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Phenomenology from Lattice QCD
In the Standard Model and Beyond
Dipartimento di Fisica di Roma I
Guido Martinelli
Frascati October 10 2002
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Quark masses Generation Mixing
e-
?-decays
Vud 0.9735(8) Vus 0.2196(23) Vcd
0.224(16) Vcs 0.970(9)(70) Vcb
0.0416(16) Vub 0.00355(36) Vtb
0.99(29) (0.999)
W
?e
down
up
Neutron
Proton
Vud
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B0 - B0 mixing
?B2 Transitions
(
)
H11
H12
H
H22
W
H21
b
d
B0
B0
t
Heff?B2
b
d
W
Hadronic matrix element
( d ?? (1 - ?5 ) s )2
?
CKM
G2F M2W
m2t
lt O gt
A2 ?6 Ftt ( )
?md,s
16 ?2
M2W
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semileptonic decays
B0d,s - B0d,s mixing
K0 - K0 mixing
Unitary Triangle
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B0 - B0 Mixing depends on the Hadronic Matrix
element of a local operator
?md (bunch of known factors) ? Vtb Vtd 2 ?
? B0 (sLA
?? dLA) (sLB?? dLB) B0 ?
?md measured with 4 of accuracy ?md only a
lower bound but is expected to be
measured soon
Lattice QCD is a method to measure this matrix
element
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1) We introduce an ultraviolet cutoff by
defining the fields on a (hypercubic) four
dimensional lattice ?(x) -gt ?(a n) where
n( nx , ny , nz , nt ) and a is the lattice
spacing ?? ? (x) -gt ?? ? (x) (?(xa
n?) - ?(x)) / a The momentum p is cutoff at
the first Brioullin zone, p ? p / a
2) We introduce an infrared cutoff by working
in a finite volume, that is ni 1, 2, , L
and pi 2p ki / L with ki 0, 1, ,
L - 1 At finite volume the Green functions are
subject to finite size effects
The physical theory is obtained in the limit a
? 0 renormalizability L ? ?
thermodinamic limit
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Continuum limit
a
Formal lim a-gt0 SLattice(?) -gt SContinuum(?)
a/ ? m a 1 The size of the object is
comparable to the lattice spacing
? 1/ m
a/ ? ltlt1 i.e. m a -gt 0 The size of the object
is much larger than the lattice spacing
Similar to a ?n -gt ? dx
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LATTICE QCD
Leptonic decay constants f? , fK , fD , fDs, fB
, fBs, f?, ..
Electromagnetic form factors F?(Q2) , GM(Q2) ,
...
Semileptonic form factors f,0(Q2) ,
A0,..3(Q2), V(Q2) K -gt ?, D -gt K, K, ?, ?, B -gt
D, D, ?, ? B -gt K ? The Isgur-Wise function
B-parameters
? K0 Q ?S2 K0 ? and ? B0 Q ?B2 B0 ?
? ? Q ?S1 K ? and ? ? ? Q ?S1 K ?
Weak decays
etc. etc. etc. ...
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A prototype to discuss statistical and systematic
errors in lattice calculations
? B0 (sLA ?? dLA) (sLB?? dLB) B0 ? 8/3
f2Bd M2Bd BBd (?)
Let us start from a simpler quantity, namely the
decay constant fBd
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Yamada Lattice 2002
quenched
my averages fquenBd177(4)(21)
MeV funqBd/fquenBd1.09(4) funqBd202(25)
MeV systematic (see the following discussion)
See also Lubicz CKM Workshop 2002
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Statistical errors at the 10 level or less
it is necessary to control systematic errors at
the same level of accuracy
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Quenching errors
QUENCHED
UNQUENCHED

• MH/M? almost right
• Kaon B-parameter essentially the same
• effect on fD estimated at 10 level
• nucleon ?-term and polarized structure
  functions wrong
• problems with chiral logarithms

REAL UNQUENCHING STILL TO COME (QUARK MASSES TOO
HEAVY)
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fquenDs 230 ?14 MeV funqDs / fquenDs 1.03 -
1.10 funqDs 250 ? 30 MeV Ryan 2001 funqDs
241 ? 5 ? 27 MeV MILC 2002
fexpDs 264 ? 15 ? 33 MeV P. Roudeau _at_
Lepton-Photon 2001 hep-ph/0110397 S.
Soldner-Rembold _at_ EPS-HEP 01 hep-ph/0108023
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Extrapolation in the heavy quark mass
DISCRETIZATION ERRORS
THE ULTRAVIOLET PROBLEM
1/MH gtgt a
mq a ltlt 1
O(a) errors
p a ltlt 1
Typically a-1 2 5 GeV mcharm 1.3 GeV
mcharm a 0.3 mbottom 4.5 GeV
mbottom a 1
For a good approximation of the continuum
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Extrapolation in the Heavy Quark Mass
Quark mass dependence not clear between
relativistic quarks NRQCD
BBd 1.32(10) Yamada 1.30(12)(13) Lubicz
1.00(15) Colangelo Khodjamirian
QCDSR
no sign of quenching effects
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SYSTEMATIC ERRORS
FINITE VOLUME EFFECTS
THE INFRARED PROBLEM
BOX SIZE
L gtgt ? 1/MH gtgt a
To avoid finite size effects
For a good approximation of the continuum
Finite size effects are not really a problem for
quenched calculations potentially more
problematic for the unquenched case
L ? 4 5 ? is sufficient
O(exp- ? /L)
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Particularly in the unquenched case, because
of the limitations in computer resources VOLUMES
CANNOT BE LARGE ENOUGH TO WORK AT THE PHYSICAL
LIGHT QUARK MASSES (min. pseudoscalar mass is
MK, needed M?
an extrapolation in mlight to the physical point
is necessary
Test if the quark mass dependence is described by
Chiral perturbation Theory (?PT), Then the
extrapolation with the functional form suggested
by ?PT is justified
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SU3 SYMMETRY BREAKING
unquenched
fBs vBBs fBd vBBd
fBs/fBd 1.15(3) BBs/BBd 1.00(3)
?
1.15(5)
but for the chiral extrapolation
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Chiral Extrapolation
fB
JLQCD
MILC
Large effects due to chiral logs As suggested by
Kronfeld Ryan ? (see also Sharpe) ?
1.15(5) -gt 1.32(10) JLQCD 1.24 - 1.38
No significant effect in MILC data ?
1.18(1)
(4)
(1)
are chiral logs relevant in the range of masses
covered by present simulations ?
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chiral behaviour of simple quantities
For the pseuodscalar meson mass the curvature is
opposite to the expected one !!
decay constant perfectly described by a straight
line !!
for fBd chiral logs invisible !!!
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1) It has been argued that chiral logs
substantially increase the value (and
uncertainty) on ? 2) Kronfeld Ryan ?
1.15(5) -gt 1.32 (10) Yamada (JLQCD) ?
1.24 - 1.38 MILC does not have an
observable effect 3) In the range of masses
explored, for fBd there is no sign of chiral logs
in the quenched case (where the effect should be
larger) or for the meson masses (for which even
the sign of the curvature is opposite to what
expected) 4) If chiral effects are so large
that ?(? -1) 100 , then we are not in the
chiral regime and there is no reason to use the
functional form suggested by ?PT to make the
extrapolation.
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SUMMARY OF THE RESULTS
with smallish differences these numbers have been
agreed with L. Lellouch, ICHEP 2002
(0)
fBd 202 ? 25 MeV
fBd 192 ? 30 MeV
(-20)
(0)
fBd vBBd 232 ? 30 MeV
(-23)
fBd vBBd 220 ? 33 MeV
(8)
? 1.17(4)
? 1.21(6)
(- 0)
The uncertainty in the chiral extrapolation is
much smaller for fBs vBBs 271 ? 35 MeV
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NEW RESULTS FOR BK
BNDRK(2 GeV) BK
World Average by L.Lellouch at Lattice 2000 and
GM 2001 0.63 ? 0.04 ?0.10
0.86 ?0.06 ?0.14 CP-PACS perturbative renorm.
0.575 ?0.006 0.787
?0.008 (quenched) DWF
0.5746(61)(191) RBC non-perturbative renorm.
0.538 ?0.008 0.737
?0.011 (quenched) DWF SPQcdR
0.66 ? 0.07
0.90 ?0.10 Wilson Improved NP
renorm. NNC-HYP Overlap Fermions 0.66
? 0.04 0.90
?0.06 perturbative Garron al. Overlap
Fermions 0.61 ? 0.07
0.83 ?0.10 Non-perturbative
Lattice 2002 preliminary
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High level accuracy discretization effects
non-negligible quenching error OSU(5 ? 4)
APE(4 ? 4) consistent with theoretical
estimates by Sharpe lt 15 .
KS Fermions
Wilson Fermions
BK(2GeV)
DW Fermions
OV Fermions
a/r0
low value by RBC not confirmed
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Results for ? and ? related quantities
Allowed regions in the ?-? plane (contours at
68 and 95 C.L.)
With the constraint from?ms
? 0.203 ? 0.040 ? 0.355 ? 0.027
0.124 - 0.278 0.302 - 0.410
at 95 C.L.
sin 2 ? - 0.20 (0.23)(-0.20 ) sin 2 ?
0.734 (0.045)(-0.034) -0.58
- 0.22 0.67 -
0.81
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Mixing effects induced by SUSY when the Squark
Mass Matrix is not diagonal in the CKM basis
(m2Q )ij m2average 1ij ?mij2 ?ij ?mij2
/ m2average
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New local four-fermion operators are generated
Q1 (sLA ?? dLA) (sLB?? dLB)
SM Q2 (sRA dLA) (sRB dLB) Q3 (sRA dLB) (sRB
dLA) Q4 (sRA dLA) (sLB dRB) Q5 (sRA dLB)
(sLB dRA) those obtained by L ? R
Similarly for the b quark e.g. (bRA dLA)
(bRB dLB)
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TYPICAL BOUNDS FROM ?MK AND ?K x
m2g / m2q x 1 mq 500 GeV
Re (?122)LL lt 3.9 ? 10-2 Re
(?122)LR lt 2.5 ? 10-3 Re (?12)LL
(?12)RR lt 8.7 ? 10-4
from ?MK