Volume 82B, number 1 PHYSICS LETTERS 12 March 1979
THRESHOLD PION PHOTOPRODUCTION AND RADIATIVE PION CAPTURE ON 3He:
DETERMINATION OF THE lr -3He-3H COUPLING CONSTANT
Claude LEROY Physics Department, Mc Gill University, Montreal, Qukbec, Canada
Jean PESTIEAU lnstitut de Physique Th~orique, Universit~ Catholique de Louvaine, Louvain-la-Neuve, Belgium
Received 17 October 1978
From threshold pion photoproduction on 3tie and the Panofsky ratio in 3He, we determine the n-3He-3H coupling constant and the charge exchange combinations of the n-3He scattering lengths: j2 = 0.042 0.002 and la 1 - a3l = (0.262 0.008)(1/mn-).
During the past twenty years or so, considerable attention has been dedicated to the application of low- energy theorems [ 1 ] to threshold pion photoproduc- tion on nucleons. These low-energy theorems due to Kroll Ruderman , l ow  and Fubini Fur lan- Rossetti , allow to express the threshold pion pho- toproduction amplitudes in a series expansion with re- spect to/~, the ratio of pion mass to nucleon mass m~/ m N and to determine  the first two or three terms of the expansion. The knowledge of these first terms is based to a considerable extent  on electromag- netic gauge invariance and partially conserved axial current (PCAC) hypothesis. These terms can be ob- tained directly by considering the Born diagrams of pion-photoproduction with pseudovector pion nu- cleon coupling . Then from the measured 3' + P n + 7r + cross-section, the pion-nucleon coupling con- stant is determined [5,8] in good agreement with its current values . From the experimental proton Panofsky ratio  we get the charge exchange com- bination of 7r-N scattering lengths la 1 - a 3 [  in good agreement with other determinations .
As far as we are concerned by low energy theorems, nuclei can be considered  like elementary objects because, in the low energy range, we probe their global
structure. In our opinion, symmetry laws (gauge inva- riance, CVC, PCAC, etc.) are explicitly realized at the quark field level and, in that respect, nuclei are no more composite than nucleons, pion or A-resonances. With that point of view, threshold pion photoproduc- tion on nuclei and nucleus Panofsky ratio are unique ways for determining pion-nucleus coupling constant and charge exchange combination of pion-nucleus scattering lengths, free of the considerable difficulties involved in more traditional methods [11,12]. In the present letter, we will apply these ideas to the 3He case as it has been done for 6Li  previously.
As it has been shown in ref. , the main differ- ence between our results and previous works is that we satisfy Low's theorem at each step of our treatment while previous works [13,14] are taking little care of it. Take for example eq. (C 7) in ref. , the second term in the curly bracket is dropped without real jus- tification. This second term is a local object,
[J~'m'(0), fd3x ~xa'~-~(x, 0)1,
sandwiched between initial and final nuclei and de- pends just like the first term on the size of the nucleus. Therefore it cannot be dropped. Indeed the relation
Volume 82B, number l PHYSICS LETTERS 12 March 1979
between threshold photoproduction and the axial form factor F A [(q - k) 2 -~ -m 2] in the space-like region, obtained in ref.  and  violate kow's theorem in contradistinction with our relation between thres- hold photoproduction and the pion-nucleus coupling constant.
1 Let us consider A and B, two spln-~ nuclei, belong- and I B = - 1 ing to the same isodoublet with I A = 5 g.
(Examples: A = p or 3He, B = n or 3H). Qi(i = A or B) is their charge in proton charge units and Y, their hy- percharge (Y = QA + QB)" In the center of mass system the differential cross section at threshold for unpolar- ized spin - ~ nucleus and photon is
,. I kl do (7 + Ni -+ Nt + 7r j) alNi Uln - - - Iql+0 Iql dS2 ~ C~Vf
gi 2 1 (1 +1//)2 IMI2rff'~g (1) =c~47r4m 2 (1+//)3
where g~ is the pion-nucleus (nucleon) coupling con- stant (grrNN/4rr ~ 14),//is the ratio of pion mass to the average nucleus or nucleon mass mTr+/m, o~ =-
1 e2/47r ~ i5;~- is the electromagnetic fine structure con- stant, k and q are the photon and pion momentum. M is the reduced matrix element (see below) and C~f is the final state Coulomb correction (if necessary). In the approximation of point charges and because the pion production near threshold is a pure S-wave, we have 
b/ Iq l __ (2) C]Nf - exp(b/Iql) - 1'
1 b = 27r~Qt.Qjm,+ 1 +//'
where Qf is the charge of Nf and Qj the charge of TrY (Q+-I = + 1, Q0 = 0). Now, applying the low energy theorems as in refs.  and  we get
1 M('), + A-+ B + rr+)=-V'22 [1 - i Y//
//A +//B ) + 4 +x //2 + 0(//3)] , (3a)
1 M(y+B-+A+rr )=+X/~'[ I +~ Y//
( uA +//B ) 4 x //2 +0(//3)] ,
! 2 M(7 + A -+ A + 7r 0) = + [QA/ / - 2/-IAU + 0(/23)] , (3C) 1 2 M(7 + B -+ B + rr 0) = - [QBU - ~//B/-t + 0(/ /3)] , (3d)
where//i (i = A or B) is the total magnetic moment in units eh/2rnic and x is an unknown parameter, unob- tainable from low energy theorems. However, if we compute the Born diagrams of pion-photoproduction amplitudes with pseudovector pion-nucleus (nucleon) coupling, we will obtain x = 0 . Note that eqs. (3a-d) satisfy the isospin requirement
M(7 + A-+ B + rr+) +M(7 + B-* A + 7r )
=N/~-[M(7+A-+A+Tr 0) M(7+B-+B+rr0) ] (4)
From eqs. (1), (3a-d), we get
a~/a+A = 1 +2Y//+[2Y 2 - - ( / /A +/ /B) ] / /2 +0( / /3 ) ' (5a)
aO /a+A = 1,132 ,,2 2t~At-, +QA(Y--//A)//3 +0(//4)] , (5b)
aB/a AO + = 5' [Q2//2 +QB(y_//B)U3 +0(//4)] , (5c)
except if QB = 0 i.e. if B = n
an/a pO +=lg//n//2 4+0(/.tS)=2XlO 4+0(/ /5) , (5d)
independently of the unknown parameter x. Note that Yu "~ rnrr/rn = 0.15 and then, using eq. (5a), we get aB/a+a ~ 1.3 for any Y. From the experimental value of a~, the experimental Panofsky ratio
p- lim r - l (v , - +A-~B+Tr 0), (6) q~0 r- l (rr +A-+B+7)
and eq. (5a), it is possible to determine the charge ex- change combination of rr-nucleus (nucleon) scattering l engths [a I -- a3[ thiough the relation
P-= lira o(rr- +A-+B+rr ) qrr--+0 a(rr- +A-+B+' ) ')
Iq,ol 47r lal -a312 - (7)
Ik.rl 9 a B
Applications. (1) A = p, B = n with Y = 1 ;//p = 2.793 and/l n = --
1.913 in eh/2mNc units; m N = 938.9 MeV, the aver- age nucleon mass, m~r+ = 139.6 MeV, the charged pion mass,/ /= mrr+/mN = 0.149" qrro = 28 MeV, k v = 130.5 MeV in eq. (7). Eqs. (5) give 
Volume 82B, number 1 PHYSICS LETTERS 12 March 1979
an/a; = 1.32 + 0(At3),
0 + ap/ap = 0.8 X 10 .2 + 0(/~ 4) ,
0 + an/a p=2X 10 4 +00/5) ,
ap + = 15.4 - + 0.5 ~b/sr , (8)
r-l(Tr +p-+n+Tr 0) P = = 1.546 + 0.009 [10l . (9)
r l(n- +p~n+7)
Comparison of eqs. (8) and (9) with eqs. (1), (3a) with x = 0, (Sa') and (7) gives
f2 _g2 m2. p =~ 4m2N = 0.080 + 0.003,
la 1 a31=(O.258+-O.OO4)(I/m~-),
in very good agreement with the most current values . (2) A = 3He, B = 3H with Y= 3;/33H e = --6.367
and/23H = 8.918 in e1~/2m3He c units; m3H e = 2.808 GeV, m3H = 2.809 GeV,/a -= mTr+/m3H e = 0.050; q~ro = 32 MeV; k. r = 135 MeVin eq. (7).
Eqs. (5) give
- - +
a3H/a3H e = 1.34 + 0(/23) ,
0 + a3He/a3tte = 0.6 X 10 -2 + 0(At 4) ,
0 + a3H/a3H e = 0.9 X 10 .3 + 0(/14) . Experimentally,
Recently, by extrapolating the differential cross- section of the elastic 3He-3H scattering to the charged pion pole, Dumbrajs [11 ] has obtained f2He = 0.050 neglecting the Coulomb effects. Previous analyses [ 12, 20] give higher values for f2He(~0.08--0.16) but are not very reliable [11 ]. Eq. (15) is in good agreement with the current algebra prediction [21 ]
3 1 m~r+ al --a 3 - (1 +0(U2)) ~0.25 ~
4n f2 1 +/a m~r_ '
where f . = 0.96 m~. Determination of the on-shell n -3He-3H coupling constant, as given in eq. (14), is useful to estimate the corrections  to the naive Goldberger Treiman relation  which predicts f]fte = 0.085 and, supplemented with eq. (15), to study the forward dispersion relation substracted at thres- hold when applied for the antisymmetric n -+ 3 He for- ward scattering amplitude , On the same lines  we have determined n -6L i -6He  and 7r 12C-12B coupling constants, using corresponding ex- perimental threshold Non photoproduction cross-sec- tions [25,26] : f2Li ~ 0.020 +' and f22c ~ 0.006. Clearly, the reductions in f2He, f2Li and f22c com- pared to f2 are manifestations of the binding forces (e.g. pion exchanges) between the nucleons in nuclei.
We would like to thank J.P. Deutsch, D. Favart and B. Van Oystaeyen for helpful discussions. One of us (C.L.) wants to thank the financial support of the National Research Council of Canada and the Quebec Department of Education.
1 g2 m~ , l f62Li=24n4 2 as given inref..
+ + a3He/a p = 0.62 + 0.02  , (12)
P=r- l (n- +3He-+3H+n0) -2 .83+-0.07  . (13) r-l(n - +3He~3H+7)
Comparison of eqs. (12) and (13) with eqs. (1), (3a) with x = 0, (5a"), (7) and (8) gives
f2He g2He m2+ = = 0.042 + 0.002 , (14) 4rr 4m2He
Ja 1 --a31 = (0.262 +- 0.008) (1/m~-). (1 5)
[1 ] V. De Alfaro, S. Fubini, G. Furlan and C. Rossetti, Currents in hadron physics (North-Holland, Amsterdam, 1973), see previous references therein.
 N.M. Kroll and M.A. Ruderman, Phys. Rev. 93 (1954) 233.
 F.E. Low, Phys. Rev. 110 (1958) 974: S.L.