NUCLEAR PHYSICS A

ELSEVIER Nuclear Physics A 601 (1996) 425-444

S- and P-state annihilation in interactions at rest

C.J. Batty Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OXl l OQX, UK

Received 31 October 1995; revised 18 January 1996

Abstract

Two body branching ratios for ~p annihilation at rest are used to calculate the fraction of P-state annihilation as a function of target density. In the presence of Stark mixing it is necessary to take into account an enhancement of annihilations from fine structure states over that expected from a statistical population. The enhancement factors are obtained using an atomic cascade calculation which fits available ~p atom X-ray measurements.

PACS: 13.75.Cs; 36.10.Gv

Keywords: Antiproton-proton annihilation; Exotic atoms; Atomic cascade

1. Introduct ion

Following a suggestion by Madansky, the important role of the Stark effect in the cascade of 7r-p and K - p atoms formed in liquid hydrogen was first discussed by Day,

Snow and Sucher [1 ]. In particular, they pointed out that since the 7r-p or K - p atom is small and electrically neutral, in collisions with neighbouring atoms it will move through

regions of high atomic electric field and undergo Stark effect mixing between states of different angular momentum. This leads to absorption from low angular momentum states at high n values and so prevents the (~p) atom from reaching low-lying levels by

radiative transitions. Since the Stark mixing is proportional to the collision rate, and hence target density, the effects are largest in liquid hydrogen. Consequently, atomic physics processes, some of which are dependent on the target density, play a key role in determining the properties of the cascade, such as X-ray yields and cascade time, as well as the orbital angular momentum of the final meson-proton hadronic interaction which terminates the cascade.

These conclusions of Day, Snow and Sucher were placed on a more quantitative basis by the work of Leon and Bethe [2]. They carried out a detailed study of the atomic cascade and showed, for a given choice of values for the K - p strong interaction, that in

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426 C.J. Batty~Nuclear Physics A 601 (1996) 425-444

liquid hydrogen less than 1% of K mesons will interact from atomic P states. An impact parameter method was used to calculate the Stark mixing. These calculations were later modified and extended by Borie and Leon [3] and also carried out for antiprotonic hydrogen atoms. Of particular interest for the present work, calculations using the Borie and Leon model [3] have been used [4] to fit the measured yields as a function of target

density for K and L X-rays from antiprotonic hydrogen (~p) atoms. Here we are concerned with the fraction of ~p atoms which annihilate from atomic S-

and atomic P-states. Annihilation from states l/> 2 can safely be ignored due to the negligible overlap of the p and ~ atomic wavefunctions. The work is particularly relevant to current studies at LEAR of ~p annihilation channels and the production of meson resonances using antiprotons stopped in liquid or gaseous hydrogen. The results, usually presented in the form of a Dalitz plot, are generally analysed using partial wave analyses with some assumptions as to the angular momentum states involved in the initial ~p annihilation; generally S-state annihilation only in liquid hydrogen, P-state in low pressure gas and roughly equal mixtures of S- and P-states in hydrogen gas at STP. However the observation [5] in liquid hydrogen of the annihilation process ~p ~ ~r°Tr °, which is forbidden from initial states with even orbital angular momentum, indicates that there must be some P-wave annihilation. Similarly the observation [6] in low pressure targets of the process pp ~ KsK L, which proceeds only from even 1 states, shows that there must be some S-wave annihilation in low density gas targets.

Measured branching ratios for the production of two-body final states have been used [7], together with similar measurements using an L X-ray trigger to preferentially select 2p-states, to directly obtain quantitative estimates of the relative S- and P-state contribu- tions to ~p annihilation at rest as a function of target density. However this method either ignores any differences between the annihilation widths of the fine (and hyper- fine) states of the ~p atom or assumes that the relative populations of the atomic fine structure states are independent of the target density. The present paper particularly addresses these last two problems.

The layout of the paper is as follows. Section 2 discusses the atomic cascade and the role of the S- and P-wave fine structure states. Section 3 discusses the use of two-body branching ratios to determine the S- and P-state annihilation probabilities as a function of target pressure, both without and with the inclusion of effects due to the differing populations and widths of the fine structure levels. The calculation of these latter effects using the Borie and Leon model [3] is described in Section 4 and the resulting S- and P-state annihilation fractions derived from two-body branching ratios are presented in Section 5. The final section discusses the method used and results obtained together with a short summary.

2. The atomic cascade

The general features of the ~p atomic cascade have been discussed elsewhere; see Ref. [4] for a review of processes involved in the formation of the ~p atom and its

C.J. Ba t t y ~ Nuc l e ar Physics A 601 (1996) 4 2 5 - 4 4 4 427

n

3O

l 0 l 2

"•AnC•nihilatio7

- ' ~ ,! i "7--

g ; ,(,,, '2,,' /

3 4 29 / /

/ /

5g

4f 3d

Annihilation

I Is

Fig. 1. Level scheme for pp atoms, showing X-ray transitions between low lying states, Stark transitions at high n and annihilation from S- and P-states.

subsequent cascade. Briefly, the antiproton is captured by the proton into a high Bohr

orbit with typically principal quantum number n--~ 30. The highly excited atom then

de-excites by a number of processes including radiative transitions, with the emission of

X-rays and the external Auger effect involving the ionisafion of a neighbouring H 2

molecule. Once the antiproton reaches a state of low angular momentum (either S- or

P-state), annihilation due to the strong interaction can occur and the cascade is

terminated. Fig. l illustrates some features of the atomic cascade which are of particular

importance in the present work.

The Stark effect gives mixing of the angular momentum states at high-n which allows

the antiprotons to transfer to S- and P-states where they can annihilate before reaching the low-n states. In liquid hydrogen the probability of Stark transitions from P- to

S-states is a factor of ~ 104 larger than the probability of P-state capture for n = 30, a

factor ~ 200 larger at n = 20, ~ 5 at n = 10 and ~ 0.1 at n = 5. As a result the X-ray yields of transitions between low-n states are reduced, the cascade time is also reduced

and annihilation from S-states is enhanced. At a target density of 0.001 PSTP (where PSTP is the density of H 2 gas at STP) the corresponding factor varies between 10 -3 and 10-7 indicating that Stark mixing now plays only a minor role in the atomic cascade.

428 CJ. Batty~Nuclear Physics A 601 (1996) 425-444

Table 1 Predicted widths for S- and P-states in pp. Values predicted by the DRI, DR2 and KW optical potentials are taken from Ref. [8]

Width DR 1 DR2 KW

2 So (keV) 1.02 1.04 1.26 t S 1 (keV) 0.90 0.92 0.98 l Pi (meV) 26 28 26 3P 0 (meV) 112 80 96 3P 1 (meV) 20 18 22 3P 2 (meV) 30 32 36

So far we have ignored the effect of the fine (and hyperfine) structure of the atomic states. For ~p atoms the states with l < 2 are ~S o, 3S1, IP 1, 3P o, 3p! a n d 3P 2 with corresponding jPc = 0-+, 1 - - , 1 +-, 0 ++, 1 f+ and 2 ++. No direct measurements of

the strong interaction (annihilation) widths of these states are at present available. Here we use the predictions of Carbonell et al. [8], given in Table 1, which were obtained using optical potentials for the ~ -p interaction due to Dover and Richard (DR1 and DR2) and Kohno and Weise (KW). Whilst these models only describe the overall features of the antinucleon-nucleon interaction, more recent analyses generally do not give detailed predictions for the widths of the fine structure states relevant to the present work. However a particular feature of all these predictions is the very large width of the 3P 0 state due to the dominance of tensor forces. These relatively large differences in the P-state widths result in two effects which are particularly relevant to the present work.

The yield of radiative (X-ray) transitions from P- to S-states is given by

r ( x ) X-ray yield = ot i F ( X ) + / ' ( P i ) ' (2.1)

where ai is the population of the state, F(X) is the width for X-ray transitions and F (P i) is the strong interaction width of the state Pi with total angular momentum J~. For the case where Stark effects are small and the population of P-states is according to their statistical weight

4

o,, = (2J, + 1 ) / E (2J, + 1), (2.2) i = l

the competition between radiative transitions and annihilations (see Fig. 2) will lead to the ~S 0 and 3S1 states having populations which are not statistical. For example using the strong interaction widths given in Table 1 for the DR1 model results in the ~S 0 population being enhanced by a factor 1.047 above the statistical value and that of the 3Sj state depleted by a factor 0.984. For the DR2 model the corresponding factors are 0.968 and 1.011, whilst for the KW model they are 1.151 and 0.950, respectively.

This effect is most important for states at small n-values where radiative transitions play a significant role. At high n, Stark mixing is larger and can be dominant as we have illustrated earlier. Here the fine structure levels are continually and rapidly repopulated according to their statistical weight. A fine structure level with a large

C.J. Batty/Nuclear Physics A 601 (1996) 425-444 429

1P t

~ r(IPl )

'so/ r(~pp

r(~sd ~ 3P 1

rdP,)

3p °

3 S I /

F(~Sj) ~(

Fig. 2. Level scheme illustrating how differing annihilation widths / ' (P) of P-states affect the population of S-states by X-ray transitions.

strong interaction width will therefore contribute more to annihilation than would be expected from its statistical weight only. This effect is particularly important for

annihilations from the 3P 0 level where the annihilation width (Table 1) is much larger than for other P-states. On the other hand if the repopulation of fine structure states due

to the Stark effect is small, as is the case in low pressure gas targets or at low n, the states will annihilate according to their statistical weight.

Both of these effects can be important when considering the analysis of two-body annihilations in terms of their initial state. In particular the fraction of annihilations from the two S- or four P-wave fine structure levels will no longer necessarily be according to their statistical weight and also may well be a function of the principal quantum number n of the atomic state. As a consequence the fraction of ~p annihilations arising from a particular fine structure level may well be "enhanced" over that expected from a purely statistical population of the level. In the present work we describe this enhancement in terms of an "enhancement factor" E which is a function of the target density p. Values

of E < 1 correspond to a fraction of annihilations less than that expected on the basis of a purely statistical population of the level.

The role of initial S- and P-states in the annihilation process has been discussed in detail by Gastaldi et al. [9]. The relationship between the enhancement factors E used here and their work is discussed in the appendix. Here for convenience we note that for either S or P states only i.e., either L = 0 or 1 then

]~w(SLs) = 1 (2.3) S,J

430

Table 2 Two-body reactions

C.J. Batty ~Nuclear Physics A 601 (1996) 425-444

Allowed initial states

~p' ~ Ir°rr ° 3p 0 3p 2 __, ~-+ ~-- 3S l 3P o 3p 2 --~ KsK L aS l --* KsK s 3P o 3p 2

K + K - 3S 1 3P 0 3P 2

and

~',E(SLj, p)oj(SLj) = 1, (2.4) S,J

where to(SLj) is the statistical population of the state SLj and is separately normalised

to unity for S- and P-states.

3. Use of two-body branching ratios

The two-body reactions of particular interest for the determination of the fractions of S- and P-wave annihilations in ~p annihilations at rest are listed in Table 2, together with the allowed initial states. Table 3 gives the available branching ratios for these reactions and for the ratio of branching ratios BR(~p-- ,K+K-) /BR(~p~ ~-+~--),

Table 3 Measurements of two-body branching ratios. The branching ratios are in units 10 -a. Measured values are taken from Refs. [5,10-17] as indicated

Reaction Density (PSTP)

0.002 0.005 1 15 liquid

"z'+ 7r - 4.26-1-0.11 4.30+-0.15 [10] [111

7r% -° 1.27+0.21 [12]

KsK s 0.03 +- 0.01 [13]

KsK L 0.36+-0.06 [13]

K + K - 0.46+-0.03 0.69+-0.04 [10] [ l l l

(~'+ ~'- )x 4.81 +-0.49 [11]

(K + K- )x 0.29+-0.05 [II1

(K s Ks) x 0.037 4- 0.014 [13]

Ratio of branching ratios (K+ K-/Tr+Tr - ) 0.10+-0.02 0.11+-0.01

[lO1 (K + K-/ , rr+~r- )x

0.16+-0.01

0.06 +- 0.01

0.21 +0.02 [141

3.07+0.13 [15] 0.69 + 0.04 [5] 0.004 + 0.003 [13,16] 0.9 + 0.06 [17] 0.99 + 0.05 [151

0.32+0.01

C.J. Batty/Nuclear Physics A 601 (1996)425-444 431

which has been directly measured in some cases, for values of the target density in the range from 0.002 PsvP to liquid. The subscript X e.g. (~-+ 7r-) x indicates those cases when the branching ratio was measured for the reaction in coincidence with L X-rays so that annihilation only occurs from 2p states. The value for the branching ratio BR(~p KsK L) = (9.0 + 0.6) × 10 - 4 is taken from a recent Crystal Barrel collaboration paper [17], where they point out that the result is in agreement with the result (9.2 + 0.3) × 10 - 4

obtained in an early unpublished bubble chamber measurement. Both these results are significantly larger than the value (7.6 + 0 .4)x 10 - 4 quoted in Ref. [13]. However using this latter value would not significantly alter the conclusions of the present paper.

3.1. Analysis without enhancement effects

We first consider the analysis of the two body branching ratios to give relative S- and P-state annihilation without including enhancement effects. This ignores differences in the annihilation widths of the fine structure components for the S- and P-states and assumes that the only effect of Stark mixing is to vary the relative magnitudes of the total S- and P-state contributions to the annihilation as a function of target density. Here we follow the notation used in an earlier paper of Reifenrother and Klempt [7] describing a similar analysis of the more limited range of data available at that time.

Following Table 2, the branching ratios listed in Table 3 are then given by:

BR(~'+ 7r - ) = (1 - f p ) B 0 +fpB l, (3.1)

BR(TrOTrO) = l 7fpB,, (3.2)

BR(~-+ Zr)x = B,, (3.3)

BR(K+K - ) = (1 - f p ) C 0 +fpC l, (3.4)

BR( K+K- )x = C1, (3.5)

BR(K+K-) /BR(Tr+Tr - ) = {(1 - f p ) C 0 +fpCl}/{(1 - f p ) B o +fpBx}, (3.6)

BR(KsKs) = ½fpD,, (3.7)

BR(KsKL) = (1 - f p ) D 0, (3.8)

B R ( KsKs ) x = 2D,,1 (3.9)

where fp is the fraction of annihilations from P-states. The branching ratios from S and P states are denoted by B 0, C 0, D O and Bt, Ci, D~ respectively. Note that fp and the measured branching ratios are a function of the target density, whilst the coefficients B o, B~, C 0, C~, D 0, D~ are independent of density and /> 0. Eqs. (3.3), (3.5) and (3.9) assume that for the branching ratio measurement with L X-rays (nd--, 2p radiative transitions) in coincidence, fp = 1 and the partial branching ratios BI, C~ and D~ for annihilation from P-states are independent of the principal quantum number n of the state from which annihilation occurs.

4 3 2 C.l. Batty/Nuclear Physics A 601 (1996) 425-444

The results from an analysis of this type will be briefly described in Section 5.1.

3.2. Analysis with enhancement effects

When the fine structure components of the P-state are taken into account, Eq. (3.2) can be written in the form

BR(~'°Tr°) =fp{~2BR(3po ~ ¢r%r °) + 5BR(3p2 ~ 7r°Tr°)} (3.10)

and when the enhancement factors are included

BR( o o) = p{ E(3po)BR(3p o E(3p2) R(3p2 (3.11)

Here the factors ~2 and ~ are the statistical weights of the states, E(3Po ), etc., are the enhancement factors and BR(3po ---) 7r°qr°), etc., are the partial branching ratios. The general expression for any process of the type ~p --* yy can then be written

BR(~p -0 y y ) = {1-fp}(¼E(ISo)BR( 'So -o y y ) + 3E(3SI)BR(3S I --~ yy)}

+ fo{3E( ip, )BR( 'Pl ~ YY) + T-2E(3Po)BR(3po -o yy)

+3 (3p I )BR(3p I -o yy) + ~E(3p2)BR(3p2 ~ yy)}. (3.12)

The partial branching ratios depend on SLj whilst the enhancement factors E(SLj,p) depend both on SL s and on the H 2 target density in which the ~p atoms are formed. Note that the statistical weights of the states are separately normalised to unity for S- and P-states.

Since the lP 1 and 3P 1 states do not contribute to the annihilation processes considered in this paper we can, for simplicity and without ambiguity, write the enhancement factor E(SLs,p) in the form ELj and the partial branching ratio BR(SLs --* yy) as BLj. Using Eq. (3.12) the equations corresponding to Eqs. (3.1) to (3.9) can then be written in the general form

BR(qr + "rr-) = (1 -fp)3EolB m +fp(~EloBlo + 5EI2BI2), (3.13)

I l BR( 7r%r °) = 7fp(~E,oB,o + q~El2 bl2 ), (3.14)

BR( 7r + 7r- ) x = T2BI0 + ~2B,2, (3.15)

BR(K+K - ) = (1-fp)3Eo,Co, +fp(~2E, oC,o + -~E12C,2), (3.16)

BR(K+K- ) x = ~C,o + ~C12, (3.17)

B R ( K + K - ) / B R ( . + I t - ) = {(1 -fp)3Eo,C m +fp(~E,oClo + ~E12C,2)}

/{(1 -4) eo,So, +

(3.18)

BR(KsKs) = ½fp(~2E,oDio + ~EI2D,2 ), (3.19)

C.J. Batty~Nuclear Physics A 601 (1996) 425-444 433

BR(KsKL) = (1 3 - fp )'~ gol Ool , (3.20)

= , 1 + ~2D,2), (3.21) B R ( K s K s ) x 7(q~Dj0

where both fp and ELj depend on the target density. The partial branching ratios are independent of the target density but Btj, CLj, DLj >/ 0. With only a limited range of branching ratio measurements available it is not possible to determine the enhancement factors E ( S L s , p) and BR(SL~) experimentally. In the next section we discuss the determination of the enhancement factors using a cascade calculation.

4. Cascade calculations

The enhancement factors discussed in the previous sections were calculated using a cascade calculation based on the method of Borie and Leon [3]. In this method the effects of Stark mixing are calculated using an impact parameter technique in which, in its interaction with the electric field of the neighbouring H 2 atoms, the exotic atom is treated as moving along a definite straight-line trajectory. If an S state is involved, the removal of the degeneracy of the energy levels due to the energy shift caused by the strong interaction hinders the Stark mixing, since the electric field must overcome the energy difference between S and P states. This is taken into account by using a smaller impact parameter. Because of uncertainties in the absolute value of the rate for Stark mixing, an overall normalisation parameter ksT n is usually used and its value deter- mined by fits to X-ray yield values.

An alternative method, the so-called "Mainz model", has been used by Reifenr~ther and Klempt [18]. Here the collisions of the ~p atoms with neighbouring H atoms are simulated using the classical-trajectory Monte Carlo method. This method only allows Stark mixing when the ~p atom experiences strong electric fields during collisions with neighbouring hydrogen atoms, so that the Stark mixing varies with time. On the other hand the Borie and Leon model treats Stark mixing as a continuous process but the calculations are simpler.

Once the parameter ksT K in the Borie and Leon model has been adjusted to fit the X-ray yield data, the two cascade models give similar predictions for the overall variation in the X-ray yields and cascade time as a function of target density [4]. However the calculated P-wave annihilation probabilities differ significantly [4] with the Borie and Leon model predicting values of fp ~ 50% in liquid hydrogen.

It seems likely that the unrealistically large value of fp predicted by the Borie and Leon model could be associated with the approximations used in the calculation of Stark mixing, particularly for transitions between S and P states where the effects of the strong interaction shift need to be included and where the Mainz model obtains very high rates. With these considerations in mind, in the present work an additional normalisation parameter K 0 was used for the rate of Stark transitions between S and P states only i.e., S ~ P and P ~ S transitions. As will be shown later, the inclusion of this additional parameter gives significantly improved fits to the X-ray yield data and much reduced values for fp ~ 15% in liquid hydrogen.

434 C.J. Batty / Nuclear Physics A 601 (1996) 425-444

4.1. Fit to X-ray yield measurements

In any treatment of the ~p atomic cascade it is necessary to include effects due to the

strong interaction. Measured values for the energy shift and width of the 1S ground state

AEls = - 7 3 0 eV, F~s = 1130 eV and for the width of the 2P state F2p = 34 meV were

used. These are the average of values summarised in Table 11 of Ref. [4] and the later

values of Ref. [19]. The rates for higher n states were then assumed to scale as

finS = / ' I s / n 3 ' (4.1)

32 [ n 2 - 1 F r p = - - ~ - [ ~ } F 2 p . (4.2)

These formulae, which were originally derived by West [20] for pionic atoms, have

been used extensively in exotic atom cascade calculations. Their validity has recently

been discussed [21] where it is shown that very good agreement is obtained with more

exact calculations for ~p atoms.

The X-ray yield data to be fitted were taken from the summary of measurements

given in Tables 8 and 9 of Ref. [4] and the more recent values of Ref. [19]. In all there

are 9 values of the K,~ yield, 6 of K s yield and 8 of K ~> r yield (K r and higher K

transitions) spread over 7 different target densities; in some cases there is more than one

measurement at the same pressure. For the L X-rays there are 14 values for the L~ yield

and 13 each for the L~ and L~ ~, X-ray yields at a total of 11 different densities.

The starting value for the atomic cascade was n = 30 and the population of the

angular momentum states was assumed to be statistical. The kinetic energy of the 15p

atom was taken to be T = 1 eV as used in many earlier calculations. It has been checked

that the final results in the present work are insensitive to the value assumed for T.

In the first set of calculations the value K 0 = 1 was used. That is the Stark transitions

between S and P states were taken as in the original formulation of Borie and Leon [3].

Adjusting ksv n then gave a best least squares fit to the yield data with ksv K = 1.28 ___ 0.05 and X 2 = 170.1 which for 63 data points gives a X 2 per degree of freedom X 2/N = 170.1/62 = 2.74. Allowing the additional Stark mixing parameter K 0 to vary, gave a

significantly improved fit with K 0 = 7.6 + 2.6, ksv K = 1.19 ___ 0.06 and X 2/N = 130.6/61 = 2.14. The marked decrease in X 2 from 170.1 to 130.6 is largely due to a

much improved fit to the higher K ~ v X-ray yields where the partial X 2 decreases from

55.1 to 18.6 for 8 data points. The fits to the experimental yield values with both K 0 = 1 and with K 0 = 7.6 are

shown in Fig. 3. The rather poor x 2 / N = 2.14 obtained in the best fit, is seen to be due

to the somewhat inconsistent measurements from different experiments, reflecting the

difficulties in measuring absolute X-ray yields in the energy region from 1.5 to 12 keV.

4.2. Calculation of enhancement factors

In order to calculate the enhancement factors in the cascade calculation it is necessary

to include the differing populations and strong interaction widths for the ~S 0, 3S l, ~PI,

CJ. Batty/Nuclear Physics A 601 (1996) 425-444 435

3p0, 3P 1 and 3P 2 fine structure states as a function of the principal quantum number n.

At n = 30, a statistical ( 2 J + l ) population of the fine structure states was assumed. For

all n, fine structure effects for l > 1 were ignored.

Stark transitions from D ~ P states were assumed to populate the P states according

to a statistical distribution. From P to S states there is no spin-flip and the al lowed

A

0.01

0.0O9

0.O08

O.007

O.OO6

t~ 005

0.004

0.003

O.002

0.001

0

I0.2

KaX-ray K~X-ray

• PSI71

" • PS174

~ I & PSI75

i i i ~J

io .t t /o 1o 2

Psrr

x I0 -2

0.2

0.18

0.16

0.14

0.12

0.1

~' 0.08

O.O6

O.O4

0.02

0 I l i l l l l l [

io -2 io -1

• PSI 74

/ 1o 1o 2

Psre

0.01

0.oo9

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

K_> r X-ray

• PSI 74

& P S I 75

0.02

0 .018

0.016

0.014

0.012

0.01

~' o.0o8

0.006

0.004

O.OO2

0

1o .z io 4 I to io z

Ps'rp

Kr~ , X-ray

• P S I 7 4

& PS175

1o ~ /o "~ i /o /o 2

Psn,

Fig. 3. Predicted (a) K-series and (b) L-series X-ray yields for ~p atoms as a function of gas density using best fit parameters with K o = 1 (dashed curve) or K o = 7.5 (full line). Measured values are from Refs. [4,19] (see text).

436 C.J. Batty/Nuclear Physics A 601 (1996) 425-444

B L a X-ray I.~ X-ray I E 0.1 --

0.9 I IP$171 0.18 "--- ePSI74 • PS174 -- a PSIT5

0,8 A P$I75 0.16

0.7 0.14

0.6 0.13

0.5 0.1

0.3 0.06

0.2 0.04

0.1 0.02

0 0

1o .2 1o "l I 1o 1o z 1o ~ io "s I 1o 1o 2 Psre Psre

L~ X-ray Lto ~ X-ray 'I O. 18 • P $ 1 7 4 & 9 • P ~ I 7 !

i t~ PSI 7S ~ • PSI 74

0.16 0.8

0.14 0.7

0.12 0.6

0.1 0.5

1106 0.3

0.04 0.,2

O.O2 0.1

0 0

io .2 Jo .i 1 i o 1o z 1o 4 1o "J ! /o io 2

P~rp Psn,

Fig. 3 (continued).

transitions are IP 1 ~ 1S 0 and 3P 0, 3pl , 3P 2 ~ 35 I. Transitions from the 351 state were taken to populate the triplet P states statistically and I S o ~ t Pv Radiative transitions from P to S states were calculated according to the scheme outlined in Fig. 2 and similarly for S to P radiative transitions. Annihilation from S and P states was calculated

1.5

1.4

1.3

1.2

1.1

I

0.9

O.8

0.7

0,6

0.5

C-L Batty~Nuclear Physics A 601 (1996) 425-444

S-state enhancement Ko=Z6 (DR1)

437

- - 3S!

. . . . . . 1So

10-3 102 10 4 1 10 lO 2 10 3 Density ( P sre)

Fig. 4. S-state enhancement factors from a cascade calculation using K o = 7.6 and annihilation widths from the DR1 potential.

taking into account the differing populations and strong interaction widths of the fine

structure states. The relative fractions of annihilation from these states, summed over all

n values, were then used to calculate the enhancement factors.

All calculations used K 0 = 7.6, ksT K = 1.19 and the predicted strong interaction

widths from the three models listed in Table 1. For the DR1 potential, calculated enhancement factors as a function of target density for S states and P states are shown in

Figs. 4 and 5, respectively. Calculations using K 0 = 1 and ksT n = 1.28 gave very similar results.

Using the strong interaction widths from the DR2 and KW models gave enhancement

factors as a function of density similar to those of Figs. 4 and 5. The calculated

enhancement factors for these three potentials at the five target densities of interest for

the present work are listed in Table 4.

As expected the enhancement factors for S-states are close to 1. The difference from

the value 1 at low pressures is due to the differential feeding of the S states by P ~ S radiative transitions discussed in Section 2 and illustrated in Fig. 2. For the P states, the

biggest enhancement factors are for 3P 0 where the strong interaction widths are also

large in all three models. Enhancement factors rather smaller than 1 are obtained for the

3P t and IP l states which however do not contribute to the two-body reactions listed in Table 2. The factor for the 3P 2 state varies only slowly with target density, from 1.0 in

low pressure gas to values in the range 0.95 to 1.06 in liquid hydrogen.

438 CA. Batty~Nuclear Physics A 601 (1996) 425-444

P-state enhancement Ko= Z6 (DR1)

3 - -

2.75

2.5

2.25

~ 2

~ 1.75

~ 1.5

1.25

1

0.75

O.5

SPo

. . . . . . 3pi

........... 3p2

. . . . . . IPi

lO-S 10"2 10"1 1 10 10 2 l0 s Density ( P sre)

Fig. 5. P-state enhancement factors from a cascade calculation using K o = 7.6 and annihilation widths from the DR1 potential.

Table 4 Calculated enhancement factors

State Density (PSTP) Model

0.002 0.005 1 15 liquid

1.044 1.046 1.020 1.012 1.032 DR 1 t S o 0.973 0.975 1.007 1.005 1.028 DR2

1.138 1.139 1.030 1.020 1.060 KW

0.985 0.985 0.993 0.996 0.989 DR 1 3S 1 1.009 1.008 0.998 0.998 0.991 DR2

0.954 0.954 0.989 0.993 0.980 KW

0.999 0.999 0.974 0.966 0.856 DR 1 I Pl 1.000 1.001 0.991 0.991 0.933 DR 2

0.998 0.997 0.960 0.941 0.809 KW

1.011 1.016 1.288 1.399 2.556 DRI 3 P0 1.010 1.014 1.205 1.302 2.076 DR2

1.009 1.013 1.227 1.318 2.176 KW

0.995 0.993 0.929 0.886 0.685 DR 1 3p! 0.993 0.990 0.914 0.856 0.541 DR2

0.995 0.993 0.932 0.891 0.703 KW

1.001 1.002 1.000 1.009 0.964 DR I 3P 2 1.002 1.003 1.016 1.032 1.041 DR2

1.002 1.003 1.019 1.037 1.058 KW

CJ. Batty~Nuclear Physics A 601 (1996) 425-444 439

Enhancement factors were also calculated for P-state annihilation from n = 2 states only, as observed in experiments using an L X-ray trigger. In all cases the values obtained were within + 1% of the value 1.0 assumed in the derivation of Eqs. (3.15),

(3.17) and (3.21).

5. Results

5.1. Without enhancement factors

To allow a comparison with earlier work and to study the effect of including the enhancement factors, a series of calculations were first made without enhancement factors, following the treatment of Section 3.1. The two body branching ratios fitted are those given in Table 3, with the exception of the ratio of branching ratios BR

(K+K - ) / B R (Tr + ~r-) at densities 0.005psTp, 1 PsvP and in liquid where the values are directly derived from the absolute (K + K - ) and (Tr + 7r-) branching ratios and so are not statistically independent. This gives a total of 17 data points to be fitted.

The data were fitted with the least squares method using Eqs. (3.1)-(3.9) by adjusting the value of fp at each of the 5 different pressures, together with the parameters B 0, B~, Co, C1, D O and D~, so making a total of 11 variables. A best fit is obtained with the values of fp given in the first line of Table 5 and x E / N = 24.1/6 where N = 6 is the number of degrees of freedom. The fit to the data is relatively poor with the branching

ratios for the reactions 7r°zr°(PSTP), 7r + 7r-(0.005PSTP) and K+K-(PSTP) being badly fitted ( X 2 > 5). A point of particular note in this fit, is the rather large value obtained for the P-wave fraction in liquid hydrogen, fp = 0.27 + 0.02. This value is largely determined by the 7r°Tr ° branching ratio measurement of the Crystal Barrel collabora- tion [5]. It was this apparently large value for the BR(Tr°er °) and the derived fp in liquid hydrogen which stimulated the present work.

It is of interest to look at similar fits to sub-sections of the data. Fitting the branching

ratios for the reactions ¢r + zr-, ¢r°¢r °, (~-+ 7r- )x at PSTP and in liquid gives fp(PSTP) =

0.64 _ 0.08 and fp (liq) = 0.26 _ 0.02, x2/N = 6.49/1. The branching ratio for zr°zr°(PSTP) is rather poorly fitted with X 2 = 4.8. Omitting this latter point causes fp (PSTP) to increase to 0.79 + 0.15.

Table 5 Fraction of P-wave annihilation fv

Density (PsTP) ,v 2/tN Enhancement

0.002 0.005 1 15 liquid factors

0.81 + 0.06 0.77 d: 0.05 0.69 ± 0.04 0.50 ± 0.05 0.27 ± 0.02 24.1/6 None 0.82±0.08 0.80±0.06 0.58±0.06 0.43±0.06 0.13±0.04 7.1/5 DR1 0.83 ± 0.08 0.80!0.06 0.58±0.04 0.43+0.05 0.14±0.01 7.3/5 DR2 0.825:0.08 0.80±0.06 0.57±0.06 0.43+0.06 0.13 ±0.04 7.3/5 KW

440 C.I. Batty ~Nuclear Physics A 601 (1996) 425-444

Fitting the branching ratios for the reactions K s K s , K s K L and (KsKs ) x at PSTP and in liquid, gives fp (PSTP) = 0.65 + 0.07 and fo (liq) = 0.10 + 0.08 with x 2 / N =

0.21/1. The much smaller value obtained for fp (liq) is surprising. However the kaon branching ratios are relatively inaccurate and fixing fp (liq) = 0.27 still gives a good fit

to the data with f p ( P S T P ) = 0.72 _+ 0.05 and x2/N = 2.83/2.

5.2. With enhancement factors

The analysis of the previous section was repeated using Eqs. (3.13)-(3.21) together

with the enhancement factors of Table 4. This gives three additional partial branching

ratios to be fitted. It was found important to include the constraint BLj, CLj, DLj >/0

since in some cases parameters with small negative values, consistent with zero within

errors, were obtained. In these cases the appropriate partial branching ratios were set

equal to zero.

In a fit to all available measurements using the enhancement factors derived using the

DR1, DR2 and KW potential models, the parameters Cz2 and Dl0 were both set to zero

following the above considerations. The derived values of the fraction of P-wave annihilation are given in Table 5. Good consistency between the results from the three models is observed and a markedly better fit to the data is obtained with X 2 / N dropping from 4.02 to 1.46 when the enhancement factors are included in the analysis. The only

branching ratio measurement not well fitted is that for ~p ~ ~r%r ° at PSTP for which X 2 = 3.9. The partial branching ratios (i.e. coefficients B u , CLj , DLj) obtained with the three potential models used in these fits to the data, are listed in Table 6. The best fit

using the DR2 potential required B12 = 0 whilst for the other two cases the values of

B~2 are positive but consistent with zero.

Fitting the branching ratios for the reactions 7r + 7r-, ~°Tr° and (Tr+Tr-)x at RSTP

and in liquid gives fp (PsvP) = 0.49 + 0.06, fp (liq) = 0.11 + 0.01 and requires Bl2 = 0. The value of fe (PSTP) is smaller than that obtained from the fit to all data, although the error bars just overlap, and x 2 / N = 3.35/1 with again the major contribution to X 2 coming from the branching ratio measurement for Op ~ ~.%.0 at PsvP-

Table 6 Partial branching ratios. Values are in units 10 -3

State Potential

DR 1 DR2 KW

3S l Boj 2.68 ± 0.27 2.72 ± 0.23 3P o B m 44.9 ±21.5 57.0 +2.46 3P 2 B 12 2.42 ± 4.44 0.0

351 Col 1.40+0.09 1.44±0.08 3P o C lo 3.68 ± 0.54 3.67 ± 0.53 3P 2 C 12 0.0 0.0

3S l Doj 1.34±0.11 1.36± 0.09 3po Dio 0.0 0.0 3P 2 D n 0.20±0.05 0 .19±0.04

2.73 + 0,28 55.8 + 36.0 0.35 + 7,32

1.44 + 0.09 3.70 + 0.55 0.0

1.36+0.13 0.0 0.19+0.05

C.J. Batty~Nuclear Physics A 601 (1996) 425-444 441

Fitting the branching ratios for K s K s, K s K L and (KsKs ) x at RSTP and for liquid, gives fp(PSTP) = 0.62 + 0.07, fp (liq) = 0.04 + 0.03 and x2/N = 4" 10-4 /1 . As in the previous section the value of fp (liq) seems low and the fit in this case requires

DI2 = 0. However fixing fp (liq) = 0.13, as in Table 5, gives a very good fit to the data

with fp (PSTP)= 0.66 + 0.06, xZ/N = 0 .29/2 and now D10 = 0 as in the fit to the complete set of branching ratio measurements discussed above.

6. Discussion and conclusions

An analysis of two body branching ratios for ~p annihilation at rest has been presented and values for the fraction of P-state annihilation derived as a function of

target density. It has also been shown that in the presence of Stark mixing it is necessary

to take into account an enhancement of annihilation from fine structure states over that

expected from a purely statistical population of the levels. Fine structure widths

predicted by the DR1, DR2 and KW potentials have been used, together with a cascade

calculation based on the Borie and Leon model [3], to calculate the "enhancement

factors". A fit to the two body data, using these "enhancement factors" gives a significantly improved X 2 than for the case where the enhancement effects are omitted.

The derived P-state fractions ( fp) are given in Table 5 and also plotted in Fig. 6 as a function of target density. Also plotted in the figure are the predictions of the Mainz

model and of the Borie and Leon model using parameters obtained from a best fit to the X-ray data with either K 0 = 1 or 7.6 (see Section 4). The values of fp at higher target densities are seen to be in general accord with the predictions of the Borie and Leon

model with K 0 = 7.6. However at densities less than 0.01PSTP the derived values for fp are lower than the predictions from any of the models, although in this case the Mainz model is closest. On the other hand the finite branching ratio [6] for the annihilation ~p ~ K s K L, which can only occur from initial 3SI states, indicates that there must be significant S-state annihilation at these low densities.

The CPLEAR collaboration have reported [22] a measurement of the ratio of the

branching ratios for the processes ~p ~ f 2 ~ -° and ~p ~ pTr ° at a gas density 16PsTP

and derive a P-state fraction fp = 0.38 + 0.07. For these reactions the initial states are IS0 ' 3S~ ' 3p~, 3p2 and 3S~, tP~, respectively. With the 3P 0 state not contributing to either

reaction, the enhancement factors for the relevant states are close to 1.0 and should have little effect on the derived fp. The analysis uses branching ratios for these reactions for pure S-wave and pure P-wave measured earlier where assumptions were made as to the P-wave fraction in liquid H 2 and in gas at PSTP" We have confirmed that using the values of fp for liquid and gas obtained in Table 5 makes a negligible difference to their derived fp at 16pSTP which is in good agreement with the values presented in Fig. 6 from the present work.

Finally we should comment on the partial branching ratios presented in Table 6. The values obtained using fine structure widths from the three models DR1, DR2 and KW are in excellent agreement apart from BI2 , which was discussed earlier, where the

442 C.J. Batty/Nuclear Physics A 601 (1996) 425-444

P-state annihilation (DRI )

1 --

0.9 ~ "

o.s ",,, --- r0=J.0

0.7

0.6

.~. 0.5

t~ 0.4

0.3

0.2

0.I

10 J 10-2 10 -! 1 10 102 10 J Density ( P sre)

Fig. 6. Fraction of P-state annihilation as a function of gas density predicted by the Borie-Leon model [3] giving a best fit to the X-ray data with K 0 = 1 (dashed line) or K o = 7.6 (full line). Also shown are the predictions of the Mainz [18] model (dotted line) and the values derived in the present work from measured branching ratios and annihilation widths from the DR1 potential (Table 5).

values are all consistent with zero. A particularly striking feature of these results is the

near equality of the S- and P-state branching ratios for ~p ~ 7r~" after al lowing for the

differing statistical weights of the states. This correction is particularly important for the

3P 0 state which has a statistical weight of ~2- For ~p ~ KK the P-state branching ratios

are much smaller than those for the S-state, with P-state annihilation occurring from 3P o

states in ~p ~ K ÷ K - and from 3P 2 states in ~p ~ K ° K --6.

A large variety of models have been used to calculate two body branching ratios (see

Ref. [15] for references to some of these). In some cases [23] the effects of initial state

interactions have been specifically considered. Comparisons between theoretical calcula-

tions of branching ratios and measured values may need, of course, to take into account

the enhancement factors discussed in this work. However any further discussion of the

results of Table 6 would be beyond the scope of the present paper.

Acknowledgements

I wish to thank the members of the Crystal Barrel collaboration for helpful discus-

sions and comments on the contents of this paper. The work was supported by the

British Particle Physics and Astronomy Research Council.

CA. Batty~Nuclear Physics A 601 (1996) 425-444 443

Appendix

Some of the features of the atomic cascade which are of particular relevance to the annihilation process have been discussed in detail by Gastaldi et al. [9] where they suggest that the fractions of annihilations from the initial states of ~p atoms could be determined by measurements of six different two-body branching ratios at four separate target densities.

Using the prefix G to denote equation numbers in Ref. [9], Gastaldi et al. show that the fraction of annihilations into channel ch

Pfch: EPf~h (G.15) jec

and

pf.~h¢ = B~hcPFgpc. (G.16)

B~ h is the branching ratio for annihilations into the channel ch from the initial state jPc. In the notation of the present paper, used in eq. (3.12)

pf~h = BR(~p ~ ch) (A.1)

and

B ~ : BR(SLj ~ ch). (A.2)

PFjPc is the fraction of all annihilations for all initial states of jPC (i.e. summed over all atomic n values).

Again using the notation of Eq. (3.12)

PFjPc =fp(p)E(sCy ,p)og(Sc j ) for P-states (A.3)

and

PFjPc = ( 1 - f p ( p ) ) E ( S L j , p ) w ( S L j ) for S-states, (A.4)

where we have explicitly indicated the density dependence of fp. Using Eq. (G.16) and Eqs. (A.I) to (A.4), Eq. (G.15) can then be written in the form

of Eq. (3.12). Summing Eqs. (A.3) and (A.4) over all j r c separately for S-states and for P-states, using Eq. (2.4) gives

PFs = E PFjP¢ = (1 - f v ( p ) ) , (A.5) L=0

PFp = ~.. PFjPc = f p ( p ) . (A.6) L=I

Adding (A.5) and (A.6) then gives Ejpc PFjPc = I as in Eq. (G.17).

References

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C. Balmy et al., Phys. Rev. Lett. 15 (1965) 532. [17] Crystal Barrel Collaboration, C. Amsler et al., Phys. Lett. B 346 (1995) 363. [18] G. Reifenrbther and E. Klempt, Nucl. Phys. A 503 (1989) 885. [19] K. Heitlinger, R. Bacher et al., Z. Phys. A 342 (1992) 359. [20] D. West, Rep. Prog. Phys. 32 (1958) 271. [21] C.J. Batty and R.E. Welsh, Nucl. Phys. A 589 (1995) 601. [22] CPLEAR Collaboration, R. Adler et al., Z. Phys. C 65 (1995) 199. [23] E. Klempt, Phys. Lett. B 244 (1990) 122.