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Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Pion formfactor in time-like region in the framework of relativistic composite quark model.
A. F. Krutov1 , M. A. Nefedov1 , V. E. Troitsky2

QFTHEP 2010

1 2

Samara State University D. V. Skob eltsyn institute of nuclear physics, MSU


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Definition of a pion formfactor.
e
-

e

-



Formfactor is a scalar function, that parametrises a current matrix element.
- -

e

-



i |^µ (0)| j

j

(pj + pi )µ F (pi - pj )2

(1)

- e - - e
+

t = q 2 = -Q 2 = (pi - pj )2 < 0 - Space-like region
+

i j |^µ (0)|0 (pj - pi )µ F (pi + pj )2 j t = q 2 = -Q 2 = (pi + pj )2 > 4µ2 - Time-like region

(2)

e
- -



-

e + e - +


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Fundamental analytical properties of F .

Im t

Analiticity region. Normalization condition F (0) = 1 Physical value of formfactor
( Fphys ) ( ) = lim F ( + i ) +0

4µ

2

Ret

QCD asympthotics |t | F (t ) 1 t


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Pion formfactor experimental data in space-like region.


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Pion formfactor experimental data in time-like region.

Experiment gives us an absolute value of F .


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Expression for the F in constituent quark model.
1 4 (-t )(s + s - t ) (s - 4M )(s - 4M ) 3/2 (s , t , s ) (s )(s )
2 2

F (t ) =

dsds

ss

Region of integration: = {(s , s )|s [4M 2 , +); s [s1 (s , t ), s2 (s , t )]} ts 1 (s , t ) = s + t - ± (-t )(4M 2 - t )s (s - 4M 2 ) 2 2M 2M 2 (s , t , s ) = s 2 + t 2 + s 2 - 2(ss + st + s t )
ss

s

1,2

Wavefunction of constituent in instant form of RHD: (s ) = 4 s
s -4M 2
2

u(



s -4M 2

2

)

Where u (k ) - wavefunction depends of a three-momentum.


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Typically using wawefunctions.

Harmonic oscillator wavefunction. -k 2 2b 2

u (k ) = NHO exp
1.0

0.8

Wavefunction with p ower-b ehaviour (QCD-motivated). k2 b2
-n

0.6

u (k ) = NPL
0.4

1+

0.2

WF with linear confinement and Coulomb b ehaviour in a small scale.
0 1 2

2

1

u (r ) = NT exp -r

3/2

- r


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

New symmetrical variables.
x =s +s -t y = 4ss Formfactor in new variables: (-t ) F (t ) = 82
+ x
min

y2 (x ,t )

(x , y , t )
y1 (x ,t ) 4M 2 4M 2 -t

xy (t + x )2

3/2

dydx
3/2

(t )

- y 2 (x 2 - y 2 )

, y2 (x , t ) = x + t , xmin (t ) = (4M - t ) + 2M 4M 2 - t Function (x , y , t ) satisfies the relation: s - 4M 2 s - 4M 2 u = (s + s - t , 4ss , t ) u 2 2
2

where: y1 (x , t ) = x


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Statement about analitical properties of a model.

Theorem If special requirements on a -function satisfied, integral representation defines an analytical function in a plane t with a cut from 4M 2 to infinity.


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Proof of analiticity.
Absolute convergence Uniform convergence Analiticity. Integration paths in and planes: x = {[xmin (t ) + ] | [0, +)} y = {[(1 - )y1 (xmin (t ) + , t ) + y2 (xmin (t ) + , t )] | [0, 1]} Absolute value of under integral expression: | (x , y , t )| xy
3/2

(t + x )2 - y 2 (x 2 - y 2 )3/2 = | (, , t )||K (, , t )|

(y2 (x , t ) - y1 (x , t ))

=
x =x (),y =y (, )


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Proof of analiticity.
|K (, , t )| where µ() = max
[0,1] [0,1]

For absolute convergence is enough that:

{

|xmin (t ) + |µ() 1-

|(y2 +y1 )+ (y2 -y1 )||x 2 -(y1 + (y2 -y1 )2 )|

|y2 -y1 ||y1 (1- )+y2 |

3/2 3/2

}

max | (, , t )||xmin (t ) + |µ()d <
1

0

Asympthotycs: |xmin (t ) + |µ()
[0,1]

For AC is enough that 1

: max | (, , t )| o


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Proof of the existence of a cut.
Im We need to prove that: F ( - i 0) = (F ( + i 0)) Limits of integration:
xmin ( ±i 0) = (4M - ) 2Mi
2

xmin (t ) 2M

t - 4M 2

y2 ( x , t ) = x + t x

- 4M 2

A
4M - t
2

B Re

4M 2 y1 (x , ± i 0) = ±ix 4M 2 - 4M 2

y2 (x , ± i 0) = x + When we crossing the cut in a plane t, all picture of singularities and contours changing to is's complex conjugate picture.

C

y1 ( x , t )


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Results of numerical calculations.
8 n3 M 0.25 b 0.4 0

Calculations with power-behaviour WF: u (k ) = NPL 1 + k b
2 2 -n

6

4

2

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

(x , y , t ) =

2 NPL (4b2 )2n y2 4

+ (x + t ) + 2

n

12

n3 M 0.25 b 0.25 0

10

8

6

4

2

Where = 4(b2 - M 2 ). Hypotesis: Behaviour of F in time-like region depends of analytical properties of wavefunction u (k ) in complex plane k.
0.2 0.4 0.6 0.8 1.0

0.4

0.2

0.0


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Criteria of existence of a resonance.
New variable: t = 4M 2 - z Im z zs zs
zs 2

xmin (t ) x

y2 -singularity y1

zs
xmin (t ) -singularity x y2

Re z -zs
y1


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Criteria of existence of a resonance.
y1 (xmin (4M -zs ), 4M -zs ) = y2 (xmin (4M -zs ), 4M -zs ) = 2M (zs +2M ) Criteria: (xmin (4M - zs ), 2M (zs + 2M )) = (xmin (4M - zs ), 2M (zs + 2M )) =
2 2 2 2 2 2 2 2 2 2 2 2

xmin (t ) -singularity x

y2

1 where K = 4 (Mzs - 2M 2 )
y1

Criteria in terms of u(k): u( K ) = u( K ) =


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Minimal rational model
Conditions for u(k): u(k) will be even. Cryteria for poles in z plane. k : u (k ) Minimal mo del:
uk

N0 2 (1+k 2 /b0 )n

14 12 N0 u0 k 1
k
2 2 b0

n

u (k ) = N

k +a (k - K )(k 2 - K )
2

2

2

n

10 8 6 4 2

Normalization condition:
+ -
1.0 0.5

0.0

0.5

1.0

k 2 u 2 (k )dk = 1


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Definition of parameters.

Equation defines parameter a:
+ -

k 2 + a2 (k 2 - K )(k 2 - K )

n 3 k 2 dk = b0 -4n B (2n - 3/2, 3/2) 1/2

3 1 2 Where K = 4 (iMzs - 2b0 ), and N = (b0 -4n B (2n - 3/2, 3/2))- Parameters of a mo del: M , b0 , zs , n

.

Spacelike timelike


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Problem of choosing of contour.
3

2

1 singularity y 3 2 1
1

1 y 1
2

2

3

2

3


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Numerical results n=2.

20 M z1 0.2; 0.5 4 M ^ 2 ; b0 0.27; 15 n1 2;

10

5

0.4

0.2

0.0

0.2

0.4

0.6

0.8


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Numerical results n=3/2.

3 M z1 b0 n1 0.145; 8 0.57 4 M ^ 2 ; 0.151; 3 2; 6 2

t

0.57

4 1 2

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0


Pion formfactor in time-like region in the framework of relativistic comp osite quark mo del.

Thank you for your attention!