Vector meson production, pion photoproduction, and vector meson dominance

Nuclear Physics B35 (1971) 541-555. North-Holland Publishing Company

VECTOR MESON PRODUCTION,

PION PHOTOPRODUCTION,

A N D V~ECTOR M E S O N D O M I N A N C E B.H.KELLETT Department of TheoreticalPhysics, University of Manchester Received 20 July 1971 (Revised manuscript received 23 August 1971)

Abstract: The vector meson dominance hypothesis can be formulated covariantly in terms of invariant amplitudes. This method, which has been successfully used to predict n-p --* pOn from pion photoproduction, is extended to include to production and charged p production. With the introduction of only two further parameters to describe the mass dependence of the pton coupling, a unified description of vector meson production and pion photoproduction is obtained.

1. INTRODUCTION The successful prediction o f the differential cross section and density matrix elements for neutral p meson production from a model fit to pion photoproduction [1 ] (referred to as I hereafter) suggests the application of the same vector dominance model to 6o production and charged p production. One of the difficulties encountered in the relation of pion photoproduction and vector meson production via the vector meson dominance (VMD) hypothesis arises from the fact that the photon does not have definite isospin. The relative strengths o f the isoscalar and isovector photon amplitudes are different in different processes, so that in direct comparisons of the data for the related reactions, combinations for which isoscalar-isovector interference terms are absent or unimportant must be taken [2]. The separate determination of the isoscalar and isovector amplitudes obtained in a model fit to pion photoproduction eliminates these problems, and allows the direct comparison of the amplitudes with definite isospin in the various processes. In this paper we make use of this possibility, and starting from the same basic fit to all the photoproduction data [3], extend the vector dominance model developed in I to give a unified description of p± and ¢o production, as well as po production and pion photoproduction. The isospin decomposition of the pion photoproduction amplitudes is given as

542

B.H.Kellett, Vector meson production

usual by A = 8~3 A(+) + { [rd, r3] A ( - ) + r a A (°) ,

(1)

where a is the isospin index of the pion. Isovector photons couple to A (±) and isoscalar photons to A(°). The various charge states correspond to the following combinations of isospin amplitudes: 3'P ~ zr÷n : x/~ (.4 (°) + A ( - ) ) ,

(2)

7n ~ 7r-p : X/~-(A(°)- A ( - ) ) ,

(3)

7P -~ n°P : A (°) +A(+),

(4)

7n --> rr°n : A ( ° ) - A (+) •

(5)

If we denote the t-channel isospin in vector meson production by the suffix 0 or 1, we obtain the following isospin decomposition. rr-p ~ p°n : x/~A1 ,

(6)

rrOp ~ pOp : A o ,

(7)

rr-p--> p-p : A o - A 1 ,

(8)

rr+p ~ p + p : A o + A 1 ,

(9)

n+n--> cop : v ~ B 1 .

(10)

From these equations it can readily be seen that the vector meson dominance hypothesis relates A(-) to A 1, A(+) to Ao, and A (°) to B 1. In I, a model fit to the photoproduction data [3] was used to isolate the isovector amplitude A(-). The same model for photoproduction obviously yields the amplitudes A (°) and A (+), and it is of some interest to utilize this additional information within the VMD framework in order to predict reactions (8)-(10).

2. THE MODEL In order to predict vector meson production from photoproduction it is necessary to define an extrapolation in the vector mass in such a way that the longitudinal polarization states of the massive vector particles are obtained from the purely transverse photon. The separation of the polarization states of a massive vector par-

B.H.Kellett, Vector meson production

543

ticle into longitudinal and transverse components is in general a Lorentz-frame dependent procedure, and has given rise to considerable ambiguity in the application of VMD. It was suggested in I, however, that if VMD is formulated in terms of the explicitly Lorentz covariant invariant amplitudes, this ambiguity of frame is reduced to the well defined kinematic question o f the relation between the invariant and helicity amplitudes in any given frame. The invariant amplitudes must be free of kinematic singularities, and the assumption is then made that they do not depend kinematically on the extrapolation in the external mass. This freedom from kinematical dependence on k 2, where k is the four-momentum of the vector particle, does not exclude dynamical dependence on k 2 through form factors associated with particular exchange mechanisms. Such dynamical mass dependence is not connected with frame ambiguity, however. It depends on the dynamical model employed, and can be treated in an explicitly covariant manner. Vector meson production can be described in terms of the six independent invariant amplitudes B i introduced by Ball [4]. Gauge invariance at k 2 = 0 reduces the number of independent amplitudes to four, and it is convenient to use the explicitly gauge invariant CGLN [5] amplitudes A i. As is discussed in more detail in I, at k 2 = 0 these sets of amplitudes are related in the following way: B 1 =A 1-2mA 4 , B 2 = _ q. kA 2 , B 3 = _p.

kA 2 ,

(11) B 5 = q.kA

3 + 2P. kA 4 ,

B6 = _ 2A 4 , B8 =A 3 . These relations are strictly valid only in the zero mass limit, and the kinematic factors must be evaluated at k 2 = 0. Thus we obtain the B i at k 2 = 0 from the photoproduction amplitudes, and since the assumption o f the model is that these amplitudes do not depend kinematically on the extrapolation in k 2, we can use them at the vector mass. Use o f the appropriate kinematic factors then gives the vector meson production amplitudes. If the VMD hypothesis is to be useful, the possible dynamical or form factor k 2 dependence must be related to particular couplings that the dynamical model may allow to vary. Complete freedom, amounting to the possibility of multiplying each B i b Y an arbitrary number, would mean that VMD is not a useful hypothesis. Since the kinematic relations (11) from gauge invariance must be satisfied at k 2 = 0, vector dominance necessitates the retention o f this basic structure of the B i for k 2 :/: 0. It has been suggested by Schildknecht [2] that

B.H.Kellett, Vector meson production

544

basing VMD on invariant amplitudes merely transfers the frame ambiguity to the ambiguity of choosing a set of invariant amplitudes. In the present model, however, it is explicitly assumed that the invariant amplitudes do not have any kinematic singularities, which means that this ambiguity is not present. The singularity-free CGLN and Ball amplitudes are essentially unique in that any other singularity-free set of amplitudes must be related to these by a simple linear transformation with constant coefficients. In this context it is clear than external particle masses are to be treated as kinematic variables rather than as constants. Our formulation of VMD is invariant under such a linear transformation, so that there is no ambiguity inherent in the choice of invariant amplitudes. The relations between the invariant and helicity amplitudes are given in I. In this paper we work directly with the t-channel helicity amplitudes in order to avoid the necessity of crossing the density matrices to the Jackson frame. In terms of the gXNX~,Xp of I, the vector meson cross section is given by dOdt-

1 128rr k2s

~

hN,h~l,hO

IgxNx~,xp 12 •

(12)

Similarly, the density matrix in the Jackson frame is given by

Pxx'-

XN,h~

gXNhR,hg~Nh~,X'

hN,hR,hp

(13) Ig~,NXN,xol2

3. COUPLING CONSTANTS AND DYNAMICAL MASS DEPENDENCE The relative normalization of the vector meson and photoproduction invariant amplitudes is given by the photon-vector meson coupling constants g'rp, g~¢o, and gT~. We shall ignore the ¢ contribution in the following work since, although g'r~ is comparable in magnitude to g~to, the ¢ is further suppressed by its weak coupling to the other states. The cross section for rr-p ~ Cn is of the order of only 1 or 2% of the ~o cross section [6]. For the p meson we take, as in I,f2/4rr = 2.0, and from the storage ring experiments [7] we take f~/47r 2 v = 14.0. The photoproduction amplitudes are then multiplied by g~v = fv/e, where e2/4~r = 1/137. The coupling fp andft o are measured at the vector mass, so these are taken as fixed, and any apparent departure from these values is interpreted as a variation of the strong vertex form factor between the photon and vector meson mass. For instance, it is clear that the pcorr coupling gp¢o, must be sensitive to extrapolation off the mass shell since charged p production is apparently related to po production by the simple isospin factor of x/~ [8]. From eqs. (6)-(9) this can be seen to imply

B.H.Kellett, Vector meson preduction

545

that the isoscalar amplitude A o is not important. However, this amplitude is dominated by co exchange, and it is well known that high energy 7r° photoproduction proceeds primarily by co exchange [9]. Since co exchange couples to the isovector photon in lr° photoproduction, we have the same g'ro photon-vector coupling for both isospin amplitudes in p production. Furthermore, the normalization of the pO predictions in I shows that the storage ring value for f2/4rt is reliable in this VMD p model. It follows that the couplinggpto~r must be sensitive to the extrapolation from the photon to the p mass. The possibility that the pwlr coupling may depend on the external particle mass has been observed independently in connection with the Gell-Mann, Sharp and Wagner [ 10] model for electromagnetic decays of neutral mesons. Investigations of this vertex within the framework of current algebra [ I 1-14] have shown that in the absence of dependence on the external mass, gpto~r --- 0. Mannheim [15] has interpreted this as a breakdown of VMD, but it is clear that mass dependence of this coupling can be quite simply accommodated in a vector dominance framework. In the comparison of photoproduction and vector production, we can investigate gpto,r at only two points in the external mass extrapolation. Little insight into the functional form of this mass dependence can be gained in this way. It would be necessary to investigate co electroproduction over a range of k 2 as suggested by Fraas [16], or possibly pion electroproduction ~rt larger angles where the isoscalar (co-exchange) amplitude is more important, in order to understand the details of the mass extrapolation. In the present work we assume that the relevant p and co Regge pole exchanges factorize so that gpto~r can be isolated in each of the natural parity helicity amplitudes. In principle, the associated Regge cut contributions do not factorize, but since it was found that changing the relative strengths of pole and cut contributions did not affect the results to any great extent, we made the simplifying assumption that the cut contributions depend linearly on gpwTr as well. Consequently, any dependence of the 0co~r vertex on the mass extrapolation is parametrized simply by an overall weight factor for the co-exchange contribution to p-+ production and the p-exchange contribution to co production. Since different extrapolations are involved in each case, these weights are independent. The possibility of additional k 2 dependence in the other exchange contributions is ignored. In particular, the pion contribution is defined by the electric Born term for photoproduction and its generalization for/9 production as discussed in I. Thus the relative strengths in photoproduction and vector meson production of contributions independent of the pcolr coupling are assumed to be given solely by the photon-vector meson coupling g~v- In the absence of independent evidence for additional k 2 dependence, this is the simplest possible assumption, and its reliability is borne out by the success of the present model:

546

B.H.Kellett, Vector meson production

4. PREDICTIONS FOR p AND co PRODUCTION The input for the prediction of vector meson production from the VMD model of I is the detailed model fit to pion photoproduction of ref. [3]. The modification of this original fit to include the more recent photoproduction data has been described in I, and the reader is referred to these papers for a discussion of the model. Performing the kinematic extrapolation by using the explicit formulae of I, and the dynamical gpto~r extrapolation by introducing a free parameter as described in the previous section, reasonable results were obtained for p-+ production, but the differential cross section for n+n --+ cop was not well reproduced. Since co production exposes the isoscalar amplitude for pion photoproduction, this failure was not considered catastrophic. Although the ratio of 7r- to n + photoproduction is quite accurately known over a wide energy range, the ratio of 7r° photoproduction off neutrons to that off protons is very poorly determined, so that considerable flexibility in the isoscalar amplitude is allowed by the present photoproduction data. The gross features, such as the absence o f a dip in the isoscalar amplitude around t ~ - 0.6 (GeV/c) 2 indicated by the smooth behaviour of the lr-/Tr÷ ratio in this region, correlate well with the observed absence of a dip in n+n ~ cop, but the finer details are not well determined by the photoproduction data alone. The absence of a dip at the wrong signature nonsense point of the p trajectory in Ir÷n ~ cop has aroused considerable interest in this process. Also, the fact that the density matrix element Poo is non-vanishing means that simple p exchange alone is unable to account for the data. The solution to the difficulty has been sought in the introduction of the unnatural parity B trajectory [ 17-21 ]. The slight indication of a dip in Poo at t ~ - 0.2 (GeV/c) 2 has been associated with the nonsense zero o f the B [19, 21], but the evidence for this structure in Poo is not conclusive at all energies, and for the nonsense zero to be in this region, the slope of the trajectory must be very small. In general, because of the large mass of the B, important B contributions at higher energies require an abnormally flat trajectory. An alternative explanation has been proposed by Henyey et al. [22] who obtained qualitative fits to the data with strongly absorbed p exchange without the necessity for a B contribution. This explanation of the unnatural parity exchange in terms of the p cut may be preferable to an abnormal B trajectory, but the data can be understood in either model. The results of ref. [3] indicate that the B is not necessary in order to fit pion photoproduction, which is consistent with its having a low-lying trajectory. Instead of including additional terms such as the B in the fit, therefore, we use the co production data as an additional constraint on the p pole and cut contribution to the photoproduction isoscalar amplitude. The idea is then to see if a simultaneous fit is possible within the framework of the present VMD model. We use the same photoproduction data as in our earlier work [1, 3], and the co production data are taken from refs. [ 2 3 - 2 7 ] . The parameters are defined in refs. [1] and [3], and the values resulting from the present fit are shown in table 1. Comparison o f these numbers

547

B.H.Kellett, Vector meson production Table 1 Revised values of the parameters of the model for pio'n photoproduction of ref. [3]. F~+)

F(1-)

F(2+)

(ub~" GeV-1)

0zb½. GeV - 3 )

(#b~ : GeV -2)

#p

16.7

10.4

#to

Cut discontinuity t dependence (GeV-2)

- 108.1

- 164.8

#A2

29.8

-95.0

Cpp

- 356.7

- 8.6

37.8

2.86

Crop

23.4

28.0

127.3

0.72

- 140.9

311.8

- 222.4

2.03

CA2P

Born term form factor: A = 1.134, b = 1.45 GeV-2; ap = 0.51 + 0.8 t, ato = 0.53 + 0.8 t, aA2 = 0.4 + t. with the earlier results shows that the general features of the fit to photoproduction are not significantly altered. We have simply made use o f the fact that photoproduction does not give an accurate determination of the isoscalar amplitude, and have adjusted the p pole and cut contributions to reproduce to production. The isovector amplitude is essentially unchanged within the errors on the fitted parameters. The fit to the p h o t o p r o d u c t i o n data is very similar to that of ref. [3], and the reader is referred to that paper for figures and further discussion. The 60 production results are shown in figs. 1 and 2 for incident pion m o m e n t a between 2.7 and 6.95 GeV/c. The rt÷n ~ top cross section normalization is determined by the variation of gp~o~r discussed above. We found

gpw~(k 2 = mL)/gpwTr(k 2 = 0) = 0 . 4 5 ,

(14)

where k is the external 60 momentum. This value is dependent on our choice of f2/41r = 14.0, and since it gives an overall normalization, the to density matrix is independent of this parameter. In fig. 1 the to production differential cross section data is compared with the resuits of the VMD calculation. The deviation from the measured cross section at 2.7 GeV/c is not o f great significance since this corresponds to a centre of mass energy of 2.4 GeV, which is lower than the energies considered in photoproduction, and it is possible that there are residual direct channel resonance effects at such energies. The energy dependence a n d apparent absence o f shrinkage are well, reproduced, although the range o f energies for which data are available is rather restricted. From fig. 2 it can be seen that the general features of the to-decay density matrices in the Jackson frame are reproduced b y the model. The data at the different energies are not completely consistent, although the small magnitude o f p I - 1, and the fact that

548

B.H.Kellett, Vector meson production

m ~

,

IT+n .-), wp

.

,.ot

1T°:

~-

I

2!"0'

r~

0.1"

o0,

5.1 GeV/c

. . . . . . . . 0"2

0"4

t 0"0 0"0 - t (GeV//c);

+\1"0

1"2

1.4

Fig. 1. Differential cross section for 7r+n-+ top between 2.7 and 6.95 GeV/c. The data axe from refs: [23-27]. the p lo is small and negative are natural features of the model. We predict a large Poo for small t in accordance with the data, b u t Poo falls off rather too quickly with increasing - t. Overall, however, the p pole plus cut model o f the photoproduction isoscalar amplitude gives a reasonable description of co production. No doubt a more accurate fit to ¢r+n --* cop is possible, just as alternative fits to the isoscalar amplitude in pion photoproduction are possible with a more sophisti-

549

B.H.Kellett, Vector meson production

Tr+n ,~ ~p po:'0I~

1 " 0 ~ Poe

2.'/GeV/c

4-19GeV/c I

0'5

o,

, "t~,~-~ : 0'2

0.4

J , 01

ole

0"6

iI i

- t

i

!

04

•

/

i

1~2 -'t

0-8

o5

r~.

P1-1 '

t'

,

T

-0"5

-0.55L

o t ~ T ~ !

,

Rep1o ~ -0.5

F '

'

1.0~

*

,

J

'

'

-

.

i

1"0r~

5.1 GeV/c

i

i

i

6.95GeV/c

poe

o.s-,~,~ i

t

1

i

0'2

0.4

0.6

i ~ 0.8

- t

~ 0.4

0.8

lr.2

- t

,

,

0-5

-0'5 [

-0.8

~

0

Rep,I

Replo -0.5

,

,

,

,

,

-0.5 L

~ ~

,

,

i J

i

Fig. 2. Density matrices for ~r+n--, top between 2.7 and 6.95 GeV/c. cated parametrization and the inclusion o f additional small contributions. On the other hand, the present results show that w production is perfectly compatible with existing fits to pion photoproduction via vector meson dominance, with the introduction o f only one further free parameter. Turning now to p production, we can use the model of I to predict p± as well as #o production. In order to incorporate this into the general pattern with w production, we use the p h o t o p r o d u c t i o n parameters of table 1 determined in the simultaneous fit described above. We retain -f214~r = 2.0, and again introduce a further p-

B.H.Kellett, Vector meson production

550

parameter to allow gp~,r to vary off mass shell. Because the isovector amplitudes are essentially unaltered in the new photoproduction fit, the pO predictions are the same as in I, and the reader is referred to that paper for figures and further discussion. The predictions for the charged p production data of refs. [28-34] are shown in figs. 3 - 6 . As discussed above, the w-exchange contribution to p± production is expected to be small, and we find 2 gp<.o~(k 2 = mp)lgpoo.(k

2

= O) --- 0 . 1 2 ,

(15)

lr- p.,.p*p

10"0

1.0

i

l.0

1.0

o.1. o.oi

~ e ~ . v / o

~

~

~

o

a.v/,=

o:1 o'.l o:3 o~4 o'.s o'.0 o'.7 -t (GeV/c)2

Fig. 3. Differential cross section for *r-p - , p - p at 2.7, 4.16, and 8.0 GeV/c. The data axe from refs. [ 2 8 - 3 0 ] .

551

B.H.Kellett, Vector meson production

10-0

n~p ..~ p~'p

1.0"

4'0 GeV/¢

,.-, >

0.1

E

0"01

8"0 GeV/e

0,001

0.5

t-0

1.5

Fig. 4. Differential cross section for ~r+p~ p+p at 4.0 and 8.0 GeV/c. The data are from refs. [32] and [33]. where k is now the external 0 momentum. The zr-p -+ p - p data do not extend to large enough values of - t for the presence of a dip in the cross section due to 60 exchange to show in the data, and in fact the model predictions of fig. 3 shown a completely smooth behaviour. The zr+p ~ p+p cross section data, on the other hand, extend to large - t at 4 and 8 GeV/c, and do show some indication of a dip at t ~ - 0.5 (GeV/c) 2. Because of the suppression of co exchange in the model, however, a definite dip is not reproduced, although there are signs in fig. 4 of a change

552

B.H.Kellett, Vector meson production

w-p--)p-p 2-?GeV/c : ~

1"0

3.0GeV/c

,' ~1

~2

0"4

~3

I

"

P

I

0"2

I

0"4

010

-t

!

,o

-0.2 L

-0.2 ,

I

I

'

'

f

ReP+o -0.5

,

~

,

,-0.5

1.0[+ 4.16GeV/c P" ' -T~<'~-.~ -'r

01

I 0-1

I 0"2

l

,

1'0 -~ f

I

°+I

-015L

-t

I

01

,

8"OGeV/c

S°*

0"3

i

0!1

0!2

0i.3

-t

I

-0.5 t

~--""~Rep Io l

i-O. 5 - -

J

I

I

I

Fig. 5. Density matrices for n-p --, p-p at 2.7, 3.0, 4.16 and 8.0 GeV/c. The data are from refs. [28-31]. Notice that P l - 1 is essentially zero for - t < 0.6. of slope, which is not inconsistent with the data. Figs. 5 and 6 show that the model gives a good representation of the p± decay Jackson frame density matrices. In particular, the fact that the correct magnitude and slope are obtained for Poo indicates that the model is able to predict the longitudinal polarization states o f the vector mesons from the purely transverse photon.

B.H.Kellett, Vector meson production

553

w+p ,.-) p÷p

"°k

1 P

I r

8"0 GeV/c

0.5

ol

p,.°'if~, '

-0.5 L

-0.5~.

'=.,V :

,

0-2

',

0.4

'

~ 0.6

-t

[ ,

!

Fig. 6. Density matrices for 7r+p ~ p+p at 4.0 and 8.0 GeV/c. The data are from refs. [33] and

[34].

5. DISCUSSION A unified description of pion photoproduction and vector meson production by pions has been one of the major objectives for the application of the hypothesis of vector meson dominance of the electromagnetic current. The extension of the covariant formulation of VMD introduced in I to include co production and charged p production in this paper shows that such a description is possible. Starting from a detailed parametrization of pion photoproduction, we can isolate the pure isospin states, and from these obtain a description of the hadronic vector meson reactions in terms of the same basic set of parameters. The extrapolation from the photon to the vector meson is to be seen as primarily kinematical, and the minimal dependence of the dynamics on the external mass can be parametrized in a very simple and economical way. At first sight it might seem surprising that the continuation ofgoto~ to the P pole should give a result approximately a factor of four smaller than that obtained on continuation to the co pole. The errors on the ratio in eqs. (14)and (15) are difficult to estimate since these values are dependent on the relative normalizations of different experiments and on our chosen values for fp and fro. Furthermore, the continuation to the p and 6o poles affects the co and p Regge contributions to pion photoproduction respectively, so that the ratios we obtain are to some extent dependent on the model used for pion photoproduction. The various current algebra calculations of the mass dependence of the peon vertex [11-14] do not give

554

B.H.Kellett, Vector meson production

a consistent picture. The calculations of Perrin [ 12] and Nagy [ 14] would indicate that when the pion is on the mass shell, the continuations to the p and co poles should be similar, whereas the model of Brown and West [11] predicts that the ratio (14) is approximately a factor of two greater than the ratio (15) for the pion on its mass shell and t = 0 for the exchanged vector particles. The relevance of these results to the present work where one of the particles is Reggeized is not, however, completely clear. Our phenomenological results for the mass dependence clearly come from a rather simple-minded parametrization of a complicated dynamical situation. For instance, it can be seen from the parameters of table 1 that the relative strengths of the pole and cut contributions for p and co exchange are quite different, so that p and co exchange are important over different ranges of t. Consequently, it is difficult to know how far off the mass shell to take the Reggeized particle when comparing with the current algebra predictions. An additional factor of two could be easily accommodated in the model of Brown and West by a judicious choice of virtual mass for the reggeon. Because of these uncertainties in the comparison with current algebra, we do not feel that it would be useful to pursue the matter further at this stage. The phenomenological results for the mass dependence contained in eqs. (14) and (15) are presented as an indication of a physical effect that should be taken into account in any attempt to construct a more complete dynamical theory. The present VMD calculations of vector meson production from pion photoproduction cannot be considered to be a complete dynamical theory because of the important role played by a particular model fit to the photoproduction data. This model is not unique, and the photoproduction fit is to be understood as a parametrization of the data rather than as a complete dynamical theory. We have, in fact, made use of the flexibility of the model in order to fit all the data. On the other hand, the formulation of VMD is essentially unique in that the kinematics are well defined, and there is no frame ambiguity associated with the polarization states of the massive vector mesons. In the context of the VMD model, we find that the possibility of a simultaneous description of many different reactions provides significantly greater constraints on the basic amplitudes. Therefore, compatability with the rest of the data should be considered an important test of the reliability of any model for an individual reaction. The present model for VMD has been shown to be very successful in the extrapolation of the photon mass to the timelike region. In principle, the model should also be applicable to the extrapolation to spacelike values of k 2, and thus the incorporation of single pion electroproduction into the same basic framework. This possibility will be investigated in a separate publication. The author should like to thank Professor A.Donnachie for a critical reading of the manuscript. Use of the computing facilities at the Daresbury Nuclear Physics Laboratory is also gratefully acknowledged.

B.H.Kellett, Vector meson production

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