Regge behavior of electroproduction structure functions

Volume 40B, number 4

PHYSICS LETTERS

24 July 1972

REGGE BEHAVIOR OF ELECTROPRODUCTION S T R U C T U R E F U N C T I O N S * R . A . BRANDT ** and P. VINCIARELLI *** New York UniversiO', Department of Physics, New York, New York 10012, USA

and M. BREIDENBACH $ CERN, Geneva, Switzerland Received 2 December 1971 The Regge behavior F 2 (p)p'-"~.~C 1 +

C2p

1/2 of the electroproduction scaling function for protons and neutrons

is studied theoretically and experimentally. The values of Cl and C2 are obtained from Regge fits to uW2(Q2, u) for smaller Q2. Regge behavior is found to set in at about p = 16. The results are used to check theoretical predictions for f dpF2(p)p - 2 and for C1 and to give an accurate estimate of the residue of the expected fixed pole in o-photoproduction.

The scaling behavior of the nucleon electroproduction structure functions W2(Q 2, v) measured in the SLAC-MIT [1, 7] experiments is well confirmed experimentally: uW2(Q 2, u) ~ F2(P ) A

(1)

in the A-limit u ~ oo with p = 2u/Q 2 fixed (Q2 = _q2 > 0, u = q . p, m 2 = 1). * W e assume the usual behavior in the Regge (R) limit u ~ 0% Q2 fixed:

pWv(Q 2, u) ~ /31 (Q2) + (2u) 1/2/32(Q2 ) . -

(2)

R

Here fll(Q 2) is the Pomeron residue function and /32(Q 2) is the sume of the fo and A 2 residue functions. We further assume the correctness of the expected [2] connections between the A and R limits:

* Supported in part by tile National Science Foundation, Center for Theoretical Physics under Grant No. NSF GU 2061. ** A.P. Sloan Foundation Fellow. *** Also Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 • :t: National Science Foundation Postdoctoral Fellow. * Our variable p is called co in ref. [1], 20 in ref. [2], and co 1 in refs. [9] and [10].

fil(Q 2) -+ c 1 , /32(Q2) -+ C2(Q2)1/2 ,

(3)

for Q2 -, 0% and

F2(P)_>C1 +p 1/2 C2-~(p)

(4)

for p-+ ~o. For many theoretical reasons, it is important to know the values of the constants (~1 and C 2. Two difficulties are encountered in attempting to extract these values from the experimental data: (i) The available data for large p corresponds to values of Q2 too small for the scaling behavior (1) to hold. (ii) It is not a priori clear how large p must be for the behavior (4) to dominate. Difficulty (i) can be avoided by indirectly obtaining knowledge of the asymptotic behavior of the F 2's from the smaller Q2 data. This will be done by using the explicit Regge residue functions/3q (Q2) suggested by Breidenbach and Kuti [3], which give Q2 Q2 uW2(Q2, u)-+ c 1 - + c2(2u ) 1/2 R Q2 + u2 (Q2+/22)1/2 "

(5) This form satisfies (3) and will be discussed below. To overcome difficulty (ii), we will use DGS-representations [2] in order to estimate the value Po o f p at which (4) begins to dominate. The combined use of (5) and 49 5

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PHYSICS LETTERS

the knowledge of Po will enable us to reliably estimate the Ci's. We will then use these results to check some theoretical predictions and to give a more accurate numerical estimate of the residue of the expected fixed pole in p-photoproduction. The photon structure function Y2(v)(i.e., the absorptive part of the Compton scattering amplitude) is given by Y2(v)=

lim (Q2) 1 W2(Q2, v). 02~0

(6)

24 July 1972

where o(a, b) -(O/Ob) o(a, b). The behavior of (11) in the Regge limit is controlled by the behavior of o(a, b) for b -~ 0. To obtain eq. (2), we take

o(a, b)

= a 1 (a)ln

b + o2(a)bl/2+o3(a)b + . . . .

(14)

and so

o(a, b)= ol(a)b-l+ l 02 (a)b-1/2+o3(a) + ....

(15)

It follows from (12) that the coefficient functions the conditions

oi(a) satisfy

It satisfies

vY 2(v) = 03, (v)/47rZa' , where o~,(v) is the total

(7)

vY2(v ) -~ 71

(8)

photon-nucleon cross section and a ~ 1/137. The Regge behavior ÷ (2v)-1/2 72

is consistent with experiment [4]. This implies the linear vanishing of the fii(Q2)'s for 02 ~ 0, a behavior incorporated into (5):

"Yl = C1/u2, ")'2= C2/'tl "

(9)

The integral representation to be used is [2] W2(Q2, v) = 0 2 ; 0

daai(a ) = 0 .

(16)

0 The 0 3 terms have been included in order to estimate the next leading contributions to eq. (2). They explicitly give rise to v -1 terms, but these need not be taken seriously as trajectory contributions. They only mean that the next leading term decreases at least as fast as v -1 . They can be considered as representing ttle effects of all the remaining terms. With eq. (14), (11) gives in the R-limit

Q2 [ vW2-+R ~ f d a al(a)

a+Q2

(a+Q2] 1/2

ln v ~ - + o 2 ( a ) l k 2v ]

da

a+e 2] + 0 3 (a) 2-/--" _1"

(17)

1

× fl- dba(a'b)a(-Q2+2bv-a)e(v+b)"

(10)

vYa-+ ~ da

For sufficiently large (positive) v, this gives

1 Q2 PW2(Q2 , v) - ~-

dao

-

In particular, the behavior of the photon amplitude is

R

l .

(11)

We follow ref. [21 and implement scaling by taking o(a, b) to vanish rapidly for large a. The behavior (1) then requires the vanishing of the integral of o(a, b): = 0.

+o3(a)

.

(is) , ~u-

Similarly, with (15), (13) gives in the p ~ ,,~ limit

0

dr o(a, b)

l(a) lna+o2(a) ~

(12)

0

F2(P ) ~

fd a [o 1 (a)p +½ a2(a)p 1/2 + o3(a)]. (19)

The connections (3) and (4) between eqs. (17) and (19) immediately follow provided use is made of eq. (16). Comparison of eqs. (2) and (17) gives the representations [2]

Eq. (11) then gives [2]

1 ; daS(a,1)a,

F2 (P) = 2PP 0 4c~6

(02) =

fd= 01 (a)ln(= + Q2)

(20)

~2(Q 2) =

Q2 -sfda o2(a)(a + 0 2 ) 1/2

(21)

(13)

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PHYSICS LETTERS

for the residue functions. The specific form of eq. (5) corresponds to an assumed saturation of the a-integrals at a required mass a =//2. In view of eq. (16), this amounts to taking o 1 (a) ~ 2C 1 6 (a

//2),

(22)

02 (a) = 4 C 2 ~ ( a - - , u 2 ) .

(23)

Substitution of eqs. (22) and (23) into (20) and (21) gives precisely eq. (5). We might only add here that the precocity of the limits (1) and (3), i.e., file fact that they are experimentally valid for Q2 as small as 2.5 GeV 2, means that the a integrals in all of the above equations effectively goes up to a maximum value of about 0.2 GeV 2, since then for Q2 = 2.5, we can approximate (a+Q2)/2p by Q2/2u in eq. (11), etc., and obtain the behavior (1). This short integration range suggests the approximations (22), (23) with//2 ~ 0.2. We are now in a position to estimate the value Po of p above which (4) is valid. Let u o be the value of u for which the Regge behavior (8) sets in for the Compton scattering. Analytically, this means that the contribution of v2 to Y2(uo) in (18) is much greater than the contribution of o 3 : i a ~1/21 fdao3(a)~v fda 0 2 (a)~2uo] I >~ a

.

(24)

Similarly, Po is defined so that the contribution of o 2 to F 2 (Po) in (19) is much greater than the contribution of 03 :

fdaa} o2(a)Po 1/2

>>

fdaao3 (a)

dimensionless. It is, therefore, not a priori clear what effective mass 2 Q2 should be used to relate the value Po - 2Uo/Q2 of p for which Regge behavior sets in for F2(p ) to the value uo of u for which Regge behavior sets in for Y2(u). One might expect Q2o to be a typical hadronic mass 2 (say 1 GeV2)I We find instead that Q2 is the small mass 2//2 which determines when scaling is valid. Numerically, this means that Po is rather large. Ref. [4] gives vo ~ 2 GeV 2 and ref. [3] (see also below) gives//2 ~ 0.25 GeV 2 (consistent with precocity). Thus Po ~ 16.

(27)

We feel that (27) should be valid to within 30%. We turn next to the determination of the parameters C1, C 2 and//2 for the proton (p) and neutron (n). The procedure is the same as in ref. [3]. Fits to the data of the form (5) were made for varying values of Po, where Po is the lowest value of p of data allowed in the fitting sample. The proton data was the 6 ° measurement of Poucher et al. [8]. Including values o f E o up to 19.5 GeV and p up to 60. For values of Po between 6 and 30,//2 was constant and equal to 0.248 -+ 0.05. C 1 and C 2 had, however, (not surprisingly) a large negative correlation coefficient, so that they could not be reliably separated above Po = 15, which, as we have seen, should be about the threshold for Regge behavior o f F 2 (see fig. l a). 04

O3 i

--XLm

.

(25)

We now want to compare (24) and (25). The a integrals on the right-hand sides of (24) and (25) are the same, which is fortunate since nothing is known about o3(a ). The a integrals on the left sides are different, but they can be related because of the short effective integration range involved (precocity), keeping in mind (16). The relation can be most simply obtained by using the approximation (23). Eqs. (24) and (25) give in this way P o = 2Vo///2 •

24 July 1972

02

,,

OI

0

•

CI

• x

C2 p.Z t 8

-ol

L

I 12

i

h 16

20

24

28

Fig. 1a. The Regge-fit parameters as functions of Po" 04

%

Po

02

C 2 = 0.275

(26)

The relation (26) is perhaps a bit surprising. Unlike the enerhy variable v, the scaling variable p - 2v/Q 2 is

8

I

16

20 Po

Fig. lb. C2(0o) for fixed C1 = 0.285 and fixed ~2 = 0.248. 497

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PHYSICS LETTERS

Consequently, the value for C 1 was picked at Oo = 16 from these fit5 to be 0.285. This n u m b e r is consistent with all of the fits for Po > 12.5. Using these fixed values Of C 1 and/32, C 2 was redetermined as a f u n c t i o n o f p o. It was n o w flat to within the error of the (1 parameter) fit, and was 0.275 -+ 0.03 (see fig. lb). The X2 of both the 1 and 3 parameter fits was 15,9 for 25 data points and Po = 15. It should be n o t e d that the data points that c o n t r i b u t e to this sample all conre from near the end of the data line * and have systematic errors of a b o u t 10(S from uncertainties of radiative corrections. Thus we insert a 10% normalization uncertainty in the values o f C 1 and C 2. Note also that our expectation/3 ~ 16 is nicely c o n f i r m e d by fig. 1. In s u m n l a r y * * CliP) ~ 0 . 2 8 5 ,

C~p ) ~ 0 . 2 7 5 .

Proceeding sinrilarly for the n e u t r o n gives c]n) ~ 0 . 1 8 5 ,

24 July 1972 3

2+i

i=1 to all of the available e-p inelastic data having values of missing mass W ~> 2 GeV and O 2 ~> 2 ( G e V / c ) 2. These requirements imply that there are few data points in the fit with p > 7, although tire region below P ~ 7 is well populated. We therefore make a piecewise linear c o n n e c t i o n ~ ( 0 ) between 7~(7) and F ( I 6) in order to carry out the integration (see fig. 2). The fits of the form (28) made here using the p-variable give a 1 = 1.650, a 2 = 1.325, a 3 = 1.339 with X2 = 619 for 397 degrees of freedom. A similar fit using p' = p + rn2/O 2 gives a 1 = 0.6868, a 2 = 1.773, a 3 = - 2 . 2 4 2 with X2 = 336 for the same 397 degrees of freedom. The data used in the fit comes from refs. [1] and [7]. We use the p ' - f i t in our analyses. For the n e u t r o n scaling function, we use the relation [81 (29)

c(2n) ~ 0.465 .

Perhaps u n e x p e c t e d features of these results are C ~ ) CI(P) , C{n) #= C~p), and qn)~> C~p) The scaling curve was d e t e r m i n e d by fitting a function of the Miller [7] form

The first theoretical results we wish to check are the sum rules of Brandt and Preparata [9] ~. I 1 .labc (~ Lab ~ 9" af ~dpF ab B (m-Tu E B.

1

(30)

P-

Here F~ b (p) is the scaling f u n c t i o n for B(p) + j a ( q )

*Data are taken at fixed incident energy and angle, varying E', the energy of the scattered electron. These points with large p are those with small E', thus having relatively large radiative corrections and attendant uncertainties. **A direct fit by Pagels [5] to the scaling data gives F2(P) = 0.28 + 0.18p 112 assuming Po ~ 8. The scaling data uncertainties are certainly large enough to make this result consistent with ours. Close and Gunion [6] have shown that a fit of the form F2 (p) = 0.12 + 0.462 0 -1/2 + 4.02 × p-3/2 also fits the scaling data and changes the sign of the scaling fixed pole (see eq. (40))! Such a result cannot, however be made consistent with the smaller (more accurate) Q2 data using fits of the form (5). Furthermore, allowing such huge residues for nonleading contributions (4.02/0.12 40) would render asymptotic phenomenology impossible. (What about a 50 o - 5 / 2 contribution?) We hope that Nature is not so perverse, but the point that one can never be sure is well taken, Finally, we mention the interesting calculation of Preparata (unpublished) who directly computes C1 and C2 from the Compton and photoproduction data using mass dispersion relations. Within the experimental uncertainties encountered in his approach, his results are consistent with ours.

408

-~ B(p) + ] b ( q ) , where{ j a : a = 1 - 8 } are the SU(3) currents and B is a m e m b e r of the 51+ b a r y o n octet, c is the coefficient of pip i in the equal and d abe' E B time c o m m u t a t o r f d 3 x (B(p)[ [Ji(0, x ) , 4 ( 0 ) ] IB(p)). E~ is a good SU(3) m o n e t described b y f ~ 1/7, d 1/3, E ° ~ 1/2 2x/~3.. F o r e l e c t r o p r o d u c t i o n (ja = j b = j Q = j 3 + _~v~jS) o f f p r o t o n s (B = p) and n e u t r o n s (B = n), (30) gives LpOQ = l ( 6 f +

2d) + (2)3/2 E o ~ 0.31

(31)

and E o ~ 0.24 LQQ = - 74 d + (2_13/2 3;

(32)

Numerically, the evaluation of (30) proceeds according to the d e c o m p o s i t i o n [ F ( p ) - = FQQ p , n (p)]

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PHYSICS LETTERS

y doF(p)P 2 ~ 2 i dPF'(P)P -2 + 2 ) 6 dp/C'(P)O- 2 1

1

+

7

2f dpF(p)p -2 ,

(33)

16

24 July 1972

is the residue of the J = 0 fixed pole in deep inelastic electroproduction, b has been evaluated in ref. [4] to be b ~ - 1 . Using

--a = f dpFS(p) 0

where

7

/~(p) = F' (7) + IF (16) - P (7)] (p - 7)/( 16 - 7). (34) (Numerically, we find the unexpected result that/~*(7) = F ( 1 6 ) to 3 decimal places for both p and n.) We find L e×p p ~ 0.23 + 0.06 + 0.04 = 0.33

(35)

16

l

16

7

0

we find - a p ~ 1.56 + 3.18 - 6.75 = -2.01

(43)

and - a n ~ 1.20 + 2.71 - 6.68 = - 2 . 7 7 .

and

(44)

Thus the prediction for the proton residue is L exp ~ 0.14 + 0.05 + 0.03 = 0.22 n

(36)

in view of the experimental uncertainties, the agreement with (31) and (32) is quite good. We next wish to give a more accurate determination of the residue of the fixed J = 0 pole in p-photoproduction predicted by Brandt et al. [ 10]. This prediction is

130 ~ 23'0 M2(b- a)/(M 2 - m2),

(37)

where 3'0 ~ 2.5 is the usual y-p junction, M 2 ~ 2GeV 2,

b =-1-

1

3o . J dv o~(v)

(38)

27r2 c~ 0

13~'1 ~ - 2 1 .

(45)

This rather large value is a consequence of the largeness o f f relative to F" (see fig. 2). The final thing to be checked is the very speculative prediction of Brandt [2] that the asymptotic behavior of F~3b(p) has the highly symmetric universal form

yf3b (p)_+ -ld abe D B c =_W~3b, 7"g

(46)

where Dr3 ~- (B(p) iS c IB(p)), with SC(x) the scalar current in U(I 2), is a good SU(3) nonet described by F ~ 3/7, D ~ - 1 / 7 , D ° ~- 1/2(3/2) 1/2. For electroproduction off protons and neutrons, (46) gives

wQQ=(1/97r)(6F + 2D + 12(2/3)1/2D°) ~ 0.29 (47)

where and

l

os (P) = vY~ (p) = PY2 (v) - "71 - (2v)-l/2

72

2zr2c~ (39) is the residue of the J = 0 fixed pole in Compton scattering, and

oBr 05 ~

(

p

)

04~

03

a = - ? dpF~(p)

(40)

0

02 017

where

F~(p)=F2(P)O(p- I)-C 1

\'~(p)

?(p)

0 ~-~

p 1/2C 2

(41)

3

J 5

J ?

J 9

1_ II

L 13

1 15 P

~ 17

1 19

2 21

J 23

Fig. 2. The scaling function F2(o), equal to F2(P) for 1 ~/ 16. 499

Volume 40B, number 4

wQQ=(1/97r)(-4D + 12(2/3)1/2D °) -~ 0 . 2 3 .

PHYSICS LETTERS (48)

The e x p e r i m e n t a l results are C~P) ~ 0.29 and C~n ) ~ 0.19, in agreement with (47) and (48) within the experimental errors. One o f us (R.A.B.) thanks the C E R N T h e o r e t i c a l Studies Division for its hospitality during the e x e c u t i o n of some o f this work.

R gfcr~'FICCS I lJ M. Breidenbach et al., Phys. Rev. Lett. 23 (1969) 935; E.D. Bloom et al., Phys. Rev. Let*. 23 (1969) 930;

500

[2] [3] [4]

[5] [6] [7] [8] [9] [101

24 July 1972

M. Breidenbach, MIT Thesis, 1970, unpublished; E.D. Bloom et al., 15th Intern. Conf. on High Energy Physics, Kiev, U.S.S.R. (1970). R.A. Brandt, Phys. Rev. D1 (1970) 2808. M. Breidenbach and J. Kuti, to be published. M. Damashek and F.J. Gilman, Phys. Rev. D1 (1970) 1319; C.A. Dominguez, C. Ferro Fontan and R. Suaya, Phys. Lett. 31B (1970) 365. H.R. Pagels, Phys. Lett. 34B (1971) 299. Close and Gunion, Phys. Rev. D4 (1971) 742. G. Miller, Stanford Thesis and SLAC-REPORT-129 (1970), unpublished. Preliminary data as reported by H. Kendal at the Cornell Conference 1971. R,A. Brandt and G. Preparata, Phys. Rev. D1 (1970) 2577. R.A. Brandt, W.C. Ng, G. Preparata, P. Vineiarelli, N.Y.U. preprint 9/71, to be published.