Protonium annihilation into two mesons

Nuclear Physics B (Proc. Suppl ) 8 (1989) 225-227 North-Holland, Amsterdam

PROTONIUM

ANNIHILATION

L. Mandrup,

A.S.

INTO

Jensen,

225

TWO

MESONS

A. M i r a n d a

I n s t i t u t e of Physics, U n i v e r s i t y D K - 8 0 0 0 A a r h u s C, Denmark. The a n n i h i l a t i o n

of a p r o t o n - a n t i p r o t o n

into a p s e u d o s c a l a r analyzed for

in terms

this

final

meson

Keeping

state we g e n e r a l i z e mesons.

unit

solid

The

angle

Oades

atomic

and its a n t i p a r t i c l e

of t h r e s h o l d

p r o c e s s I)

the

and G.C.

of A a r h u s

values

probability

from an atomic

bound

state

has r e c e n t l y

of the h e l i c i t y

the s i m p l i c i t y

to allow n o n - z e r o

emission

bound

been

amplitudes

of the t w o - p a r t l c l e intrinsic

W per unit

state

I~> =

spins

of

time and per In2S+ILj>

is

g i v e n by

W =

k 32~2E

~ k k 0

U gm(~)l<~,TitikilVl~m>l 2 m

12

w h e r e ~ is the m o m e n t u m protonium angular

mass,

of

projection

the i s o s p i n

of the i'th m e s o n Fourier W(u)

k

m,

4= 0

k 1

of the i'th meson,

kk

m

0

2

i

k

p

the z-axis,

Re(p)

the

protonium by p a r i t y

of

third

is the h e l i c i t y

causing

the reaction.

2

"radial"

wavefunction

(2)

(3)

Sk>'~pkp

are p and p helicities,

related

l

2

P

where kp,k~ of

k.

and

1 2 I IO>'ALIsjI 2

R~(P)
is the

the

2

~ < 16~ 2 k kP P M

half

then leads to

" I[
h k pi k ~2 ~

is

0

Z g m ( a ) i d mJ* k _k(o ) 12

~ k

E

for the s u b s t a t e

(Ti,t i) are the total

wave e x p a n s i o n

2L+I

8~ 4 E

ALS J -

number

and V is the i n t e r a c t i o n

and p a r t i a l =

of one of the mesons,

gm is the o c c u p a t i o n

momentum

component

(i)

>dp

e is the angle b e t w e e n ~ and

part of the F o u r i e r and the h e l i c i t y

conservation

0920-5632/89/$03.50 (D Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

transform

amplitudes

are

226

L. Mandrup et al./ Protonium annihilatiml

< k l k 2 iv i j l ~ p ~ >

= qlq2 (_i)

where qi and s i are p a r i t y part

of

V

with

isospin

amplitudes

do

not

integration

interval

S +S i 2<_kl_k2

i V i j l _ k p -~-> P

and spin of m e s o n I and angular

vary

significantly

of eq.(4)

i and

momentum

Vij

is

the

J. The h e l i c i t y

across

and their

(5)

the

threshold

effective

behaviour

can

we

six

be used. Specifying independent angular

be

computed

W(IPI)

0

0-

and

i-

meson,

have

a (IJ) for each I and J. The n of the m e s o n s w i t h r e s p e c t to the d i r e c t i o n in the dipole

for such mesons.

from the 2P-state

=

one

amplitudes

emitted

annihilation

W(3p

one

distribution

of the X - r a y then

to

helicity

transition

leading

to

For the i m p o r t a n t

lu> can

example

we find

6Ku [(13ja(1)j2+28ja(1)121 s )+(ja(1)j2-41a(1)j21 5 )c°s2e]

) = 0

W(3PI)

of

(6)

= 3Ka [(27ja(1)j2+52ja(1)j23 6 )-(ja(1)12-4ja(1)I23 6 )c°s2e]

W(3P2 ) = 135K~k 2 [a 2( 2 ) 1 2 [ l - ~ T1c o s 2 e ] 1 k 2 ~ 2115~ s M0 Ix I

K where

~a is the

weighted

(L+2)'th

sum

obtained

a (IJ) n eqs.(6),

0

zero

eq.(6),

both

final fat1,

in terms

More

angular

distribution,

la21

Helicity and

structure. of

only

the

la31.

this

meson

obtained

lasI

from the a n g u l a r

Two c o m b i n a t i o n s

the

and

without

become

is no longer n e e d e d

Since

la61

distribution and

in of

related

can then be o b t a i n e d

Thus it seems that

should

if

can be detected.

involves

all five a m p l i t u d e s If

is

+i and -i are identical

distribution.

sible to d e t e r m i n e

two

a (J). A measurement of n combinations of the five

vector meson

can be e x t r a c t e d

state.

of

information

(k I = k 2 = O)

from the a n g u l a r

fine

total

isospin

(8)

of the o u t g o i n g

helicity

helicity

an

n

is then of the form

therefore

values.

polarization

to

The

and a (J) is

1

determines

threshold

this

of R(p)

= C +C cos2e

with C o and C I e x p r e s s e d W(2P)

moment

over

by adding

W(2P)

(7)

it

is

resolution

possible,

imposof the

then

as each a n g u l a r

the di-

L. ]t/landrup et al./ Protonium annihilation

227

stribution completely determines the constants. The

partial widths are found by integrating over all angles.

Comparison with

the

total

annihilation

width

leads

to

the

branching ratio which can be measured for some processes. An accurate calculation of the total width then provides on

the

helicity

amplitudes.

constraints

For S-states we find the partial

widths

rc3sl) = k

ix 121a(1)12

8~ 4 M

2

0

I o, 1:'

c9)

r(iSo) ~ l a ~ 0)I =

2

4~ 4 M

0

k

For emission of ~+~"

from S-states, only the triplet state

con-

tributes

r ÷.-(3sx) = 4~4--~kix 121a~l)(=. -)[2

(i0)

0

The ratio F(S)/F ÷=-(S), of these measured partial determines

a

combination

then

of a (I) and a (0) for known values of 2

la~l)(~+~-) I (= 0.045 fm) and In conclusion,

widths

Ix(iS0)/x

(3 5

$I)12(= 1.40).

appropriate measurements of atomic bound state

annihilation widths can be used to constrain and test

theoreti-

cal models of the annihilation.

REFERENCES

1. G.C. Oades et al., Nucl.Phys.A464(1987)538

DISCUSSION

E. Klempt: Measuring the angular correlation between two charged pions and Balmar X-rays lung are of the same

the uncertainties due to Bremsstrah-

order as the small angular

correlation

for pp(3P2 ) ~ ~* "-" L. Mandrup: Measurement of the angular distribution of neutral mesons like 0 0, 0 0 and ~0Q0 would avoid the problem of Bremsstrahlung.