Anomalous colour transparency

Physics Letters B 269 ( 1991 ) 439-444 North-Holland

PHYSICS L E T T E R S B

Anomalous colour transparency J o h n P. R a l s t o n 1 Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland

Received 22 July 1991

Colour transparency is not expected to occur in a conventional treatment of exclusive nuclear n° photoproduction yA-*n°A. I argue that a measurement may nonetheless show a signal of colour transparency due to the anomalous 73,~° vertex involved in the production. Estimates of the dependence on energy and nuclear number A are presented. The signals of anomalous transparency are a rise in the cross section with energy, and a weaker than expected fall with A from attenuation, in a specific kinematic region.

The axial a n o m a l y has a long and interesting history. W h e n first discovered by Steinberger [ 1 ] it m a y have a p p e a r e d to be a peculiarity o f a m o d e l in which pions were b o u n d states o f nucleons. In the quark basis the a n o m a l y reappears as a consequence o f the ultraviolet regularization. However, in a massless theory the a n o m a l y s o m e t i m e s seems to be coming from the infrared region [ 2 ]. The 't Hooft conditions [ 3 ] offer a n o t h e r example o f how the a n o m a l y ' s interpretation is subtle in a h a d r o n i c basis. Thus, while everyone agrees the a n o m a l y is inevitable, there is a history o f the physical picture d e p e n d i n g on the basis o f q u a n t u m mechanical states used in the calculation. This p a p e r proposes a direct e x p e r i m e n t a l probe o f the spatial characteristics o f an a n o m a l o u s interaction, the rt°yy vertex. This interaction will be studied in the n e i g h b o u r h o o d of a large nucleus with charge Z and nuclear n u m b e r A. I claim there is a possibility o f observing a n o m a l o u s l y small a t t e n u a t i o n o f n°'s p h o t o p r o d u c e d on the nucleus. The argument goes as follows. F o r m a n y years, 7t° p h o t o p r o d u c t i o n at small mom e n t u m transfer t has been recognized to proceed in the nucleus' C o u l o m b field by the n°yy vertex [4]. F r o m a 1950's p o i n t o f view, we could predict n ° photoproduction by measuring the n ° decay rate (and vice versa). F r o m such a p h e n o m e n o l o g i c a l viewpoint, the on-shell 7t° p r o d u c e d in p h o t o p r o d u c t i o n Permanent address: Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA.

should have an utterly normal, 20 m b or so, inelastic cross section to interact with the protons and neutrons in the target. F o r a wide range o f reasonably high n ° energies, the production and attenuation cross section is fairly fiat, and the elastic 7t° p h o t o p r o d u c tion should be similarly i n d e p e n d e n t o f energy. G o i n g b e y o n d this picture, I consider the same exp e r i m e n t using the quark triangle graph to calculate the vertex. On must keep in m i n d that this is a m o d e l for the interaction, and a very peculiar m o d e l indeed. One replaces the full ~o wave function by insertion o f the local o p e r a t o r OuJ5u in the loop. The numerical value o f the vertex is d o m i n a t e d by large quark mom e n t a in the loop. That is, most o f the a m p l i t u d e in the calculation comes from a configuration o f the quarks separated by a small s p a c e - t i m e region, approaching a single point. While this mathematical fact m a y be physically misleading, I will a d o p t it provisionally, qualifying it at the p r o p e r m o m e n t . A n y quarks p r o d u c e d at a point are not really a n ° but have an a m p l i t u d e ( p a r a m e t r i z e d byf~) o f evolving into a n °. Other states to which the qdl system could evolve are excluded by the experiment asking for elastic n ° production. With this qualification understood, in the triangle graph model I say we begin with a "small n °''. This emphasizes that the triangle graph m o d e l for the vertex has m o r e i n f o r m a t i o n than the p h e n o m e n o l o g y o f using the vertex alone. The small n ° state is associated with a time scale ro to evolve the p h o t o p r o d u c e d q(t pair into a normal

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

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~o. Such a time scale is non-perturbative, but should be o f order A ~ D , say. Moreover, once the small ~o is p h o t o p r o d u c e d it is moving quickly in the lab and the time scale should a p p e a r extended by a boost factor o f 7=E~o/m~o. N o w if the energy E~o is large enough, one should see little expansion o f the small 7to until it has m o v e d considerably, and even b e y o n d the vicinity of the nucleus. F o r example, if E~0 = 5 GeV and Zo~A~cI,D~--0.2 GeV, then the small ~o would stay small for up to 30 fro. The range of the small ~o is about 14.6 fm (E~o / 10 G e V ) ( r o / G e V - L), while the size of a nucleus is o f o r d e r 1.1 fm A ~/3 Thus the concept o f the small ~o m a y be relevant prov i d e d we have E~o/GeV >> 0.75A ~ / 3 / ( r 0 / G e V - i )

( 1)

which is very feasible, technically. U n d e r these conditions we expect the small r¢° to exhibit colour transparency [5,6] ~ . That is, destructive interference o f gluon r a d i a t i o n from the closely separated qq pair should be nearly complete, extinguishing the system's " o l d - f a s h i o n e d " strong interactions. The large nucleus still acts as a source o f photons to produce the rt°, but disappears as an att e n u a t o r o f small rt°'s, for high enough energy satisfying ( 1 ). The possibility of measuring " a n o m a l o u s " transparency as a test o f the p r o d u c t i o n o f small n°'s is the point o f this paper. The term " a n o m a l o u s " for this kind of interaction certainly applies, because one would not expect colour transparency here on the basis of conventional arguments. Previous discussions o f colour transparency have focused on using large m o m e n t u m transfer (Q2) to enhance the exclusive p r o d u c t i o n o f spatially small, colour singlet states. In p h o t o p r o d u c t i o n there is no large Q2: we are interested in I tl > m 2 to get the p h o t o n pole. W i t h no large Q2 there is no apparent " s h o r t t i m e " coming from the off-shellness of quarks being p r o d u c e d under the kinematic conditions o f the experiment. That is, without invoking at all the triangle graph picture, a light-cone calculation of Ko p h o t o p r o d u c t i o n would not necessarily lead to colour transparency. Instead, the vertex might come from details of an especially tuned " o m n i s c i e n t " soft wave function capable o f giving the correct a n o m a ~' For data on colour transparency in pA~p'p" (A-- l ) see ref. [7]. 440

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1OUS value from an ordinary small transverse mom e n t u m region ~2. Physically, the small and large n ° distinction has probably been idealized so far. It staggers the imagination to think that the n°?? vertex could literally be in the ultraviolet, and the triangle graph picture could literally be true. if it were true, we would not need the smoothness properties o f PCAC at all; other experiments, such as inserting a far off-shell photon, or even [ 8 ] an on-shell Z °, into the graph would show experimentally that the vertex acted pointlike. Assuming this is not the case, how do we interpret the success of the triangle graph phenomenology? It must have some truth, because it mysteriously produces the right answer, absolutely normalized, including the colour factors and phase. A reasonable interpretation is that the process is i n d e e d small, but only small on the scale of chiral symmetry breaking. The characteristic scale being (q~u), this means that the small n °, at the m o m e n t of production, needs only to be small c o m p a r e d to a n o r m a l h a d r o n size. Then the m a t h e m a t i c s o f the triangle graph could be reasonably true: the system looks a little bit ultraviolet, but it is not so ultraviolet as to cause other problems. This concept can be m a d e more definite by a thought calculation. Take one's model for the complete light-cone qq pair wave function in the n °, constructed so that a t t a c h m e n t o f photons will give a correct n°7? vertex. To separate the regions that contribute, a d d and subtract a real n u m b e r 2 times the usual PCAC pointlike n o wave function (fig. 1 ). We have a " t w o - c o m p o n e n t " n ° model if we crudely identify the subtracted a m p l i t u d e with the large n ° t42 I thank AI Mueller for insisting that a soft light-cone wave function could be sufficiently omniscient.

. . . . . .

+ (l-K)

....

V

Fig. 1. Decomposition of a realistic Ty~° vertex using the PCAC triangle model (weight 2) and everything else (weight 1-2). The symbol "x" indicates that the short-distance contribution has been subtracted.

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and the P C A C one as the small n °. Varying p a r a m e t e r 2, I consider the sum over c o m p o n e n t s as a sort o f q u a n t u m mechanical superposition. ( T h e key ass u m p t i o n is that the small n o, once small, can safely be replaced by the local p r o d u c t i o n since the calculation is smooth, as I will show. ) That is, for the p r o d u c t i o n vertex without accounting for interaction with the nucleus I write

V

Itu .u --2 Vsmal I + ( 1 - 2 ) V~arge,

It~ __

(2)

where # and u are p h o t o n indices. This d e c o m p o s i t i o n becomes meaningful when we i m p l e m e n t the different time evolution o f the two components due to interaction with the nucleus. I will estimate the survival a m p l i t u d e o f the qq to emerge as a n o in the two cases. Some m i n i m a l F e y n m a n diagrams, useful for setting up the m o m e n t u m flow, are shown in fig. 2. Note there are three loops, m i n i m u m . Various techniques exist to sum rescatterings if production is via the small n ° vertex, resulting in exponentiation. This is not an i m p o r t a n t issue. The interesting question is whether the small n ° concept is relevant at all, and whether it can be measured in an experiment. First I must specify the kinematics. Let qU be the m o m e n t u m difference between the outgoing n ° and incoming photon, so Iq2l ~q'~ ~ It], where _L de-

Z, A

(a)

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notes the transverse coordinate. To make sure the n o penetrates the nucleus we need q i > A -~/3/fm. To see the one-photon exchange is an experimental question, resolved by fitting the region q l ~ A - ~ / 3 is d e m a n d e d , the region ofq~ allowed can only range up to q 2 . Thus, the production is p r e d o m i n a n t l y outside, and I will not consider production inside; one can easily see that this just makes a small correction to the measured value o f 2. The interesting physics is in the growth with time o f the small n °. The transverse coordinate b o f the quark separation is the most important. Suppose the typical b 2 grows linearly with time in the qCl rest frame, as the system evolves in free space. As observed in the lab, we must correct for the time evolution with a boost factor 7To model this, I use an ansatz for the a m p l i t u d e ~u(b, x - x ' ) for the quarks which have travelled a distance x - x ' after being created at x ' :

v / ( b , x - x ' ) - #Texp[-½ybZ#/(x-x')]

(3)

X--X'

z, A

(b)

Fig. 2. Two processes producing n°'s near a nucleus. (a) Production outside the nucleus via the usual anomalous vertex and scattering in the target. (b) A graph involving production inside the nucleus. The nucleus is shown as the heavy spectator lines, interacting with a qq pair in the loop.

for x - x ' < 7To. The normalization is such that V/begins as 6(b 2) at x - x ' . The choice o f a delta function is a simplification, since the same result will be obtained no m a t t e r what small size for the initial system is chosen, so long as the size is much smaller than a fermi. The mass scale # is a parameter. The ansatz ( 3 ) , reminiscent o f " q u a n t u m diffusion" [6], is a model o f the free s p a c e - t i m e evolution. It would not apply inside the nucleus. Inside the nucleus, filtering effects o f attenuation become important. Let the coordinates be chosen so the nucleus sits between x = 0 and x=A ,/3. The ef441

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fects of transmission through the nucleus are a factor

f. of fA ~exp( -- ~bZA 1/3nty' ) , where a'b 2 is the cross section, and n the nuclear number density. After transmission through the nucleus one must take the overlap with the soft n o wave function ¢~(b2), which can be represented by

O ~ ( b 2 ) ~ e x p ( - ½ m Z b 2) . The product ~f~0~ needs to be integrated over transverse separation b, and then integrated over the production point x'. Let the x-space amplitude to find the Coulomb photon of the nucleus be d ( x ' ). We need, now, 7r0

oo

A l/3fm

0

(4)

stood in a simple way ~3. Integrating over the production point, the typical attenuation cross section including expansion is set by the duration of the expansion in units of the effective lifetime, A 1/3/7. The attenuation depends on the effective cross section times the target length, which is A 2/3/7. The approximation ( 5 ) is complemented by evaluating (4) and numerically studying it as a function of the several parameters. I then estimate the cross section by summing the amplitudes and squaring, [~//tot ]2= [J.J[srnau +exp(i~) ( 1 --2)dllarge 12 ,

1

1 {roT~

J/'/sma,l~ 1+A2/3/7 n!,,-~5 ) •

(5)

The logarithm is an infrared singularity, set by the closest and farthest points of production, as can be anticipated from the q' loop in fig. 2. Such singularities are general, slowly varying, and will cancel out when I take the ratio of cross sections to study the transparency. The only dependence of the transparency then is a "scaling" dependence on the variable 7/A 2/3. This can be compared to another scaling result predicted [9 ] in ordinary, large Q2 and large A processes such as electroproduction. In that case, deliberately averaging over any hadron expansion in the high energy limit, the dependence is 7 / a 1/3 (since

7...QZ). The two results differ, because the free space expansion of the system has been put in for ~o photoproduction. The new scaling result can be under442

(6)

where d¢~argeis the case of the large ~o. This should he exponentially attenuated with an energy independent cross section of about 20 mb. The relative phase 6 of the two amplitudes is too delicate for a crude estimate so I will treat it as a parameter. The transparency ratio T(E, A) will be defined as

T(E,A)The x' integral is essentially one-dimensional, since production is taking place over a long tube with transverse size x'~ ~A 1/3 fm, the target radius. The x' integral is the q' integral of fig. 2 and the Coulomb potential ~¢(x' ) ~ 1 / ~ ~ 1/x'. Now all the dependence on large scales, namely the boost u and the nuclear size A, can be extracted. The asymptotic limit of J//~H for 7>> 1, A >> 1 is

31 October 1991

1 da(TA~ 7t°A)/dt Z2fr d a ( y A ~ nOp)/dt ,

(7)

where Z~ff is the square of the photon-nucleus form factor at the t-value of the experiment, which can be obtained from electron scattering experiments. I set my normalization by dividing (4) by 2 + In (7/A ~/ 3 ) and requiring that T(E, A ) ~ 1 for 2 = 1, A = 1 and E~, in which limit the model gives zero attenuation. For parameters 1/to, #, and m I use 300 MeV. The most important behaviour is the energy dependence, which should reveal a rise in production with increasing energy. This is shown in fig. 3. The curves show the result of setting the small ~o amplitude 2 to different values. As 2 is varied from one to zero the energy dependence becomes flatter; above 2 = 0.1, it would become possible to detect a small no component. In these plots, 3= 0. In figs. 4a, 4b, I explore the effects of the relative phase 3. An objective procedure is to let ~ vary over its full range, 0 < 6 < 2~, and present the extreme range of the predictions. I also show a curve for the "typical" value J = re/2. For 2 below 0.5, say, the range includes the possibility of transparency decreasing with

~3 The same argument can be applied to the intermediate energy region in ordinary electroproduction or other transparency measurements such as pp~pp. This will be considered elsewhere.

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0.4

1.0

0.3

T

O.5

0.2

0.3 0.1 0

0.1

5

10

15

20

25

E(GeV)

Fig. 3. Energy dependence of the no production at fixed A = 200. Tis the transparency ratio [eq. (7) ]. Curves are shown for different values of the parameter 2, which is the amplitude for the initial q~t pair to be small. For 2 > 0.5 there is a substantial rise of production with energy, and even for 2>0.1 there is a signal.

=

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energy, i.e., d e s t r u c t i v e i n t e r f e r e n c e . H o w e v e r , one d o e s not really expect a phase shift c o n s p i r a c y to occur, a n d for 2 > 0.5 the c h a r a c t e r i s t i c rise o f t r a n s p a r e n c y with energy should not be c o n c e a l e d by interfere n c e effects. T h e A d e p e n d e n c e is v e r y interesting. Plots i n c l u d e the region o f small A ( w h e r e the a p p r o x i m a t i o n s o f neglecting f e w - b o d y effects are too c r u d e ) to s h o w that in this region n o t h i n g d e p e n d s o n 2. F o r 2 = 0 a n d large A the A d e p e n d e n c e falls e x p o n e n t i a l l y (fig. 5). H o w e v e r , for 2 # 0 this b e h a v i o u r changes to a p o w e r law for large A, at fixed energy. T h e a s y m p totic p o w e r A -2/3 is not r e a c h e d for A < 200 because one needs A2/3/7>> 1 to see it; still, there is a clear d e p a r t u r e f r o m the e x p o n e n t i a l fall. Sensitively o f the p r o d u c t i o n to the r e l a t i v e phase ~ begins to d i s a p p e a r at large A, because the large n o a m p l i t u d e is attenu a t e d away. Thus, the A d e p e n d e n c e can also be used

e i0

0.5 T

e in/2

o.1

/

~

-

:.....---

0.4

T 03 _

02 1 o1. }

_

e'"

-7

,0

1'5

;0

~"=1o 05 0:3 0.1

2'5

E(GeV)

50

100

150

200

A 0.5

8 = e i0

= 0.3

0.15

0.1

0.4

~

eig/2

T

0.05

T

~

~

0.3 8 = e i0

0.2 ~

e

0.1

e

(b) (b)

~ 5

10

15

, 2O

, 25

E(GeV)

Fig. 4. Effects of a relative phase d [eq. (6) ]. At each energy the extreme range of variations of the predicted production for any phase 0 < ~< 2n is shown, as well as the choice ~= n/2. For 2 > 0.5 there is a signal whatever the phase A = 200.

i~/2

5o

i

a

,i

100

150

200

A Fig. 5. (a) Nuclear number A dependence of the transparency ratio [ eq. ( 7 ) ] at fixed energy E,o = 10 GeV; 2 values as shown. For 2S 0 the attenuation with A is much less rapid than the exponential attenuation conventionally expected (shown for 2 = 0). (b) Effects of the phase &. Varying over all &, 0<&<2n, the production is less sensitive to the effects of phases at large A. 443

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as a signal o f a n o m a l o u s t r a n s p a r e n c y . T h e e s t i m a t e s p r e s e n t e d h e r e are n o t the last w o r d on this subject. O n e a n t i c i p a t e s t h a t s o m e experim e n t a l w o r k m a y be r e q u i r e d to m o r e precisely locate the region o f energy a n d m o m e n t u m transfer o v e r w h i c h the signal c o u l d be o b s e r v e d . C e r t a i n l y m u c h w o r k can be d o n e to i m p r o v e the calculations, b u t they are p r o b a b l y a d e q u a t e r e p r e s e n t a t i o n s o f the physics. O t h e r a p p l i c a t i o n s o f the s a m e " a n o m a l o u s " physics - such as 11, q ' p h o t o p r o d u c t i o n - w o u l d also be interesting. I t h a n k B e r n a r d Pire, A1 M u e l l e r a n d D i c k N o r t o n for useful discussions a n d the T h e o r y G r o u p at C E R N for hospitality. A p r e l i m i n a r y v e r s i o n o f this w o r k was p r e s e n t e d at the C E N Saclay M e e t i n g h e l d in J u n e 1991. I t h a n k P. G u i c h o n a n d D P h N for h o s p i t a l i t y at Saclay. T h i s w o r k was s u p p o r t e d in part u n d e r D O E grant N o . D E - F G 0 2 8 5 E R 4 0 1 2 4 . A 0 0 7 a n d N S F C N R S I n t e r n a t i o n a l P r o g r a m s G r a n t No. 8914626.

References [ 1 ] J. Steinberger, Phys. Rev. 76 (1949) 480; H. Fukuda and Y. Miyamoto, Prog. Theor. Phys. (Kyoto) 4 (1949) 347;

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J.S. Bell and R. Jackiw, Nuovo Cimento 60 A ( 1969 ) 47; S. Adler, Phys. Rev. 177 (1969) 2426. [2] A.D. Dolgov and V.I. Zakharov, Nucl. Phys. B 27 (1971) 525. [ 3 ] G. 't Hoofl, in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980); Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, Nucl. Phys. B 177 (1981) 157; S. Coleman and B. Grossman, Nucl. Phys. B 203 (1982) 205. [4] H. Primakoff, Phys. Rev. 81 (1951) 899; for a review, see: A. Donnachie and M. Shaw, Electromagnetic interactions of hadrons, Vol. 1 (Plenum, New York, 1978) sect. 3.2.3. [ 5 ] S.J. Brodsky and A.H. Mueller, Phys. Lett. B 206 ( 1985 ) 685, and references therein; S.J. Brodsky and G. de Terramond, Phys. Rev. Lett. 60 (1988) 1924; J.P. Ralston and B. Pire, Phys. Rev. Lett. 61 (1988) 1823; 65 (1990) 2343; B. Jennings and G. Miller, Phys. Lett. B 236 (1990) 209; Triumf preprint (/990). [6] G.R. Farrar, H. Liu, L. Frankfurt and M. Strikman, Phys. Rev. Lett. 61 (1988) 61. [7] A.S. Carroll et al., Phys. Rev. Lett. 61 (1988) 1698. [8] M. Jacob and T.T. Wu, Phys. Lett. B 232 (1989) 529. [9] B. Pire and J.P. Ralson, Phys. Lett. B 256 (1991) 523.