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Relativistic (NN∗) wave function of the deuteron
Nuclear Physics A264 (1976) 365-3'78; ~
North-Holland Publishiny Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
RELATIVISTIC (NN*) WAVE FUNCTION OF T H E DEUTERON t H. J. WEBER
Department of Physics. University of Virginia, Charlottesville, Va. 22901 Received 2 March 1976 Abstract: The (NN*) wave function in the deuteron ground state is studied in a relativistic and threedimensional linear, three-body formalism involving two nucleons and one pion as basic particles. The off-shell extrapolation used is unambiguously prescribed by time reversal invariance. For the NN*(1520) configuration off-shell effects are found to be sizable.
1. Introduction
At present there is considerable interest in the role of mesons and nucleon resonances (N*) as constituents of nuclei t). It is known that magnetic moments of light nuclei and the photodisintegration 2) of the deuteron near threshold and just below the pion production threshold 3) cannot be understood without directly taking meson and N* degrees of freedom into account. Particularly, the A(1236) isobar plays an important role in generating the strong medium range attraction in the nucleon-nucleon force 4) which may be considered as the driving force for the A in nuclei. Thus for the two-nucleon system there are now several processes involving mostly large momentum transfer where isobars seem to play an important though indirect role as they are usually far off their mass shell. A bound isobar contributes to the short range nuclear structure regime predominantly. Baryon resonances are in general a more efficient source of high momentum components than the short range repulsion in the nucleon-nucleon force. A high baryon spin, in particular, admits or requires a correspondingly high orbital angular momentum and the associated power of the momentum transfer in the relevant NN ~ NN* or NN --* N ' N * transition interaction; yet small distances are avoided by the angular momentum barrier. Nonetheless, such arguments may not be a sufficient justification for using nonrelativistic potential theory at high momentum transfer when the isobars or nucleons involved are substantially ( < 50 ~ ) off their mass shell. Thus relativistic and off-shell corrections can be expected to be nonnegligible. Furthermore, there is usually a plethora of "background" reaction mechanisms, such as non-resonant nN exchange, which compete with the N* signal. The situation is fairly typical and applies to the inelastic high energy pick-up 5), t Supported in part by the National Science Foundation. 365 July 1976
366
H.J. WEBER
quasi-elastic scattering and spectator experiments 6.7) and inclusive processes s,9) that have recently been designed to show more directly the presence of isobars in the deuteron. For two of these isobar-spectator experiments 7,8) there are indications that to some extent the N* spectator signal may have been experimentally separated from the background of nonresonant zrN systems. As a consequence, there is now some evidence for a total (AA) and (NN*) admixture probability of about 0.4 ~o to 3 ~o in the deuteron ground state. In each case, however, a nonrelativistic isobar wave function was used as a crude estimate for the AAd vertex involved in the A-spectator signal. This includes taking the A to have equal energy and the physical mass rna = 1.236 GeV/c 2. For the spectator experiments with the observed recoiling A(k) on its mass shell and the exchanged A of energy m a - k o and mass (m2+rnA2 - 2 m ~ k o ) ~ = < m d - m d < ½m j, this nonrelativistic approximation is a large extrapolation, seems questionable and should bc investigated in more detail. To this end we study an off-shell NN* system in the deuteron ground state in terms of a linear, covariant, and three-dimensional formalism of the Lippmann-Schwinger type. The large off-mass-shell extrapolations which are a characteristic feature of isobar admixtures to bound many-nucleon wave functions have motivated us choosing a covariant three-particle equation 1o) involving two nucleons and one pion so that elastic and three-body unitarity are preserved in conjunction with an unambiyuous off-shell extrapolation procedure. As a consequence the formalism is well suited to study off-shell effects. However, its essential restriction to three particles N, N, 7z prevents us from introducing p-meson exchange in addition to the pion exchange. The former is known ~1) to generate a core in a natural way and thus stabilize the unphysical dependence of the pion-exchangc on the short range cut-off form factor or regularization procedure to some extent. Using a quasi-particle approximation, an off-shell nucleon and isobar N* of isospin ½are admitted as (~, N) composites in a relativistic scattering problem of an N from a quasi-particle N or N* proceeding via intermediate three-particle N, N, rr states. In this context, it turns out to be essential to use the relative Wightman-G~rding momenta 12) instead of the usual Jacobi variables for enforcing correct cluster properties and extrapolating to the bound-state deuteron pole. The general formalism is described in sect. 2. The partial-wave helicity analysis, transition potentials and coupled-channel equations for the radial wave functions are presented in sect. 3. Numerical results are obtained for the NN*(1520) configuration and discussed in sect. 4.
2. The (NN*) configuration in relativistic three-body theory Our starting point is the relativistic Lippmann-Schwinger equation ~'dakl (,piT(s) [q) = (Pl V(s)lq> + j ~ ) ~ (p[ V(s)lk I k2k 3>G(k 1k2k31T(s)lq),
(1)
(NN*) WAVE FUNCTION
"~-'" N*(N,)
367
~ N(I,,)
.... = N*(K,)
~ { ~ ~
(~II)
N(k3)
Fig. 1. Schematic representation of NN* scattering, cqs. (1), (24) and (34), at the deuteron pole s = rn~ with intermediate NNn states in the quasi-particle approximation. for off-shell N N and N N * scattering leading to a final N N or N N * state p. Using a quasi-particle a p p r o x i m a t i o n , one off-shell N and N* are admitted as (n, N) composites. As shown in fig. 1 for the final quasi-two-body state IN(k3), N*(K3) = (n, N)], the intermediate states are restricted to the q u a s i - t w o - b o d y states [N(kl), N(*)(KI) = (rt, N)], where 13) "I" K I = k 2 + k 3,
K 3 = k l + k 2,
s i = K z.
(2)
Thus the separable t w o - t o - t h r e e - b o d y I N N (*) ~ N N n ] production amplitude (NNrtlT(s)lq)
= F q(sl)(N(kt),
N(*)(K1) = (rc(k2), N ( k 3 ) ) l T ( s ) l q ) ,
(3)
involves the p r o p a g a t o r g(K~) o f the (n, N) quasi-particle and its decay vertex F ( N (*) ~ nN). This system, including the initial N N or N N * state q, and eq. (1) are extrapolated from the on-shell invariant energy g=F~.g=W
z,
g=~r
+k- z+k-3,
(4)
to the bound-state deuteron pole at s = W Z = m ~ . In the following such on-shell quantities are generally denoted by a bar. W h e n the intermediate three-particle states are described in terms o f the usual Jacobi relative variables in conjuction with a spectator a s s u m p t i o n for the nucleon N(kt), say, i.e. k~ = m~ = m 2, a spurious singularity a t s = 0 is k n o w n to occur 10). In this context, therefore, it is essential to use the W i g h t m a n - G f i r d i n g relative m o m e n t u m variables 12), e.g. ql = ~k2 - k 3 ) - ( m ~ - m Z 3 ) K ~ / 2 K ~ ,
(5)
QI = ~(k~-Kt)-(mZ-g,)K/2~,
with m I = m 3 = m,
m 2 = m~,
K = kl+k2+k3,
(6)
to enforce proper clustering and eliminate the relative energies in a covariant way. In fact, q; and Qi (i = l, 2, 3) reduce to space-like three-vectors in the two- and t We use the conventions and metric of ref. ~3).
368
H..1. WEBER
three-body c.m. systems, respectively, because qi" Ki = 0 = Qi" K,
(7)
when k 2 = m,.2. The off-energy-shell continuation is designed so that all energy components of momentum four-vectors in the c.m.s, become proportional to the total three-particle c.m.s, energy W, e.g.
k° = ~, w / ¢ ¢ =
1+
.....
w,
(8) K ° = K,°W/ff / = :2 1
W.
As a consequence, time-reversal invariance is preserved as it is related to a reflection in the total energy W ---, - W because only the total energy W is nonzero while all relative energies QO are zero in the c.m.s. The cluster variables satisfy si = Ki 2 = 02 + s(.~-O2)/g, which is the covariant version of eq. (8); in particular, they no longer introduce any spurious singularity at s = 0. I f Q 1 and 0 3 are chosen as independent variables,
f f = (m2+Q2)'~+(m2+(Ql +Q3)2)½+(m32 +¢~2~I~,~3,,
(9)
where 0 2 = - O 2 in the three-body c.m.s. The three-particle propagator G of eq. (I) and the transition potential V are determined from dispersion relations ",i la Blankenbecler and Sugar ~4) using their discontinuities which follow from imposing elastic and three-particle unitarity. The discontinuity of T resulting from eq. (1) may be written in operator form as
T +-T-
= T+(G ÷ - G - ) T -
+T+G+(V +- V-)G-T-.
(10)
The general unitarity relation for T,
T +-T-
=i
Z
T+P, T - ,
(11)
n=2,3
where p, is the n-body phase-space volume element, is restricted to two- and threebody, intermediate N N ~*)and N N n states• Inserting eq. (3) in eq. (11), and comparing the latter with eq. (10) yields G +-G-
• 1,) = 41mlm (2rt) 2 ~ + (s, l -
m(,)2
)~ + ( k2l - - m l2 ) + n
-
1
~ 21mlm3.q(sl ).q(st)
x f d , , q , ( r 2 ) ~i
+( k2j - m j )2,
(12)
j=l
for the discontinuity of the intermediate [N, N t*) = (nN)] propagator, where involves the proper summation over spins and q~ is the W G relative momentum of eq. (5) for rr(k2) and N(k3). The first term in eq. (12) is due to elastic
(/-,2•
(NN*) WAVE FUNCTION
N(k,)
369
N(k,)
(a)
(b)
Fig. 2. Direct and exchange contribution of intermediate NNrt states from three-particle unitarity to the discontinuities of (a) the three-body propagator G and (b) the transition potential V.
unitarity, i.e. n = 2 in eq. (11), and the second to (rtN) quasi-particle break-up such that rt(k2) forms a quasi-particle N (*~ with the same N(k3) in T and T + (see fig. 2a). The exchange term of eq. (10) (see fig. 2b), 3
G" (k3 X V + - V- )G- (ki) = 4mxm3 i(2rt)3 I-[ 6 +(k~ - m 2j)F~s 3+)g(s 1-)I',
(13)
j=l
is generated from the three-particle contribution ofeq. (I 1) when the pion combines with a different nucleon in T ~ and T- ofeq. (10). The spectator assumption G(ki)~3c 6+(k~-rn~) is ruled out by the off-shell extrapolation ofeq. (8), i.e. time reversal invariance. But eqs. (12) and (13) admit a solution of the spectator form
G(k,)
=
4rtmtq(s,)6 +( ~ - m 2)
=
(W/l,~)2rcm, q(s,)6(k ° - W~lW)ll~,,
(14)
where 9 is the renormalized N (*~ propagator of eq. (3). This reduces eq. (1) to threedimensional form. In eq. (12), we now use the spectator eq. (14) for i = I in conjunction with + 2 + ,.-:-2 2 (q,'Kt)l 6 (k-~2-m2)6 ( k 3 - m 3 ) = fix, x/s, / 2 0203 6(sl-~l),
(15)
and o i = (m,.2 - ?/~)~, which is readily verified in the quasi-particle rest frame where 02 = _qZ. This yields t"
#(s~)--q(s?) = 4nim(*)f(sl --rat*)2)+g(s?)y(s?).j| ~
--m3g~l(/-'2>2/ri6(sl--gl),
(16)
O2093
for the discontinuity of the dressed quasi-particle N ~*) propagator to be used as input in a dispersion relation for 9 in the cluster variable sl. The resulting propagator takes the form
g(Sl) = 2m(,)/Is _m(,)2 + f d3 qt 2m3m(*) ( ° 2 ~ ° 3 ) ( F 2 > ] + J
(17)
Next, upon using the quasi-particle approximation, we also write the exchange con-
370
H.J. WEBER
tribution in eq. (10) in terms of the Wightman-G~rding variables Q~, Q3, i.e. we use (~(s,--m(*)2)a+ ( ~ - - m ~ )
=
¢5
(%-) .
r.U,(4-OhJ' l
6(s'--m¢*)z) 2,
(18)
~, = [(m~*'~-O~)~+(m,~-O,%q ~
(i = 1, 3),
so that G +(k3X V + - V - ) G ( k 1) = (27z)2mlm3.q(s3)g(s, X v + - v - )
×'~t,
4s
"/ ( ,KIF )'~t,~)L~,(~
........2----~--2 -Q,)s3(m~-Q3)]
(19)
which is substituted into eq. (13). For the three-particle term in the unitarity relation (11) we use
i:
(20)
I
= ( m ~ - C Q , +Q3)2)~,
~
= (.q-f#)%
i = 1, 3,
which we substitute into the r.h.s, of eq. (13). As a result, the discontinuity of V becomes V + - V - = 2 ~ i 6 ( s - g)FpF,
I P =
s'Y%s
(3, - ~ x ~ 3 - Q ~ - ( Q ,
+ ¢?,)~)
1½ '
(21)
which serves as input for a dispersion relation in the total invariant three-particle energy s to which the three basic particles contribute equally. This implies V(s) =
(rpr)/(s- ~),
(22)
and says that the quasi-two-body interaction V in eq. (1) consists of pion exchange between (~, N) = N (*) isobars. Clearly then, a direct NN* interaction would be inconsistent with unitarity and proper clustering in three-body theory. The energy denominator of V also reveals that unitarity for the intermediate N, N, rc system of positive energy in conjunction with the cluster properties eliminate retardation effects in this relativistic three-body model. In other words, the exchanged pion can emerge only from a (r~, N) composite and, then, will have positive energy. We are now ready to extrapolate eq. (1) to the deuteron pole at s = rn,]. First we introduce relativistic wave functions @N~,,,(Q,) =
2x/,~W(-mz _ Q2)~ g(s,)
(i = 1, 3);
(23)
then substitute eq. (14) for the propagator G in the homogeneous version of eq. (1)
(NN*) WAVE FUNCTION
371
which we transform to the Wightman-G'~rding variable Q1 upon using eq. (18). As a result, we obtain the following set of (covariant) coupled channel equations, written in the deuteron rest frame for a final N N and NN* state, respectively, , fd3Q, [(NN,,)I rlNN)~NN(Q,) - - ,aNt*)(S3)I//NNt*)(Q3) ~" J(2~)3 m3 +(NN(*) 1ITINN*)~NN,(Q 1)],
(24)
with the effective two-body interaction for s = m 2 [7 = -/~.-!$3S/('~3+ 02))½F
(25)
~(s-~)
For
two final nucleons
and
Q~, Q3 ~ 0,
~ has the correct
static
limit
F2/(m2~+ (Ql + Q3) 2) and a Yukawa-like spatial part. We note also that the cluster variable s 3 in eq. (24) depends only on Q3, and not on Q~ or the angle ( ~ - Q3, provided ~ of eq. (18) is used instead of # in the off-shell extrapolation eq. (8) in conjunction with g~ = m¢*), which is consistent with the quasi-particle approximation. 3. Partial-wave helicity analysis When the (nNW *)) vertices, the transition potentials and wave functions are expressed in the helicity representation, eq. (24) reduces to one-dimensional coupled integral equations involving the radial wave functions only. To this end, we choose the quantization axis along /(3 = :~ in the three-body c.m.s, and take QI, Q3 as independent variables so that k i = Qi, i = 1, 3. The intermediate nucleon N(k~) then has the momentum k I = (k°; k I sin 0, 0, k~ cos 0) in the (x, z) plane. The (nNN) vertex (in fig. 1) is the usual pseudoscalar coupling F~tNN = gu(K3, A3)ysU(kl, zl),
(26)
with ,q2/4rr = 14.5 and the standard spinor normalization, e.g. Nk, = ((k ° + m)/2m) ½. Hence 15) FnNN = gNktNK3[(-- l)~-a'kl/(k ° + m ) - ( - 1)½-A3K3/(K°+m)]d~,A3(O),
(27)
where the lambda's denote the helicities. The (nNN*) vertex for the N*(1520) has the form F,tNN* = y'/~(g3, A3)" (K3 -kl)ysu(kt, ';,1),
(28)
where g ' 2 / 4 x = 0.54/m 2 is determined from the N* ~ nN decay width, and the N* is described in terms of a Rarita-Schwinger spinor. Thus --2 ½ F~NN. = g , Nx3 Nk,x/~[alda~,~3(O) + b ldLA3(O)],
(29)
372
H.J.
WEBER
where, e.g., b I = 2k I [-(- 1)"r- a'kl(3-IA31)g°/m * + IA31)/(k°l + m) + ( - 1)~- a~g3((Ia31-3)g°/m* + [A31)/(K ° + m*)].
(30)
The other vertex functions, such as r~N*(KI)N(k3) in fig. 1 involving the coefficients a 1, and b~, are obtained from eqs. (26) to (30) by symmetry considerations. In the following the partial-wave expansion is carried out in some detail only for the NN* --+ N*N exchange interaction shown in fig. 1 ; the NN ~ NN* transition is treated similarly. We start from eq. (25), where we substitute the results of eqs. (27) and (29), including the isospin matrix element for the T = 0 channel. Hence F 2 in eq. (25) becomes = - 2Nk3NK3Nk, NK, • [S]dS,x(O), s [S]=
(31)
,~i . 21, - AI, /'. , X~2 S , A3, - ;,3, )~) alal(2-~S, 1,
+ b l b l ( ~3 3 S , z. l4 ,
.
33 . - A 1, 2 , XI~S, A 3,
--23, 2)
t 13 + a I bl(2~S ,. 21, - A 1, z X ~1 3 S , . A3, -)~3, 2) ",
• 31 . +albl(-i~S, 21, - A 1,/.4 , X3~1 S , . A3, - 23, 2),
and the Clebsch-Gordan coefficients result from coupling of the rotation matrices in the two 7tNN* vertices. Inserting the partial-wave expansion for the helicity-free factors of V in eq. (25) and combining the d-matrices yields (note that J = l here) _
= ~ = 2rr
f_
2J+l
47z dSa'a(O)'
d(cos O)dJa,a(O)
1
-
4n ,2 -2 2 g Nk~NK~Nk,NK,x/~af(-Q~ )f(-(~3 ) ~ (2E'+I)[S"](E'S"J;02'2') 2J + 1 L"s'" x
f'.
J_, ox
f
s
- ~
"~*
PL,.(x)
----
, (32)
Where x = cos 0 = al~3 • ~1 = - 0 3 " 01" The short,range cut-off form factors f ( Q 2 ) = A2/(A 2 _ Q2) with the regulator mass A take into account the finite pion interaction size. When V is properly symmetrized in N and N*, F is a real and symmetric matrix in the space of the N N and NN* channels. If we now write the (NN (*)) wave functions in terms of radial components/~ in the
(NN*) WAVE FUNCTION
373
LS representation, viz. @A3,;,,~.(Q3)NN*.
=
~-x/-4-~L(LSJ;O22X3 IS;
a3,
(33)
_ ;~3, ')~I~NN*'/"I'! LS ,Y-.3/,'
the coupled channel eqs. (24) take the form dQ 1 ~ KL.t,(Q3, Q,)RL, ~NN(Q,)+
~N(Q3) =
dQl ~ KL, L's'(Q3,
L'
NN*
~LS (Q3) =
;0
~NN*
L'S"
NN
dQI~_,KLs.L,(Q3, Q1)I~L, (Q1)+ L'
fo
(34) ~NN*
dQl ~_,KLs, I=s,(Q3, QI)RL's'(QI) • L'S'
The NN* exchange kernel is given by
KI.S.L's'(Q3, QI)
m3 ~.. ~L' . , ., (27t) -3 ~3 gN,(s3)O~z~.~. ~ (Q3,tJ, N N[VIQI/~ J, NN*)
x(3~S;A3, - - ' ~ 3 ,
/.X½~S " 3 , ; 2~, - A 1,/~ " . XLSJ; . . . 0z2XES . . . J; 0a 2 ). (35)
When the results in eqs. (32), (31) and (30) are inserted into eq. (35), each kernel can be written as a sum of terms each of which contains a helicity-free coefficient depending on momenta only and a sum over all helicities ofproducts of eight ClebschG o r d a n coefficients. Each sum over helicities factorizes into one over those of the intermediate and final NN ~*~states, respectively. As a typical example we display the term
2 (LSI', 0)~2X£'S" I ; 0AJ.X~5S; a 3, -J-3, )~)(½2"-'K"'., 1 "'3,A --'/'3, )-X-- 1)-~+33 A3A3A
(ES''l;02z)(ES'"' " "l;0,t,tX½~S"" 3 ';).I,-A1,2'X½~S"',"ll,-AI,
x ~
J-X- ) a ' - ½
AI~.I 3.'
= - [(LS 1 ; o00XE'S"I ; O00X 1 + ( - 1)s +s")(3 ½S; 5, - ½, OX5 5S"; ½, - 5, O) +(LS1 ; O11XE'S"I ; O11X1 + ( - liL+L"~t3-1S" , ~ ,½ ~ 1X55S"; 55 t)] x [(ES'I ; 000XE'S" 1 ; 000X 1 + ( - l)S' +s,'X½~S'; ~, - 5, 0X½½s" ;5, - 5, 0)
-(ES'I;OI1XE'S"I;OIlX1 + ( - 1 )
L'+L"
1 3~ s ,., 551X½~S"; ½51)]. )(-2
(36)
For the N*(1520) which has spin ~ and negative parity, we recall that (L, S) takes on the values (1, I), (1, 2), (3, 2) only; the same holds for (L', S') in the exchange kernel. Therefore, in the remaining sum over (L", S") the selection rules leave only a few values which L", S" can take on; e.g. in eq. (36) only L " = 1 and S" = 0, 1. The other terms and kernels are treated similarly.
374
H.J. WEBER
4. Results and discussion
For the NN*(1520) configuration there are three (L, S) states (1, 1), ( I, 2) and (3, 2) as a consequence of selection rules involving the spin 3 and the negative parity of the N*(1520)*. Since the (np) deuteron wave function is only slightly modified by the coupling to the N* channels, most of which is due to a renormalization that is included here, we have chosen two different phenomenological (np) deuteron wave functions as input and solve the resulting three coupled inhomogeneous integral equations numerically using Gaussian integration with twenty mesh points in conjunction with matrix inversion. Both McGee's parameterization ~6) of the Hamada-Johnston (np) deuteron wave function and that of Gourdin et al. ~) which fits medium and low energy deuteron photo-disintegration data, are displayed in fig. 3.
601\ '
~
"
,
,
5O e,l
i
>; u
40
o z
° vO
30
Z-J
ZV) ZJ
~rr
20
0
I0 -2 0.!
0.2 0.3 k (GeVlc )
0.4
0.5
Fig. 3. The 3S and 3D deuteron (np) radial wave functions (6.5 " D-state) from ref. ~') (dashed) and ref. ; ~) (solid): for parameters see also ref. t s)
0.1
0.3 0.5 0.7 Q (GeV/c)
0.9
1.1
Fig. 4. Radial NN*(1520) wave functions using (np) wave functions of ref. ;v) in the impulse approximation (dashed) and couplcd channel calculation (solid).
It is instructive to study how sensitive the N* wave functions are to the high momentum components of different (np) parametrizations. As is shown in figs. 4--6 * It is in agreement with the work in ref. 19) that the 3P l NN*(1520) state is not removed by the antisymmetry o f the N N * wave function: Besides the usual coordinate spin-isospin component ~b., each N N * state contains also an intrinsic N* component ~b,. The intrinsic wave function ~b, is symmetric under exchange o f " n u c l e o n s " in the same intrinsic N or N* state, while ~O, is antisymmetric. Under exchange of nucleons in different intrinsic N (*) states, however, various NN* permutation symmetries can occur in ~, and q~, subject only to the condition that the total NN* wave function q~ = A,@,(I,2)~,(I,2) be antisymmetric.
(NN*) WAVE F U N C T I O N
375
JSp\ 8
~.vu 4 °~
z~ zr~
Z-J
(.9
2
oV
z~ z.J ale
-2 (3.1
0.3
~
0.7
0.9
1.1
13
Q(GeV/c) Fig. 5. Same as in fig. 4, except for (np) wave functions from ref. 16).
0.1
0.3
0..5
0.7
O,9-
U
Q(GeV/c) Fig. 6. Radial NN*(1520) wave functions using (np) wave functions from ref. 16) (dashed) and ref. ~7) (solid) and a dipole finite-sizeform factor with regulator mass A = 0.5 GeV.
the dominantSF N* component is stable while the 5p state is more sensitive to the (np) high m o m e n t u m components; the latter is three times smaller for the Gourdin (np) wave function which has smaller high m o m e n t u m components than the McGee version. This is corroborated by variation of the regulator mass A [ = 0.5 GeV in eq. (32) and figs. 4 and 5] in the finite-size form factor which, to some extent, simulates p- and w-meson exchange. The results in fig. 6, which have been calculated with a dipole form f a c t o r f ( Q 2) = A * / ( A 2 - Q z) z in eq. (32) and A = 0.5 GeV, are qualitatively similar to those of figs. 4 and 5. The results of figs. 4 and 5 also show that the impulse approximation works reasonably well; i.e. modifications due to the N N * exchange potential amount to less than 30 %. The peak position o f the radial N* wave functions is only weakly dependent on the finite-size parameter A. It is a consequence of the quite different kinematic off-shell extrapolation used in this relativistic three-body theory as compared to the one inherent in the nonrelativistic Schr6dinger equation that the covariant results of figs. 4 and 5 differ in magnitude, though not qualitatively, from the nonrelativistic results in figs. 7 and 8. Again, the impulse approximation is seen to be valid. The nonrelativistic results for
376
H.J. WEBER L5
1.0
~0
G5 Z~
zJ
Z~ ZJ
0
0
~rr
-0.5
Q (GeV/c) Fig. 7. Nonrelativistic radial N N * ( 1 5 2 0 ) w a v e functions using (np) wave functions of ref. ~7) in
0.2
0.4
0.6
~
~,0 •
Fig. 8. Same as in tig. 7, except for (np) wave function from ref. 16).
the impulse approximation (dashed) and coupled channel calculation (solid) and a monopole finite-size form factor with A = 0.5 GeV.
a dipole form factor with regulator mass A = 0.5 GeV are practically identical with those of figs. 7 and 8. For the purpose of studying the kinematic off-shell extrapolation of eq. (8), we have used a spectator assumption for N(kl) and N(k 3) instead so that s/ = (m d (m2+ Q2)~)2_ Q2 in the three-body c.m.s. The corresponding covariant calculation in fig. 9, using the monopole finite-size form factor with A = 0.5 GeV, is closer in magnitude to (within 25 ~o) the nonrelativistic results of figs. 7 and 8 than to the relativistic ones in figs. 4 and 5 which are based on the correct off-shell extrapolation. It appears that off-shell effects are more important than other relativistic corrections.
,sp
1 ~
~'
s ~
Q
-1
0.1
0..3
0,5
0.7
0.g
1.1
Q(GeV/c) Fig. 9. Radial NN*(1520) wave functions using (np) wave functions from ref. ,6) (dashed) and ref. 1T) (solid), a monopole finite-size form factor with A = 0.5 GeV and a spectator assumption for one nucleon.
(NN*) WAVE FUNCTION
377
In conclusion, we emphasize that the relativistic three-body isobar model used here has several advantages over covariant perturbation theory or the Bethe-Salpeter equation (BSE). First it is a three-dimensional formalism and thus readily amenable to unitarity-preserving numerical calculations in contrast to the BSE. Second, fundamental principles such as (two- and three-body) unitarity and time reversal invariance constrain the reduction to three dimensions and the off-shell continuation of each particle's momentum to be essentially unique; furthermore via dispersion relations they allow a determination of the interaction V and the two- and threeparticle propagators. Thus, there are only two basic model assumptions. The quasiparticle approximation simplifies the calculations considerably but may be removed; it ties up the covariant elimination of all relative energies with the cluster properties of the system. The restriction to three basic particles is essential, though, and entails several shortcomings among which we list that it prevents us from (i) inclusion of p- and e~-meson exchange besides pion exchange, and (ii) similarly treating AA (and N ' N * ) configurations except in the impulse approximation upon taking ~z, N, A as the basic particles. Among its interesting dynamical consequences we recall the elimination of(a) a direct NN* interaction and (b) retardation effects, and the closely related (spurious) pole in the physical region of the exchanged pion's Feynman propagator caused by the large N-N* mass difference and (c) negative energy N and N* states and the corresponding short range components in the deuteron groundstate wave function. It is a pleasure to thank I. Afnan, H. Arenh6vel, J. Eisenberg and J. Namyslowski for useful discussions. References I) H. Arenh6vel and H. J. Weber, Springer Tracts in Modem Physics 65 (1972"1 58: H. Arenh6vel, Isobar configurations, in Proc. Symp. on interaction studies in nuclei, ed. H. Jochim and B. Ziegler, Mainz (North-Holland~ Amsterdam, 1975) p. 727; H. J. Weber, Spectator experiments as probes of nucleon-resonance admixtures, ibid, p. 749 and Proc. 7th Int. Conf. on few body problems in nuclear and particle physics, Delhi, ed. A. N. Mitra (North-Holland, Amsterdam, 1976); L. S. Kisslinger, Current research on N* in nuclei, Proc. Topical Meeting on high energy collisions, Trieste, Italy, 1974; R. Beurtey, Proc. 6th Int. Conf. on high energy physics and nuclear structure, ed. D. E. Nagle et al., Santa Fe (AIP, 1975) p. 653 2) D. O. Riska and G. E. Brown, Phys. Lett. 38B (1972) 193; M. Gari and A. H. Huffman, Phys. Rev. C"/(1973) 994 3) H. Arenh6vel, W. Fabian and H. G. Miller, Phys. Lett. 52B (1974) 303; Z. Phys. 271 (1974) 93 4) H. Sugawara and F. von Hippel, Phys. Rev. 172 (1968) 1764; R. Vinh Mau, Proc. 7th Int. Conf. on few body problems in nuclear and particle physics, ed. A. N. Mitra, Delhi (North-Holland, Amsterdam, 1976) 5) H. J. Weber, Phys. Rev. C9 (1974) 1771 6) C. P. Home et al., Phys. Rev. Lett. 33 (1974) 380; M. J. Emms et al., Phys. Lett. 52B (1974) 372;
378
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
H.J. WEBER H. Braun et al., Phys. Rev. Lett. 33 (1974) 312; B. S. Aladashvili et al., Nucl. Phys. B89 (1975) 405 C. F. Perdrisat et al., Phys. Lett. (1976); N. R. Nath, H. J. Weber and P. K. Kabir, Phys. Rev. Lett. 26 (1971) 1404 P. Bcnz and P. S6ding, Phys. Lett. 52B (1974) 367 N. R. Nath, P. K. Kabir and H. J. Weber, Phys. Rev. DI0 (1974) 811 A. Aaron, R. D. Amado and J. E. Young, Phys. Rev. 174 (1968) 2022 P. Haapakoski and M. Saarela, Phys. Lett. 53B (1974) 333; H. Arenh6vel, ibid., p. 224 J. M. Namyslowski, Nuovo Cim. 57A (1968) 355; U. Weiss, Nucl. Phys. 1344 (1972) 573 J. D. Bjorken and S. D. Drell, Relativistic quantum mechanics (McGraw-Hill, NY, 1964) R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051 K. Erkelenz, Phys. Reports CI3 (1974) 191 I. McGee, Phys. Rev. 151 (1966) 772 M. Gourdin, M. LeBellac, F. M. Renard and J. Tran Thanh Van, Nuovo Cim. 37 (1965) 524 N. R. Nath and H. J. Weber, Phys. Rev. I)6 (1972) 1975 G. J. Klem and H. J. Weber, Z. Phys. 268 (1974) 299; J. S. Sharma, V. S. Bhasin and A. N. Mitra, Nucl. Phys. B35 (1971) 466;
North-Holland Publishiny Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
RELATIVISTIC (NN*) WAVE FUNCTION OF T H E DEUTERON t H. J. WEBER
Department of Physics. University of Virginia, Charlottesville, Va. 22901 Received 2 March 1976 Abstract: The (NN*) wave function in the deuteron ground state is studied in a relativistic and threedimensional linear, three-body formalism involving two nucleons and one pion as basic particles. The off-shell extrapolation used is unambiguously prescribed by time reversal invariance. For the NN*(1520) configuration off-shell effects are found to be sizable.
1. Introduction
At present there is considerable interest in the role of mesons and nucleon resonances (N*) as constituents of nuclei t). It is known that magnetic moments of light nuclei and the photodisintegration 2) of the deuteron near threshold and just below the pion production threshold 3) cannot be understood without directly taking meson and N* degrees of freedom into account. Particularly, the A(1236) isobar plays an important role in generating the strong medium range attraction in the nucleon-nucleon force 4) which may be considered as the driving force for the A in nuclei. Thus for the two-nucleon system there are now several processes involving mostly large momentum transfer where isobars seem to play an important though indirect role as they are usually far off their mass shell. A bound isobar contributes to the short range nuclear structure regime predominantly. Baryon resonances are in general a more efficient source of high momentum components than the short range repulsion in the nucleon-nucleon force. A high baryon spin, in particular, admits or requires a correspondingly high orbital angular momentum and the associated power of the momentum transfer in the relevant NN ~ NN* or NN --* N ' N * transition interaction; yet small distances are avoided by the angular momentum barrier. Nonetheless, such arguments may not be a sufficient justification for using nonrelativistic potential theory at high momentum transfer when the isobars or nucleons involved are substantially ( < 50 ~ ) off their mass shell. Thus relativistic and off-shell corrections can be expected to be nonnegligible. Furthermore, there is usually a plethora of "background" reaction mechanisms, such as non-resonant nN exchange, which compete with the N* signal. The situation is fairly typical and applies to the inelastic high energy pick-up 5), t Supported in part by the National Science Foundation. 365 July 1976
366
H.J. WEBER
quasi-elastic scattering and spectator experiments 6.7) and inclusive processes s,9) that have recently been designed to show more directly the presence of isobars in the deuteron. For two of these isobar-spectator experiments 7,8) there are indications that to some extent the N* spectator signal may have been experimentally separated from the background of nonresonant zrN systems. As a consequence, there is now some evidence for a total (AA) and (NN*) admixture probability of about 0.4 ~o to 3 ~o in the deuteron ground state. In each case, however, a nonrelativistic isobar wave function was used as a crude estimate for the AAd vertex involved in the A-spectator signal. This includes taking the A to have equal energy and the physical mass rna = 1.236 GeV/c 2. For the spectator experiments with the observed recoiling A(k) on its mass shell and the exchanged A of energy m a - k o and mass (m2+rnA2 - 2 m ~ k o ) ~ = < m d - m d < ½m j, this nonrelativistic approximation is a large extrapolation, seems questionable and should bc investigated in more detail. To this end we study an off-shell NN* system in the deuteron ground state in terms of a linear, covariant, and three-dimensional formalism of the Lippmann-Schwinger type. The large off-mass-shell extrapolations which are a characteristic feature of isobar admixtures to bound many-nucleon wave functions have motivated us choosing a covariant three-particle equation 1o) involving two nucleons and one pion so that elastic and three-body unitarity are preserved in conjunction with an unambiyuous off-shell extrapolation procedure. As a consequence the formalism is well suited to study off-shell effects. However, its essential restriction to three particles N, N, 7z prevents us from introducing p-meson exchange in addition to the pion exchange. The former is known ~1) to generate a core in a natural way and thus stabilize the unphysical dependence of the pion-exchangc on the short range cut-off form factor or regularization procedure to some extent. Using a quasi-particle approximation, an off-shell nucleon and isobar N* of isospin ½are admitted as (~, N) composites in a relativistic scattering problem of an N from a quasi-particle N or N* proceeding via intermediate three-particle N, N, rr states. In this context, it turns out to be essential to use the relative Wightman-G~rding momenta 12) instead of the usual Jacobi variables for enforcing correct cluster properties and extrapolating to the bound-state deuteron pole. The general formalism is described in sect. 2. The partial-wave helicity analysis, transition potentials and coupled-channel equations for the radial wave functions are presented in sect. 3. Numerical results are obtained for the NN*(1520) configuration and discussed in sect. 4.
2. The (NN*) configuration in relativistic three-body theory Our starting point is the relativistic Lippmann-Schwinger equation ~'dakl (,piT(s) [q) = (Pl V(s)lq> + j ~ ) ~ (p[ V(s)lk I k2k 3>G(k 1k2k31T(s)lq),
(1)
(NN*) WAVE FUNCTION
"~-'" N*(N,)
367
~ N(I,,)
.... = N*(K,)
~ { ~ ~
(~II)
N(k3)
Fig. 1. Schematic representation of NN* scattering, cqs. (1), (24) and (34), at the deuteron pole s = rn~ with intermediate NNn states in the quasi-particle approximation. for off-shell N N and N N * scattering leading to a final N N or N N * state p. Using a quasi-particle a p p r o x i m a t i o n , one off-shell N and N* are admitted as (n, N) composites. As shown in fig. 1 for the final quasi-two-body state IN(k3), N*(K3) = (n, N)], the intermediate states are restricted to the q u a s i - t w o - b o d y states [N(kl), N(*)(KI) = (rt, N)], where 13) "I" K I = k 2 + k 3,
K 3 = k l + k 2,
s i = K z.
(2)
Thus the separable t w o - t o - t h r e e - b o d y I N N (*) ~ N N n ] production amplitude (NNrtlT(s)lq)
= F q(sl)(N(kt),
N(*)(K1) = (rc(k2), N ( k 3 ) ) l T ( s ) l q ) ,
(3)
involves the p r o p a g a t o r g(K~) o f the (n, N) quasi-particle and its decay vertex F ( N (*) ~ nN). This system, including the initial N N or N N * state q, and eq. (1) are extrapolated from the on-shell invariant energy g=F~.g=W
z,
g=~r
+k- z+k-3,
(4)
to the bound-state deuteron pole at s = W Z = m ~ . In the following such on-shell quantities are generally denoted by a bar. W h e n the intermediate three-particle states are described in terms o f the usual Jacobi relative variables in conjuction with a spectator a s s u m p t i o n for the nucleon N(kt), say, i.e. k~ = m~ = m 2, a spurious singularity a t s = 0 is k n o w n to occur 10). In this context, therefore, it is essential to use the W i g h t m a n - G f i r d i n g relative m o m e n t u m variables 12), e.g. ql = ~k2 - k 3 ) - ( m ~ - m Z 3 ) K ~ / 2 K ~ ,
(5)
QI = ~(k~-Kt)-(mZ-g,)K/2~,
with m I = m 3 = m,
m 2 = m~,
K = kl+k2+k3,
(6)
to enforce proper clustering and eliminate the relative energies in a covariant way. In fact, q; and Qi (i = l, 2, 3) reduce to space-like three-vectors in the two- and t We use the conventions and metric of ref. ~3).
368
H..1. WEBER
three-body c.m. systems, respectively, because qi" Ki = 0 = Qi" K,
(7)
when k 2 = m,.2. The off-energy-shell continuation is designed so that all energy components of momentum four-vectors in the c.m.s, become proportional to the total three-particle c.m.s, energy W, e.g.
k° = ~, w / ¢ ¢ =
1+
.....
w,
(8) K ° = K,°W/ff / = :2 1
W.
As a consequence, time-reversal invariance is preserved as it is related to a reflection in the total energy W ---, - W because only the total energy W is nonzero while all relative energies QO are zero in the c.m.s. The cluster variables satisfy si = Ki 2 = 02 + s(.~-O2)/g, which is the covariant version of eq. (8); in particular, they no longer introduce any spurious singularity at s = 0. I f Q 1 and 0 3 are chosen as independent variables,
f f = (m2+Q2)'~+(m2+(Ql +Q3)2)½+(m32 +¢~2~I~,~3,,
(9)
where 0 2 = - O 2 in the three-body c.m.s. The three-particle propagator G of eq. (I) and the transition potential V are determined from dispersion relations ",i la Blankenbecler and Sugar ~4) using their discontinuities which follow from imposing elastic and three-particle unitarity. The discontinuity of T resulting from eq. (1) may be written in operator form as
T +-T-
= T+(G ÷ - G - ) T -
+T+G+(V +- V-)G-T-.
(10)
The general unitarity relation for T,
T +-T-
=i
Z
T+P, T - ,
(11)
n=2,3
where p, is the n-body phase-space volume element, is restricted to two- and threebody, intermediate N N ~*)and N N n states• Inserting eq. (3) in eq. (11), and comparing the latter with eq. (10) yields G +-G-
• 1,) = 41mlm (2rt) 2 ~ + (s, l -
m(,)2
)~ + ( k2l - - m l2 ) + n
-
1
~ 21mlm3.q(sl ).q(st)
x f d , , q , ( r 2 ) ~i
+( k2j - m j )2,
(12)
j=l
for the discontinuity of the intermediate [N, N t*) = (nN)] propagator, where involves the proper summation over spins and q~ is the W G relative momentum of eq. (5) for rr(k2) and N(k3). The first term in eq. (12) is due to elastic
(/-,2•
(NN*) WAVE FUNCTION
N(k,)
369
N(k,)
(a)
(b)
Fig. 2. Direct and exchange contribution of intermediate NNrt states from three-particle unitarity to the discontinuities of (a) the three-body propagator G and (b) the transition potential V.
unitarity, i.e. n = 2 in eq. (11), and the second to (rtN) quasi-particle break-up such that rt(k2) forms a quasi-particle N (*~ with the same N(k3) in T and T + (see fig. 2a). The exchange term of eq. (10) (see fig. 2b), 3
G" (k3 X V + - V- )G- (ki) = 4mxm3 i(2rt)3 I-[ 6 +(k~ - m 2j)F~s 3+)g(s 1-)I',
(13)
j=l
is generated from the three-particle contribution ofeq. (I 1) when the pion combines with a different nucleon in T ~ and T- ofeq. (10). The spectator assumption G(ki)~3c 6+(k~-rn~) is ruled out by the off-shell extrapolation ofeq. (8), i.e. time reversal invariance. But eqs. (12) and (13) admit a solution of the spectator form
G(k,)
=
4rtmtq(s,)6 +( ~ - m 2)
=
(W/l,~)2rcm, q(s,)6(k ° - W~lW)ll~,,
(14)
where 9 is the renormalized N (*~ propagator of eq. (3). This reduces eq. (1) to threedimensional form. In eq. (12), we now use the spectator eq. (14) for i = I in conjunction with + 2 + ,.-:-2 2 (q,'Kt)l 6 (k-~2-m2)6 ( k 3 - m 3 ) = fix, x/s, / 2 0203 6(sl-~l),
(15)
and o i = (m,.2 - ?/~)~, which is readily verified in the quasi-particle rest frame where 02 = _qZ. This yields t"
#(s~)--q(s?) = 4nim(*)f(sl --rat*)2)+g(s?)y(s?).j| ~
--m3g~l(/-'2>2/ri6(sl--gl),
(16)
O2093
for the discontinuity of the dressed quasi-particle N ~*) propagator to be used as input in a dispersion relation for 9 in the cluster variable sl. The resulting propagator takes the form
g(Sl) = 2m(,)/Is _m(,)2 + f d3 qt 2m3m(*) ( ° 2 ~ ° 3 ) ( F 2 > ] + J
(17)
Next, upon using the quasi-particle approximation, we also write the exchange con-
370
H.J. WEBER
tribution in eq. (10) in terms of the Wightman-G~rding variables Q~, Q3, i.e. we use (~(s,--m(*)2)a+ ( ~ - - m ~ )
=
¢5
(%-) .
r.U,(4-OhJ' l
6(s'--m¢*)z) 2,
(18)
~, = [(m~*'~-O~)~+(m,~-O,%q ~
(i = 1, 3),
so that G +(k3X V + - V - ) G ( k 1) = (27z)2mlm3.q(s3)g(s, X v + - v - )
×'~t,
4s
"/ ( ,KIF )'~t,~)L~,(~
........2----~--2 -Q,)s3(m~-Q3)]
(19)
which is substituted into eq. (13). For the three-particle term in the unitarity relation (11) we use
i:
(20)
I
= ( m ~ - C Q , +Q3)2)~,
~
= (.q-f#)%
i = 1, 3,
which we substitute into the r.h.s, of eq. (13). As a result, the discontinuity of V becomes V + - V - = 2 ~ i 6 ( s - g)FpF,
I P =
s'Y%s
(3, - ~ x ~ 3 - Q ~ - ( Q ,
+ ¢?,)~)
1½ '
(21)
which serves as input for a dispersion relation in the total invariant three-particle energy s to which the three basic particles contribute equally. This implies V(s) =
(rpr)/(s- ~),
(22)
and says that the quasi-two-body interaction V in eq. (1) consists of pion exchange between (~, N) = N (*) isobars. Clearly then, a direct NN* interaction would be inconsistent with unitarity and proper clustering in three-body theory. The energy denominator of V also reveals that unitarity for the intermediate N, N, rc system of positive energy in conjunction with the cluster properties eliminate retardation effects in this relativistic three-body model. In other words, the exchanged pion can emerge only from a (r~, N) composite and, then, will have positive energy. We are now ready to extrapolate eq. (1) to the deuteron pole at s = rn,]. First we introduce relativistic wave functions @N~,,,(Q,) =
2x/,~W(-mz _ Q2)~ g(s,)
(i = 1, 3);
(23)
then substitute eq. (14) for the propagator G in the homogeneous version of eq. (1)
(NN*) WAVE FUNCTION
371
which we transform to the Wightman-G'~rding variable Q1 upon using eq. (18). As a result, we obtain the following set of (covariant) coupled channel equations, written in the deuteron rest frame for a final N N and NN* state, respectively, , fd3Q, [(NN,,)I rlNN)~NN(Q,) - - ,aNt*)(S3)I//NNt*)(Q3) ~" J(2~)3 m3 +(NN(*) 1ITINN*)~NN,(Q 1)],
(24)
with the effective two-body interaction for s = m 2 [7 = -/~.-!$3S/('~3+ 02))½F
(25)
~(s-~)
For
two final nucleons
and
Q~, Q3 ~ 0,
~ has the correct
static
limit
F2/(m2~+ (Ql + Q3) 2) and a Yukawa-like spatial part. We note also that the cluster variable s 3 in eq. (24) depends only on Q3, and not on Q~ or the angle ( ~ - Q3, provided ~ of eq. (18) is used instead of # in the off-shell extrapolation eq. (8) in conjunction with g~ = m¢*), which is consistent with the quasi-particle approximation. 3. Partial-wave helicity analysis When the (nNW *)) vertices, the transition potentials and wave functions are expressed in the helicity representation, eq. (24) reduces to one-dimensional coupled integral equations involving the radial wave functions only. To this end, we choose the quantization axis along /(3 = :~ in the three-body c.m.s, and take QI, Q3 as independent variables so that k i = Qi, i = 1, 3. The intermediate nucleon N(k~) then has the momentum k I = (k°; k I sin 0, 0, k~ cos 0) in the (x, z) plane. The (nNN) vertex (in fig. 1) is the usual pseudoscalar coupling F~tNN = gu(K3, A3)ysU(kl, zl),
(26)
with ,q2/4rr = 14.5 and the standard spinor normalization, e.g. Nk, = ((k ° + m)/2m) ½. Hence 15) FnNN = gNktNK3[(-- l)~-a'kl/(k ° + m ) - ( - 1)½-A3K3/(K°+m)]d~,A3(O),
(27)
where the lambda's denote the helicities. The (nNN*) vertex for the N*(1520) has the form F,tNN* = y'/~(g3, A3)" (K3 -kl)ysu(kt, ';,1),
(28)
where g ' 2 / 4 x = 0.54/m 2 is determined from the N* ~ nN decay width, and the N* is described in terms of a Rarita-Schwinger spinor. Thus --2 ½ F~NN. = g , Nx3 Nk,x/~[alda~,~3(O) + b ldLA3(O)],
(29)
372
H.J.
WEBER
where, e.g., b I = 2k I [-(- 1)"r- a'kl(3-IA31)g°/m * + IA31)/(k°l + m) + ( - 1)~- a~g3((Ia31-3)g°/m* + [A31)/(K ° + m*)].
(30)
The other vertex functions, such as r~N*(KI)N(k3) in fig. 1 involving the coefficients a 1, and b~, are obtained from eqs. (26) to (30) by symmetry considerations. In the following the partial-wave expansion is carried out in some detail only for the NN* --+ N*N exchange interaction shown in fig. 1 ; the NN ~ NN* transition is treated similarly. We start from eq. (25), where we substitute the results of eqs. (27) and (29), including the isospin matrix element for the T = 0 channel. Hence F 2 in eq. (25) becomes
(31)
,~i . 21, - AI, /'. , X~2 S , A3, - ;,3, )~) alal(2-~S, 1,
+ b l b l ( ~3 3 S , z. l4 ,
.
33 . - A 1, 2 , XI~S, A 3,
--23, 2)
t 13 + a I bl(2~S ,. 21, - A 1, z X ~1 3 S , . A3, -)~3, 2) ",
• 31 . +albl(-i~S, 21, - A 1,/.4 , X3~1 S , . A3, - 23, 2),
and the Clebsch-Gordan coefficients result from coupling of the rotation matrices in the two 7tNN* vertices. Inserting the partial-wave expansion for the helicity-free factors of V in eq. (25) and combining the d-matrices yields (note that J = l here) _
f_
2J+l
47z dSa'a(O)
d(cos O)dJa,a(O)
1
-
4n ,2 -2 2 g Nk~NK~Nk,NK,x/~af(-Q~ )f(-(~3 ) ~ (2E'+I)[S"](E'S"J;02'2') 2J + 1 L"s'" x
f'.
J_, ox
f
s
- ~
"~*
PL,.(x)
----
, (32)
Where x = cos 0 = al~3 • ~1 = - 0 3 " 01" The short,range cut-off form factors f ( Q 2 ) = A2/(A 2 _ Q2) with the regulator mass A take into account the finite pion interaction size. When V is properly symmetrized in N and N*, F is a real and symmetric matrix in the space of the N N and NN* channels. If we now write the (NN (*)) wave functions in terms of radial components/~ in the
(NN*) WAVE FUNCTION
373
LS representation, viz. @A3,;,,~.(Q3)NN*.
=
~-x/-4-~L(LSJ;O22X3 IS;
a3,
(33)
_ ;~3, ')~I~NN*'/"I'! LS ,Y-.3/,'
the coupled channel eqs. (24) take the form dQ 1 ~ KL.t,(Q3, Q,)RL, ~NN(Q,)+
~N(Q3) =
dQl ~ KL, L's'(Q3,
L'
NN*
~LS (Q3) =
;0
~NN*
L'S"
NN
dQI~_,KLs.L,(Q3, Q1)I~L, (Q1)+ L'
fo
(34) ~NN*
dQl ~_,KLs, I=s,(Q3, QI)RL's'(QI) • L'S'
The NN* exchange kernel is given by
KI.S.L's'(Q3, QI)
m3 ~.. ~L' . , ., (27t) -3 ~3 gN,(s3)O~z~.~. ~ (Q3,tJ, N N[VIQI/~ J, NN*)
x(3~S;A3, - - ' ~ 3 ,
/.X½~S " 3 , ; 2~, - A 1,/~ " . XLSJ; . . . 0z2XES . . . J; 0a 2 ). (35)
When the results in eqs. (32), (31) and (30) are inserted into eq. (35), each kernel can be written as a sum of terms each of which contains a helicity-free coefficient depending on momenta only and a sum over all helicities ofproducts of eight ClebschG o r d a n coefficients. Each sum over helicities factorizes into one over those of the intermediate and final NN ~*~states, respectively. As a typical example we display the term
2 (LSI', 0)~2X£'S" I ; 0AJ.X~5S; a 3, -J-3, )~)(½2"-'K"'., 1 "'3,A --'/'3, )-X-- 1)-~+33 A3A3A
(ES''l;02z)(ES'"' " "l;0,t,tX½~S"" 3 ';).I,-A1,2'X½~S"',"ll,-AI,
x ~
J-X- ) a ' - ½
AI~.I 3.'
= - [(LS 1 ; o00XE'S"I ; O00X 1 + ( - 1)s +s")(3 ½S; 5, - ½, OX5 5S"; ½, - 5, O) +(LS1 ; O11XE'S"I ; O11X1 + ( - liL+L"~t3-1S" , ~ ,½ ~ 1X55S"; 55 t)] x [(ES'I ; 000XE'S" 1 ; 000X 1 + ( - l)S' +s,'X½~S'; ~, - 5, 0X½½s" ;5, - 5, 0)
-(ES'I;OI1XE'S"I;OIlX1 + ( - 1 )
L'+L"
1 3~ s ,., 551X½~S"; ½51)]. )(-2
(36)
For the N*(1520) which has spin ~ and negative parity, we recall that (L, S) takes on the values (1, I), (1, 2), (3, 2) only; the same holds for (L', S') in the exchange kernel. Therefore, in the remaining sum over (L", S") the selection rules leave only a few values which L", S" can take on; e.g. in eq. (36) only L " = 1 and S" = 0, 1. The other terms and kernels are treated similarly.
374
H.J. WEBER
4. Results and discussion
For the NN*(1520) configuration there are three (L, S) states (1, 1), ( I, 2) and (3, 2) as a consequence of selection rules involving the spin 3 and the negative parity of the N*(1520)*. Since the (np) deuteron wave function is only slightly modified by the coupling to the N* channels, most of which is due to a renormalization that is included here, we have chosen two different phenomenological (np) deuteron wave functions as input and solve the resulting three coupled inhomogeneous integral equations numerically using Gaussian integration with twenty mesh points in conjunction with matrix inversion. Both McGee's parameterization ~6) of the Hamada-Johnston (np) deuteron wave function and that of Gourdin et al. ~) which fits medium and low energy deuteron photo-disintegration data, are displayed in fig. 3.
601\ '
~
"
,
,
5O e,l
i
>; u
40
o z
° vO
30
Z-J
ZV) ZJ
~rr
20
0
I0 -2 0.!
0.2 0.3 k (GeVlc )
0.4
0.5
Fig. 3. The 3S and 3D deuteron (np) radial wave functions (6.5 " D-state) from ref. ~') (dashed) and ref. ; ~) (solid): for parameters see also ref. t s)
0.1
0.3 0.5 0.7 Q (GeV/c)
0.9
1.1
Fig. 4. Radial NN*(1520) wave functions using (np) wave functions of ref. ;v) in the impulse approximation (dashed) and couplcd channel calculation (solid).
It is instructive to study how sensitive the N* wave functions are to the high momentum components of different (np) parametrizations. As is shown in figs. 4--6 * It is in agreement with the work in ref. 19) that the 3P l NN*(1520) state is not removed by the antisymmetry o f the N N * wave function: Besides the usual coordinate spin-isospin component ~b., each N N * state contains also an intrinsic N* component ~b,. The intrinsic wave function ~b, is symmetric under exchange o f " n u c l e o n s " in the same intrinsic N or N* state, while ~O, is antisymmetric. Under exchange of nucleons in different intrinsic N (*) states, however, various NN* permutation symmetries can occur in ~, and q~, subject only to the condition that the total NN* wave function q~ = A,@,(I,2)~,(I,2) be antisymmetric.
(NN*) WAVE F U N C T I O N
375
JSp\ 8
~.vu 4 °~
z~ zr~
Z-J
(.9
2
oV
z~ z.J ale
-2 (3.1
0.3
~
0.7
0.9
1.1
13
Q(GeV/c) Fig. 5. Same as in fig. 4, except for (np) wave functions from ref. 16).
0.1
0.3
0..5
0.7
O,9-
U
Q(GeV/c) Fig. 6. Radial NN*(1520) wave functions using (np) wave functions from ref. 16) (dashed) and ref. ~7) (solid) and a dipole finite-sizeform factor with regulator mass A = 0.5 GeV.
the dominantSF N* component is stable while the 5p state is more sensitive to the (np) high m o m e n t u m components; the latter is three times smaller for the Gourdin (np) wave function which has smaller high m o m e n t u m components than the McGee version. This is corroborated by variation of the regulator mass A [ = 0.5 GeV in eq. (32) and figs. 4 and 5] in the finite-size form factor which, to some extent, simulates p- and w-meson exchange. The results in fig. 6, which have been calculated with a dipole form f a c t o r f ( Q 2) = A * / ( A 2 - Q z) z in eq. (32) and A = 0.5 GeV, are qualitatively similar to those of figs. 4 and 5. The results of figs. 4 and 5 also show that the impulse approximation works reasonably well; i.e. modifications due to the N N * exchange potential amount to less than 30 %. The peak position o f the radial N* wave functions is only weakly dependent on the finite-size parameter A. It is a consequence of the quite different kinematic off-shell extrapolation used in this relativistic three-body theory as compared to the one inherent in the nonrelativistic Schr6dinger equation that the covariant results of figs. 4 and 5 differ in magnitude, though not qualitatively, from the nonrelativistic results in figs. 7 and 8. Again, the impulse approximation is seen to be valid. The nonrelativistic results for
376
H.J. WEBER L5
1.0
~0
G5 Z~
zJ
Z~ ZJ
0
0
~rr
-0.5
Q (GeV/c) Fig. 7. Nonrelativistic radial N N * ( 1 5 2 0 ) w a v e functions using (np) wave functions of ref. ~7) in
0.2
0.4
0.6
~
~,0 •
Fig. 8. Same as in tig. 7, except for (np) wave function from ref. 16).
the impulse approximation (dashed) and coupled channel calculation (solid) and a monopole finite-size form factor with A = 0.5 GeV.
a dipole form factor with regulator mass A = 0.5 GeV are practically identical with those of figs. 7 and 8. For the purpose of studying the kinematic off-shell extrapolation of eq. (8), we have used a spectator assumption for N(kl) and N(k 3) instead so that s/ = (m d (m2+ Q2)~)2_ Q2 in the three-body c.m.s. The corresponding covariant calculation in fig. 9, using the monopole finite-size form factor with A = 0.5 GeV, is closer in magnitude to (within 25 ~o) the nonrelativistic results of figs. 7 and 8 than to the relativistic ones in figs. 4 and 5 which are based on the correct off-shell extrapolation. It appears that off-shell effects are more important than other relativistic corrections.
,sp
1 ~
~'
s ~
Q
-1
0.1
0..3
0,5
0.7
0.g
1.1
Q(GeV/c) Fig. 9. Radial NN*(1520) wave functions using (np) wave functions from ref. ,6) (dashed) and ref. 1T) (solid), a monopole finite-size form factor with A = 0.5 GeV and a spectator assumption for one nucleon.
(NN*) WAVE FUNCTION
377
In conclusion, we emphasize that the relativistic three-body isobar model used here has several advantages over covariant perturbation theory or the Bethe-Salpeter equation (BSE). First it is a three-dimensional formalism and thus readily amenable to unitarity-preserving numerical calculations in contrast to the BSE. Second, fundamental principles such as (two- and three-body) unitarity and time reversal invariance constrain the reduction to three dimensions and the off-shell continuation of each particle's momentum to be essentially unique; furthermore via dispersion relations they allow a determination of the interaction V and the two- and threeparticle propagators. Thus, there are only two basic model assumptions. The quasiparticle approximation simplifies the calculations considerably but may be removed; it ties up the covariant elimination of all relative energies with the cluster properties of the system. The restriction to three basic particles is essential, though, and entails several shortcomings among which we list that it prevents us from (i) inclusion of p- and e~-meson exchange besides pion exchange, and (ii) similarly treating AA (and N ' N * ) configurations except in the impulse approximation upon taking ~z, N, A as the basic particles. Among its interesting dynamical consequences we recall the elimination of(a) a direct NN* interaction and (b) retardation effects, and the closely related (spurious) pole in the physical region of the exchanged pion's Feynman propagator caused by the large N-N* mass difference and (c) negative energy N and N* states and the corresponding short range components in the deuteron groundstate wave function. It is a pleasure to thank I. Afnan, H. Arenh6vel, J. Eisenberg and J. Namyslowski for useful discussions. References I) H. Arenh6vel and H. J. Weber, Springer Tracts in Modem Physics 65 (1972"1 58: H. Arenh6vel, Isobar configurations, in Proc. Symp. on interaction studies in nuclei, ed. H. Jochim and B. Ziegler, Mainz (North-Holland~ Amsterdam, 1975) p. 727; H. J. Weber, Spectator experiments as probes of nucleon-resonance admixtures, ibid, p. 749 and Proc. 7th Int. Conf. on few body problems in nuclear and particle physics, Delhi, ed. A. N. Mitra (North-Holland, Amsterdam, 1976); L. S. Kisslinger, Current research on N* in nuclei, Proc. Topical Meeting on high energy collisions, Trieste, Italy, 1974; R. Beurtey, Proc. 6th Int. Conf. on high energy physics and nuclear structure, ed. D. E. Nagle et al., Santa Fe (AIP, 1975) p. 653 2) D. O. Riska and G. E. Brown, Phys. Lett. 38B (1972) 193; M. Gari and A. H. Huffman, Phys. Rev. C"/(1973) 994 3) H. Arenh6vel, W. Fabian and H. G. Miller, Phys. Lett. 52B (1974) 303; Z. Phys. 271 (1974) 93 4) H. Sugawara and F. von Hippel, Phys. Rev. 172 (1968) 1764; R. Vinh Mau, Proc. 7th Int. Conf. on few body problems in nuclear and particle physics, ed. A. N. Mitra, Delhi (North-Holland, Amsterdam, 1976) 5) H. J. Weber, Phys. Rev. C9 (1974) 1771 6) C. P. Home et al., Phys. Rev. Lett. 33 (1974) 380; M. J. Emms et al., Phys. Lett. 52B (1974) 372;
378
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
H.J. WEBER H. Braun et al., Phys. Rev. Lett. 33 (1974) 312; B. S. Aladashvili et al., Nucl. Phys. B89 (1975) 405 C. F. Perdrisat et al., Phys. Lett. (1976); N. R. Nath, H. J. Weber and P. K. Kabir, Phys. Rev. Lett. 26 (1971) 1404 P. Bcnz and P. S6ding, Phys. Lett. 52B (1974) 367 N. R. Nath, P. K. Kabir and H. J. Weber, Phys. Rev. DI0 (1974) 811 A. Aaron, R. D. Amado and J. E. Young, Phys. Rev. 174 (1968) 2022 P. Haapakoski and M. Saarela, Phys. Lett. 53B (1974) 333; H. Arenh6vel, ibid., p. 224 J. M. Namyslowski, Nuovo Cim. 57A (1968) 355; U. Weiss, Nucl. Phys. 1344 (1972) 573 J. D. Bjorken and S. D. Drell, Relativistic quantum mechanics (McGraw-Hill, NY, 1964) R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051 K. Erkelenz, Phys. Reports CI3 (1974) 191 I. McGee, Phys. Rev. 151 (1966) 772 M. Gourdin, M. LeBellac, F. M. Renard and J. Tran Thanh Van, Nuovo Cim. 37 (1965) 524 N. R. Nath and H. J. Weber, Phys. Rev. I)6 (1972) 1975 G. J. Klem and H. J. Weber, Z. Phys. 268 (1974) 299; J. S. Sharma, V. S. Bhasin and A. N. Mitra, Nucl. Phys. B35 (1971) 466;