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Coupled channel appooach for pionic atoms 1H and 3He
NUCLEAR PHYSICS A Nuclear Physics A 609 (1996) 377-390
ELSEVIER
Coupled channel approach for pionic atoms 1H and 3He A. Ciepl3) a,b, R. M a c h a a Nuclear Physics Institute, CZ-250 68 {¢ei, Czech Republic b Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel t
Received 28 November1995; revised 15 July 1996
Abstract
The momentum-space method by Vincent and Phatak is extended to deal with a two-particle bound state, coupled to other open channels. The present approach is applied to two systems of coupled channels represented by fr-p, fr°n, Tn and fr-3He, fro 3H, T3H, respectively. The energy shifts and absorption widths of the ls levels of pionic hydrogen and the 3He fr-atom are calculated and compared with existing experimental data.
1. I n t r o d u c t i o n
The increasing power of present computers has stimulated the formulation and practical use of many computation techniques. One of them is represented by the treatment of combined Coulomb and strong interaction in momentum space. The classical method for solving the scattering and bound-state problems is based on coordinate-space formulation and utilizes the mathematical apparatus of differential equations. On the other hand, the strong interaction of particles with nuclei is much more conveniently expressed in momentum space. This is mostly due to its rather complicated dynamics that includes absorption, non-local effects, the finite range of interacting particles, etc. In addition, the relativistic formulation is also much simpler in the momentum-space representation. Unfortunately, in this representation the Coulomb term has a well-known singularity of the 1/q 2 type. Analytic solutions of the point-Coulomb problem are available, but they I Address until 31 July 1996. 0375-9474/96/$15.00 Copyright(~) 1996 Elsevier Science B.V. All rights reserved PII S0375-9474 (96) 00269-2
378
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
do not help when one wishes to include the complex and non-local strong-interaction optical potential. The regularization methods proposed by various authors differ in both the original ideas and in their applications. First of all, there have been many approaches that used an extraneous cutoff to constrain the singular behaviour of the Coulomb interaction. These methods are, in principle, exact; however the cutoff can never truly be taken to the limit. Quite recently, Chinn et al. [ 1] presented a method in which the limits of the cutoff are taken analytically (by folding the T-matrix and the optical potential operators into the Coulomb eigenfunctions) and no cutoff parameters appear in the numerical calculations. The corresponding computation procedure is, however, very time consuming. Another drawback of the approach is its restriction only to scattering, while the application to bound-state problems remains unsolved. On the other hand, almost 20 years ago, Kwon and Tabakin developed a momentum-space formalism based on the subtraction technique suggested by A. Lande [2]. Their approach was originally applied to hadronic atoms without any coupling to other channels, but later extended by R.H. Landau, who presented the coupled channel version and calculated the bound-state characteristics of the K - p system [3]. The computer code which implements the algorithm by Kwon and Tabakin is, however, not fully satisfactory due to its sensitivity to the choice of the integration grid. This grid should compromise the requirements dictated by two very different scales, atomic and nuclear, which is hard to maintain with a relatively small number of grid points. Consequently, it makes the numerical performance less effective. Another regularization method was proposed by Vincent and Phatak [4] for the case of pion-nucleus scattering. Some time ago, their formalism was extended to the coupled channel processes that include inelastic modes [5]. Recently, a version suitable for the solution of the pion-nucleus bound-state problem was presented as well [6]. Certainly, the method can also be used when dealing with the scattering of other charged particles by nuclei and with hadronic atoms in general. In comparison with the method by Kwon and Tabakin, the new approach is numerically more stable due to the separation of the nuclear and atomic scales that are treated separately (see Ref. [6] for details). In this paper we present the momentum-space formalism that extends the Vincent and Phatak method to the case in which the hadron-nucleus bound state is coupled to other reaction channels. Furthermore, the algorithm is applied to the description of the ls level characteristics of the lightest pionic atoms 1H and 3He. The impact of the coupled channels on these 7"r-atomic systems is discussed in detail and the results are compared with the present experimental data. Some concluding remarks are drawn in the end.
2. Method
Let us start from the multi-channel formulation of non-relativistic two-body reactions as described extensively, e.g. in the classical textbook by Taylor [7]. The /th components of the time-independent solution ~p,~(r) satisfy the system of coupled Schr6dinger equations
A. Ciepl2~, R. Mach/Nuclear Physics A 609 (1996) 377-390
[ d~r22
/ ( / +l ) r 2
+pZ~]~la(r)-Z2./M,~Vt~,(r)~pta,(r)=O
379
(1)
ot¢
for the channels a = 1 . . . . . N'. The matrix elements V,~,~,(r) describe the transition from the od to the a channels, and the relative momenta p,~ and reduced masses A.4,~ are defined via the relation p,~ = 2.Aria ( E
-
E ° ~ ) 1/2 ,
(2)
where E is the total energy and E°~ denotes the threshold of the o~th channel. The channels are numbered in order of their threshold energies, so that E°~i < E°~j for ai < 09. The threshold energies E°~ < E correspond to bound states of the interacting particles, while E°~ > E holds for the scattering channels. The asymptotic limit of the wave functions ~0t ( r ) must satisfy the standard condition
rlirnoo@/(r)
= Ft(p,,o r, rico ) ~3,,~oO(ao - aAc. ) + p,~odJ~,~o W-i.q~,l+l/2( - 2 i p . r )
, (3)
where ~b,~.0 stands for the amplitude describing the transition from the state la0, P,~0) to [a, p,~>. There is no incoming wave in the channels corresponding to the bound states of the interacting particles. (A/'B is the number of bound-state channels involved in Eqs. ( 1 ) . ) The Coulomb function Ft and Whittaker function W_i,., l+l/2 become the standard Bessel and Hankel spherical functions [8] for r / = 0, where r / = Z 1 Z 2 a P v l / p is the Sommerfeld parameter, z Following Vincent and Phatak [4] we cut the transition operators into long-distance and short-distance parts, V.., = V<., + V>,,, = V.,~, O ( R - r) + V.., O ( r - R ) ,
(4)
where O ( x ) = 1 for x > 0 and O ( x ) = 0 otherwise. The cut-off parameter R is chosen to be so large that the strong-interaction term and corrections due to the finite size of the interacting particles do not contribute to the long-distance potentials Vo~,~,, > Normally, a value slightly larger than the nuclear radius (i.e. R .~ 5-10 fm) provides a good choice. If only the Coulomb term survives (if there is any) in Voto~ > t then the solution of Eq. (1) coincides with the asymptotic limit (3) in the region r > R. Let us denote the function on the r.h.s, of Eq. (3) by u >,,~0 (P) (In what follows, we omit the index l and use p = par to simplify the notation.) The solution u<~,.(p) corresponding to the short-range potential Vo, R acquires the form < l , ! -4I u~,~(p ) = P'Jt(P') 6,~,,~ - 2tp,~.T.,. p h t ( p )
(5)
Notice the difference between the transition amplitudes ~b,~,,~ and .T,ea. The latter can be calculated from the system of coupled Lippman-Schwinger equations 2 In fact, the relations read Ft(x, r/= 0) = x jr(x) and W-in~,/+1/2(-2ix) = x hi (x).
A. Ciepl~, R. Mach/Nuclear PhysicsA 609 (1996) 377-390
380
~ . , . ( Q ' , Q ; E ) =9.,.(Q',Q;E) + ~
• -] ~p.,, v.,~,,,(Q ,a I I .,E).~'~,,.(Q",Q;E)
ettt >~fB
;
ee,,e,,2 att=l
0
,q,,2
_2 Pa"
,
(6)
--
in which we also suppressed the index l and denoted =
2~r
vg,.
(7)
The system of equations (6) can easily be symmetrized and solved by the inverse matrix method [9]. Matching smoothly at the point r = R the short- and long-range solutions, one obtains the conditions Z
Z
< > b,,~oU~,,(pa, R)= Ua,,~o(p,~,R),
du<'a du a>,~o ( p,~,R) . b~o "-d-7 (p'~'R) = dp'
( 8)
ot
The coefficients baao characterize the overlap of the wave functions < t u,,,,(p ) in separate channels. They satisfy the system of equations
Z
{W~,, u.<,.} b~o = {IV.,, F~, } 8.,~oO(ao - aNn )
( _ t x ) and a'ao~P
U>
(9)
t/
for a t = 1. . . . . .Af. Here {u,w} = u(dw/dp) - (du/dp)w is the Wronskian of the functions u and w taken at the point r = R. If the initial channel a0 represents the bound state (i.e. a0 ~< A/n), the non-trivial solution of the system (9) can exist only under the condition det ff~,,~= det[ {W~,, u,~,, < } I =0.
(10)
In other words, the determinant of the Jost matrix has to vanish if the particles form a bound state in the a0 channel. Eq. (10) generalizes the one-channel formula obtained in Ref. [ 6]. The only difference is that the long-range potential defined in [6] includes also a long-range tail of vacuum polarization corrections. They can easily be incorporated into the many-channel approach presented here just by modifications of the pure Coulomb functions Ft and W_ m at short distances comparable with the Compton wavelength. Then, the correct wave function u,~,,~(p) > is to be obtained numerically by solving the proper differential equation starting from the well-known asymptotics defined by Eq. (3). Coming back to the .Af = 1 case, we see that Eq. (10) reduces just to the condition required for the Jost function, which equals zero for the energies corresponding to bound states (Eq. (10) in [6] ). The discrete energies E = Ent, which determine the bound states in the a0 channel, are found as the iterative solutions of Eq. (10) solved in the complex energy plane.
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381
3. Transition amplitudes The matrix elements V,,,,~ describe the interaction of particles in a specific channel (for d = a ) and define the coupling of all channels taken into our consideration. In fact, they represent the transition amplitudes in the Born approximation. In what follows, we introduce the transition amplitudes for a simple system consisting of three coupled channels: the bound-state 7r-X (pionic atom), 7r°X ~ and yX ~. Here X stands for either JH (--- p) or 3He and X ~ is the corresponding isotopic partner (n or 3H, respectively). The strength of the transition to the electromagnetic channel yX' is rather small in comparison with that of the charge-exchange reaction. Nevertheless, this channel has to be included to preserve the unitarity of the S-matrix at low energies. Of course, unitarity is not satisfied exactly due to the existence of other possible channels (e.g. e+e-X'), but their contribution to the total inelastic flux is negligible. The most simple case is represented by the triplet of coupled channels 7r-p, 7r°n and yn. This system was studied thoroughly within a hamiltonian approach in Ref. [ 10]. Its authors included also the single baryons d and N in their channel space and applied the model to pion photoproduction on the nucleon. The single-baryon channels can be omitted in our approach, since they do not contribute to the s-wave 7rN interaction. The strong-interaction part of the matrix elements V ~ reads
UzrN',~.N(Qt, Q) = hi (O') A hi (Q) + h2(Q') a h2(Q) ,
( 11 )
where
hk(Q)-
ak Qmk
(Q2q_b~)n,,
k=l,2.
(12)
Here ,~ = + 1 specify the character of rrN interaction in a given state ( + 1 for attraction and - 1 for repulsion) and mk and nk define the off-shell dependence of the form factors hk(Q). Finally, the parameters ak and bk characterize the strength and effective range of the interaction. All the parameters were fitted to reproduce the 7rN phase shifts and their values were presented in Ref. [ 10]. At very low energies, the s-wave photoproduction amplitudes can be written in the form
tg~N,rN( Q t, Q) = Fcut( O') ~o+ ( TrN) Fcut( Q ) .
(13)
The threshold values of the Born amplitudes are [ 11 ] /~0+ (Tr-P) = - 3 1 . 9 × 10 -3 m - l , ~0~ (Tr°n) =0.4 × 10 -3 m -1 ,
(14) (15)
where m is the pion mass and the off-shell form factor is usually chosen in the simple form Feut(Q) =
1 + A2 j
,
v = 1 or 2.
(16)
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382
The value of the cut-off parameter A will be discussed later. Of course, the elementary photoproduction amplitude /~0+(TrN) is an energy-dependent quantity. Nevertheless, it appears reasonable to neglect this dependence and use the threshold values, as far as the interaction energy is very close to the threshold. The last matrix element v~,n.z,n describes the Compton scattering of photons by the neutron. Because this channel is very weak, it is quite sufficient to approximate it by v~,n,y, ~ (a/g)v~,-p,yn ~ lO--3V~r-p,~,n. Here the quotient a/g is the ratio of the electromagnetic and strong-interaction constants. Finally, the Coulomb term and corrections due to the finite size and vacuum polarization have to be added to the transition amplitudes in channels describing the interaction of charged particles. For the channel system under discussion it means that
V~<,~(Q',Q) =v~,~(Q',Q) ÷ [Vc(q) + Vvp(q) + VFs(q)]~ 6~,~6,l ,
(17)
where q = Q - Q' and the form of the electromagnetic terms Vc, Vve and VFS was presented, e.g., in Ref. [6]. The construction of matrix elements V~,~ for nuclei composed of many nucleons is based on the impulse approximation and involves the treatment of Fermi motion, nuclear structure effects and many-body processes. The standard approach to that problem was developed by Watson and coworkers in late 50's [ 12]. The momentum-space formulation of the method was applied to pion scattering off nuclei, charge-exchange reactions and to pionic atoms as well [5,13-15]. Very recent work by Kamalov et al. [ 16] presented an extensive analysis of the pion interaction with the three-nucleon system up to the ~7 production threshold. Further details on the construction of the pion-trinucleon optical potential can also be found in Ref. [ 17]. Finally, the momentum-space expression of the nuclear photoproduction amplitude was given, e.g., in Ref. [ 18]. In the present paper we merely show the resulting form of the transition operators v,~,,~ and refer the reader to previously published papers for more details. In the pion sector, the strong-interaction optical potential can be written as
2tZ (Q']V~'A'~AIQ) = A [a°°F°°(q) + 2At " 7" aol Fol ( q) ] 47r +A(A - 1) [B0 + COO" Q] Go(q),
(18)
where /z stands for the 7rN reduced mass, t and 7" are the pion and nuclear isospin operators, respectively, and the isoscalar and isovector nuclear form factors Fsr(q) were specified in [ 17]. The optical potential (18) consists of a first-order term expressed as a coherent sum of the elementary 7rN amplitudes oo
L--O
+L (fL3/2 + (--1)T fLI/2)] P L ( q ' q ' ) ,
(19)
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
383
where PL(x) are the Legendre polynomials and the amplitudes f ~ l = arLt q q" e) were constructed from the experimental rrN phase shifts [ 17]. The second-order term is introduced phenomenologically and parametrized by the two complex parameters B0 and Co. They were fitted in [ 15] to the ls and 2p level characteristics of pionic atoms 160 and 4°Ca, and reproduce the experimental data for 4He quite well too. For heavy nuclei the function Go(q) converges to the Fourier transform of nuclear density squared. The explicit form of Go(q) involving the two-particle nuclear correlations and a more general structure of the second-order term were investigated thoroughly in Ref. [ 15]. However, the construction of Vrr-He,Tr-He requires some additional care. Since the first-order optical potential is expressed through the elementary ~-N amplitudes (i.e. the experimentally measured quantities), it already accounts for the process r r - p 7r°n , 7 r - p (and also for yn in the intermediate state). As far as the coupled channels rr°3H and T3H contribute to the real part of Y,~- He,, r He just via the aforementioned subprocesses, one has to subtract their contribution from aoo to avoid double counting. A consistent way to do so is the multiple scattering expansion of the pion-nuclear optical potential up to the second order in t,~t¢. The appropriate second-order term is standardly referred to as Pauli correlations and expressed as a repulsive correction to the s-wave part of am (denoted by bo). The derivation is given in Appendix A where we omitted the y3H channel, which is justified for its relatively weak coupling to the 7r- 3He channel. Considering only the s-shell nuclei 3He and 4He we arrived at
Abo=
M
IX
b2+
2+
~aa,3
b
},
(20~
where (r--{7.) denotes the nuclear matrix element of the pion propagator in the intermediate state. Note that Eq. (20) differs from the standard rescattering correction [191 obtained for one-channel formulation within the closure approximation. Absorbing the A-independent part of Eq. (20) into the second-order optical potential (term proportional to B0 and estimating (7~2) for 3He in the q = 0 limit, the contribution to b0 is IAbo(A) I ~ (43.6 MeV) bl2 = (2.8:t:0.3) 10-3m -I, where bl stands for the s-wave part of a01 and the error estimate is an educated guess. This correction is to be subtracted from b0, increasing the s-wave repulsion of the isoscalar term proportional to a00. Within the impulse approximation the photoproduction matrix element acquires the form 27"/"
(Q'IVrrA,rAIQ) = - - -
Ix
v~rN,r u ( Q ' , Q ) Frx~r~x'(q) ,
(21)
so that ~/rgA,yA is equal to the elementary photoproduction amplitude multiplied by a nuclear form factor and some correction factors arising due to the transformation from the rrN to the pion-nucleus centre-of-mass frame. The form factor F~x--+~x,(q) takes account of the overlap between the initial and final state nuclear wave functions. It can be written as
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384
Table 1 Energy shifts AEN and absorption widths flABS of the l s levels of 1H and 3He pionic atoms..IV" denotes the number of channels included in the calculation. The results are presented in eV IH
.Af
3He
AEN( Is)
flABS(Is)
AEN( Is)
flABS(Is)
1
-5.00
18.9
--6.14 -6.67
0.00 0.53 0.74
- 37.7
2 3
-47.9 -47.8
26.8 27.9
Exp.
- 7 . 1 -4- 0.3
-
- 3 2 4- 3
28 -t- 7
A
F~,x--,~,x,(q) = (!/*,,, I ~--~(r+)ro'j • ~a e-iq'rjlaP'3} "~ F i r ( q ) .
(22)
j=l
The explicit form of the spin-flip form factors FIT(q) is given in [ 17]. In Eq. (22), ~a is the photon polarization vector and the operator ¢rj • ~a flips the spin of the target nucleon, which is in correspondence with the conservation of the total angular momentum. Since the Pauli exclusion principle forbids the existence of two identical fermions in one quantum state, the photon can be absorbed only by a nucleon the final state of which differs from the states already occupied. For that reason the reaction 3H ~ 7r- 3He can proceed only on the neutron that has the same spin projection as the proton spectator (and also as the whole 3H nucleus). The Compton scattering matrix element can be treated in the same way as earlier, i.e. V),A,ra ,.~ 10 -5 U?,H,zr-He, and the electromagnetic terms Vc, VFS and Vw are to be added to V~-He,,r-He. We also mention that the matrix elements (Q'Iv~,~IQ) defined by Eqs. (18) and (21) must be decomposed into partial waves to get the required quantities
(z ,~(Q,, Q ).
4. Results The numerical precision of the method presented in Section 2 was tested by comparison with the analytical solution of the bound-state problem corresponding to a pure Coulomb potential in the c~ = 1 channel and no coupling to the other channels (vzl = Vc and vl ~ ~ l = v~ ÷ l 1 = 0). The relative accuracy I(E - En )/EBI achieved in reproducing the Bohr energy E8 o f the ls state in pionic hydrogen was at the level of 10 -6 when 40 grid points were used to cover the momentum distribution. This result is, of course, the same as the one we obtained by testing the one-channel version of the computer code [6]. The structure o f the present code allows the user to switch off any channel or to include them successively. Our results for IH and 3He pionic atoms are shown in Table 1. Since the matrix element V~r-p,~r- p is real, the absorption width of the ls level of pionic hydrogen is equal to zero within the one-channel approximation. Nevertheless, the existence of the
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385
Table 2 The characteristics of pionic hydrogen calculated by using different models of low-energy ~'N interaction. The values are given in eV
AEN ( I s) FAns(Is)
v v=l
v~2
vI [201
vII [21]
a~rN [21]
a~rN [24]
K [25]
--6.67 0.74
--6.61 0.74
--6.25 0.89
--7.77 1.01
--7.81 1.00
--6.66 0.90
--6.45 0.78
strong interaction causes a shift of the energy level with respect to its electromagnetic value. The pion charge-exchange reaction gives a contribution to the width of the Is level and affects (by some 20-30%) the strong-interaction shift AEN(Is) as well. To include the electromagnetic channel ce = 3, one has to fix the cut-off parameter A first. We performed our calculation with A = 1 400 MeV for a monopole form factor (v = 1) defined by Eq. (16). This value was chosen in the middle of a region in which the calculated quantities AEN and FABS remained practically constant. In that region the change of A by 300 MeV causes the deviations 6(AEN) = 0.2 eV and 6(FABs) = 0.01 eV. For comparison, we calculated the characteristics of pionic hydrogen using also the dipole form factor (v = 2, A = 2000 MeV) and different low-energy ¢rN potentials. The results of our calculations are presented in Table 2, where the predictions of other authors are shown as well. The potential v I is local and of the Yukawa-type (see Ref. [20] for its definition), while vn was chosen in the separable form defined in [21]. They both reproduce well the low-energy 7rN scattering data and the threshold value of the Panofsky ratio. The results obtained with various types of matrix elements v,~,~ differ from each other at the level of 20%. These deviations are caused mainly by the fact that the 7rN potentials give different 7rN scattering lengths when extrapolated to zero energy. To some extent, the results are also sensitive to the off-shell form of the potential matrix, but they are almost independent of the behaviour of v~,~ at higher energies. In fact, the characteristics of pionic hydrogen can easily be expressed through the scattering lengths by means of formulas derived by Deser et al. [22] :
4EB(ls) 4EB (ls) AEN(Is) = - a~-p - - (2al + a 3 ) , rB 3rB /'ABs(ls) = 8p21gB(ls) l
1+
lac~xl 2
rB
-
16p2IEB(ls)I (1 + ~-~) 9rB (a3 -- al) 2 •
(23)
Here a2t denotes the s-wave 7rN scattering lengths for the total isospin I = ½, I = ~ and Pr is the threshold value of the Panofsky ratio. (Experiment gives Pz, = 1.546 5:0.009 [23].) The derivation of Eq. (23) is based on the expansion of AEN/EB in powers of a~r-p/rB, where rB is the Bohr radius of the pionic orbit. Its validity is guaranteed by the smallness of the ratio a~r-p/rB ~ 10 -3. In Table 2 we show the ~'-atomic characteristics calculated for the scattering lengths that correspond to vn and for the values of a~N
386
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
taken from the analysis of the 7rN scattering data [24]. Finally, the results presented in the last column of Table 2 were obtained by Rasche and Woolcock within the K-matrix coupled channel approach [25]. At this point, one should mention that the experimental data on the low-energy 7rN scattering and charge-exchange reactions are not quite consistent and different authors give different values of a,~N (compare, e.g., the results of the analyses presented in Refs. [21,24] ). For example, the isovector combination of the scattering lengths usually denoted by bl (aCEX = v ~ b l ) varies within a rather wide range from bl = -0.086 m -I to bl -- - 0 . 0 9 7 m -1 [21]. In addition, the extrapolation of the elementary 7rN amplitudes to zero energy is not quite well defined, since they are derived from experimental data that are available at much higher energies than required for going to the limit P~,N --~ 0. The only way for getting the scattering lengths in a model-independent way is a direct measurement of the 1s-level characteristics of pionic hydrogen. Unfortunately, only the energy shift dEN(Is) = ( - 7 . 1 + 0.3) eV was obtained in a recent experiment [26]. This value corresponds to a scattering length a~,-p = (0.086 4-0.004) m -1, which is in agreement with the phase-shift analysis of the Karlsruhe-Helsinki collaboration [24]. At present, the experimental precision is not sufficient enough to distinguish among the various models of low-energy zrN interaction. A new, more precise data on both the energy shift dEN(Is) and the absorption width FA~s(ls) are needed to determine the values of the scattering lengths al and a3 (or their combinations a~-p and aCEX). We mention that these quantities are of fundamental importance for testing the low-energy structure of QCD through the calculation of the pion-nucleon g-term [27]. The 3He ~'-atom represents a much more complicated system. It is the lightest pionic atom, the description of which requires a knowledge of the nuclear structure (the characteristics of the 3H pionic atom have not been measured yet). However, the nuclear structure of the trinucleon system is very well known and its interaction with the pion field is well understood too. It is quite surprising then that the ls-level energy shift of the 3He ~--atom cannot be reproduced (cf. Table 1) by using the standard pionnucleus optical potential within the full .A/= 3 coupled channel calculation. It was also shown recently [ 15] that the problem cannot be solved by correctly considering the spin-isospin structure of the elementary 7rNN amplitude. Certainly, one could add a sufficiently strong isovector term to the second-order optical potential as Kamalov et al. did in [ 16] when dealing with similar problems for the charge-exchange reaction at higher energies. Unfortunately, such a modification of the optical potential at zero energy is ruled out, because it would spoil the good description of isospin effects observed for heavier nuclei. We note that it is the isovector term of the first-order optical potential that plays the dominant role in the description of the energy shift A E N ( l s ) in pionic 3He. Since the isovector term is proportional to bl, the results are strongly dependent on the value chosen for this combination of the ~ N scattering lengths. Our calculations were performed with the elementary ¢rN amplitudes that give bl = -0.094 m -l at zero energy. For comparison, we calculated also the 1s-level characteristics of the 3He zr-atom suppressing the isovector term by a factor 0.9 (bl = -0.085 m -1). The corresponding
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387
value A E N ( I s ) = - 3 9 . 2 eV (for A/" = 3) is much closer to the experimental data, while the absorption width FABs(ls) = 26.0 eV remains almost unchanged. We remind the reader that the characteristics of heavier pionic atoms (with A t> 4) are dominated by isoscalar terms that contribute coherently to the optical potential, while the isovector term is suppressed by the factor ( N - Z ) / A . Leaving apart the problematic region 4 < A < 10, a change of bl at the level of 10% cannot affect significantly the good description of the characteristics of heavier 7r-atoms presented in Ref. [ 15]. Nevertheless, only a precise measurement of ~r-atomic transitions in pionic hydrogen can fix the value of bl and the strength of the isovector term in the optical potential. The measurement of rratomic characteristics on tritium can provide a valuable information too. Unfortunately, with the current value of b~ = - 0 . 0 9 4 m -~ our present level of understanding of the 7rN interaction at zero energy does not allow a satisfactory reproduction of the energy shift of the ls level in pionic 3He. The coupling to the 3H channel contributes to the discrepancy by some 30% of the experimental value. The correction due to the electromagnetic channel is negligible, which justifies the assumption made above for estimating Abo. On the other hand, the corrections due to coupled channels bring the absorption width of the ls level into a nice agreement with the data. However, this improvement is not as significant, since the experimental error bars are quite large in this case. Finally, we mention that our results for the corrections due to the coupling to the 7r° 3H channel agree well with the previous estimates made by Pilkuhn and Wycech [281.
5. Summary We have presented the extension of the method by Vincent and Phatak to the coupled channel processes involving the bound states of two interacting particles. The boundstate energies can be found as the solutions of Eq. (10), which is quite similar to the condition given by Landau [3]. In our approach it reflects the well-known fact that the determinant of the Jost matrix vanishes for the complex energies corresponding to bound states. The method represents a generalization of an earlier approach formulated to deal with the Coulomb singularity when solving the bound-state problem in the onechannel approximation. The present method retains good numerical characteristics of the previous one and makes possible the inclusion of coupling to other open channels. We demonstrated its feasibility by solving the system of three coupled channels and calculating the bound-state characteristics of pionic atoms 1H and 3He. Our results are numerically stable and show a remarkable degree of numerical precision. The calculated characteristics of pionic hydrogen do not differ significantly from the results presented by other authors. Here, our aim was to demonstrate the applicability of our method and to study the influence of the coupling to other channels on the characteristics of both IH and 3He pionic atoms treated on the same footing. At present, the experimental data on electromagnetic transitions in pionic hydrogen are not precise enough to distinguish between the various models of ~ N and TN interactions at low
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
388
energies. A more precise measurement of not only the energy shift AEN(lS) but also of the absorption width is needed to determine the 7rN scattering lengths al and a3 in a model-independent way. The failure of the coupled channel optical potential approach to reproduce the energy shift of the ls level of pionic 3He is in stark contrast to the good description of the ~'-atomic characteristics in 4He [ 15]. Since the low-energy interaction of pions with the trinucleon system is well understood, the origin of the observed discrepancy is obscure. Clearly, the solution of this problem remains open to new ideas.
Acknowledgements We are indebted to A. Gal for alerting us to the problem of double counting mentioned in the text. It is also our pleasure to thank him for many useful comments on our work and for carefully reading the manuscript. This work was supported by grant agency of the Academy of Sciences of Czech Republic, grant No. 148401. The research of A.C. was partially supported by the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities, Jerusalem.
Appendix A Here we give a brief derivation of the second-order correction Ab0, Eq. (20). We follow the standard Watson formulation of the multiple scattering theory [ 12] in which the second-order optical potential is expressed as
V(2) ( E) = A 2 [--~ f, rN( E) ] 27r-~G(E) [--~ f~rN( E) l -A
f~rN( E) -'~
where f~N(E) stands for the free ¢rN amplitude, G(E) and g(E) are the pion-nuclear and pion-nucleon Green functions, respectively. The operator 79 = 10){01 projects on single nuclear ground state for A --- 4 while
79 = ~ IJJzrrz)(TzTJzJI
(A.2)
J= T:
is a four-term expression when the coupled channels 7r- 3He and 7r° 3H are considered. The elementary zrN amplitude is taken in the form [ 17] M
- - f , rN(E) = aoo + aolt" ~"+ [aoo + aolt" ~'] i~r. [qi × qf] . The low-energy limit of the s-wave part of our coefficients aoT is a0r ~ where b0 and bl are standard combinations of 7rN scattering lengths.
(A.3)
(A~/mu)br,
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
389
In closure approximation, the matrix elements of Eq. (A.1) read
(Q'OIV~2)(E)IOQ) =A
Z
/ d3a '' (a , las'r'la ,, )(Q ,, lasrlQ) 27r2
Q--~S Q-~-
S,T, S~.T~
X [A(Olei(Q'-Q"I'rOs,T,~ ei(Q"-Q)'rOsTIO) - ( A - 1) {Olei(~-Q'')'rl O(1) "~S'T' ei(Q"-Q)'r2 0(2s~)10)]
,
(A.4)
where we introduced the operators Osr = (~rz)S(7"z) r. Taking further into account the symmetry in the coordinate-space part of the nuclear wave function of 1s-shell nuclei and restricting ourselves only to the pion-nuclear s-wave we arrive at
(Q'OIV(2)(E)IOQ)=a
(7)
Z b r , br A(Ol(t. "r)r'7 ~ (t. r)r[0) T,T'
- ( A - 1)(Ol(t.~'(~))r'(t.z(2))rlO)] I ( Q ' , Q ) ,
(A.5)
where we have denoted
I(Q',Q)
27r2
Q02-Q '7~
=
L , ) F(q) .
(A.6)
A direct calculation of isovector and isotensor matrix elements involved in Eq. (A.5) for 3He and 4He nuclear ground states leads to Eq. (20) used in the text. The isovector part of Eq. (A.5) corresponds to
Abl = ---~-dk4 2b0bl( ~1 ),~,_0.001rn -1 ,
(a.7)
which represents a negligible correction when compared with the total value of bl -0.090
m -1 .
References 11] [2] [3] I4] [5] [6] 17] [8] [9] [ 10] [ 11 ] [ 12] [13]
C.R. Chinn, Ch. Elster and R.M. Thaler, Phys. Rev. C 44 (1991) 1569. Y.R. Kwon and F. Tabakin, Phys. Rev. C 18 (1978) 932. R.H. Landau, Phys. Rev. C 27 (1983) 2191. C.M. Vincent and S.C. Phatak, Phys. Rev. C 10 (1974) 391. M. Gmitro, J. Kvasil and R. Mach, Phys. Rev. C 31 (1985) 1349. A. Ciepl~', M. Gmitro, R. Mach and S.S. Kamalov, Phys. Rev. C 44 (1991) 713. J.R. Taylor, Scattering Theory - The Quantum Theory on non-relativistic collisions (Wiley, New York, 1972). M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions, National bureau of standards (1964). M.I. Haftel and F. Tabakin, Nucl. Phys. A 158 (1970) 1. S. Nozawa, B. Blankleider and T.-S.H. Lee, Nucl. Phys. A 513 (1990) 459. D. Drechsel and L. Tiator, J. Phys. G - Nucl. Part. Phys. 18 (1992) 449. M.L. Goldberger and K.M. Watson, Collision Theory (Wiley, New York, 1964). R.H. Landau, S.C. Phatak and E Tabakin, Ann. Phys. 78 (1973) 299.
390 [14] [15] [16] [ 171 [18] 1191 [20] [21] [22] [23] [241 [25] [26] [27] [28]
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390 R.H. Landau and A.W. Thomas, Nucl. Phys. A 302 (1978) 461. A. Ciepl2~ and R. Mach, Phys. Rev. C 49 (1994) 1454. S.S. Kamalov, L. Tiator and C. Bennhold, Phys. Rev. C 47 (1993) 941. I.V. Falomkin et al., Nuovo Cimento 57A (1980) 111. R.A. Eramzhyan, M. Gmitro, S.S. Kamalov and R. Mach, J. of Phys. G 9 (1983) 605. M. Ericson and T.E.O. Ericson, Ann. of Phys. 36 (1966) 32. W.B. Kaufmann and W.R. Gibbs, Phys. Rev. C 35 (1987) 838. EB. Siegel and W.R. Gibbs, Phys. Rev. C 33 (1986) 1407. S. Deser, M.L. Goldberger, K. Baumann and W. Thirring, Phys. Rev. 96 (1954) 774. J. Spuller, D. Berghofer, M.D. Hasinoff, R. Macdonald, D.E Measday, M. Salomon, T. Suzuki, J.-M. Poutissou, R. Poutissou and J.K.P. Lee, Phys. Lett. B 67 (1977) 479. R. Koch and E. Pietarinen, Nucl. Phys. A 336 (1980) 331. G. Rasche and W.S. Woolcock, Nucl. Phys. A 381 (1982) 405. W. Beer et al., Phys. Lett. B 261 (1991) 16. J. Gasser, H. Leutwyler and M.E. Sainio, Phys. Lett. B 253 (1991) 252. H. Pilkuhn and S. Wycech, Phys. Lett. B 76 (1978) 29.
ELSEVIER
Coupled channel approach for pionic atoms 1H and 3He A. Ciepl3) a,b, R. M a c h a a Nuclear Physics Institute, CZ-250 68 {¢ei, Czech Republic b Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel t
Received 28 November1995; revised 15 July 1996
Abstract
The momentum-space method by Vincent and Phatak is extended to deal with a two-particle bound state, coupled to other open channels. The present approach is applied to two systems of coupled channels represented by fr-p, fr°n, Tn and fr-3He, fro 3H, T3H, respectively. The energy shifts and absorption widths of the ls levels of pionic hydrogen and the 3He fr-atom are calculated and compared with existing experimental data.
1. I n t r o d u c t i o n
The increasing power of present computers has stimulated the formulation and practical use of many computation techniques. One of them is represented by the treatment of combined Coulomb and strong interaction in momentum space. The classical method for solving the scattering and bound-state problems is based on coordinate-space formulation and utilizes the mathematical apparatus of differential equations. On the other hand, the strong interaction of particles with nuclei is much more conveniently expressed in momentum space. This is mostly due to its rather complicated dynamics that includes absorption, non-local effects, the finite range of interacting particles, etc. In addition, the relativistic formulation is also much simpler in the momentum-space representation. Unfortunately, in this representation the Coulomb term has a well-known singularity of the 1/q 2 type. Analytic solutions of the point-Coulomb problem are available, but they I Address until 31 July 1996. 0375-9474/96/$15.00 Copyright(~) 1996 Elsevier Science B.V. All rights reserved PII S0375-9474 (96) 00269-2
378
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
do not help when one wishes to include the complex and non-local strong-interaction optical potential. The regularization methods proposed by various authors differ in both the original ideas and in their applications. First of all, there have been many approaches that used an extraneous cutoff to constrain the singular behaviour of the Coulomb interaction. These methods are, in principle, exact; however the cutoff can never truly be taken to the limit. Quite recently, Chinn et al. [ 1] presented a method in which the limits of the cutoff are taken analytically (by folding the T-matrix and the optical potential operators into the Coulomb eigenfunctions) and no cutoff parameters appear in the numerical calculations. The corresponding computation procedure is, however, very time consuming. Another drawback of the approach is its restriction only to scattering, while the application to bound-state problems remains unsolved. On the other hand, almost 20 years ago, Kwon and Tabakin developed a momentum-space formalism based on the subtraction technique suggested by A. Lande [2]. Their approach was originally applied to hadronic atoms without any coupling to other channels, but later extended by R.H. Landau, who presented the coupled channel version and calculated the bound-state characteristics of the K - p system [3]. The computer code which implements the algorithm by Kwon and Tabakin is, however, not fully satisfactory due to its sensitivity to the choice of the integration grid. This grid should compromise the requirements dictated by two very different scales, atomic and nuclear, which is hard to maintain with a relatively small number of grid points. Consequently, it makes the numerical performance less effective. Another regularization method was proposed by Vincent and Phatak [4] for the case of pion-nucleus scattering. Some time ago, their formalism was extended to the coupled channel processes that include inelastic modes [5]. Recently, a version suitable for the solution of the pion-nucleus bound-state problem was presented as well [6]. Certainly, the method can also be used when dealing with the scattering of other charged particles by nuclei and with hadronic atoms in general. In comparison with the method by Kwon and Tabakin, the new approach is numerically more stable due to the separation of the nuclear and atomic scales that are treated separately (see Ref. [6] for details). In this paper we present the momentum-space formalism that extends the Vincent and Phatak method to the case in which the hadron-nucleus bound state is coupled to other reaction channels. Furthermore, the algorithm is applied to the description of the ls level characteristics of the lightest pionic atoms 1H and 3He. The impact of the coupled channels on these 7"r-atomic systems is discussed in detail and the results are compared with the present experimental data. Some concluding remarks are drawn in the end.
2. Method
Let us start from the multi-channel formulation of non-relativistic two-body reactions as described extensively, e.g. in the classical textbook by Taylor [7]. The /th components of the time-independent solution ~p,~(r) satisfy the system of coupled Schr6dinger equations
A. Ciepl2~, R. Mach/Nuclear Physics A 609 (1996) 377-390
[ d~r22
/ ( / +l ) r 2
+pZ~]~la(r)-Z2./M,~Vt~,(r)~pta,(r)=O
379
(1)
ot¢
for the channels a = 1 . . . . . N'. The matrix elements V,~,~,(r) describe the transition from the od to the a channels, and the relative momenta p,~ and reduced masses A.4,~ are defined via the relation p,~ = 2.Aria ( E
-
E ° ~ ) 1/2 ,
(2)
where E is the total energy and E°~ denotes the threshold of the o~th channel. The channels are numbered in order of their threshold energies, so that E°~i < E°~j for ai < 09. The threshold energies E°~ < E correspond to bound states of the interacting particles, while E°~ > E holds for the scattering channels. The asymptotic limit of the wave functions ~0t ( r ) must satisfy the standard condition
rlirnoo@/(r)
= Ft(p,,o r, rico ) ~3,,~oO(ao - aAc. ) + p,~odJ~,~o W-i.q~,l+l/2( - 2 i p . r )
, (3)
where ~b,~.0 stands for the amplitude describing the transition from the state la0, P,~0) to [a, p,~>. There is no incoming wave in the channels corresponding to the bound states of the interacting particles. (A/'B is the number of bound-state channels involved in Eqs. ( 1 ) . ) The Coulomb function Ft and Whittaker function W_i,., l+l/2 become the standard Bessel and Hankel spherical functions [8] for r / = 0, where r / = Z 1 Z 2 a P v l / p is the Sommerfeld parameter, z Following Vincent and Phatak [4] we cut the transition operators into long-distance and short-distance parts, V.., = V<., + V>,,, = V.,~, O ( R - r) + V.., O ( r - R ) ,
(4)
where O ( x ) = 1 for x > 0 and O ( x ) = 0 otherwise. The cut-off parameter R is chosen to be so large that the strong-interaction term and corrections due to the finite size of the interacting particles do not contribute to the long-distance potentials Vo~,~,, > Normally, a value slightly larger than the nuclear radius (i.e. R .~ 5-10 fm) provides a good choice. If only the Coulomb term survives (if there is any) in Voto~ > t then the solution of Eq. (1) coincides with the asymptotic limit (3) in the region r > R. Let us denote the function on the r.h.s, of Eq. (3) by u >,,~0 (P) (In what follows, we omit the index l and use p = par to simplify the notation.) The solution u<~,.(p) corresponding to the short-range potential Vo,
(5)
Notice the difference between the transition amplitudes ~b,~,,~ and .T,ea. The latter can be calculated from the system of coupled Lippman-Schwinger equations 2 In fact, the relations read Ft(x, r/= 0) = x jr(x) and W-in~,/+1/2(-2ix) = x hi (x).
A. Ciepl~, R. Mach/Nuclear PhysicsA 609 (1996) 377-390
380
~ . , . ( Q ' , Q ; E ) =9.,.(Q',Q;E) + ~
• -] ~p.,, v.,~,,,(Q ,a I I .,E).~'~,,.(Q",Q;E)
ettt >~fB
;
ee,,e,,2 att=l
0
,q,,2
_2 Pa"
,
(6)
--
in which we also suppressed the index l and denoted =
2~r
vg,.
(7)
The system of equations (6) can easily be symmetrized and solved by the inverse matrix method [9]. Matching smoothly at the point r = R the short- and long-range solutions, one obtains the conditions Z
Z
< > b,,~oU~,,(pa, R)= Ua,,~o(p,~,R),
du<'a du a>,~o ( p,~,R) . b~o "-d-7 (p'~'R) = dp'
( 8)
ot
The coefficients baao characterize the overlap of the wave functions < t u,,,,(p ) in separate channels. They satisfy the system of equations
Z
{W~,, u.<,.} b~o = {IV.,, F~, } 8.,~oO(ao - aNn )
( _ t x ) and a'ao~P
U>
(9)
t/
for a t = 1. . . . . .Af. Here {u,w} = u(dw/dp) - (du/dp)w is the Wronskian of the functions u and w taken at the point r = R. If the initial channel a0 represents the bound state (i.e. a0 ~< A/n), the non-trivial solution of the system (9) can exist only under the condition det ff~,,~= det[ {W~,, u,~,, < } I =0.
(10)
In other words, the determinant of the Jost matrix has to vanish if the particles form a bound state in the a0 channel. Eq. (10) generalizes the one-channel formula obtained in Ref. [ 6]. The only difference is that the long-range potential defined in [6] includes also a long-range tail of vacuum polarization corrections. They can easily be incorporated into the many-channel approach presented here just by modifications of the pure Coulomb functions Ft and W_ m at short distances comparable with the Compton wavelength. Then, the correct wave function u,~,,~(p) > is to be obtained numerically by solving the proper differential equation starting from the well-known asymptotics defined by Eq. (3). Coming back to the .Af = 1 case, we see that Eq. (10) reduces just to the condition required for the Jost function, which equals zero for the energies corresponding to bound states (Eq. (10) in [6] ). The discrete energies E = Ent, which determine the bound states in the a0 channel, are found as the iterative solutions of Eq. (10) solved in the complex energy plane.
A. Cieplf,, R. Mach/Nuclear Physics A 609 (1996) 377-390
381
3. Transition amplitudes The matrix elements V,,,,~ describe the interaction of particles in a specific channel (for d = a ) and define the coupling of all channels taken into our consideration. In fact, they represent the transition amplitudes in the Born approximation. In what follows, we introduce the transition amplitudes for a simple system consisting of three coupled channels: the bound-state 7r-X (pionic atom), 7r°X ~ and yX ~. Here X stands for either JH (--- p) or 3He and X ~ is the corresponding isotopic partner (n or 3H, respectively). The strength of the transition to the electromagnetic channel yX' is rather small in comparison with that of the charge-exchange reaction. Nevertheless, this channel has to be included to preserve the unitarity of the S-matrix at low energies. Of course, unitarity is not satisfied exactly due to the existence of other possible channels (e.g. e+e-X'), but their contribution to the total inelastic flux is negligible. The most simple case is represented by the triplet of coupled channels 7r-p, 7r°n and yn. This system was studied thoroughly within a hamiltonian approach in Ref. [ 10]. Its authors included also the single baryons d and N in their channel space and applied the model to pion photoproduction on the nucleon. The single-baryon channels can be omitted in our approach, since they do not contribute to the s-wave 7rN interaction. The strong-interaction part of the matrix elements V ~ reads
UzrN',~.N(Qt, Q) = hi (O') A hi (Q) + h2(Q') a h2(Q) ,
( 11 )
where
hk(Q)-
ak Qmk
(Q2q_b~)n,,
k=l,2.
(12)
Here ,~ = + 1 specify the character of rrN interaction in a given state ( + 1 for attraction and - 1 for repulsion) and mk and nk define the off-shell dependence of the form factors hk(Q). Finally, the parameters ak and bk characterize the strength and effective range of the interaction. All the parameters were fitted to reproduce the 7rN phase shifts and their values were presented in Ref. [ 10]. At very low energies, the s-wave photoproduction amplitudes can be written in the form
tg~N,rN( Q t, Q) = Fcut( O') ~o+ ( TrN) Fcut( Q ) .
(13)
The threshold values of the Born amplitudes are [ 11 ] /~0+ (Tr-P) = - 3 1 . 9 × 10 -3 m - l , ~0~ (Tr°n) =0.4 × 10 -3 m -1 ,
(14) (15)
where m is the pion mass and the off-shell form factor is usually chosen in the simple form Feut(Q) =
1 + A2 j
,
v = 1 or 2.
(16)
A. Ciepl2~,R. Mach/Nuclear Physics A 609 (1996) 377-390
382
The value of the cut-off parameter A will be discussed later. Of course, the elementary photoproduction amplitude /~0+(TrN) is an energy-dependent quantity. Nevertheless, it appears reasonable to neglect this dependence and use the threshold values, as far as the interaction energy is very close to the threshold. The last matrix element v~,n.z,n describes the Compton scattering of photons by the neutron. Because this channel is very weak, it is quite sufficient to approximate it by v~,n,y, ~ (a/g)v~,-p,yn ~ lO--3V~r-p,~,n. Here the quotient a/g is the ratio of the electromagnetic and strong-interaction constants. Finally, the Coulomb term and corrections due to the finite size and vacuum polarization have to be added to the transition amplitudes in channels describing the interaction of charged particles. For the channel system under discussion it means that
V~<,~(Q',Q) =v~,~(Q',Q) ÷ [Vc(q) + Vvp(q) + VFs(q)]~ 6~,~6,l ,
(17)
where q = Q - Q' and the form of the electromagnetic terms Vc, Vve and VFS was presented, e.g., in Ref. [6]. The construction of matrix elements V~,~ for nuclei composed of many nucleons is based on the impulse approximation and involves the treatment of Fermi motion, nuclear structure effects and many-body processes. The standard approach to that problem was developed by Watson and coworkers in late 50's [ 12]. The momentum-space formulation of the method was applied to pion scattering off nuclei, charge-exchange reactions and to pionic atoms as well [5,13-15]. Very recent work by Kamalov et al. [ 16] presented an extensive analysis of the pion interaction with the three-nucleon system up to the ~7 production threshold. Further details on the construction of the pion-trinucleon optical potential can also be found in Ref. [ 17]. Finally, the momentum-space expression of the nuclear photoproduction amplitude was given, e.g., in Ref. [ 18]. In the present paper we merely show the resulting form of the transition operators v,~,,~ and refer the reader to previously published papers for more details. In the pion sector, the strong-interaction optical potential can be written as
2tZ (Q']V~'A'~AIQ) = A [a°°F°°(q) + 2At " 7" aol Fol ( q) ] 47r +A(A - 1) [B0 + COO" Q] Go(q),
(18)
where /z stands for the 7rN reduced mass, t and 7" are the pion and nuclear isospin operators, respectively, and the isoscalar and isovector nuclear form factors Fsr(q) were specified in [ 17]. The optical potential (18) consists of a first-order term expressed as a coherent sum of the elementary 7rN amplitudes oo
L--O
+L (fL3/2 + (--1)T fLI/2)] P L ( q ' q ' ) ,
(19)
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
383
where PL(x) are the Legendre polynomials and the amplitudes f ~ l = arLt q q" e) were constructed from the experimental rrN phase shifts [ 17]. The second-order term is introduced phenomenologically and parametrized by the two complex parameters B0 and Co. They were fitted in [ 15] to the ls and 2p level characteristics of pionic atoms 160 and 4°Ca, and reproduce the experimental data for 4He quite well too. For heavy nuclei the function Go(q) converges to the Fourier transform of nuclear density squared. The explicit form of Go(q) involving the two-particle nuclear correlations and a more general structure of the second-order term were investigated thoroughly in Ref. [ 15]. However, the construction of Vrr-He,Tr-He requires some additional care. Since the first-order optical potential is expressed through the elementary ~-N amplitudes (i.e. the experimentally measured quantities), it already accounts for the process r r - p 7r°n , 7 r - p (and also for yn in the intermediate state). As far as the coupled channels rr°3H and T3H contribute to the real part of Y,~- He,, r He just via the aforementioned subprocesses, one has to subtract their contribution from aoo to avoid double counting. A consistent way to do so is the multiple scattering expansion of the pion-nuclear optical potential up to the second order in t,~t¢. The appropriate second-order term is standardly referred to as Pauli correlations and expressed as a repulsive correction to the s-wave part of am (denoted by bo). The derivation is given in Appendix A where we omitted the y3H channel, which is justified for its relatively weak coupling to the 7r- 3He channel. Considering only the s-shell nuclei 3He and 4He we arrived at
Abo=
M
IX
b2+
2+
~aa,3
b
},
(20~
where (r--{7.) denotes the nuclear matrix element of the pion propagator in the intermediate state. Note that Eq. (20) differs from the standard rescattering correction [191 obtained for one-channel formulation within the closure approximation. Absorbing the A-independent part of Eq. (20) into the second-order optical potential (term proportional to B0 and estimating (7~2) for 3He in the q = 0 limit, the contribution to b0 is IAbo(A) I ~ (43.6 MeV) bl2 = (2.8:t:0.3) 10-3m -I, where bl stands for the s-wave part of a01 and the error estimate is an educated guess. This correction is to be subtracted from b0, increasing the s-wave repulsion of the isoscalar term proportional to a00. Within the impulse approximation the photoproduction matrix element acquires the form 27"/"
(Q'IVrrA,rAIQ) = - - -
Ix
v~rN,r u ( Q ' , Q ) Frx~r~x'(q) ,
(21)
so that ~/rgA,yA is equal to the elementary photoproduction amplitude multiplied by a nuclear form factor and some correction factors arising due to the transformation from the rrN to the pion-nucleus centre-of-mass frame. The form factor F~x--+~x,(q) takes account of the overlap between the initial and final state nuclear wave functions. It can be written as
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
384
Table 1 Energy shifts AEN and absorption widths flABS of the l s levels of 1H and 3He pionic atoms..IV" denotes the number of channels included in the calculation. The results are presented in eV IH
.Af
3He
AEN( Is)
flABS(Is)
AEN( Is)
flABS(Is)
1
-5.00
18.9
--6.14 -6.67
0.00 0.53 0.74
- 37.7
2 3
-47.9 -47.8
26.8 27.9
Exp.
- 7 . 1 -4- 0.3
-
- 3 2 4- 3
28 -t- 7
A
F~,x--,~,x,(q) = (!/*,,, I ~--~(r+)ro'j • ~a e-iq'rjlaP'3} "~ F i r ( q ) .
(22)
j=l
The explicit form of the spin-flip form factors FIT(q) is given in [ 17]. In Eq. (22), ~a is the photon polarization vector and the operator ¢rj • ~a flips the spin of the target nucleon, which is in correspondence with the conservation of the total angular momentum. Since the Pauli exclusion principle forbids the existence of two identical fermions in one quantum state, the photon can be absorbed only by a nucleon the final state of which differs from the states already occupied. For that reason the reaction 3H ~ 7r- 3He can proceed only on the neutron that has the same spin projection as the proton spectator (and also as the whole 3H nucleus). The Compton scattering matrix element can be treated in the same way as earlier, i.e. V),A,ra ,.~ 10 -5 U?,H,zr-He, and the electromagnetic terms Vc, VFS and Vw are to be added to V~-He,,r-He. We also mention that the matrix elements (Q'Iv~,~IQ) defined by Eqs. (18) and (21) must be decomposed into partial waves to get the required quantities
(z ,~(Q,, Q ).
4. Results The numerical precision of the method presented in Section 2 was tested by comparison with the analytical solution of the bound-state problem corresponding to a pure Coulomb potential in the c~ = 1 channel and no coupling to the other channels (vzl = Vc and vl ~ ~ l = v~ ÷ l 1 = 0). The relative accuracy I(E - En )/EBI achieved in reproducing the Bohr energy E8 o f the ls state in pionic hydrogen was at the level of 10 -6 when 40 grid points were used to cover the momentum distribution. This result is, of course, the same as the one we obtained by testing the one-channel version of the computer code [6]. The structure o f the present code allows the user to switch off any channel or to include them successively. Our results for IH and 3He pionic atoms are shown in Table 1. Since the matrix element V~r-p,~r- p is real, the absorption width of the ls level of pionic hydrogen is equal to zero within the one-channel approximation. Nevertheless, the existence of the
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
385
Table 2 The characteristics of pionic hydrogen calculated by using different models of low-energy ~'N interaction. The values are given in eV
AEN ( I s) FAns(Is)
v v=l
v~2
vI [201
vII [21]
a~rN [21]
a~rN [24]
K [25]
--6.67 0.74
--6.61 0.74
--6.25 0.89
--7.77 1.01
--7.81 1.00
--6.66 0.90
--6.45 0.78
strong interaction causes a shift of the energy level with respect to its electromagnetic value. The pion charge-exchange reaction gives a contribution to the width of the Is level and affects (by some 20-30%) the strong-interaction shift AEN(Is) as well. To include the electromagnetic channel ce = 3, one has to fix the cut-off parameter A first. We performed our calculation with A = 1 400 MeV for a monopole form factor (v = 1) defined by Eq. (16). This value was chosen in the middle of a region in which the calculated quantities AEN and FABS remained practically constant. In that region the change of A by 300 MeV causes the deviations 6(AEN) = 0.2 eV and 6(FABs) = 0.01 eV. For comparison, we calculated the characteristics of pionic hydrogen using also the dipole form factor (v = 2, A = 2000 MeV) and different low-energy ¢rN potentials. The results of our calculations are presented in Table 2, where the predictions of other authors are shown as well. The potential v I is local and of the Yukawa-type (see Ref. [20] for its definition), while vn was chosen in the separable form defined in [21]. They both reproduce well the low-energy 7rN scattering data and the threshold value of the Panofsky ratio. The results obtained with various types of matrix elements v,~,~ differ from each other at the level of 20%. These deviations are caused mainly by the fact that the 7rN potentials give different 7rN scattering lengths when extrapolated to zero energy. To some extent, the results are also sensitive to the off-shell form of the potential matrix, but they are almost independent of the behaviour of v~,~ at higher energies. In fact, the characteristics of pionic hydrogen can easily be expressed through the scattering lengths by means of formulas derived by Deser et al. [22] :
4EB(ls) 4EB (ls) AEN(Is) = - a~-p - - (2al + a 3 ) , rB 3rB /'ABs(ls) = 8p21gB(ls) l
1+
lac~xl 2
rB
-
16p2IEB(ls)I (1 + ~-~) 9rB (a3 -- al) 2 •
(23)
Here a2t denotes the s-wave 7rN scattering lengths for the total isospin I = ½, I = ~ and Pr is the threshold value of the Panofsky ratio. (Experiment gives Pz, = 1.546 5:0.009 [23].) The derivation of Eq. (23) is based on the expansion of AEN/EB in powers of a~r-p/rB, where rB is the Bohr radius of the pionic orbit. Its validity is guaranteed by the smallness of the ratio a~r-p/rB ~ 10 -3. In Table 2 we show the ~'-atomic characteristics calculated for the scattering lengths that correspond to vn and for the values of a~N
386
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
taken from the analysis of the 7rN scattering data [24]. Finally, the results presented in the last column of Table 2 were obtained by Rasche and Woolcock within the K-matrix coupled channel approach [25]. At this point, one should mention that the experimental data on the low-energy 7rN scattering and charge-exchange reactions are not quite consistent and different authors give different values of a,~N (compare, e.g., the results of the analyses presented in Refs. [21,24] ). For example, the isovector combination of the scattering lengths usually denoted by bl (aCEX = v ~ b l ) varies within a rather wide range from bl = -0.086 m -I to bl -- - 0 . 0 9 7 m -1 [21]. In addition, the extrapolation of the elementary 7rN amplitudes to zero energy is not quite well defined, since they are derived from experimental data that are available at much higher energies than required for going to the limit P~,N --~ 0. The only way for getting the scattering lengths in a model-independent way is a direct measurement of the 1s-level characteristics of pionic hydrogen. Unfortunately, only the energy shift dEN(Is) = ( - 7 . 1 + 0.3) eV was obtained in a recent experiment [26]. This value corresponds to a scattering length a~,-p = (0.086 4-0.004) m -1, which is in agreement with the phase-shift analysis of the Karlsruhe-Helsinki collaboration [24]. At present, the experimental precision is not sufficient enough to distinguish among the various models of low-energy zrN interaction. A new, more precise data on both the energy shift dEN(Is) and the absorption width FA~s(ls) are needed to determine the values of the scattering lengths al and a3 (or their combinations a~-p and aCEX). We mention that these quantities are of fundamental importance for testing the low-energy structure of QCD through the calculation of the pion-nucleon g-term [27]. The 3He ~'-atom represents a much more complicated system. It is the lightest pionic atom, the description of which requires a knowledge of the nuclear structure (the characteristics of the 3H pionic atom have not been measured yet). However, the nuclear structure of the trinucleon system is very well known and its interaction with the pion field is well understood too. It is quite surprising then that the ls-level energy shift of the 3He ~--atom cannot be reproduced (cf. Table 1) by using the standard pionnucleus optical potential within the full .A/= 3 coupled channel calculation. It was also shown recently [ 15] that the problem cannot be solved by correctly considering the spin-isospin structure of the elementary 7rNN amplitude. Certainly, one could add a sufficiently strong isovector term to the second-order optical potential as Kamalov et al. did in [ 16] when dealing with similar problems for the charge-exchange reaction at higher energies. Unfortunately, such a modification of the optical potential at zero energy is ruled out, because it would spoil the good description of isospin effects observed for heavier nuclei. We note that it is the isovector term of the first-order optical potential that plays the dominant role in the description of the energy shift A E N ( l s ) in pionic 3He. Since the isovector term is proportional to bl, the results are strongly dependent on the value chosen for this combination of the ~ N scattering lengths. Our calculations were performed with the elementary ¢rN amplitudes that give bl = -0.094 m -l at zero energy. For comparison, we calculated also the 1s-level characteristics of the 3He zr-atom suppressing the isovector term by a factor 0.9 (bl = -0.085 m -1). The corresponding
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
387
value A E N ( I s ) = - 3 9 . 2 eV (for A/" = 3) is much closer to the experimental data, while the absorption width FABs(ls) = 26.0 eV remains almost unchanged. We remind the reader that the characteristics of heavier pionic atoms (with A t> 4) are dominated by isoscalar terms that contribute coherently to the optical potential, while the isovector term is suppressed by the factor ( N - Z ) / A . Leaving apart the problematic region 4 < A < 10, a change of bl at the level of 10% cannot affect significantly the good description of the characteristics of heavier 7r-atoms presented in Ref. [ 15]. Nevertheless, only a precise measurement of ~r-atomic transitions in pionic hydrogen can fix the value of bl and the strength of the isovector term in the optical potential. The measurement of rratomic characteristics on tritium can provide a valuable information too. Unfortunately, with the current value of b~ = - 0 . 0 9 4 m -~ our present level of understanding of the 7rN interaction at zero energy does not allow a satisfactory reproduction of the energy shift of the ls level in pionic 3He. The coupling to the 3H channel contributes to the discrepancy by some 30% of the experimental value. The correction due to the electromagnetic channel is negligible, which justifies the assumption made above for estimating Abo. On the other hand, the corrections due to coupled channels bring the absorption width of the ls level into a nice agreement with the data. However, this improvement is not as significant, since the experimental error bars are quite large in this case. Finally, we mention that our results for the corrections due to the coupling to the 7r° 3H channel agree well with the previous estimates made by Pilkuhn and Wycech [281.
5. Summary We have presented the extension of the method by Vincent and Phatak to the coupled channel processes involving the bound states of two interacting particles. The boundstate energies can be found as the solutions of Eq. (10), which is quite similar to the condition given by Landau [3]. In our approach it reflects the well-known fact that the determinant of the Jost matrix vanishes for the complex energies corresponding to bound states. The method represents a generalization of an earlier approach formulated to deal with the Coulomb singularity when solving the bound-state problem in the onechannel approximation. The present method retains good numerical characteristics of the previous one and makes possible the inclusion of coupling to other open channels. We demonstrated its feasibility by solving the system of three coupled channels and calculating the bound-state characteristics of pionic atoms 1H and 3He. Our results are numerically stable and show a remarkable degree of numerical precision. The calculated characteristics of pionic hydrogen do not differ significantly from the results presented by other authors. Here, our aim was to demonstrate the applicability of our method and to study the influence of the coupling to other channels on the characteristics of both IH and 3He pionic atoms treated on the same footing. At present, the experimental data on electromagnetic transitions in pionic hydrogen are not precise enough to distinguish between the various models of ~ N and TN interactions at low
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
388
energies. A more precise measurement of not only the energy shift AEN(lS) but also of the absorption width is needed to determine the 7rN scattering lengths al and a3 in a model-independent way. The failure of the coupled channel optical potential approach to reproduce the energy shift of the ls level of pionic 3He is in stark contrast to the good description of the ~'-atomic characteristics in 4He [ 15]. Since the low-energy interaction of pions with the trinucleon system is well understood, the origin of the observed discrepancy is obscure. Clearly, the solution of this problem remains open to new ideas.
Acknowledgements We are indebted to A. Gal for alerting us to the problem of double counting mentioned in the text. It is also our pleasure to thank him for many useful comments on our work and for carefully reading the manuscript. This work was supported by grant agency of the Academy of Sciences of Czech Republic, grant No. 148401. The research of A.C. was partially supported by the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities, Jerusalem.
Appendix A Here we give a brief derivation of the second-order correction Ab0, Eq. (20). We follow the standard Watson formulation of the multiple scattering theory [ 12] in which the second-order optical potential is expressed as
V(2) ( E) = A 2 [--~ f, rN( E) ] 27r-~G(E) [--~ f~rN( E) l -A
f~rN( E) -'~
where f~N(E) stands for the free ¢rN amplitude, G(E) and g(E) are the pion-nuclear and pion-nucleon Green functions, respectively. The operator 79 = 10){01 projects on single nuclear ground state for A --- 4 while
79 = ~ IJJzrrz)(TzTJzJI
(A.2)
J= T:
is a four-term expression when the coupled channels 7r- 3He and 7r° 3H are considered. The elementary zrN amplitude is taken in the form [ 17] M
- - f , rN(E) = aoo + aolt" ~"+ [aoo + aolt" ~'] i~r. [qi × qf] . The low-energy limit of the s-wave part of our coefficients aoT is a0r ~ where b0 and bl are standard combinations of 7rN scattering lengths.
(A.3)
(A~/mu)br,
A. Ciepl~, R. Mach/Nuclear Physics A 609 (1996) 377-390
389
In closure approximation, the matrix elements of Eq. (A.1) read
(Q'OIV~2)(E)IOQ) =A
Z
/ d3a '' (a , las'r'la ,, )(Q ,, lasrlQ) 27r2
Q--~S Q-~-
S,T, S~.T~
X [A(Olei(Q'-Q"I'rOs,T,~ ei(Q"-Q)'rOsTIO) - ( A - 1) {Olei(~-Q'')'rl O(1) "~S'T' ei(Q"-Q)'r2 0(2s~)10)]
,
(A.4)
where we introduced the operators Osr = (~rz)S(7"z) r. Taking further into account the symmetry in the coordinate-space part of the nuclear wave function of 1s-shell nuclei and restricting ourselves only to the pion-nuclear s-wave we arrive at
(Q'OIV(2)(E)IOQ)=a
(7)
Z b r , br A(Ol(t. "r)r'7 ~ (t. r)r[0) T,T'
- ( A - 1)(Ol(t.~'(~))r'(t.z(2))rlO)] I ( Q ' , Q ) ,
(A.5)
where we have denoted
I(Q',Q)
27r2
Q02-Q '7~
=
L , ) F(q) .
(A.6)
A direct calculation of isovector and isotensor matrix elements involved in Eq. (A.5) for 3He and 4He nuclear ground states leads to Eq. (20) used in the text. The isovector part of Eq. (A.5) corresponds to
Abl = ---~-dk4 2b0bl( ~1 ),~,_0.001rn -1 ,
(a.7)
which represents a negligible correction when compared with the total value of bl -0.090
m -1 .
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