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Experimental test of three-body pion absorption
Volume 208, number 3,4
PHYSICS LETTERS B
21 July 1988
E X P E R I M E N T A L T E S T O F T H R E E - B O D Y P I O N A B S O R P T I O N ¢r L.L. SALCEDO 1,2, E. OSET 3, D. S T R O T T M A N Los Alamos National Laboratory, Los Alamos, NM 8 7545, USA
and E. H E R N A N D E Z Departamento de Fisica Te6rica, Facultad de Fisica, Universidad de Valencia, E-46100 Burjassot (Valencia), Spain
Received 17 November 1987; revised manuscript received 11 May 1988
A special method to analyse the three-nucleon spectra after pion absorption is proposed, which allows for the separation of quasielastic scattering followed by two-body absorption from other genuine three-body absorption mechanisms.
Recent experiments on pion absorption which determine the kinematics o f three nucleons [ 1-4 ] are giving evidence of mechanisms for the absorption that require the collaboration of three nucleons in a different way than two-body absorption with initial- or final-state interaction. The present work suggests a method to analyse the experimental data in order to disentangle the reaction mechanisms. We shall start from a realistic model o f p i o n - n u cleus scattering around the resonance region, depicted diagramatically in fig. 1, which provides a good description of integrated elastic and inclusive cross sections [ 5 ] as well as differential cross sections for inclusive [ 6 ] and elastic scattering [ 7 ]. The model evaluates the pion self-energy in infinite nuclear matter with the F e y n m a n diagrams o f fig. 1, and applies ¢~Work supported in part by CAICYT, the Joint Spanish-American Commitee for Scientific and Technological Cooperation and the DOE. Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA. 2 On leave from 1FIC, Centro Mixto, Universidad de Valencia, CSIC, E-46100 Burjasot (Valencia), Spain. 3 Departamento de Fisica Te6rica, Facultad de Fisica, Universidad de Valencia and IF1C, Centro Mixto, Universidad de Valencia, CSIC, E-46100 Burjassot (Valencia), Spain. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
it to finite nuclei by means o f the local density approximation, considering however finite-range effects [6 ]. The probability o f a reaction in an element o f longitude dz is given by [ 5,6 ], Pdz= - ( 1/q)Im H(q)dz ,
( 1)
where H ( q ) is the pion self-energy. The imaginary part of H ( q ) comes from placing on-shell the intermediate states cut by the dotted lines in fig. 1. Thus, through eq. ( 1 ), the upper cut in fig. lb provides the absorption probability by two nucleons, while the cuts in diagrams I d, I e correspond to three body absorption. The cut in diagram 1a corresponds to quasielastic pion scattering and the lower one in diagram lb plus the one in diagram 1c provide corrections to the quasielastic scattering. The wavy line in the diagram stands for the induced spin-isospin interaction, constructed from n and P exchange, modified by the effect of short-range correlations, which excite ph and Ah components in a RPA series. Now, while the pions in the induced interactions in diagram 1e are necessarily off-shell (a real pion does not excite a ph component in infinite nuclear matter, and only a negligibly small one in finite nuclei), and so are the pions o f the interaction labelled p in diagram 1d, the pions o f the interactions labelled q in diagram I d can appear onshell, in which case the diagram could be interpreted 339
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PHYSICS LETTERS B
II
ii
id
/
21 July 1988
with diagram I a, can be calculated by evaluating diagram 1a but with a pion renormalized. Thus in such a case we have d4q . --ill(k)
/ a)
/ b) /
iI
,' e)
~
×~ ~ ( - i T ) ( - i T ) i D ( q ) ,
(2)
s,t,2t
with D (q) the renormalized pion propagator
q
1 I
+..
2
3÷.
D(q)=qO2
I
/:
d)
e)
Fig. 1. Feynman diagramsenteringthe evaluationof the pion selfenergy in refs. [5,6 ]. The dotted lines indicate the analytical cuts giving rise to the imaginary part of the pion self-energy,when the particles cut by these lines are placed on-shell. as representing a quasielastic collision followed by two-body absorption. However, the same diagram contains a large contribution from the transverse part of the induced interaction [ 8 ] in the line labelled q, which contains p meson exchange and other pieces accounting for the effect of short correlations and which do not interfere with n exchange. These pieces, as well as diagram 1e, contribute to three-body absorption and cannot be cast into the classification of quasielastic scattering followed by two-body absorption (initial-state interaction) or nucleon induced knock out after pion absorption (final-state interaction). We then adopt the terminology of genuine three body absorption for such mechanisms which cannot be branded as initial- or final-state interaction preceding or following two-body absorption, respectively. The question is whether experimentally one can distinguish these situations. This work suggests a method of analysing the three-nucleon spectra, after pion absorption, which distinguishes quasielastic scattering followed by two-body absorption from genuine three-body absorption. Let us evaluate these two mechanisms from the model of fig. 1. First we concentrate on quasielastic followed by two-body absorption. If the lines with m o m e n t u m q in diagrams lb, lc, ld account for just one pion exchange, these last three pieces, together 340
q2 it2 H(q),
(3)
U(q) the Lindhard function for ph excitation and T the nN~nN amplitude (a delta pole dominance is assumed in the model of fig. 1 ). We sum T 2 over spin, isospin of the final nucleons and pion and average over spin, isospin of the initial nucleons (s, t, 2 variables). Now we use Cutkowsky rules [ 9 ] for the imaginary part of H ( H ( k ) ~, 2i Im H ( k ) ); U(p)--,2iO(p°)Im U(p); D(q)--,2iO(q°)ImD(q); G(k)-,2ilmG(k) for a positive energy fermion propagator and conjugate lines and vertices above the cut). Thus (we cut, or equivalently place on-shell, the intermediate ph excitation and the pion) t" d4q
Im H(k) = - J ~
G(k,q) 2ImD(q) ,
(4)
with
G(k, q) =O(k°-q°)O(q°) Im U(K-q)~ • IT[ 2 . (4') As a particular case we can use this formula to evaluate the quasielastic channel given by diagram la, which we obtain from eq. (4) by replacing D(q) by the free pion propagator, or equivalently Im D(q) by _ r r r ( q O 2 q 2 ~2). We shall call I m / / Q the results ofeq. (4) with this substitution. Now let us use the diagrams la and lb as input for H(q) in D(q) of eqs. (2), (3); we shall call these input pieces H Q ( q ) (as we have just discussed and H<2)(q), respectively. This last one accounts for higher order quasielastic scattering (lower cut in diagram lb, Im H 2Q) and two-body absorption (upper cut in diagram lb, I m H2A).We call/-/~3) the results of eq. (2) with this input. H (3) contains then diagrams 1c and 1d with the line q being a pion. We have
Volume208, number 3,4 Im H~3)(k) = × 2 02
2
PHYSICS LETTERSB
d4q 44 G(k, q) f (--~-~) 2
IV - q - l z - H
1Q
(q)-H(Z)(q)
× [Im HQ(q) + I m HZQ(q) + I m ll2A(q) ],
(5)
where the part proportional to I m / / Q would account for diagram lc and hence represents higher order quasielastic events while the terms proportional to Im H 20 and Im H 2A would account for diagram 1d. The term proportional to Im H 2Q would again account for higher order quasielastic corrections. On the other hand the term proportional to Im H 2A would account for the cut shown explicitly in diagram ld and hence represents three-body pion absorption, but with the line q as a pion. This latter term, which we shall call Im H oA, is what we have tentatively associated with quasielastic scattering followed by twobody absorption, but let us elaborate this point further. The probability per unit length for a single quasielastic scattering is given by - Im H Q (k)/k, neglecting Im/-/2Q, which would correspond mostly to two sequential quasielastic steps. After this scattering the pion comes up with momentum q. In the lifetime of the pion in the elastic channel, more concretely, before it has a quasielastic scattering or two-body absorption, the probability that the scattered pion is absorbed by two nucleons is given by i - -1 Im 1-/2A(q)dz , q 2
z'
×exp - z - q [Ira HO(q)+Im H2a(q)]dz"
,
(6) which can be integrated immediately. Hence the probability per unit length of a quasielastic collision followed by two-body absorption is given by 1 f d4q
PQA( k ) = - -~ j ~
G( k, q ) 27tS(qO2-q2_ iz2 )
Im H2A(q) × I m HQ(q) + I m H2A(q) "
(7)
21 July 1988
It is now rather instructive to realize that taking Re ( H Q+ H ~z) ) = 0 in the pion propagator ofeq. (5) and letting I m H Q + I m H ~2) tend to zero, pQA and - I m HOA/k coincide. We thus associate this latter magnitude to the probability of a quasielastic collision followed by the two-body absorption of the renormalized pion in the medium. Eq. (7) would be a first approximation to this term which ignores the effect of the renormalization of the intermediate pion. In some approaches, like cascade calculations, having Im H Q, Im H 2A as input, the steps of quasielastic scattering followed by two-body absorption are generated automatically. Hence - I m HQA/k, which is approximated in that scheme by pQa in an automatic way, should be omitted. At most one can include the difference between - I m HQA/k and pOA as a genuine three-body absorption piece, coming from the contribution of off-shell pions. However, the most genuine contribution to threebody absorption comes from pieces like in diagrams le, or diagram ld but with other ingredients in the ph interaction than one-pion exchange for the line q. Let us now look at the experimental signature of these mechanisms. In order to distinguish experimentally between an intermediate pion on-shell after a quasielastic scattering and other mechanisms which would qualify as genuine three-body absorption we can look at the variable
AT[~-~(Ti'4"Tj)2-(pi-Fpj)2,
i = 1 , 2 , 3,
(8)
where the Ti, Pi are kinetic energy and momentum of the outgoing nucleons labelled in diagram l d. h~r0 plays the role of an invariant mass except that it is defined with kinetic energies rather than total energies. If one neglects the momentum of the nucleons in the Fermi sea the variable M~3 should be equal to qO2q2, and thus equal t o / t z for the case of a quasielastic scattering followed by two-body absorption. The probability distribution as a function of ~r23 should then be a 0 function. Actually as soon as the pion is renormalized and has a width as we have seen, one would find a peak instead of the 0 function. This would be a signature that such a mechanism has occurred. But let us make these ideas more quantitative. We evaluate ImHQA(k) by means ofeq. (5) with only HZA(q) in the right-hand bracket, as we have said. H~2)~q) is taken from ref. [8]. We perform the daq integration and the dap needed to evaluate 341
Volume 208, number 3,4
PHYSICS LETTERS B
/7 (2) (q). Each point of this integrand will have a certain weight P(q, p) and determines the kinematics of the outgoing particles: T~=k°-q°; T2=q°-p°; Ta=p°; p~ = k - q ; p2=q-p; p3=p. Then one can simultaneously evaluate the probability distribution o f the pseudoinvariant masses PiE(h~t22), PI3(~QI23), In order to perform the integrations we have m a d e use of the useful approximation
O(k°-q°)Im U ( k - q ) = -~z6(k°-q°-e(k-q) ) , hence neglecting again the Fermi motion and the Pauli blocking factor, which are not too relevant for the present study. On the other hand we take an angular average I TI 2 and make it depend on the Mandelstam variables s. Hence,
21 July 1988
~ ITI2--,4n~22 ~,
(9)
S,t,~
with d a spin-isospin averaged rcN cross section (we use Arndt's nN phase shifts for the evaluation [8]. Following with the approximations we have only included x exchange to evaluate H (2~ and have also neglected Im H 20. The results for P23 (2Q~23) can be seen fig. 2 (dashed line). We have taken a nuclear density p = ½Po (Po is the normal nuclear matter density) which should be suited for a problem like x absorption in 3He. One can see a narrow peak around/~2 (slightly displaced because of the energy dependence of H). However, experimentally, one has no means to distinguish the pair o f particles (23) from the (12) or (13). Thus one would be observing ~ [ P t 2 ( M 2 ) + P I 3 ( M 2) +P23(Jl~ 2) ], which we plot in the same figure by a
p 0.56_
0.48_
0.40_
0.32_
0.24.
0.16.
I I I
l | t
0,08.
o.oo
~
-65.00
, -55.00
-',~.oo
"~.oo
I
-85.00
-,s.oo
L...
"~°°~/12[MeV °''t' 2~'°4 ]
Fig. 2. Dashed line: P23(~2) ; continuous line: ~(P~2+ P13+ P23 ). All of them are for one-pion exchange in the line of momentum q in diagram ld. T~= 165 MeV. 342
Volume 208, number 3,4
PHYSICS LETTERS B
continuous line. We can see in the figure that the quasielastic peak (we shall call it that way) emerges very clean over a very broad background. The area of the peak is now ] of the total area of the distribution. In order to see the shape of the distribution for a genuine three-body absorption mechanism we have taken the same formula, eq. (5) and have exchanged and object of mass 5/~ (like a p meson). In fig. 3 we see P 2 3 ( j ~ 2 3 ) which exhibits a broad bump around 5 X 10 4 MeV 3, quite distinct from the Breit-Wigner shape of the quasielastic peak. In the same figure (continuous line) we plot ~[PI2(~t2)-I-Pt3(~ f2) +/23 (~t2) ] which is what an experiment should see in the case of a pure genuine three-body absorption mechanism. Finally, we plot in fig. 4 a situation which would correspond to 50% quasielastic followed by two-body absorption and 50% genuine three-body absorption. p
21 July 1988
The lower curve is the contribution of the genuine three-body absorption plus the background of quasielastic followed by two-body absorption, ] (PI 2+ P~3 )T h e quasielastic peak emerges clearly over this background. The area of this peak over the background provides the amount of quasielastic scattering followd by two-body absorption of the total three-body absorption. More precisely, this probability is given by (3 × area of peak/total area). Note also that the separation this method offers is of quasielastic scattering followed by two-body absorption from the rest. This rest could also include pieces of two-body absorption followed by final-state interaction of one of the nucleons with a third one. This separation is a different problem which most probably can be treated in a similar way, but which we do not attempt here. Energy and momentum conservation in pion ab-
0.56
0.48
O.40
0.32
0.24
/
0.16
0.08
/
/
I
I
I
I
I
//
--£-
0.00
-65.00
-55.00
745.00
-'35.00
-'25.00
~15.00
J5.00
5.00 x I04
Fig. 3. Same as fig. 2 but for the exchange of an object of mass 5/t. T== 1.65 MeV.
343
Volume 208, number 3,4 p
PHYSICS LETTERS B
21 July 1988
o.14 _
0.12 -
0.10 _
0.08_
0.06_
0.04 _
0.02.
0.00
f - 65.00
-155.00
-145.00
735.00
-125.00
-=15.00
-~5.00
• ~5.00
xlO4
[M,,v']
Fig. 4. Situation with 50% quasielastic scattering followedby two-body absorption and 50% of genuine three-body absorption of the type of fig. 3. The probability of quasielastic scattering followedby two-body absorption is given by three times the area of the peak divided by the total area. T~= 165 MeV.
sorption allows to define Mij in terms of the variables of the third particle, b u t the experiment still requires the m e a s u r e m e n t of three nucleons in coincidence to ensure that they carry the energy a n d m o m e n t u m of the pion. The quasielastic peak has got its width from the imaginary part of the pion self-energy. It would also get an additional smearing from F e r m i m o t i o n which has been omitted here. The effective density could also be different to what we have taken, a n d in addition we have done several approximations in the evaluation of//~2). All these things would change the width of the quasielastic peak but not the area below it, which is the relevant element in the present discussion for what the method exposed, is sufficiently 344
realistic a n d simple to be encouraged as a way of disentangling the reaction m e c h a n i s m s in pion absorption.
References [ 1] R. Tacik, E.T. Boschitz, W. Gyles, W. List and C.R. Ottermann, Phys. Rev. C 32 (1985) 1335. [2] K.A. Aniol, A. Altman, R.R. Jonnson, H.W. Roser, R. Tacik, U. Wienands, D. Ashery, J. Alster, M.A. Moinester, E. Piasetzky, D.R. Gill and J. Vincent,Phys. Rev. C 33 ( 1986) 1714. [ 3] G. Backenstoss, M. Izycki, P. Salvisberg,M. Steinacher, P. Weber, H.J. Weyer, C. Cierjacks, S. Ljungfelt, H. Ullrich, M. Furic and T. Petkovic, Phys. Rev. Lett. 55 (1985 ) 2782;
Volume 208, number 3,4
PHYSICS LETTERS B
G. Backenstoss, M. Izycki, P. Salvisberg, M. Steinacher, P. Weber, H.J. Weyer, S. Cierjacks, B. Rzehorz, H. Ullrich, M. Furic, T. Petkovic and N. Simicevic, Phys. Rev. Lett. 59 (1987) 767. [4] D. Ashery, lecture, in: Proc. Workshop on pions in nuclei (Los Alamos, August 1987); R. Tacik, lecture, in: Workshop on pions in nuclei (Los Alamos, August 1987); H.J. Weyer, lecture, in: Proc. Workshop on pions in nuclei (Los Alamos, August 1987); J.D. Silk, University of Maryland preprint, Phys. Rev. C, to be published.
21 July 1988
[5] E. Oset, L.L. Salcedo and D. Strottman, Phys. Lett. B 165 (1985) 13. [6] L.L. Salcedo, E. Oset, M.J. Vicente and C. Carcia-Recio, Nucl. Phys. A 484 (1988) 557. [7] C. Carcia-Recio, E. Oset, L.L. Salcedo and D. Strottman, to be published. [8] E. Oset, Y. Futami and H. Toki, Nucl. Phys. A 448 (1986) 597. [ 9 ] C. Itzykson and J.B. Zuber, Quantum field theory (Mc Graw Hill, New York, 1980). [ 10] R.A. Arndt, private communication.
345
PHYSICS LETTERS B
21 July 1988
E X P E R I M E N T A L T E S T O F T H R E E - B O D Y P I O N A B S O R P T I O N ¢r L.L. SALCEDO 1,2, E. OSET 3, D. S T R O T T M A N Los Alamos National Laboratory, Los Alamos, NM 8 7545, USA
and E. H E R N A N D E Z Departamento de Fisica Te6rica, Facultad de Fisica, Universidad de Valencia, E-46100 Burjassot (Valencia), Spain
Received 17 November 1987; revised manuscript received 11 May 1988
A special method to analyse the three-nucleon spectra after pion absorption is proposed, which allows for the separation of quasielastic scattering followed by two-body absorption from other genuine three-body absorption mechanisms.
Recent experiments on pion absorption which determine the kinematics o f three nucleons [ 1-4 ] are giving evidence of mechanisms for the absorption that require the collaboration of three nucleons in a different way than two-body absorption with initial- or final-state interaction. The present work suggests a method to analyse the experimental data in order to disentangle the reaction mechanisms. We shall start from a realistic model o f p i o n - n u cleus scattering around the resonance region, depicted diagramatically in fig. 1, which provides a good description of integrated elastic and inclusive cross sections [ 5 ] as well as differential cross sections for inclusive [ 6 ] and elastic scattering [ 7 ]. The model evaluates the pion self-energy in infinite nuclear matter with the F e y n m a n diagrams o f fig. 1, and applies ¢~Work supported in part by CAICYT, the Joint Spanish-American Commitee for Scientific and Technological Cooperation and the DOE. Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA. 2 On leave from 1FIC, Centro Mixto, Universidad de Valencia, CSIC, E-46100 Burjasot (Valencia), Spain. 3 Departamento de Fisica Te6rica, Facultad de Fisica, Universidad de Valencia and IF1C, Centro Mixto, Universidad de Valencia, CSIC, E-46100 Burjassot (Valencia), Spain. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
it to finite nuclei by means o f the local density approximation, considering however finite-range effects [6 ]. The probability o f a reaction in an element o f longitude dz is given by [ 5,6 ], Pdz= - ( 1/q)Im H(q)dz ,
( 1)
where H ( q ) is the pion self-energy. The imaginary part of H ( q ) comes from placing on-shell the intermediate states cut by the dotted lines in fig. 1. Thus, through eq. ( 1 ), the upper cut in fig. lb provides the absorption probability by two nucleons, while the cuts in diagrams I d, I e correspond to three body absorption. The cut in diagram 1a corresponds to quasielastic pion scattering and the lower one in diagram lb plus the one in diagram 1c provide corrections to the quasielastic scattering. The wavy line in the diagram stands for the induced spin-isospin interaction, constructed from n and P exchange, modified by the effect of short-range correlations, which excite ph and Ah components in a RPA series. Now, while the pions in the induced interactions in diagram 1e are necessarily off-shell (a real pion does not excite a ph component in infinite nuclear matter, and only a negligibly small one in finite nuclei), and so are the pions o f the interaction labelled p in diagram 1d, the pions o f the interactions labelled q in diagram I d can appear onshell, in which case the diagram could be interpreted 339
Volume 208, number 3,4 I
PHYSICS LETTERS B
II
ii
id
/
21 July 1988
with diagram I a, can be calculated by evaluating diagram 1a but with a pion renormalized. Thus in such a case we have d4q . --ill(k)
/ a)
/ b) /
iI
,' e)
~
×~ ~ ( - i T ) ( - i T ) i D ( q ) ,
(2)
s,t,2t
with D (q) the renormalized pion propagator
q
1 I
+..
2
3÷.
D(q)=qO2
I
/:
d)
e)
Fig. 1. Feynman diagramsenteringthe evaluationof the pion selfenergy in refs. [5,6 ]. The dotted lines indicate the analytical cuts giving rise to the imaginary part of the pion self-energy,when the particles cut by these lines are placed on-shell. as representing a quasielastic collision followed by two-body absorption. However, the same diagram contains a large contribution from the transverse part of the induced interaction [ 8 ] in the line labelled q, which contains p meson exchange and other pieces accounting for the effect of short correlations and which do not interfere with n exchange. These pieces, as well as diagram 1e, contribute to three-body absorption and cannot be cast into the classification of quasielastic scattering followed by two-body absorption (initial-state interaction) or nucleon induced knock out after pion absorption (final-state interaction). We then adopt the terminology of genuine three body absorption for such mechanisms which cannot be branded as initial- or final-state interaction preceding or following two-body absorption, respectively. The question is whether experimentally one can distinguish these situations. This work suggests a method of analysing the three-nucleon spectra, after pion absorption, which distinguishes quasielastic scattering followed by two-body absorption from genuine three-body absorption. Let us evaluate these two mechanisms from the model of fig. 1. First we concentrate on quasielastic followed by two-body absorption. If the lines with m o m e n t u m q in diagrams lb, lc, ld account for just one pion exchange, these last three pieces, together 340
q2 it2 H(q),
(3)
U(q) the Lindhard function for ph excitation and T the nN~nN amplitude (a delta pole dominance is assumed in the model of fig. 1 ). We sum T 2 over spin, isospin of the final nucleons and pion and average over spin, isospin of the initial nucleons (s, t, 2 variables). Now we use Cutkowsky rules [ 9 ] for the imaginary part of H ( H ( k ) ~, 2i Im H ( k ) ); U(p)--,2iO(p°)Im U(p); D(q)--,2iO(q°)ImD(q); G(k)-,2ilmG(k) for a positive energy fermion propagator and conjugate lines and vertices above the cut). Thus (we cut, or equivalently place on-shell, the intermediate ph excitation and the pion) t" d4q
Im H(k) = - J ~
G(k,q) 2ImD(q) ,
(4)
with
G(k, q) =O(k°-q°)O(q°) Im U(K-q)~ • IT[ 2 . (4') As a particular case we can use this formula to evaluate the quasielastic channel given by diagram la, which we obtain from eq. (4) by replacing D(q) by the free pion propagator, or equivalently Im D(q) by _ r r r ( q O 2 q 2 ~2). We shall call I m / / Q the results ofeq. (4) with this substitution. Now let us use the diagrams la and lb as input for H(q) in D(q) of eqs. (2), (3); we shall call these input pieces H Q ( q ) (as we have just discussed and H<2)(q), respectively. This last one accounts for higher order quasielastic scattering (lower cut in diagram lb, Im H 2Q) and two-body absorption (upper cut in diagram lb, I m H2A).We call/-/~3) the results of eq. (2) with this input. H (3) contains then diagrams 1c and 1d with the line q being a pion. We have
Volume208, number 3,4 Im H~3)(k) = × 2 02
2
PHYSICS LETTERSB
d4q 44 G(k, q) f (--~-~) 2
IV - q - l z - H
1Q
(q)-H(Z)(q)
× [Im HQ(q) + I m HZQ(q) + I m ll2A(q) ],
(5)
where the part proportional to I m / / Q would account for diagram lc and hence represents higher order quasielastic events while the terms proportional to Im H 20 and Im H 2A would account for diagram 1d. The term proportional to Im H 2Q would again account for higher order quasielastic corrections. On the other hand the term proportional to Im H 2A would account for the cut shown explicitly in diagram ld and hence represents three-body pion absorption, but with the line q as a pion. This latter term, which we shall call Im H oA, is what we have tentatively associated with quasielastic scattering followed by twobody absorption, but let us elaborate this point further. The probability per unit length for a single quasielastic scattering is given by - Im H Q (k)/k, neglecting Im/-/2Q, which would correspond mostly to two sequential quasielastic steps. After this scattering the pion comes up with momentum q. In the lifetime of the pion in the elastic channel, more concretely, before it has a quasielastic scattering or two-body absorption, the probability that the scattered pion is absorbed by two nucleons is given by i - -1 Im 1-/2A(q)dz , q 2
z'
×exp - z - q [Ira HO(q)+Im H2a(q)]dz"
,
(6) which can be integrated immediately. Hence the probability per unit length of a quasielastic collision followed by two-body absorption is given by 1 f d4q
PQA( k ) = - -~ j ~
G( k, q ) 27tS(qO2-q2_ iz2 )
Im H2A(q) × I m HQ(q) + I m H2A(q) "
(7)
21 July 1988
It is now rather instructive to realize that taking Re ( H Q+ H ~z) ) = 0 in the pion propagator ofeq. (5) and letting I m H Q + I m H ~2) tend to zero, pQA and - I m HOA/k coincide. We thus associate this latter magnitude to the probability of a quasielastic collision followed by the two-body absorption of the renormalized pion in the medium. Eq. (7) would be a first approximation to this term which ignores the effect of the renormalization of the intermediate pion. In some approaches, like cascade calculations, having Im H Q, Im H 2A as input, the steps of quasielastic scattering followed by two-body absorption are generated automatically. Hence - I m HQA/k, which is approximated in that scheme by pQa in an automatic way, should be omitted. At most one can include the difference between - I m HQA/k and pOA as a genuine three-body absorption piece, coming from the contribution of off-shell pions. However, the most genuine contribution to threebody absorption comes from pieces like in diagrams le, or diagram ld but with other ingredients in the ph interaction than one-pion exchange for the line q. Let us now look at the experimental signature of these mechanisms. In order to distinguish experimentally between an intermediate pion on-shell after a quasielastic scattering and other mechanisms which would qualify as genuine three-body absorption we can look at the variable
AT[~-~(Ti'4"Tj)2-(pi-Fpj)2,
i = 1 , 2 , 3,
(8)
where the Ti, Pi are kinetic energy and momentum of the outgoing nucleons labelled in diagram l d. h~r0 plays the role of an invariant mass except that it is defined with kinetic energies rather than total energies. If one neglects the momentum of the nucleons in the Fermi sea the variable M~3 should be equal to qO2q2, and thus equal t o / t z for the case of a quasielastic scattering followed by two-body absorption. The probability distribution as a function of ~r23 should then be a 0 function. Actually as soon as the pion is renormalized and has a width as we have seen, one would find a peak instead of the 0 function. This would be a signature that such a mechanism has occurred. But let us make these ideas more quantitative. We evaluate ImHQA(k) by means ofeq. (5) with only HZA(q) in the right-hand bracket, as we have said. H~2)~q) is taken from ref. [8]. We perform the daq integration and the dap needed to evaluate 341
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/7 (2) (q). Each point of this integrand will have a certain weight P(q, p) and determines the kinematics of the outgoing particles: T~=k°-q°; T2=q°-p°; Ta=p°; p~ = k - q ; p2=q-p; p3=p. Then one can simultaneously evaluate the probability distribution o f the pseudoinvariant masses PiE(h~t22), PI3(~QI23), In order to perform the integrations we have m a d e use of the useful approximation
O(k°-q°)Im U ( k - q ) = -~z6(k°-q°-e(k-q) ) , hence neglecting again the Fermi motion and the Pauli blocking factor, which are not too relevant for the present study. On the other hand we take an angular average I TI 2 and make it depend on the Mandelstam variables s. Hence,
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~ ITI2--,4n~22 ~,
(9)
S,t,~
with d a spin-isospin averaged rcN cross section (we use Arndt's nN phase shifts for the evaluation [8]. Following with the approximations we have only included x exchange to evaluate H (2~ and have also neglected Im H 20. The results for P23 (2Q~23) can be seen fig. 2 (dashed line). We have taken a nuclear density p = ½Po (Po is the normal nuclear matter density) which should be suited for a problem like x absorption in 3He. One can see a narrow peak around/~2 (slightly displaced because of the energy dependence of H). However, experimentally, one has no means to distinguish the pair o f particles (23) from the (12) or (13). Thus one would be observing ~ [ P t 2 ( M 2 ) + P I 3 ( M 2) +P23(Jl~ 2) ], which we plot in the same figure by a
p 0.56_
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0.40_
0.32_
0.24.
0.16.
I I I
l | t
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o.oo
~
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, -55.00
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"~.oo
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-,s.oo
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Fig. 2. Dashed line: P23(~2) ; continuous line: ~(P~2+ P13+ P23 ). All of them are for one-pion exchange in the line of momentum q in diagram ld. T~= 165 MeV. 342
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continuous line. We can see in the figure that the quasielastic peak (we shall call it that way) emerges very clean over a very broad background. The area of the peak is now ] of the total area of the distribution. In order to see the shape of the distribution for a genuine three-body absorption mechanism we have taken the same formula, eq. (5) and have exchanged and object of mass 5/~ (like a p meson). In fig. 3 we see P 2 3 ( j ~ 2 3 ) which exhibits a broad bump around 5 X 10 4 MeV 3, quite distinct from the Breit-Wigner shape of the quasielastic peak. In the same figure (continuous line) we plot ~[PI2(~t2)-I-Pt3(~ f2) +/23 (~t2) ] which is what an experiment should see in the case of a pure genuine three-body absorption mechanism. Finally, we plot in fig. 4 a situation which would correspond to 50% quasielastic followed by two-body absorption and 50% genuine three-body absorption. p
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The lower curve is the contribution of the genuine three-body absorption plus the background of quasielastic followed by two-body absorption, ] (PI 2+ P~3 )T h e quasielastic peak emerges clearly over this background. The area of this peak over the background provides the amount of quasielastic scattering followd by two-body absorption of the total three-body absorption. More precisely, this probability is given by (3 × area of peak/total area). Note also that the separation this method offers is of quasielastic scattering followed by two-body absorption from the rest. This rest could also include pieces of two-body absorption followed by final-state interaction of one of the nucleons with a third one. This separation is a different problem which most probably can be treated in a similar way, but which we do not attempt here. Energy and momentum conservation in pion ab-
0.56
0.48
O.40
0.32
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/
0.16
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/
/
I
I
I
I
I
//
--£-
0.00
-65.00
-55.00
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-'35.00
-'25.00
~15.00
J5.00
5.00 x I04
Fig. 3. Same as fig. 2 but for the exchange of an object of mass 5/t. T== 1.65 MeV.
343
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o.14 _
0.12 -
0.10 _
0.08_
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0.04 _
0.02.
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f - 65.00
-155.00
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735.00
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Fig. 4. Situation with 50% quasielastic scattering followedby two-body absorption and 50% of genuine three-body absorption of the type of fig. 3. The probability of quasielastic scattering followedby two-body absorption is given by three times the area of the peak divided by the total area. T~= 165 MeV.
sorption allows to define Mij in terms of the variables of the third particle, b u t the experiment still requires the m e a s u r e m e n t of three nucleons in coincidence to ensure that they carry the energy a n d m o m e n t u m of the pion. The quasielastic peak has got its width from the imaginary part of the pion self-energy. It would also get an additional smearing from F e r m i m o t i o n which has been omitted here. The effective density could also be different to what we have taken, a n d in addition we have done several approximations in the evaluation of//~2). All these things would change the width of the quasielastic peak but not the area below it, which is the relevant element in the present discussion for what the method exposed, is sufficiently 344
realistic a n d simple to be encouraged as a way of disentangling the reaction m e c h a n i s m s in pion absorption.
References [ 1] R. Tacik, E.T. Boschitz, W. Gyles, W. List and C.R. Ottermann, Phys. Rev. C 32 (1985) 1335. [2] K.A. Aniol, A. Altman, R.R. Jonnson, H.W. Roser, R. Tacik, U. Wienands, D. Ashery, J. Alster, M.A. Moinester, E. Piasetzky, D.R. Gill and J. Vincent,Phys. Rev. C 33 ( 1986) 1714. [ 3] G. Backenstoss, M. Izycki, P. Salvisberg,M. Steinacher, P. Weber, H.J. Weyer, C. Cierjacks, S. Ljungfelt, H. Ullrich, M. Furic and T. Petkovic, Phys. Rev. Lett. 55 (1985 ) 2782;
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G. Backenstoss, M. Izycki, P. Salvisberg, M. Steinacher, P. Weber, H.J. Weyer, S. Cierjacks, B. Rzehorz, H. Ullrich, M. Furic, T. Petkovic and N. Simicevic, Phys. Rev. Lett. 59 (1987) 767. [4] D. Ashery, lecture, in: Proc. Workshop on pions in nuclei (Los Alamos, August 1987); R. Tacik, lecture, in: Workshop on pions in nuclei (Los Alamos, August 1987); H.J. Weyer, lecture, in: Proc. Workshop on pions in nuclei (Los Alamos, August 1987); J.D. Silk, University of Maryland preprint, Phys. Rev. C, to be published.
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[5] E. Oset, L.L. Salcedo and D. Strottman, Phys. Lett. B 165 (1985) 13. [6] L.L. Salcedo, E. Oset, M.J. Vicente and C. Carcia-Recio, Nucl. Phys. A 484 (1988) 557. [7] C. Carcia-Recio, E. Oset, L.L. Salcedo and D. Strottman, to be published. [8] E. Oset, Y. Futami and H. Toki, Nucl. Phys. A 448 (1986) 597. [ 9 ] C. Itzykson and J.B. Zuber, Quantum field theory (Mc Graw Hill, New York, 1980). [ 10] R.A. Arndt, private communication.
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