Finite volume study of the magnetic moments using ... · Finite volume study of the magnetic...
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Finite volume study of the magnetic moments using dynamical Clover
fermionsChristopher Aubin
College of William & MaryJuly 14, 2008
with K. Orginos (WM)V. Pascalutsa (Mainz)
M. Vanderhaeghen (Mainz)
!
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Outline
Delta Magnetic Moments (PDG)
Background Field:Periodicity issuesFinite Volume effects
Quenched Results
Preliminary Dynamical Results
Conclude
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From the PDG
µ!+ = (2.7+1.0!1.3
± 1.5 ± 3)µN
µ!++ = (4.52 ± 0.50 ± 0.45)µN
Theoretical input
Best way to get this theoretically is on the lattice!
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Difficult to reach low q2 values,due to the finite volume of the lattice
pmin =2!
aL
Solutions:Bigger Volumes
Twisted Boundary Conditions
Using Form Factors
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In a background magnetic field, Bthe ground-state energy is shifted to
2pt function: CB,s(t) ∼ As,Be!mBt, t " 1
mB = m! ± µB
Alternative: Background Field method
One can also calculate other EM properties(see all the other talks in this session)
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Studies with this technique go all theway back to C. Bernard, et al., PRL 49:1076 (1982)
Up until now, studies are quenchedand do not use magnetic fields which satisfy the necessary
periodicity constraint...
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The U(1) gauge fields take the form
Uµ(x, y, z, t) = exp [ieaAµ(x, y, z, t)]
For constant B-field in the z-direction:
Aµ(x, y, z, t) =
{
aBx µ = y
0 µ = x, z, t
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(naive) Periodicity
We wish to make sure the particles don’t see a discontinuity on the boundaries of
the lattice.
Thus, the link at x=L-1 must equalthe link at x=0 for all y, so
! B =2!n
L
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(naive) Periodicity
For current lattices, the minimallyallowed field is often too large
L = 20, a−1= 2 GeV
This field is large enoughto distort our hadrons
⇒ B = 314 MeV
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Again, we can solve this by using largerlattices, but this is expensive
Try smaller B fields (non-periodic), place the baryon far from the boundaries.
(and hope the discontinuity is notnoticeable)
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Or, modify implementation of B field to change the constraint [Damgaard, Heller, NPB309 (1988)]
First, recognize that it is not the vector potential that must be periodic.
We want the magnetic flux, or plaquette to be continuous over the boundary
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In other words, on the boundary, the x-links become:
Ux(L, y, z, t) = e!iaBy(L+1)
B =2!n
L2
or
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But is this sufficient?
But we want more than one field tosimulate (higher n means much stronger field)
In fact, we need at least three (ifwe want two fields and to do the )!
+
Bpatchedmin =
1
LB
unpatchedmin
Is it safe not to satisfy the periodicity?
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Answer: Yes, sort of
The worry is that the particle maypropagate to the boundary and see
the discontinuity
Safe, for large enough volume!
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First a quenched test(Clover quarks)
Volume as, atPion Mass
(MeV)
163x128 0.1 fm, 0.03fm 750 0.39 0.025
243x128 0.1 fm, 0.03fm 750 0.26 0.011
2!
L
2!
L2
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0 0.005 0.01 0.015 0.02 0.025 0.03
q a2B
1
2
3
4
5
!!
patched, 163x128
unpatched, 163x128
patched, 243x128
unpatched, 243x128
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Preliminary Dynamical Results
Volume as, atPion Mass
(MeV) #confs
163x128 0.1 fm, 0.036 fm 366 39
243x128 0.1 fm, 0.036 fm 366 120 (s)
147 (u)
See talks by R. Edwards & M. Peardon
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0 0.005 0.01 0.015 0.02 0.025 0.03
e a2 B
2.8
3.2
3.6
4
4.4
4.8
5.2
5.6
!!
163, !
++
243, !
++
mΔ = 1.408 GeV
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0 0.01 0.02 0.03
e a2 B
1.8
2.1
2.4
2.7!!
163, !
+
243, !
+
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0 0.005 0.01 0.015
e a2 B
2
2.5
3
3.5
4
4.5
5
5.5!!
243, !
++
243, !
+
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0.005 0.01 0.015 0.02
e a2 B
-2.2
-2
-1.8
-1.6
!!
243, !
-
m!
= 1.65 GeV
0.005 0.01 0.015 0.02
e a2 B
-2.2
-2
-1.8
-1.6
!!
243, !
-
m!
= 1.65 GeV
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Using the “patched” B-field, FV errorsfrom BF are not very large
Must patch!
Need more 163 statistics
Also: diff. masses for chiral extrap.
Other moments with BF? No..Cost is comparable with FF approach,
and theoretically challenging
To conclude...
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Computing resources
NERSC cyclades cluster@ WM
JLab