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Asymptotic behavior of form factor of the composite system at large momentum transfer.
Gamzova E.S., Krutov A.F., Troitsky V.E.
Samara State University D.V. Skobeltsyn Institute of Nuclear Physics

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XIX International Workshop "QFTHEP'2010" Golitsyno, Moscow, Russia

Gamzova E.S., Krutov A.F., Troitsky V.E., Tsirova N.A. Relativistic Constituent Quark Model and Exp eriments at JLab. Physics of Atomic Nuclei. 2010. 73. p. 1063­1068.

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Motivations

· The relativistic description of comp osite systems, on example

the pion, as quark-antiquark system;

· Mo dernization of JLab will allow for measurement of the pion

form factor at the large momentum transfer and check the predictions of pQCD and CQM; region of future experiments.

· Comparison of predictions CQM and pQCD for the pion in the


Instant form of relativistic quantum mechanics
The electromagnetic current matrix element for pion: p | jµ |p


= (p + p )µ F (Q 2 )

F (Q 2 ) ­ the electromagnetic form factor of the pion, p , p ­ the four-momentum of the pion. In RQM the Hilbert space of composite particle states is: Hq q Hq Hq ¯ ¯

As a basis in Hqq : one can cho ose the following set of vectors: ¯ | p1 , m1 ; p2 , m
2

= | p1 , m

1

| p2 , m

2

,

p , m | p , m = 2p0 (p - p ) mm , Here p1 , p2 are particle momenta, m1 , m
2

spin projections.


Instant form of relativistic quantum mechanics

· The natural basis is one with separated center-of-mass motion:

s , J , L, S , mJ , 2 the invariant mass of with Pµ = (p1 + p2 )µ , Pµ = s , s two-particle system , L the angular momentum in the center-of-mass frame, S the total spin, J the total angular momentum, mJ the projection of the total angular momentum. · Wave function of the comp osite system in RQM: P , s , J , l , S , mJ | p = N (P - pc )JJ mJ mJ ll SS JS (k ) l s = 4(k 2 + M 2 ) , M is the quark mass , N , NCG are factors due to normalization.

| P,




Electromagnetic structure of the pion within the instant form of RQM

· The pion electromagnetic form factor:

F (Q 2 ) =

d s d s (k )g0 (s , Q 2 , s )(k )

(k ) ­ the pion wave function within relativistic quantum mechanics (RQM) g0 (s , Q 2 , s ) ­ free two-particle form factor.


Electromagnetic structure of the pion within the instant form of RQM

u d g0 (s , Q 2 , s ) = a(s , Q 2 , s )(GE (Q 2 ) + GE (Q 2 )) + b(s , Q 2 , s ) u d (GM (Q 2 ) + GM (Q 2 )) q q where GE (Q 2 ) and GM (Q 2 ) ­ the constituent quark electric and magnetic form factors 2 q GE (Q 2 ) = eq fq (Q 2 ), q GM (Q 2 ) = (eq + q )fq (Q 2 ) ¯

¯

eq the quark charge, q magnetic moment
2

the constituent quark anomalous

A. F. Krutov, V. E. Troitsky, Relativistic instant­form approach to the structure of two­b ody comp osite systems // Phys. Rev. C 65, 045501 (2002)


Electromagnetic structure of the pion within the instant form of RQM

fq (Q 2 ) =

1 2 1 + ln(1 + rq Q 2 /6)

2 where rq ­ the ro ot-mean-square radius of the quark. Prediction pQCD quark counting rules: 3

F (Q 2 ) Q

-2

V. A. Matveev, R. M. Muradyan, and A. N. Tavkhelidze,Lett. Nuovo Cimento 7, 719 (1973); 15, 907 (1973); S. Brodsky and G. Farrar, Phys. Rev. Lett. 31, 1153 (1973).

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Asymptotic estimation of some multiple integrals
In the following we will consider integrals of the kind: F () =


f (, x )e

S (,x )

dx

where is a domain in Rn , x = (x1 ; ...; xn ), is a large positive parameter. Then at the following asymptotic expansion is valid:4


F () exp[S (, x )] hkm () ­ known function.
4

0

hkm ().
k =0 m =0

A. F. Krutov, V. E. Troitsky, and N. A. Tsirova, J. Phys. A 41, 255401 (2008) [nucl-th/0709.2312].


Asymptotic behavior of the pion form factor for Q 2

· Pointlike quarks:

d d u u GE (Q 2 ) + GE (Q 2 ) = 1, GM (Q 2 ) + GM (Q 2 ) = 1

¯

¯

F (Q 2 ) quarks:

25/2 M - e Q

QM 4b 2

1+

7b2 2MQ

.

· In case of the electromagnetic structure of the constituent

F (Q 2 ) â 1+
b2 2MQ

25/2 M - Qe

QM 4b 2 ¯

u d GE (Q 2 ) + GE (Q 2 ) â
¯ ¯

¯

u d u d 16 GM (Q 2 )+GM (Q 2 ) -9 GE (Q 2 )+GE (Q 2 ) u d GE (Q 2 )+GE (Q 2 )


Asymptotic behavior of the pion form factor for Q 2

In the limit case M /b 0
· Pointlike quarks:

14 2b F (Q ) Q2
2

2

· In case of the electromagnetic structure of the constituent

quarks:

d d u u â 16 GM (Q 2 ) + GM (Q 2 ) - 9 GE (Q 2 ) + GE (Q 2 )

F (Q 2 )
¯

23/2 b Q2

2

â

¯


Asymptotic behavior of the pion form factor and relevant present-day experiments

Figure: Results of asymptotic calculations for the pion form factor.


Conclusions

· The result obtained in limit M /b 0 in the constituent quark

mo del for the pion coincides with the F (Q 2 )Q 2 = const behavior predicted by pQCD. in the relativistic constituent quark mo del.

· The region of exp eriments at JLab is asymptotic for the pion