Fit Mass and Width of r
Sheldon Stone
Jianchun Wang
12/08/00
Introduction
 A wide wp resonance (mass ~ 1.4 GeV) observed
in BD(*)wp decays ( CBX 00-16, 00-31 )
 The resonance is identified to be 1- state r
 We want to determine the mass and width
 More details in CBX 00-68
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Jianchun (JC) Wang
2
Simple Breit-Wigner function
D*wp-
BW 
( M wp

- M r' )2  2 / 4
Fit Parameters:
M = 1434±35 MeV
 = 457±88 MeV
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Jianchun (JC) Wang
3
More Considerations
JP
B  D* r
0 - 1- 1-
r  w p1- 1- 0-
 P-wave decay r  wp-  Running width
 Kinematic limits from the two decays
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Jianchun (JC) Wang
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Differential Decay Distribution
 General expression:
d( B  D * wp) 
1
| A( B  D * r' )  BW (r' )  A(r'  wp) |2
2M B
dM 2wp
 dP( B  D * r' )  dP(r'  wp)
2p
 Assume two decay stages are independent and
can be factorized:
dM 2wp
d( B  D * wp)  ( B  D * r' )  2M wp (r'  wp) | BW (r' ) | 
2p
2
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Jianchun (JC) Wang
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Estimation of the B Decay Width
 ( B  D * r' ) 
1
| A( B  D * r' ) |2 dP( B  D * r' )
2M B
dP( B  D * r' ) 
1  2p D* 


8p  M B 
A( B  D * r' ) ~ G F Vbc  Lorentz Structure  g( M wp )
Use knowledge of BD*l decay (in HQET framework)
with factorization
( B  D * r' )  6p 2 | Vud |2 f r2' | a 1 |2 
 p D* M
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2
wp

2
d( B  D * l)
dq 2
q 2  M 2w p
2
 H  (M )  H - (M )  H 0 (M )
2
wp
Jianchun (JC) Wang
2
wp
2
wp
2

6
Helicity Amplitude
The effects of interaction are parameterized in terms of three form
factors A1, A2 and V
H  (q 2 )  ( M B  M D* ) A1 (q 2 ) 
H o (q ) 
2
1
2 M D*
2 M B p D*
V(q 2 )
M B  M D*
 2
4 M 2B p 2D*
2
2
2
2 
(
M
M
q
)(
M

M
)
A
(
q
)
A
(
q
)
B
D*
B
D*
1
2
2 
M

M
q 
B
D*

In the HQET framework, the form factors are:

 M B  M D*
q2
A1 (q )  1 h A1 ( w )
2
(
M

M
)
B
D*

 2 M B M D*
M  M D*
A 2 (q 2 )  R 2  B
h A ( w)
2 M B M D* 1
2
V(q 2 )  R 1 
M B  M D*
h A (w)
2 M B M D* 1
Heavy Quark Symmetry limit  R1, R2  1 hA1 Isgur-Wise function
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Jianchun (JC) Wang
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The B Decay Width
 hA1(w) = 1 - rA12 (w-1)
( CLEO: rA12 (0) = 0.910.150.06 )
 R1, R2
 Neubert:
R1(w) = 1.35 - 0.22(w-1) + 0.09(w-1)2
R2(w) = 0.79 + 0.15(w-1) - 0.04(w-1)2
 Close-Wambach
R1(w) = 1.15 - 0.07(w-1)
R2(w) = 0.91 + 0.04(w-1)
 CLEO Measurement:
R1(0) = 1.18  0.30  0.12
R2(0) = 0.71  0.22  0.07
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Jianchun (JC) Wang
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The r Decay Width
(r'  wp) 
1
| A(r'  wp) |2 dP(r'  wp)
2 M wp
dP(r'  wp) 
1  2p w 


8p  M wp 
A(r'  wp)  Lorentz Structure  h( M 2wp )
 Construction of P-wave Lorentz structure:

 r' w pr' pw
used for this case

(r'  pw )( w  pr' )
violates Parity conservation
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Jianchun (JC) Wang
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Lorentz Structure
| r'  w p r' pw |2  M r2 ' p 2w
(r'  wp)  h 2 ( M 2wp )  p 3w
r polarization w polarization Lorentz Structure
Non
Long
Long
Trans
Trans
Non
Long
Trans
Long
Trans
2 Mr2 pw2
0
Mr2 pw2 (1-cos2)
0
Mr2 pw2 (1+cos2)
Note: terms for longitudinal w is 0 due to parity conservation
Long/Tot measured: 109% (D*wp), -0.422%(Dwp)
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Jianchun (JC) Wang
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The Breit-Wigner Term
BW 
( M 2wp
1
- M r2' ) - iM wp tot ( M wp )
 Total width tot appears in the denominator
 Assume the mass dependence of tot can be approximated
by the mass dependence of wp
2
 h( M )   p w ( M wp ) 
wp ( M wp )
 

tot ( M wp )  tot ( M r ' )
 o  
 h( M )   p ( M ) 
wp ( M r ' )

  w r' 
2
wp
2
r'
12/08/00
Jianchun (JC) Wang
11
3
Decay Form Factor
 The dimension of h(Mwp) is Mass-1
 We try h(Mwp)  Mwp-n
 Blatt-Weisskopf factor (barrier penetration factor)
can also be included
1  R  p w ( M r' ) 
2
FF( M ) 
2
wp
12/08/00
1  R  p w ( M wp ) 
2
, R ~ 1 fm / c
Jianchun (JC) Wang
12
The Differential Distribution
  M 2wp
d( B  D * wp)
 C   ( B  D * r' ) 
dM wp
M 2wp - M r2' 2  M 2wp   2
 M r'

 M wp
 1  R  p w ( M r ' ) 

2


1

R

p
(
M
)

w
wp
Jianchun (JC) Wang
13
 p w ( M wp ) 

( M wp )  ( M r ' ) 
 p (M ) 
 w r' 
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3
n
2
Fit to the Spectrum
 (BDwp) slightly differ from (BD*wp)
 Weighted sum of all D(*)wp modes
M r '  1336  24 MeV
r '  510  58 MeV
12/08/00
Jianchun (JC) Wang
R
35
 1.90 -17..03
fm / c
n
 1.04  0.63
14
Systematic Error
Parameters
Mass (MeV) Width (MeV)
Neubert
1  
 
10  
 
 10
 
- 
- 10
Close-Wambach - 0
 
CLEO D*l 
 
Systematic Error  
 1
R  1
N  1
M = MD
12/08/00
Jianchun (JC) Wang
 1
15
Summary
 A more sophisticated form is used to fit the Mwp
mass spectrum from B  D*wp- decays
 The mass and width are measured to be
1336±24±9 MeV and 510±58±61 MeV
 Draft of paper is ready
 Acknowledge: Alan Weinstein, Deirdre Black
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Systematic Error Study
Mass (MeV)
Width (MeV)
R (fm)
n
2 / ndof
Prob
Neubert
Fit error
1
 

 
0
0
111
0

 00
- 0
Fit error (fix R, n)
Fit error (only R,n)
 
 
- 11
 1
- 101
 0
- 0
Fit up to 2 GeV
Fit up to 1.8 GeV
1 -1
1 -
 -
1 -
00
00



0
0
0
R-
R
R=0
1 -
1 
1 11
 1
1 -0
 
00
00
0000
100
00
0
11
1101
11
0
0
0
n-
n
n=0
11 
1 -1
1 0
1 1
 10
 -
1
110
0
-0
10
0000
11
111
101
00
01
0
R, n = 0
R,n contour 1 

1 -
1 

1 -0
 
M = MD
Close-Wambach
CLEO D*l 
Systematic Error
1 -0
11 -1
1 

1 -0
  -
1 

12/08/00
Jianchun (JC) Wang
11
1

01
0
01
0
101
101
11
0
00
0
17
R-n Contour Study
 400 points on R-n contour with 1 standard deviation
 Mass and width are fit with selected R and n
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Jianchun (JC) Wang
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