

 01 Aug 1995

 fl

\Lambda p ! \Delta FORM FACTORS IN QCD

V.M. Belyaev

\Lambda

Continuous Electron Beam Accelerator Facility

Newport News, VA 23606, USA

E-mail: belyaev@cebaf.gov

and ITEP, 117259, Moscow, Russia

and A.V. Radyushkin Physics Department, Old Dominion University

Norfolk, VA 23529, USA

and Continuous Electron Beam Accelerator Facility

Newport News, VA 23606, USA

ABSTRACT We use local quark-hadron duality to estimate the purely nonperturbative soft contribution to the fl

\Lambda p ! \Delta form factors. Our results are in agreement with

existing experimental data. We predict that the ratio G

\Lambda

E(Q

2)=G\Lambda

M(Q

2) is small

for all accessible Q2, in contrast to the perturbative QCD expectations that G

\Lambda

E(Q

2) ! \Gamma G\Lambda

M(Q

2).

1.

There are two competing explanations of the experimentally observed power-law behaviour of hadronic form factors: hard scattering 1 and the Feynman mechanism 2. At sufficiently large momentum transfer, the hard scattering mechanism dominates3. However, there is increasing evidence that, for experimentally accessible momentum transfers, the form factors are still dominated by the soft contribution corresponding to the Feynman mechanism 4.

In this talk, we consider the soft mechanism contribution to the fl

\Lambda p ! \Delta form

factors. The relevant hard scattering contribution was originally considered in ref. 5.

Here, we use the local quark-hadron duality to estimate the soft contribution for the G

\Lambda

E(Q2) and G

\Lambda M (Q2) form factors of the fl

\Lambda p ! \Delta transition.

The starting object for a QCD sum rule analysis 6 of the fl

\Lambda p ! \Delta transition is

the 3-point correlator:

T_* (p; q) = Z h0jT fj_(x)J*(y)_j(0)gj0ieipx

\Gamma iqyd4xd4y (1)

\Lambda Contribution to the International Symposium and Workshop on Particle Theory and Phenomenology at Iowa State University, Ames, May 22-24

of the electromagnetic current

J* = eu _ufl* u + ed _dfl* d (2) and two Ioffe currents 7

j = "abc iuaCflaeubj flaefl5dc ; j_ = "abc i2 iuaCfl_dbj uc + iuaCfl_ubj dcj : (3) On the hadronic level, the contribution of fl

\Lambda p ! \Delta transition to (1) is:

T fl

\Lambda p!\Delta

_* = l

N l\Delta

(2ss)4

X_ff(p) p2 \Gamma M 2 \Gamma ff* (p; q)fl5

^p \Gamma ^q + m (p \Gamma q)2 \Gamma m2 ; (4)

where \Gamma ff* (p; q)fl5 is the fl

\Lambda p ! \Delta vertex function

\Gamma ff* (p; q) = G1(q2) (qfffl* \Gamma gff* ^q) + G2(q2) (qffP* \Gamma gff* (qP ))

+G3(q2) iqffq* \Gamma gff*q2j (5)

(P j p \Gamma q=2 ), lN and l\Delta are the residues of nucleon and \Delta of the quark currents (3), X_ff(p) the projector onto the isobar state

X_ff(p) = `g_ff \Gamma 13 fl_flff + 13M (p_flff \Gamma pfffl_) \Gamma 23M 2 p_pff' (^p + M ): (6) The form factors G1; G2; G3 are related to a more convenient set G

\Lambda

E; G

\Lambda M ; G

\Lambda C by

G

\Lambda

M (Q2) = m3(M + m) ((3M + m)(M + m) + Q2) G

1(Q2)

M

+(M 2 \Gamma m2)G2(Q2) \Gamma 2Q2G3(Q2)j ; (7)

G

\Lambda

E(Q2) = m3(M + m) (M 2 \Gamma m2 \Gamma Q2) G

1(Q2)

M

+(M 2 \Gamma m2)G2(Q2) \Gamma 2Q2G3(Q2)j ; (8)

G

\Lambda

C (Q2) = 2m3(M + m) `2M G1(Q2) + 12 (3M 2 + m2 + Q2)G2(Q2)

+(M 2 \Gamma m2 \Gamma Q2)G3(Q2)j ; (9)

2.

To incorporate the local quark-hadron duality, we write down the dispersion relation for each of the invariant amplitudes:

Ti(p21; p22; Q2) = 1ss2 Z

1

0 ds1 Z

1 0 ds2

aei(s1; s2; Q2) (s1 \Gamma p21)(s2 \Gamma p22) + "subtractions" ; (10)

where p21 = (p \Gamma q)2, p22 = p2. The perturbative contributions to the amplitudes Ti(p21; p22; Q2) can also be written in the form of eq.(10). The local quark-hadron duality assumes, that the two spectral densities are in fact dual to each other:Z

s0 0 ds1 Z

S0 0 ae

pert: i (s1; s2; Q2) ds2 = Z

s0

0 ds1 Z

S0 0 aei(s1; s2; Q

2) ds2 ; (11)

For the tensor structures of T_* , it is convenient to use the basis in which fl_ is placed at the leftmost position. Then, the invariant amplitudes corresponding to the structures with q_ and g_* are free from the contributions due to the spin-1/2 isospin-3/2 states.

The number of independent amplitudes can be diminished by taking some explicit projection of the original amplitude T_* (p; q). In particular, the invariant amplitude corresponding to the structure q_[^q; ^p] for the amplitude T_* p* is proportional to the quadrupole form factor G

\Lambda

C (Q2):Another possibility is to take the trace of

T fl

\Lambda p!\Delta

_* . The result is proportional tothe magnetic form factor

G

\Lambda

M (Q2): However, one should remember that the trace of

T_* is not free from contributions due to spin-1/2 isospin-3/2 states.

Though the invariant amplitude related to the trace of T_* is contaminated by the transitions into spin-1/2 isospin-3/2 states, it makes sense to consider this amplitude because it has the simplest perturbative spectral density:

1 ss2 ae

pert: M (s1; s2; Q2) = Q

2

8^3 (^ \Gamma (s1 + s2 + Q

2))2(2^ + s1 + s2 + Q2) ; (12)

where

^ = q(s1 + s2 + Q2)2 \Gamma 4s1s2 : (13) Imposing the local duality prescription, we get

G

\Lambda

M (Q2) = 2mlN l\Delta (M + m) Z

s0

0 ds1 Z

S0 0

aepert:M (s1; s2; Q2)

ss2 ds2

= 6m(M + m) F (s0; S0; Q2) ; (14)

where F (s0; S0; Q2) is a universal function

F (s0; S0; Q2) = s

30S30

9lN l\Delta (Q2 + s0 + S0)3 i1 \Gamma 3oe + (1 \Gamma oe)p1 \Gamma 4oej (15)

and oe = s0S0=(Q2 + s0 + S0)2. We fix the nucleon duality interval s0 at the standard value s0 = 2:3 GeV 2 extracted from the analysis of the two-point function and used

0 5 10 15

Q2 (GeV2)

-1.0

-0.5

0.0

0.5 1.0

G*

E/G

* M(

Q2 )

G*E and G*M from eqs.(16,19) and G+ G*E and G*M from eqs.(17,18) and G+

Experimental data

Fig. 1. Ratio of Form Factors: G

\Lambda

E(Q

2)=G\Lambda

M(Q

2)

earlier in the nucleon form factor calculations. To fine-tune the S0 value, we consider two independent sum rules for the G1 form factor

mG1(Q2) = 2 3 + Q2 ddQ2 ! F (s0; S0; Q2) \Gamma 2Q2 ddQ2 !

2 Z S

0

0 F (s0; s2; Q

2) ds2 (16)

and

M G1(Q2) = 32 Q2 ddQ2 !

2 Z S

0

0 F (s0; s2; Q

2) ds2 (17)

corresponding to the structures q_[fl*; ^p] and q_[fl* ; (^p \Gamma ^q)].

Taking the ratio of these two relations, one can investigate their mutual consistency and test the overall reliability of the quark-hadron duality estimates. The best agreement is reached for S0 = 3:5 GeV 2, and we will use this value as the basic isobar duality interval in further calculations.

From eqs.(7) and (8), it follows that G1 is proportional to the difference of the magnetic G

\Lambda

M and electric G

\Lambda E transition form factors:

G(

\Gamma )(Q2) j G\Lambda

M (Q2) \Gamma G

\Lambda E (Q2) = 2m3M (M + m) i(M + m)2 + Q2j G1(Q2) : (18)

0 5 10

Q2 (GeV2)

0

1 2

Q4 G * M(

Q2 )

(Ge V4 )

G*M from eq.(14), S0=3.7 GeV2G

*M from eq.(14), S0=3.5 GeV2G *M from eqs.(16,19) and G+G *M from eqs.(17,19) and G+

StolerKeppel

Fig. 2. Form Factor G

\Lambda

M(Q

2)

The sum G(+)(Q2) j G

\Lambda

M (Q2) + G

\Lambda E (Q2) of these form factors can be obtainedfrom the invariant amplitude corresponding to the structure g

_* [^p; ^q]:

G(+)(Q2) = 8mM + m 24F (s0; S0; Q2) \Gamma Q

2

12

d dQ2 !

2 Z s

0

0 F (s1; S0; Q

2)ds135 : (19)

An important observation is that G

\Lambda

E(Q2) is predicted to be much smaller than

G

\Lambda

M (Q2) (see Fig.1). It should be noted that pQCD approach predicts, that G

\Lambda M (Q2) '\Gamma

G

\Lambda

E(Q2) for asymptotically large Q2.One should realize, that

G

\Lambda

E (Q2) is obtained in our calculation as a small differencebetween two large combinations

G(+)(Q2) and G(

\Gamma )(Q2), both dominated by G\Lambda

M (Q2),so we restrict ourselves to a conservative statement that the electric form factor

G

\Lambda

E(Q2) is small compared to G

\Lambda M (Q2).Experimental points for

G

\Lambda

M shown in Fig.2 were taken from the results for the

GT (Q2) form factor obtained from analysis of inclusive data 8, 9, 10. One can see

that, in the Q2?,3 GeV 2 region, the local duality predictions G

\Lambda

M (Q2) are close to theresults of the recent analysis by C. Keppel (see 10).

The quadrupole (Coulomb) form factor G

\Lambda

C (Q2) has been calculated also. Weobtained that G \Lambda C (Q2) is essentially smaller than G

\Lambda M (Q2) and has an extra 1=Q2

suppression compared to G

\Lambda

M (Q2).

3.

We applied the local quark-hadron duality prescription to estimate the soft contribution to the fl

\Lambda p ! \Delta transition form factors. We observed a reasonable agreement

between the results obtained from different invariant amplitudes. We found that the transition is dominated by the magnetic form factor G

\Lambda

M (Q2) while electric G

\Lambda E (Q2)and quadrupole

G

\Lambda

C (Q2) form factors are small compared to G

\Lambda M (Q2) for all experi-mentally accessible momentum transfers. Numerically, our estimates for

GT (Q2) are

close to those obtained from a recent analysis of inclusive data 10. Hence, there is no need for a sizable hard-scattering contribution to describe the data. Furthermore, if future exclusive measurements at CEBAF would show that the ratio G

\Lambda

E(Q2)=G

\Lambda M (Q2)is small above

Q2 , 3 GeV 2, this would give an unambiguous experimental proof of

the dominance of the soft contribution.

4. Acknowledgements

We are very grateful to P.Stoler, N.Isgur, V.Burkert and C.E.Carlson for discussions which strongly motivated this investigation. We thank C. Keppel for providing us with the results of her analysis 10.

This work was supported by the US Department of Energy under contract DEAC05-84ER40150.

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