Lattice form factors: An update

524

Nuclear Physics B (Proc. Suppl.) 4 (1988) 524-530 North-Holland, Amsterdam

LATTICE FORM FACTORS: AN UPDATE* Terrence Draper and Richard Woloshyn TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3

Walter Wilcox Department of Physics, Baylor University, Waco, TX, USA 76798

Keh-Fei Liu Department of Physics, University of Kentucky, Lexington, KY, USA 40506

A calculation of the SU(3) pion electric form factor on an 103x29 lattice at 1~=5.9 using a technique that treats the zero momentum pion field as a secondary source is described. 1. INTRODUCTION Lattice hadron

the nonzero momentum

form

factors

internal

comparable

structure

with

a

useful

and

are

ultimately

In

previous

experiment.

publications,

systematic

SU(2)

pion

lattice

are

studies

form

of

probe

the

factor have

of

quenched

been carried

lattice two or three point

functions. We

will

next

explain

some

of

the

relevant

technical aspects of this study. This will be followed by

an

evaluation

discussion and

of

results

and

a

concluding

summary.

out for the Wilson 1 and staggered fermion 2,3 cases. We

report

quenched

here

on

SU(3)

some

pion

partial

form

results

factor

using

for

the

Since SST has been explained in other contexts5,

present

simply state how this technique has been applied.

fermions. There

2. TECHNICAL ASPECTS

Wilson

we will not elaborate on it further here, but will are

several

calculation besides

new

the use

aspects

to

of SU(3).

the

Since

it is

In

previous

workl'3, 7

an

actual

source

necessary to extract matrix elements of the lattice

exponentiation was carried out. That is, the current

charge

density

density,

one

must

devise

a

calculational

scheme to evaluate lattice three point functions. A technique

to

do

this

"exponentiation"

in

it here

sequential

as the

the

has

been

called

literature 4, but we refer to source technique (SST).

operator at

was

included

derivative three

explore higher momenta as well as to use a variety

momentum

of operators to probe the pion,

numerical

limit

ourselves to

examining the conserved vector

position

and

in the

of

the

fermion

two

points

action.

A numerical

function

was

then

taken with respect to a, which produced the desired

The manner in which we employ SST allows us to

although here we

some fixed time

spatial momentum, multiplied by a parameter a << 1,

point

exactly

function transfer. derivative

the

desired

at

a

SST

single

value

is

superior

method

since

combination

of

it

of

spatial to

this

produces two

point

charge density. Our method of analysis is also new

functions. The use of SST to simulate the effect of,

and reveals an unexpected and useful behavior of

for example, axial or vector current densities, has

* This research was supported in part by the Natural Sciences and Engineering Research Council of Canada and in pan by the U.S. Department of Energy under grant DE-FG05-84ER40154.

0920 5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

T. Draper et al. / Lattice form factors

the

advantage

of

hadronic

state,

functions

for

not

fixing

so that

the

one can

various

initial

or

final

form three point

interpolating

fields

from

525

>> tl >> 1, we have that6 Gnn(t;p) ~ Ns IC(p)I 2 e "Et

(3)

a

single set of quark propagators. On the other hand, it

limits

some

the

possible

largest

source

value

probe.

four momentum

and

These

fixes

the

limitations

transfer to

nature are

of

avoided

probes

then at

transfer.

reconstruct

any

Of

desired

course

one

the

effects

of

lattice

spatial

momentum

then

has

various

propagators

e'm(t2"tl)e-Etl

(4)

if

as the SST secondary source. In a separate analysis can

x

the

instead one uses the initial or final hadronic state

one

Gnjn(tl;q) ~ N 2 C(0)C*(q)

Our

method

is

based

upon

the

analysis

of

the

normalized two and three point functions: G~jn(tl;q) Gnn(t2;0)

_~'-~" e "(E'm)tl C~(0)

(5)

Gnn(t;p) Gnu(t;0) ~

C e_(E.m)t cC(~0) [2

(6)

in

which not the probe but one of the sources (and it's associated momentum) is fixed. We have chosen to examine the consequences of the use of SST in the latter

situation in order to explore lattice electric

Fitting

the

normalized propagator

data

to Eqs.(5)

form factors at higher momentum tranfers. In this

and (6) yields a value for the matrix element of the

study we do not necessarily expect to be in a regime

charge

density

where lattice artifacts are not detectable. However,

electric

form

we

continuum

hope

to

begin

to

understand

some

of

the

operator, factor

limit.

in

We

is

related

to

the

which

usual

manner

in

the

prefer

to

with

the

work

the

systematics involved in attempting to reach such a

normalized propagators since we will see that this

regime.

data is numerically quite well determined. In order

The

lattice

two

and three

point

functions that

to

guide

us

in

the

choice

of

the

two

point

propagator data to use in our fits, we define the

we measure are

local mass and energy by ' Gnn(t;p) = Z e -ip-x

<01T(0(x)0l(0))10>

(I)

x

Gnjn(tl;q)=

,'Gnn(t-1;0)X n ~ -)

(7)

• ~(t-1;p)-~ E(t) = m~'"~nn(t:p) ~

(8)

m(t) = l Z e - i q 'xl <01T(¢(x2)J4(xl)¢'l(0))10> Xl ,x2

(2)

Of course, the right hand side of Eq.(2) is also a function

of

t 2 , the

interpolation

time

field,

position

¢(x2),

but

of in

pion

and look for a time region where these quantities

present

have leveled off. An alternate measurement of the

the

the

t2 will be fixed. The spatial momentum

local energy minus mass, which we use to check

transfer in (2) is q, In calculating (2) on the lattice

consistency, is also available from the use of the

it is necessary to leave off the effect of current

normalized three point function, Eq.(5).

application

self-contractions, which

do

in

they

the

SU(3)

case

as

not do

necessarily for

SU(2).

vanish This

Our 103x20

original

quenched

configurations

were

and were calculated using the Monte Carlo

represents the interaction of the probe with a sea

Cabbibo-Marinari

rather than a valence quark. In the formal limit t2

fundamental Wilson action was used with periodic

p s e u d o h e a t b a t h 8.

The

SU(3)

T. Draper et al. / Lattice form factors

526

boundary

conditions

and

13 = 5.9. W e do not have

justification for this choice in that a certain type of

e n o u g h m a s s data yet to h a v e set the lattice scale or

total

measure

in

idea

K c r at this 13. However, in order to give an

o f the

physical

box

size,

although

we

realize

derivative

(9)

term c a n c e l s b e t w e e n

in the

combination

classical

we are not in a perturbative scaling regime, the 13 =

zero

6.0 results of Ref. 9 scaled to 5.9 give a w --- .12fm. The

s u p p o s e d l y located

g a u g e field was thermalized for 5000 s w e e p s after a cold

start

and

sweeps

10 c o n f i g u r a t i o n s

were

1/3Re,

saved. equals

Our

region

evolution

characteristics

configurations

in

In

which

were

by

average

.5822(2).

adequate

separated

of

plaquette,

order to

1000

to

have

examine

propagators,

lengthened

to

an

momentum

We

used

quark

were

by

criterion

103x29

gradient

we

gauge

At p r e s e n t

this

profile

at

current

we

demanded

technique

As

a

test

checked invariant

parameters. we

Also,

time

conserved

evaluation.

propagators accuracy.

local

conjugate

program

hopping

a

limit.

correct

exactly at time t.

the

propagator

these

the

for

computer

time

continuum

possesses

the two t e r m s

final

For

our

that

the

of

the

that

up

have

for

to

pion machine

results for two convergence

maximum

change

duplicating part o f the g a u g e fields in time. For the.

in the absolute s u m of squares o f the quark or SST

quarks

propagators

we u s e d periodic

spatial

directions

conditions,

boundary

and

which

conditions

"fixed"

consist

of

time

setting

the

couplings

across the time edge to zero.

boundary

conditions

exact,

and

this

inversion also

the

provides

algorithm.

preserve

propagators.

lattice

gauge check

These

exactly The

a

the

origin

of

With

on

the

fixed is

quark

conditions

condition

all

quark

invariance

boundary

CPT

in the

boundary

quark

on

quark

propagators

As

one

be

less

check

than

of

the

propagator origin

was

SST

lattice

seen

to

propagator

Ward

conservation

convergence

accuracy.

pion

jackknife

25.

We

that

far

from

nonvacuum In

these positions

lattice

time

are

sufficiently

boundaries

to

avoid

contaminations.

extracting

results

In

our

the

normalized

two

10

propagators

with

O(10-4).

an

origin

to

In addition,

the

responsible

is

with

another In

technique

fits

functions single

from

to

guarantees,

zero m o m e n t u m

the

secondary

for

our

charge boundary

conditions, that the pion charge is one. W e m e a s u r e which

expect

5 iterations.

agree

identity

1+O(1 0-4),

fixed at time site

SST

over

made possible by exact CPT, an origin to origin pion

was chosen to be at lattice time site 5; the secondary source was

5x10 5

of

over

all

for

cases

a

given

we

estimating

normalized

exponential

indication

two

time

and

interval,

chi-squared

of

the

used

the

error

bars.

three

point

we

used

a

fit. 1 1

3. RESULTS

and three point functions, it is n e c e s s a r y in the fits to

assign

time

to

the

location,

conserved time.

We

which

is

labelled lattice

tl

such

operator

in Eq.(2). density

of the

symmetrical

currents

density

charge

u s e a form a

nonlocal

charge

a unique

However, is

nonlocal

the in

vector current, j 4 ( t ) ,

combination

of

conserved

Some

of

presented referred

to

Figs.

at

that (9)

is defined t

and

t-l.

at sites t + l There

a consecutive

1 and 2 were

mass

from the two momentum

j+

of

this

investigation

1-8. R e s u l t s in these labelling

of

are

figures are lattice

time

useful

as a

guide

in our

choice o f propagator time data. T h e y show the local

j4(t) = 1/2( j+ + j-)

defined

results

sites from 1 to 29.

energy,

where

the

in Figs.

is

and

t and j- is

some

theoretical

energy

point

transfers

(~/5,n/5,n/5), equivalent

and

and lattice

minus

functions. were

mass

The

q values

the pion

measured

q = (n/5,0,0),

(2n/5,0,0).

for

We and

spatial

(n/5,n/5,0),

averaged

over

treated

the

T. Draper et al. / Lattice form factors

averaged

results as a single d a t u m

routines.

The

that

(E-m)

alone,

lower

is statistically

i.e.,

normalized The

portion

that two

(E-m)

point

also

place

examined

for

and

at both

allows us to reliably time

which

values

improves

the

from

site

9,

which

time

site

error

bar

8

onwards

in

these

relation

figures

results

for

m a s s e s for a 17 to 25 fit.

become

flat

individually.

This

to

transfers values.

or

m

It

individually,

estimates.

that

lines

dispersion

mass. Table 1 gives the results for the pion and rho

In

all

of

results are seen to plateau at

means

horizontal

out.

parameter

for E

The

continuum

cancel

to

momentum

than

the

the

seen

hopping

show

in

tend

all

individually.

E

fit the n o r m a l i z e d f u n c t i o n s at

our g r a p h s the (E-m) time

are

show than E

fluctuations

s o o n e r than E or m

took

graphs

determined

functions

behavior

smaller

these

better

statistical

results

significantly

of

in the j a c k k n i f e

527

the p r o p a g a t o r falls

like

a

data single

exponential. W e selected time steps 8 through

and

(E-m)

given

the

lattice

result

for

the pion

Figs. 3 and 4 s h o w the local (E-m) v a l u e s from the

time

Eq.(4). the

two

aspect

behavior

of

the

three

point

function,

T h e s e v a l u e s are seen to be c o n s i s t e n t with point

function

o f the

results.

normalized

A

three

quite

point

remarkable

function

seen

in these figures is the fact that it falls like a single exponential

from

time

site

6 onwards

in all cases.

(Remember, time site 5 is the location of one of the local pion fields.) T h u s , an 8 to 12 fit of this data is quite

12 in

conservative

A

as far as e s t i m a t i n g error bars.

comparison

of

lattice

and

continuum

all of our exponential fits of the normalized two and

dispersion,

three point

functions, is carried out in Figs. 5 and 6. Since we

functions, Eqs.(5)

and (6).

One can see

using

the

8 to

12

time

fit

two

point

in Figs. 1 and 2 that such a time interval would not

find

be

determined than E alone, we have plotted (E-m) as a

appropriate

if

we

were

to

fit

E

and

m

I I I I I I I I I I 1 1 1

2.6

that

(E-m)

Eolol

2.6

oo

K=O. 140

2.4-

t

2.4-

D

2.01.8 1.6

0 A

0 °

•

Es- ~

o E a

•

E2- ~

n

•

Ej-~

Re o E~

I

2.0-

1.6-

t

I

I

I

I

and

I

reliably

1

I

E a

K=0.148

c

o

~0

easily

o •

Ea-.vn,

c

A

Ee-~ ~

a

* E~ m

n o~

1.8-

r.q 1.4

I

2.2-

O~

0

more

ao ao o

2.2-

is

o Eo=132

u~

zx o

1.4-

~

T

nZ~I~

~ 1.2

a; 1.o

~4 1.0 0.8--

0.8

0.6--

0.6

0.4--

0.4

•

0.2 -' 0.0

I

I

3 5

[

I

I

I

I

I

I

I

I

7 9 11131517192123252729 TIME SLICE

FIGURE 1 T h e local e n e r g y , m a s s and e n e r g y m i n u s m a s s in lattice u n i t s f r o m the n o r m a l i z e d pion two p o i n t function as a function of time for K=.140.

0.0

I 3

I 5

I 7

I 9

I I 1 I I I I I I 1113151719212325272~ TIME SLICE

FIGURE 2 Same as Fig.1 but for K=.148.

--

Re o E, --

528

T. Draper et al. / Lattice f o r m factors

.8

[

[

.7-

I

[

[

[

I

I

I

K=0.140

I

[

[

I

• Es-rre

1 f

.8

-

*7 --

-

.6-

I

I

I

[

L I

I

K=0.148

I

t

I

• E3--TTb

• Ee-va

• E2-,tre .6-

I

• Et-,rn

,5

E1--m

•

--

~.4-

.3

,~

-

- t

,2

-

--

1][

.0

I

I

E

I

I

I

I

I

I

I

I

J

.0

I

I

FIGURE 3 L o c a l e n e r g y m i n u s m a s s f r o m the n o r m a l i z e d p i o n three point f u n c t i o n as a f u n c t i o n o f t i m e for K=.140. .8

.6 -

I

[

I

I

I

I

I

I

I

I

I

I

I

I

I

L

1

I

3 5 7 9 11131517192123252729 TIME SLICE

3 5 7 9 11f31517192123252729 TIME SLICE

FIGURE 4 S a m e as Fig. 3 but for K=.148.

]

.8

K=O.140

I

.6 -

~.4-

I

[

I

I

I

I

1

K=0.148

~.4-

r

.2

.19 T I t 0.0 0.2 0 . 4 0 . 6 0 . 8

I 1.0

p/m

I I 1.2 1.4

I 1.6

.0 L8

FIGURE 5 Pion energy minus mass versus momentum, both s c a l e d by m a s s f r o m the 8 to 12 t i m e fit of the n o r m a l i z e d two p o i n t f u n c t i o n at K = . 1 4 0 . T h e p i o n m a s s w a s m e a s u r e d s e p a r a t e l y in a 17 to 25 t i m e fit o f the zero m o m e n t u m two point function.

-

-

I

I

[

I

0.0 0.2 0 . 4 0 . 6 0 . 8

I

I

I

l

1.0

1.2

1.4

1.6

FIGURE 6 S a m e as Fig. 5 but for K=.148.

1.0

T. Draper et aL / Lattice form factors

529

function of the momentum, both scaled by mass. We

higher m o m e n t u m transfers.

see that the K=.140

found

violate

result seems to

continuum

dispersion.

systematically

However,

the

In so doing,

we have

a useful behavior of normalized lattice

and three point

two

functions. This behavior allows us

comparison has significantly improved at K=.148, as

to

one would

The error bars on the form factor data are also seen

expect for lower lattice mass, with the

exception of the highest m o m e n t u m point. Our

final

results

for

the

pion

reliably

check

the

lattice

dispersion

relation.

to be quite reasonable.

electric

form

We have

seen that

the form

factor results are

factor as a function of Minkowski four m o m e n t u m

systematically higher than would

squared are given in Table 1 and shown in Figs. 7

vector dominance. This is not surprising at K=.140

and

where our accurate (E-m) measurements allow us to

8.

results

Also

shown

from

are

vector

the

dominance

expected using

monopole the

lattice

be expected

from

see a significant violation of continuum dispersion,

measurement of the rho meson mass. The accuracy

but is more puzzling for

of our form factor determinations allows us to see

dispersion seems to

that the lattice

larger error bars. A violation in the same direction

results are probably deviating from

vector dominance by being systematically too high.

One

of

demonstrate

the

major

the

implementation

in

goals

in

this

of

the

I

1.2

1.0

1

(which

form

study

was

present

factor

data

to

SST at

I

are

consistent

single

monopole

1,0

0.8

0.8

0,6

r~ 0.6

0.4

0.4

0.0

where

respected,

spatially

continuum albeit with

form.

vector q=n/5

Of

doubled

spatial

course,

103x20

m o m e n t u m points

dominance, point

lies

restoration

I

but

the

above

the

(within

I

--

--

}(=0.140

0.0

a

with

calculated

1.2

\

0,2 -

uses

lattice) where the q=n/10

usefulness extracting

K=.148

well

also appears in the SU(2) Wilson form factor data of Ref.

4. SUMMARY AND DISCUSSION

be

K=0.148

fit: 8-12

fit: 8-12

0.2

[

I

0.5

1.0

&

0,0 1.5

FIGURE 7 The pion electric form factor as a function of lattice four m o m e n t u m transfer at K=.140.

I

0.0

0.5

[

qe

FIGURE 8 The same as Fig. 7 but for K=.148.

1.0

1,5

T. Draper et al. / Lattice form factors

530

TABLE 1 masses

momenta

K

mn

mp

(n/5,0,0)

(n/5,n/5,0)

(n/5,n/5,n/5) (2n/5,0,0)

.140 .148

1.121(10) .797(13)

1.139(12) .829(16)

.83(2) .73(3)

.73(3) .62(6)

.68(4) .58(6)

.59(5) .42(8)

error bars) of the physical dispersion relation does not

guarantee

that

the

measured

form factors

are

also physical. However, we think it more likely that the

K=. 148

fortuitous

dispersion

and

violation

that

results

we

really

of both

continuum

for

range

dominance

our

are

have

somewhat

a

factor

results

at quite

the

dispersion and vector

of momentum

two

K

similar.

values,

By

which

delving

transfer,

improvement.

The

there

pion

to

should

vector

be

dominance

However,

we

have

also

an

We

have on

succeeded form

in

reducing

factors

and

the

lattice

seen

that

statistical dispersion

enough for it to be apparent where future efforts must be directed.

One can probe to smaller four

momentum transfer on the present size lattices by choosing

the

current

density

(at

lowest

boosting

statements

the

concerning

pion.

Of

course

the

usual

improved

results

on

larger

lattices apply as well. We intend to investigate these matters

further

in

future

publications.

We thank George Hockney for providing us with a copy of his SU(3) gauge field Monte Carlo code. One of us (WW) would also like to thank the theory group

at

SLAC,

where

Lattice Gauge Theory: A Challenge in Large-Scale C o m p u t i n g , eds. B. Bunk, K.H. Mutter and K. Schilling (Plenum, New York, 1986) pp. 199-207. 6. W. Wilcox and Keh-Fei Liu, Phys. Rev. D35 (1987) 2056. 7. W. Wilcox and R.M. Woloshyn, "Lattice Hadron Structure and the Electric Form Factor," in: Advances in Lattice Gauge Theory, eds. D.W. Duke and J.F. Owens (World Scientific, Singapore, 1985) pp. 136-147. 8. N. Cabibbo and E. Marinari, Phys. Lett. l 1 9 B (1982) 387. 9. H.W. Hamber, Phy. Lett. B178 (1986) 277.

ACKNOWLEDGEMENTS

performed,

5. T. Draper, "Lattice Evaluation o f Strong Corrections to Weak Matrix Elements -- AI=I/2 Rule," Ph.D. thesis, University of California, Los Angeles, 1984 (unpublished); C. Bernard, T. Draper, G. Hockney, and A. Soni, "Calculation of Weak Matrix Elements: Some Technical Aspects," in:

spatial

momentum) as the secondary SST source and then Lorentz

4. C. Bernard, "Lattice Calculation of Hadronic Weak Matrix Elements: The AI=l/2 Rule," in: Gauge Theory on a Lattice: 1984, eds. C. Zachos, W. Celmaster, E. Kovacs, and D. Sivers (National Technical Information Service, Springfield, VA) pp. 85-101.

result

statistical errors increase in this direction.

errors

3. W. Wilcox and R.M. Woloshyn, Phys. Rev. Lett. 54 (1985) 2653

look

smaller

may indeed be recovered, and this would be quite dramatic.

2. R.M. Woloshyn and A.M. Kobos, Phys. Rev. D33 (1986) 222.

transfers.

lattice hadron mass, and therefore also smaller four momentum

1. R.M. Woloshyn, Phys. Rev. D34 (1986) 605.

systematic

This is suggested also by a comparison of the form qualitatively

REFERENCES

part

for their hospitality.

of

this

work

was

10. B. Efron, SIAM Rev. 21 (1979) 460. 11. We used the CURFIT program from: P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).