The Muon g-2 and its Impact for New Physics
F. J EGERLEHNER
Instytut Fizyki Ja̧drowej PAN, Krakow
DESY Zeuthen/Humboldt-Universität zu Berlin
Colloquium, May 9, 2008, University of Silesia, Katowice
supported by EU projects TARI and CAMTOPH
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Outline of Talk:
① The Anomalous Magnetic Moment of the Muon
② Standard Model Prediction for aµ
③ Theory vs Experiment; do we see New Physics?
④ Outlook
A BSTRACT :
T HE
MUON ANOMALOUS MAGNETIC MOMENT IS ONE OF THE MOST PRECISELY MEASURED QUANTITIES IN
PARTICLE PHYSICS . I N A RECENT EXPERIMENT AT
14- FOLD
IMPROVEMENT OF THE PREVIOUS
B ROOKHAVEN
CERN
IT HAS BEEN MEASURED WITH A REMARKABLE
EXPERIMENT REACHING A PRECISION OF
PRESENT A “ DISCREPANCY ” BETWEEN THEORY AND EXPERIMENT OF
IS THE LARGEST “ ESTABLISHED ” DEVIATION FROM THE
OBSERVABLE AND THUS COULD BE A HINT FOR
3.3
STANDARD DEVIATIONS IS OBSERVED. I T
S TANDARD M ODEL
N EW P HYSICS
0.54 PPM . AT
SEEN IN A “ CLEAN ” ELECTROWEAK
TO BE AROUND THE CORNER .
T HIS
PERSISTING
DEVIATION TRIGGERED NUMEROUS SPECULATIONS ABOUT THE POSSIBLE ORIGIN OF THE “ MISSING PIECE ”.
DISCUSS THE PRESENT STATUS OF THEORY AND EXPERIMENT AND WHAT THE MUON G -2 TELLS US ABOUT
POSSIBLE
F. Jegerlehner
N EW P HYSICS
AT THE
LHC
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
I
① The Anomalous Magnetic Moment of the Muon
µ
~ = gµ
e~
~s ; gµ = 2 (1 + aµ )
2mµ c
Dirac: gµ
Stern, Gerlach 22: ge
= 2 , aµ muon anomaly
= 2; Kusch, Foley 48: ge = 2 (1.00119 ± 0.00005)
µ(p0 )
γ(q)
0
h
µ
2
= (−ie) ū(p ) γ F1 (q ) +
µ(p)
µν
i σ2mqµν F2 (q 2 )
i
u(p)
F1 (0) = 1 ; F2 (0) = aµ
aµ responsible for the Larmor precession
directly proportional at magic energy ∼ 3.1 GeV
CERN, BNL g-2 experiments
ω
~a =
F. Jegerlehner
e
m
h
~ − aµ −
aµ B
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
1
γ 2 −1
~
β~ × E
– May 9, 2008 –
iE∼3.1GeV
at ”magic γ”
'
e
m
h
~
aµ B
i
Basic principle of experiment: measure Larmor precession of highly polarized muons
circulating in a ring
aµ = 0 would mean no rotation of spin relative to muon momentum!
µ
⇒
⇒
⇒
⇒
⇒
eB
mc
⇒
⇒
spin
ωa = a µ
⇒
⇒
Storage
Ring
momentum
⇒
actual precession × 2
Spin precession in the g
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
− 2 ring (∼ 12◦ /circle)
The BNL muon storage ring
www.g-2.bnl.gov
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Principle of CERN and BNL muon g
− 2 experiment:
• orbital and spin motion of highly polarized muons in a magnetic storage ring
• improvements with E821
– very high intensity of the primary proton beam from the proton storage ring AGS
(Alternating Gradient Synchrotron)
– the injection of muons instead of pions into the storage ring
– a super–ferric storage ring magnet
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
How the muon g
− 2 experiment at Brookhaven works:
Protons
from AGS
Polarized Muons
Inflector
Pions
p=3.1 GeV/c
Injection Point
π + → µ + νµ
Target
Injection Orbit
Storage Ring Orbit
Kicker
Modules
In Pion Rest Frame
⇒
νµ
π+
⇐
µ
+
spin
momentum
Storage
Ring
“Forward” Decay Muons
are highly polarized
The schematics of muon injection and storage in the g
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
− 2 ring
• in horizontal plane → orbit the muons: relativistic cyclotron motion angular frequency ωc
• motion of the muon magnetic moment in homogeneous magnetic field: → Larmor precession, muon’s spin
axis changes by 12’ (arc seconds) per circle [modulus of muon momentum unchanged]: ω a
~ + electric quadrupole field E
~ in
• collimating the beam: electrostatic focusing needed → magnetic field B
plane normal to particle orbit
→ angular frequency
e
ω~a =
mµ c
~ − aµ −
aµ B
1
γ2 − 1
~
~v × E
c2
!
γmagic
'
e
~
aµ B
mµ c
• possibility: choose γ [energy] such that aµ − 1/(γ 2 − 1) = 0 ⇒ magic momentum
• at the magic energy the muons are highly relativistic [amost speed of light]: Emagic = γmµ c2 ' 3.098 GeV
• muons are rather unstable and decay spontaneously: lifetime of muon at rest 2.19711 µs (micro seconds)
p
• at the magic energy γ -factor γ = 1 + 1/aµ = 29.3 → lifetime in ring 64.435 µs
• muons decay µ+ → e+ + νe + ν¯µ ; electrons follow direction of muon spin
. 1 − 2xe
A(Ee ) =
, xe = 2Ee /mµ
3 − 2xe
asymmetry factor reflecting parity violation
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
µ+ → e+ + νe + ν̄µ
e+
Calorimeter
⇒
~s
PMT
Wave Form
Digitizer
Signal (mV)
·
p
~
µ+
450
400
350
300
250
200
150
100
50
0
0 10 20 30 40 50 60 70 80
Time (ns)
Decay of µ+ and detection of the emitted e+ (PMT=Photomultiplier)
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Decay spectrum: electrons of energy >
Distribution of counts
versus time
for the 3.6 billion decays
in the 2001
Million Events per 149.2ns
N (t) = N0 (E) exp
−t
γτµ
E yields very precise ωa
[1 + A(E) sin(ωa t + φ(E))] ,
10
1
10
negative muon
10
-1
-2
data-taking period
10
-3
0
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
20
40
60
80
100
Time modulo 100 µ s [µ s]
The magnetic field B is measured by Nuclear Magnetic Resonance (NMR) using a
standard probe of H2 O. This standard can be related to the magnetic moment of a free
proton by
B = ~ωp /2µp ,
where ωp is the Larmor spin precession angular velocity of a proton in water.
Using this, the frequency ωa and µµ
= (1 + aµ ) e~/(2mµ c), one obtains
aµ =
where
R
λ−R
R = ωa /ωp and λ = µµ /µp .
The quantity λ appears because the value of the muon mass mµ is needed, and also
because the B field measurement involves the proton mass mp . Measurements of the
microwave spectrum of ground state muonium (µ+ e− ) at LAMPF at Los Alamos, in
combination with the theoretical prediction of the Muonium hyperfine splitting ∆ν (and
references therein), have provided the precise value
µµ /µp = λ = 3.183 345 39(10) (30 ppb) ,
which is used by the E821 experiment to determine aµ .
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Final BNL determined R
= 0.0037072063(20), which yields new world average value
aµ = 11659208.0(5.4)(3.3) · 10−10 ,
240
µ+
µ
+
µ
+
µ−
Theory (2006)
290
Theory KNO (1985)
Anomalous Magnetic Moment
(aµ -11659000)× 10−10
with a relative uncertainty of 0.54 ppm.
290
240
190
190
140
140
µ+
1979
CERN
1997
1998
1999
2000
BNL Running Year
2001 Average
Results of individual E821 measurements, together with last CERN result and theory
values quoted by the experiments
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
② Standard Model Prediction for aµ
❏ QED Contribution
The QED contribution to aµ has been computed (or estimated) through 5 loops
mµ
Growing coefficients in the α/π expansion reflect the presence of large ln m
e
terms coming from electron loops.
New: aexp
e
' 5.3
= 0.001 159 652 180 73(28) Gabrielse et al. 2008
α−1 (ae ) = 137.035999084(51)[0.37ppb]
based on work of Kinoshita, Nio 04
aQED
= 116 584 718.104 (0.043) (0.014) (0.025) (0.137) ×10−11
µ
| {z } | {z } | {z } | {z }
αinp
me /mµ
α4
α5
The current uncertainty is well below the ±60 × 10−11 experimental error from E821
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Ci [(α/π)n ]
# n of loops
1
+0.5
2
+0.765 857 410(26)
3
+24.050 512 28(46)
4
+130.8105(85)
5
+663.0(20.0)
tot
aQED
× 1011
µ
116140973.289 (43)
413217.620 (14)
30141.905 (1)
380.807 (25)
4.483 (137)
116584718.104 (0.147)
❶ 1 diagram
Schwinger 1948
❷ 7 diagrams
Peterman 1957, Sommerfield 1957
❸ 72 diagrams
Lautrup, Peterman, de Rafael 1974, Laporta, Remiddi 1996
❹ about 1000 diagrams
Kinoshita 1999, Kinoshita, Nio 2004
❺ estimate of leading terms Karshenboim 93, Czarnecki, Marciano 00, Kinoshita, Nio 05
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
❏ Weak Contributions
Brodsky, Sullivan 67, ...,
Bardeen, Gastmans, Lautrup 72
W
W
Higgs contribution tiny!
+
νµ
+
Z
H
weak(1)
aµ
= (194.82 ± 0.02) × 10−11
Kukhto et al 92
potentially large terms ∼
µ
µ
Z
•γ
+
W
νµ
W
νµ
W
+
Z
e,u,d,· · ·
Peris, Perrottet, de Rafael 95
γ
+ · · · quark-lepton (triangle anomaly) partial
µ
Czarnecki, Krause, Marciano 96
Heinemeyer, Stöckinger, Weiglein 04, Gribouk, Czarnecki 05 final full 2–loop result known
weak(2)
aµ
= (−42.08 ± 1.5[mH , mt ] ± 1.0[had]) · 10−11
Most recent evaluations: improved hadronic part (beyond QPM)
aweak
= (153.2 ± 1.0[had] ± 1.5[mH , mt , 3 − loop]) × 10−11
µ
(Knecht, Peris, Perrottet, de Rafael 02, Czarnecki, Marciano, Vainshtein 02)
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
MZ
GF m2µ α
π ln mµ
– May 9, 2008 –
❏ Hadronic Contributions
General problem in electroweak precision physics:
contributions from hadrons (quark loops) at low energy scales
Leptons
e, µ, τ <
γ
γ
Quarks
α : weak coupling
pQED✓
u, d, s, · · <
·
γ
γ
>
(a)
µ
(b)
u,d,· · ·
γ γ
γ
Z
+
+
γ γ(Z)
µ
•
γ
>
αs : strong coupling
pQCD
✗
(c)
µ
g
u,d,· · ·
γ
+ ···
µ
(a) Hadronic vacuum polarization O(α2 ), O(α3 )
(b) Hadronic light-by-light scattering O(α3 )
(c) Hadronic effects in 2-loop EWRC O(αGF m2µ )
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Light quark loops → Hadronic “blob”
ahad
µ
❏ Evaluation of
Leading non-perturbative hadronic contributions
ahad
µ can be obtained in terms of
2
Rγ (s) ≡ σ (0) (e+ e− → γ ∗ → hadrons)/ 4πα
data via dispersion integral:
3s
2
ahad
µ
αm 2 µ
=
3π
+
Ecut
Z
Rγdata (s) K̂(s)
ds
s2
γ
4m2π
Z∞
RγpQCD (s) K̂(s)
ds
s2
2
Ecut
• Low energy contributions enhanced: ∼ 67% of error
2
on ahad
comes from region 4m2π < m2ππ < MΦ
µ
had(1)
aµ
e+ e− –data based
F. Jegerlehner
ρ, ω
= (692.0 ± 6.0) 10−10
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
0.0 GeV, ∞
3.1 GeV
φ, . . .
– May 9, 2008 –
2.0 GeV
γ
··
γ
µ
Data:...,CMD2,KLOE,SND
Key role: VEPP-2M/Novosibirsk, DAFNE/Frascati
• Experimental error implies theoretical uncertainty!
1.0 GeV
µ
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Recent evaluations of ahad
µ
had(1)
data
aµ
e+ e−
× 1010
Ref.
hep-ph
694.8[8.6]
FJ,GJ 03
0310181
e+ e−
692.0[6.0]
FJ 06
0608xxx∗∗
e+ e−
696.3[7.2](6.2)exp (3.6)rad
DEHZ 03
0308213
e+ e−
690.9[4.4](3.9)exp (1.9)rad (0.7)QCD
DEHZ 06
ICHEP06∗∗
e+ e−
692.4[6.4](5.9)exp (2.4)rad
HMNT 03
0312250
e+ e−
699.6[8.9](8.5)exp (1.9)rad (2.0)proc
ELZ 03
0312114
τ
711.0[5.8](5.0)exp (0.8)rad (2.8)SU (2)
DEHZ 03
0308213
e+ e−
693.5[5.9](5.0)exp (1.0)rad (3.0)`×`
TY 04
0402285
e+ e− + τ
701.8[5.8](4.9)exp (1.0)rad (3.0)`×`
TY 04
0402285
Differences in errors mainly by utilizing more or less theory: pQCD, SR, low energy QCD methods
∗∗
most recent data SND, CMD-2, BaBar included
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Contributions to ahad
µ
× 1010 with relative (rel) and absolute (abs) error in percent.
10
ahad
µ [%](error) × 10
rel [%]
abs [%]
ρ, ω (E < 2MK )
538.58 [ 77.8](3.84)
0.7
37.9
2MK < E < 2 GeV
102.33 [ 14.8](4.72)
4.6
57.3
Energy range
F. Jegerlehner
2 GeV < E < MJ/ψ
22.13 [
3.2](1.23)
5.6
3.9
MJ/ψ < E < MΥ
26.39 [
3.8](0.59)
2.2
0.9
MΥ < E < Ecut
1.40 [
0.2](0.09)
6.2
0.0
Ecut < E pQCD
1.53 [
0.2](0.00)
0.1
0.0
E < Ecut data
690.83 [ 99.8](6.23)
0.9
100.0
total
692.36 [100.0](6.23)
0.9
100.0
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Key issue limiting accuracy of theoretical prediction:
• Isospin symmetry allow to include τ –decay spectral data =⇒ unfortunately: large
discrepancy above ρ–resonance 10 − 20 % not understood
• Dominating (72 %) low energy e+ e− → π + π − cross section only from one
experiment (before 2004); who is right ALEPH/CLEO τ data or CMD-2 e+ e− –data
• New approach e+ e− → π + π − by RADIATIVE RETURN at Φ–factory
DAPHNE/Frascati
• Key theory background: PHOKARA Monte Carlo
Katowice–Karlsruhe–Valencia collaboration
• 2004: KLOE/Frascati resolves τ vs. e+ e− in favor of Novosibirsk (CMD-2 and SND)
• cannot use τ data to reduce the hadronic uncertainty as we do not understand either
isospin breaking or problem with the ALEPH data (new: τ –spectra from B factory
Belle agrees in shape better with e+ e− )
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
had(2)
• Higher order hadronic contributions aµ
= −(100 ± 6) × 10−11 (Krause 96)
had(2)
aµ
= −( 98 ± 1) × 10−11 (Hagiwara et al. 03)
+
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
+
– May 9, 2008 –
③ Hadronic Light-by-Light Scattering Contribution to g
−2
−11
• Hadronic light–by–light scattering albl
(Knecht&Nyffeler 02)
µ = ( 80 ± 40) × 10
−11
albl
(Melnikov & Vainshtein 03)
µ = (136 ± 25) × 10
shift by +56 × 10−11
83(12) × 10−11
−19(13) × 10−11
+62(3) × 10−11
q = (u, d, s, ...)
π±, K ±
π 0 , η, η 0
L.D.
L.D.
Low energy effective theory: e.g. ENJL
S.D.
Kinoshita-Nizic-Okamoto 85, Hayakawa-Kinoshita-Sanda 95, Bijnens-Pallante-Prades 95
Evaluation of aLbL
in the large-Nc framework
µ
• Knecht & Nyffeler and Melnikov & Vainshtein were using pion-pole approximation
together with large-Nc π 0 γγ –formfactor
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
• FJ & Nyffeler: relax from pole approximation, using KN off-shell LDM+V formfactor
Fπ0∗ γ ∗ γ ∗ (p2π , q12 , q22 )
Fπ P(q12 , q22 , p2π )
=
3 Q(q12 , q22 )
P(q12 , q22 , p2π ) = h7 + h6 p2π + h5 (q22 + q12 ) + h4 p4π + h3 (q22 + q12 ) p2π
+h2 q12 q22 + h1 (q22 + q12 )2 + q12 q22 (p2π + q22 + q12 ))
Q(q12 , q22 ) = (q12 − M12 ) (q12 − M22 ) (q22 − M12 ) (q22 − M22 )
(1)
all constants are constraint by SD expansion (OPE), except for h3
cV T
+ h4 = 2 cV T with
= MV21 MV22 χ/2 and ΠV T (0) = −(hψ̄ψi0 )/2 χ with evaluations of χ[GeV−2 ]
-2.7 (Ball et al. ’03)
-3.3 (LMD)
-8.2 (Ioffe&Smilga ’84)
New estimate together with A. Nyffeler:
X
aµ (LbL; X) × 1011
π 0 , η, η 0
96.63 ± 4.47
-8.9 (Vainshtein ’03)
h3 ∈ [−10, 10] GeV−2
a1 , f10 , f1
28.13 ± 20%
a0 , f00 , f0
⇒ aµ (LbL) ' (118.76 ± 40) × 10−11
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
−5.98 ± 20%
③ Theory vs Experiment; do we see New Physics?
Contribution
QED incl. 4-loops+LO 5-loops
Value
Error
Reference
11 658 471.81
0.02
Remiddi et al., Kinoshita et al. ...
Leading hadronic vacuum polarization
692.3
6.0
2008 update
Subleading hadronic vacuum polarization
-10.0
0.2
2006 update
Hadronic light–by–light
11.9
4.0
new evaluation (J&N)
Weak incl. 2-loops
15.32
0.22
CMV06
Theory
11 659 181.3
7.2
–
Experiment
11 659 208.0
6.4
BNL
-26.7
9.6
–
The. - Exp.
2.8 standard deviations
Standard model theory and experiment comparison [in units 10−10 ]
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
- 11659000
10
a µ× 10
230
220
µAvg.
µ+
210
[ τ]
200
+
[e e -]
190
aµ = 116592080(63) · 10−11
180
2.9 σ (e
170
160
Experiment
+ −
e ) [ 1.4 σ (τ ) ]
BNL - E821
Muon (g-2) Collaboration
hep-ex/0401008/0602035
Theory
150
Uncertainties:
F. Jegerlehner
experiment
0.5ppm
=
6 × 10−10
theory
0.6ppm
=
7×10−10
new BNL 969 prop.
0.2ppm
=
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
2.4 × 10−10
– May 9, 2008 –
[ 5 × 10−10 stat, 4 × 10−10 syst ]
(e + e− )
[0.6ppm = 7 × 10−10 (τ ) ]
New physics contributions: (examples)
a)
γ
M
M
f
f
m M0 [S,P] m
b)
c)
d)
H+
H−
X0
X+
X−
M0 [V,A]
X0
Possible New Physics contributions: neutral boson exchange: a) scalar or pseudoscalar and c) vector or
axialvector, flavor changing or not, new charged bosons: b) scalars or pseudoscalars, d) vector or axialvector
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
(a) Case:
F. Jegerlehner
mµ = M M0
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
(b) Case:
m µ M0 = M
Most promissing New Physics scenario: SUSY
a)
b)
χ̃
χ̃
µ̃
χ̃0
ν̃
Leading SUSY contributions to g
− 2 in supersymmetric extension of the SM.
• m̃ lightest SUSY particle; SUSY requires two Higgs dublets
• tan β =
v1
v2 ,
vi =< Hi > ; i = 1, 2
2
100GeV
SUSY
• ∆aµ
/aµ ' 1.25ppm
tan β
m̃
• tan β ∼ mt /mb ∼ 40 [4 − 40]
Most NP models yield
|aµ (New Physics)| ' m2µ /M 2
F. Jegerlehner
α
π
⇒ M ' 1 − 2 TeV
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
µ̃
– May 9, 2008 –
80.70
experimental errors 68% CL:
LEP2/Tevatron (today)
Tevatron/LHC
MW [GeV]
80.60
LEP/Tevatron
ILC/GigaZ
SY
light SU
MSSM
80.50
USY
80.40
yS
heav
eV
80.30
MH
SM
G
= 114
MH =
SM
MSSM
both models
eV
400 G
80.20
Heinemeyer, Hollik, Stockinger, Weber, Weiglein ’06
160
165
170
175
180
185
mt [GeV]
F. Jegerlehner
Figure
Fizyki, Uniwersytet
Katowice
– May the
9, 2008
– as a function of
1: Institut
Prediction
for M Śla̧ski,
in the
MSSM and
SM
m in comparison with the present
There are a lot of “SUSY’s”
• General MSSM has > 100 free parameters
– Advantage: well. we do not know them, frankly speaking
– Disadvantage: Not predictive, but experiments can “restrict” parts of the
multidimensional parameter space
– Beware of claims of “Ruling out SUSY” !
• CMSSM – “constrained” and, related but even more constrained MSUGRA, and
others
– These models assume many degeneracies of masses and couplings in order to
restrict the number of parameters
– Typically, mo , m1/2 , sign(µ), tan β, A (or even more)
• Then there is R–parity – is sparticle number conserved (dark matter candidate?)
• An, many ways to describe EW symmetry breaking
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Role for LHC searches:
3 σ deviation in muon g-2 (if real) requires sign(µ) and tan β preferable large
Note:
• sign(µ) cannot be obtained from LHC (hadron collider)
• tan β can’t be pinned down by LHC (hadron collider)
so muon g-2 important hint for constraining SUSY parameter space (is SUSY)
More scenarios:
• sequential fermions (4th family?)
• Grand Unified Theories: Z 0 , W 0 , leptoquarks etc
• extra dimensions, graviton excitations etc
• Little Higgs Models of EW symmetry breaking
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
tan β = 10 , µ > 0
800
mh = 114 GeV
600
Ωh
mχ± = 104 GeV
mh = 114 GeV
2
m0 (GeV)
500
2σ
400
300
mχ± = 104 GeV
g-2
b→sγ ex
100
0
100
1σ
excluded for
neutral dark matter
200
300
400
500
600
700
800
900
1000
0
100
g-2
1σ
excluded for
neutral dark matter
1000
1500
m1/2 (GeV)
m1/2 (GeV)
The (m0 , m1/2 ) plane for µ
Ω h2
2σ
b→sγ ex
m0 (GeV)
700
200
tan β = 40 , µ > 0
1000
> 0 for (a) tan β = 10 and (b) tan β = 40 in the constrained MSSM (mSUGRA)
scenario. The allowed region by the cosmological neutral dark matter constraint is shown by the black parabolic
shaped region. The disallowed region where mτ̃1
< mχ has dark shading. The regions excluded by b → sγ
have medium shading (left). The gµ − 2 favored region at the 2 σ [(287 ± 182) · 10−11 ] (between dashed lines
the 1 σ [(287 ± 91) · 10−11 ] band) level has medium shading. The LEP constraint on mχ± = 104 GeV and
mh = 114 GeV are shown as near vertical lines. Plot courtesy of K. Olive
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
aµ Summary: experiment vs. theory
World Ave
CERN (79)
KNO (85) Theory
E821 (00) µ+
E821 (01) µ+
E821 (02) µ+
E821 (04) µ−
Average
E969 goal
100
208.0 ± 6.3
aµ ×1010-11659000
200
EJ 95 (e+ e− )
 (e+ e− )
DEHZ03 
(+τ )
+ −
GJ03 (e e )
SN03 (e+ e− TH)
HMNT03
(e+ e− incl.)

 (e+ e− )
TY04 
(+τ )
DEHZ06 (e+ e− )
HMNT06 (e+ e− )
+ −
FJ06 (e e )









LbLBPP,HK,KN
LbLFJ
LbLMV
181.3 ± 16.
180.9 ± 8.0
195.6 ± 6.8
179.4 ± 9.3
169.2 ± 6.4
183.5 ± 6.7
180.6 ± 5.9
188.9 ± 5.9
180.5 ± 5.6
180.4 ± 5.1
300
[1.6
[2.7
[1.3
[2.5
[4.3
[2.7
[3.2
[2.2
[3.3
[3.4
177.6 ± 6.4
179.3 ± 6.8
182.9 ± 6.1
Theory (e+ e− )
3.2 σ
Theory (τ )
1.3 σ
−150 −50 0
100
200
QED
σ]
σ]
σ]
σ]
σ]
σ]
σ]
σ]
σ]
σ]
EW 1–loop
EW 2–loop
L.O. had
H.O. had
LbL 1995
LbL 2001
in units
+ −
had τ -e e
10−10
aµ × 1010 − 11659000
[3.3 σ]
[3.2 σ]
[2.9 σ]
had(1)
Given theory results only differ by aµ
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
!
④ Outlook
• BNL g − 2 experiment 14–fold improvement (vs CERN 1979) to 0.54ppm
aµ = 116592080(54)(33) · 10−11
• SM Test:
– substantial improvement of CPT test
– confirms for the first time weak contribution:
2σ − 3σ
– very sensitive to precise value of the hadronic contribution: limits theoretical
precision 2σ
→ 1σ , now ∼ δ LBL
– Light-by-Light (model-dependent) not far from being as important as the weak
contribution, may become the limiting factor for future progress (theory?)
• New Physics ??? 2.9 [3.4] σ
SM
−10
|aexp
µ − aµ | = (28 ± 9)[(27 ± 8)] × 10
Small discrepancy persists, however, established deviation from the SM, in contrary:
very strong constraint on many NP scenarios
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
• On “prediction” side:
– big experimental challenge: attempt cross-section measurements at 1% level up
to J/ψ [Υ] !!! crucial for g
− 2 and αQED (MZ ) at GigaZ
– new ideas on how to get model–independent hadronic LbL contribution
– theory of pion FF below 1 GeV allows for further improvement (Colangelo, Gasser,
Leutwyler); constraints from ππ scattering phase shifts
• Plans for new g − 2 experiment!
Present:
BNL E821
0.5 ppm
Future:
BNL E969
0.2 ppm
J-PARC
0.1 ppm
Hunting for precision goes on!!!
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
History of sensitivity to various contributions
BNL
2004
CERN III
1976
CERN II
1968
CERN I
1961
4th
QED 6th
8th
hadronic VP
hadronic LBL
weak
???
New Physics
SM precision
10−1
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
1
10
102
aµ uncertainty [ppm]
– May 9, 2008 –
103
104
The Muon g
−2
“the closer you look the more there is to see”
New Book: F. Jegerlehner,
The Anomalous Magnetic Moment of the Muon,
Springer Tracts in Modern Physics,
Vol. 226, November 2007
Thanks for your attention!
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Hadrons in h0|T {Aµ (x1 )Aν (x2 )Aρ (x3 )Aσ (x4 )}|0i
γ(k)
kρ
+ 5 permutations of the qi
had
q3λ
q2ν
q1µ
µ(p0 )
µ(p)
Key object full rank-four hadronic vacuum polarization tensor
Πµνλρ (q1 , q2 , q3 ) =
Z
d4 x1 d4 x2 d4 x3 ei (q1 x1 +q2 x2 +q3 x3 )
×h 0 | T {jµ (x1 )jν (x2 )jλ (x3 )jρ (0)} | 0 i .
• non-perturbative physics
• general covariant decomposition involves 138 Lorentz structures of which
• 32 can contribute to g − 2
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
• fortunately, dominated by the pseudoscalar exchanges π 0 , η, η 0 , ... described by the effective Wess-Zumino
Lagrangian
• generally, pQCD useful to evaluate the short distance (S.D.) tail
• the dominant long distance (L.D.) part must be evaluated using some low energy effective model which
includes the pseudoscalar Goldstone bosons as well as the vector mesons play key role
u, d
+
+
g
+
+ ···
u, d
π0
→
|
+
π±
{z
L.D.
+···+
}
+ ···
|
{z
S.D.
}
Hadronic light–by–light scattering is dominated by π0 exchange in the odd parity channel, pion loops etc.
at long distances (L.D.) and quark loops incl. hard gluonic corrections at short distances (S.D.)
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
The spectrum of invariant γγ masses obtained with the Crystal Ball detector. The three rather
pronounced spikes seen are the γγ
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
→ pseudoscalar (PS) → γγ excitations: PS=π 0 , η, η 0
– May 9, 2008 –
Pion-pole contribution
dominating hadronic contributions = neutral pion exchange
γ
π 0 , η, η 0
1
2
µ
3
Leading hadronic light–by–light scattering diagrams
The key object here is the π 0 γγ form factor Fπ 0 γ ∗ γ ∗ (m2π , q12 , q22 ) which is defined by the matrix element
i
Z
d4 xeiq·x h 0|T {jµ (x)jν (0)}|π 0 (p)i = εµναβ q α pβ Fπ0 γ ∗ γ ∗ (m2π , q 2 , (p − q)2 ) .
Properties:
• bose symmetric Fπ0∗ γ ∗ γ ∗ (s, q12 , q22 ) = Fπ0∗ γ ∗ γ ∗ (s, q22 , q12 )
• need it off-shell in integrals; if (s, q12 , q22 ) 6= (m2π , 0, q 2 ) in fact not known
• pion pole approximation in mind (pion pole dominance) s = m2π
• for generality use vertex function (not matrix element)
Z
i d4 xeiq·x h 0|T {jµ (x) jν (0) ϕ̃π0 (p)}|0i = εµναβ q α pβ Fπ0∗ γ ∗ γ ∗ (p2 , q 2 , (p − q)2 ) ×
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
i
,
p2 − m2π
with ϕ̃(p)
=
R
d4 y eipx ϕ(y) the Fourier transformed π 0 –field. ⇒ representation in terms of form factors
LbL;π 0
To compute aµ
≡ FM (0)|pion pole , we need i
∂
∂kρ
(π 0 )
Πµνλσ (q1 , q2 , k
− q1 − q2 ) at k = 0 where
p3 = −(p1 + p2 ). Computing the Dirac traces yields
Z 4
d q1 d 4 q2
1
0
LbL;π
6
aµ
= −e
(2π)4 (2π)4 q12 q22 (q1 + q2 )2 [(p + q1 )2 − m2 ][(p − q2 )2 − m2 ]
Fπ0∗ γ ∗ γ ∗ (q22 , q12 , q32 ) Fπ0∗ γ ∗ γ (q22 , q22 , 0)
×
T1 (q1 , q2 ; p)
q22 − m2π
Fπ0∗ γ ∗ γ ∗ (q32 , q12 , q22 ) Fπ0∗ γ ∗ γ (q32 , q32 , 0)
+
T2 (q1 , q2 ; p) ,
q32 − m2π
where T1 (q1 , q2 ; p) and T2 (q1 , q2 ; p) are scalar kinematics factors; two terms unified by bose symmetry.
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
The pion-photon-photon transition form factor
• Form factor function Fπ0∗ γ ∗ γ ∗ (s, s1 , s2 ) is largely unknown
• Fortunately some experimental data is available:
• The constant Fπ0 γγ (m2π , 0, 0) is well determined by the π 0 → γγ decay rate
The invariant matrix element reads (follows from Wess-Zumino Lagrangian)(fπ
Mπ0 γγ
Information on
∼ F0 )
e2 N c
α
−1
= e Fπ0 γγ (0, 0, 0) =
=
≈
0.025
GeV
,
12π 2 fπ
πfπ
2
Fπ0 γ ∗ γ (m2π , −Q2 , 0) comes from experiments e+ e− → e+ e− π 0 where the electron
(positron) gets tagged to high Q2 values.
0
e − (pt )
e− (pb )
γ
Q2 > 0
∗
γ
π0
q2 ∼ 0
0
e+
e+
Measurement of the π 0 form factor Fπ 0 γ ∗ γ (m2π , −Q2 , 0) at high space–like Q2
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Data for Fπ 0 γ ∗ γ (m2π , −Q2 , 0) is available from CELLO and CLEO. Brodsky–Lepage interpolating formula
Fπ0 γ ∗ γ (m2π , −Q2 , 0) '
1
1
4π 2 fπ 1 + (Q2 /8π 2 fπ2 )
gives an acceptable fit to the data.
0970597-008
0.30
CELLO
2
2
Q IIF(Q ) II (GeV)
CLEO
0.20
2f
0.10
0
2.5
5.0
2
7.5
10.0
2
Q (GeV )
Fπ0 γ ∗ γ (m2π , −Q2 , 0) data from CLEO and CELLO. Shown is the Brodsky–Lepage prediction (solid
curve) and the phenomenological fit by CLEO (dashed curve)
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Assuming the pole approximation this FF has been used by all authors (HKS,BPP,KN) in the past, but has
been criticized recently (MV).
In fact in g
− 2 we are at zero momentum such that only the FF
Fπ0∗ γ ∗ γ (−Q2 , −Q2 , 0) 6= Fπ0 γ ∗ γ (m2π , −Q2 , 0) is consistent with kinematics. Unfortunately, this
off–shell form factor is not known and in fact not measurable.
An alternative way to look at the problem is to use the anomalous PCAC relation and to relate π 0 γγ to
directly the ABJ anomaly.
High energy behavior of Fπ 0∗ γ ∗ γ ∗ is best investigated by OPE of
h 0 | T {jµ (x1 )jν (x2 )jλ (x3 )} | γ(k) i
taking into account that the external photon is in a physical state. Looking at first diagram:
❏ q1 and q2 as independent loop integration momenta
❏ most important region q12 ∼ q22 q32 , ⇔ short distance expansion of T {jµ (x1 )jν (x2 )} for x1 → x2 .
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
x
y
x
y
=
x
×
+···
×
+···
y
=
Usual structure of OPE
“perturbative hard short distance coefficient function” × “soft long distance matrix element”
Surprisingly, the leading possible term in is just given by the ABJ anomaly diagram known exact to all
orders and given by the lowest order (one–loop) result. This requires of course that the leading operator
in the short distance expansion must involve the divergence axial current. Indeed,
i
Z
d4 x1
Z
d4 x2 ei (q1 x1 +q2 x2 ) T {jµ (x1 )jν (x2 )X} =
Z
2i
d4 z ei (q1 +q2 ) z 2 εµναβ q̂ α T {j5β (z)X} + · · ·
q̂
µ
= q̄ Q̂2 γ µ γ5 q the relevant axial current and q̂ = (q1 − q2 )/2 ≈ q1 ≈ −q2 ; the momentum
flowing through the axial vertex is q1 + q2 and in the limit k → 0 of our interest q1 + q2 → −q3 which is
assumed to be much smaller than q̂ (q12 − q22 ∼ −2q3 q̂ ∼ 0).
with j5
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
After the large q1 , q2 behavior has been factored out the remaining soft matrix element to be calculated is
Tλβ = i
Z
d4 z ei q3 z h0|T {j5β (z)jλ (0)}|γ(k)i ,
which is precisely the well known VVA triangle correlator
(a)
Tλβ
= −
i eNc
×
4π 2
wL (q32 ) q3β q3σ f˜σλ
+
wT (q32 )
−q32 f˜λβ
+
q3λ q3σ f˜σβ
−
q3β q3σ f˜σλ
Both amplitudes the longitudinal wL as well as the transversal wT are calculable from the triangle
fermion one–loop diagram. In the chiral limit they are given by
(a)
(a)
wL (q 2 ) = 2wT (q 2 ) = −2/q 2 .
as discovered by (Vainshtein 03). So everything is perturbative with no radiative corrections at all!
At this stage of the consideration it looks like a real mystery what all this has to do with π 0 exchange, as
everything looks perfectly controlled by perturbation theory. The clue is that as a low energy object we
may evaluate this matrix element at the same time perfectly in terms of hadronic spectral functions by
saturating it by a sum over intermediate states.
Melnikov, Vainshtein: vertex with external photon must be non-dressed! i.e. no VDM damping ⇒ result
increases by 30% !
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Basic problem:
(s, s1 , s2 )–domain of Fπ0∗ γ ∗ γ ∗ (s, s1 , s2 ); here (0, s1 , s2 )–plane
???
pQCD
RχPT/ENJL
???
Compare VP: s-axis
RχPT/ENJL
F. Jegerlehner
pQCD
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Originally VMD model (Kinoshita et al 85):
❏ ρ mesons play an important role in the game (see hadronic VP) ⇒ looks natural to apply a vector
meson dominance (VDM) model
❏ Naive VDM
m2ρ
i
i
i
i
→
=
−
.
q2
q 2 q 2 − m2ρ
q2
q 2 − m2ρ
❏ provides a damping at high energies, ρ mass as an effective cut-off (physical version of a Pauli-Villars
cut-off) ⇒ photons → dressed photons!
❏ naive VDM violates electromagnetic WT-identity
Correct implementation in accord with low energy symmetries of QCD:
• vector meson extended CHPT (EχPT) model (Ecker et al 89)
• hidden local symmetry (HLS) model (Bando et al 85 )
• extended NJL (ENJL) model (Dhar et al 85, Ebert, Reinhardt 86, Bijnens 96)
to large extend equivalent
Problem: matching L.D. with S.D. ⇒ results depend on matching cut off Λ ⇒ model dependence
(non-renormalizable low energy effective theory vs. renormalizable QCD)
Novel approach: refer to quark–hadron duality of large-Nc QCD, hadrons spectrum known, infinite series of
narrow spin 1 resonances (’t Hooft 79) ⇒ no matching problem (resonance representation has to match quark
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
level representation) (De Rafael 94, Knecht, Nyffeler 02)
⇒ lowest meson dominance (LMD) plus one vector state (V) approximation to large-Nc QCD
allows for correct matching
⇒ “LMD+V” parametrization of π 0 γγ form–factor (see below)
Summary of most recent results
1011 aµ
π 0 , η, η 0
BPP
HKS
KN
MV
83 ± 12
114 ± 10
85 ± 13
82.7 ± 6.4
π, K loops
−19 ± 13
−4.5 ± 8.1
0 ± 10
axial vector
2.5 ± 1.0
1.7 ± 0.0
22 ± 5
scalar
−6.8 ± 2.0
-
quark loops
21 ± 3
9.7 ± 11.1
total
83 ± 32
89.6 ± 15.4
-
-
-
-
80 ± 40
136 ± 25
Note: MV and KN utilize the same model LMD+V form factor:
Fπγ ∗ γ ∗ (q12 , q22 )
where M1
F. Jegerlehner
4π 2 Fπ2 q12 q22 (q12 + q22 ) − h2 q12 q22 + h5 (q12 + q22 ) + (Nc M14 M24 /4π 2 Fπ2 )
,
=
Nc
(q12 + M12 )(q12 + M22 )(q22 + M12 )(q22 + M22 )
= 769 MeV, M2 = 1465 MeV, h5 = 6.93 GeV4 .
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
with two modifications:
❏ form factor Fπ0∗ γ ∗ γ (q22 , q22 , 0) = 1: undressed soft photon (non-renormalization of ABJ) Note: to have
anomaly correct doesn not imply that there is no damping!
P V V anomaly quark loop is counter example; it has
correct πγγ in chiral limit (anomaly) and goes like 1/qi2 up to logs in all directions
❏ Fπ0∗ γ ∗ γ ∗ (q22 , q12 , q32 ) ' Fπ0 γ ∗ γ ∗ (m2π , q12 , q32 ) = KN
with h2
= 0 ± 20 GeV2 (KN) vs. h2 = −10 GeV2 (MV) fixed by twist 4 in OPE (1/q 4 )
❏ a1 [f1 , f1∗ ] different mixing scheme
Criticism: KN Ansatz only covers (0, q12 , q22 )–plane, with consistent kinematics depends on 3 variables → 2–dim
integral representation no longer valid.
Is this the final answer? How to improve? A limitation to more precise g
− 2 tests?
Looking for new ideas to get ride of model dependence
In principle lattice QCD could provide an answer [far future (“yellow” region only)]
Ask STRING THEORY ? (see below)
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –
Summary of contributions:
aµ in units 10−6 , ordered according to their size (L.O. lowest order, H.O. higher order,
LBL. light–by–light); (α−1 = 137.0359991100)
L.O. universal
e–loops
1161.409 73 (0)
6.194 57 (0)
H.O. universal
−1.757 55 (0)
L.O. hadronic
0.069 20 (56)
L.O. weak
0.001 95 (0)
H.O. hadronic
−0.001 98 (1)
LBL. hadronic
0.001 00 (39)
τ –loops
0.000 43 (0)
H.O. weak
−0.000 41 (2)
e+τ –loops
0.000 01 (0)
theory
1165.917 93 (68)
experiment
1165.920 80 (63)
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
“New Physics” ???
the
−11
aexp
[3.1 σ]
µ − aµ = 287 ± 91 · 10
– May 9, 2008 –
aµ : type and size of contributions
had(2)
weak(1)
+ aµ
weak(2)
new physics
+ albl
)
µ (+aµ
+ aµ
&
$
$
!
.
*
"!
$
#
$
&
%
,
.
!
/
#
, +
-
*
%
+ aµ
)('
!
had(1)
aµ = aQED
+ aµ
µ
All kind of physics meets !
Almost all EURODAFNE & EURIDICE nodes involved in one or the other problem !!!
F. Jegerlehner
Institut Fizyki, Uniwersytet Śla̧ski, Katowice
– May 9, 2008 –