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Form factors and wave functions in Coulomb gauge
I~ Lg[I I f-q ,'i N-"k'k~[ L 1 "!
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 386--389 North-Holland
Form factors and wave functions in Coulomb gauge G. M. Hockney a aTheoretical Physics Group Fermilab Batavia, I1 60500, USA In the static approximation, the charge density distribution of the light quark in a heavy-light system is a physical observable. This form factor ((BIj°(g)tB)) may be extracted from the lattice by putting the current at the light propagator source point and tying it to the static line a distance g away at some positive and negative time slice. When the heavy line connection is smeared at both ends using a source function derived from the Coulomb-gauge wave function, the result is a stable extraction of the charge density.
1. I N T R O D U C T I O N Even though the Coulomb-gauge wave function couples very well to heavy-light mesonic states[i][2], it does not give a very good idea of the form factors. There are two reasons for this: the wave function ignores both the non-singlet contributions and the relativistic effects that enter into the form factors. In a heavy-light system the charge distribution (Blj°(~)lS) m a y be easily extracted and compared to the square of the wave function.
2. E X T R A C T I O N
OF
3. E X T R A C T I O N RADIUS
THE
CHARGE
Figure 4 plots r 2(BIj ° (~)IB) for the 2-2 separation to give an idea of the charge radius. The vertical scale of these graphs is normalized to show the fraction of the total charge at each distance. The table shows the charge radius extracted from the graph, in both physical and lattice units. For the lightest state there is a significant noisy contribution at large distances which leads to the large error, but the peak is clearly visible on the graph.
~;
0.151
Table 1 0.154
0.1545
M
0.287 580
0.160 400
0.080 200
5.9q-0.6 .474-.05
7.7 4- 1.4 .614- .11
9.2 4- 2.7 .744- .21
(BIj°(~)IB)
Figures 1, 2 and 3 show the charge density as a function of radius extracted at source distances of 2, 3, and 4 time-slices to either side of the operator for three ~ values corresponding roughly to light quark masses of 0.080, 0.160, and 0.237 in lattice units of a b o u t 2.4 GeV. For this 243 x 48,f~ = 6.1 lattice a is approximately 0.08fm. The graphs are the average over 50 configurations. The source smearing function was a slight modification of the Coulomb-gauge wave function optimized to couple to the lower-lying (B I state. As the graphs show, with this smeared source the extraction is stable as a function of distance even as it becomes noisy as the source distance increases.
MeV (r)
fm
4. C H A R G E FUNCTION
DENSITY
VS.
WAVE
Finally the charge distribution, which is a physical quantity, is compared to the Coulomb-gauge wave function. Figure 5 shows the charge distribution from this c o m p u t a t i o n plotted with the square of the source wave function (which is in some sense the singlet-state density in Coulomb gauge), and the difference between the two. The
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All fights reserved.
SSD10920-5632(94)00290-C
OF
G.M. Hockney /Form factors and wave functions in Coulomb gauge
scales have been adjusted so the curves match at the origin, where one might naively expect the singlet piece to dominate (and the wave function at the origin is, after all, a gauge-invarient object). On these plots it is clearly visible that the charge distribution is wider than the the square of the wave function. REFERENCES 1. A. Duncan, E. Eichtcn, H. Thacker, Phys. Lett. B303 (1993) 109. 2. A. Duncan et al., these proceedings.
387
0.01
.
.
.
.
0.001
0.0001
10 -6
+ -trim+ lO-a
]
10 -?
,
J
~
10 r (lattice
0
i
I 30
20 units)
0.01 . . . .
I
. . . .
3-3
I
. . . .
i
ic 0 . 1 5 1
0.00
0.000
10 -6
+ ~
+
10 -s
I
10 -v
,
,
L
10 r (lattice
i
I
J
J
i
20 units)
30
0.01 . . . .
I
. . . .
4-4
tc
]
. . . .
0.151
\+
0.001
0.0001
10-6 .......... +; +41-~- 4-t4~
10-s
10-"
. . . . 0
. . . .
I
I0 r (lattice
I
20 units)
. . . .
30
Figure 1. Charge density for s -- 0 . 1 5 1 at separation of 4, 6, and 8 lattice spacings.
388
G.M. Hockney/Form factors and wave functions in Coulomb gauge
0.01
0.01
. . . .
I
. . . .
2-2
[
. . . .
r[
. . . .
'
'
0.001
0.0001
i0-6
I
'
1
'
'
I
.
'
'
0.1545
%
O.OOl
0.0001
'
~
2-2
K: 0 . 1 5 4
0 -6
++ ' '~g-'4-'
+
10 -8
10 -e
I0
. . . .
-7
I
. . . .
I
10
. . . .
lO-V
20
.
.
.
P ....
.
10
30
0.01 j
. . . .
3-3
I
0.01
. . . .
. . . .
I
. . . .
O.OOl %
0.001
0.0001
[
*
3-3
~: 0 . 1 5 4
10 -5
÷ + .
.
.
I
.
0
. . . .
I
%b+ 1+1
+
0,001
~-++ ÷
10 -8
+
30
0.01
I
,
I
]
,
I
&
I
[
I
I
I
[
10 20 r (lattice units)
....
30
I .... I .... 4-4 /c 0 . 1 5 4 5
0.154
%
000lL~-~
÷
0.0001
,
0
°°iE ~
. . . .
I0 -e
10-7
. . . .
10 20 r ( l a t t i c e units)
4-4
30
0.1545
+
1o I ~
.
~ - ~
0.0001
I 0 -a
.
r (lattice units)
r (lattice units)
. . . .
.
20
++ o.oooi
~+ ~ =
L-
J~.,+
+++
+ + l 0 -5
÷,÷:,if_+ ~
,
÷
~F +4-b"H-
10-6 10-" ,
++
l
i
l
l I0
l
i
l
I
I
20
l
l
l
10-7~,
i
30
r (lattice units)
Figure 2. Charge density for ~ - 0.154 at separation of 4, 6, and 8 lattice spacings.
0
,
,
,
I
10
,
,
,
,
I
20
,
,
,
,
30
r (lattice units)
Figure 3. Charge density for R = 0.1545 at separation of 4, 6, and 8 lattice spacings.
G.M. Hockney /Form factors and wave functions in Coulomb gauge
I ....
o.oo6~ . . . . q
0.20o. 15 [- .... +'~-+~'w~+**-~ ] '~....= 0.151 I ....
-
389
vs.
I ....
~*-2
aL
0.151
0.004
0.002 ~ .
+
0.10 -- ++ 0.000
o o
-0.002 0.00
EO
iO r (lattice
....
i
. . . .
l
+~++$+
¢
,
i
I
. . . .
I
10
,
,
i
20
3O
30
units)
0.004 0.15
i
. . . . = 0.154
I
. . . .
I q
. . . .
. . . . ~*~2 at
vs.
I . . . . 0.154
0.003 0.002
0.10
0.001 x ~ x ~ 0.05 -+
+ + l -0.001
0.00
. . . .
I
. . . .
I
10
0
20
r (lattice
~
. . . .
r
I
L
,
i
i
10
i
i
i
2O
L
30
units)
I . . . . I ~ = 0.1545
+
,
30 0,004
0.15
,
0
. . . .
. . . .
L q vs.
. . . .
. . . . @~2 at
i . . . . 0.1545
0'003 I 0.002
o. o
+
:~
0.001 - - ~ 005
+
+-~
+$+
+
o.ooo +
+
+
r
3
+ + -0.001
o.oo
%~j~t~--
+
,
,
,
,
I
,
,
10 r (lattice
,
,
I
20
,
,
,
~
~
i
i
Ii
10
,
i
i
i
i , i i
i
20
30
30
units)
Figure 4. Charge radius extraction from charge density.
2-2
Figure 5. Comparison of charge density and square of wave function at 2-2 separation. + is charge density, x is square of wave function, and o is the difference.
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 386--389 North-Holland
Form factors and wave functions in Coulomb gauge G. M. Hockney a aTheoretical Physics Group Fermilab Batavia, I1 60500, USA In the static approximation, the charge density distribution of the light quark in a heavy-light system is a physical observable. This form factor ((BIj°(g)tB)) may be extracted from the lattice by putting the current at the light propagator source point and tying it to the static line a distance g away at some positive and negative time slice. When the heavy line connection is smeared at both ends using a source function derived from the Coulomb-gauge wave function, the result is a stable extraction of the charge density.
1. I N T R O D U C T I O N Even though the Coulomb-gauge wave function couples very well to heavy-light mesonic states[i][2], it does not give a very good idea of the form factors. There are two reasons for this: the wave function ignores both the non-singlet contributions and the relativistic effects that enter into the form factors. In a heavy-light system the charge distribution (Blj°(~)lS) m a y be easily extracted and compared to the square of the wave function.
2. E X T R A C T I O N
OF
3. E X T R A C T I O N RADIUS
THE
CHARGE
Figure 4 plots r 2(BIj ° (~)IB) for the 2-2 separation to give an idea of the charge radius. The vertical scale of these graphs is normalized to show the fraction of the total charge at each distance. The table shows the charge radius extracted from the graph, in both physical and lattice units. For the lightest state there is a significant noisy contribution at large distances which leads to the large error, but the peak is clearly visible on the graph.
~;
0.151
Table 1 0.154
0.1545
M
0.287 580
0.160 400
0.080 200
5.9q-0.6 .474-.05
7.7 4- 1.4 .614- .11
9.2 4- 2.7 .744- .21
(BIj°(~)IB)
Figures 1, 2 and 3 show the charge density as a function of radius extracted at source distances of 2, 3, and 4 time-slices to either side of the operator for three ~ values corresponding roughly to light quark masses of 0.080, 0.160, and 0.237 in lattice units of a b o u t 2.4 GeV. For this 243 x 48,f~ = 6.1 lattice a is approximately 0.08fm. The graphs are the average over 50 configurations. The source smearing function was a slight modification of the Coulomb-gauge wave function optimized to couple to the lower-lying (B I state. As the graphs show, with this smeared source the extraction is stable as a function of distance even as it becomes noisy as the source distance increases.
MeV (r)
fm
4. C H A R G E FUNCTION
DENSITY
VS.
WAVE
Finally the charge distribution, which is a physical quantity, is compared to the Coulomb-gauge wave function. Figure 5 shows the charge distribution from this c o m p u t a t i o n plotted with the square of the source wave function (which is in some sense the singlet-state density in Coulomb gauge), and the difference between the two. The
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All fights reserved.
SSD10920-5632(94)00290-C
OF
G.M. Hockney /Form factors and wave functions in Coulomb gauge
scales have been adjusted so the curves match at the origin, where one might naively expect the singlet piece to dominate (and the wave function at the origin is, after all, a gauge-invarient object). On these plots it is clearly visible that the charge distribution is wider than the the square of the wave function. REFERENCES 1. A. Duncan, E. Eichtcn, H. Thacker, Phys. Lett. B303 (1993) 109. 2. A. Duncan et al., these proceedings.
387
0.01
.
.
.
.
0.001
0.0001
10 -6
+ -trim+ lO-a
]
10 -?
,
J
~
10 r (lattice
0
i
I 30
20 units)
0.01 . . . .
I
. . . .
3-3
I
. . . .
i
ic 0 . 1 5 1
0.00
0.000
10 -6
+ ~
+
10 -s
I
10 -v
,
,
L
10 r (lattice
i
I
J
J
i
20 units)
30
0.01 . . . .
I
. . . .
4-4
tc
]
. . . .
0.151
\+
0.001
0.0001
10-6 .......... +; +41-~- 4-t4~
10-s
10-"
. . . . 0
. . . .
I
I0 r (lattice
I
20 units)
. . . .
30
Figure 1. Charge density for s -- 0 . 1 5 1 at separation of 4, 6, and 8 lattice spacings.
388
G.M. Hockney/Form factors and wave functions in Coulomb gauge
0.01
0.01
. . . .
I
. . . .
2-2
[
. . . .
r[
. . . .
'
'
0.001
0.0001
i0-6
I
'
1
'
'
I
.
'
'
0.1545
%
O.OOl
0.0001
'
~
2-2
K: 0 . 1 5 4
0 -6
++ ' '~g-'4-'
+
10 -8
10 -e
I0
. . . .
-7
I
. . . .
I
10
. . . .
lO-V
20
.
.
.
P ....
.
10
30
0.01 j
. . . .
3-3
I
0.01
. . . .
. . . .
I
. . . .
O.OOl %
0.001
0.0001
[
*
3-3
~: 0 . 1 5 4
10 -5
÷ + .
.
.
I
.
0
. . . .
I
%b+ 1+1
+
0,001
~-++ ÷
10 -8
+
30
0.01
I
,
I
]
,
I
&
I
[
I
I
I
[
10 20 r (lattice units)
....
30
I .... I .... 4-4 /c 0 . 1 5 4 5
0.154
%
000lL~-~
÷
0.0001
,
0
°°iE ~
. . . .
I0 -e
10-7
. . . .
10 20 r ( l a t t i c e units)
4-4
30
0.1545
+
1o I ~
.
~ - ~
0.0001
I 0 -a
.
r (lattice units)
r (lattice units)
. . . .
.
20
++ o.oooi
~+ ~ =
L-
J~.,+
+++
+ + l 0 -5
÷,÷:,if_+ ~
,
÷
~F +4-b"H-
10-6 10-" ,
++
l
i
l
l I0
l
i
l
I
I
20
l
l
l
10-7~,
i
30
r (lattice units)
Figure 2. Charge density for ~ - 0.154 at separation of 4, 6, and 8 lattice spacings.
0
,
,
,
I
10
,
,
,
,
I
20
,
,
,
,
30
r (lattice units)
Figure 3. Charge density for R = 0.1545 at separation of 4, 6, and 8 lattice spacings.
G.M. Hockney /Form factors and wave functions in Coulomb gauge
I ....
o.oo6~ . . . . q
0.20o. 15 [- .... +'~-+~'w~+**-~ ] '~....= 0.151 I ....
-
389
vs.
I ....
~*-2
aL
0.151
0.004
0.002 ~ .
+
0.10 -- ++ 0.000
o o
-0.002 0.00
EO
iO r (lattice
....
i
. . . .
l
+~++$+
¢
,
i
I
. . . .
I
10
,
,
i
20
3O
30
units)
0.004 0.15
i
. . . . = 0.154
I
. . . .
I q
. . . .
. . . . ~*~2 at
vs.
I . . . . 0.154
0.003 0.002
0.10
0.001 x ~ x ~ 0.05 -+
+ + l -0.001
0.00
. . . .
I
. . . .
I
10
0
20
r (lattice
~
. . . .
r
I
L
,
i
i
10
i
i
i
2O
L
30
units)
I . . . . I ~ = 0.1545
+
,
30 0,004
0.15
,
0
. . . .
. . . .
L q vs.
. . . .
. . . . @~2 at
i . . . . 0.1545
0'003 I 0.002
o. o
+
:~
0.001 - - ~ 005
+
+-~
+$+
+
o.ooo +
+
+
r
3
+ + -0.001
o.oo
%~j~t~--
+
,
,
,
,
I
,
,
10 r (lattice
,
,
I
20
,
,
,
~
~
i
i
Ii
10
,
i
i
i
i , i i
i
20
30
30
units)
Figure 4. Charge radius extraction from charge density.
2-2
Figure 5. Comparison of charge density and square of wave function at 2-2 separation. + is charge density, x is square of wave function, and o is the difference.