[PPT] - Non-p erturbative lo w energy amplitudes in non-lo al hiral PowerPoint Presentation

SLIDE 1 Non-p erturbative lo w energy amplitudes in non-lo al hiral qua rk mo del Piotr K

tk
Jagiellonian

Universit y , Krak

w

49 Cra o w S ho

l
f

Theo reti al Physi s, Zak

pane

2009

SLIDE 2 OUTLINE

Non-p

erturbative input to amplitudes fo r ex lusive p ro esses is analyzed within full non-lo al hiral qua rk mo del

T

w

examples:
Photon

Distribution Amplitudes

Pion-photon

T ransition Distribution Amplitudes

Sp

e ial attention is paid to the question

f

inheriting QCD p rop erties b y

bje ts

al ulated in the ee tive mo del

SLIDE 3 T ABLE OF CONTENTS 1 Intro du tion 2 Chiral Qua rk Mo del and its p roblems 3 Photon Distribution Amplitudes 4 T ransition Distribution Amplitudes 5 Con lusions

SLIDE 4 INTRODUCTION F a to rization

f

the amplitudes fo r ex lusive p ro esses in the p resen e

f

the ha rd s ale 1, 2, 3

M = ( HARD) ⊗ (

SOFT)

⇒

HARD pa rt an b e al ulated in p erturbation theo ry

⇒

SOFT pa rt is a subje t to the non-p erturbative treatment Examples: Distribution Amplitudes (D A)

Generalized

P a rton Distributions (GPD)

T

ransition Distribution Amplitudes (TD A) Ho w to a ess the SOFT pa rt ?

⇒

extra tion from the exp eriment

⇒

latti e al ulations

⇒

lo w energy ee tive mo dels 1 Efremov, Radyushkin; 2 Bro dsky , Lepage; 3 Collins, F rankfurt, Strikman

SLIDE 5 INTRODUCTION ( ontinued...) SOFT pa rt pa rametrizes matrix elements

f

ertain non-lo al qua rk (gluon)

p

erato rs

n

the light- one, e.g.

˙

H′ ˛

˛ ¯ ψ (y) Oψ (x) ˛ ˛

H

¸ ⇒

they should p

ssess

p rop erties

riginating

from QCD symmetries (e.g. Lo rentz inva rian e, W a rd identities, axial anomaly)

= ⇒

it is not

bvious

that ee tive mo dels do inherit all QCD symmetries

= ⇒

p

ssible

p roblems with

rre t

p rop erties

f

SOFT pa rt in the ee tive mo dels

SLIDE 6 CHIRAL QUARK MODEL (χ QM) F

r

simpli it y w e

nsider

pions

nly

. In

rder

to

btain
nsidered

matrix elements w e need the mo del

f

qua rk-pion intera tions. A t lo w energy s ales sp

ntaneous

hiral symmetry b reaking (χSB) pla ys very imp

rtant

role

⇒

the mo del should in o rp

rate χSB

⇒

there app ea r

nstituent

qua rk mass M ∼ 350 Me V The simplest mo del is the semi-b

sonized

Nambu-Jona-Lasinio mo del 1 ( hiral limit) S lo =

ˆ

d 4 x ¯

ψ (x) (i

D − MUγ 5) ψ (x) where Uγ 5 (x) = exp

n

i Fπ τ aπ a (x) γ 5

,

with Fπ = 93 Me V .

In
rder

to get nite qua rk lo

ps

w e need to imp

se

some kind

f

regula rization (but w e annot remove the uto pa rameter at the end)

Ho

w ever to get

rre t

results fo r anomalous p ro esses w e have to remove regula rization

P

a rti ula r regula rization s heme la ks motivation in terms

f

QCD... 1 see e.g. S.P . Klevansky

SLIDE 7 NON-LOCAL χ QM The most natural w a y

f

regula rizing qua rk lo

ps

= ⇒

momentum dep endent

nstituent

qua rk mass M ≡ M ( k) S Int =

ˆ

d 4 k d 4 l

(2π)

8

¯ ψ(k) p

M (k) Uγ 5(k − l)

p

M(l)ψ(l) where usually

ne

denes M (k) = M F 2 (k) , and F (0) = 1, F (k → ∞) → . This a tion w as derived from QCD instanton va uum theo ry , with Eu lidean analyti al exp ression fo r M (k) 1 . Problem: momentum dep endent mass

⇒

naive ve to r urrent ¯

ψγµψ

is not

nserved

⇒

lo al vertex γµ has to b e repla ed b y the non-lo al

ne Γµ

The p re ise fo rm

f

the vertex is un onstrained and has to b e mo deled 2, 3, 4 . One

f

the simplest solution is

Γµ (k,

p) = γµ − kµ + pµ k 2 − p 2 (M (k) − M (p)) The

n rete

mo del is sp e ied b y giving M (k) and the fo rm

f

the verti es. 1 Diak

nov,

P etrov; 2 Bo wler, Birse; 3 B. Holdom, R. Lewis; 4 A. Bzdak, M. Praszalo wi z

SLIDE 8 ONE LOOP CALCULA TIONS As the mass dep enden e

n

momentum w e tak e F (k) =

„ −Λ

2 k 2 − Λ 2 + iǫ

«

n

al ulations

in Eu lidean as w ell as Mink

wski

spa e

analyti al

solutions

1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

Finstk n 3 n 1

k GeV Fk

The ansatz ab

ve

leads to set

f

p

les

in the

mplex

plane. Using some tri ks w e an exp ress the lo

p

integral with N p ropagato rs as 4n+1

X

i 1,...,i N f i 1 . . . f i N η M 1 i 1

. . . η

M N i N

ˆ

d Dκ

(2π)

D g (κ, η i 1, . . . , η N) where g is a fun tion

ntaining
nly

N p

les, η

i a re solutions

f

z 4n+1 + z 4n − (M/Λ) 2 = and f i a re some numb ers

mp
sed

from η i . Higher t wist light

ne

amplitudes

⇒

delta t yp e singula rities in the b

unda

ries

f

physi al supp

rt

Praszalo wi z, Rost w

ro

wski, Bzdak, P .K.

SLIDE 9 APPLICA TION I Simplest SOFT

bje ts =

⇒

Distribution Amplitudes (D A) Example: radiative ve to r meson de a y V → Sγ and Photon D A. Relevant

rdinates

a re dened b y t w

light-lik

e ve to rs n = ( 1, 0, 0, − 1), ˜ n = ( 1, 0, 0, 1). Then w e an de omp

se

any ve to r v as v µ = v + 2 ˜ nµ + v − 2 nµ + v µ T PSfrag repla ements V (q) S(q + P)

γ(P)

General denition:

˙ ˛ ˛ψ (λ n) Oψ (−λ n) ˛ ˛ γ (P) ¸ ∼

FO

`

P 2´ ˆ 1 du e i( 2u−1)λ P+ φO

“

u, P 2” where O = {σµν, γµ, γµγ 5} and FO a re relevant fo rm fa to rs. P . Ball, Bro wn; Arriola, Bronio wski, Do rokhov;

SLIDE 10 As an example

nsider

ve to r photon D A φ V

`

u, P 2´

⇒

urrent

nservation

in QCD: F V (0) = 0. Nχ QM al ulations:

Using

full non-lo al photon-qua rk vertex and leaving pure QCD ve to r urrent

p

erato r w e re over F V ( 0) = 0.

Diagram

in the right has t w

ntributions:

hadroni and innite p erturbative pa rt

rresp
nding

to ele tromagneti ingredient

f

the photon.

Amplitudes

up to t wist-4 in all hannels have b een al ulated. PSfrag repla ements

¯ ψγµψ γ(P)

nonlo al urrent

0.2 0.4 0.6 0.8 1.0 P2GeV2 0.2 0.1 0.1 0.2 0.3 0.4

FVP2

nonlocal vertex local vertex

M = 350 Me V, n = 1

SLIDE 11 APPLICA TION I I Mo re demanding

bje ts

to study in Chiral Qua rk Mo dels: T ransition Distribution Amplitudes app ea ring fo r example in π+π− → γ∗γ in the fo rw a rd region 1 . Kinemati s:

high

virtualit y Q 2

f

the upp er photon

lo

w momentum transfer to the lo w er blob

∆

2 = ( P 2 − P 1) 2 = t ≪ Q 2 PSfrag repla ements TD A

γ∗(q

2)

π−(q

1)

π+(P

1)

γ(P

2) u

¯

d e− e− Example: V e to r TD A (VTD A)

ˆ

dλ 2π e iλ Xp+ < γ(P 2)| d(−λ 2 n)γµ u(λ 2 n)|π+(P 1) >∼ εµναβε∗

ν

P 1 α P 2 β V (X, ξ, t) where ξ = − 2∆+/ p+ with p = 1 2 (P 1 + P 2) is so alled sk ew edness. 1 Pire, Szymano wski

SLIDE 12 Prop erties

f

VTD A

riginating

from QCD:

p
lynomialit

y

´

dX X n V (X, ξ, t) = a nξ n + a n−1ξ n−1 + . . . + a

no

rmalization is xed b y axial anomaly

´

dX V (X, ξ, t = 0) = 1/ 2π 2 Nχ QM al ulations:

P
lynomialit

y is satised.

W

e

btain
rre t

no rmalization

nly

when b

th

ve to r urrents a re non-lo al. VTD A is related to pion-photon transition fo rm fa to r

ˆ

dX V (X, ξ, t) ∼ Fπγ (t)

ntrolling γ∗γ → π

rea tion. New BaBa r data a re available (29 Ma y) whi h ast some new light

n

pion Distribution Ampli- tudes...

⇒

this is urrently under investigation... PSfrag repla ements

¯ ψγµψ π+(P

1)

γ(P

2) nonlo al urrents

1.0 0.5 0.5 1.0 X 0.05 0.05 0.10

VX local model full model

M = 350 Me V, n = 1, ξ =

0. 5

SLIDE 13 SUMMARY

Non-lo

al hiral qua rk mo del allo ws fo r analyzing lo w energy matrix elements

Ho

w ever, b efo re using in real p ro esses they have to b e evolved (s ale

f

ee tive mo dels is lo w)

not

dis ussed

In
rder

to mak e al ulations

nsistent

w e have to use mo died urrents

The

fo rm

f

the full urrents is not restri ted and has to b e mo delled

Ho

w ever, in general it is not lea r y et whi h urrents w e should mo dify and when

Case
f

full axial urrent is mo re di ult

not

dis ussed

First

analysis

f

pion-photon T ransition Distribution Amplitudes in non-lo al mo del

SLIDE 14 BA CKUP

SLIDE 15 PHOTON D A DEFINITIONS

tenso

r hannel

D ˛ ˛ ˛ψ (λ n) σαβψ (−λ n) ˛ ˛ ˛ γ (P, ε) E =

i 2 e

˙ ¯ ψψ ¸

F T

`

P 2´

( “ εα

T˜ nβ − εβ T˜ nα” P+ 2 χ m

ˆ

1 du e iξλ P+ φ T

`

u, P 2´

+

1 2P+

“ ˜

nα nβ − ˜ nβ nα”

ε+ ˆ

1 du e iξλ P+ ψ T

`

u, P 2´

+

1 P+

“ εα

T nβ − εβ T nα” ˆ 1 du e iξλ P+ h T

`

u, P 2´

) ,

ve to

r hannel

˙ ˛ ˛ψ (λ n) γµψ (−λ n) ˛ ˛ γ (P, ε) ¸ =

ie f 3γ F V

`

P 2´

(

1 2 ˜ nµε+

ˆ

1 du e iξλ P+ φ V

`

u, P 2´

+ εµ

T

ˆ

1 du e iξλ P+ ψ V

`

u, P 2´

−

1 2 P 2

(P+)

2 nµε+

ˆ

1 du e iξλ P+ h V

`

u, P 2´

) ,

SLIDE 16

axial

hannel

˙ ˛ ˛ψ (λ n) γµγ

5ψ (−λ n)

˛ ˛ γ (P, ε) ¸ =

i 1 2 e f 3γ F A

`

P 2´

ǫµναβεν

T ˜ nα nβ P+λ

ˆ

1 du e iξλ P+φ A

`

u, P 2´

, ξ =

2u − 1,

¯ ψψ

is

qua rk

ndensate, χ

m is the magneti sus eptibilit y

f

the qua rk

ndensate.

SLIDE 17 AXIAL TD A

ˆ

dλ 2π e iλ Xp+ fi

γ (P

2, ε)

˛ ˛ ˛ ˛

d

„ −λ

2 n

« γµγ

5 u

„λ

2 n

«˛ ˛ ˛ ˛ π+ (P

1)

fl =

ie 2

√

2 Fπ p+ Pµ 2 (q · ε∗) A (X, ξ, t) + . . .

1 0.5 0.5 1 X 0.02 0.02 0.04 0.06 0.08 AX local model n 5 n 1 1 0.5 0.5 1 X 0.01 0.02 0.03 0.04 AX local model n 5 n 1

SLIDE 18 LEADING TWIST PHOTON D A

local model local vertex P 2 = 0 φT (u) pion DA u 1 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8