hadronic corrections to muon anomaly within the instanton vacuum model
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Hadronic corrections to muon anomaly within the instanton vacuum model
A.E. Dorokhov (JINR, Dubna)
I. Introduction
Data from BNL (g-2) Collaboration (+/-0.5 ppm)
SM prediction from QED, EW and Strong sectors (2.7s)
II. Leading order Hadronic contribution: phenomenologye+e- - annihilation vs – decay (shape discrepancy,
as(MZ))
III. LO and NLO Hadronic contribution from theory (main source of theoretical error)
Conclusions
SND 05’
2
2Sg
a
Muon Anomaly (Current status)
Magnetic AnomalyQED HadronicWeak SUSY... ... or other new
physics ?
QED 10
1
11614098.1 41321.810
3014.2 38.1 0.6
n
n
a
EW 10one two loops 15.4 3 10a
Had 10700 10 10a
The hadronic contribution to the muon anomaly, where the dominant contribution comes from (a). The hadronic light-by-light contribution is shown in (e)
Z*interference
Hadronic polarization Light-by-light process
Nonlocal Chiral Quark model SU(2) nonlocal chirally invariant action describing the interaction of soft quarks
4231
4
1
44
54
,,,,2
xXXQXxXQxXXQXxXQxfxdXdG
xqmxAxVixqxdS
iin
nini
Spin-flavor structure of the interaction is given by matrix products ii
,,),(
)(),('),11(
5555,
555555
1
faa
AV
Taaaa
GGG
iiGiiGiiG
Instanton interaction: G'=-G
For gauge invariance with respect to external fields V and A the delocalized quark fields are defined (with straight line path)
yqzAzVdziTPyxQy
x
aaa
5exp,
Quark and Meson Propagators
1,ˆ1 pZpMppS
kMk
kMkf
kdpfNGNmpM cf 22
24
42
24
22
2
at exp pp
pM
The dressed quark propagator is defined as
The Gap equation
has solution pfmMmpM q2
0.35
7.14 106
M p2
30 p
qGq 2Mconst.
Mcurr.
From existence of derivativesof the quark condensate
pMp
pMp
pdNqDq n
Cn
222
4
42
400
222 at )/exp( pppM
aaa T
kk
kMkMkkTqkkqk
22'
''',,
Conserved Vector and Axial-Vector currents.
The Vector vertex
Nonlocal part
s in pQCD
AF
2q
The Iso-Triplet Axial-Vector vertex has a pole at 02 q
22
2
2
2
255
'
''
1'2',,
kk
kfkfkk
qJG
qJmGMkfkf
q
qTqkkqk
PPP
PPaa
Pion pole
The Iso-Singlet Axial-Vector vertex has a pole at '22mq
522
2
2
2
250
'
''
'1
1'2
'',,
kk
kMkMkk
qJG
qGJ
q
qkMkM
G
Gqkkqk
PP
PP
1-G’JPP(q2)
’ meson pole
III. LO Hadronic contribution to gIs expressed via Adler function as
1
0
22
2hvp)2(
1
2/11
3
8 m
x
xD
x
xxdxa V
, 4 , 2 2V ab iqx V ab ab VTq i d xe x q q g q Q
t
Qt
Qdt
dQ
QdQQD V
VV
022
2
2
222
Adler function is defined as
, 0 0 0J ab a bx T J x J
NQM Adler function and ALEPH data
NJL
ALEPHILMAS
Quark loop
Quark loop
Mesons
Meson loop
M(p)
pQCD(NNNLO)
QCD
The instanton model estimates
,,,10076.0932,6
,,10077.0849,68
8hvp)2(
eee
eeea
and from phenomenology one gets
,1077.0,1013.0,1033.5
,105.023,68hvp)2(
Mloop,8hvp)2(
mesons-,8hvp)2(
Qloop,
8hvp)2(QMN,
aaa
a
LO contribution to a
1q
2q
3q
4q
OPE
22
24
23
21 ,0
qqq
q
1q
2q
LBL, OPE and Triangle diagram VAV
3q
4q
AX
XX
...ˆˆ2
, 52444
21
43
21
43 zjqq
izedyjxjTyedxdi zqqiyiqxiq
(Similar to amplitude at large photon virtualities)
The structure of V*AV amplitudeFor specific kinematics 0h photon wit real is ,arbitrary - 112 qqqq
Only 2 structures exists in the triangle amplitude
21 2 2 1 2 1 2 2 1 2 2 1 2, ,T LT q q w q q q q q q q q w q q q
The amplitude is transversal with respect to vector current
021
TqTq
but longitudinal with respect to axial-vector current
21
222 qqqwTq L Thus Adler-Bell-Jackiw anomaly
Operator Product Expansion and V*AV amplitude
In local theory for massive quarks one gets
1
0221
1
3
22
mqxx
xxdx
Nww C
TL
Which in chiral limit (m=0) becomes
2
1
3
22
q
Nww C
TL
Perturbative nonrenormalization of wL (Adler-Bardeen theorem, 1969)Nonperturbative nonrenormalization of wL (‘t Hooft duality condition, 1980)Perturbative nonrenormalization of wT (Vainshtein theorem, 2003)Nonperturbative corrections to wT at large q are O(1/q6) (De Rafael et.al., 2002)Absence of Power corrections to wT at large q in Instanton model (this work)
NnLO QCD =0 for all n>0
Anomalous wL structure (NonSinglet)
Diagram with Local vertices
5
X
Diagram with NonLocal Axial vertices
X
+ rest
5
2
3 1
3
2
q
Nw C
L
In accordance with Anomalyand ‘t Hooft duality principle(massless Pion states in triplet)
+
Anomalous wL structure (Singlet)
Diagram with Local vertices
5
X
Diagram with NonLocal Axial vertices
X
+ rest
5
0)(0
2022
qL qwq
In accordance with Anomalyand ‘t Hooft duality principle(no massless states in singlet)
wLT in the Instanton Model (NonSinglet)
2
0
2
23
GeV 6.42
q
LT
q
qw
TLLT www 2
Perturbative nonrenormalization of wL (Adler-Bardeen theorem, 1969)Nonperturbative nonrenormalization of wL (‘t Hooft duality condition, 1980)Perturbative nonrenormalization of wT (Vainshtein theorem, 2003)Nonperturbative corrections to wT at large q O(1/q6) (De Rafael et.al., 2002)Absence of Power corrections at large q in Instanton model (this work)
0)3( LTw In perturbative QCD
wLT in the Instanton Model (Singlet)
220 GeV 6.00 qw LT
TLLT www 2
Z*contribution to a
TZ
ZT
Z
ZL
EW
wkm
mw
km
mw
mk
kp
kpk
kdimGa
22
2
22
2
22
2
24
42
21
3
1
2
1
222
VMD + OPE(Czarnecki, Marciano, Vainshtein, 2003)
11EW 1002.2 a
Instanton model:(Our work, 2005)
11EW 1048.1 a
Perturbative QCD(Anomaly cancelation)
0EW a
Light-by-Light contribution to muon AMM
Vector Meson Dominance like model:(Knecht, Nyffeler 2002)
80LbL)3( 10058.0 a
VMD + OPE(Melnikov, Vainshtein 2003)
8LbL)3( 1007.0136.0PVPS, a
• This contribution must be determined by calculation.
• the knowledge of this contribution limits knowledge of theoretical value.
Instanton model:(Dorokhov 2005)
8LbL)3( 1001.0105.0, PVPSa
Conclusions• There are plans to increase accuracy of (g-2) experiment by factor 2
(BNL) or even 10 (Japan)• The standard model calculations has to be at the same level of
accuracy• The weakest part of calculations is strong sector• At the moment effective (Instanton) model provide the approach
which competes with other (Lattice QCD, etc)• The accuracy of calculations is of order 10-20% for NLO corrections• The longitudinal structure of triangle diagram is norenomalized by
nonperturbative corrections in agreement with ‘t Hooft arguments• Transverse structure is calculated for arbitrary q.• At large q the transversal amplitude has exponentially decreasing
corrections, that reflects nonlocal structure of QCD vacuum in terms of instantons
• Instanton model is a way to extrapolate the results of OPE and PT to the regions not achievable by these methods.
2
2had
2
4
( )
( )3
m
a R sK
ss
ds
”Dispersion relation“, uses
unitarity (optical theorem)
and analyticity
(Bouchiat and Michel, 1961)
had
had
Still it is impossible to calculate LO contribution from Theory with required accuracy
II. Leading Order Hadronic contributions
ee
eesR
hadrons
(0)
Better agreement between exclusive and inclusive (2) data
Agreement between Data (BES) and pQCD (within correlated systematic errors)
use QCD
use data
use QCD
Evaluating the Dispersion Integral
About 91% of a comes from s<(1.8GeV)2,
73% is covered by final 2 state
The ratio of the e+e− → +− cross section.
The shaded area shows the systematic error of the SND measurements.
e+e− → p+p− cross section in the SND experiment at theVEPP-2M collider (June,2005)measured at 400 < √s < 1000 MeV
New SND agrees well with CMD2 and -data.KLOE is in conflict with SND and CMD2.
SND Data on R(s)
2 contribution to ahadr
• SND has evaluated the Dispersions Integral for the 2-Pion-Channel in the Energy Range 0.39 <s<0.97 GeV2
a= (488.7 2.6stat 6.6syst ) 10-10
• Comparison with CMD-2 and KLOE in the Energy Range 0.37 <s<0.93 GeV2
(375.6 0.8stat 4.9syst+theo) 10-10
(378.6 2.7stat 2.3syst+theo) 10-10
KLOECMD2
10SND ( ) 10385.6 5 (2005.2 )
(2004)
(2003)
LO vs NLO Hadron corrections to Muon Anomalous magnetic moment (Theory)
1% from phenomenology,10% from the model
NO phenomenology,50% from the existing model,The aim to get 10% accuracy
Lowest Order(EM and Tau data;Adler function)
Higher Order(OPE and Triangle diagram)
Vacuum Polarization Light-by-Light scattering
The hadronic light-by-light contribution is likelyto provide the ultimate limit to the precision of the standard-model value of aμ.
Triangle diagram
Vector and Axial-Vector correlators.V and A correlators are fundamental quantities of the
strong-interaction physics, sensitive to small- and large-distance dynamics. In the limit of exact isospin symmetry they are
000,
2
22,4,
22,4,
baabJ
AL
ab
AT
ababAiqxabA
VT
ababViqxabV
JxJTx
Qqq
Qqgqqxxediq
Qqgqqxxediq
where the QCD currents are
,qTqJ aa ,5
5 qTqJ aa
bJaJ
Current-current correlators
Current-current correlators are sum of dispersive and contact terms
kSkQQkTrkd
MQS
kSkQkkSkQkTrkd
QK
QSQKQQ
Jq
J
JJJ
JJJ
',,,2
2
,,,''',,2
,
4
42
4
42
2222
The transverse and longitudinal part of the correlators are extracted by projectors
22 ,3
1
q
qqP
q
qqgP LT
~
Dispersive term
I
Contact term
Current-current correlators in ILMV correlator
kMkkMkD
kMkdN
kMMMkkkMkkkkMMDD
kdNQQ
C
CV
)2(24
4
2)1(2)1(224
422
3
4'
24
3
4
3
21
22
and the difference of the V and A correlators
22)1()1(2
4
422
3
41
24 kMkMkMkMMMkMM
DD
kdNQQ C
AV
One may explicitly verify that the Witten inequality is fulfilled and that at Q2=0 one gets the results consistent with the first Weinberg sum rule
0,0 2,222 QfQQ AVL
AVT
Above we used definitions
kMkk
MM
kkkM
kk
MMkM '
1, 2222
)2(22
)1(
Other model approaches
eeea ,,10077.0849,6 8hvp)2( Phenomenological estimate:
NQM: 8hvp)2(QMN, 103.053,6 a
Extended Nambu-Iona-Lasinio:(Bijnens, de Rafael, Zheng)
8hvp)2(ENJL, 105.7 a
Minimal hadronic approximation(Local duality):(Peris, Perrottet, de Rafael)
8hvp)2(MHA, 107.17.4 a
Lattice simulations:(Blum;Goeckler et.al. QCDSF Coll.)
8hvp)2(Lattice, 1023.046.4 a
VAV* correlator
kSqkkqkSqkqkqkSkqkTrkd
NqqT VAVC 2221
5114
4
21 ,,,2
2,
It is transverse with respect to vector currents
5 V
Triangle diagram
0,, 212211 qqTqqqTq
V
~
X
wLT structure
kMMkqkMMk
q
kqkMMM
DD
Mkd
q
Nqw C
LT
)1()1(2
22
22
4
22
')(23
4
)(2'2
3
4
Above we used definitions
,
, , ,
22)1(
kk
MMkM
kMMqkkkk
TLLT www 2 0LTPTw
22 qfqwLT Exponentially suppressed, no power corrections!
InstantonVector Meson
Dominance
Inclusive v-a spectral function, measured by the ALEPH collaboration
ALEPH data on decays
Inclusive v+a spectral function, measured by the ALEPH collaboration
pQCD
pQCD
a1 a1
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