Pion form factors and decays.

Introduction

g-2

Transversity pion form factor: models vs Lattice

Conclusions

A.E. Dorokhov

in collaboration with

W. Broniowski and E. Ruiz Arriola

Cosmology tell us that 95% of matter is not described in text-books yet

Two search strategies:

1) High energy physics to excite heavy degrees of freedom. No any evidence till now. We live in LHC era!

2) Low energy physics to produce Rare processes in view of huge statistics.

There are some rough edges of SM.

Muon anomaly(g-2)is the most famous example

That’s intriguing

m

ehga

sm

eg

BL

S

SS

SI

2a1 ,

2

2

,2

,

Gyromagnetic ratio

Anomaly

Some Definitions

mμ =105.6583692(94) MeV,

mτ = 1776.99 (29) MeV

mμ/me = 206.768 2838(54)

PDG

A charged particle with spin S has a magnetic moment

The general form of the ffvertex is

qepuqFm

qqF

m

qiqFpuie

ll

235

22

21 22

'

•F1 is the electric charge distribution el=eF1(0) •F2 corresponds to Anomalous Magnetic Moment (AMM) al=(gl-2)/2=F2(0)

• F3 corresponds to Anomalous Electric Dipole Moment dl=-el/(2ml )F3(0)

However, in SM al is not zero due to Radiative Corrections

dl=0 due to T- and P symmetries

Magnetic Anomaly

QED Hadronic Weak SUSY... ... or other new physics ?

Basic of Standard Model

• Electron anomaly is measured extremely accurately. QED test.

• It is the best for determining • For a lepton L, Mass Scale contributes to aL as

• Tau anomaly is difficult to measure since its fast decay• Muon anomaly is measured to 0.5 parts in a million

(ppm) SM test.

• Thus muon AMM leads to a (mme)2~ 40 000 enhancement of the sensitivity to New Physics versus the electron AMM.

Lepton Anomalies

22 / Lm

exp -12ea 1 159 652 180.73(0.28) · 10 [0.24 ppb] Harvar d 2008

-1 137.035 999 084(51) [0.37 ppb]

Electron AMM

QED is at the level of the best theory ever built to describe nature

To measurable level ae arises entirely from virtual electrons and photons

The theoretical error is dominated by the uncertainty in the input value of the QED coupling α ≡ e2/(4π)

Das ist fantastisch!

5

21

SM (QED) (hadron) (weak),

(QED) ...n

nn

a a a ae e e e

a Ce

Tau anomaly•Tau due to its highest mass is the best for searching for New Physics,•But Tau is short living particle, so the precession method is not perspective•The best existing limits -0.052<a

Exp<0.013are obtained at OPAL, L3 and DELPHI (LEP, CERN) from the high energy process e+e- e+e- •While the SM estimate is a

SM=1.17721(5) 10-3

SM Contributions to Muon AMM from BNL

BNL 10(6.3)11 659 208.9 10 0.54 ppm a

From BNL E821 g-2 experiment (1999-2006)

From Standard Model A. Hoecker Tau2010 Update

SMQED EW Strong ???a a a a

SM 1011 659 180 (4.9).2 10 a

exp SM 1028.7(8.0) 10 3.6 ! a a a

New Prop. E989 at Fermilab 0.14 ppm

2

2

ga

QED 10a = 11 658 471. (0.015809 1) 0

plusEW 10(0.2)a = 15.4 10

plus

the Hadronic Contribution estimated asStrong 10a = 693.0 1(4.9) ( 1% accur0 ac !) y

The main question how to get such accuracy from theory.

Kinoshita&Nio 2004, 2006

Czarnetski&Marciano&Vainshtein 2003

M. Davier, A. Hoecker, B. Malaescu, Z. Zhang 2010; F. Jegerlehner, Robert Szafron 2011

h

he

HVP 10692.3 4.2 10a LbL 1010.5 2.6 10a

LbL to g-2Strong contributions to Muon AMMM

Hadronic Vacuum polarization

(Davier, Hoecker, Zhang)

Hadronic Light-by-Light Scattering

(Dubnicka, Bartos, KuraevAED, Radzhabov, Zhevlakov)

2

2had

2

4

( )

( )3

m

a R sK

ss

ds

(0)

hadrons

W hadrons

e+

e –

CVC: I =1 & V W: I =1 & V,A : I =0,1 & V

Structure of hadronic LbL contribution

2 2 21 2; ,MF p q q

LbL 1010.5 2.6 10a

QL,LbL 102.1 0.3 10a

PS,LbL 109.5 1.3 10a

PV,LbL 101.5 1.0 10a

S,LbL 100.7 0.7 10a

L,LbL 101.9 1.9 10a

Phenomenological and QCDConstraints are used to reduceModel Dependence

Interpretations

SUSY mSUSY ≈100-500 GeVMulti-Higgs ModelsExtra Dimensions<2TeVDark Photons 10-150MeV, ∼ α’=10-8

Light Higgs <10MeV?

Davier etal 2010

g-2 is the most important constraint (for SUSY), even more important than dark matter

Since the pion has spin zero, its longitudinal spin structure interms of quark and gluon degrees of freedom is trivial.

An instructive quantity describing the transverse spin structure of hadrons is the probability density (x,ktr,str)

In terms of moments one has the Generalized Form Factors (GFFs)

2

20

11

1

20

, , ,

1

2

nn

i ij j

b nn T

b s dxx x b s

s

mb

bA b B

2 2 2 20 00, , 0,n n Tn TnH b A b E b B b Connection with Generalized

Parton Distributions (GPDs)

The Generalized Form Factors of the Pion

Introduce auxiliary vectors 2 2, 0, 1a a b b

11

0,

11 1

,

2

,

0,

' 2 ,

' 2 , ' 2 ,

1where , ,

2, '1

' , ' , .2 , '

nn n k

keven

nnu

Tnk

n k

keven

unk

B t

p ui a iDa u p aP

p ui a b iDa u p a

A t

P ap bp

D igA

a p pP p p t p p

a p p

��

��

������

,10u FA t t

, ,1n

Tnk Tnuk

dB t B t

Matrix Elements in the Chiral Models

The Quark Propagator

1 2

2 2 2

,

where is dynamical quark massq

S k k m k

m k M f k

And the Quark-Pion vertex

225

2 2

2 2

2 2

instanton mod

, ,

el' where , '

1chiral model'

2

a aip F k k p

f

m k m kF k k

m k m k

In the local (NJL) model one has

2,

2 2, '

q

q

m k M

F k k M

Transversity pion form factor (momentum space)

Transversity pion form factor (impact parameter space)

Forward: distribution function

Transverse size distribution function

Normalized density dT2(x)=bT

2(x)/f(x)

=log(1/x)One can interpret it as an evolution of the probability density for a stochastic motion of a particle in the transverse plane

QCD evolutionGFFs BT evolve multiplicatively

0/2

, ,0 0 0

0

1 2

02 20

; ;

8with the anomalous dimensions , 8

34

and , where 226 MeV, 9.log /

Tk

u uk k

T T

QCD

QCD

B t B t

For Local Model one has for the Normalizations

,10 0 2 2

8/27,20,10 0

0; ,4

0; 1

0; 3

u CT q

uTu

T

NB t M

f

B t

B t

0

Lattice QCD (Brommel etal., Phys Rev Lett., 2008)

Chiral Model (Broniowski, A.D.,

2 GeV

Arrio

,

320 MeV, la, 2010)

The results: Lattice vs Local Model

Brommel etal., Phys Rev Lett., 2008

The results: Lattice vs Nonlocal Models

The transversity form factors in the chiral model (solid line) and in the instanton-motivated model (dashed line) for Mq = 300 MeV.

Rare Pion Decay 0→e+e-- from KTeV

From KTeV E799-II EXPERIMENT at Fermilab experiment (1997-2007)

99-00’ set,

The result is based on observation of 794 candidate 0 e+e- events usingKL 30 as a source of tagged 0s.

PRD (2007)

One of the simplest process for THEORY

8KTeV 1025.029.048.7

eeB

CLEO+QCD

CLEO

3 diff

What is next? It would be very desirable if Others will confirm KTeV resultAlso, pair decay is very perspective

AED, M. Ivanov PRD (2007)

The anomalous 511 keV -ray signal from Galactic Center observed by INTEGRAL/SPI (2003) is naturally explained

* 10 MeVU

M

Enhancement in Rare Pion Decays from a Model of MeV Dark Matter (Boehm&Fayet)was considered by Kahn, Schmitt and Tait (PRD 2008)

excluded

allowed

Conclusions

High statistics Low-Energy experiments are

important independent way to search for effects of New Physics,

They complement to possible signals from High-Energy experiments (LHC,…) allowing to get combined restrictions on the parameters of hypothetical interactions

They sensitive to New particles with low masses

New experiments are urgent

Theory has to be in good shape

Photon-pion transition form factorin nonperturbative QCD approach

The theory of hard exclusive processes was formulated within the factorization approach to perturbative quantum chromodynamics (pQCD) in 1977-1981

The photon-pion transition γ*γ*→π⁰ is of special interesta)it is the simplest process for theory;b)related to the axial anomaly when both photons are real;c)At large photon virtualities it was studied in [6,7]

Photon-pion transition form factor:In the factorization approach

* ²²

2( ,0) f =92.4 , wh MeV

3ere pQCD f

F Q JQ

1

0

J= dxx

x

*

, 1( ,0) 2 , 6fo² 1r

²pQCD As AsF Q f x x x

Q

* * ²21

( ²²

, )3

pQCD fF Q Q

Q

Symmetric kinematics

Asymmetric kinematics(measured by BABAR)

Factorization f J* 1/Q2

and it settle in at ~ 1 GeV2

BABAR disaster 2009 (and CELLO-1991, CLEO-1997) data

Passive modeActive mode

A. Data higher than BL Asymptotic limit,

B. They GROW!

* *

*'

BABAR puzzles

'

'

/

/

1.10 0

1.6 to

.17,

2.3

BAB

THEOR

AR

R

R

Nonperturbative Nonlocal QCD Approach to Form FactorAnd its asymptotic properties AED JETP Lett. (2003)

2

22 22

2

1 1

5 5

Quark Propagator

Quark-Photon Vertex

where

guarantes WT

, ' , , '

'

Quark-Pion

'

ve

1 1

I :

r

, '

texk

k m kk kS k k

k kD k

k k k k k

k k q q S k S k

F k kf f

��������������

��������������

2

2'

0k

22 2 is related to nonlocal quark condensate and th us

aC k

qm k m k M e

Main properties

2'

2 2

0, ', ' 1

, ' 2

Instanton Model

Chiral Model

kf k f k

F k kg k f k f k

The vertex F is equivalent of the light-cone pion WF

representationF decays as 1/k2 or faster

The main property for asymptotic analysis

N.N.Bogolyubov, D.V. ShirkovO.I. Zavialov

2

1(0;0,0)

4A

f

Photon-pion transition form factor: Symmetric Kinematics

Take chiral limit p2=0 and symmetric kinematics q12= q2

2= q2

Leading Asymptotics as q2∞ corresponds to γ0 thus

, dm0, d1+…

Factorization – Yes!

Photon-pion transition form factor: Symmetric Kinematics

representation – URA!Factorization – URA!!Brodsky-Lepage - URA!!!

Distribution AmplitudePion DA is obtained if we perform substitution

then

AED JETP Lett. (2003)

2

2

2

2300 MeV, =0.64

135 MeV, =

exp

0.02

2

GeV ,

G

,

eVq

q

q

m k

M

M

M k

Distribution AmplitudeFor Instanton model

For chiral model

At x=0 one gets

Distribution Amplitude 2 2exp 2qm k M k

Take chiral limit p2=0 and symmetric kinematics q12= q2 , q2

2=0

Leading Asymptotics as q2∞ corresponds to γ0 or 0 thus, F

m0, F1+…, but keep Exp small terms!Gm,0

0, G0,mg

m , GF

Factorization – Not always!

Photon-pion transition form factor: Asymmetric Kinematics

Small

Small

Factorization – Not at all!

Small

Small

Small

Small

Photon-pion transition form factor: Asymmetric Kinematics

Photon-pion transition form factor: Asymmetric Kinematics in Instanton Model

A) Confining Propagator

2

2

22

1 exp1

kL

kD k

B) Chiral Propagator 2 2 2 2

1 1

D k k m k

*

212

2 0

2

22 2

2,0 1 exp

3

1~ ln

Q

L Q

f dx xQF Q

Q x L

*

2

2

12

2 20 22

2,0

3

1~ ln

0

dxF Q f

xQ M xQ

Q

M Q

If 1x

Photon-pion transition form factor: Asymmetric Kinematics in Chiral Model

Thus one finds three possible Asymptotic regimes:1)1/Q2 Pion DA strongly suppressed at endpoints2)“lnQ2/Q2” Pion DA vanishes at endpoints, but almost flat otherwise3)lnQ2/Q2 Pion DA does not vanishes at endpoints

Two last regimes may be responsible for BABAR data!BABAR puzzle is cracked!!

Exact asymptotic expressionIt is nonfactorizable, no representation in terms of pion DA

2

2

2

135 MeV,

exp 2

=0.02 GeV

q

qM

m k M k

2 2

1, ' ' ,

2

Chiral Model

F k k m k m k

2 2

2 2

' , , ' 1

' , 2

Instanton Model

Chiral Model

m k m kF k k

m k m k

2 2exp 2qm k M k

Mq = 300 MeV

Mq = 300 MeV

Mq = 135 MeV

CONCLUSIONS1. BABAR measured photon-pion form factor at large Q2 in wide kinematical regionand found that the data for Q2F(Q2) exceed the asymptotic Brodsky-Lepage constant limitand, moreover, continue to growth – BABAR puzzle

2. We show that depending on the properties of the quark-pion vertex there are two possible shapes of the pion distribution amplitude:

vanishing or not vanishing at the endpoints

3. These different cases provide different possibilities for the asymptotic behavior of the form factor

Q2F(Q2) ~ constOr

Q2F(Q2) ~ln(Q2)

4. BABAR data, if will be confirmed, point out on specific properties of quark dynamics in the pion and of the underlying QCD vacuum

AntiBABAR Shock therapy 2009

135 MeV, AED hep-ph/0905.45 Ametller etal NPB 7 (7; 1 3) 98qM

A)

B)

2 0.96 GeV, AVR hep-ph/0906.032 Musatov and AVR, PRD (13; 997)

C)

Normalization by anomaly A“Factorization” B,COPE BValue of Parameters Mq, and M and their meaning ?fis external parameter, it is not definedAll are based on flat (local) pion DA (x)=1. How to justify that?? No QCD Evolution

0.776 GeVM

2 2 2 2~ ln / /qQ M Q

2 2~ ln / 2 /Q Q

2 2 2~ ln / / ,Q M Q

AntiBABAR Shock therapy 2009

BABAR hep-ex/0905

AV Rad

AED hep-ph/0905

.4778

yushkin hep-ph/0906.

.4577

0323

Nonfactorizable

“Factorizable”

Chiral Model Mq =125 MeV

Mq =135 MeV Mq =115 MeV