Radiative capture of negative pions in deuterium

Received 1 December f975 Ahstract:Tbe transition rate for radiative capture of negative pions in deuterium is calculated with the Hamiltonian deduced from pion photoproduction data and with the transition amplitude obtained an the basis of the soft-pion theory. A large contribution to the physical amplitude from the intermediate states with the p-meson and n~~~n”~t~~~~n pairs in the soft~pion approach brings both results into agreement within 5 %.

The rad~tive pion capture reaction

6’ -t-d -+ Zn-ky,

(Q is of great importance. It has played a considerable role in providing us with the value of the Textron-neutron (n-n) scattering length arm. For this purpose, an extensive analysis of the n-n and y-ray spectra was performed [see ref. ‘) and references therein]. Besides the spectra, the .ratio D of the transition rates for non-radiative [Qd+ z- -+ 2n) = f,] and radiative [f(d+ n- -+ 2n+r) 3 I’J pion capture in deuterium is measured. The experimental data are summarized in table 1. TABLE1 Experimentai values of L) = ~~~~~ Ref. D

3 2.36kO.S 3.38+0.7

? 1.5f0.8

?

3

3.i6;frO.lO

2.89+0.09

3 2.35kO.35 2.39kO.36

The iower ftgures are corrected for the Panofsky ratio P = 1.53 [ref. “1).

The ratio D enters the important relation between the cross section for the reaction p-k p -P z’ i-d at low energies and the cross section gyp for the reaction y-i-p -+ ~+~nne~t~esho~d’~

401

x-+d

Here T = T,/r(V +p + n + y), R = o(y +n -+ rc- +p)/o,,, and m and M are the pion and nucleon masses, respectively. Eq. (2) was obtained from the detailed balance, charge independence and using the extrapolation to zero energy. Therefore, it provides a means of testing the fundamental physical principles. In order to verify the validity of eq. (2), several authors 7, ‘, lo) have studied the reaction rc+ + d + 2p near threshold. The cross section of this reaction is 7,

where p, is the proton c.m. momentum and q is the pion c.m. momentum in units of me. The functions C, and C, reflect the existence of Coulomb forces. Near the threshold, the cross section of the reaction is determined by a. The original value of GI obtained by Crawford and Stevenson ‘) was tl = 0.138 f 0.015 mb which from eq. (2) gives arP = (0.111 _t 0.020)~ mb. The cross section gyp is found experimentally to be crYyp = (0.201+ 0.005)~ mb [ref. 9)]. Rose 7, reinvestigated the problem. He obtained CI= 0.24kO.02 mb, which for pion photoproduction gives gyp = (0.193 i=O.O30)~ mb, in good agreement with the direct experimental result. However, the subsequent study of the reaction rcf + d + 2p by Richard-Serre et al. lo) returned the inconsistency in low-energy pion physics again. In fact, their value is a = 0.18+0.02 mb, which yields gyp= (0.145&0.03O)y mb. We note that in the analysis mentioned above, the values of R = 1.23 +0.06 [ref. ‘I)], T = 0.83fO.O18’[ref. 12)] and D = 3.02kO.l [the average values of D from refs. 5, ‘j)] were chosen. The ratio D was calculated by Reitan 13). Evaluating Tr, he made use of the low energy limit of the amplitude for a photon emission obtained by Chew et al. ‘“). Reitan has also taken into account the pion rescattering effects in the lowest order in the frame of the theory with a semiphenomenological s-wave pion-nucleon interaction quadratic in the pion field operator. The value of To was obtained by Koltun and Reitan 15, according to the same theory. The results are l”) I’0 = 1.00x 1015 set-‘,

Tr = 3.32x 1014 set-‘,

D = 3.02.

This value of Tr was obtained by Reitan for the rcNy coupling constant A = ,/Z g with g = 0.0234 m-l. In Reitan’s formula for rY, the kinematic factor (l+m/M)2/ (1 +-m/2&f) is lacking, however. With this factor rY = 4.0x 1014 set-‘,

D = 2.5.

In table 2 we present the existing calculations of To. The inconsistency in the calculations of r. is apparent. We note that the nuclear wave functions employed in all three calculations of To

402

M. SOTONA AND E. TRUHLfK TABLE 2

The values of To (in see-I) for reaction n- + d --t 2n from different calculations To x lo-l5

Ref.

1.oo 0.72 11.0

13 16 18)

Strong XN interaction semi-phenomenological s-wave ~cNinteraction Miyazawa ‘) dispersion relations approach as above

All three results were obtained using the HJ potentials in the 3P, and 3S1-3D, NN states. “) Ref. 17).

and also in rY by Reitan were generated from the Hamada-Johnston potentials (HJ) lg) in corresponding nuclear states. Here we present another calculation of the transition rate rY for the reaction (1). Recently, progress was made in derivation of the effective Hamiltonian for radiative pion capture by nuclei from pion photoproduction data ‘O, ‘I). In sect. 2, using this Hamiltonian, we evaluate the rate Tr. The strong rrN interaction is considered in the pseudo-potential model (optical potential) “, 23) of the Kisslinger type. The ‘So n-n interaction is studied in a variety of potentials including the Reid soft-core (RSC) ““) and de Tourreil-Sprung (DTS) 25) super soft-core potentials. In sect. 3, we discuss the problem in the framework of the soft-pion approach 26) and calculate the corrections to the soft-pion amplitude from the coherent, incoherent and genuine intermediate nuclear states. In sect. 4 we present the main results and conclusions.

2. The x- +d -P 2nfy 2.1. TRANSITION

reaction: phenomenological approach

MATRIX ELEMENT

The transition matrix element with two nucleons in the initial (i) and final (f) states is taken following ref. 21) as (fP’, kAlTli; m) = - 2$7t s

(2n)36(P’ + k)

In eq. (3), P’ is the c.m. momentum of two neutrons, k (jkl = co> is the photon momentum, P(K) is the n-n relative coordinate (momentum), 4k-‘(r)(~,) is the twoneutron coordinate (spin) wave function, cl is the Pauli spin operator of the first nucleon, E _,(A = f 1) is the photon polarization vector, ‘p, is the pion wave function and +d(r)(~;) stands for the deuteron coordinate (spin) wave function. The transition amplitude [see ref. 2’) eq. (8)] describing emission of a photon of polarization 1 is taken at threshold, as we deal with pions captured from the 1s

403

z-+d

atomic state. The value of constant A is [see ref. ‘O), table 21 A = -(0.0320f0.0012)m-1.

(4)

2.2. STRONG xN INTERACTION

The strong nN interaction was taken into account in the framework of the optical potential approach to the problem. We have constructed the optical potential V, explicitly, on the basis of the equation “) v, = (, 01i t,l, 0).

(5)

i=l

Here the amplitude ti is the low-energy elastic scattering amplitude 27) of the pion on the ith nucleon, the state vector 1,0) describes the ground state of the deuteron. Here we do not consider the strong nuclear absorption of the pion. This effect will be evaluated in the soft-pion approach. With the potential defined in eq. (5), the Schrodinger equation reads

(&d+E-vc-vN )q,=

0,

(6)

where V, is the Coulomb potential, l/&’ = l/m+ 1/2M and

v, = & [q(r)-

Va(r)V].

(7)

The functions a(r) and q(r) are as follows “a(r)+

s __ l+$

s 1 dcr - - + __-2r dr l+$s

1 d2a 4 dr”

s = m/M,

b, = $a, +2a,),

(8)

co = ~4~,,f2a,,+2~,,+~,,).

are radial deuteron wave functions, and a2T (T = i, $) and are the 7tN scattering lengths 27). ‘2T, The optical potential presented in eq. (8) is similar to the standard one 23). Moreover, it correctly takes into account the c.m. motion of the pion-deuteron system and also contains a surface term. The value of co is calculated with uZT, 2j from free rcN scattering (table 3). Another set of aZT, zi [ref. “‘)I provides co = 0.208 rnV3. With this value of co, eq. (6) becomes singular, because of the factor (1 +a(r))- l.

In eq. (Q u. 2.j

tT,j

=

f2

2

$3

404

M. SOTONA AND E. TRUHLfK TABLE

3

The values of ~~~~~~‘) and c0 (in nze3) used in present caicuiations

0.089

-0.016

"13

a31

-0.13

-0.13

a33

0.201

“) Ref. 28),

The value of b, is rather uncertain and the recommended 3mb, x lo3 = -20+,6$ I

value is ““) (9)

Solving eq. (6) for the pion wave f~~tjon, we also get the energy of the pionic 1s atomic state. For b, = -0.01 m-l we obtain E = -3.4558 keV. This yields the strong shift LIE,, = 3.2 eV. The ex~~rne~ta~ value is J1) d;E”;,P= 4.8+$x eV. The energy shift LSE,, is related to the pion-deuteron scattering length aled [ref. 32)]. For LIE,, = 3.2 eV we have axd = -0.050 fm. This result for alcd lies between the value ax4 = -0.047 fm obtained by Kolybasov and Kudryavtsev 33) with graph su~atio~ technique and us& = -0.06 fm from the Faddeev-type calculation by Petrov and Peresypkin “*). 2.3. NU~R~CAL

RESULTS FOR f,

The transition probability is

Here the transition matrix element is given in eq. (3), E,,,, is the total energy of initial {final) state and ~~ is the momentum of the ith neutron. The summation goes over photon polarization A and the spin directions not observed. With the method of ref. 35, we calculate Tr for a set of potentials which correctly describe the ‘S, n-n final state [e.g. the local potentials used in ref. 36) and the non-local potentials from ref. 37)]. The RSC deuteron wave function was ~lculated by direct solution of the corresponding coupled differential equations 38). For the RSC potential both in the ‘S, and 3S,-3D, two-nucleon channels, with the constant A from eq. (4) and pion wave function from subsect. 2.2,

r Y=375x1014sec-‘. *

(11)

For the potential considered, the effect of the %, n-n interaction is 2 % to 3 76. Tt is found that the ~o~tribu~io~ from the IS, n-n interaction to f, strongly depends on the type of this interaction, on the deuteron wase function and also on the value

405

x-+d

of the effective range Y,, (and less on the unn). This contribution is too small however to change Tr considerably. We have also found that the influence of the n-n interaction in the 3P,(J = 0, 1,2) states and the D-state admixture in the deuterium wave function on the value of Tr is negligible. At the same time, the strong rcN interaction (subsect. 2.2) lowers the value of r,, by = 5 %. Comparing our result eq. (11) for r, with the corrected value TV= 4.0 x 1014 set-’ obtained by Reitan we see that both results agree within 6 %. We also note that our calculation of the ratio T (see eq. (2)) yields T = 0.78 which is in agreement with T = 0.83 +0.08 obtained by Traxler “).

3. The a- + d + 2n + y reaction: soft-pion approach 3.1. SOFT-PION AMPLITUDE

The problem of radiative pion capture in nuclei in the soft-pion approximation together with its corrections was discussed in detail by Ericson and Rho ‘“). So we confine ourselves to present only the formulae necessary for the calculations. Our notations and convention in this section agree with those of ref. ‘“). Reaction (1) is ideal for the calculation in this method as all the pions are captured from the 1s atomic state. The transition amplitude is Jr

where Ed is the photon polarization

(12)

= A;&“,

vector and

M; =J,(O) + emC GW3&P, -PiI n

(fI~gm.(0)ln>(nljl;(O)li> _cross (E _ Eij(m _ E + E,j n ” 1

term

(13l

In our case the Ii) and If) nuclear states correspond to the deuteron and twoneutron systems with the energies Ei and E,, respectively. In eq. (13), the soft-pion amplitude&‘; (0) is &i(O) = it

(fJA;(O)li), x

(14)

with e2/4n = I/137 and f, x 0.94 m. There exist some transitions for which the nuclear matrix element of the axial current A; is known for two values of the momentum transfer; at t = (pi-pf)' x 0 from P-decay, and t z -mZ from p-capture. Then the extrapolation of the matrix element to the value t M -m2 provides the most interesting test of the soft-pion approach 26). As the P-decay of deuteron is not observed and the muon capture in deuterium is not a pure Gamow-Teller transition we make use of the single nucleon

406

approximation

M. SOTONA AND E. TRUHLfK

to the axial current A;. In the non-relativistic

approximation

$

A-(O) = .n”l,(O)? = -ie-- g;O (fl (bj* s_,),jli), R j=l

(15)

where gA is the weak axial-vector coupling constant and 7; is the isospin-lowering operator for the ith nucleon. The amplitude M-(O) is of the same form as the transition matrix element eq. (3), but with a somewhat different value of A. The transition rate forA!eq. (15) is r,(O) = 2.96 x 1014 see-’ ,

(16)

not including the ‘S, n-n interaction. To obtain this result, we have used gA = 1.25. The corresponding piece of Tr eq. (11) is Tr = 3.63 x 1014 set-r .

(17)

The results in eqs. (16) and (17) differ by z 20 %. If we employed the GoldbergerTreiman relation and f /47c = 0.08 we would obtain&-(O) with the equivalent A close to A = -0.032 me1 [ref. ‘l)] and r,(O) = 3.36 x 1014 set-‘, much closer to rY in eq. (17). Because of the approximate character of the Goldberger-Treiman relation 3g), this result is however incorrect. 3.2. CORRECTIONS

DUE TO FINITE PION MASS

The corrections due to finite pion mass are given by the last two terms in eq. (13). The intermediate states saturating the sum in eq. (13) are as follows 26): (i) The state which consists of the ground state Ii) with a pion, i.e. Iin). This state reflects the distortion effect of the pion wave and can be taken into account by employing the optical potential. For reaction (1) we have 4; where A;(O)

=A?‘; (O)(l + S) + other corrections,

is given in eq. (15) and

(1% In eq. (19), ID: = m2 +q2 and the non-relativistic amplitude j&, is given by

( > 1+ 2

pion-deuteron

fqo= -2~2
scattering

(20)

with V, from eq. (7); lx,) is the plane wave state for the red system and the scattering

401

n--l-cl

state I#(+)) satisfies the equation

I@+‘>= Ix>f

vNk#++9. E+i;_H 0

(21)

For the optical potential V, from subsect. 2.2 6 = -2x1o-2.

(22)

The result for 6 with c0 = 0.208 PZ-~ is about the same. (ii) The inte~ediat~ state without pion but containing a nucleon-antinucleon In the impulse approximation this correction is 26)

pair.

with A+(O) from eq. (15). (iii) The genuine nuclear intermediate state. The term considered here corresponds to the n-n intermediate state without pion, i.e. we evaluate the two-nucleon absorption correction. In ref. ‘@j)this correction is omitted as an uncertainty in the calculations. The contribution to the physical amplitude is

s

(fll”.m(o).

Q, +~#wbz

c ____

Elq142v’)(4142v’l~~(0)li) -.__

+,’(2A4)_ I($ +&cm-

(4:. -l-q32.M +

i&l

(24)

where qi (qf G q:) is the momentum of the ith neutron and v’ is the spin projection of the n-n system. Neglecting the off-energy-shell effects, we calculate the matrix elements in eq. (24) as follows. The matrix element (q~q~~~~O)li) is considered in Eckstein’s model 40) for the absorption of the pion by the two-nucleon system. In the LSZ reduction formalism for the S-matrix element of this process we have Sei = (qlq2v’/n-d>

= i s

d’xe -‘P*Yo,_tpn2)(qlq2v’/~-(x)Id)

= X2n)4d(4)(qr+q2-~-~i)
(25)

On the other hand, ,!& = - 2niS(E,, - Ei)Tni,

E,, = E,,+E42,

Ei =Ep+Ed.

(26)

In Eckstein’s model r,i = cp,(O)g,

s

d~d~~(~~[~V~~~~(~,p)l- (~1 +~&#&)x:“.

In eq. (27) g; is the coupling constant determined from the reaction p f p -+ d +

(27)

xs,

408

M. SOTONA AND E. TRUHLiK

f(r) is the form factor describing the short-range behaviour of the two-nucleon absorption mechanism 41) and II/,, is the two-neutron wave function. In the amplitude @$“.“(O) . ~lq~q~v’), the electromagnetic current is taken in the impulse approximation, i.e. $P”(v)

= - 1.91i &

$’ (bj x k)d(r-

rj).

j-l

With Ig; I x 3 fm4 [ref. 41)], the contribution of the term eq. (24) to the transition rate is x 0.1 ‘? and may be safely neglected. (iv) Incoherent scattering. This correction arises from the intermediate excited nuclear states accompanied by one pion. We write down the corresponding term with negative pion in the intermediate state in the form em Sdxdyepik'y

j;

(fl$“.m(y, 0). 4qIq2v’,4)(4pp’, (ilj;k

W

(E,-E,)(m-E,+Ei+b)

dq,dq,dq x (27c)9(2M)220, *

(29

We take (q1q2v’, qlj;(x,O)li)

= 4rc 1+ $

(

>

(q1q2V’1 f:


e-iq’S(UN++~j3~,)~(~-~j)li), (30)

j=l

t eiq’Ya,~~6(y-y,)lq,q,v’).

(31)

k=l

In eq. (31) aN are the 7rN scattering lengths. With a; z 0 and (1 + m/M& w 0.1 m- ’ [ref. 26)] we find that the contribution from the incoherent scattering lowers the transition rate by w 2 %. This small change in rY is due to the fact that the contributions from the intermediate proton-neutron singlet and triplet states cancel each other for the singlet n-n final states. Employing the closure approximation and the Fermi gas model for the nuclei, Ericson and Rho 2”) obtained a contribution of +5 % to the amplitude from the incoherent scattering. (v) Contribution due to intermediate states with a vector meson, INp). The amplitude d-(O) eq. (15) is modified by the factor (1 +a,) with 26) 6, = 0.1.

(32)

Adding all the corrections to r,(O) in eq. (16), we obtain the following value of the transition rate for reaction (1) r Y= 383x1014sec-1. not including the ‘S, n-n interaction.

(33)

409

n-i-d

Comparison of Tr eq. (33) with Tr eq. (17) shows an agreement within 5 % in the value of the transition rate obtained by two different approaches to the problem. The contributions from the intermediate states with a vector meson and the nucleonantinucleon pairs to the physical amplitude in the soft-pion method seem to be quite large. However, only with them was it possible to bring into reasonable agreement the results for r, in both types of calculations. 4. Conclusions and discussion The most important results we have obtained are: (i) The inclusion of the strong rcN interaction lowers the radiative transition rate rr by z 5 %. Our prediction of the 1s energy level shift AE,, = 3.2 eV is in good agreement with the experimental value 31)AE”;,”= 4.8?:$, eV. The corresponding pion-deuteron scattering length is alrd = - 0.05 fm. (ii) The realistic potentials which correctly describe the ?&, proton-proton phase shifts in a broad energy range (e.g. RSC or DTS potentials) make almost the same (2 % to 3 %) contributions to r,. These contributions to rY from the potentials of different shape but with the same arm and Y,, may differ from one another considerably (up to 60 %). However, since they are only smail corrections to r,, they do not change rY appreciably. The same conclusion also holds for the dependence of the r, upon the values of r,, and an,,. (iii) The influence on the I’, of the n-n interaction in the higher partial waves and also of the D-state admixture in the deuteron wave function is negligible. (iv) The different calculations of r, with the phenomenological Hamiltonian obtained from the photoproduction data agree within 6 ‘A. These results are also

TABLET

Values of ry (in set- ‘) for reaction (1)

ry x 10-14

Ref.

Transition amplitude

Strong sN interaction

3.75 3.63

this work

from pion photoproduction data

optical potential

2.96

this work

in soft-pion limit

3.83

this work

soft-pion limit + corrections

optical potential

3.32 4.0 b)

11)

from pion photoproduction

semi-phenomenological s-wave interaction

Potentials in 3S,-3D, and IS, NN states RSC RSC “) RSC *)

‘) The rS, n-n interaction not included. ‘) Corrected for the factor (I+ m/M)‘/( I+ m/2&Q.

RSC =) HJ

410

M. SOTONA AND E. TRUHLIK

in good agreement with the r, obtained in the framework of the soft-pion theory of the radiative pion capture in nuclei (table 4). As already mentioned, in reaction (1) all the pions are captured from the 1s atomic state. Since the un~~inties due to the nuclear strncture are small (with regard to heavier nuclei), an experimental knowledge of the I; would be very useful for testing the soft-pion expansion of the physical amplitude. In tables 2 and 4 we present the results for the r0 and r, obtained by different methods and approbations. From these tables we conclude that existing calculations of To and r, do not reasonably reproduce the experimental value of the ratio D = ~~1~~ = 3,02+0.1. The reason seems to lie in the remarkable sensitivity of the transition rate To to the D-state probability P, of the deuteron. In fact, Koltun-Reitan-type calculations by Thomas and Afnan 42) of the cross section for reaction p+ p 4 a+ + d show that the value of the parameter cxin the cross section (Coulomb forces not included) o(p+p

-+ n+ +d) = a~+&‘,

(34)

varies signi~~ntly with the change in PD. Particul~ly, using the three-term separable potential in the 3S,- ‘D, NN channel with PD = 6.5 % (4.31 %), they obtained CI= 0.229 mb (0.285 mb). With the same cl, in Eckstein’s model ‘*),

With c1= 0.240 mb [ref. 7)j, this gives To = I .14 x lOI5 set- ’ and using Fy = 3.75 x 1014 set-‘, we obtain D = 3.05.

(35)

At the same time, employing the RSC potential (PD = 6.47 %), Thomas and Afnan obtained CI= 0.171 mb, giving r0 = 0.812x lOi set-’ and D = 2.17.

(36) For more quantitative conclusions, consistent calculations of F,, and F, with a better ex~rimentally dete~ined nN s-wave interaction are needed.

Appendix

Here we present a formula for the Fy obtained from eq. (10) which is suitabIe for practical calculations :

X (I c p[Z,(r~)]~ + 2 c iz[l,(k-)I2 -t 8n Re [1,,(rc)Y~(~)] + 16~~~Y~(7c)~~).(A.l) 1=0,2 1=1,3

x- +d

411

In eq. (A.l) P = 2M(-l+t),

r=(l+$-$)i

64.2)

P = 21-l-l.

Ll = m+M,-M,-I&J,

Here MpCa,is the proton (neutron) mass and sd is the deuteron binding energy. Further

y,(4 = ~-,(4

m

(rc’)2drc’

J

0 s(x)+ iv-E(7c’)

~,(G rM(~‘, x).

In eqs. (A.3) R,($ r) is the pion radial wave function of the 1s atomic state and uo(r) is the S-wave component of the deuteron wave function. The on-shell twonucleon scattering amplitude Y,(K) is related to the half-off-shell functionf, by Y-o(K) =

l-47c

Vo(K,")

[

m

J

(rc’)2drx’

l@, W@‘, 0 F(K)+ iy - &(Ic’)

1 -1

K)

9

(A-4)

and the equation for the functionf,(lc’, K) is given in ref. 33). Our convention is such that the Fourier-Bessel transform of the potential V is

m J

V(r)j,(ar)j,(br)r”dr.

(A-5)

0

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

W. R. Gibbs, B. F. Gibson and G. J. Stephenson, Jr., Phys. Rev. Cl1 (1975) 90 W. K. H. Panovsky, R. L. Aamodt and J. Hadley, Phys. Rev. 81 (1951) 565 W. Chinowski and J. Steinberger, Phys. Rev. 95 (1954) 1561 J. A. Kuehner, A. W. Merrison and S. Tornabene, Proc. Phys. Sot. 73 (1959) 551 J. Ryan, Phys. Rev. 130 (1963) 1554 P. K. Kloeppel, Nuovo Cim. 34 (1964) 11 C. M. Rose, Jr., Phys. Rev. 154 (1967) 1305 F. S. Crawford, Jr. and M. L. Stevenson, Phys. Rev. 97 (1955) 1305 J. D. Simpson, Ph.D. thesis, Uhiversity of Illinois, 1964 (unpublished) C. Richard-Serre, W. Hirt, D. F. Measday, E. G. Michaelis, M. J. M. Saltmarsh and P. Skarek, Nucl. Phys. B20 (1970) 413 T. S. Yoon, Ph.D. thesis, University of Illinois, 1964 (unpublished) R. H. Traxler, preprint UCRL-10417, 1962 (unpublished) A. Reitan, Nucl. Phys. 87 (1966) 232 G. F. Chew, M. L. Goldberger, F. E. Low and Y. Nambu, Phys. Rev. 106 (1957) 1345 D. S. Koltun and A. Reitan, Phys. Rev. 141 (1966) 1413 K. Nishimoto, H. Ohtsubo and K. Narumi, Prog. Theor. Phys. 46 (1971) 135

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