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Nuclear electroproduction of negative pions below threshold
Volume 59B, number 1
PHYSICS LETTERS
NUCLEAR ELECTROPRODUCTION
13 October 1975
OF NEGATIVE PIONS BELOW THRESHOLD e J.H. KOCH
Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 3 July 1975 We calculate the electroproduction of negative pions into bound atomic states. The effect of the strong n-nucleus interaction is included in the calculation. Several recent papers have discussed the nuclear photo- and electroproduction of charged pions near threshold [ 1 - 3 ] . Such experiments provide information about the production amplitude, the isobaric analog states of the target nucleus and the low energy pion-nucleus interaction. In addition to the production of pions into the continuum, it is also possible to produce negative pions directly into bound lr-mesic states below threshold. This was pointed out first by Tzara [4] for photoproduction. These rt-mesic states are very interesting for two reasans. First, they are built on top of isobaric analog states of the target nucleus. For example, a pion produced from a 12C target will be bound by the 12N(JnT = 1+ 1) nucleus. With the usual method to form mesic atoms by capturing a slow ~r- into a Bohr orbit, it is only possible to obtain atomic states based on the 12C ground state. Second, while in the observed lr-mesic cascade primarily l = n - 1 orbits are reached, the final states in a direct production are not limited to circular orbits. For a given n the pion will preferably be produced into an 1 = 0 orbit, since this state has the largest wave function inside the nucleus. However, it would be extremely difficult to actually measure the photoproduction of a r r - below threshold. The x-rays from atomic transitions of the produced pion or the nuclear fragments resulting from nuclear pion capture would have to be detected against a background that is several orders of magnitude larger. On the other hand such bound states can also be reached in inelastic electron scattering. By measuring the energy distribution of the scattered electrons at a fixed scattering angle, these states could be seen as peaks on top of the quasi¢r This work is supported in part through funds provided by ERDA under Contract AT(11-1)-3069.
elastic background. Such an experiment might be feasible with the high resolution of the new electron scattering facilities and since the signal to noise ratio is more favorable in electro-production. Below, we therefore derive the differential cross section for electrons scattering inelasticaUy from a nucleus by producing a pion in a given atomic state. Some of the features of this reaction are discussed in detail. The amplitude for electroproduction from a nucleus, mediated by a one photon exchange, can be written as [1,51
Tn= - x / 2 e~2 (2ff-~) f dx (f, ¢nl(X)leuHUlexp(ik'x),i), (1) with
e 2 = 4~rt~,
(G2/41r)(mn/2M)2=f 2= 0.08,
and
eta = -~(s2)~/t~U(Sl) , where s 1 and s 2 are the initial and final momentum of the electron. k = ( k 0 , k ) =s 1 - s 2 , is the four-momentum of the virtual photon. In the matrix element, i and f denote the initial and final nuclear wavefunction and ~nl(X) is the normalized wavefunction for the pion in the atomic n,/-orbit. For the light nuclei used as examples below, one expects an appreciable cross section only for electroproduction into the ls state. Therefore, only l = 0 is considered in the following analysis. Using the amplitude of Fubini, Nambu and Wataghin [6] for electroproduction from a single nucleon and neglecting terms proportional to the pion momentum, one obtains in the impulse ap45
Volume 59B, number 1
'•
10-3
10-4
PHYSICS LETTERS
13 October 1975
10-3
sl = 2 5 0 M e V
s~ = 2 5 0
MeV
Ill'
\
10 -4
\
e~
::k
2 10-5
-o
\
\
b
\\
b 13
10-5
\ -
I0"
I
I
I0 °
20*
I 30 °
I
I
40 °
50 °
I 60*
I
\.
-0 ~'\. ~c~ I' I I'Xbl
~
I
10-6
I0.°
I ~ 20 °
0
300
40"
50*
60*
0
Fig. 1. Electroproduction from a 12C target: differential cross section do/ds22 of scattered electron for production of a bound ls and 2s pion with 12N(l+l) as final nuclear state. (a): Is production for Vn = 0. b): ls production without longitudinal multipoles.
Fig. 2. Electroproduction from a 11B target with 11C(3/2-1/2) as.final nuclear state. Labelling for (a) and (b) same as in fig. 1. (c): Is production with l l c ( 1 / 2 - 1/2) as final nuclear state.
p r o x i m a t i o n [ 1]
Ji is the target spin and the electron m o m e n t u m s 2 is for a given scattering angle 0 defined through
1
A
euHU=~]~=l ~" [~(] ) k(~(])'k)-] r+(]) ,
s 2 + s 22 - 2 1s 2 c o s o
s 1 =s2+gnl+COex +
:o
where r+ = ½ ( r l + i r 2 ) , and ~o is the total pion energy. Since the final n-mesic states in light nuclei have a width o f only a few keV, we can for all practical purposes assume that the density o f final states for the pion is given b y
2MT
where M T is the mass and COex the excitation energy of the target. By expanding t exp ( i k "x), eq. (1), into vector spherical harmonics, the transition m a t r i x element in eq. (3) can be written as [7] [M[2 = l l J ~ l
[[(Jf[lT~15r+[lJi)12
+ I(Jfll T~nax r+ll Ji)l 2 ]
(¢o - Enl), where En, ! is the
+12 ~ J~0
I(1-K2/k2)(JfllL5%llJi)12,
energy o f the n, I b o u n d state. S u m m i n g over the spins and integrating over the energy of the pion and scattered electron, the differential cross section for the electron becomes
t 1 = ½ (s I ' s 2 - (s 1 • k) (s 2 • k)/K 2 ) ,
d o _ 16n c~2f2 1 s(~_T ) d~22 m2 k4Enl ~
t~=lk[,
46
~
4zt IM[ 2 .
(3)
l 2 = (s t • k) (s 2 • k)/K 2 _ k 2 / 2 ,
k2=k2-K 2.
(4)
Volume 59B, number 1
PHYSICS LETTERS
The operators in eq. (4) and their parities are
TmagS=~o(X)[jg(Kx) J,h
~ll'a
'
11=(-1)
g
'
(Sa)
it = ( - 1 ) J+l ,
(Sb) "J,h
" II,
n = ( - 1) J+t ,
(5c)
While in threshold photoproduction only the transverse multipoles, eq. (5a, b), with J t> 1 contribute, in electroproduction there are also transitions through the longitudinal mflltipoles, eq. (5c), starting with J = 0. The atomic wavefunction, ¢n0, of the pion is distorted by the strong nuclear interaction. This is taken into account by including in the Schr6dinger equation an optical potential obtained from n-mesic atoms
V~=
4n
2m~ [boP(r) + b l ( P n ( r ) - pp(r))
+Bop2(r) - Vq(r). V], q(r) = CoP(r ) + Cl(Pn(r) - p p ( r ) ) + i Im Cop2(r) . The s-wave parameters were taken from the work of Tauscher and Schneider [81. For the p-wave part we used [91 c 0 = 0 . 1 7 m ; 3,
c 1 = 0 . 2 2 m ~ -3,
I m C 0 = 0 . 0 3 6 m ~ 6.
Fig. 1 shows the cross section for electroproduction from a 12C target with 12N(J~T= 1+1) as the final nuclear state. For the evaluation of the nuclear matrix element we used the harmonic oscillator single particle wavefunctions of O'Connell et al. [7]. To emphasize the sensiti~,ity of this process to the final state interactions, the differential cross section for electroproduction into the pionic ls state is shown with and without the optical potential V~r, eq. (6), which reduces the sstate production by about 70%. The contribution from the longitudinal matrix element vanishes in the forward direction and grows only slowly with increasing momentum transfer. Electroproduction into the 2s state, which lies 130 keV above the 1s state, is down by an order of magnitude. Similar curves for a llB target are shown in fig. 2. Here the nucleus remains within the ground state isotopic multiplet by making an "isoelastic" transition [10, 11] from 11B(3/2- 1/2)to 11C(3/2-1/2). The
13 October 1975
nuclear matrix elements were evaluated with the harmonic oscillator single particle wavefunctions of Donnelly and Walecka [11 ], who truncate the model space to the p-sheU and fit the parameters to the observed electromagnetic and semileptonic weak processes in the 11B.11C isodoublet. The production into the 1s orbit is again strongly suppressed (60%) by the repulsive n-nucleus s-wave interaction. The longitudinal multipoles are much more important in this case, contributing about one half of the cross section at 0 = 45 °. It is also possible to reach final states where the nmeson is bound to an excited state of the residual nucleus. If the additional energy, which is required to excite the residual nucleus, is larger than the binding energy of the pion, the production of such states will be accompanied by pions produced above threshold with the residual nucleus in its ground state. This is the case for t i c , where the first excited state, J~rT= 1/2- 1/2, lies 2 MeV above the 11C ground state and the binding energy for a ls pion is 127 keV. Neglecting the coupling between these pion bound states and pion continuum channels, eq. (3) yields a ls cross section for t i c * (curve c in fig. 2) that is of the same magnitude as for 11C (g.s.). This estimate was obtained by using a lh model for llB, a lp-2h model for l l c * and adjusting the parameters to fit the M1 7-decay rate in liB* 11B + 3' [12j. With these model wavefunctions, the cross section becomes even larger for the excited state than for the ground state of I1C if0 < 30 ° (K < 167 MeV). The cross sections obtained above for t 1B and 12C are very small and only production into the ls state will be observable. To be able to detect production into higher orbits, one has to use a heavier target nucleus, since the cross sections rise fast with increasing Z. However, for a larger nucleus the pion states become broader and the background due to the radiation tail and quasielastic scattering increases. Therefore even for a heavy target nucleus, the number of final pion states that can be resolved experimentally will remain small. Up to now, most of the information about the low energy n-nucleus interaction has come from the analysis of n-mesic X-rays. While in the usual n-mesic atom the cascade takes place through circular orbits around the target nucleus in its ground state, in electroproduction of bound n - one reaches orbits built on top of one of the isobaric analog states of the target. This makes it possible to study the n-nucleus interaction for different 47
Volume 59B, number 1
PHYSICS LETTERS
spin-isospin configurations of the target and for specific angular m o m e n t u m states of the pion. Inelastic electron scattering to bound states of negative pions, together with photo- and electroproduction of pions above threshold, will therefore provide more detailed information about the strong nuclear potential and the isobaric analog states of the target nucleus. It is a pleasure to acknowledge stimulating discussions with W. Bertozzi, E.J. Moniz and W. Turchinetz.
References [1] E. Borie, H. Chandra and D. Drechsel, Physics Lett. 47B (1973) 291; Nucl. Phys. A226 (1974) 58.
48
13 October 1975
[2] F. Cannata et al,, Can. Jour. Phys. 52 (1974) 1405; Phys. Rev. Lett. 33 (1974) 1316. [3] J.H. Koch and T.W. DonneUy, Nucl. Phys. B64 (1973) 478. [41 C. Tzara, Nucl. Phys. BI8 (1970) 246. [51 W. Czyz and J.D. Walecka, Nucl. Phys. 51 (1964) 312. [6] S. Fubini, Y. Nambu and V. Wataghin, Phys. Rev. 111 (1958) 329. [7] J.S. O'Connell, T.W. Donnelly and J.D. Walecka, Phys. Rev. C6 (1972) 719. [8] L. Tauscher and W. Schneider, Z. Physik 271 (1974) 409. [9] L. Tauscher, Proc. Int. Seminar on n-Meson-nucleusinteractions, Strasbourg (1971). [10] T.W. Donnelly and J.D. Waleeka, Nucl. Phys. A201 (1973) 81. [11] T.W. Donnelly and J.D. Walecka (to be published); J.D. Walecka, Proc. Int. Conf. on Nuclear structure and spectroscopy, Amsterdam (1974). [12] T.W. Donnelly, private communication.
PHYSICS LETTERS
NUCLEAR ELECTROPRODUCTION
13 October 1975
OF NEGATIVE PIONS BELOW THRESHOLD e J.H. KOCH
Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 3 July 1975 We calculate the electroproduction of negative pions into bound atomic states. The effect of the strong n-nucleus interaction is included in the calculation. Several recent papers have discussed the nuclear photo- and electroproduction of charged pions near threshold [ 1 - 3 ] . Such experiments provide information about the production amplitude, the isobaric analog states of the target nucleus and the low energy pion-nucleus interaction. In addition to the production of pions into the continuum, it is also possible to produce negative pions directly into bound lr-mesic states below threshold. This was pointed out first by Tzara [4] for photoproduction. These rt-mesic states are very interesting for two reasans. First, they are built on top of isobaric analog states of the target nucleus. For example, a pion produced from a 12C target will be bound by the 12N(JnT = 1+ 1) nucleus. With the usual method to form mesic atoms by capturing a slow ~r- into a Bohr orbit, it is only possible to obtain atomic states based on the 12C ground state. Second, while in the observed lr-mesic cascade primarily l = n - 1 orbits are reached, the final states in a direct production are not limited to circular orbits. For a given n the pion will preferably be produced into an 1 = 0 orbit, since this state has the largest wave function inside the nucleus. However, it would be extremely difficult to actually measure the photoproduction of a r r - below threshold. The x-rays from atomic transitions of the produced pion or the nuclear fragments resulting from nuclear pion capture would have to be detected against a background that is several orders of magnitude larger. On the other hand such bound states can also be reached in inelastic electron scattering. By measuring the energy distribution of the scattered electrons at a fixed scattering angle, these states could be seen as peaks on top of the quasi¢r This work is supported in part through funds provided by ERDA under Contract AT(11-1)-3069.
elastic background. Such an experiment might be feasible with the high resolution of the new electron scattering facilities and since the signal to noise ratio is more favorable in electro-production. Below, we therefore derive the differential cross section for electrons scattering inelasticaUy from a nucleus by producing a pion in a given atomic state. Some of the features of this reaction are discussed in detail. The amplitude for electroproduction from a nucleus, mediated by a one photon exchange, can be written as [1,51
Tn= - x / 2 e~2 (2ff-~) f dx (f, ¢nl(X)leuHUlexp(ik'x),i), (1) with
e 2 = 4~rt~,
(G2/41r)(mn/2M)2=f 2= 0.08,
and
eta = -~(s2)~/t~U(Sl) , where s 1 and s 2 are the initial and final momentum of the electron. k = ( k 0 , k ) =s 1 - s 2 , is the four-momentum of the virtual photon. In the matrix element, i and f denote the initial and final nuclear wavefunction and ~nl(X) is the normalized wavefunction for the pion in the atomic n,/-orbit. For the light nuclei used as examples below, one expects an appreciable cross section only for electroproduction into the ls state. Therefore, only l = 0 is considered in the following analysis. Using the amplitude of Fubini, Nambu and Wataghin [6] for electroproduction from a single nucleon and neglecting terms proportional to the pion momentum, one obtains in the impulse ap45
Volume 59B, number 1
'•
10-3
10-4
PHYSICS LETTERS
13 October 1975
10-3
sl = 2 5 0 M e V
s~ = 2 5 0
MeV
Ill'
\
10 -4
\
e~
::k
2 10-5
-o
\
\
b
\\
b 13
10-5
\ -
I0"
I
I
I0 °
20*
I 30 °
I
I
40 °
50 °
I 60*
I
\.
-0 ~'\. ~c~ I' I I'Xbl
~
I
10-6
I0.°
I ~ 20 °
0
300
40"
50*
60*
0
Fig. 1. Electroproduction from a 12C target: differential cross section do/ds22 of scattered electron for production of a bound ls and 2s pion with 12N(l+l) as final nuclear state. (a): Is production for Vn = 0. b): ls production without longitudinal multipoles.
Fig. 2. Electroproduction from a 11B target with 11C(3/2-1/2) as.final nuclear state. Labelling for (a) and (b) same as in fig. 1. (c): Is production with l l c ( 1 / 2 - 1/2) as final nuclear state.
p r o x i m a t i o n [ 1]
Ji is the target spin and the electron m o m e n t u m s 2 is for a given scattering angle 0 defined through
1
A
euHU=~]~=l ~" [~(] ) k(~(])'k)-] r+(]) ,
s 2 + s 22 - 2 1s 2 c o s o
s 1 =s2+gnl+COex +
:o
where r+ = ½ ( r l + i r 2 ) , and ~o is the total pion energy. Since the final n-mesic states in light nuclei have a width o f only a few keV, we can for all practical purposes assume that the density o f final states for the pion is given b y
2MT
where M T is the mass and COex the excitation energy of the target. By expanding t exp ( i k "x), eq. (1), into vector spherical harmonics, the transition m a t r i x element in eq. (3) can be written as [7] [M[2 = l l J ~ l
[[(Jf[lT~15r+[lJi)12
+ I(Jfll T~nax r+ll Ji)l 2 ]
(¢o - Enl), where En, ! is the
+12 ~ J~0
I(1-K2/k2)(JfllL5%llJi)12,
energy o f the n, I b o u n d state. S u m m i n g over the spins and integrating over the energy of the pion and scattered electron, the differential cross section for the electron becomes
t 1 = ½ (s I ' s 2 - (s 1 • k) (s 2 • k)/K 2 ) ,
d o _ 16n c~2f2 1 s(~_T ) d~22 m2 k4Enl ~
t~=lk[,
46
~
4zt IM[ 2 .
(3)
l 2 = (s t • k) (s 2 • k)/K 2 _ k 2 / 2 ,
k2=k2-K 2.
(4)
Volume 59B, number 1
PHYSICS LETTERS
The operators in eq. (4) and their parities are
TmagS=~o(X)[jg(Kx) J,h
~ll'a
'
11=(-1)
g
'
(Sa)
it = ( - 1 ) J+l ,
(Sb) "J,h
" II,
n = ( - 1) J+t ,
(5c)
While in threshold photoproduction only the transverse multipoles, eq. (5a, b), with J t> 1 contribute, in electroproduction there are also transitions through the longitudinal mflltipoles, eq. (5c), starting with J = 0. The atomic wavefunction, ¢n0, of the pion is distorted by the strong nuclear interaction. This is taken into account by including in the Schr6dinger equation an optical potential obtained from n-mesic atoms
V~=
4n
2m~ [boP(r) + b l ( P n ( r ) - pp(r))
+Bop2(r) - Vq(r). V], q(r) = CoP(r ) + Cl(Pn(r) - p p ( r ) ) + i Im Cop2(r) . The s-wave parameters were taken from the work of Tauscher and Schneider [81. For the p-wave part we used [91 c 0 = 0 . 1 7 m ; 3,
c 1 = 0 . 2 2 m ~ -3,
I m C 0 = 0 . 0 3 6 m ~ 6.
Fig. 1 shows the cross section for electroproduction from a 12C target with 12N(J~T= 1+1) as the final nuclear state. For the evaluation of the nuclear matrix element we used the harmonic oscillator single particle wavefunctions of O'Connell et al. [7]. To emphasize the sensiti~,ity of this process to the final state interactions, the differential cross section for electroproduction into the pionic ls state is shown with and without the optical potential V~r, eq. (6), which reduces the sstate production by about 70%. The contribution from the longitudinal matrix element vanishes in the forward direction and grows only slowly with increasing momentum transfer. Electroproduction into the 2s state, which lies 130 keV above the 1s state, is down by an order of magnitude. Similar curves for a llB target are shown in fig. 2. Here the nucleus remains within the ground state isotopic multiplet by making an "isoelastic" transition [10, 11] from 11B(3/2- 1/2)to 11C(3/2-1/2). The
13 October 1975
nuclear matrix elements were evaluated with the harmonic oscillator single particle wavefunctions of Donnelly and Walecka [11 ], who truncate the model space to the p-sheU and fit the parameters to the observed electromagnetic and semileptonic weak processes in the 11B.11C isodoublet. The production into the 1s orbit is again strongly suppressed (60%) by the repulsive n-nucleus s-wave interaction. The longitudinal multipoles are much more important in this case, contributing about one half of the cross section at 0 = 45 °. It is also possible to reach final states where the nmeson is bound to an excited state of the residual nucleus. If the additional energy, which is required to excite the residual nucleus, is larger than the binding energy of the pion, the production of such states will be accompanied by pions produced above threshold with the residual nucleus in its ground state. This is the case for t i c , where the first excited state, J~rT= 1/2- 1/2, lies 2 MeV above the 11C ground state and the binding energy for a ls pion is 127 keV. Neglecting the coupling between these pion bound states and pion continuum channels, eq. (3) yields a ls cross section for t i c * (curve c in fig. 2) that is of the same magnitude as for 11C (g.s.). This estimate was obtained by using a lh model for llB, a lp-2h model for l l c * and adjusting the parameters to fit the M1 7-decay rate in liB* 11B + 3' [12j. With these model wavefunctions, the cross section becomes even larger for the excited state than for the ground state of I1C if0 < 30 ° (K < 167 MeV). The cross sections obtained above for t 1B and 12C are very small and only production into the ls state will be observable. To be able to detect production into higher orbits, one has to use a heavier target nucleus, since the cross sections rise fast with increasing Z. However, for a larger nucleus the pion states become broader and the background due to the radiation tail and quasielastic scattering increases. Therefore even for a heavy target nucleus, the number of final pion states that can be resolved experimentally will remain small. Up to now, most of the information about the low energy n-nucleus interaction has come from the analysis of n-mesic X-rays. While in the usual n-mesic atom the cascade takes place through circular orbits around the target nucleus in its ground state, in electroproduction of bound n - one reaches orbits built on top of one of the isobaric analog states of the target. This makes it possible to study the n-nucleus interaction for different 47
Volume 59B, number 1
PHYSICS LETTERS
spin-isospin configurations of the target and for specific angular m o m e n t u m states of the pion. Inelastic electron scattering to bound states of negative pions, together with photo- and electroproduction of pions above threshold, will therefore provide more detailed information about the strong nuclear potential and the isobaric analog states of the target nucleus. It is a pleasure to acknowledge stimulating discussions with W. Bertozzi, E.J. Moniz and W. Turchinetz.
References [1] E. Borie, H. Chandra and D. Drechsel, Physics Lett. 47B (1973) 291; Nucl. Phys. A226 (1974) 58.
48
13 October 1975
[2] F. Cannata et al,, Can. Jour. Phys. 52 (1974) 1405; Phys. Rev. Lett. 33 (1974) 1316. [3] J.H. Koch and T.W. DonneUy, Nucl. Phys. B64 (1973) 478. [41 C. Tzara, Nucl. Phys. BI8 (1970) 246. [51 W. Czyz and J.D. Walecka, Nucl. Phys. 51 (1964) 312. [6] S. Fubini, Y. Nambu and V. Wataghin, Phys. Rev. 111 (1958) 329. [7] J.S. O'Connell, T.W. Donnelly and J.D. Walecka, Phys. Rev. C6 (1972) 719. [8] L. Tauscher and W. Schneider, Z. Physik 271 (1974) 409. [9] L. Tauscher, Proc. Int. Seminar on n-Meson-nucleusinteractions, Strasbourg (1971). [10] T.W. Donnelly and J.D. Waleeka, Nucl. Phys. A201 (1973) 81. [11] T.W. Donnelly and J.D. Walecka (to be published); J.D. Walecka, Proc. Int. Conf. on Nuclear structure and spectroscopy, Amsterdam (1974). [12] T.W. Donnelly, private communication.