Elastic scattering slope and total cross section relations

Nuclear Physics B97 (1975) 452-460 © North-Holland Publishing Company

E L A S T I C S C A T T E R I N G SLOPE A N D T O T A L C R O S S S E C T I O N RELATIONS V. B A R G E R *

Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 USA R.J.N. PHILLIPS

Rutherford Laboratory, Chilton, Didcot, Oxon, England Received 2 June 1975 The emperical equality of B2/~t is noted, for the pomeron terms in NN, 7rN, KN, pN, t~N, and @N scattering, where B is the elastic slope parameter and o t is the total cross section. This ratio increases slowly with energy, but remains the same in all channels. This is equivalent to a relation between the diffractive interaction radii and opacities; the opacity scales with the square of the radius. We conjecture that this systematics extends to ~N scattering. We contrast B2/ot universaility with the slope predictions of an f-coupled pomeron model. Some other predictions of the f-dominance hypothesis are tested against data.

The systematics o f the p o m e r o n exchange amplitudes have been intensively studied, especially in c o n n e c t i o n with geometrical scaling ideas. In such a study [1] it was noticed that the p o m e r o n terms in NN, 7rN and KN scattering are united by the empirical relation

B2/o t (NN) = B2/ot QTN) = B 2 / o t (KN),

(1)

where B is the elastic slope parameter: do~dr ~ exp(Bt) at small t. The ratios in eq. (1) in fact change slowly with energy, but such as to preserve the equality at each value o f s . The experimental numbers [2,3] at s -~ 375 GeV 2 are B 2 / o t (pp) = 3.0 + 0.1 GeV - 4 mb -1 ,

(2a)

B2/ot(Tr*p) = 3.2 + 0.1,

(2b)

BZ/at(K+p) = 3.3 + 0.7.

(2c)

* Supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the US Atomic Energy Commission under contract AT(I 1-1 )-881, C00-881-445.

V. Barger, R.J.N. Phillips /Elastic scattering

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Fig. 1. B2/at comparison of NN, nN and KN scattering data. A common interpolating curve is shown. Further comparison of the experimental B2/ot ratios for NN, 7rN and KN scattering is made in fig. 1 over the complete s-range. The B2/ot ratios for the different reactions remain quite similar, even at low s where secondary trajectories could produce deviations from eq. (1). In the present letter we point out that eq. (1) also holds for pN, coN, and ~N scattering, i.e. for mesons in the same SU(6) multiplet as ~r and K. We conjecture that this systematics may also extend to ~N scattering assuming a ~ = gc interpretation in terms of charmed quarks (and indeed to any charmed particles in the same supermultiplet), qvN scattering is a particularly clean case of pomeron exchange, since normal Regge exchanges are forbidden. On the other hand, since the pomeron is built from the shadow of inelastic processes that should include significant charm production in the ~N case, the pomeron should not be fully developed here until well above charm threshold [4] (s >> 40). Preliminary ~N scattering results, via photoproduction at Fermilab [5] (s ~ 160), are not in contradiction with eq. (1). Results from SLAC [6] are not yet available, but the maximum SLAC energy of s ~ 40 may well be too low to be relevant to the pomeron in ~N. Elastic scattering information for vector mesons on nucleons is deduced from photoproduction data through the vector dominance model (VDM) relation do 3[' 2 eBt dt- (TP ~ Vp) - 16Trarnv [o t (Vp)] ,

(3)

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V. Barger, R.J.N. Phillips/Elastic scattering

Table 1 Comparison of data on BZ/ot ratios at s ~ 19 GeV 2 Elastic reaction

Slope parameter [s ~- 19 GeV2I B(GeV)-2

o t (mb)

B2/ot

pp pp ~-p ~+p K-p K÷p pOp ~p @p

11.8±3.0 9.8±0.2 7.5~0.8 8.2±0.9 7.8±0.2 5.6±0.2 6.5±0.2 6.6±1.1 4.6±0.7

54.7±0.6 39.9±0.6 26.9±0.1 24.8±0.2 22.7±0.6 17.3±0.1 23 ±3 24 ~3 9 ±1

2.5~1.3 2.4±0.1 2.1±0.5 2.7±0.6 2.7±0.2 1.8±0.2 1.8±0.3 1.8±0.7 2.3±0.8

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s (GeV) 2

Fig. 2. B2/ot comparison of oN and @N scattering data. The common interpolation curve of the data of fig. 1 is shown. w h e r e P is t h e V ~ e+e - partial w i d t h * and c~ the fine structure c o n s t a n t . These VDM estimates assume Q2 i n d e p e n d e n c e o f 3'V couplings. The values o f B2/ot for pp, cop and 4bp at s -~ 19 GeV 2, d e d u c e d f r o m p h o t o p r o d u c t i o n data [7] using eq. (3), are c o m p a r e d in table 1 with pp, 7rp, and K p data. The results are c o n s i s t e n t with a universal B2/ot ratio. A m o r e extensive c o m p a r i s o n w i t h p p and (bp versus s is m a d e in fig. 2, w h e r e the curve r e p r e s e n t s the c o m m o n i n t e r p o l a t i o n o f the data * We use I'(,o0 ~ e÷e-) = 6.45 keV, l'(to -~ e+e-) = 0.76 keV and 1"(0-~ e+e-) ~ 1.34 keV.

V. Barger, R.J.N. Phillips / Elastic scattering

455

in fig. 1. This agreement of the ON data with the B2/ot hypothesis is significant, since ot(qSN) is over a factor o f 2 smaller than the other total cross sections. The accomodation of existing elastic data on the basis of a universal B2/ot ratio leads us to conjecture that this will also be the case for giN scattering. Measurements ofTN ~ f N at Fermilab [5] yield o t ( ~ N ) ~ 1 mb

(4a)

for an average photon energy of ~ 8 0 GeV. From figs. 1 and 2, we would correspondingly expect B(~N) ~ 2 GeV --2.

(4b)

The present Fermilab data do not precisely determine the ~N slope parameter, but seem to be consistent with a shallow slope. Why and how do two-particle cross sections differ? Since the allowed exchange mechanism differ, between various two-body systems, it is easy to believe that the interaction ranges differ. Opacity is another parameter: if the opacity were fixed and only the range altered, B/o t would be a universal constant, contrary to experiment. Clearly both range and opacity are changing together, from one two-particle system to another. In the absence of a complete theory, we turn to experiment for guidance. The present work is an attempt to extract from data an empirical regularity for the range/opacity correlation. Universality of B2/ot would imply that the opacity (measured by Oel/Ot) scales with the square of the interaction range (measured by B), since B = o2/(167r%1) for the usual diffraction peaks. (Universality of 3 2 ). The striking decrease in central B2/ot can also be stated as universality of ot/Oel opacity for reactions with smaller o t is illustrated in fig. 3 by the impact parameter

I°I

Imfel (b) : ~

e-b2/2B

A r-, v

wE

05

0

0 5

1.0 b (fm)

1.5

20

Fig 3. hnpact par~/meter profiles of pp, rip, and ~p elastic amplitudes at s ~ 19 GeV2. The total cross sections at this energy are ot(pp) = 40 nab, ot(rrp) = 25 mb, and ot(Op) = 9 mb.

456

I/.. Barger, R.J.N. Phillips / Elastic sea ttering

profiles of pp, rip, and q~p, calculated in an exponential approximation to the forward diffraction peaks.at s ~ 19 GeV 2. Future (final) results from 7N -~ ~N experiments will test these ideas more stringently. It is interesting to contrast B2/ot universality with the predictions of an f-coupled pomeron model [8,9]. In this model, the couplings of the pomeron to external particles are mediated by the tensor trajectories f, f', and fc (constructed from quark states ~p + fin, ~.X, and gc, respectively). The pomeron residue in ab elastic scattering is given by

p ~iaa (t) ~fbb (t) Aab(t ) = ~ BP.(t) i! ap(t) - ~i(t) Otp(t) -- O~](t)'

(5)

where i,j = f, f', fc and ~p(t) is the pomeron trajectory. With exact SU(4) symmetry for the f, f', fc couplings, symmetry breaking in the pomeron couplings to external particles is determined by the trajectory ratios av(t) - af(t) ap(t) - c@t) rx(t) - ap(t) - af,(t) ' rc(t) - ap(t) - afc(t ) " (6) For the pomeron contributions in meson-nucleon elastic amplitudes, the f-coupling model gives AP(Kp)/AP(np)

= ½(1 + rx(t)),

AP(f@)/AP(pOp) = rx(t),

(7) AP(wp)/AP(pOp) = 1,

AP(¢p)/AP(Op) = rc(t)/rx(t).

For linear and exchange-degenerate w-- f, ~b- f' and ~ - fc trajectories, with slopes t t t a~, c%, c%, the predictions for the pomeron total cross sections reduce to ot(KP)/Ot(np)

n l L O~r

ot((gp)/ot(pOp) -

1

1 m 2 o~;

a~

O/

co_0.59 w , , , m 2 a0 s0 I

°t(~"P)/°t (~bp) - m 29 a¢~ , m~ a~

(8)

!

-0'11--7-

a~ a~

"

For o~o = a'~ = o~'~ -= a ,' we have

ot(Kp)/ot(rrp)

Theory (for P) 0.8

Experiment [3,5,7] 0.84 (s = 400)

ot(~bp)/ot(p0p)

0.6

0.4 (s -~ 19)

ot(~p)/at(qSp)

0.1

~0.1

V.Barger,R.J.N.Phillips/ Elasticscattering

457

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Fig. 4. Comparison of at(qsp) and ot(pp ) deduced from q,p ~ Vp data using the vector-meson dominance model. The ratio ot(q~p)/ot(pp) provides a test of f-dominance; see eq. (8). In the comparison of ~ p and q~p, the approximate constancy of o t is assumed inasmuch as the data are at different energies. Since only ~p and ~ p are exclusively pomeron exchange, the above predictions are in reasonable accord with the data; see fig. 4 for a comparison o f a t ( p a p ) and at(~,bp) data. The observed degree of supt t pression of a t ( ~ p ) relative to at(~,bp) requires that a+ -~ ae; therefore we hereafter take a common slope for the tensor trajectories. The f-coupled pomeron also leads to relations [8] between elastic slope parameters B. From eqs. (6) and (7), the predictions at t = 0 are

BTrp-BKp =2X(r~2 12) \m2+m2/=O.5xGeV-2, (9)

Bop - B~p

\rn~

= 1.7x,

where t

t

X = 1 -- (C~p/Ot), !

and O~p is the slope of the pomeron trajectory. Hence constant slope differences are predicted for the pomeron contribution, independently of the energy dependence of the total cross sections. At finite t the predicted differences of the slopes are smaller than above differences at t = 0. The maximal value of x is 1. Typical slope estimates, c~' -~ 1, @ ~ 0.3, yield x = 0.7. Comparisons for eq. (9) of experimental slope parameters at small t are made in fig. 5.

458

V. Barger, R.J.N. P h i l l i p s / E l a s t i c scattering

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Fig. 5. Data on forward elastic slopes for comparison with tile f-coupled-pomeron asymtotic predictions Bnp - BKp ~ 0.35 GeV -2,

BoO p - Bdpp ~- 0.9 GeV -2.

The crucial test of the constant slope difference prediction will be B4~p - B q j p, where only p o m e r o n exchange contributes to the amplitudes. Existing q~p forward slope data for 4 < s < 34 GeV 2 can be represented by the I n s dependence B~hp = 3.5 + 0.32 Ins.

(10)

Assuming this extrapolation for higher s, the f-coupling model would predict BOp ~ 3.5 - 1.7x + 0.32 In s.

(1 l)

V. Barger, R.J.N. Phillips / Elastic scattering

0"35f

459

o-t (Tr*p) + o-t(Tr-p)

o't(pp) + o't(~p) + o't(pn) + o't( ~n I

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Fig. 6. Experimental test of the prediction from the f-dominance model of pomeron couplings that the ratio [atOr'p) + otOr-p)]/[ot(pp) + at(Up) + ot(Pn) + at(Pn) ] is energy independent. For x = 0.7 the slope predictions are Bq~p-~3.5

at

s = 4 0 G e V 2,

Bqjp -~ 3.9

at

s = 160 GeV 2.

The latter number differs markedly from the prediction of B2/ot universality in eq. (4b), but is within the range suggested by the Fermilab ~-photoproduction experiment [5]. One of the earlier successes of the f-dominance model was the prediction that the ratio of f-reggeon exchange to pomeron exchange in total cross sections is reaction independent, for example Af(pp) Ap(pp)

A f(np) Ap(np) "

(12)

Accordingly the ratio ot(rr-p) + ot(rr+p) ot(pp) + ot(pp) + at(Pn) + ot(Pn ) should be independent of energy. This prediction is not borne out by the Fermilab a t data [3], see fig. 6. Consequently f-dominance predictions for VN scattering must be viewed with some caution.

460

K Barger, R.J.N. Phillips / Elastic scattering

We are g r a t e f u l to J. L u t h e for valuable assistance and discussion.

References [1] [2] [3] [4]

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

V. Barger, J. Luthe and R.J.N. Phillips, Nucl. Phys. B88 (1975) 237. C.W. Akerlof et al., University of Michigan report 74 -20. A.S. Carroll et al., Phys. Rev. Letters 33 (1974) 928,932. M.K. Gaillard et al., Rev. Mod. Phys. 47 (1975) 277; S. Okubo et al., Phys. Rev. Letters 34 (1975) 38,236; A. de Rujula and S.L. Glashow, Phys. Rev. Letters 34 (1975) 46. B. Knapp et al., Phys. Rev. Letters 34 (1975) 1040. R. Prepost et al., SLAC-Wisconsin experiment. J.F. Martin et al., Phys. Rev. Letters 34 (1975) 288. R. Carlitz, M.B. Green and A. Zee, Phys. Rev. D l l (1971) 3439. C.E. Carlson and P.G.O. Freund, Phys. Letters 39B (1972) 349; Enrico Fermi Institute report 7 4 - 6 0 ; R. Brower and J. Primack, UCSC 74/103 (1974); L. Clavelli, University of Maryland report 7 5 - 0 5 8 ; T. Inami, Phys. Letters 56B (1975) 291.