- Email: [email protected]
On the differential cross section and the slope of vector meson electroproduction
Nuclear Physics B88 (1975) 301-317 © North-Holland Publishing Company
ON THE DIFFERENTIAL CROSS SECTION AND THE SLOPE OF VECTOR MESON ELECTROPRODUCTION H. FRAAS Physikalisches lnstitut der Universitiit, Wi~rzburg, Germany B.J. READ Daresbury Laboratory, Warrington, England D. SCHILDKNECHT Deutsches Elektronen-Synchrotron, DESY, Hamburg, Germany Received 31 October 1974 The effect of higher mass vector meson contributions (e.g. transitions p'p ~ pp, p"p -~ pp etc.) on diffractive vector meson electroproduction is investigated. The ansatz for these off-diagonal transitions is motivated from diffractive dissociation results in 7rp and pp interactions_and our recent treatment of the total electroproduction cross section with inclusion of off-diagonal terms. For t = 0 and large q2 (co' > 10) we find, comparing with simple pO dominance, a change in normalization, but the same power of q2 dependence. From results on slopes in diffraction dissociation in ~rp and pp interactions we predict some flattening of the p0 electroproduction slope with increasing q2 in the region of small -t.
1. Introduction The production of vector mesons by photons is quite successfully described [1,2] by vector meson dominance: the photon is viewed as virtually dissociating into the vector meson V, which then scatters off the nucleon. Vector meson production by photons is thus closely related to elastic vector meson nucleon scattering. Off-diagonal transitions, in which the photon virtually converts into a more massive spin 1 state, say V', and subsequently produces the vector meson V via V'p ~ Vp, are usually neglected. They are assumed to be suppressed due to the smaller coupling of the photon to higher mass states, and in addition by the smaller hadronic amplitudes for diffraction dissociation V'p ~ Vp (V' :/: V) compared with the ones for elastic processes Vp ~ Vp. Indeed, the agreement with experiment [1 ] of the relations between p0 (oJ, 6) photoproduction at t = 0 and the corresponding vector meson nucleon total cross sections (as experimentally determined from vector meson production from complex nuclei [3]) seems to justify the neglect of such terms in photoproduction.
302
H. Fraas et aL /Elektroproduction
Nevertheless, there are at least three motivations for studying such off-diagonal transitions more carefully. Firstly, a more reliable, quantitative estimate o f the magnitude of off-diagonal transitions now becomes possible by exploiting the increased knowledge of diffraction dissociation in hadron hadron interactions [4] and the coupling of the photon in e+e - annihilation to more massive hadron states [5]. Secondly, although off-diagonal transitions in p0, co, ~ photonproduction seem to have but little effect at q2 = 0, their relative importance increases strongly with growing space-like values o f q 2 due to the propagator factor 1/(q2/m 2 + 1) in the amplitude. This is of consequence for the q2 region explored in electroproduction [2,6] at present and in the near future. Finally, there is a third and in our opinion the most important reason to carefully analyse off-diagonal transitions in vector meson electroproduction. It is the importance which these terms apparently have in contributing to the virtual forward Compton amplitude. In fact, as we have recently shown [7], it is exactly these off-diagonal contributions to the Compton amplitude which allow one to derive within the generalized vector dominance framework a convergent expression and scaling behaviour of the transverse part of the proton structure function vW 2 starting from a 1Is behaviour of the e+e - annihilation cross section. Vector meson electroproduction provides an additional test of the general features of the model recently proposed [7]. In the present paper we shall consider electroproduction of the low-lying and best known vector mesons pO, c.o, q~only. For these low-lying states we can best hope to use our empirical knowledge of diffraction dissociation [4] in 7rp and pp interactions to infer the correct ansatz for the necessary off-diagonal Vp -+ VNP (or rather VNP ~ Vp) transition amplitudes. Here V stands for p0 (co, ~) and V N for some more massive vector state. Thus from a reasonable Ansatz for these diffraction dissociation amplitudes and with vector photon couplings as suggested from e+e - annihilation, we shall obtain essentially two results in this paper: (i) The first result concerns forward (or rather t = 0) production of vector mesons by transverse virtual photons. We shall show that under rather general conditions, to be specified below, we expect the amplitude for vector meson production in the diffractive region of w' = 1 + W2/q 2 ~ 10 to fall off with q2 for q2 _+ ,~ as a simple pole in q2 just as in p0 dominance. Quite simply, the influence of the off-diagonal transitions may be expressed by the (approximately valid) interpolation formula
(co'> lO) ~-/(w,da q2, t = 0 ) = ( ~ 2 ~2 Iqreall dO (w, q 2 = O , t = O ) ~q2 + ~/2] Iqvirtl dt ~4 Iqreall do (W, q2 = 0 t = O) q4 Iqvirtl dt ' '
(1)
where e.g. for p0 production ~ is expected to be smaller than the p0 mass mp, the exact value depending upon the magnitude of the diffraction dissociation amplitudes. Relation (1) with ~ < mp then implies a suppression of the cross section relative to
H. Fraas et al./Elektroproduction
303
p0 dominance. As will be pointed out, this suppression effect comes about through the alternating sign assumption necessary for convergence o f the virtual Compton forward amplitude. If all off-diagonal transitions were to contribute with positive sign, ~ > mp would be obtained. (ii) Our second result concerns the non-forward (t 4: 0) production o f vector mesons b y virtual, space-like photons. To predict the q2 dependence in this region o f t, we need information on the slope of diffractive processes, e.g. PoP ~ PNP (thinking o f a series of discrete states PN, N = 0, 1 , 2 , . . . . . ). F r o m np and pp interactions it appears [4,8] that diffraction dissociation into low-lying states such as pp ~ pN*(1470) or rrp -~ A l P shows a somewhat steeper slope than elastic processes. Assuming this effect to be also present in PoP ~ PNP, we are led, in electroproduction o f vector mesons p0, w, ~b, to expect within our off-diagonal framework a slight flattening of slope with increasing q2.
2. Model assumptions: q2 dependence o f f o r w a r d (t = O) vector meson production Let us start b y stating our main assumptions. As in [7] we shall work with a Veneziano type mass spectrum for the vector mesons 2 _- m 02 (1 + XN) , rnN
N=0,
1,2, . . . . .
,
(2)
where m 0 is equal to the p 0 (co, 4) mass. The choice k = 2 together with the vector meson photon coupling [9] 1 m2
1 -
7N 2
(3)
72m 2
for the N t h vector state leads to a total e+e - annihilation cross section o(e+e - -~ hadrons) = (47r2/k72) o(e+e - ~/1+/~-). For k = 2 this is roughly 2.5 times that of pair production, quite consistent with experiment [5] up to an e+e - centre-of-mass energy of about 3.5 GeV. Due to suppression of still higher mass contributions by their photon coupling and b y the small hadronic diffraction dissociation into very high mass states, the recent deviations from a 1/s law indicated b y CEA [5] and SPEAR [10] data are not expected to change our results for the modestly large values of q2 to be explored in electroproduction in the foreseeable future. In addition to the vector meson photon couplings we have to specify the mass dependence o f the hadronic diffraction dissociation amplitude VoP -> VNP. In contrast to our treatment [7] o f the total virtual photoproduction cross section, where mainly for simplicity of presentation (effective) transitions to next neighbours only were included, we shall in this paper actually consider contributions due to the complete diffraction dissociation mass spectrum as observed [4] in 7rp and pp interactions. Experimentally, as a function o f the mass m o f the system X diffractively pro-
304
H. Fraas et al./Elektroproduction
duced, one finds for these reactions a dependence for the cross section o f roughly [4] do 1 -(4) dm 2 m 2 For rrp interactions such a behaviour seems to be valid [4] down to rather small values o f m 2. A difficulty for inferring from (4) the correct diffraction dissociation law for V0P ~ VNP arises from the fact that the system X diffractively produced in rrp ~ pX contains in general spins quite different from the spin of the initial particle, the contribution of the different spins actually not being too well known. If, e.g., the number o f different spin states available for population were to increase linearly with m 2, then assuming equal population, (4) should just be modified by an additional power of 1/m 2. Motivated by such arguments we shall for diffraction dissociation with spin conservation replace (4) by the more general power law (for PoP ~ PNP transitions) TooP ~ p N p = TPoP -*pip ( m l 12p+ 1 Introducing the ratio c o of the first off-diagonal amplitude PoP -> PIP to the elastic one PoP ~ PoP, CO = TOoP ~ P lP/TPOP ~ PoP,
(6)
we may rewrite (5) as To°P-~PNP = CO T P O p ~ o o p (\ ~mNl t]2 p + l
(7)
This power law is assumed to be valid in the forward direction (t = 0), the exponent p assumed to be p > 0, with the limiting case p = 0 corresponding to the observed behaviour (4)*. It is likely that from further theoretical analysis and further data on diffractive production more will be learned on how well justified our simple ansatz (7) for the spin conserving part of diffraction dissociation is. Let us also note that the exact numerical value o f p in (7) is of importance for our conclusions only in so far as the detailed numerical results to be presented are concerned, but certainly not as regards the general conclusions to be drawn on the q2 dependence o f vector meson electroproduction. The relative normalization c O o f first diffraction dissociation transition to elastic one is known [4] to be substantially smaller thanl, the exact value again being of relevance for the detailed numerical predictions only. As a check, from our ansatz (7) let us also estimate the total diffraction dissociation cross section for production o f a spin 1 state, i.e. P0 + P " P + E~= 1 ON. With* We expect 0 < p < 2, say. For larger values ofp the off-diagonal terms are tiny and have negligible effect.
H. Fraaset al./Elektroproduction
305
Out taking into account the necessary energy cut-off and also any dependence of the slope o f diffraction dissociation on the mass of the produced system, one obtains by straightforward summation O0oP-* P~2ON = C2 O0oP~PoP
) 2p+ 1 ~(2p) ( ~ ) ( - 1 ) 2p+ 1( 1 (2p)! ~ +1 -- + 1 ,
(8)
where ~(2P)(x) is the 2pth derivative of the diagamma function ~(x). Numerically, 2 21 = ½, one then obtains e.g., for p = ~, h = 2 and c 0 = mo/m Ooop ~
pzpN ~--0.23 Ooop _ poP '
i.e., a small fraction of the elastic cross section, a result which is reasonable when compared with our experimental knowledge [4]. From the above assumptions (2), (3) and (7) we now have the following ansatz for the amplitude of O0 production (an analogous one holding for ~ and ~b) by transverse virtual photons and protons, T+I
")'vP---~P
-
0
P [w ,2 t - m - Vt~'r p0 P--+P0 Pt'[ap ~",~t , ' - " J - - - ~ - o
m2
X[_q2+m------~o+COk 0
~
~
m2
*+1
~",t =0)
. m 0 [m 1~2p+1],
q2+m2(--1)~Vm----Ntm---~)
(9)
0
where T_+°lp o p (W, t = 0) is the transverse (helicity -+ 1) helicity conserving pOp scattering amplitude. The ( - 1 ) N factor is the alternating sign assumption mainly motivated* from the requirement of convergence of the virtual forward Compton amplitude. (For completeness, we shall later on also briefly discuss the result obtained from having positive couplings throughout, although we do not consider this as a realistic alternative in view of the negative contributions required for the Compton amplitude. Nevertheless, positive signs will provide us with an upper limit for the amplitude (9).) The infinite sum over N in (9) should actually be cut off at a value N, roughly corresponding to the available c.m. energy W, as ansatz (7) breaks down for large vector meson masses, i.e. mN.~ W. In practice, however introducing such a cut off in (9) does not to a good approximation change the result obtained compared with the infinite sum over N to be evaluated subsequently, provided one stays in the diffraction region of co' large, i.e. w' ~ 10 say. With this restriction the large q2 behaviour in the ensuing discussion should be understood as the limit q2 ~ oo with co' fixed (co' ~ 10). For integer values of p the sum in (9) may be easily evaluated, as we shall now show. First let us rewrite (9) as * See ref. [7]. Negative signs seem also to be required for the nucleon form factors. Alternating signs, and relation (3) have in addition been obtained in quark model calculations [11 ].
H. Fraaset aL/Electroproduction
306
2 ~'rvP ~ pOp l . . . . 2a N/c~ "v'°°p ~ O°P[" mo 1+-1 tvv'q ' =-"~-'0 "-+1 [.q2 + m02
(mli2p+l -- CO \ - ~ 0 ]
1 S(p)( l+q2/m2
+1,1+1)1 '
~tP + 1
(lO)
where by definition oo
S(P)(x,y) = ~
(-1)n n=O (x +n) (y +n)P
For integer have
(11)
p, S(P)(x,y) reduces to known functions.
In fact, for p = 0 we simply
s(0)(x, y ) - s(O)(x) = ~ ( x ) ,
(12)
/3(x) = ~ [4 (½(x + 1)) - ~(½x)l ,
(13)
with
where ~k(x) = (d/dx) In F(x) is the digamma function tabulated in [12]. is easily obtained to be S(1)(x, y) = ~ and
(14)
[~(x) -/3(y)l ,
S(P)(x,y) for p >
S(1)(x,y)
1 is related to S(1)(x, y) by differentiation
(-1)p-1 dp-1
S(P)(x'Y) = -~- ~ ~yp--1 S(1)(x'Y)'
p = 2, 3 . . . .
(15)
Thus explicitly
p(-1)p~tp .fl ( 1 + S(P)( 1 + q2/mgx + 1,X+I 1) = (q2/m2) +(-1)°-1
p-1
1
xk+l
k=0 ( P - 1 - - k ) ! , 2, 2,k+1
q2x/m2
I-1)
~(p_l_k)/l+l\.(_~)
(16)
tq /m O)
This relation holds for p = 0, 1, 2 , . . . where for p = 0 the second term on the righthand side should ignored. In (16), (3(r)(x)denotes the rth derivative of the function
H. Fraaset al./Electroproduction
307
/3(x). Let us also note the asymptotic behaviour of S(P) in (16) for q2 -+ oo. Using the asymptotic expansion of the function/3(x),
1+
(17)
3(x) ~ 2x
we obtain for q2 ~ oo (1 + q2/m2 S (°)
+1,1+
1
~
(18)
- + 0
2q 2
and S(p)( 1 +q2/m2
1 1) +1,-~+
+O( 1 ~ , \q4!
Xm2~ --p q2 U[--l) ~
1 /3(p_ 1)
(1+1)
(p= 1,2 . . . . . ).
(19)
The sum of poles S(P) appearing in (9), (10) thus sums up to a simple pole for q2 ~oo. From (17) and (18) the asymptotic (q2 -+ oo) behaviour of the p0 production amplitude is now easily obtained to be 2 "+1 -
~ 3, 0
~+1 -
"--:",1 q \
- - CO o--Z:-"
+
,
(p=0),
(20)
and 2 ,r3,vp~O°p %/~ 7'+01P o ~ OoP mO ~+-1 ~ ~ 70 q2
X I l _ c o ( m l ] 2p+1 (-1)P -1 \m0/
(p= 1 , 2 , . . . . ) .
~
i~! t3(p-
1)(~+ 1)1
+
0(~4)
,
(21)
Thus even with inclusion of off-diagonal transitions subject to the power law (5) the P0 production amplitude asymptotically (i.e. q2 ~ 0% keeping co' large) behaves as a simple pole in q2. (The only exception to this general rule would occur when
H. Fraaset al./Electroproduction
308
2mo=. 2 ml X / ] - ~ '
(p=0) ; (22)
C O ='
. 1 (--1)P-1 •P(p-- 1)! (1 +X)P+~ ~(P-1)(1/X+I) '
(p=l,2 ..... ).
From (22), for ?~= 2 we would obtain c o = 1.15 for p = 0, and c o = 0.9 for p = 1, and thus a diffractive forward amplitude of roughly the magnitude of the elastic one Tpop_~poP' whereas experimentally the diffraction dissociation cross section into a particular state close in mass to the incoming one seems strongly suppressed [4] compared with the eleastic cross section. Thus we may exclude the possiblility (22), and from (20), (21) obtain quite generally a pole behaviour for large q2, just as in simple p0 dominance.) From (20) and (21) there is a change in normalization, however, compared with simple p0 dominance, which we wish to discuss next. Let us first consider the normalization of the amplitude (10) at q2 = 0. From (11) one obtains
lim0 S(P)
q2
+
~k
1,~
+1 =
.
~(P)
+1
,
(23)
and thus by substitution into (10) ( m l ]2p+1
_
T+'rt~-~p0p (W, q 2 : 0 ) - N / ~ T+O~p~p°p I1 _ Co
70
[ml]2p+l + c0 ~'~00)
\mo/
(--1)P /3(p)(1)] Xp+ lp!
= v/-~ T°+~P--"P°PKp(X) .
(24)
In simple p0 dominance, the corrrection factor Kp on the right-hand side of (24) would be replaced by 1, the deviation from 1 thus being due to off-diagonal transitions. As a numerical example, let us simply substitute
(mO]2p+l_{ 1.~2p+l=[O.58(p=O), CO=\ml/ -~TJ [0.19 (p 1), into (24), which values give reasonable suppression of diffractive dissociation " PoP -+ PlP for 0 ~
(25)
H. Fraaset aL/Electroproduction -~t3
,
309
p--0; (26)
G=
which numerically for X = 2 amounts to K 0 = 0.79 (p = 0) and K 1 = 0.92 (p = 1). Although the pure p0 dominance prediction (neglecting the small real part)
do~O-*P°P(w, q2=O,t=O)=~_~ d'-t
1
3,0 167r [aoop] 2
(27)
is satisfied by the data [1,3], systematic and statistical errors in the determination of the O0 photoproduction and the total pOp cross sections and in 3,2/41r are large enough to allow for a 10% to 20% variation required from off-diagonal contributions. As we are mainly interested in the q2 dependence o f o 0 production, which is more directly accessible to experiment than (27), let us next express O0 production by virtual photons in terms of the photoproduction amplitude (24). From (10) and (24) we then obtain
T[[p-~o°p (W, q2, t = 0 ) [ml]2p+l + c0 t--~00)
= 1--~-F m2 ( 1 [ml~2p+l~ KpLq 2 + mg c0k-~00) ]
__S(p)(l+Xq2/m'gl)lT~P-'P°P(W'q2=O't=O)' 1
xP+l
(2S) where we have used
S(P) (x + 1, y + 1) =
xyP
- S(P) (x, y ) ,
(29)
which directly follows from the definition (11). Finally, let us give the cross section following from (28), for simplicity restricting ourselves to the realistic cases p = 0 and p = 1 and to the reasonable choice (25) for c0, which we shall consider in our numerical evaluation. The t = 0 differential cross section for electroproduction doO/dt for p = 0 is then given by (co' ~ 10) do 0 "W q2) =IJ 3((1 + q2/m~)/X)~ 2 ~ [qreall d°0"W q2 0) dt ( ' l_ ~ -] Iqvirtl-d~ ( ' =
A m
•2 Iqreallda 0 q4 4/32(1/X)Iqvirtl dt ( W ' q 2 = 0 ) '
(30)
310
H. Fraas et al./Electroproduction
and for
p = 1
we obtain
d°O~-f(,'W q 2 ) = ~ -4 ~.2f3((1
+ q21m2)lX) - 3(1/X)]2 Iqr~l doO
~iS(l/~)
_ Iqv~l dt
(W, q 2 = 0 ) .
(31) The flux factor !qreal[ _
W2 - m2
(32) 1
Iqvirtl
[(W2 _ m 2 _ q2)2 + 4W2q2]
is well-known [13] and arises from the definition of the virtual photon absorption cross section. Numerically, (30) and (31) may be evaluated from the definition (13) and the tables [12] for the diagram function ~ (x). As mentioned, the asymptotic dependence of (30) and (31) is the same power o f q 2 as obtained from simple p 0 dominance. There is an asymptotic suppression factor for the normalisation, however, which for X = 2 from 3(~) = 1.57 and -3(1)(½) = 3.66 one calculates to be 0.41 (for p = 0) and 0.74 (for p = 1) respectively, relative to simple p0 dominance. The results (30) and (31) may be described approximately* by the much simpler interpolation formula (co' > 10) d°0(W ' q2) - ( ~n 2 dt - \ ~ ]
~2 Iqreall do 0 "W Iqvirtl ~- ( ' q2 = 0 ) ,
(33)
with 23(1/X)'
(p= 0), (34)
~=m o -~3(1/~) 3(1)(1/~)
( p - - 1). '
For ~ = 2 we have ~ 2 = 0.41 m 2 (for p = O) and ~ 2 = 0.74 m 2 (for p = 1). Our result may thus most simply be stated by saying that the consequence of introducing off-diagonal transitions is to replace the vector meson mass by a somewhat smaller effective mass ~ , which sets the scale for the q2 dependence of diffractive vector meson electroproduction. It is clear that in view of our limited knowledge concerning hadronic diffraction dissociation with spin conservation we cannot exactly pre* The formula (33) has the correct normahsatlon and asymptotic q behaviour but can differ from the exact expressions by up to 30% at worst. We would emphasise this simple result (33) as an abstraction of rather general validity derived from our various (and perhaps perforce a little artificial) model calculations. •
.
.
2
H. Fraas et al./Electroproduction
311
dict the value of m. We do predict, however, because o f destructive interference from the higher vector mesons (arising in our calculation from the alternating sign assumption), ~ to be smaller than mp for p0 production (smaller than m~o' ~ for w, q~production). We suggest that accurate data be fitted using expression (33) and varying m. Before comparing our result with the data, let us add a comment at this point on what would have happened, had we not made the alternating sign assumption in (9), but rather had kept the positive sign throughout. We shall briefly deal with this possibility, as it will give us an upper limit for the cross section at large q2. With positive contributions throughout, S(P) in (11) must be replaced by oO
~(p)(x,y)
- ~
1
(35)
.
n =0 (x + n) (y + n)P For p = 0, S(P) diverges logarithmically. For p --- 1 we obtain ~(1)(x ' y) = ~
1
_ qJ(x) - if(y)
n=O (x + n) (y + n)
x - y
(36)
'
and the differentiation employed for S(P) may be used to obtain s(P) for p > 1. Let us restrict ourselves to the case p = 1, and also let us consider the choice (25) for c o to simplify our discussion. The 7P -+ pOp amplitude is then obtained to be
_, o
m2 [ ( 1 + ~ / m 2 ) __
T 7vp P P([M X / ~ Tp°p~p°p "-0 j/ --± 1 ~--, q2) = 7p +-1 )tq2
- ~
(-~)1
(37)
For q2 = 0 we obtain a correction factor of ff(1)(1/X)/X2 which for X = 2 amounts to ~rr 2 ~ 1.23. This seems somewhat large to be accommodated in experimental uncertainties in Opop and 72/41r in contrast to the case of alternating signs where K 1 equalled 0.92. Still, normalised to photoproduction, the amplitude now becomes
_ o
,-r")'vP 1-+1 O PfrAp ~ , , , q 2 ) : T')'p~o°p(w, 0)+-1
~ T ~'p~p°p (IV, O) -+-1 q2
- qJ(1/X)
Ix
- ~(1)(1/X )
].
Xm2
1
F (l+q2/m20~
--
L~¢
q2 ~(1)(1/X ) n
X
[l~q
)-=¢WJ
X
(38)
1 + q21m2 The asymptotic q2 dependence is thus changed to a In (q2/Xm2)/q 2 behaviour for the amplitude compares with 1/q2 from simple p 0 dominance. Numerically the cross section is enhanced relative to p0 dominance.
H. Fraaset aL/Elecrroproduction
312 J.
i
i
i
I
i
i
200 t
i
i
I
1
i
i
[
I
i
l
do ° ~'(YP -'P'P}
]
-
%
100- ~ ' - .
2"2
_
~, "*...
mp
...... 50-
\ x~.'~ %. ........... \ "~.~ .......
- - - -2 1 2 ~q +mp; t 2
"'" ...... ............ o
20-- ~
,o-
,
l
,
.
Positive .......... / .....
•.............
'
.
_
Alternating s i g n s / / " ~
0
0.5
1-0 q2(GeV2)
1.5
Fig. 1. The q2 dependence of the forward (t = 0) differential cross section for rho electroproduction. (Data from the DESY streamer chamber [14], with the SBT [18] photoproduction point.) The dashed curve is the simple rho pole squared;while the continuous curve [from eq. (30)] shows how inclusion of off-diagonal terms (with alternating signs) predicts a faster decrease with q2. (The upper, dotted curve [eq. (38)] illustrates the consequence of entirely positive higher vector Meson contributions to the electroproduction amplitude.) Our predictions (30) and (38) are c o m p a r e d w i t h experimental data [14] for da/dt (?rift P --> pOp) in fig. 1. The data, although rather limited, favour off-diagonal terms with alternating signs. In our specific calculation though, even w i t h p = 0 there is still a t e n d e n c y to overestimate the present experimental results, bearing in mind the small longitudinal c o n t r i b u t i o n has been subtracted f r o m the data points*.
3. Non-forward vector meson production We c o m e n e x t to our expectations for the q2 d e p e n d e n c e o f non-forward/90 production. The generalization o f (9) to t =/=0 is simply given b y * In this paper we have concentrated on the introduction of off-diagonal terms, and hope to come back later to deal explicitly with the longitudinal coupling.
313
H. Fraas et al./Electroproduction
,~'~vP ~-+1
--~ 0 O P Jill t " ' q 2 ' t) = " ~ O
~=1 + Co
m2
2
'r00p~o0P l +-1 (W, t = 0 )
e-lboNrLq2m0 +mg
(mo)(mll2p+le-½(bN-b°)'l
q2 + m~q (--1)N ~NN \m---NN!
(39)
where we have introduce'd the slopes bN for the PoP ~ PNP processes. Clearly, if b N is independent of N, i.e., b N =-bo, the results for the q2 dependence are independent of t, or in other words, the slope in t of p0 electroproduction does not change with q2. If, however, the slopes b N for the next few neighbours of the P0, whose contributions are the most important ones in (39), are significantly different from the elastic PoP ~ PoP slope, then the effect of the off-diagonal terms in (39) will be different for different values of t. This results in a change of the t distribution relative to simple p0 dominance. Clearly, since the effect of the off-diagonal terms increases with q2 we expect the t-distribution for ~/vP -->pOp to change with increasing q2. In particular, we consider the case b N > b 0 for N = 1 , 2 , . . . , since there is empirical evidence that diffraction dissociation into states not far away in mass from the projectile mass does indeed show a larger slope than the elastic reaction. The effect is present [4] in 7rp -+ AlP and is particularly pronounced in pp -+ pN*. Indeed, a recent analysis [8] of diffraction dissociation in pp -+ pX at 12 and 24 GeV shows slopes b as large as ~ 16 to 20 GeV - 2 f o r M 2 < 2 GeV 2 with a rapid drop at larger masses M x to or slightly below the elastic pp slope of b ~- 9 GeV - 2 in this energy range. From this empirical evidence, let us thus conjecture that such an effect is a genuine effect of diffraction dissociation quite independent of the projectile particle and thus also present in PoP ~ P,NP. The qualitative effect of b N > b 0 on the t distribution can be quite easily read off from (39). Indeed, whereas'off-diagonal terms at t = 0 yield a suppression effect relative to (diagonal) p0 dominance, for b N sufficiently large the off-diagonal terms give a vanishingly small contribution and the diagonal term remains as the only important one at larger [tl. We thus expect a flattening of the p0 distribution, with the effective slope decreasing as q2 increases (i.e. the off-diagonal terms have most effect at reducing the cross section at small [tl and large q2). For a quantitative example, guided by what was observed in pp interactions, let us assume a rather strong variation of slope parameterized by
\-~--~-N/
1
(40) .....
,
with b 0 = 8 GeV - 2 from 7rp interactions and X = 2. It would be preferable to motivate our ansatz (40) for the slope change with the mass of the diffractively produced system from rrp interactions, but not as much information seems to be available as
314
H. Fraas et al./Electroproduction
Table 1 Effective slopes beff (GeV_2)
q2 = 0
q2 = rnp2
q2 = 5 m 2
p =0 t2
6.4 7.1
5.6 6.6
1
7.5
7.2
4.2 5.9 6.8
This table gives the effective exponential slope parameter for po electroproduction in the range 0 < - t < 0.2 GeV 2 when the basic diagonal term has a slope b0 = 8 GeV -2 as in ~rp ~ ~rp, and the slopes b N of the off-diagonal interactions are given by expression (40). in t h e p p case*. N u m e r i c a l e v a l u a t i o n o f (39) ( w i t h c o f r o m ( 2 5 ) ) and c o m p u t a t i o n o f the cross s e c t i o n is t h e n straight f o r w a r d . It t u r n s o u t t h a t changes in slope set in r a t h e r slowly as a f u n c t i o n o f q 2 a n d are also restricted t o relatively small values o f Jtl, say, It[ ~ 0.4 G e V 2. A l t h o u g h t h e resulting t d i s t r i b u t i o n is n o t a strict e x p o n e n tial, it m a y b e c o n v e n i e n t l y p a r a m e t e r i z e d b y effective slopes. E x a m p l e s a s a funcI
I
I
I
I
I
I
I
Slope parameter b 12-
v(aev • Cornell spec (preliminary') 5.9 o Cornell sp.chl -,,5 a SLAC sp.ch. *,,10.5
10-
•" - "
B
8-
"Q 6-
4-
2-
O0
0~-2
04
016
d8
110
1!2
1!4
I'.6
1.8
q2 [GeV2) Fig. 2. A compilation, from ref. [6], suggesting that the slope parameter b of the pO electroproduction cross section tends to decrease as q2 increases. * Furthermore, we do not here consider the possibility that diffraction dissociation might in np -* Xp show a variation with energy which would depend upon M x, so then the slopes could also be energy dependent
315
H. Fraas et al./Electroproduction 1000
I
I
i
i
I
I
t
I
I
,
1000
lO0
I00'
10
I0
10
-I
>~ .6
1
0.1
0.'
.0.1
0.0 0
-t
0.5 (GeV 2)
.0.01 l-O
Fig. 3. The t dependence of the differential cross section for rho photoproduction and electroproduction, showing how the inclusion (taking p = 0) of higher vector mesons can flatten the slope - especially for small Itl or large q2. (DESY data points [ 14] with 0.3 < q2 < 0.5 GeV 2 and 0.5 < q2 < 1.5 GeV2; SBT [18] photoproduction data points (open circles) at W = 2.48 GeV.)
tion o f q 2 and p are given in table 1. These slopes m a y be compared w i t h the experim e n t a l l y d e t e r m i n e d slopes given in fig. 2 taken from Talman's talk at the B o n n Conference. The c o n t e n t o f fig. 2 m a y be summarized by saying that there are some
316
H. Fraas et al./Electroproduction
indications for the p0 slope to change with q2 from something like b ~ 6 to 8 GeV -2 in photoproduction to b ~- 4 to 6 GeV -2 in electroproduction at q2 ~ 1.5 GeV 2. Although we consider our calculations as exploratory ones, in the sense of what one might reasonably expect in p0 electroproduction from our knowledge of hadron hadron interactions, it is satisfying that the effect predicted has roughly the magnitude observed experimentally. In fig. 3 we directly compare our predictions with the DESY streamer chamber data, which are reasonably well reproduced. Let us at this point add a general remark on slope changes. A flattening of slope with increasing q2 had first been conjectured by Cheng and Wu [15] in analogy to results from their QED calculations. Light-cone arguments have been added [16] and photon shrinkage with q2 seemed to appear as a rather universal property of spacelike photons. Such a universal property, once verified in vector meson production, would then allow one to draw far-reaching conclusions for other photon induced processes, as, e.g. n 0 electroproduction [17]. We think that our interpretation of a possible slope change by linking it with properties of hadron diffraction dissociation is less universal. Clearly, whether similar "radius type" effects also appear in different reactions, if once verified accurately in vector meson electroproduction, is in our interpretation very much dependent upon the hadron dynamics of the specific reaction under consideration.
4. Summarizing and concluding remarks We have attempted in this paper to explore the effect of off-diagonal transitions on p0 (~, ~) electroproduction in the large co' diffraction region. Our ansatz for the relevant pp ~ PNP diffraction dissociation amplitudes has been motivated by our empirical knowledge of diffraction dissociation in 7rp and pp interactions; and we have tried to keep the ansatz as general as possible. The ansatz for the photon vector meson coupling has been motivated from e+e - annihilation. For t = 0, under rather general conditions as regards the hadron physics input, we obtained the result, abstracted from detailed calculations, that the effect of off-diagonal transitions may approximately be described by replacing the vector meson mass appearing in the simple vector dominance propagator by a somewhat smaller one, m < mp (m~,~). The simple pole in q2 dependence of p0(w, 4) dominance (apart from the replacement of rap(w, ~b) by m) may thus be treated with increased confidence, even at very large q2 (keeping w' large). (A possible exception to this simple result has been excluded, as it would, e.g., require a PoP ~ PIP amplitude as large as the elastic PoP ~ PoP one.) The numerical value of ~h depends on the precise magnitude of the spin conserving hadronic diffraction dissociation amplitudes which are not at present experimentally known. We have thus suggested that electro-production data be fitted by varying r~ (rather than by varying the power law for the q2 dependence). As regards the slope in electroproduction of vector mesons, we conjecture that possible slope changes with q2 are related to differences in slope between elastic and
H. Fraas et al./Electroproduction
317
d i f f r a c t i o n d i s s o c i a t i o n processes as o b s e r v e d in 7rp a n d p p i n t e r a c t i o n s . More specifically, w i t h t h e a l t e r n a t i n g sign a s s u m p t i o n for t h e p h o t o n v e c t o r m e s o n couplings m o t i v a t e d f r o m o u r t r e a t m e n t o f d e e p inelastic ep scattering, and t h e s t e e p e r slopes o f d i f f r a c t i o n d i s s o c i a t i o n c o m p a r e d w i t h elastic scattering in lrp a n d pp, we obt a i n e d a f l a t t e n i n g o f slope w i t h increasing q 2 in e l e c t r o p r o d u c t i o n . I n d e e d , experim e n t i n d i c a t e s s u c h a n effect. O n e o f us (D.S.) wishes t o t h a n k Professor A. D o n n a c h i e a n d Dr. G. Shaw for a visit to D a r e s b u r y w h i l e ' t h i s w o r k was c o m p l e t e d .
References [ 1 ] G. Wolf, Proc. 1971 Int. Symp. on electron and photon interactions at high energies, Cornell, Ithaca, N.Y., ed. N.B. Mistry, p. 189. [2] K. Moffeit, Proc. 1973 Int. Symp. on electron and photon interactions at high energies, Bonn, ed. H. Rollnik and W. Pfeil (North-Holland, Amsterdam, 1974) p. 313; SLAC report SLAC-PUB-1314 (1973). [3] K. Gottfried, Proc. 1971 Int. Symp. on electron and photon interactions at high energies, Cornell, Ithaca, N.Y., ed. N.B. Mistry, p. 221. [4] L. Fo~i, Int. Conf. on elementary particles, Aix-en-Provence, 1973 J. de Phys. Suppl. 34 (1973) 317. [5] V. Silvestrini, Proc. 16th Int. Conf. on high-energy physics, Chicago, ed. J.D. Jackson and A. Roberts, 4 (1972) 1; K. Strauch, Proc. 1973 Int. Syrup. on electron and photon interactions at high energies, Bonn, ed. H. Rollnik and W. Pfeil (North-Holland, Amsterdam, 1974) p. 1. [6] R. Talman, Proc. Bonn Conf., p. 145; Cornell report CLNS-249 (1973). [7] H. Fraas, B.J. Read and D. Schildknecht, Nucl. Phys. B86 (1975) 346. [8] J. Benecke et al., Max-Planck-Institut report MPI-PAE/Exp. El. 38 (1974). [9] A. Bramon, E. Etim and M. Greco, Phys. Letters 41B (1972) 609; M. Greco, Nucl. Phys. B63 (1973) 398. [10] B. Richter, Invited talk at APS Chicago meeting, February, 1974, see CERN Courier 14 (1974) 41. [11] M. BShm, H. Joos and M. Krammer, Acta Phys. Austriaca 38 (1973) 123. [12] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York 1965) p. 267. [13] H. Fraas and D. Schildknecht, Nucl. Phys. B14 (1969) 543. [14] V. Eckardt et al., DESY report 74/5 (1974). [15] H. Cheng and T.T. Wu, Phys. Rev. Letters 22 (1969) 1409; Phys. Rev. 183 (1969) 1324. [16] J. Bjorken, Proc. Cornell Conf., p. 281. [17] H. Harari, Phys. Rev. Letters 27 (1971) 1028. [18] J. Ballam et al., Phys. Rev. D5 (1972) 545.
ON THE DIFFERENTIAL CROSS SECTION AND THE SLOPE OF VECTOR MESON ELECTROPRODUCTION H. FRAAS Physikalisches lnstitut der Universitiit, Wi~rzburg, Germany B.J. READ Daresbury Laboratory, Warrington, England D. SCHILDKNECHT Deutsches Elektronen-Synchrotron, DESY, Hamburg, Germany Received 31 October 1974 The effect of higher mass vector meson contributions (e.g. transitions p'p ~ pp, p"p -~ pp etc.) on diffractive vector meson electroproduction is investigated. The ansatz for these off-diagonal transitions is motivated from diffractive dissociation results in 7rp and pp interactions_and our recent treatment of the total electroproduction cross section with inclusion of off-diagonal terms. For t = 0 and large q2 (co' > 10) we find, comparing with simple pO dominance, a change in normalization, but the same power of q2 dependence. From results on slopes in diffraction dissociation in ~rp and pp interactions we predict some flattening of the p0 electroproduction slope with increasing q2 in the region of small -t.
1. Introduction The production of vector mesons by photons is quite successfully described [1,2] by vector meson dominance: the photon is viewed as virtually dissociating into the vector meson V, which then scatters off the nucleon. Vector meson production by photons is thus closely related to elastic vector meson nucleon scattering. Off-diagonal transitions, in which the photon virtually converts into a more massive spin 1 state, say V', and subsequently produces the vector meson V via V'p ~ Vp, are usually neglected. They are assumed to be suppressed due to the smaller coupling of the photon to higher mass states, and in addition by the smaller hadronic amplitudes for diffraction dissociation V'p ~ Vp (V' :/: V) compared with the ones for elastic processes Vp ~ Vp. Indeed, the agreement with experiment [1 ] of the relations between p0 (oJ, 6) photoproduction at t = 0 and the corresponding vector meson nucleon total cross sections (as experimentally determined from vector meson production from complex nuclei [3]) seems to justify the neglect of such terms in photoproduction.
302
H. Fraas et aL /Elektroproduction
Nevertheless, there are at least three motivations for studying such off-diagonal transitions more carefully. Firstly, a more reliable, quantitative estimate o f the magnitude of off-diagonal transitions now becomes possible by exploiting the increased knowledge of diffraction dissociation in hadron hadron interactions [4] and the coupling of the photon in e+e - annihilation to more massive hadron states [5]. Secondly, although off-diagonal transitions in p0, co, ~ photonproduction seem to have but little effect at q2 = 0, their relative importance increases strongly with growing space-like values o f q 2 due to the propagator factor 1/(q2/m 2 + 1) in the amplitude. This is of consequence for the q2 region explored in electroproduction [2,6] at present and in the near future. Finally, there is a third and in our opinion the most important reason to carefully analyse off-diagonal transitions in vector meson electroproduction. It is the importance which these terms apparently have in contributing to the virtual forward Compton amplitude. In fact, as we have recently shown [7], it is exactly these off-diagonal contributions to the Compton amplitude which allow one to derive within the generalized vector dominance framework a convergent expression and scaling behaviour of the transverse part of the proton structure function vW 2 starting from a 1Is behaviour of the e+e - annihilation cross section. Vector meson electroproduction provides an additional test of the general features of the model recently proposed [7]. In the present paper we shall consider electroproduction of the low-lying and best known vector mesons pO, c.o, q~only. For these low-lying states we can best hope to use our empirical knowledge of diffraction dissociation [4] in 7rp and pp interactions to infer the correct ansatz for the necessary off-diagonal Vp -+ VNP (or rather VNP ~ Vp) transition amplitudes. Here V stands for p0 (co, ~) and V N for some more massive vector state. Thus from a reasonable Ansatz for these diffraction dissociation amplitudes and with vector photon couplings as suggested from e+e - annihilation, we shall obtain essentially two results in this paper: (i) The first result concerns forward (or rather t = 0) production of vector mesons by transverse virtual photons. We shall show that under rather general conditions, to be specified below, we expect the amplitude for vector meson production in the diffractive region of w' = 1 + W2/q 2 ~ 10 to fall off with q2 for q2 _+ ,~ as a simple pole in q2 just as in p0 dominance. Quite simply, the influence of the off-diagonal transitions may be expressed by the (approximately valid) interpolation formula
(co'> lO) ~-/(w,da q2, t = 0 ) = ( ~ 2 ~2 Iqreall dO (w, q 2 = O , t = O ) ~q2 + ~/2] Iqvirtl dt ~4 Iqreall do (W, q2 = 0 t = O) q4 Iqvirtl dt ' '
(1)
where e.g. for p0 production ~ is expected to be smaller than the p0 mass mp, the exact value depending upon the magnitude of the diffraction dissociation amplitudes. Relation (1) with ~ < mp then implies a suppression of the cross section relative to
H. Fraas et al./Elektroproduction
303
p0 dominance. As will be pointed out, this suppression effect comes about through the alternating sign assumption necessary for convergence o f the virtual Compton forward amplitude. If all off-diagonal transitions were to contribute with positive sign, ~ > mp would be obtained. (ii) Our second result concerns the non-forward (t 4: 0) production o f vector mesons b y virtual, space-like photons. To predict the q2 dependence in this region o f t, we need information on the slope of diffractive processes, e.g. PoP ~ PNP (thinking o f a series of discrete states PN, N = 0, 1 , 2 , . . . . . ). F r o m np and pp interactions it appears [4,8] that diffraction dissociation into low-lying states such as pp ~ pN*(1470) or rrp -~ A l P shows a somewhat steeper slope than elastic processes. Assuming this effect to be also present in PoP ~ PNP, we are led, in electroproduction o f vector mesons p0, w, ~b, to expect within our off-diagonal framework a slight flattening of slope with increasing q2.
2. Model assumptions: q2 dependence o f f o r w a r d (t = O) vector meson production Let us start b y stating our main assumptions. As in [7] we shall work with a Veneziano type mass spectrum for the vector mesons 2 _- m 02 (1 + XN) , rnN
N=0,
1,2, . . . . .
,
(2)
where m 0 is equal to the p 0 (co, 4) mass. The choice k = 2 together with the vector meson photon coupling [9] 1 m2
1 -
7N 2
(3)
72m 2
for the N t h vector state leads to a total e+e - annihilation cross section o(e+e - -~ hadrons) = (47r2/k72) o(e+e - ~/1+/~-). For k = 2 this is roughly 2.5 times that of pair production, quite consistent with experiment [5] up to an e+e - centre-of-mass energy of about 3.5 GeV. Due to suppression of still higher mass contributions by their photon coupling and b y the small hadronic diffraction dissociation into very high mass states, the recent deviations from a 1/s law indicated b y CEA [5] and SPEAR [10] data are not expected to change our results for the modestly large values of q2 to be explored in electroproduction in the foreseeable future. In addition to the vector meson photon couplings we have to specify the mass dependence o f the hadronic diffraction dissociation amplitude VoP -> VNP. In contrast to our treatment [7] o f the total virtual photoproduction cross section, where mainly for simplicity of presentation (effective) transitions to next neighbours only were included, we shall in this paper actually consider contributions due to the complete diffraction dissociation mass spectrum as observed [4] in 7rp and pp interactions. Experimentally, as a function o f the mass m o f the system X diffractively pro-
304
H. Fraas et al./Elektroproduction
duced, one finds for these reactions a dependence for the cross section o f roughly [4] do 1 -(4) dm 2 m 2 For rrp interactions such a behaviour seems to be valid [4] down to rather small values o f m 2. A difficulty for inferring from (4) the correct diffraction dissociation law for V0P ~ VNP arises from the fact that the system X diffractively produced in rrp ~ pX contains in general spins quite different from the spin of the initial particle, the contribution of the different spins actually not being too well known. If, e.g., the number o f different spin states available for population were to increase linearly with m 2, then assuming equal population, (4) should just be modified by an additional power of 1/m 2. Motivated by such arguments we shall for diffraction dissociation with spin conservation replace (4) by the more general power law (for PoP ~ PNP transitions) TooP ~ p N p = TPoP -*pip ( m l 12p+ 1 Introducing the ratio c o of the first off-diagonal amplitude PoP -> PIP to the elastic one PoP ~ PoP, CO = TOoP ~ P lP/TPOP ~ PoP,
(6)
we may rewrite (5) as To°P-~PNP = CO T P O p ~ o o p (\ ~mNl t]2 p + l
(7)
This power law is assumed to be valid in the forward direction (t = 0), the exponent p assumed to be p > 0, with the limiting case p = 0 corresponding to the observed behaviour (4)*. It is likely that from further theoretical analysis and further data on diffractive production more will be learned on how well justified our simple ansatz (7) for the spin conserving part of diffraction dissociation is. Let us also note that the exact numerical value o f p in (7) is of importance for our conclusions only in so far as the detailed numerical results to be presented are concerned, but certainly not as regards the general conclusions to be drawn on the q2 dependence o f vector meson electroproduction. The relative normalization c O o f first diffraction dissociation transition to elastic one is known [4] to be substantially smaller thanl, the exact value again being of relevance for the detailed numerical predictions only. As a check, from our ansatz (7) let us also estimate the total diffraction dissociation cross section for production o f a spin 1 state, i.e. P0 + P " P + E~= 1 ON. With* We expect 0 < p < 2, say. For larger values ofp the off-diagonal terms are tiny and have negligible effect.
H. Fraaset al./Elektroproduction
305
Out taking into account the necessary energy cut-off and also any dependence of the slope o f diffraction dissociation on the mass of the produced system, one obtains by straightforward summation O0oP-* P~2ON = C2 O0oP~PoP
) 2p+ 1 ~(2p) ( ~ ) ( - 1 ) 2p+ 1( 1 (2p)! ~ +1 -- + 1 ,
(8)
where ~(2P)(x) is the 2pth derivative of the diagamma function ~(x). Numerically, 2 21 = ½, one then obtains e.g., for p = ~, h = 2 and c 0 = mo/m Ooop ~
pzpN ~--0.23 Ooop _ poP '
i.e., a small fraction of the elastic cross section, a result which is reasonable when compared with our experimental knowledge [4]. From the above assumptions (2), (3) and (7) we now have the following ansatz for the amplitude of O0 production (an analogous one holding for ~ and ~b) by transverse virtual photons and protons, T+I
")'vP---~P
-
0
P [w ,2 t - m - Vt~'r p0 P--+P0 Pt'[ap ~",~t , ' - " J - - - ~ - o
m2
X[_q2+m------~o+COk 0
~
~
m2
*+1
~",t =0)
. m 0 [m 1~2p+1],
q2+m2(--1)~Vm----Ntm---~)
(9)
0
where T_+°lp o p (W, t = 0) is the transverse (helicity -+ 1) helicity conserving pOp scattering amplitude. The ( - 1 ) N factor is the alternating sign assumption mainly motivated* from the requirement of convergence of the virtual forward Compton amplitude. (For completeness, we shall later on also briefly discuss the result obtained from having positive couplings throughout, although we do not consider this as a realistic alternative in view of the negative contributions required for the Compton amplitude. Nevertheless, positive signs will provide us with an upper limit for the amplitude (9).) The infinite sum over N in (9) should actually be cut off at a value N, roughly corresponding to the available c.m. energy W, as ansatz (7) breaks down for large vector meson masses, i.e. mN.~ W. In practice, however introducing such a cut off in (9) does not to a good approximation change the result obtained compared with the infinite sum over N to be evaluated subsequently, provided one stays in the diffraction region of co' large, i.e. w' ~ 10 say. With this restriction the large q2 behaviour in the ensuing discussion should be understood as the limit q2 ~ oo with co' fixed (co' ~ 10). For integer values of p the sum in (9) may be easily evaluated, as we shall now show. First let us rewrite (9) as * See ref. [7]. Negative signs seem also to be required for the nucleon form factors. Alternating signs, and relation (3) have in addition been obtained in quark model calculations [11 ].
H. Fraaset aL/Electroproduction
306
2 ~'rvP ~ pOp l . . . . 2a N/c~ "v'°°p ~ O°P[" mo 1+-1 tvv'q ' =-"~-'0 "-+1 [.q2 + m02
(mli2p+l -- CO \ - ~ 0 ]
1 S(p)( l+q2/m2
+1,1+1)1 '
~tP + 1
(lO)
where by definition oo
S(P)(x,y) = ~
(-1)n n=O (x +n) (y +n)P
For integer have
(11)
p, S(P)(x,y) reduces to known functions.
In fact, for p = 0 we simply
s(0)(x, y ) - s(O)(x) = ~ ( x ) ,
(12)
/3(x) = ~ [4 (½(x + 1)) - ~(½x)l ,
(13)
with
where ~k(x) = (d/dx) In F(x) is the digamma function tabulated in [12]. is easily obtained to be S(1)(x, y) = ~ and
(14)
[~(x) -/3(y)l ,
S(P)(x,y) for p >
S(1)(x,y)
1 is related to S(1)(x, y) by differentiation
(-1)p-1 dp-1
S(P)(x'Y) = -~- ~ ~yp--1 S(1)(x'Y)'
p = 2, 3 . . . .
(15)
Thus explicitly
p(-1)p~tp .fl ( 1 + S(P)( 1 + q2/mgx + 1,X+I 1) = (q2/m2) +(-1)°-1
p-1
1
xk+l
k=0 ( P - 1 - - k ) ! , 2, 2,k+1
q2x/m2
I-1)
~(p_l_k)/l+l\.(_~)
(16)
tq /m O)
This relation holds for p = 0, 1, 2 , . . . where for p = 0 the second term on the righthand side should ignored. In (16), (3(r)(x)denotes the rth derivative of the function
H. Fraaset al./Electroproduction
307
/3(x). Let us also note the asymptotic behaviour of S(P) in (16) for q2 -+ oo. Using the asymptotic expansion of the function/3(x),
1+
(17)
3(x) ~ 2x
we obtain for q2 ~ oo (1 + q2/m2 S (°)
+1,1+
1
~
(18)
- + 0
2q 2
and S(p)( 1 +q2/m2
1 1) +1,-~+
+O( 1 ~ , \q4!
Xm2~ --p q2 U[--l) ~
1 /3(p_ 1)
(1+1)
(p= 1,2 . . . . . ).
(19)
The sum of poles S(P) appearing in (9), (10) thus sums up to a simple pole for q2 ~oo. From (17) and (18) the asymptotic (q2 -+ oo) behaviour of the p0 production amplitude is now easily obtained to be 2 "+1 -
~ 3, 0
~+1 -
"--:",1 q \
- - CO o--Z:-"
+
,
(p=0),
(20)
and 2 ,r3,vp~O°p %/~ 7'+01P o ~ OoP mO ~+-1 ~ ~ 70 q2
X I l _ c o ( m l ] 2p+1 (-1)P -1 \m0/
(p= 1 , 2 , . . . . ) .
~
i~! t3(p-
1)(~+ 1)1
+
0(~4)
,
(21)
Thus even with inclusion of off-diagonal transitions subject to the power law (5) the P0 production amplitude asymptotically (i.e. q2 ~ 0% keeping co' large) behaves as a simple pole in q2. (The only exception to this general rule would occur when
H. Fraaset al./Electroproduction
308
2mo=. 2 ml X / ] - ~ '
(p=0) ; (22)
C O ='
. 1 (--1)P-1 •P(p-- 1)! (1 +X)P+~ ~(P-1)(1/X+I) '
(p=l,2 ..... ).
From (22), for ?~= 2 we would obtain c o = 1.15 for p = 0, and c o = 0.9 for p = 1, and thus a diffractive forward amplitude of roughly the magnitude of the elastic one Tpop_~poP' whereas experimentally the diffraction dissociation cross section into a particular state close in mass to the incoming one seems strongly suppressed [4] compared with the eleastic cross section. Thus we may exclude the possiblility (22), and from (20), (21) obtain quite generally a pole behaviour for large q2, just as in simple p0 dominance.) From (20) and (21) there is a change in normalization, however, compared with simple p0 dominance, which we wish to discuss next. Let us first consider the normalization of the amplitude (10) at q2 = 0. From (11) one obtains
lim0 S(P)
q2
+
~k
1,~
+1 =
.
~(P)
+1
,
(23)
and thus by substitution into (10) ( m l ]2p+1
_
T+'rt~-~p0p (W, q 2 : 0 ) - N / ~ T+O~p~p°p I1 _ Co
70
[ml]2p+l + c0 ~'~00)
\mo/
(--1)P /3(p)(1)] Xp+ lp!
= v/-~ T°+~P--"P°PKp(X) .
(24)
In simple p0 dominance, the corrrection factor Kp on the right-hand side of (24) would be replaced by 1, the deviation from 1 thus being due to off-diagonal transitions. As a numerical example, let us simply substitute
(mO]2p+l_{ 1.~2p+l=[O.58(p=O), CO=\ml/ -~TJ [0.19 (p 1), into (24), which values give reasonable suppression of diffractive dissociation " PoP -+ PlP for 0 ~
(25)
H. Fraaset aL/Electroproduction -~t3
,
309
p--0; (26)
G=
which numerically for X = 2 amounts to K 0 = 0.79 (p = 0) and K 1 = 0.92 (p = 1). Although the pure p0 dominance prediction (neglecting the small real part)
do~O-*P°P(w, q2=O,t=O)=~_~ d'-t
1
3,0 167r [aoop] 2
(27)
is satisfied by the data [1,3], systematic and statistical errors in the determination of the O0 photoproduction and the total pOp cross sections and in 3,2/41r are large enough to allow for a 10% to 20% variation required from off-diagonal contributions. As we are mainly interested in the q2 dependence o f o 0 production, which is more directly accessible to experiment than (27), let us next express O0 production by virtual photons in terms of the photoproduction amplitude (24). From (10) and (24) we then obtain
T[[p-~o°p (W, q2, t = 0 ) [ml]2p+l + c0 t--~00)
= 1--~-F m2 ( 1 [ml~2p+l~ KpLq 2 + mg c0k-~00) ]
__S(p)(l+Xq2/m'gl)lT~P-'P°P(W'q2=O't=O)' 1
xP+l
(2S) where we have used
S(P) (x + 1, y + 1) =
xyP
- S(P) (x, y ) ,
(29)
which directly follows from the definition (11). Finally, let us give the cross section following from (28), for simplicity restricting ourselves to the realistic cases p = 0 and p = 1 and to the reasonable choice (25) for c0, which we shall consider in our numerical evaluation. The t = 0 differential cross section for electroproduction doO/dt for p = 0 is then given by (co' ~ 10) do 0 "W q2) =IJ 3((1 + q2/m~)/X)~ 2 ~ [qreall d°0"W q2 0) dt ( ' l_ ~ -] Iqvirtl-d~ ( ' =
A m
•2 Iqreallda 0 q4 4/32(1/X)Iqvirtl dt ( W ' q 2 = 0 ) '
(30)
310
H. Fraas et al./Electroproduction
and for
p = 1
we obtain
d°O~-f(,'W q 2 ) = ~ -4 ~.2f3((1
+ q21m2)lX) - 3(1/X)]2 Iqr~l doO
~iS(l/~)
_ Iqv~l dt
(W, q 2 = 0 ) .
(31) The flux factor !qreal[ _
W2 - m2
(32) 1
Iqvirtl
[(W2 _ m 2 _ q2)2 + 4W2q2]
is well-known [13] and arises from the definition of the virtual photon absorption cross section. Numerically, (30) and (31) may be evaluated from the definition (13) and the tables [12] for the diagram function ~ (x). As mentioned, the asymptotic dependence of (30) and (31) is the same power o f q 2 as obtained from simple p 0 dominance. There is an asymptotic suppression factor for the normalisation, however, which for X = 2 from 3(~) = 1.57 and -3(1)(½) = 3.66 one calculates to be 0.41 (for p = 0) and 0.74 (for p = 1) respectively, relative to simple p0 dominance. The results (30) and (31) may be described approximately* by the much simpler interpolation formula (co' > 10) d°0(W ' q2) - ( ~n 2 dt - \ ~ ]
~2 Iqreall do 0 "W Iqvirtl ~- ( ' q2 = 0 ) ,
(33)
with 23(1/X)'
(p= 0), (34)
~=m o -~3(1/~) 3(1)(1/~)
( p - - 1). '
For ~ = 2 we have ~ 2 = 0.41 m 2 (for p = O) and ~ 2 = 0.74 m 2 (for p = 1). Our result may thus most simply be stated by saying that the consequence of introducing off-diagonal transitions is to replace the vector meson mass by a somewhat smaller effective mass ~ , which sets the scale for the q2 dependence of diffractive vector meson electroproduction. It is clear that in view of our limited knowledge concerning hadronic diffraction dissociation with spin conservation we cannot exactly pre* The formula (33) has the correct normahsatlon and asymptotic q behaviour but can differ from the exact expressions by up to 30% at worst. We would emphasise this simple result (33) as an abstraction of rather general validity derived from our various (and perhaps perforce a little artificial) model calculations. •
.
.
2
H. Fraas et al./Electroproduction
311
dict the value of m. We do predict, however, because o f destructive interference from the higher vector mesons (arising in our calculation from the alternating sign assumption), ~ to be smaller than mp for p0 production (smaller than m~o' ~ for w, q~production). We suggest that accurate data be fitted using expression (33) and varying m. Before comparing our result with the data, let us add a comment at this point on what would have happened, had we not made the alternating sign assumption in (9), but rather had kept the positive sign throughout. We shall briefly deal with this possibility, as it will give us an upper limit for the cross section at large q2. With positive contributions throughout, S(P) in (11) must be replaced by oO
~(p)(x,y)
- ~
1
(35)
.
n =0 (x + n) (y + n)P For p = 0, S(P) diverges logarithmically. For p --- 1 we obtain ~(1)(x ' y) = ~
1
_ qJ(x) - if(y)
n=O (x + n) (y + n)
x - y
(36)
'
and the differentiation employed for S(P) may be used to obtain s(P) for p > 1. Let us restrict ourselves to the case p = 1, and also let us consider the choice (25) for c o to simplify our discussion. The 7P -+ pOp amplitude is then obtained to be
_, o
m2 [ ( 1 + ~ / m 2 ) __
T 7vp P P([M X / ~ Tp°p~p°p "-0 j/ --± 1 ~--, q2) = 7p +-1 )tq2
- ~
(-~)1
(37)
For q2 = 0 we obtain a correction factor of ff(1)(1/X)/X2 which for X = 2 amounts to ~rr 2 ~ 1.23. This seems somewhat large to be accommodated in experimental uncertainties in Opop and 72/41r in contrast to the case of alternating signs where K 1 equalled 0.92. Still, normalised to photoproduction, the amplitude now becomes
_ o
,-r")'vP 1-+1 O PfrAp ~ , , , q 2 ) : T')'p~o°p(w, 0)+-1
~ T ~'p~p°p (IV, O) -+-1 q2
- qJ(1/X)
Ix
- ~(1)(1/X )
].
Xm2
1
F (l+q2/m20~
--
L~¢
q2 ~(1)(1/X ) n
X
[l~q
)-=¢WJ
X
(38)
1 + q21m2 The asymptotic q2 dependence is thus changed to a In (q2/Xm2)/q 2 behaviour for the amplitude compares with 1/q2 from simple p 0 dominance. Numerically the cross section is enhanced relative to p0 dominance.
H. Fraaset aL/Elecrroproduction
312 J.
i
i
i
I
i
i
200 t
i
i
I
1
i
i
[
I
i
l
do ° ~'(YP -'P'P}
]
-
%
100- ~ ' - .
2"2
_
~, "*...
mp
...... 50-
\ x~.'~ %. ........... \ "~.~ .......
- - - -2 1 2 ~q +mp; t 2
"'" ...... ............ o
20-- ~
,o-
,
l
,
.
Positive .......... / .....
•.............
'
.
_
Alternating s i g n s / / " ~
0
0.5
1-0 q2(GeV2)
1.5
Fig. 1. The q2 dependence of the forward (t = 0) differential cross section for rho electroproduction. (Data from the DESY streamer chamber [14], with the SBT [18] photoproduction point.) The dashed curve is the simple rho pole squared;while the continuous curve [from eq. (30)] shows how inclusion of off-diagonal terms (with alternating signs) predicts a faster decrease with q2. (The upper, dotted curve [eq. (38)] illustrates the consequence of entirely positive higher vector Meson contributions to the electroproduction amplitude.) Our predictions (30) and (38) are c o m p a r e d w i t h experimental data [14] for da/dt (?rift P --> pOp) in fig. 1. The data, although rather limited, favour off-diagonal terms with alternating signs. In our specific calculation though, even w i t h p = 0 there is still a t e n d e n c y to overestimate the present experimental results, bearing in mind the small longitudinal c o n t r i b u t i o n has been subtracted f r o m the data points*.
3. Non-forward vector meson production We c o m e n e x t to our expectations for the q2 d e p e n d e n c e o f non-forward/90 production. The generalization o f (9) to t =/=0 is simply given b y * In this paper we have concentrated on the introduction of off-diagonal terms, and hope to come back later to deal explicitly with the longitudinal coupling.
313
H. Fraas et al./Electroproduction
,~'~vP ~-+1
--~ 0 O P Jill t " ' q 2 ' t) = " ~ O
~=1 + Co
m2
2
'r00p~o0P l +-1 (W, t = 0 )
e-lboNrLq2m0 +mg
(mo)(mll2p+le-½(bN-b°)'l
q2 + m~q (--1)N ~NN \m---NN!
(39)
where we have introduce'd the slopes bN for the PoP ~ PNP processes. Clearly, if b N is independent of N, i.e., b N =-bo, the results for the q2 dependence are independent of t, or in other words, the slope in t of p0 electroproduction does not change with q2. If, however, the slopes b N for the next few neighbours of the P0, whose contributions are the most important ones in (39), are significantly different from the elastic PoP ~ PoP slope, then the effect of the off-diagonal terms in (39) will be different for different values of t. This results in a change of the t distribution relative to simple p0 dominance. Clearly, since the effect of the off-diagonal terms increases with q2 we expect the t-distribution for ~/vP -->pOp to change with increasing q2. In particular, we consider the case b N > b 0 for N = 1 , 2 , . . . , since there is empirical evidence that diffraction dissociation into states not far away in mass from the projectile mass does indeed show a larger slope than the elastic reaction. The effect is present [4] in 7rp -+ AlP and is particularly pronounced in pp -+ pN*. Indeed, a recent analysis [8] of diffraction dissociation in pp -+ pX at 12 and 24 GeV shows slopes b as large as ~ 16 to 20 GeV - 2 f o r M 2 < 2 GeV 2 with a rapid drop at larger masses M x to or slightly below the elastic pp slope of b ~- 9 GeV - 2 in this energy range. From this empirical evidence, let us thus conjecture that such an effect is a genuine effect of diffraction dissociation quite independent of the projectile particle and thus also present in PoP ~ P,NP. The qualitative effect of b N > b 0 on the t distribution can be quite easily read off from (39). Indeed, whereas'off-diagonal terms at t = 0 yield a suppression effect relative to (diagonal) p0 dominance, for b N sufficiently large the off-diagonal terms give a vanishingly small contribution and the diagonal term remains as the only important one at larger [tl. We thus expect a flattening of the p0 distribution, with the effective slope decreasing as q2 increases (i.e. the off-diagonal terms have most effect at reducing the cross section at small [tl and large q2). For a quantitative example, guided by what was observed in pp interactions, let us assume a rather strong variation of slope parameterized by
\-~--~-N/
1
(40) .....
,
with b 0 = 8 GeV - 2 from 7rp interactions and X = 2. It would be preferable to motivate our ansatz (40) for the slope change with the mass of the diffractively produced system from rrp interactions, but not as much information seems to be available as
314
H. Fraas et al./Electroproduction
Table 1 Effective slopes beff (GeV_2)
q2 = 0
q2 = rnp2
q2 = 5 m 2
p =0 t2
6.4 7.1
5.6 6.6
1
7.5
7.2
4.2 5.9 6.8
This table gives the effective exponential slope parameter for po electroproduction in the range 0 < - t < 0.2 GeV 2 when the basic diagonal term has a slope b0 = 8 GeV -2 as in ~rp ~ ~rp, and the slopes b N of the off-diagonal interactions are given by expression (40). in t h e p p case*. N u m e r i c a l e v a l u a t i o n o f (39) ( w i t h c o f r o m ( 2 5 ) ) and c o m p u t a t i o n o f the cross s e c t i o n is t h e n straight f o r w a r d . It t u r n s o u t t h a t changes in slope set in r a t h e r slowly as a f u n c t i o n o f q 2 a n d are also restricted t o relatively small values o f Jtl, say, It[ ~ 0.4 G e V 2. A l t h o u g h t h e resulting t d i s t r i b u t i o n is n o t a strict e x p o n e n tial, it m a y b e c o n v e n i e n t l y p a r a m e t e r i z e d b y effective slopes. E x a m p l e s a s a funcI
I
I
I
I
I
I
I
Slope parameter b 12-
v(aev • Cornell spec (preliminary') 5.9 o Cornell sp.chl -,,5 a SLAC sp.ch. *,,10.5
10-
•" - "
B
8-
"Q 6-
4-
2-
O0
0~-2
04
016
d8
110
1!2
1!4
I'.6
1.8
q2 [GeV2) Fig. 2. A compilation, from ref. [6], suggesting that the slope parameter b of the pO electroproduction cross section tends to decrease as q2 increases. * Furthermore, we do not here consider the possibility that diffraction dissociation might in np -* Xp show a variation with energy which would depend upon M x, so then the slopes could also be energy dependent
315
H. Fraas et al./Electroproduction 1000
I
I
i
i
I
I
t
I
I
,
1000
lO0
I00'
10
I0
10
-I
>~ .6
1
0.1
0.'
.0.1
0.0 0
-t
0.5 (GeV 2)
.0.01 l-O
Fig. 3. The t dependence of the differential cross section for rho photoproduction and electroproduction, showing how the inclusion (taking p = 0) of higher vector mesons can flatten the slope - especially for small Itl or large q2. (DESY data points [ 14] with 0.3 < q2 < 0.5 GeV 2 and 0.5 < q2 < 1.5 GeV2; SBT [18] photoproduction data points (open circles) at W = 2.48 GeV.)
tion o f q 2 and p are given in table 1. These slopes m a y be compared w i t h the experim e n t a l l y d e t e r m i n e d slopes given in fig. 2 taken from Talman's talk at the B o n n Conference. The c o n t e n t o f fig. 2 m a y be summarized by saying that there are some
316
H. Fraas et al./Electroproduction
indications for the p0 slope to change with q2 from something like b ~ 6 to 8 GeV -2 in photoproduction to b ~- 4 to 6 GeV -2 in electroproduction at q2 ~ 1.5 GeV 2. Although we consider our calculations as exploratory ones, in the sense of what one might reasonably expect in p0 electroproduction from our knowledge of hadron hadron interactions, it is satisfying that the effect predicted has roughly the magnitude observed experimentally. In fig. 3 we directly compare our predictions with the DESY streamer chamber data, which are reasonably well reproduced. Let us at this point add a general remark on slope changes. A flattening of slope with increasing q2 had first been conjectured by Cheng and Wu [15] in analogy to results from their QED calculations. Light-cone arguments have been added [16] and photon shrinkage with q2 seemed to appear as a rather universal property of spacelike photons. Such a universal property, once verified in vector meson production, would then allow one to draw far-reaching conclusions for other photon induced processes, as, e.g. n 0 electroproduction [17]. We think that our interpretation of a possible slope change by linking it with properties of hadron diffraction dissociation is less universal. Clearly, whether similar "radius type" effects also appear in different reactions, if once verified accurately in vector meson electroproduction, is in our interpretation very much dependent upon the hadron dynamics of the specific reaction under consideration.
4. Summarizing and concluding remarks We have attempted in this paper to explore the effect of off-diagonal transitions on p0 (~, ~) electroproduction in the large co' diffraction region. Our ansatz for the relevant pp ~ PNP diffraction dissociation amplitudes has been motivated by our empirical knowledge of diffraction dissociation in 7rp and pp interactions; and we have tried to keep the ansatz as general as possible. The ansatz for the photon vector meson coupling has been motivated from e+e - annihilation. For t = 0, under rather general conditions as regards the hadron physics input, we obtained the result, abstracted from detailed calculations, that the effect of off-diagonal transitions may approximately be described by replacing the vector meson mass appearing in the simple vector dominance propagator by a somewhat smaller one, m < mp (m~,~). The simple pole in q2 dependence of p0(w, 4) dominance (apart from the replacement of rap(w, ~b) by m) may thus be treated with increased confidence, even at very large q2 (keeping w' large). (A possible exception to this simple result has been excluded, as it would, e.g., require a PoP ~ PIP amplitude as large as the elastic PoP ~ PoP one.) The numerical value of ~h depends on the precise magnitude of the spin conserving hadronic diffraction dissociation amplitudes which are not at present experimentally known. We have thus suggested that electro-production data be fitted by varying r~ (rather than by varying the power law for the q2 dependence). As regards the slope in electroproduction of vector mesons, we conjecture that possible slope changes with q2 are related to differences in slope between elastic and
H. Fraas et al./Electroproduction
317
d i f f r a c t i o n d i s s o c i a t i o n processes as o b s e r v e d in 7rp a n d p p i n t e r a c t i o n s . More specifically, w i t h t h e a l t e r n a t i n g sign a s s u m p t i o n for t h e p h o t o n v e c t o r m e s o n couplings m o t i v a t e d f r o m o u r t r e a t m e n t o f d e e p inelastic ep scattering, and t h e s t e e p e r slopes o f d i f f r a c t i o n d i s s o c i a t i o n c o m p a r e d w i t h elastic scattering in lrp a n d pp, we obt a i n e d a f l a t t e n i n g o f slope w i t h increasing q 2 in e l e c t r o p r o d u c t i o n . I n d e e d , experim e n t i n d i c a t e s s u c h a n effect. O n e o f us (D.S.) wishes t o t h a n k Professor A. D o n n a c h i e a n d Dr. G. Shaw for a visit to D a r e s b u r y w h i l e ' t h i s w o r k was c o m p l e t e d .
References [ 1 ] G. Wolf, Proc. 1971 Int. Symp. on electron and photon interactions at high energies, Cornell, Ithaca, N.Y., ed. N.B. Mistry, p. 189. [2] K. Moffeit, Proc. 1973 Int. Symp. on electron and photon interactions at high energies, Bonn, ed. H. Rollnik and W. Pfeil (North-Holland, Amsterdam, 1974) p. 313; SLAC report SLAC-PUB-1314 (1973). [3] K. Gottfried, Proc. 1971 Int. Symp. on electron and photon interactions at high energies, Cornell, Ithaca, N.Y., ed. N.B. Mistry, p. 221. [4] L. Fo~i, Int. Conf. on elementary particles, Aix-en-Provence, 1973 J. de Phys. Suppl. 34 (1973) 317. [5] V. Silvestrini, Proc. 16th Int. Conf. on high-energy physics, Chicago, ed. J.D. Jackson and A. Roberts, 4 (1972) 1; K. Strauch, Proc. 1973 Int. Syrup. on electron and photon interactions at high energies, Bonn, ed. H. Rollnik and W. Pfeil (North-Holland, Amsterdam, 1974) p. 1. [6] R. Talman, Proc. Bonn Conf., p. 145; Cornell report CLNS-249 (1973). [7] H. Fraas, B.J. Read and D. Schildknecht, Nucl. Phys. B86 (1975) 346. [8] J. Benecke et al., Max-Planck-Institut report MPI-PAE/Exp. El. 38 (1974). [9] A. Bramon, E. Etim and M. Greco, Phys. Letters 41B (1972) 609; M. Greco, Nucl. Phys. B63 (1973) 398. [10] B. Richter, Invited talk at APS Chicago meeting, February, 1974, see CERN Courier 14 (1974) 41. [11] M. BShm, H. Joos and M. Krammer, Acta Phys. Austriaca 38 (1973) 123. [12] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York 1965) p. 267. [13] H. Fraas and D. Schildknecht, Nucl. Phys. B14 (1969) 543. [14] V. Eckardt et al., DESY report 74/5 (1974). [15] H. Cheng and T.T. Wu, Phys. Rev. Letters 22 (1969) 1409; Phys. Rev. 183 (1969) 1324. [16] J. Bjorken, Proc. Cornell Conf., p. 281. [17] H. Harari, Phys. Rev. Letters 27 (1971) 1028. [18] J. Ballam et al., Phys. Rev. D5 (1972) 545.