

⇥(13D2)

⇤ ⇥(1S) + ⇥+⇥� 0.188± 0.046 [63] 0.07 [529]

⇤b1(2P )

⇤ ⇤b1(1P ) + ⇥+⇥� 0.83± 0.33 [523] 0.54 [530]⇤ ⇥(1S) + ⇧ 1.56± 0.46

⇤b2(2P )

⇤ ⇤b2(1P ) + ⇥+⇥� 0.83± 0.31 [523] 0.54 [530]⇤ ⇥(1S) + ⇧ 1.52± 0.49

⇥(3S)

⇤ ⇥(1S) + ⇥+⇥� 0.894± 0.084 1.85 [528]⇤ ⇥(1S) + � < 3.7� 10�3 0.012 [521]⇤ ⇥(2S) + ⇥+⇥� 0.498± 0.065 0.86 [528]

⇥(4S)

⇤ ⇥(1S) + ⇥+⇥� 1.64± 0.25 4.1 [528]⇤ ⇥(1S) + � 4.02± 0.54⇤ ⇥(2S) + ⇥+⇥� 1.76± 0.34 1.4 [528]

⇥(5S)

⇤ ⇥(1S) + ⇥+⇥� 228± 33⇤ ⇥(1S) +K+K� 26.2± 8.1⇤ ⇥(2S) + ⇥+⇥� 335± 64⇤ ⇥(3S) + ⇥+⇥� 206± 80

Heavy quarkonium: progress, puzzles, and opportunitiesN. Brambilla et.al. [arXiv:1010.5827]


• η transitions

– Large deviations from model expectations. Transitions are very poorly understood.

• Υ(3S) ->Υ(1S) ππ andΥ(4S) ->Υ(2S) ππ transitions

– Mππ distributions NOT the expected S-wave behaviour

– Likely explanation - same as overlap dynamically suppressed in Υ(3S) -> χb(1P ) γ EM transitions

– Further study would be useful. Look at polarization.

30

Some Puzzles

BELLE

FIG. 8: Plots overlaying projections of the data (points with error bars) and the fit result (his-

tograms) onto the M!! and cos !X variables. The plots are summed over electrons and muons, butare di!erentiated by pion charge. The neutral modes (open symbols, dashed lines) show only apositive distribution in cos !X because the two pions are indistinguishable. For the charged modes

(solid symbols, solid lines) the angle is that of the "+.

and proportional to 1/!

ai, where ai is the Monte Carlo phase space yield in bin i. Hence,

!i =!

di + d2i /ai.

The bins for which di = 0 require special treatment, and !i is modified appropriately. Tominimize the e!ect of such bins with zero yield, we sum over muon and electron final states.This takes a weighted average over the distributions, rather than taking account of the

14

CLEO

Why it works so well below threshold


• Lattice calculation V(r), then SE

– What about the gluon and light quark degrees of freedom in QCD?

– Two thresholds

• Usual (Qq) + (qQ) decay thresholds

• Exciting the string - hybrids

– Hybrid states will appear in the spectrum associated with the potentials Πu, ...

– In the static limit this occurs at separation r ≈ 1.2 fm.

• Between the 3S and 4S in (cc) system

• Just above the 5S in the (bb) system

31

SPECTROSCOPY

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0.2 0.4 0.6 0.8 1 1.2 1.4[V

(r)-

V(0

.5 f

m)]

/Ge

V

r/fm

Σg+

Πu

2 m0-

m0- + m1

+

quenchednf = 2

Fig. 3.6: The singlet static energy (quenched and unquenched data) from Ref. [51], see also [143]

2.3.3 The QCD static spectrum and mechanism of confinement18

The spectrum of gluons in the presence of a static quark–antiquark pair has been extensively studied with

high precision using lattice simulations. Such studies involve the calculation of large sets of Wilson loops

with a variety of different spatial paths. Projections onto states of definite symmetries are done, and the

resulting energies are related to the static quark–antiquark potential and the static hybrids potentials. With

accurate results, such calculations provide an ideal testing ground for models of the QCD confinement

mechanism.

The singlet static energy

The singlet static energy is the singlet static potential V (0)s .

In the plot3.6, we report simulation results both with and without light quark–antiquark pair cre-

ation. Such pair creation only slightly modifies the energies for separations below 1 fm, but dramatically

affects the results around 1.2 fm, at a distance which is too large with respect to the typical heavy quarko-

nium radius to be relevant for heavy quarkonium spectroscopy. At finite temperature, the so-called string

breaking occurs at a smaller distance (cf. corresponding Section in Chapter 7,Media).

One can study possible nonperturbative effects in the static potential at short distances. As it has

already been mentioned in the ”static QCD potential” subsection, the proper treatment of the renormalon

effects has made possible the agreement of perturbation theory with lattice simulations (and potential

models) [78,88–92]. Here we would like to quantify this agreement assigning errors to this comparison.

In particular, we would like to discern whether a linear potential with the usual slope could be added to

perturbation theory. In order to do so we follow here the analysis of Ref. [90, 144], where the potential

is computed within perturbation theory in the Renormalon Subtracted scheme defined in Ref. [81]. The

comparison with lattice simulations [145] in Fig. 3.7 shows that nonperturbative effects should be small

and compatible with zero, since perturbation theory is able to explain lattice data within errors. The

systematic and statistical errors of the lattice points are very small (smaller than the size of the points).

Therefore, the main sources of uncertainty of our (perturbative) evaluation come from the uncertainty in

the value of !MS (±0.48 r!10 ) obtained from the lattice [146] and from the uncertainty in higher orders

in perturbation theory. We show our results in Fig. 3.7. The inner band reflects the uncertainty in !MSwhereas the outer band is meant to estimate the uncertainty due to higher orders in perturbation theory.

We estimate the error due to perturbation theory by the difference between the NNLO and NNNLO

evaluation. The usual confining potential, !V = "r, goes with a slope " = 0.21GeV2. In lattice units

18Authors: N. Brambilla, C. Morningstar, A. Pineda

91

LQCD calculation of static energy

The leading Born-Oppenheimer approximation

In the leading Born-Oppenheimer approximation, one replaces the covariant Lapla-

cian DDD2 by an ordinary Laplacian ∇∇∇2, which neglects retardation effects. The spin in-

teractions of the heavy quarks are also neglected, and one solves the radial Schrödinger

equation:

!1

2µ

d2u(r)

dr2+

!

"LLL2QQ

#

2µr2+VQQ(r)

"

u(r) = E u(r), (2)

where u(r) is the radial wavefunction of the quark-antiquark pair. The total angularmomentum is given by

JJJ = LLL+SSS, SSS= sssQ+ sssQ, LLL= LLLQQ+ JJJg, (3)

where sssQ is the spin of the heavy quark, sssQ is the spin of the heavy antiquark, JJJg is the

total spin of the gluon field, and LLLQQ is the orbital angular momentum of the quark-

antiquark pair. In the LBO, both L and S are good quantum numbers. The expectation

value in the centrifugal term is given by

"LLL2QQ

# = "LLL2#!2"LLL · JJJg#+ "JJJ2g#. (4)

The first term yields L(L+1). The second term is evaluated by expressing the vectors interms of components in the body-fixed frame. Let Lr denote the component of LLL along

the molecular axis, and Lξ and Lζ be components perpendicular to the molecular axis.

Writing L± = Lξ ± iLζ and similarly for JJJg, one obtains

"LLL · JJJg# = "LrJgr#+12"L+Jg! +L!Jg+#. (5)

Since Jg± raises or lowers the value of Λ, this term mixes different gluonic stationarystates, and thus, must be neglected in the leading Born-Oppenheimer approximation. In

the meson rest frame, the component of LLLQQ along the molecular axis vanishes, and

hence, "LrJgr# = "J2gr# = Λ2. In summary, the expectation value in the centrifugal termis given in the adiabatic approximation by

"LLL2QQ

# = L(L+1)!2Λ2+ "JJJ2g#. (6)

We assume "JJJ2g# is saturated by the minimum number of allowed gluons. Hence, "JJJ2g#= 0

for the Σ+g level and "JJJ

2g#= 2 for theΠu and Σ

!u levels.Wigner rotations are used as usual

to construct |LSJM;λη# states, where λ = JJJg · rrr and Λ = |λ |, then JPC eigenstates arefinally obtained from

|LSJM;λη#+ ε|LSJM;!λη#, (7)

where ε = 1 for Σ+ levels, ε = !1 for Σ! levels, and ε = ±1 for Λ $ 1 levels. Hence,the JPC eigenstates satisfy

P= ε(!1)L+Λ+1, C = ηε(!1)L+S+Λ. (8)


Crossing the Threshold

• Many new degrees of freedom influence spectrum and transitions

– Normal strong decay channels - strong coupled channel effects

– New four quark states possible:

• molecules

• diquark-antidiquark

• hadrocharmonium

– Hybrid states:

• exciting the gluon degrees of freedom

• valence gluons picture

• string picture

• Not hopeless. Two handles to understand systematics:

– lattice QCD

– known scaling from (cc) to (bb) systems

• May provide insight into strong dynamics of light hadron spectroscopy.

32

_ _

★


New States Above Threshold

State EXP M + i Γ (MeV) JPC Decay Modes Production ModesX(3872) Belle, CDF,

D0, BaBar3871.67±0.17 + i(<1.3) 1++ or

2-+π+π-J/ψ, ωJ/ψ, ΥJ/ψ, Υψ’, D0D*0

B decays, ppbar

χc(23P2) Belle, BaBar 3927.2±2.6 + i(24±6) 2++ D0D0, D+D-, ωJ/ψ ϒϒ, B decays

X(3940) Belle 3942+7-6±6 + i(37+26-15±8) JP+ DD* e+e- (recoil against J/ψ)

Y(4008) Belle

BaBar4008±40+72-28 + i(226±44+87-79)

(not seen)

1-- π+π-J/ψ e+e- (ISR)

Y(4140) CDF

LHCb4143.0±2.9±1.2 + i(11.7+8.3-5.0±3.7) (not seen)

JP+ ϕ J/ψ ppbar

ppψ(4160) CLEO 4153±3 + i(103±8) 1-- π+π-hc(1P) e+e-

X(4160) Belle 4156+25-20±15+ i(139+111-61±21) JP+ D*D* e+e- (recoil against J/ψ)

Y(4260) BaBar, CLEO, Belle

4263 +8-9 + i(95±14) 1-- π+π-J/ψ, π0π0J/ψ,K+K-J/ψ

e+e- (ISR), e+e-

Y(4360) BaBar, Belle 4361±9±9 + i(74±18±10) 1-- π+π-ψ(2S) e+e- (ISR)

Y(4660) Belle 4664±11±8 + i(48±15±3) 1-- π+π-ψ(2S), ΛcΛcbar e+e- (ISR)

ϒ(5S) Belle, BaBar

10,876±11 + i(55±28) 1-- π+π-ψ(nS) n=1,2,3

π+π-hb(nP) n=1,2

e+e-

Charged states: X±(4250) -> π±χc1(1P), X±(4430) -> π±ψ(2S), Zb±(10,610)-> π±hb(nP) and Zb± (10650)-> π±hb(nP)

★

✔

✔

✔

★★

• The ψ(4160) -> hc(1P) + π+π- transition observed by CLEO

• ψ(4160) in the 23D1 charmonium state. Unexpectedly large transition rate.

• Spin flip transition: E1 M1

• Gross violation of QCDME for hadronic transitions !


ψ(4160) Transitions

34



Υ(5S) Transitions

• Large rates

– Υ(5S): m=10,876 ± 11 MeV and Γ= 55 ± 23 MeV

– BR(Υ 5S) -> Υ(2S) + π+π-) = (0.78 ± 0.13) %

– π+π- system I= 0 – total branching ratio for known hadronic transitions (3.9 ± 0.7)% => Γ = 2.1 ± 0.9 MeV

• Clear violation of QCDME expectations:

– the transitions Υ(5S) -> hb(1P,2P) + π+π- requires a heavy quark spin flip (M1)(E1)

• The usual formulation of QCDME needs modification.

• New ηtransitions:

35

BELLE [arXiv:1103.3419]

P. Krokovny at La Thuille 2012 [BELLE Collaboration]

(n=1) (n=2)


Why the QCDME Assumptions Fail Above Threshold

• Four options exist for breakdown of the QCDME

1. Because states above threshold are not compact the expansion becomes unreliable.

2. The model of hybrid intermediate states is insufficient as hybrid thresholds are crossed.

3. The coupling to decay channels adds new contributions. Transitions in the two meson channels (breaks the factorization assumption)

4. There are new exotic states (that are not hybrids) which appear in the intermediate state. (Again breaks the factorization assumpion)

• These options are ranked from least surprising (1) to most extreme (4). But Belle has observed new states that if confirmed will show that at least in the Υ(5S) transitions the breakdown is caused by option (4).

36


Zb±(10,610) and Z b± (10,650)

• BELLE has observed two new charged states in the Υ(5S) -> Υ(nS) + π+π- (n=1,2,3) and

the Υ(5S) -> hb(nP) + π+π- (n=1,2) transitions [arXiv:1105.4583]

• Υ(5S) -> Zb++ π- and Zb -> hb(nP) + π+ .

• Explicitly violates the factorization assumption of the QCDME.

• Clear evidence for four quark molecular states if confirmed

• The Zb± (10610) is a narrow state (Γ= 15.6 ± 2.5 MeV) at the BB* threshold (10605).

• The Zb± (10650) is a narrow state (Γ= 14.4 ± 3.2 MeV) at the B*B* threshold (10650).

37

_

_


Zb±(10,610) and Z b± (10,650)

• Υ(5S) -> Υ(nS) + π+π- (n=1,2,3)

• Zb in Υ(2S), hb(1P) and hb(2P) pion transitions

• JP consistent with 1+

• Systematics of the interference between the Zb states in the hb(1P,2P) pion transitions [M. Voloshin arXiv:1201.1222]

38


Working Hypothesis

• Zb±(10,610) and Z±b(10650) are JP = 1+ threshold molecular states of BB* and B*B* respectively.

• Comments:

– I=1 -> partner states Zb0 observable inΥ(5S) -> π0 Zb0 -> π0π0 hb(1P,2P)

– Possible I=0 BB* and B*B* molecular states.

– Analogy states in the cc system at DD* and D*D* thresholds.

– Observable in ψ(4S) -> π Zc -> ππ hc(1P)

39

__

__ _

__

• The X(3872) was the first BIG surprise

• Discovered by BELLE in 2003. It is firmly established (BELLE, BaBar, CDF, D0, LHCb, CMS)– Discovery mode X(3872) -> π+π-J/ψ

– Seen in B decays and direct hadronic production

• Mass at D0 D*0 threshold. Very narrow state Width < 1.2 MeV (90% cl)


X (3872)

40

BELLE (Phys.Rev.Lett.91:262001,2003)

_

• Large isospin breaking observed in the detailed study of X(3872) -> π+π-J/ψ decays.

– (cc) states have isospin zero (I =0) but the mass spectrum of π+π- fits ρ resonance I = 1

– X(3872) -> ω(off shell) J/ψ = π+π-π0 J/ψ also observed

– The charged D meson pair threshold far away. Virtual D0D*0 contribution to X(3872) much larger than the D+D*- component. This can induce large isospin breaking in X(3872).


X (3872)

41

BELLE CollaborationPR D84, 052004 (2011)

Fit to ρJ/ψ including

ω-ρinterference.

L=0 (solid) L=1 (dash)

_

_


X(3872)

• JPC = 1++ or 2-+ CDF, BaBar, Belle

– difficult to distinguish from M(π+π-)

– BaBar favors 2-+ from fit to 3 pion (“ω”) transitions

– 1++ favored by molecular or charmonium interpertation of X(3872)

42

P. del Amo Sanchez et al. (BABAR), PR D 82, 011101(R) (2010) (2-=) C. Hanhart, et al. [arXiv:1111.6241v2] (1++) [fig] R. Faccini, t al. [arXiv:1204.1223] (2-+)

M(π+π-) M(π+π-π0)

(1++) (2-+) fit 1 (2-+) fit 2

• Strong decays to D0 D*0 (D0 D0 π0) seen– Observed at both BELLE and BaBar. Mass consistent

with the mass determined from the ππ J/ψ decays.

– Much theoretical work has gone in to understanding the detailed behaviour of the X(3872) as a state right at threshold.

– Panda is the only experiment that has the energy resolution to map out the line shape.


X (3872)

43

---- π πψ

___ D0 D*0

BoundState

VirtualState

_ _

• Υ J/ψ and Υ ψ’ decays

– A significant ratio of ψ(2S) to J/ψ photon transitions is expected for a charmoniun 23P1 state

• What theoretical interpertations survive?

– Tetraquarks - No expected isopsin partner state observed. How to explain extremely small binding?

– Hybrid state - How to explain extremely small binding?

– Charmonium state - only 23P1 remains viable possibility.

– Molecule D0 D*0 - Remains the most likely. For threshold states like the X(3872) many features are universal (E. Braaten, M. Kusunoki). In fact for binding energies as small as the experimental central mass value. The D0 and D*0 have an average separation > 6 fm !!!!


X (3872)

44

[BELLE][BaBar]


Working Hypothesis

• The X(3872) state is a JPC = 1++ state which is a mixture of a 23P1 charmonium state and a D0D*0 molecule: |X> = α|23P1(cc)> + (1-α) |D0D*0>.

• Comments:

– The mixing parameter α can be determined by detailed measurements careful study of the relative transitions to charmonium states.

– Whether the X(3872) is bound or not will require a measurement of the lineshape in both the ππJ/ψ and D0 D0* decay channels

45

_ _

_

_


Four Quark States

• The Zb±(10,610), Z±b(10650) and Xc(3872) states all occur at threshold for associated two body decay.

• This suggests necessary conditions for such threshold states T <-> A+B:

1. There is a strong attractive QCD force between A and B. (Nearby QQ state or attractive exchange interaction)

2. The two body final states (A, B) are both narrow. (For light mesons: A,B=0-; for charmed mesons: A,B= 0- or 1- ; charmed strange mesons = charmed mesons + 0+ or 1+).

3. The quantum numbers (JP) of T allow an S-wave decay into A + B. examples: (T=1+, A=0-, B=1-), (T=1+, A=1-, B=1-), (T=0+, A=0-, B=0-), (T=2+, A=1-, B=1-), ...

46

_

• In e+e- collisions (ISR process) the JPC = 1-- state Y(4260) is observed in the Y(4260) -> J/ψ + π+π- transition by Belle, BaBar and CLEO

• The Y(4260) not seen in D(*)D(*) final states. At dip in R. Large rate.


Y(4260)

47

JPC=1--

_

• Belle and CLEO observe Y(4260) -> J/ψ + π0π0 consistent with I=0

• Hint of Y(4260) -> hc π+π-


Y(4260)

48


• Additional 1-- states: Y(4360), Y(4660) with transitions ψ(2S) + π+π-

• Clear evidence for large ππ transitions rates. But these Y states are not conventional charmonium states. No available 1-- states

• Working Hypothesis: The Y(4260) is a non-exotic hybrid state.


Y(4260), ...

49


Charmonium Hybrids on the Lattice

• Recent results for the charmonium in Lattice QCD Hadron Spectrum Collaboration (L. Liu et.al. arXiv:1204.5424)

50

4S(cc)Hybrid (ccg)

3S(cc)

_

_

_

• Born-Opperheimer Approimation

• Put the correct short (pNRQCD) and long distance (NG string) behaviour together using lattice QCD can determine the hybrid potentials

• Toy model - minimal parameters


3

TABLE I: Operators to create excited gluon states for smallqq separation R are listed. E and B denote the electric andmagnetic operators, respectively. The covariant derivative D

is defined in the adjoint representation [10].

gluon state J operator!+ !

g 1 R · E, R · (D !B)"g 1 R ! E, R ! (D! B)!"

u 1 R · B, R · (D! E)"u 1 R ! B, R ! (D! E)!"

g 2 (R · D)(R · B)"!

g 2 R ! ((R · D)B + D(R · B))#g 2 (R ! D)i(R ! B)j + (R ! D)j(R ! B)i

!+u 2 (R · D)(R · E)

"!

u 2 R ! ((R · D)E + D(R · E))#u 2 (R ! D)i(R ! E)j + (R ! D)j(R ! E)i

predicted short–distance degeneracies. Only the states!u and "+!

g show considerable soft breaking of the ap-proximate symmetry at the shortest R values.Crossover region. For 0.5 fm < R < 2 fm, a dramaticcrossover of the energy levels toward a string-like spec-trum as R increases is observed. For example, the states"!

u with N = 3 and "!

g with N = 4 break violently awayfrom their respective short-distance O(3) degeneracies toapproach the ordering expected from bosonic string the-ory near R ! 2 fm.

An interesting feature of the crossover region is the suc-cessful parametrization of the "+

g ground state energy bythe empirical function E0(R) = a + !R" c !

12R, with the

fitted constant c close to unity, once R exceeds 0.5 fm.The Casimir energy of a thin flux line was calculated inRefs. [11, 12], yielding c = 1, and this approximate agree-ment is often interpreted as evidence for string formation.While the spectrum, including the qualitative orderingof the energy levels, di#ers from the naive bosonic stringgaps for R < 1 fm, a high precision calculation showsthe rapid approach of ce!(R) to the asymptotic Casimirvalue in the same R range [13]. Although there is no in-consistency between the two di#erent findings, a deeperunderstanding of this puzzling situation is warranted.

We will return to this issue in a high precision study ofthe 3-dimensional Z(2) gauge model in a future publica-tion [14]. This accurate study of ce!(R) and the excita-tion spectrum of the Z(2) flux line for a wide range of Rvalues between 0.3 fm and 10 fm will clearly demonstratethe early onset of c # 1 without a well-developed stringspectrum. For now, Fig. 3 shows the lowest excitations inZ(2) for R = 0.7 fm, revealing a bag-like disorder profilesurrounding the static qq pair in the vacuum [14]. Thetwo lowest energy levels are substantially dislocated fromexact "/R string gaps and all other excitations form acontinuous spectrum above the glueball threshold. Sincethe submission of this work, a new study of Z(2) at fi-nite temperature has appeared [15], reporting very earlyonset of string behavior in support of Ref. [13].

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14

atEΓ

R/as

Gluon excitations

as/at = z*5

z=0.976(21)

β=2.5

as~0.2 fm

Πu

Σ-u

Σ+g’

ΔgΠg

Σ-gΠg’

Π’uΣ

+u

Δu

Σ+g

short distancedegeneracies

crossover

string ordering

N=4

N=3

N=2

N=1

N=0

FIG. 2: Short-distance degeneracies and crossover in thespectrum. The solid curves are only shown for visualization.The dashed line marks a lower bound for the onset of mixinge$ects with glueball states which requires careful interpreta-tion.

String limit. For R > 2 fm, the energy levels exhibit,without exception, the ordering and approximate degen-eracies of string-like excitations. The levels nearly re-produce the asymptotic "/R gaps, but an intriguing finestructure remains.

It has been anticipated that the interactions of mass-less excitations on long flux lines are described by a lo-cal derivative expansion of a massless vector field ! withtwo transverse components in four–dimensional space-time [11, 12]. Symmetries of the e#ective QCD stringLagrangian require a derivative expansion of the form

Le! = a#µ!·#µ!+b(#µ!·#µ!)2+c(#µ!·#"!)(#µ!·#"!)+...,(1)

where the dots represent further terms with four or morederivatives in world sheet coordinates. The coe$cient ahas the dimension of a mass squared and can be identifiedwith the string tension !. The other coe$cients must bedetermined from the underlying microscopic theory. Ex-amples with calculable coe$cients include the D=3 Z(2)

Fixes Mc = 1.84 GeV, √σ = .427 GeV, αs = 0.39

n(R) = [n] (string level) if no level crossing [n - 2 tanh(R0/R)] for Σ u potential (n=3)

Hybrid SpectrumThe leading Born-Oppenheimer approximation

In the leading Born-Oppenheimer approximation, one replaces the covariant Lapla-

cian DDD2 by an ordinary Laplacian ∇∇∇2, which neglects retardation effects. The spin in-

teractions of the heavy quarks are also neglected, and one solves the radial Schrödinger

equation:

!1

2µ

d2u(r)

dr2+

!

"LLL2QQ

#

2µr2+VQQ(r)

"

u(r) = E u(r), (2)

where u(r) is the radial wavefunction of the quark-antiquark pair. The total angularmomentum is given by

JJJ = LLL+SSS, SSS= sssQ+ sssQ, LLL= LLLQQ+ JJJg, (3)

where sssQ is the spin of the heavy quark, sssQ is the spin of the heavy antiquark, JJJg is the

total spin of the gluon field, and LLLQQ is the orbital angular momentum of the quark-

antiquark pair. In the LBO, both L and S are good quantum numbers. The expectation

value in the centrifugal term is given by

"LLL2QQ

# = "LLL2#!2"LLL · JJJg#+ "JJJ2g#. (4)

The first term yields L(L+1). The second term is evaluated by expressing the vectors interms of components in the body-fixed frame. Let Lr denote the component of LLL along

the molecular axis, and Lξ and Lζ be components perpendicular to the molecular axis.

Writing L± = Lξ ± iLζ and similarly for JJJg, one obtains

"LLL · JJJg# = "LrJgr#+12"L+Jg! +L!Jg+#. (5)

Since Jg± raises or lowers the value of Λ, this term mixes different gluonic stationarystates, and thus, must be neglected in the leading Born-Oppenheimer approximation. In

the meson rest frame, the component of LLLQQ along the molecular axis vanishes, and

hence, "LrJgr# = "J2gr# = Λ2. In summary, the expectation value in the centrifugal termis given in the adiabatic approximation by

"LLL2QQ

# = L(L+1)!2Λ2+ "JJJ2g#. (6)

We assume "JJJ2g# is saturated by the minimum number of allowed gluons. Hence, "JJJ2g#= 0

for the Σ+g level and "JJJ

2g#= 2 for theΠu and Σ

!u levels.Wigner rotations are used as usual

to construct |LSJM;λη# states, where λ = JJJg · rrr and Λ = |λ |, then JPC eigenstates arefinally obtained from

|LSJM;λη#+ ε|LSJM;!λη#, (7)

where ε = 1 for Σ+ levels, ε = !1 for Σ! levels, and ε = ±1 for Λ $ 1 levels. Hence,the JPC eigenstates satisfy

P= ε(!1)L+Λ+1, C = ηε(!1)L+S+Λ. (8)

�QQ(⌃r) =unl(r)

rYlm(�,⇥)

Spectroscopic notation of diatomic molecules

Πu


• Only interested in states below 4.8 GeV for cc system. Unlikely higher states will be narrow (DD, glueball+J/ψ, etc). Only Πu, Σu- , andΣg+‘ systems have sufficiently light states.

SSSSSS PPPPP DDDDState

3.8

4.0

4.2

4.4

4.6

Mas

s(G

eV)

�u Spectrum


3.8

4.0

4.2

4.4

4.6

Mas

s(G

eV)

��u Spectrum


3.8

4.0

4.2

4.4

4.6

Mas

s(G

eV)

�+�g Spectrum

• Πu (1S) = 4.132 GeV JPC = 0++, 0- -, 1+ - , 1- +

• Πu (1P) m = 4.294 GeV JPC = 1--, 1++, 0- +, 0+ -, 1+ -, 1- +, 2+ -, 2- +

• The Πu (1P), Πu (2P) and Σg +’(1S) have 1-- states with spacing seen in the Y(4260) system

• Numerous states with C=+ in the 4.2 GeV region.

Σu- Σg+’

Hybrid Spectrum


Hybrid Spectrum

53

• The spectrum of bottomonium hybrids is completely predicted as well

• For the Πu states

• Hybrid candidates for Y(4260) andΥ(5S) region

(cc) L n mass(GeV) (bb) L n mass(GeV)

0 1 4.132580 0 2 4.454556 0 3 4.752947

0 4 5.032962 0 5 5.298250 0 6 5.551412 1 1 4.293717 1 2 4.604123

1 3 4.893249 1 4 5.165793 1 5 5.424925 2 1 4.454768 2 2 4.753368

2 3 5.033384 2 4 5.298625 3 1 4.612335 3 2 4.900169 3 3 5.171746

4 1 4.765983 4 2 5.044143 5 1 4.915791

✓

0 1 10.783900 0 2 10.982855 0 3 11.172408

0 4 11.353469 0 5 11.527274 0 6 11.694851 0 7 11.856977 0 8 12.014256

1 1 10.877928 1 2 11.073672 1 3 11.259766 1 4 11.437735 1 5 11.608810

1 6 11.773931 1 7 11.933823 2 1 10.976071 2 2 11.167070 2 3 11.349124

2 4 11.523652 2 5 11.691737 2 6 11.854216 3 1 11.074034 3 2 11.260265

3 3 11.438320 3 4 11.609433 3 5 11.774550 4 1 11.170870 4 2 11.352563

4 3 11.526791 4 4 11.694614 5 1 11.266288 5 2 11.443727 5 3 11.614333

6 1 11.360209 6 2 11.533678 7 1 11.452636

✓


• The Y(4260) state is a JPC = 1-- state assosicated with the 1P state of the lowest-lying hybrid potential Πu (ccg)

• Using results from lattice for effective potentials the complete spectrum for QQ hybrid states can be prodicted (quenched). Spin dependent corrections required. (NRQCD + LQCD)

• The low lying states might be observable.

• Both exotic and non-exotic states are determined

– The Y(4350) and Y(4660) can also be understood as hybrid states.

– There should be nearby nonexotic hybrids with JPC = 1++, 0-+, 1+-, 2-+ and exotic hybrids with JPC = 0+ -, 1- +, 2+ - .

– The 1S states of the Πu potential lies in the region near 4.13 GeV and have

JPC = 0++ ,1+ - (non exotic), 0- -, 1- + (exotic)

• No free parameters in comparing c and b quark systems.– An analogy state to the Y(4260) should exist in the (bb) system near theΥ(5S)

54

Hybrid Spectrum


Summary

• Much can be learned about QCD dynamics using heavy quark (QQ) systems. The wealth of precision data solidifies our confidence in the NRQCD approach. Spectrum and EM transitions are well predicted by theory.

• For hadronic transitions the QCDME approach has many successes for states below threshold but is inadequate above threshold. Incorporation of strong thresholds and possible new degrees of freedom are required.

• Above threshold we see new states and possibly a new spectroscopy: X(3872), Y(4140), Y(4350), Y(4260), Y(4360), Y(4660), Zc+(4430), Zb+(10610), Zb+(10650), ...

– Emergence of additional threshold states and/or enhancements

– Systematic inclusion of the hybrid spectrum is possible.

• Lessons learned from heavy quark systems can be applied to light hadrons.

55

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