Leptonic decays, the nature of heavy quarkonia
and velocity counting rules
Juan-Luis Domenech-Garret a & Miguel-Angel Sanchis-Lozano b,*
a) Departament MACS, Física Aplicada, Universitat de LLeida, Spain
b) Departament de Física Teòrica & IFIC, Universitat de València – CSIC, Spain
*email: mas@ific.uv.es
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
1
Weak and strong coupling regime
of heavy quarkonium states
•
Weak coupling regime: the binding is essentially due to a Coulombic-like
potential. States below QQ threshold and not too deep
ΛQCD < mv2
are expected in the weak or perturbative regime
~
• Strong coupling regime: the binding is essentially due to the confining
potential. States deep in the potential are expected
ΛQCD > mv2
in the strong or non-perturbative regime
~
It is important to know the nature of heavy quarkonia, for example to
calculate rates of hindered transitions between (nS) and ηb(n’S) states
Brambilla, Vairo and Jia [hep-ph/0512369] relevant for hunting ηb states
In hep-ph/0511167 X. Garcia i Tormo and J. Soto employed radiative decays
to obtain important information on the nature of  resonances.
We present here a related idea using leptonic decays
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
2
Leptonic partial widths in  decays
| Rn (0) |2
1/ 2
2
2
[ (nS )  l l ]  4 Q
K
(
x
),
K
(
x
)

(
1

2
x
)
(
1

4
x
)
x

m
/
M
l

M 2
 
2
2
b
•
Leptonic partial widths are a probe of the compactness of the quarkonium system
providing important information complementary to spectroscopy.
•
NRQCD matrix (color-singlet and –octet) elements can be related to wave
functions at the origin. Potential models can provide the latter though they
can also be obtained without resorting to data fitting (e.g. lattice).
Still open questions about the accuracy (e.g. power counting in the
non-perturbative situation) and consistency within NRQCD!
There are different possible countings e.g., Brambilla et al hep-ph:0208019
Beneke hep-ph/9703429, Fleming, Rothstein and Leibovich hep-ph/0012062
•
Test of lepton universality (talk at BSM session) [arXiv:0709.3647]
BF[ee] = BF[μμ] = BF[ττ] K(x) ≈ 1
Therefore, a good knowledge of the  system is a basic ingredient
for seeking new physics if lepton universality were (slightly) broken
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
3
Partial decay width accoording to pNRQCD
Let us start with the general expression for the leptonic decay width of a vector resonance
(in the strong-coupling regime) :
 
[ (nS )  e e ] 
N C | Rn (0) |2


m2
N. Brambilla et al [hep-ph/0208019]

 En( 0) 2 3 2 3( 2, EM ) CF2 B1 
 En( 0) 1 
3
3
  Im g ee ( S1 ) 


 2 
Im f ee ( S1 ) 1 
2
2 
m
9
3
m
3
m
m
m 




Im g ee (3 S1 )   4 Q 2 2 / 9, Im f ee (3 S1 )   Q 2 2 / 3
m: heavy (bottom) quark mass
i , Bi : 6 universal (flavor and state independent) non-perturbative gluonic parameters
N. Brambilla et al hep-ph/0208019
Notably those not depending on “n”
[ (nS )  ee]

[ (rS )  ee]
Exp
QWG5 DESY
October 2007
r.h.s. (several terms drop out in the ratio)
Theory
Miguel A. Sanchis-Lozano
IFIC-Valencia
1 A
 1 A  B
1 B
4
In units of keV
Experimental values of partial*
ee[(1S)]
ee[ (2S)]
1.340 ± 0.018
ee[(3S)]
0.612 ± 0.011
0.443 ± 0.008
Radial wave functions at the origin: |Rn(0) (0)|2
Units of GeV3
Potential :
Cornell (1)
Cornell (2)
(1S)
14.05
12.23
12.22
6.477
(2S)
5.665
4.797
4.795
3.234
(3S)
4.271
3.581
3.579
2.474
Static potential:
V (r )  V
(0)
| Rn (0) |2  (mv3 ) (1  av  bv 2  )
| Rn (0) |2  | Rn( 0) (0) |2 (1  O( vq ))
QWG5 DESY
October 2007
Error in the ratios
≈ few %
Screened (2) Buchmüller-Tye (1)
V (1) (r ) V ( 2) (r )
(r ) 


m
m2
* PDG
(1) Eichten and Quigg, hep-ph/9503356
(2) P. Gonzalez et al, hep-ph/0307310
Miguel A. Sanchis-Lozano
IFIC-Valencia
5
Velocity counting rules
 nS | p 2 / m | nS   m v 2 ,  nS | V ( 0) | nS   m v 2 , r  1 / mv
perturbative regime
non-perturbative regime
s  v
V
(1)
m v
2
3
or V
(1)
m v
2
VLO
2
(1)
V
 V ( 0) 
m
V (1) (r )   s2 / r 2 or V (1)  c / r
Suggested by latice studies
Koma et al. hep-ph/0607009
V ( 2 )  m 3 v q , q  3, 4 ?
E
Spin-independent
( 0)
n
VLO  m v q , q  2, 3 ?
p2
V (1)
(0)
  nS |
V 
| nS 
m
m
VNLO / m 2  m v q , q  3, 4 ?
En( 0)  Er( 0)  ( M n  M r ) (1  O( v q )), q  1, 2
conservative
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
6
Comparing theory versus experiment
[ (nS )  e  e  ] | Rn( 0) (0) |2
q


(
1


)

(
1

O
(
v
))
nr
[ (rS )  e  e  ] | Rr( 0) (0) |2
 nr   4 3  2 3 9

  M r  M n   O( v q )



m

 ( )   ( ' ) 
Theory
0
ln
 s ( ' )
,  3 ( )   ( ) / 3
 s ( )
CF  4 / 3, C A  N C  3,  0  11C A / 3  4n f TF / 3
INPUT:
Exp
24 N cCF
?
OUTPUT:
[ (nS )  e  e  ]
[ (rS )  e  e  ]
 1   nr
( 0)
2
| Rn (0) |
 (1   nr )
( 0)
2
| Rr (0) |
O ( v q)
Collects all uncertainties from the
radial wave functions and δnr + exp error
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
7
Extraction of non-perturbative parameters
3.6
(1GeV) =
hep-ph/0109130
hep-ph/0007003
3.6 ± 2.9
3.5
5.3 ± 2.2
3(5 GeV) =
2.6
5.3
The value of 3 at μ=5 MeV can be extracted from
 nr  1 
[ (nS )  e  e  ]
[ (rS )  e  e  ]
| Rn( 0) (0) |2
| Rr( 0) (0) |2
 nr   4 3  2 3 9

Mr  Mn 



m

Experimental data from  leptonic decays, wavefunctions from
potential models and our evaluation of δnr suggest small 3 values
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
8
Results
Potential :
Δ13(%)
Δ12(%)
Δ23(%)
Cornell (1)
33
28
8
(1S,2S,3S), 3 (mb) =3.6
Cornell (2) Screened (2) Buchmüller-Tye(1)
37
37
17
31
31
12
10
10
8
Note that Δnr is systematically positive  the leptonic width ratio is overestimated
by the theory
| Rn (0) |2  (mv 3 ) (1  av  bv 2  )  | Rn (0) |2 (1  O( v 2 ))
| Rn(0) (0) |2 ?
using the VLO
in potential model calculations
 nr   4 3  2 3 9

  M r  M n   O( v 2 ) ?



m

We conclude: q = 2 is favored
unless cancellations between different sectors happen
Spectrocopy effects of V (1)/m in the lattice potential: good (better) agreement
between predicted and experimental mass levels ( ≈ several tens of MeV)
for the bottomonium family (on-going analysis)
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
9
Conclusions
In the leptonic decays of heavy quarkonium several parameters cancel out
in the ratios of partial widths
Still the ratio is sensitive to the gluonic universal parameter 3
Experimental data on (nS) leptonic decays together with the values of the
wave functions at the origin obtained from potential models favor:
• (2S) and (3S) belonging to the strong regime, while (1S) to the weak regime
• A not so conservative power counting, as naively expected from dimensional
counting in the non-perturbative regime.
• Low values for 3
More precise experimental measurements of leptonic decays should shed
light on the nature of  resonances, to be of great help in the search for
new physics effects, e.g., at high-luminosity (Super) B factories
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
10
QWG5 DESY
October 2007
Miguel A. Sanchis-Lozano
IFIC-Valencia
11