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The anomaly triangle and muon g − 2

SANTI PERIS San Francisco State U. and

• U. Autonoma de Barcelona

The anomaly triangle and muon g − 2 – p.1/8

2-loop EW contribution to g − 2

Kukhto et al. ’92 SP, Perrottet, de Rafael ’95 Czarnecki, Krause, Marciano ’95, ’96 Knecht, SP, Perrottet, de Rafael ’02 Czarnecki, Marciano,Vainshtein ’03

µ µ γ γ Z f

g−2 2

∝ α

π Gµ 8π2 m2

µ

√ 2

∞

m2

µ dQ2

• wL(Q2) +

M2

Z

M2

Z + Q2 wT (Q2)

• wL=2 wT =2 Nc

Q2

(one−loop, mf =0) g−2 2 |e,u,d ∼ 2 × 10−11

Q2 = −q2, “Gluon-irreducible” quark triangle Wµνρ(q, k) = T (3)

f

Q2

f

• wL(Q2) qν ǫµραβqα kβ +

wT (Q2) kσ q2ǫµνρσ + qνǫµρλσqλ + qµǫρνλσqλ + O(k2)

• wL(Q2) related to the chiral anomaly =

⇒

ν,e,u,d

• T (3)

f

Q2

f

• wL = 0.
• wT (Q2) not but...

The anomaly triangle and muon g − 2 – p.2/8

Theorem

Vainshtein ’02 Knecht, SP, Perrottet, de Rafael, ’03

In the massless limit, to all orders in αs: wL(Q2) = 2 wT (Q2) and, since anomaly does not get renormalized: wL = 2 Nc

Q2 exact!

(Adler,Bardeen ’69; Witten ’83)

⇒ neither does 2wT = 2 Nc

Q2 , to all orders in αs.

Using L(3)

µ

=

ℓ=ν,e ℓLγµT (3)ℓL + q=u,d qLγµT (3)qL, etc...in SU(2)L × U(1)Y :

Q2 wL(Q2) − 2wT (Q2)

• quarks ∝
• d4xd4y eiqx (y−x)λǫµνρλ

T

• L(3)

µ (x)V (Y ) ν

(y)R(Y )

ρ

(0)

• =0, (Pert. Theory)
• i.e., wL − 2wT has no pert. contributions in αs, it is like, e.g., LR = V V − AA.

The anomaly triangle and muon g − 2 – p.3/8

Non-perturbative effects

1) Adler-Bardeen-Witten : wL(Q2) = 2 Nc

Q2

(exact for all Q !) 2) However, for wT (Q2):

• Large Q2:

2wT (Q2) ≈ 2 Nc Q2

• 1 + NO αs
• + (const.) αs χ ψψ2

Q6 + O(1/Q8) Magnetic susceptibility, χ = ΠV T (0)

ψψ , very poorly known.

• Small Q2:

2wT (Q2) ≈ (const.) C(p6)

22

+ O(Q2) , C(p6)

22

∼ 1/M2

Hadron

(unknown) Chiral Pert. Theory, Leff (parity-odd):

(Ebertshauser, Fearing, Scherer ’01; Bijnens, Girlanda, Talavera ’02, Kampf, Moussallam ’09)

LO(p6) = C(p6)

22

ǫµναβ Tr

• uµ

∇γfγν

+ , fαβ +

• π,η,...

+... ; SU(NF )L×SU(NF )R → SU(NF )V

The anomaly triangle and muon g − 2 – p.4/8

Non-perturbative effects (II)

Very roughly, 2 wT (Q2) Q2 ∼

Anomaly

• 2Nc

Q2 Q2 + Λ2

Hadron

i.e.

2Nc 2w Q

T 2

The anomaly triangle and muon g − 2 – p.5/8

Conjectures

• Conjecture 1:

wL(Q2) − 2 wT (Q2) = −2 Nc

f2

π ΠLR(Q2) (Son-Yamamoto ’10)

in wide class of “AdS/QCD” models (chiral limit, Nc → ∞) (not without caveats, e.g. OPE is exponential; wrong chiral limit in pert. theory)

( Knecht, SP, de Rafael ’11)

Chiral log’s respect this relation in SU(2) × SU(2) × U(1) (mu,d = 0, ms = 0)

(Gorsky, Kopnin,Krikun, Vainshtein ’12)

− 64π2 c(p6)

13

(µ) = − Nc

f2

π

ℓ(p4)

5

(µ) However, they don’t in SU(3) × SU(3) (mu,d,s = 0)

(Knecht, SP , ’12 (unpublished))

128π2 C(p6)

22

(µ) = −

Nc f2

π

L(p4)

10

(µ)

??

• Conjecture 2:

χ = −

Nc 4π2f2

π ∼ −9 GeV−2

(Magnetic susceptibility)??

(Vainshtein ’02):

Other results: χ ∼ −3 GeV−2 , sum rules, VMD, (Ioffe, Fadin, Lipatov ’10; Balitsky et al. ’85; Belyaev et al.

’84; Ball et al. ’02)

The anomaly triangle and muon g − 2 – p.6/8

Another Perturbative Surprise

Up to now, special kinematic configuration in V V A.

Jegerlehner,Tarasov ’06

However, it has been found at two loops for arbitrary momenta that : Wµνρ(q, k) = Wµνρ(q, k)|one−loop (1 + O(αs)

=0 !!

) i.e., no renormalization, not just for the anomaly, but for the whole triangle ! Given the non-trivial momentum dependence, can this be just a coincidence ? could this be true to all orders in αs ?

The anomaly triangle and muon g − 2 – p.7/8

Summary

• VVA triangle is a very interesting theoretical laboratory for QCD
• Even though most results obtained in chiral limit: can lattice help/check ?
• The LbL ←

→ V V A connection:

( Melnikov, Vainshtein ’04; Prades, de Rafael, Vainshtein ’09)

k1 ≈ k2 ≫ k3

k k k q

1 2 3

q k3

γ γ

γ 5

H

The anomaly triangle and muon g − 2 – p.8/8