View
222
Download
2
Category
Preview:
Citation preview
Lattice 07, Regensburg, 1
Magnetic Moment of Vector Mesons in Background Field Method
• Structure of vector mesons• Background field method• Some results
x
z
polarQCD Collaborationhttp://eagle.phys.gwu.edu/~fxlee/polarQCD.html
Collaborators: Scott Moerschbacher (GWU), Walter Wilcox (Baylor University)
Thanks: U.S. Department of Energy, National Science Foundation, and computing resources from NERSC and USQCD
Frank X. Lee, GWU
Lattice 07, Regensburg, 2
Structure of Vector Mesons• Spin 1 particle, described by three form factors
),()','('2
1,||','
'
spspEE
spJsppp
22'2
32
22
1 )())(()'()(mppqqQGqgqgQGppgQG
)()()(
)()(
)()1()()()(
26
21
2
22
2
234
22
21
2
2
2
2
2
QGQGQG
QGQG
QGQGQGQG
QmQ
C
M
mQ
Q
Sachs form factors:
)0(
)0(2
)0(
2 C
M
Q
Gm
eD
Gm
e
eGq
charge
magnetic moment
quadrupole momenthep-lat/0703014, Adelaide group
Lattice 07, Regensburg, 3
Hadron Structure via Background Fields
22
22
22
11
22
12
1
12
1
2
1
2
12
1
2
1
ijMijE
jijiMjijiE
ME
BE
EBBE
BBEE
BEBH
etc ),(2
1 ,:sderivative spatial and Time ijjiij EEE
t
EE
Interaction energy of a hadron in the presence of external electromagnetic fields:
, , :
static bulk response
others :
spatial and time resolution
Probe of internal structure of the system in increasingly finer detail.
44
33
221)0()()( BcBcBcBcmBmBmMass shifts:
Lattice 07, Regensburg, 4
Compton ScatteringLow-energy expansion of real Compton scattering amplitude on the nucleon
structure characteristics: , , , 1 , 2 , 3 , 4
Lattice 07, Regensburg, 5
Introduction of an external electromagnetic field on the lattice
• Minimal coupling in the QCD covariant derivative in Euclidean space
qAgGD
• It suggests multiplying a U(1) phase factor to the links
)exp()( iagGxU • Recall that SU(3) gauge field is introduced by the link
variables
μμμ )U(iaqAxU exp)('
• This should be done in two places where the Dirac operator appears: both in the dynamical gauge generation and quark propagator generation
Lattice 07, Regensburg, 6
For Example• To apply magnetic field B in the z-direction, one
can choose the 4-vector potential
then the y-link is modified by a x-dependent phase factor
)0,,0,0(),( BxAA
yy UiqaBxU )exp(x
z
• To apply electric field E in the x-direction, one can choose the 4-vector potential
then the x-link is modified by a t-dependent phase factor
)0,0,,0( EtA
xx UiqaEtU )exp(
t
AE
AB
Lattice 07, Regensburg, 7
Computational Demands• Consider quark propagator generation
yy UiqaBxU )exp(
)det(
)( )det( 1
G
G
Sq
qS
q
emDDG
mDemDDG
• Fully dynamical: For each value of external field, a new dynamical ensemble is needed that couples to u-quark (q=1/3), d- and s-quark (q=-2/3). Quark propagator is then computed on the ensembles with matching values
• Re-weighting: Perturbative expansion of action in terms of external field (see talk by Engelhardt)
• U(1) quenched: no field in the sea, only in the valence – any gauge ensemble can be used to compute valence quark
propagators.
qAgGD
Lattice 07, Regensburg, 8
Lattice details• Standard Wilson gauge action
– 244 lattice, =6.0 (or a ≈ 0.1 fm)
– 150 configurations
• Standard Wilson fermion action =0.1515, 0.1525, 0.1535, 0.1540, 0.1545, 0.1555
– Pion mass about 1015, 908, 794, 732, 667, 522 MeV
– Strange quark mass corresponds to =0.1540 (or m~732 MeV)
– Fermion boundary conditions: periodic in y and z, fixed in x and t
– Source location (t,x,y,z)=(2,12,1,1)
• The following 5 dimensionless numbers ≡qBa2 =+0.00036, -0.00072,
+0.00144, -0.00288, +0.00576 correspond to 4 small B fields
eBa2 = -0.00108, 0.00216, -0.00432, 0.00864 for both u and d (or s) quarks.– Small in the sense that the mass shift is only a fraction of the proton mass: B/m ~ 1 to 5% at the smallest pion mass. In physical units, B ~ 1013 Tesla.
x
z
B yy UiqaBxU )exp(
Lattice 07, Regensburg, 9
What about boundary conditions?• On a finite lattice with periodic boundary conditions, to get a constant magnetic field, B has to be quantized by
to ensure that the magnetic flux through
plaquettes in the x-y plane is constant.
,3,2,1 ,22 nN
nqBa
x
x
z
• To minimize the boundary effects, we work with fixed b.c. in x-direction, so that quarks originating in the middle of the lattice has little chance of propagating to the edge.
• But, for Nx=24 and 1/a=2 GeV, the lowest allowed field would give the proton a mass shift of about 500 MeV, which is unacceptably large (proton is severely distorted). So we have to abandon the quantization condition, and work with much smaller fields.
B
yy UiqaBxU )exp(
Lattice 07, Regensburg, 10
Interpolating Field
yxyx iuid 2
1
2
1
yyxyyxxx i 2
1
For + meson:
Correlation function:
Extract interaction energies:
Other mesons similar: dssdKsuKusK
ssdduudu
0
0
, ,
, ,
Expected by symmetry: small ,
0 ,0
KKK
tEe
Lattice 07, Regensburg, 11
Magnetic moment in background field• For a particle of spin s and mass m in small fields,
where upper sign means spin-up and lower sign spin-down, and
BmE
sm
eg
2
• g factor (magnetic moment in natural magnetons) is extracted from
)()(
eBs
mEmEmg
• Look for the slope (g-factor) in the mass shift as a function of the field
)(eBgm
Lattice 07, Regensburg, 12
+ meson mass shifts
• We use the 2 smallest fields to fit the line.
)(eBgm
Lattice 07, Regensburg, 13
Effective mass plots for + mass shifts
Lattice 07, Regensburg, 14
Effective mass plots for mass
Lattice 07, Regensburg, 15
meson g-factors
mcgg 10
2210 mcmcgg
hep-lat/0703014, Adelaide group
Also agrees with that from the Charge Overlap Method byW. Andersen and W. Wilcox, Annals Phys. 255, 34 (1997)
Lattice 07, Regensburg, 16
K* meson g-factors
small
K
KK
Lattice 07, Regensburg, 17
Vector Meson Magnetic Moment
hep-lat/0703014, Adelaide group
This work
Lattice 07, Regensburg, 18
K*0 Meson Magnetic Moment
hep-lat/0703014, Adelaide group
This work
Lattice 07, Regensburg, 19
Magnetic moments for other hadrons
F.X. Lee, R. Kelly, L. Zhou, W. Wilcox, Phys. Lett. B 627, 71 (2005)
Lattice 07, Regensburg, 20
Conclusion• The background field method in lattice
QCD is a viable way of probing hadron internal structure – Magnetic moments (vector mesons in this talk)
– Electric and magnetic polarizabilities
– Neutron electric dipole moment
– Proton beta-decay
– and more
• Further calculations could improve on several fronts– discretization errors (actions, b.c effects, continuum limit)
– unquenching
Lattice 07, Regensburg, 21
Beta-decay of proton in magnetic field
• At sufficiently large B fields (1016 Tesla), proton can become heavier than neutron, allowing the ‘-decay’ of the proton:
BmE ppp
BmE nnn
B
Energy
B0
evenp
evepn
• As compared to the natural neutron -decay:
Such process can take place in stars where extremely strong magnetic field exists.
Recommended