the muon g-2 in the light of two photon physicsfjeger/mainz121.pdf · the muon g-2 in the light of...

The muon g-2 in the light of two photon physics andfirst lessons from the LHC

Fred Jegerlehner∗

[email protected]

Graduiertenkolleg und EMG Seminar des Instituts fur Physik, JG UniversitatMainz, 27. Juni 2012

AbstractI give a brief summary of the status of the muon anomaly and then focus onproblems in the determination of the hadronic light-by-light contribution. I willdiscuss how yet to be collected experimental data can constrain better thenon-perturbative amplitudes required. The impact of recent LHC results forthe interpretation of the muon g-2 ”discrepancy” is briefly discussed.

F. Jegerlehner JG Univ. Mainz June 2011

Outline of Talk:

vIntroductionvStandard Model Prediction for aµvThe hadronic LbL: setup and problemsvPseudoscalar exchanges: π0, η, η′

vAxial exchanges: a1, f ′1, f1vScalar exchanges: a0, f ′0, f0, · · ·vLbL: Present & FuturevTheory vs experiment: do we see New Physics?vImplications of low and high energy measurements on SUSY

F. Jegerlehner JG Univ. Mainz June 2011 1

IntroductionParticle with spin ~s⇒ magnetic moment ~µ (internal current circulating)

~µ = gµe~

2mµc~s ; gµ = 2 (1 + aµ)

Dirac: gµ = 2 , aµ = α2π + · · · muon anomaly

γ(q)µ(p′)

µ(p)

= (−ie) u(p′)[γµF1(q2) + i σ

µνqν2mµ

F2(q2)]

u(p)

F1(0) = 1 ; F2(0) = aµ

aµ responsible for the Larmor precession

Electromagnetic Lepton Vertex


Larmor precession ~ω of beam of spin particles in a homogeneous magnetic field ~B

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒µ

⇒spin

momentum

Storage

Ring

ωa = aµeBmc

actual precession × 2

Spin precession in the g − 2 ring(∼ 12′/circle)

Magic Energy: ~ω is directly proportional to ~B at magic energy ∼ 3.1 GeV

~ωa =em

[aµ~B −

(aµ − 1

γ2−1

)~β × ~E

]E∼3.1GeV

at ”magic γ”' e

m

[aµ~B

]CERN, BNL g-2 experiments

Stern, Gerlach 22: ge = 2; Kusch, Foley 48: ge = 2 (1.00119 ± 0.00005)


Basic principle of experiment: measure Larmor precession of highly polarizedmuons circulating in a ring

aµ = 0 would mean no rotation of spin relative to muon momentum!

Lorentz factor γ = 1/√

1 − v2/c2 = E/mc2, γmag =√

1 + 1/aµ ' 29.3⇒ muonlifetime τµ = 2.19711 µs at rest→ τµ = 64.435 µs in motion.

For the measurement of the anomalous magnetic moment we need to look at the

r equation of motion of a charged Dirac particle in an external field Aextµ (x):(

i~γµ∂µ + Q`ecγ

µ(Aµ(x) + Aextµ (x)) − m`c

)ψ`(x) = 0(

2gµν −(1 − ξ−1

)∂µ∂ν

)Aν(x) = −Q`eψ`(x)γµψ`(x) .

In the derivation of the precession equation the radiation field Aµ(x) is commonlyneglected (see below).


BNL Result [2004] and Update [2012]

aµ measured via a ratio of frequencies (measurement of aµ and B)

Cyclotron motion in transversal fields (~v · ~E = ~v · ~B = 0): d~vdt = ~ωc ×~v

Spin-precession motion in the LAB frame: d~Pdt = ~ωs × ~P ; ~P polarization vector

Spin-precession due to magnetic moment = Larmor precession ~ωL =eg2m~B

Spin-precession due to acceleration = Thomas precession ~ωT =γ−1v2

d~vdt ; relativistic

motion effect (doublet fine splitting in atom)

In a uniform magnetic field, as in muon g − 2 experimental setup:

ωa = ωs − ωc = aµ eBmµc ; ωc =

eBγmµc ; ωs = ωc + ωa = ωT + ωL


Eliminate B determined by proton nuclear resonance using:

B = ~2ωpµp

; µµ = (1 + aµ) ~2

emµc ; ωa = aµ eB

mµc ⇒

aµ =ωaγωc=

ωa/ωpµµ/µp−ωa/ωp

= R

λ−R

r ωp = (e/mpc)〈B〉 free proton NMR frequencyr R = ωa/ωp = 0.003 707 2063(20) from E-821r λ = ωL/ωp = µµ/µp = from hyperfine splitting of muonium

value used by E-821 3.18334539(10)new value 3.183345107(84) CODATA 2011: [raXiv:1203.5425v1]

⇒change in aµ: +1.10 × 10−10

aexpµ = (11 659 209.1 ± 5.4 ± 3.3[6.3]) × 10−10 updated


Standard Model Prediction for aµ

What is new?

l new CODATA values for lepton mass ratios mµ/me, mµ/mτ

l spectacular progress by Aoyama, Hayakawa, Kinoshita and Nio on 5–loop QEDcalculation (as well as improved 4–loop results)

r O(α5) electron g − 2, substantially more precise α(ae)

r Complete O(α5) muon g − 2, settles better the QED part

r QED Contribution

The QED contribution to aµ has been computed through 5 loops

Growing coefficients in the α/π expansion reflect the presence of large ln mµ

me' 5.3

terms coming from electron loops. Input:


aexpe = 0.001 159 652 180 73(28) Gabrielse et al. 2008

α−1(ae) = 137.0359991657(331)(68)(46)(24)[0.25ppb] Aoyama et al 2012

aQEDµ = 116 584 718.851 (0.029)︸︷︷︸

αinp

(0.009)︸︷︷︸me/mµ

(0.018)︸︷︷︸α4

(0.007)︸︷︷︸α5

[0.36] × 10−11

The current uncertainty is well below the ±60× 10−11 experimental error from E821

# n of loops Ci [(α/π)n] aQEDµ × 1011

1 +0.5 116140973.289 (43)2 +0.765 857 426(16) 413217.628 (9)3 +24.050 509 88(32) 30141.9023 (4)4 +130.8796(63) 381.008 (18)5 +753.290(1.04) 5.094 (7)

tot 116584718.851 (0.036)


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, Nio2004, Ayoama et al. 2009/2012

❺ estimates of leading terms Karshenboim 93,Czarnecki, Marciano 00, Kinoshita, Nio 05 all 12672 diagrams (fully automated numerical)Ayoama et al. 2012


r Weak contributions

W W

νµ

+Z

+H

µ µ

Z γ• +•νµ νµ

W W

W+

µ

e, u,d,· · ·

Z γ+ · · ·

Brodsky, Sullivan 67, ...,Bardeen, Gastmans, Lautrup 72

Higgs contribution tiny!aweak(1)µ = (194.82 ± 0.02) × 10−11

Kukhto et al 92potentially large terms ∼ GFm2

µαπ

ln MZmµ

Peris, Perrottet, de Rafael 95quark-lepton (triangle anomaly) cancellation

Czarnecki, Krause, Marciano 96

Heinemeyer, Stockinger, Weiglein 04, Gribouk, Czarnecki 05 full 2–loop resultMost recent evaluations: improved hadronic part (beyond QPM)

aweakµ = (153.2 ± 1.0[had] ± 1.5[mH,mt, 3 − loop]) × 10−11

(Knecht, Peris, Perrottet, de Rafael 02, Czarnecki, Marciano, Vainshtein 02)


r Hadronic stuff: the limitation to theoryGeneral problem in electroweak precision physics:

contributions from hadrons (quark loops) at low energy scalesLeptons Quarks

γ γγ

e, µ, τ <

>

α : weak couplingpQED

γ γ

gu, d, s, · · ·<

>α

s: strong coupling

pQCD

(a)

µ µ

γ γ(Z)• +•

(b)

µ

u,d,· · ·γ γ γ

+

(c)

µ

u,d,· · ·

Z γ+ · · ·

(a) Hadronic vacuum polarization O(α2),O(α3) Light quark loops(b) Hadronic light-by-light scattering O(α3) ↓

(c) Hadronic effects in 2-loop EWRC O(αGFm2µ) Hadronic “blobs”


Hadronic vacuum polarization effects in g − 2 [quark loops]

Role of hadronic two point correlator (non-perturbative):

· ·

γ γ

r key object 〈0|T jµ hadem (x) jν had

em (0)|0〉r hadronic electromagnetic current

jµ hadem =

∑c

(23

ucγµuc −

13

dcγµdc −

13

scγµsc +

23

ccγµcc −

13

bcγµbc +

23

tcγµtc

),

r hadronic part on photon self-energy Π′ hadγ (s)⇔ 〈0| jµ had

em (x) jν hadem (0)|0〉

r hadronic vacuum polarization due to the 5 “light” quarks q = u, d, s, c, b

r top quark [mass mt ' 173 GeV] pQCD applies [αs(mt) small]

r in fact t is irrelevant by decoupling theorem [heavy particles decouple inQED/QCD], t like τ VP loop extra factor NcQ2

t = 4/3:


Low energy effective theory: e.g. CHPT here equivalently scalar QED of pions

ρπ

γ γµ

γ

u, d

a) b) c)

Low energy effective graphs a) [ρ-exchange] and b) [π-loop] and high energygraph c) [quark-loops]

Low energy effective estimates of the leading VP effects a(4)µ (vap) × 108

For comparison: 5.8420 for µ–loop, 590.41 for e–loopdata [280,810] MeV ρ0-exchange π±-loop (u, d)-loops

4.2666 4.2099 1.4154 2.2511[449.25]∗∗ current quarks: mu ∼ 3MeV,md ∼ 8MeV

Often resorting to QPM using effective “constituent quark masses” [concept notwell-defined] e.g. mu ∼ md ∼ 300 MeV (about 1/3 of the proton mass) one gets2.2511 × 10−8 (ambiguous)


Quark and pion loops fail: missing is the pronounced ρ0 spin 1 resonancee+e− → ρ0 → π+π− almost saturates the result based on dispersion relation ande+e−-data.

Lesson:

− pQCD fails; QPM result arbitrary (quark masses)− ChPT (only knows pions) fails; reason only converge for p > 400 MeV− dominating is spin 1 resonance ρ0 at ' 775 MeV (VDM); cries for large Nc

QCD [haron spectrum = infinite series of narrow vector boson resonances]− lattice QCD now on the way to solve the problem once one can simulate at

(close to) physical quark masses− resort on sum rule type semi-phenomenological approach Dispersion Rela-

tions (DR) and experimental data (causality & unitarity⇔ analyticity & opticaltheorem).


r Evaluation of ahadµ

Leading non-perturbative hadronic contributions ahadµ can be obtained in terms of

Rγ(s) ≡ σ(0)(e+e− → γ∗ → hadrons)/4πα2

3s data via dispersion integral:

ahadµ =

(αmµ

3π

)2 ( E2cut∫

4m2π

dsRdataγ (s) K(s)

s2 +

∞∫E2

cut

dsRpQCDγ (s) K(s)

s2

)

Data: CMD-2, SND, KLOE, BaBar

0.0 GeV, ∞

ρ, ω

1.0 GeV

φ, . . . 2.0 GeV

3.1 GeV

ψ 9.5 GeVΥ0.0 GeV, ∞

ρ, ω

1.0 GeV

φ, . . .

2.0 GeV

3.1 GeV

l Experimental error implies theoretical uncertainty!l Low energy contributions enhanced: ∼ 75% come from region 4m2

π < m2ππ < M2

Φ

ahad(1)µ = (690.7 ± 4.7)[695.5 ± 4.1] 10−10

e+e−–data based [incl. BaBar MD09]


vGood old idea: use isospin symmetry to include existing high quality τ–data(including isospin corrections)

γ γ

e− u, d

e+ u, d

π+π−, · · · [I = 1]

⇑

isospin rotation

⇓

W W

νµ d

τ−u

π0π−, · · ·

Corrected data: large discrepancy [∼ 10%] persists! τ vs. e+e− problem! [manifestsince 2002]


Recent: τ (charged channel) vs. e+e− (neutral channel) puzzle resolvedF.J.& R. Szafron, ρ − γ interference(absent in charged channel):

30(s) = rργ(s) RIB(s) 3−(s)

−i Πµν (π)γρ

(q) = + .

r τ require to be corrected for missing ρ − γ mixing!

r results obtained from e+e− data is what goes into aµ


aµ[ππ], I = 1, (0.592− 0.975) GeV ×10−10

τ decays

e+e−+CVC

380 390 400

ALEPH 1997

ALEPH 2005

OPAL 1999

CLEO 2000

Belle 2008

τ combined

390.75± 2.65± 1.94

388.74± 4.00± 2.07

380.25± 7.27± 5.06

391.59± 4.11± 6.27

394.67± 0.53± 3.66

391.06± 1.42± 2.06

CMD-2 2006

SND 2006

KLOE 2008

KLOE 2010

BABAR 2009

e+e− combined

386.58± 2.76± 2.59

383.99± 1.40± 4.99

380.21± 0.34± 3.27

377.35± 0.71± 3.50

389.35± 0.37± 2.00

385.12± 0.87± 2.18

I=1 part of ahadµ [ππ]


aµ[ππ], I = 1, (0.592− 0.975) GeV ×10−10

τ decays

e+e−+CVC

380 390 400

ALEPH 1997

ALEPH 2005

OPAL 1999

CLEO 2000

Belle 2008

τ combined

385.63± 2.65± 1.94

383.54± 4.00± 2.07

375.39± 7.27± 5.06

386.61± 4.11± 6.27

389.62± 0.53± 3.66

385.96± 1.40± 2.10

CMD-2 2006

SND 2006

KLOE 2008

KLOE 2010

BABAR 2009

e+e− combined

386.58± 2.76± 2.59

383.99± 1.40± 4.99

380.21± 0.34± 3.27

377.35± 0.71± 3.50

389.35± 0.37± 2.00

385.12± 0.87± 2.18

I=1 part of ahadµ [ππ]


|Fπ(E)|2 in units of e+e− I=1 (CMD-2 GS fit)


Is our model viable?

Look at γγ → ππ


Strong tensor meson resonance in ππ channel f2(1270) with photons directlyprobing the quarks structure of the pions!

l Photons seem to see pions below 1 GeV

l Photons definitely look at the quarks in f2(1270) resonance region

l We apply the sQED model up to 0.975 GeV (relevant for aµ). This should bepretty save (still we assume a 10% model uncertainty)

l Switching off the electromagnetic interaction of pions, is definitely not a realisticapproximation in trying to describe what data we see in e+e− → π+π−


New BaBar ISR 2π result: arXiv:1205.2228v1 10 May 2012

510 530 550 570

e+e−

ee BaBar

ee KLOE

ee CMD-2

ee SND

τ

τ ALEPH

τ CLEO

τ OPAL

τ Belle

a2π,LOµ ×10

−10

γ − ρ correction: not applied


New BaBar ISR 2π result: arXiv:1205.2228v1 10 May 2012

510 530 550 570

e+e−

ee BaBar

ee KLOE

ee CMD-2

ee SND

τ

τ ALEPH

τ CLEO

τ OPAL

τ Belle

a2π,LOµ ×10

−10

γ − ρ correction: applied


'

&

$

%

a2π,LOµ [2mπ − 1.8 GeV]: (514.1 ± 3.8) × 10−10 [BaBar]

(507.8 ± 3.2) × 10−10 [ee all](508.7 ± 2.5) × 10−10 [ee + τ JS]

'

&

$

%

atheµ : 116591865(54) × 10−11 [BaBar]

116591802(50) × 10−11 [ee all]116591811(46) × 10−11 [ee + τ JS]

'

&

$

%

δaµ = aexpµ − athe

µ : (224 ± 83) × 10−11, 2.7σ [BaBar](287 ± 80) × 10−11, 3.6σ, [ee all](278 ± 78) × 10−11, 3.6σ, [ee + τ JS]


The hadronic LbL: setup and problems

Hadrons in 〈0|T Aµ(x1)Aν(x2)Aρ(x3)Aσ(x4)|0〉

µ(p)

γ(k) kρ

had

µ(p′)

q1µq2νq3λ

Key object full rank-four hadronic vacuum polarization tensor

Πµνλρ(q1, q2, q3) =∫

d4x1 d4x2 d4x3 ei (q1x1+q2x2+q3x3)

×〈 0 |T jµ(x1) jν(x2) jλ(x3) jρ(0) | 0 〉 .v non-perturbative physics

v general covariant decomposition involves 138 Lorentz structures of which

v 32 can contribute to g − 2

v fortunately, dominated by the pseudoscalar exchanges π0, η, η′, ... described bythe effective Wess-Zumino Lagrangian


v generally, pQCD useful to evaluate the short distance (S.D.) tail

v the dominant long distance (L.D.) part must be evaluated using some low energyeffective model which includes the pseudoscalar Goldstone bosons as well asthe vector mesons which play a dominant role (vector meson dominance mech-anism); HLS, ENJL, general RLA, large Nc inspired ansatze, and others

Need appropriate low energy effective theory⇒amount to calculate the followingtype diagrams

π0, η, η′

83(12)× 10−11

L.D.

−19(13)× 10−11

L.D.

π±, K±

+62(3)× 10−11

q = (u, d, s, ...)

S.D.

LD contribution requires low energy effective hadronic models: simplest case π0γγvertex


+ + + + · · ·

→ + + · · · + + · · ·

︸︷︷︸

L.D.

︸︷︷︸

S.D.

π0

π±

u, d

u, d

g

Crystal Ball 1988

Data show almost background free spikes of the PS mesons! Substantialbackground form quark loop is absent (seems to contradict large quark-loopcontribution as obtained in Schwinger-Dyson approach (SDA) Darmstadt group).Clear message from data: fully non-perturbative, evidence for PS dominance.However, no information about axial mesons (Landau-Yang theorem). Illustrateshow data can tell us where we are.

Low energy expansion in terms of hadronic components: theoretical models vsexperimental data à KLOE, KEDR, BES, BaBar, Belle, ?


Basic problem: (s, s1, s2)–domain of Fπ0∗γ∗γ∗(s, s1, s2); here (0, s1, s2)–plane

Two scale problem: “open regions”

RLA

???

???

pQCD

One scale problem: “no problem”

RLA pQCD

– Data, OPE,??? – QCD factorization,

– Brodsky-Lepage approach

Novel approach: refer to quark–hadron duality of large-Nc QCD, hadron spectrumknown, infinite series of narrow spin 1 resonances ’t Hooft 79⇒no matchingproblem (resonance representation has to match quark level representation)De Rafael 94, Knecht, Nyffeler 02

Constraints for on-shell pions (pion pole approximation)

v General form–factor Fπ0∗γ∗γ∗(s, s1, s2) is largely unknown


v The constant e2Fπ0γγ(m2π, 0, 0) = e2Nc

12π2 fπ= α

π fπ≈ 0.025 GeV−1 well determined by

π0 → γγ decay rate (from Wess-Zumino Lagrangian); experimental improvementneeded!

v Information on Fπ0γ∗γ(m2π,−Q2, 0) from e+e− → e+e−π0 experiments

π0

e−(pb)

e′−(pt)

e+ e′+

q2∼ 0

Q2 > 0

γ

γ∗

2.5 5.0 7.5 10.0

Q2 (GeV2)

Q2

F(Q

2 )(G

eV)

IIII

2 f

0970597-008

CELLO

CLEO

0.30

0.20

0.10

0

CELLO and CLEO measurement of the π0 form factor Fπ0γ∗γ(m2π,−Q2, 0) at

high space–like Q2. outdated now by BB?


Brodsky–Lepage interpolating formula gives an acceptable fit.

Fπ0γ∗γ(m2π,−Q2, 0) '

14π2 fπ

11 + (Q2/8π2 f 2

π )∼

2 fπQ2

Inspired by pion pole dominance idea this FF has been used mostly(HKS,BPP,KN) in the past, but has been criticized recently (MV and FJ07).

r Melnikov, Vainshtein: in chiral limit vertex with external photon must benon-dressed! i.e. use Fπ0γ∗γ(0, 0, 0), which avoids eventual kinematic

inconsistency, thus no VMD damping⇒result increases by 30% !

r In g − 2 external photon at zero momentum⇒ only Fπ0∗γ∗γ(−Q2,−Q2, 0) not

Fπ0γ∗γ(m2π,−Q2, 0) is consistent with kinematics. Unfortunately, this off–shell form

factor is not known and in fact not measurable and CELLO/CLEO constraint doesnot apply!. Obsolete far off-shell pion (in space-like region).


π0

e−(pb)

e′−(pt)

e+ e′+

q2∼ 0

Q2 > 0

γ

γ∗

π0

µ− µ′−

q2∼ 0

Q2

γ

γ∗

hard soft

hard hard“soft” hard

a) b)

Measured is Fπ0γ∗γ(m2π,−Q2, 0) at high space–like Q2, needed at externalvertex is Fπ0∗γ∗γ(−Q2,−Q2, 0).

r I still claim using Fπ0∗γ∗γ(0, 0, 0) in this case is not a reliable approximation!

Need realistic “model” for off–shell form–factor Fπ0∗γ∗γ(−Q2,−Q2, 0)!

Is it really to be identified with Fπ0∗γ∗γ(0, 0, 0)?

Can we check such questions experimentally or in lattice QCD?


Evaluation of aLbLµ in the large-Nc framework

v Knecht & Nyffeler and Melnikov & Vainshtein were using pion-pole approxima-tion together with large-Nc π

0γγ–form-factor

v FJ & A. Nyffeler: relax from pole approximation, using KN off-shell LDM+V form-factor

Fπ0∗γ∗γ∗(p2π, q

21, q

22) =

Fπ

3P(q2

1, q22, p2

π)

Q(q21, q

22)

P(q21, q

22, p2

π) = h7 + h6 p2π + h5 (q2

2 + q21) + h4 p4

π + h3 (q22 + q2

1) p2π

+h2 q21 q2

2 + h1 (q22 + q2

1)2 + q21 q2

2 (p2π + q2

2 + q21))

Q(q21, q

22) = (q2

1 − M21) (q2

1 − M22) (q2

2 − M21) (q2

2 − M22)

all constants are constraint by SD expansion (OPE). Again, need data to fixparameters!


Looking for new ideas to get ride of model dependence

r Need better constrained effective resonance Lagrangian (e.g. HSL and ENJLmodels vs. RLA of Ecker et al.). “Global effort” needed!

recent: HLS global fit available Benayoun et al 2010

r Lattice QCD will provide an answer [take time (“yellow” region only?)]!

r Try exploiting possible new experimental constraints:


Pseudoscalar exchanges: π0, η, η′

Leading LbL contribution from PS mesons:

aµ[π0, η, η′] ∼ (93.91 ± 12.40) × 10−11

r π0γγ form-factor: experimental facts and possibilities

l relation between the off-shell (needed for aµ) and the on-shell (measured)from-factor is all but obvious

Note: Fπ0∗γ∗γ(−Q2,−Q2, 0) is a one-scale problem. Self-energy type of problem⇒can get it via dispersion relation from appropriate data

Existing data for F(m2π,Q

2, 0): e+e− → e+e−π0 single tag data dσdQ2

àCELLO: 0.5 GeV2 < Q2 < 2.17 GeV2 [Z. Phys. C49 (1991) 401]àCLEO: 1.5 GeV2 < Q2 < 9 GeV2 [Phys. Rev. D57 (1998) 33]


àBB: 4 GeV2 < t2 < 40 GeV2 [Phys. Rev. D80 (2009) 052002]àBelle: 4 GeV2 < t2 < 40 GeV2 [arXiv:1205.3249 [hep-ex]]

à new quest for theoryl BB seems to violate Q2 F(m2

π,Q2, 0)→ 2 fπ0 (constant) in π0 channel

l BB: π0, η and η′ seem to show different behavior

àtheory: Brodsky-Lepage (BL) behavior ∼ 1/Q2 for all pseudoscalars


l asymptotic behavior ? data consistent ? BaBar conflict relaxed by Belle


KLOE, Lattice

Cross check of BB by Belle (anomalous increase not seen), BESIIIF. Jegerlehner JG Univ. Mainz June 2011 39

intermediate


Axial exchanges: a1, f ′1, f1

Axial exchangesLandau-Yang Theorem: A (axial meson→ γγ)=0

e.g. Z0 → γγ, while Z0 → γe+e−5 3

Why aµ[a1, f ′1, f1] ∼ 25 × 10−11 so large?

r untagged γγ → f () no signal!r single-tag γ∗γ → f () strong peak is Q2 m2

f


σ(γ∗γ → f1 → K0s Kπ)

CELLO 1989Sparse data so far, new measurements important; in particular momentumdependent Γ(a1 → γγ∗) etc.Expected contribution from axial mesons:

aµ[a1, f ′1, f1] ∼ (28.13 ± 5.63) × 10−11


Scalar exchanges: a0, f ′0, f0, · · ·

Mesons: M(qq), M(qqqq), glueballs mixingExperimental: Crystal Ball, Mark II, Belle!Theory: Mennessier, Pennington et al., Mousallam et al., Achasov et al., ...


Strong tensor meson resonance in ππ channel f2(1270)


So: expect usual pion-loop in HLbL plays role like pion-loop in VP. i.e. likemissing the ρ.à Need to explicitly include tensor mesons

The di-pion amplitude Mdirectres (γγ → π+π−; s) gets contribution caused by mixed

σ(600) and f0(980) resonances with the direct coupling constants of the σ(600)and f0(980) to photons, g(0)

σγγ and g(0)f0γγ

,

Mdirectres (γγ → π+π−; s) = s eiδππB (s)

×g(0)σγγ[D f0(s)gσπ+π− + Π f0σ(s)g f0π+π−] + g

(0)f0γγ

[Dσ(s)g f0π+π− + Π f0σ(s)gσπ+π−]

Dσ(s)D f0(s) − Π2f0σ

(s).

For√

s < 2mK, the phase coincides with the I=0, S wave ππ phase shiftδ0

0(s) = δππB (s) + δres(s).

Scalars everywhere. Many scalars many small contributions may sum up tosubstantial effect!


Expected contribution from qq scalars:

aµ[a0, f ′0, f0] ∼ (−5.98 ± 1.20) × 10−11

So far nobody has evaluated qqqq in S U(3) sector [u, d, s ] many possible states,which individually are expected rather small


LbL: Present & Future

l γγ, γ∗γ and γ∗γ∗ physics a mandatory input for constraining hadronic LbLamplitudes.

r Need improved hadron→ γγ and γγ → hadrons measurements for π0, η, η′ aswell for axial and scalar mesons

r Single tag form factors very much improvement to come for pseudoscalars

r Double tag form factors: experimentally not simple, requires very high luminosity

r γγ → π+π−, π0π0,K+K−,KLKS , · · · important input for scalar sector and pion andkaon loops

l Challenge for theory: radiative corrections needed

r Question of asymptotic behavior (BaBar vs Belle) important: do we understandit?


r Can we check controversial dressed/undressed (i.e damping or not?) atexternal vertex?


Phase Space – Pseudoscalar Exchange


Pseudoscalar Exchange – with Pion Form-Factor


Phase Space – Axial Meson Exchange


Phase Space – Scalar Exchange

Working out recommendations for most relevant experimental input


Theory vs experiment: do we see New Physics?

Contribution Value Error Reference

QED incl. 4-loops+5-loops 11 658 471.885 0.04 Remiddi, Kinoshita ...Leading hadronic vac. pol. 691.0 4.7 2011 updateSubleading hadronic vac. pol. -10.0 0.1 2011 updateHadronic light–by–light 11.6 3.9 evaluation (J&N 09)Weak incl. 2-loops 15.32 0.22 CMV06

Theory 11 659 179.7 6.1 –Experiment 11 659 209.1 6.3 BNL UpdatedExp.- The. 3.3 standard deviations 29.2 8.8 –

Standard model theory and experiment comparison [in units 10−10]. Whatrepresents the 3 σ deviation: r new physics? r a statistical fluctuation? runderestimating uncertainties (experimental, theoretical)?vdo experiments measure what theoreticians calculate?


Implications of low and high energy measurements on SUSYmodels

Most promising New Physics scenario: SUSY

ν

χ χ

a)

χ0

µ µ

b)

Leading SUSY contributions to g − 2 in supersymmetric extension of the SM.

v m lightest SUSY particle; SUSY requires two Higgs doublets

v tan β = 3132

, 3i =< Hi > ; i = 1, 2 ; tan β ∼ mt/mb ∼ 40 [4 − 40]

vaSUSYµ '

sign(µM2)α(MZ)8π sin2ΘW

(5 + tan2ΘW

)6

m2µ

M2SUSY

tan β(1 −

4απ

lnMSUSY

mµ

)F. Jegerlehner JG Univ. Mainz June 2011 55

with MSUSY a typical SUSY loop mass and the sign is determined by the Higgsinomass term µ ( RG improved). In general:

∆aNPµ = α

NP m2µ

M2NP

NP searches (LEP, Tevatron, LHC): typically MNP >> MW, then ∆aexp−theµ = ∆aNP

µ

requires αNP ∼ 1 spoiling perturbative arguments. Exception: 2HDM, SUSY tan βenhanced coupling!

r muon g − 2 in contrast requires moderately light SUSY masses and in thepre-LHC era fitted rather well with expectations from SUSY

r a particular role is played by the mass of the light Higgs

At tree level in the MSSM mh ≤ MZ. This bound receives large radiative correctionsfrom the t/t sector, which changes the upper bound to (Haber & Hempfling 1990)


m2h ∼ M2

Z cos2 2β +3√

2Gµ m4t

2π2 sin2 βln

(mt1 mt2

m2t

)+ · · ·

which in any case is well below 200 GeV. A given value of mh fixes the value ofm1/2 represented by mt1,mt2

r if Higgs is found at 125 GeV (CERN “rumors”) we must have m1/2 > 800GeV orhigher!


Constraint on large tan β SUSY contributions as a function of MSUSY. Thehorizontal band shows ∆aNP

µ = δaµ. The region left of MSUSY ∼ 500 GeV isexcluded by LHC searches. If mh ∼ 125 ± 1.5 GeV actually MSUSY > 800 GeVdepending on details of the stop sector (t1, t2 mixing and mass splitting) and

weakly on tan β .l if Higgs is found at 125 GeV (will know within one month from now) muong − 2 looks to me in big trouble. If δaµ is not SUSY, what else? most otherNP scenarios give likely even smaller contributions!


vmost likely for me we could have been missing some electromagnetic radiationeffects in the relation between observed and calculated quantity!

vthe other obstacle: hadronic light-by-light

vprogress in evaluating HVP: more data (BaBar, Belle, VEPP 2000, BESIII,...),Lattice QCD in progress, effective field theories etc.

The big challenge: two complementary experiments: Fermilab with ultra hotmuons and KEK with ultra cold muons (very different radiation) to come

Provided deviation is real 3σ→ 9σ possible? Provided theory and needed crosssection data improves the same as the muon g − 2 experiments!

Muon g − 2 remains a key monitor for NPand

a great challenge for theorists as well as for experimenters!


An interesting frontier of digging to understand the details of the SM and its limitations

Thank you for your attention!


Backup Slides

M. Benayoun et al. HLS effective resonance Lagrangian model global fits

150 200 250

incl. ISRDHMZ10 (e+

e−)

180.2± 4.9

[3.6 σ]

DHMZ10 (e+e−+τ)

189.4± 5.4

[2.4 σ]

JS11 (e+e−+τ)

179.7± 6.0

[3.4 σ]

HLMNT11 (e+e−)

182.8± 4.9

[3.3 σ]

excl. ISRDHea09 (e+

e−)

178.8± 5.8

[3.5 σ]

A (e+e−+τ)

173.4± 5.3

[4.3 σ]

B (e+e−+τ)

175.4± 5.3

[4.1 σ]

experimentBNL-E821 (world average)208.9± 6.3

aµ×1010-11659000

Data below E0 = 1.05 GeV (just above the φ) constrain effective Lagrangian


couplings, using 45 different data sets (6 annihilation channels and 10 partialwidth decays).


r Effective theory predicts cross sections:π+π−, π0γ, ηγ, η′γ, π0π+π−, K+K−, K0K0 (83.4%),

l Missing part:4π, 5π, 6π, ηππ, ωπ and regime E > E0

evaluated using data directly and pQCD for perturbative region and tail

l Including self-energy effects is mandatory (γρ-mixing, ρω-mixing ..., decays withproper phase space, energy dependent width etc)

l Method works in reducing uncertainties by using indirect constraints

l Able to reveal inconsistencies in data. In our case in region [1.00,1.05] GeVtension between KK and 3π data sets. All data: Solution A 71.2% CL;excluding 3π above 1 GeV: Solution B 97.0% CL. Conflict in data?, model?

r Singling out effective resonance Lagrangian by global fitis expected to help in improving EFT calculations of hadroniclight-by-light scattering (such concept so far missing)


A new representation for single particle exchange in LbL

• aµ does not depend on direction of muon momentum p⇒ may average inEuclidean space over the directions P:

〈· · ·〉 =1

2π2

∫dΩ(P) · · ·

Hadronic single particle exchange amplitudes independent of p⇒ 2 integrationsmay be done analytically: amplitudes Ti, propagators (4) ≡ (P + Q1)2 + m2

µ and(5) ≡ (P − Q2)2 + m2

µ with P2 = −m2µ

〈1

(4)1

(5)〉 =

1m2µR12

arctan( zx1 − zt

)〈(P · Q1)

1(5)〉 = − (Q1 · Q2)

(1 − Rm2)2

8m2µ

,


〈(P · Q2)1

(4)〉 = (Q1 · Q2)

(1 − Rm1)2

8m2µ

〈1

(4)〉 = −

1 − Rm1

2m2µ

〈1

(5)〉 = −

1 − Rm2

2m2µ

Rmi =

√1 + 4m2

µ/Q2i , (Q1 · Q2) = Q1 Q2 t, t = cos θ, θ= angle between Q1 and Q2.

Denoting x =√

1 − t2, we have R12 = Q1 Q2 x and

z =Q1Q2

4m2µ

(1 − Rm1) (1 − Rm2) .

• For any hadronic form-factor end up with 3–dimensional integral over Q1 = |Q1|,


Q2 = |Q2| and t = cos θ:

aµ(LbL; π0) = −2α3

3π2

∫ ∞

0dQ1dQ2

∫ +1

−1dt√

1 − t2 Q31 Q3

2

× (F1 P6 I1(Q1,Q2, t) + F2 P7 I2(Q1,Q2, t))

where P6 = 1/(Q22 + m2

π), and P7 = 1/(Q23 + m2

π) denote the Euclidean singleparticle exchange propagators. I1 and I2 known integration kernels. Thenon-perturbative factors are

F1 = Fπ0∗γ∗γ∗(q22, q

21, q

23) Fπ0∗γ∗γ(q

22, q

22, 0) ,

F2 = Fπ0∗γ∗γ∗(q23, q

21, q

22) Fπ0∗γ∗γ(q

23, q

23, 0) .


Note: SU(3) flavor decomposition of em current→ weight factors

W (a) =

(Tr [λaQ2]

)2

Tr [λ2a]Tr [Q4]

; W (3) =14, W (8) =

112

, W (0) =23.

where Tr [Q4] = 2/9 is the overall normalization such that∑

a W (a) = 1. Note(W (8) +W (0))/W (3) = 3, higher states enhanced in coupling by factor 3![Melnikov&Vainshtein] overlooked by previous analyzes [HKS,HK,BPP].

Such representations I worked out for axial exchanges as well as for scalar ones.Missing is tensor state, could play similar role as ρ exchange vs scalar QED ππcontribution.


observable is the W mass given by

M2W

1 − M2W

M2Z

= πα√

2GF

(1 + ∆r) ; ∆r = f (α,GF,MZ,mt, · · · )

l ∆r model-dependent radiative corrections

l in SUSY models MW is sensitive to the top/stop sector parameters

while

sin2Θeff =14

(1 − Re

3eff

aeff

)

remains much less affected Buchmueller et al., Heinemeyer et al.


r for MW SUSY looks favored by data!


Explanation: Precision predictions

MW : sin2ΘW = 1 −M2

WM2

Z

g2 : sin2Θg = e2/g22 =

πα√

2 Gµ M2W

a f : sin2Θ f = 14|Q f |

(1 −

v fa f

), f , ν

a f : ρ f = 11−∆ρ , independent on α

sin2Θi cos2Θi =πα

√2 Gµ M2

Z

11 − ∆ri

where

∆ri = ∆ri(α, Gµ, MZ, mH, m f,t, mt)

quantum corrections from gauge boson self-energies, vertex– and


box–corrections. for the most important cases and the general form of ∆ri reads

∆ri = ∆α − fi(sin2Θi) ∆ρ + ∆ri remainder

with a universal term ∆α which affects the predictions for MW, ALR, A fFB, Γ f , etc.

while ∆ρ is very process dependent: fW(sin2ΘW) = cos2ΘWsin2ΘW

' 3.3 ; fg(sin2Θg) = 1.

l sensitivity to top/stop sector strongly enhances in MW

l General: MSSM results merge into SM results for larger SUSY masses, asdecoupling is at work.


r SUSY does not yield better global fit!


Other strong constraints

r Data on the penguin loop induced B→ Xsγ transition

SM prediction B(b→ sγ)NNLL = (3.15 ± 0.23) × 10−4 is consistent within 1.2 σ withthe experimental result (HFAG) B(b→ sγ) = (3.55 ± 0.24 ± 0.09) × 10−4.

l it implies that SUSY requires heavier m1/2 and/or m0 in order not to spoil thegood agreement.

r Data on dark matter relict density ΩCDMh2 = 0.1126 ± 0.0081

SUSY+R-parity scenarios represent a tough constraint for the relic density ofneutralinos produced in the early universe.

l A DM neutralino is a WIMP DM candidate. The density predicted is

Ωh2 ∼0.1 pb〈σ3〉

∼ 0.1( MWIMP

100GeV

)2

,


where 〈σ3〉 is the relativistic thermally averaged annihilation cross-section.

l in most scenarios the dominating neutralino annihilation process isχ + χ→ A→ bb and the observed relict density requires the cross section to betuned to 〈σ3〉 ∼ 2 × 10−26 cm3/s. The cross section is of the form

〈σ3〉 ∝ tan2 βm2

b

M2Z

M4χ

(4M2χ − M2

A)2 + M2AΓ

2A

and has to be adjusted to Mχ ≈ 1.8 MA to 2.2 MA. On resonance the cross sectionwould be too big, too far off resonance too small. Note that except from ΩCDM allobservables prefer heavier SUSY masses such that effects are small bydecoupling.


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