Yukawa Institute for Theoretical Physics - Kyoto, 2nd March 2015

Unquenched Vector Mesons withOpen-Beauty
Susana Coito
Institute of Modern Physics, Chinese Academy of Sciences
Collaborators: George Rupp, Eef van Beveren, Xiang Liu
I. Introduction
II. The RSE Model

III. Vectors with Open Beauty
IV. Summary and Conclusions

I. Introduction
Mesons with Open-Beauty: Experimental StatusbnDecays

JPbsDecays

JPbc

DecaysJP

B[5279]weak0−

Bs [5367]

weak0−

Bc [6277]

weak0−−−∗∗

B [5325]

Bγ

1B [5415]

Bs γ1

B1 (5721)0

B ∗π1+ ?

Bs1 (5830)0

B ∗K

1− ?

BJ∗ (5732)

B ∗ π, Bπ+∗?

B2 (5747)

Bπ, B π2 ?

B2∗ (5840)

BK , B ∗ K

2+ ?

B(5970)Bπ?

∗BsJ(5850)?

?

Vectors (most recent observations):
B ∗ : PRL 110, 151803 (2013) [LHCb]
Bs∗ : PRL 102, 021801 (2009) [Belle]
Recent observation: B(5970) PRD 90, 012013 (2014) [CDF] - is it a vector?

Puzzles within the Vectors
B∗

Bs∗

D∗Ds∗ ?

K∗

exp data5325

5415

20092112892

first th (OZI)Bπ 5417

BK 5775

Dπ 2005

DK 2363

K π 634

BE-92-360+4-251+258

ρ, ωφJ/ψΥ

775/783

1020

3097

9460

first th (OZI)

ππ 275

KK 991

DD 3734

BB 10559

BE+501/+508+28-638-1099

exp data

Some lattice results concerning the B mesons:
PRD 62, 114507 (2000); PRD 69, 094505 (2004)
arXiv: 1501.01646 [hep-lat]and Phenomenological Models:
PRD 89, 074042 (2014) unitarized approach
PRD 85, 094008 (2012) one-boson-exchange.

Unquenching in the models
QED - good knowledge of energy levels in H atom. well defined potential, wellknown spin-orbit corrections.
QCD - wished the energy levels of hadrons would be quenched andperturbative. In this case a confining potential would be clearly defined!
Some quenched models, e.g., Godfrey-Isgur model, assume meson spectrum as
a bare spectrum of the underlying ”funnel” (i.e. Coulomb + linear term)
potential, with spin-orbit corrections, without considering any other relevant
hadronic degrees of freedom. All states that do not fit in this spectrum must
be considered ’exotic’.
However this approaches fail to reproduce data, which reveals many
nonperturbative effects that cannot be justified by spin-orbit corrections,
namely deformation of Breit-Wigner line shapes and mass energies very
different from the underlying spectrum.
Unquenched approaches consider that resonances are not pure states, instead
they are strongly coupled to other important components of the hadronic sea
and to the nearby decay channels.

II. The Resonance Spectrum Expansion (RSE) Model
Elastic-Scattering: AB → CD, A, B, C , D are strongly interacting states.
Here, they are non-exotic mesons M. Born expansion:

MV =MV ΩV =q q̄
MMMMq q̄q q̄M− free meson-meson MM− confined q q̄ which includes a whole radial spectrum.
Transition mechanism: q q̄ annihilation/creation at the vertices, 3 P0 model.
Decays according to the Okubo-Zweig-Iizuka OZI rule.

MEffective potential, in momentum space:
Vij (pi , pj0 ; E ) = λ2 jLi i (pi a)Rij (E )jLj j (pj0 a)
String breaking - Spherical Bessel function ⇔ spherical delta function
Free parameters:
a - “string-breaking” distanceλ - global coupling
RSE formula:
Rij =∞
XXgnli c S gnlj Sc(l )clc ,S n=0 E − En

Coupling constants gnli c S , i, j - decay channels
the g ’s are computed within the 3 P0 model using expansions on aharmonic-oscillator basis
gn = rn /4n , where rn is a polynomial - rapid convergence of the series

Separable potential, Lippmann-Schwinger is evaluated in closed form.
Transition matrix:
NL ,LjTij iqX√(pi , pj0 ; E ) = −2aλ2 µi pi jLi i (pi a)
Rim {[11 − ΩR]−1 }mj jLj j (pj0 a) µj pj0 ,m=12(1)jΩij (kj ) = −2iaλ µj kj jLj j (kj a) hLj (kj a) δij .
Harmonic-oscillator confining potential:
known solutions, good results in phenomenological applications
En = mq + mq̄ + ω(2n + lc + 3/2)
Manifest unitarity of the scattering matrix - results are unquenched.
S =1 + 2iTResonances and bound states are poles of the scattering matrix(found in the 2nd Riemann Sheet in relation to the nearest threshold)III. Vector Mesons with Open Beauty
The B ∗ spectrum within the RSE:
B ∗0 , B ∗±ChannelTh (GeV)2
gn=0,l=02
gn=0,l=2(bn+nn) PPBπBηBη 0Bρ
Bω
B ∗πB ∗ηB ∗ η0B ∗ρB ∗ρ
B ∗ω
B ∗ω
Bs K
Bs K ∗
Bs∗ KBs∗ K ∗Bs∗ K ∗

5.417
5.827
6.237
6.055
6.062
5.463
5.873
6.283
6.101
6.101
6.108
6.108
5.862
6.261
5.911
6.309
6.309

0.020830.004390.00255
0.08333
0.02778
0.08333
0.01758
0.00255
0.006940.13889
0.002310.04630
0.013890.05556
0.05556
0.00463
0.09259

0.006940.001460.00085
0.00694
0.002310.006940.001460.00085
0.002310.000460.000770.000150.004630.004630.004630.001540.00031

PVVV(bs+sn) PPPVVVThe Bs∗ spectrum within the RSE:
Bs∗0Channel

Th (GeV)2
gn=0,l=02
gn=0,l=2(bn+ns) PPPVBKB ∗K
BK ∗B ∗K ∗B ∗K ∗Bs ηBs η 0Bs φBs∗ ηBs∗ η 0Bs∗ φBs∗ φ

5.775
5.821
6.173
6.219
6.219
5.915
6.324
6.386
5.963
6.373
6.435
6.435

0.02778
0.111110.111110.009260.185190.005100.00879
0.05556
0.020400.035150.00463
0.09259

0.00926
0.00926
0.00926
0.00309
0.000620.001700.002930.004630.001700.002930.001540.00031

VVVV(bs+ss) PP

PVVV

gn,l=0 = (2n/3 + 1)/4n , gn,l=2 = (n + 1)/4n

Fixed parameters (MeV),cf. PRD 27, 1527 (1983), and for mb EPJ 32, 493 (2004):
ω = 190, mn = 406, ms = 508, mc = 1562, mb = 4877
En=0,1,2 (MeV ) :(bn) 5568

5948

RSE (HO)6328

(bs)

5670

6050

6430

Experimental Data

Quenched Models †B ∗ , 5325

(1S) 5.32-5.37

bn1S5327

r0 =2 GeV−1

2S,1D

5748 − i5

BJ∗ , 5698 − i64 ?

(2S) 5.90-5.94

λ=102S,1D5845 − i1

B2 , 5743 − i11 ?

(1D) 6.02-6.12

3S,2D6214 − i5

B(5970) ?

(3S) 6.34-6.39

3S,2D6273 − i0

[5970 − i35]

(2D) 6.47-6.54

bs

1S5417

Bs∗ , 5415

(1S) 5.41-5.45

r0 =2 GeV−12S,1D5899 − i2

λ=6.3

2S,1D5980 − i4

∗ , 5853 − i23 ?
BsJ(1D) 6.12-6.21

3S,2D6336 − i9(3S) 6.43-6.48

3S,2D6378 − i1(2D) 6.54-6.63† cf. PRD 89, 054026 (2014)(2S) 5.99-6.02

RSE (HO)Exp Data

Quenched Models †1S5327

B ∗ , 5325

(1S) 5.32-5.37

r0 =2 GeV−12S,1D5748 − i5B(5970) ?

λ=102S,1D5845 − i13S,2D6214 − i5(3S) 6.34-6.39

3S,2D6273 − i0(2D) 6.47-6.54bn

1S5347r0 =2 GeV−12S,1D

5791 − i4

λ=72S,1D

5868 − i3

1S5351

r0 =3 GeV−12S,1D

5793 − i9λ=72S,1D

5871 − i3(2S) 5.90-5.94(1D) 6.02-6.12† cf. PRD 89, 054026 (2014).
Without increasing the constituent mass mb = 4.877 GeV to around 5 GeV we do notfind any pole around 5.97 GeV.
A proper adjustment of mb should be done comparing the results of the RSE model toother quantum numbers.

B ∗ , n=0 and n=0,1Re E (GeV)
5.40
5.50

5.30
•• ∗∗0λ = 100.000

5.70

Re E (GeV)
5.80
5.85

5.75

•

λ=0r0 = 2 GeV−1

λ=7-2r0 = 2 GeV−1∗-0.005••-8-0.010BπB ∗πr0 = 3 GeV−1-0.015

r0 = 3 GeV−1

Im E (GeV)-6

Im E (10−4GeV)

-4•

∗

∗

∗

Bη

∗λ = 7.0•λ = 10.0

5.90

Bs K

B ∗ηBs∗ K5.95

Bs∗ , n=0 and n=0,1λ = 6.3• •Re E (GeV)
5.50

5.60
λ=05.90

0.000

Re E (GeV)
5.95
6.00

•

-0.0026.05

r0 = 2 GeV−1•-0.004•-0.006

r0 = 3 GeV−1-0.008-0.010

Im E (GeV)

5.40•
Bs ηBs∗ η• λ = 6.3

Re E (GeV)
5.80
5.85

5.75

•

r0 = 2 GeV−1∗-0.005•-0.010r0 = 3 GeV−1

Im E (GeV)-0.015••5.95150Bπgreen: r0 = 2 GeV−1 , λ = 7
red: r0 = 3 GeV−1 , λ = 7∗100
∗

∗50Bη

∗λ = 7.0•λ = 10.0

5.90

σ (GeV−2)

0.000

5.70Bs KB ∗η
Bs∗ K

5.75

5.80
5.85
E (GeV)

5.90

green: r0 = 2 GeV−1 , λ = 710−1

red: r0 = 3 GeV , λ = 786644σ (GeV−2)8B ∗πσ (GeV−2)

10r0 = 3 GeV−1 , λ = 7BπB ∗π225.6

5.8E (GeV)

6.0

5.6

5.8
E (GeV)

6.0

Vector Mesons with Open Charm

RSE (HO)cn1S2010

Experimental Data
D ∗ (2007)0 , 2007 − i < 1.1
D ∗ (2010)± , 2010 − i0.48−1r0 =2 GeV

2S,1D2488 − i3D(2460), 2461 − i25 ?

2S,1D2570 − i4D(2600), 2609 − i45 ?

cs1S2113

r0 =2 GeV−12S,1D

2601 − i2

λ=4.5

2S,1D

2691 − i4

λ=5.6

Ds∗ , 2112 − i < 1VI. Summary and Conclusions• The resonances with open bottom are still very poorly known from theexperimental, and from the theoretical point of view.
• The unquenched and unitarized Resonance-Spectrum-Expansion model
gives different predictions than the quenched models.
• The newly observed B(5970) is unlikely to be a vector within the RSE.
• More data is needed in the energy region between 4 and 10 GeV.
∴Domo arigato gozaimashita!