The ratio of the magnetic to the electric proton form factors in Quantum Chromodynamics

Volume 76B, number 4

PHYSICS LETTERS

19 June 1978

THE RATIO OF THE MAGNETIC TO THE ELECTRIC PROTON FORM FACTORS IN QUANTUM CHROMODYNAMICS

Robert COQUEREAUX a and Eduardo de RAFAEL 2

Centre de Physique Thdorique, C.N.R.S., F-13274 Marseille Cedex 2, France Received 4 April 1978

We propose a two parameter expression for the high momentum transfer behaviour of the ratio of proton form factors

GM(Q2)/QE(Q2) based on perturbation theory calculations in Quantum Chromodynamics. A fit to the present experimental data and its consistency with a previous fit to the high Q2 behaviour of the Dirac form factor [1 ] is also discussed.

On a recent letter [1] we have proposed a two parameter expression suggested from perturbation theory calculations in quantum chromodynamics (QCD) to describe the high momentum transfer dependence of the Dirac form factor of the proton. Encouraged by the remarkably good quality of the fit thus obtained, we have now extended our work to predict the high Q2 behaviour of the ratio GM(Q2) El(Q2) + F2(Q2 ) /~(Q2) --= , (I) GE(Q2) F I ( Q 2 ) - ( Q 2 / 4 M 2 ) F 2 ( Q 2) of the magnetic GM(Q2 ) to the electric GE(Q2 ) form factors of the proton. In eq. (1), F 1 and F 2 are the conventional Dirac and Pauli form factors. We find that asymptotically i.e., for Q2 > A 2 ..~ (0.5 GeV) 2, and with color SU(3) as the underlying gauge theory of the strong interactions, q /a(Q2) = P

A

1 ~-log

_3_~1 1

4 (2)

A2 ]

{ + 2/32!og(loge2/A t 1

/32

/31=-

11 2n 6 "3+3 2

/32=

(17.32

-

log Q 2 / A 2

I Allocataire DGRST. 2 Centre de Physique Thdorique, CNRS, Marseille.

J

(3)

1 4 n -~'5"~-U

5

n) 3"3 '

and n the number of flavors of quark constituents. In principle the two parameters p and q in eq. (2) are unknown. They are specific to the binding dynamics of quarks within the proton, much the same as the two parameters A and b which appear in the expression for the asymptotic behaviour of the Dirac form factor of the proton proposed in ref. [1 ]: ,1 F~(Q2) =

×exp

A (1 + Q2/A2)Z(1+b) ~/31

4zlogz-4z-~l

l°g2z

, (4a)

where z = ~1 log Q2/A2

where A is the renormalization group invariant scale associated to the strong effective fine coupling structure constant ~s =_/31~og Q2/A 2

with

.

(4b)

,1 Eq. (4) differs slightly from the expression given in ref. [1]. In ref. [ 1] we forgot the effect of the/~2 coefficient of the Callan-Symanzik/3-function in the exponential. This term, at the order of accuracy we solved the differential equation from which the asymptotic expression of F1 (Q2) follows, should be taken into account. Its presence is only significant in the relatively low Q2 region. We have also modified the form of the denominator. 475

Volume 76B, number 4

PHYSICS LETTERS

The reason why/I(Q 2) has a non trivial asymptotic behaviour can be seen already from its definition in eq. (1). In the denominator, the Pauli form factor F 2 appears multiplied by a power of Q2. The effect of this is to promote the a priori negligible asymptotic behaviour of F2(=)(Q2) (in perturbation theory F 2~ (1/Q 2) logmQ 2) to an asymptotic behaviour competitive with that of F~=)(Q2). We have studied the asymptotic behaviour of #(Q2) for a free quark in perturbation theory, up to two loops, in color-SU(N). Many of the calculations needed can be found in the paper by Barbieri et al. [2] on QED form factors. We find that, to the calculated order of accuracy, and with y = ~1 log

Q2/m2,

(5)

where ~s is the same effective coupling constant as the one defined (at one loop) in eq. (3), i.e., ~s

%/rr

-- =

rr

1 - (%/rr)/31 ~ log Q2/m2 -/31 log Q2/A2

and z = ~1 log Q2/A2 ,

_ 1 { .-2]

(7a)

81(%) t - 2 N ] 2~ =

%+o

(7b)

rr

rri

t.\ ~ i

'x' (1 X with

j"

{ li+,io
x

,o
Q~2) N2-1 1 (1 Q log A z ~ 2N 2131

(X 2 -

3N2-~1 ) fix

(10) ,

1t 2 3

2/31 ×

[(1

3 N2-1)logx+~l /31 22V

N2-I] 2N

}

.

(llb)

The interesting thing about eq. (10) is that all the dependence on quark mass singularities has been lumped into an overall coefficient p(x) and the power q(x) of the Q2/A2 dependence. For completeness we recall that the x dependence of the exponent b in the expression for the asymptotic behaviour of the Dirac factor of an on-shell quark is

b(x)-

The solution to eq. (6) is the following ~t(~°)(y, ac) = #(~')(0, ~s) exp -

(9b, c)

1

and 60(ac), 61 (c%) anomalous dimension type functions: 2N

x --- ~1 log m2/A 2 .

Eq. (8) can also be cast in the form

(6)

where a c = g2/4rrwith g the renormalized QCD coupling constant * 2;/3(0%) is the usual QCD CallanSymanzik/3-function

80(ac)=

;(9a)

and

--~Y +/3(ac)ac ~ c - [6 l(ac)Y + 60(ac)]

l~(ac)=( -ll--~N+~-~2n)-~+O[(-~)2];

2 =

1

where m is the quark mass, ~(~)(y, ac) is governed by the differential equation

X #(=)(y, ac) = 0 ,

19 June 1978

N2-1

2N

1 2131l o g x .

(12)

So far we have been concerned with the asymptotic behaviour for on shell quarks. To the extent that the large Q2 behaviour of the proton is governed by the large Q2 behaviour of its constituent quarks ,3 we expect eq. (10) to be an adequate description of the large Q2 behaviour of p(~)(Q2) for the proton itself. There will be however binding effects which will presumably alter the perturbation theory dependence on quark masses via the variable x into some kind of ef,3 See e.g. the review article in ref. [4] where earlier references can be found.

Volume 76B, number 4

PHYSICS LETTERS

19 June 1978

6E 1,5I GDipole Bonn 71 CEA71 DESY70 u 5LAC70

_. ;*_" ~

l

2

~

l

i

I

6

-

]1

0

I

i ~

]•

i

]8

i

i

22

i

i

26

i

I 4b

30

o, [,G,v,cl,] oo 2 Fig. 2. Plot of G ~ )/GDipole versus Q , where GDipole =

L

O.21

o.c

5

5

;

;

;

"

Q2 [(Gev/c )2] Fig. 1. Plot of u(0)/~(Q 2) versus Q2 with u(Q ~) = GM(Q2)/ GE(Q2) (see eq. (11)). The experimental points are those of refs. [6] to [9]. Curve (a) is the fit of all the experimental points with the two-parameter expression in eq. (2). Curve (b) is the same fit with omission of the two points from ref. [7].

fective x dependence, in much the same way as in the QED bound state cancellations of infrared divergences. This is why we do not insist in attributing too much significance to the explicit form o f p ( x ) and q(x) in eqs. (1 la,b) and instead, in going from the constituent quarks to the composite proton, we treat p and q as phenomenological parameters. This is the way we have obtained eq. (2) where obviously we have put N = 3. Predictions for the ratio/a(~)(Q 2) have also been made by other authors [5] from the assumption of local duality and the asymptotic freedom analysis of deep inelastic scattering. They are not contradictory with the behaviour of eq. (2). 2 M 2 There exist measurements ,4 of the ratio GE/G for values of Q2 ranging between 1 (GeV/c) 2 and 3.7 (GeV/c) 2. In fig. 1 we have plotted the corresponding experimental points for values of Q2 ~> 1 (GeV/c) 2. Since these are measurements made in different ex *4 See refs. [61 to [91.

#[1 + Q2/0.71 (GeV/e) 2 ]-2 and/~ is the magnetic moment of the proton. The experimental points are those of ref. [ 11 ]. The curve is the fit with the two-parameter expression in eq. (3).

periments we have made two different types of fits witk the two parameter expression in eq. (2). We have used A = 0.474 GeV which is the value obtained by a careful fit of QCD to deep inelastic electron scattering data [10]. The fit shown in curve (a) of fig. 1 has been obtained using all the experimental points with Q2 1 (GeV/c) 2 from refs. [6] to [9]. The values of the parameters p and q in eq. (2) which correspond to the best fit (X2 per degree of freedom = 1.05), for 13 experimental points) are p = 2.83

and

q = -0.104.

(13a,b)

The fit shown in curve (b) of fig. 1 has been obtained using the data of refs. [6, 8, 9]. Then we find (X2 per degree of freedom = 0.47, for 11 experimental points) p = 2.25

and

q = -0.011 .

(14a,b)

In fig. 2 we show the fit of the proton form factor data [11 ] with the two parameter expression in eq. (3) [see footnote 4:1] using the same value A: 0.474 GeV. The corresponding values for the parameters A and b from a fit to 10 experimental points ranging from Q2 = 3.75 (GeV/c) 2 to 25.03 (GeV/c) 2 are (X2 per degree of freedom = 1.05, for 10 experimental points)

A/It = 4.24

and

b = -0.003.

(15a,b) 477

Volume 76B, number 4

PHYSICS LETTERS

It is now tempting to check the consistency of the values obtained for the parameters + s q and b from the fit to experimental data with the perturbation theory expressions in eqs. (1 l a , b ) and (12). From the fit value b = - 0 . 0 0 3 we obtain x = 1.01 using eq. (12). Then, from eq. (1 la) and (1 lb), we get q(0.804) = - 0 . 0 7 6 ,

(16)

to be compared with the fit values in eqs. (13) and (14). The consistency is not unreasonable. We hope that these considerations on the ratio /a(Q 2) based on perturbative QCD will stimulate further experimental work on the proton form factors, in particular on the separation of the electric and magnetic form factors. , s We do not include A and p in this consistency check because they depend on other parameters than x.

478

19 June 1978

References [1] R. Coquereaux and E. de Rafael, Phys. Lett. 74B (1978) 105. [2] R. Barbieri, J.A. Mignaco and E. Remiddi, Nuovo Cimento l l A (1972) 824,865. [3] C.P. Korthals-Altes and E. de Rafael, Nuclear Phys. B125 (1977) 275. [4] D. Sivers, S.J. Brodsky and R. Blankenbecler, Phys. Reports 23C (1976) 1. [5] A. De Rtijula, H. Georgi and H.D. Politzer, Ann. Phys. 103 (1977) 315. [6] W. Barrel et al., Phys. Lett. 33B (1970) 245 (DESY). [7] J. Litt et al., Phys. Lett. 31B (1970) 40 (SLAC). [8] Ch. Berger et al., Phys. Lett. 35B (1970) 87 (Bonn). [9] L.E. Price et al., Phys. Rev. D4 (1971) 45 (CEA). [10] A.J. Buras, E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B131 (1977) 308. [11] P.N. Kirk et al., Phys. Rev. D8 (1973) 63.