axial form factorsmicrotron.iasa.gr/pavi06/talks/schindler_iic.pdfgoldberger-treiman-relation...

Axial Form Factors

M. R. Schindler

Institut für KernphysikJohannes Gutenberg-Universität, Mainz

From Parity Violation to Hadronic Structure and more . . .Milos Island

16-20 May, 2006

In collaboration with T. Fuchs, S. Scherer

M. R. Schindler Milos 2006

Outline

Definition and Properties

Axial form factor GA(q2)

Induced pseudoscalar form factor GP(q2)

GA, GP in Lorentz-invariant ChPT

Summary


Definition and Properties


Form factors

Form factors of the nucleonParametrization of nucleon structureElectromagnetic form factors F1(q2), F2(q2)[GE(q2), GM(q2)

]Vector current ∼ γµ

Well known experimentallyImportant test for theories and models

Axial form factors GA(q2), GP(q2), GT (q2)

Axial-vector current ∼ γµγ5Not very well known


Axial-vector current operator

Aµ,a(x) ≡ q(x)γµγ5τa

2q(x), q =

(ud

), a = 1, 2, 3

HermitianA†

µ,i(x) = Aµ,i(x)

Vector under isospin transformations

[Ia, Aµ,b(x)] = iεabcAµ,c(x)

Axial vector under parity

Aµ,i(x)P7→ −Ai

µ(Px)

Charge conjugation

Aµ,i(x)C7→ Aµ,i(x), i = 1, 3

Aµ,i(x)C7→ −Aµ,i(x), i = 2


Definition of axial form factors

Axial-vector current matrix element

〈N(p′)|Aµi (0)|N(p)〉

= u(p′)[GA(q2)γµ +

qµ

2MGP(q2) + i

σµνqν

2MGT (q2)

]γ5

τi

2u(p)

qµ = p′µ − pµ

GA(q2): axial form factorGP(q2): induced pseudoscalar form factorGT (q2): induced pseudotensorial form factor


Properties of axial form factors

Hermiticity for q2 ≤ 0

GA(q2)∗ = GA(q2)

GP(q2)∗ = GP(q2)

GT (q2)∗ = −GT (q2)

For mu = md : invariance under G-conjugation

G = C exp(iπI2)

⇒GT (q2) = 0


Breit-System

~p = −12~q = −~p ′, q0 = 0

〈N (~q2 ) |A0,a(0)|N (−~q

2 )〉 = 0

〈N (~q2 ) |~A

a(0)|N (−~q2 )〉 =

τa

2

[E(−~q

2 )

M(~σ − ~σ · qq)GA(−~q 2) + ~σ · qqD(−~q 2)

]



GA(0) = gADetermined from n → p + e− + νeGA(q2 ≈ 0) = gA = 1.2670± 0.0035

GA(q2)

(Quasi)elastic (anti)neutrino scatteringPion electroproductionParametrization

GA(q2) =gA

(1− q2

M2A)2

with MA: axial mass


Neutrino scattering

0.85 0.95 1.05 1.15 1.25MA [GeV]

Average

Argonne (1969)

CERN (1977)

Argonne (1977)

CERN (1979)

BNL (1980)

BNL (1981)

Argonne (1982)

Fermilab (1983)

BNL (1986)

BNL (1987)

BNL (1990)

Argonne (1973)

from V. Bernard, L. Elouadrhiri and U.-G. Meißner,

J. Phys. G 28, R1 (2002)

Global average (new)

MA = (1.001± 0.020) GeV

H. Budd, A. Bodek and J. Arrington,

arXiv:hep-ex/0308005


Pion electroproductionγ∗ + p → n + π+ at threshold

Extract MA from E0+

0.85 0.95 1.05 1.15 1.25MA [GeV]

Frascati (1970)Frascati (1970) GEn=0Frascati (1972)DESY (1973)Daresbury (1975) SPDaresbury (1975) DRDaresbury (1975) FPVDaresbury (1975) BNR

AverageMAMI (1999)Saclay (1993)Olsson (1978)Kharkov (1978)DESY (1976)

Daresbury (1976) SP

Daresbury (1976) BNRDaresbury (1976) DR

from V. Bernard, L. Elouadrhiri and U.-G. Meißner,

J. Phys. G 28, R1 (2002)

Global average

MA = (1.068± 0.017) GeV


Heavy Baryon ChPT: Modification at O(q3)

Notation for amplitude

~M∣∣∣thr

= −4πWmN

i[i(~σ − ~σ · k k)E0+(k2) + i~σ · k kL0+(k2)

]Contribution from pion loops

1 1

2

1 1

2

1

2


Heavy Baryon ChPT: Modification at O(q3)

Modification of the k2 dependence of E (−)0+

E (−)0+ (k2) =

egA

8πFπ[1 +

k2

4m2N

(κV +

12

)+

k2

6r2A︸︷︷︸

old

+M2

8π2F 2π

f(

k2

M2

)︸︷︷︸

new

+ · · · ]

Pion electroproduction: MA = (1.068± 0.017) GeV∆MA = 0.056 GeVNeutrino scattering: MA = (1.001± 0.020) GeV

⇒

Agreement

V. Bernard, N. Kaiser, and U.-G. Meißner, Phys. Rev. Lett. 69, 1877 (1992)M. R. Schindler Milos 2006

Haberzettl claims: Pion electroproduction data at thresholdcannot be interpreted in terms of GA[H. Haberzettl, Phys. Rev. Lett. 85, 3576 (2000)]

Solution: Minimal coupling does not respect constraintsdue to chiral symmetry[T. Fuchs and S. Scherer, Phys. Rev. C 68, 055501 (2003)]

L(3)eff = const. Ψγµγ5[Dν , f−µν ]Ψ + · · ·

with

f−µν = −2(∂µaν − ∂νaµ) + 2i ([vµ, aν ]− [vν , aµ])

+iF

[~τ · ~π, ∂µvν − ∂νvµ] + · · ·


Induced pseudoscalar form factor GP(q2)


Chiral limit

Possible solutions◦m= 0 and

◦GP (q2) = 0

◦GP (q2) = −4

◦m

2 ◦GA(q2)q2

GA(0) = gA ≈ 1.26 ⇒◦GP (q2) has pole for q2 → 0


Pion pole contribution

Interpretation in terms of pion pole contribution

Most general pion pole contribution

−2Fπ(q2)gπNN(q2)

q2 −M2 − Σ(q2)uqµγ5

τa

2u


Pion pole contribution: chiral limit

M2 → 0

→ −2F (q2)◦gπNN (q2)

q2−◦Σ (q2)

2◦m u

qµ

2◦m

γ5τa

2u

⇒ Contribution to◦GP

−4◦m F (q2)

◦gπNN (q2)

q2−◦Σ (q2)

with◦Σ (0) = 0


Goldberger-Treiman-Relation

Consider limit

limq2→0

q2 ◦GP,π (q2) = −4

◦m F

◦gπNN

!= −4

◦m

2 ◦GA (0)

Goldberger-Treiman-Relation

◦gA

F=

◦gπNN

◦m

Real world

Satisfied up to ≈ 2%


Experimental data

mµ

2mNGP(q2 = −0.88m2

µ) = gP

Ordinary muon capture (OMC): µ− + p → νµ + n

gP = 8.7± 1.9

Liquid hydrogen target: complications due to formation ofpµp moleculesMuCap experiment at PSI: hydrogen gas

Radiative muon capture (RMC): µ− + p → νµ + n + γ

gP = 12.3± 0.9

q2-dependence: one pion electroproduction experiment


BUT: new measurement of ortho-para transition rate in pµpmolecule

Λnewop = (11.1±1.7±0.9

0.6)×104 s−1 vs. Λoldop = (4.1±1.4)×104 s−1

OMC (Saclay)gP = 0.8± 2.7

RMC (TRIUMF)gP = 10.6± 1.1

average liquid H2gP = 5.6± 4.1

J. H. D. Clark et al., Phys. Rev. Lett. 96, 073401 (2006)


Heavy baryon ChPT

Order O(q3) calculation

GP(q2) =4mNFπgπNN(q2)

M2 − q2 − 23

m2NgA〈r2〉A

gP = 8.23

Allows for systematical corrections


GA, GP in manifestly Lorentz-invariant ChPT


Calculation of GA(q2) and GP(q2) up to and includingO(q4)

Infrared renormalization scheme

GA(q2) = gA +16

gA〈r2〉Aq2 +g3

A4F 2 H(q2)

with H(0) = H ′(0) = 0

gP = 8.09

0.1 0.2 0.3 0.4

Q2@GeV2D

0.2

0.4

0.6

0.8

1

GAHQ2L�GAH0L

-0.1 0 0.1 0.2Q2@GeV2D

-150

-100

-50

0

50

100

150

200

GPHQ2L


Electromagnetic form factors

Compare to situation for e.m. form factors

Inclusion of vector mesons (ρ, ω, Φ)

M. R. Schindler, J. Gegelia and S. Scherer, Eur. Phys. J. A 26, 1 (2005)M. R. Schindler Milos 2006

Inclusion of axial-vector meson a1

Reformulated infrared renormalization → inclusion ofaxial-vector mesons as explicit degrees of freedom

L(3)πA = fA〈AµνFµν

− 〉

L(0)AN = GANΨAµγµγ5Ψ

GAVMA (q2) = −8fAGAN

q2

q2 −M2a1

GAVMP (q2) = 32m2

N fAGAN1

q2 −M2a1

effectively only one additional coupling constant fAGAN


Summary


Summary

Axial form factors parameterize axial structure of nucleonGA(q2)

(Anti)neutrino scatteringPion electroproductionDiscrepancy explained in HBChPT

GP(q2)

Ordinary muon capture (OMC)Radiative muon capture (RMC)Results for gP do not agreeTheory agrees with OMC

Infrared renormalizationInclusion of a1 mesonOnly one additional low energy constant


axial form factorsmicrotron.iasa.gr/pavi06/talks/schindler_iic.pdfgoldberger-treiman-relation...

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