- Email: [email protected]
Pion-nucleon attraction at short distances
j
7.C
]
Nuclear Phvs,cs 18 (1960) 14--22, ~ ) North-Holland PubhsMng Co, Amsterdam Not to be reproduced by photoprmt or nncro'fllm without written permission from the pubhsher
PION-NUCLEON
ATTRACTION S DONIACH and E
AT SHORT DISTANCES YAMADA t
Department o/ Theoret,cal Physics, Umvers,ty o/ L,verpool Received 10 March 1960
A b s t r a c t : I t is suggested t h a t a t h e o r y which t r e a t s the ~ - m e s o n as a strongly b o u n d nucleona n t m u c l e o n s t a t e could promde a n explanation for t h e observed : s o b a n c - s p m dependence of t h e meson-nucleon s-wave scattering length The consistency of this proposal is tested using a simple model
1. D i s c u s s i o n of a M o d e l a n d C o n c l u s i o n s The observed attraction in the s-wave, T = ], : plon-nucleon scattermg phase shift has so far proved rather hard to reconcile wlth the usual one or two pion approximataons of the Yukawa type meson theory (Edwards and Matthews x)) In this paper we propose a quahtative explanation of this effect on the basis of a strong coupling model for the nucleon-antlnucleon system. It is supposed that a nucleon and antinucleon interact so strongly at short distances that they form a bound state - - the plon. Such an idea was first proposed b y Fermi and Yang *) and has since been taken up b y a number of other people, (See, e.g. the paper b y Maki s) and references given there). But at this stage the idea does not necessarily go beyond the usual Yukawa theory, the nucleon-antinucleon scattering matrix must certainly contain a pole corresponding to the pion state, Independently of any model for their interaction Our proposal is, however, that the bas:c nucleon-antlnucleon mteraction should not be treated as having purely pseudoscalar properties, but must involve various types of exchange of lsobanc spin, ordinary spm and p a n t y This observatxon has two main consequences: (I) the nucleon-antlnucleon system m a y have bound states other than the isobaric triplet (T ---- 1) pseudoscalar pion state (though they need not necessarily be stable ones), (fi) in addition to the usual Yukawa mechamsm for p:on-nucleon anteraction, whach m the present model proceeds via annihilation and creation of nucleon-antinucleon pairs, the basic :nteraction now provades a further mechanism for plon nucleon interaction whlch proceeds via the direct mteractlon of a constituent nucleon or antinucleon in the paon with the external nucleon (fig 1) It as this direct process which, we propose, may lead to a short range pion-nucleon attraction of the observed type *t t On leave of absence from Nagoya Umversaty, Nagoya, J a p a n tt We find t h a t S A t a k a *) has independently p u t forward a similar proposal 14
PION-NUCLEON ATTRACTION AT SHORT DISTANCES
15
This hypothesis appears plausible on order of magnitude grounds on account of the considerable strength required of an interaction which can bind the pion: for this purpose it must have total energy (well depth) of order at least 2M, where M is the nucleon mass; and if we suppose its range to be about 1/M (in umts of length), the Born approximation s-wave scattering length for the direct pion-nucleon scattering process will be of order 2/~2M(1/M) 3 m 0.1/~-1, where is the plon rest-mass, winch is comparable with the observed scattering lengths. However, for matrix elements of the type of fig. lb) to exist for s-waves, the basic interaction must contain scalar or vector components since no spin or parity IS to be exchanged at this vertex Furthermore the observed scattering is strongly dependent on the relative isobaric spin states of meson and nucleon.
a) Yukawo type
~ Fig
b) direct
ptonstate ~ b a s ~ nu~-nuck~nInteract~n 1
Pzon-nucleon mteractlon
mechamsms.
Can an interaction winch leads to strongest binding for a pseudoscalar, T = l, plon satisfy these conditxons~ To answer tins question some sort of a model is required for the basic interaction. In sect. 2 we consider the general form of an Interaction of contact (6-functlon) type between identical fermions. On account of the exclusion principle tins interaction must contain mixtures of scalar, vector, e t c , components, and is now determined by 5 real parameters as opposed to 10 for a non-antisymmetrized interaction. With this degree of freedom still remaanlng it should not be difficult to satisfy the above conditions, and we proceed to estimate in more detail what range of parameters is hkely to lead to the observed type of plon-nucleon attraction, using a Born approximation argument In sect 3 we consider the bound-state problem in a two-body covarlant approximation winch is subject to the usual strong-couphng difficulties. For tins equation the pseudoscalar state appears the most easily bound, but the condition t h a t the lowest state should have T ---- 1 Imposes a stronger restriction on the form of the basic interaction. It turns out that the latter restriction does not contradict the condition that the direct interaction mechanism can lead to the observed type of plon-nucleon attraction, so that we have shown our main hypothesis to be consistent, within the hmltatlons of our approach.
1{~
S
DONIACH
AND
•
YAMADA
2. P r o p e r t i e s of t h e D i r e c t I n t e r a c t i o n
We start this section b y considering the general form of a contact interaction between identical fermions Consider matrix elements of the form
= l¢
m
(x),
(2.1)
where / is a coupling constant and the ~v~ are nucleon wave functions, spmor suffix ~, with spinor summation imphed ~ava is a two-particle spmor and isobaric s p a operator which can generally be written in the form d~ = ¼(3@31 • 32)~g+1(1-- 31 • 32)If
(2 2)
where d and 5f are linear combinations of the Lorentz-invanant 7-matrix products
S = IXP,
V
=
7"plOp2,
T = (½~[7'j,, 7.])I(½,[7/,, 7'.]) 2,
A ---- (~7'57~,)I(z757'/,) 2,
P = 751752
(2.3)
B y the exclusion principle the expression (2.1) must be antisymmetrlc under interchange of the wave functions ~v or ~. Therefore, since ¼(3+31 • 3z), ¼(1--31" 32) are isobaric triplet and singlet projection operators, off and ~9~ should be purely antlsymmetrlc and symmetric, respectively, against particle interchange. From the Fierz identities one finds five independent hnear combinations of the 7' products, three antlsymmetric, two symmetric, satisfying the above requirements.
off1=
I(S+P--T),
,5~1 = - ~ ( S + P + ½ T ) ,
off2 ---- ¼ ( V - - A ) ,
off3 = I ( A - - T + 2 P ) ,
,9", = I ( V + A - - § T - - 4 P ) .
(2 4)
Hence a general contact interactmn between identical fermlons must be of the form gaven b y eqs. (2 1), (2.2), where
off = aloffl+a,offg.+a3off3;
~9~ = S l , ~ l - - ~ - S 2 ~ a 2
(2 5)
and a and s are arbitrary real coefficients Using this contact model for the basic nucleon-antmucleon interaction we now consider m somewhat more detail the dependence on isobanc spin, etc., of the direct plon-nucleon interaction mechanism proposed in sect. 1 We confine ourselves to a consideration of the sign of the isobaric spin dependent part of the Born approximation direct scattering matrix element (fig lb), since it so happens that this can be evaluated without any detailed knowledge of the plon wave function It is hoped that tins calculation will indicate what sort of basic interaction Is needed to gave the observed type of isobaric spin dependence for the plon-nucleon attraction, since, apart from radiative corrections, which we assume not too important in the low energy hmlt, this sign would still be
PION-NUCLEON
ATTRACTION
AT S H O R T
DISTANCES
17
preserved in an exact treatment of the static source s-wave scattering problem, as in the Wentzel pair theory Of course the usual Yukawa process (fig la) also contributes to the s-wave phase shift. The magnitude of this contribution is rather dependent on the pion-nucleon coupling type, pseudoscalar or pseudovector, which, in the present model is deduced from the details of the pion wave function and the form of the basic interaction. In the pseudovector limit, the s-wave contribution is extremely small, while in the pseudoscalar case the predicted scattering is purely repulsive 1), so that we do not consider this contribution in further detail. In the Born approximation the S-matnx element for the direct pion nucleon scattering process reads, in the forward scattering hmit,
S~ = 2,/(V~,p~,af d4x(vr(k)[eA(x)y,p(x)[z~(k) )(q[(,~,(x)y,~(x)[q),
(2.6)
where In(k)) represents a bound pion state, four-momentum k, and Iq) a free nucleon state, momentum q The interaction is as defined above. Since the nucleon number is conserved at the direct interaction vertex, the transfer of spin, isobaric spin and parity from meson to nucleon must have a boson symmetry character Therefore, by a generahzed Furry's theorm (Pals and Jost 5)) the only components of ~ B ~ to contribute to this process are S,
P,
A,
v 1. T2V,
1"1. v2T.
(27)
For s-waves, only the forward scattering limit is of interest, so that only the scalar and vector terms in eq (2.7) will contribute to this process, and of these the scalar part is independent of the isobaric spin, leaving the vector part to carry the lsobarm spin dependence"
SB(~) = 2z/l(a2--s2) f d'x(n(k)lg(x)vyt, lg(k)) " (q(~)(x)ryaW(z)lq)
(2 8)
The form of the Isobaric spin dependence of (2 8) follows from lnvarlance in isobaric spin space: since [z~(k)) is a T ---- 1 state, (~[r]n) must be proportional to matrix elements of the isobaric spin 1 vector operator ¢o This can conveniently be represented by introducing an effective plon field 6~ (x), in terms of which (2 8) is represented as a first order S-matrix element of the effective symmetrlzed Interaction
~ e r r - Fo~p I ¢ . ( x ), ( 1 ~¢B(x)) j ~(x)vT~v/(x) •
(2 9)
It remains to determine the relation between the effective couphng constant F and the constants in eq. (2.8). Fortunately the magnitude of the matrix element (2 8) can be darectly determined from current conservation since for normalized plon states (~r[, f d% (~vay,~2)Ivr) = 1.
(2.10)
18
S DONIACH AND ]~ YAMADA
Thus, the required relation is 1 ~ 7 ~___ 2~t~--1/½(~2__$2).
(2.11)
In the static limit (2.9) gives rise to Born apprommation scattering lengths proportional to F#,X/+I --
for for
T:~ T =
I
(2.12)
m the two-plon nucleon isobaric spin states, Thus, in order tohave more attraction in the T = { than T = ~- states we require a ~ - - s z > 0.
(2.13)
Hence, the right sort of isobaric spin dependence can result, provided that the basic interaction has a non-zero vector component whose sign is determined through (2.13) Before going on to consider the bound state problem, it is of interest to test the effect of the condition (2.13) on the sign of the Born approxlmatmn matrix element for nucleon-nucleon scattering. It seems reasonable that this matrix element should be either repulsive or zero, since, on account of its strength, any attractive effects of the interaction would tend to de-saturate the internucleon force In the static nucleon-nucleon limit one has
(S,
V,
T,
A,
P)---> ( P I z, I l I 2, a l . a ~, a l . a ~, 0)
(214)
and since the interaction (21), (2.2) is antlsymmetrised through (24) only the J = o, T = 1 and J -- 1, T = 0 two-nucleon states will yield non zero matrix elements of 3~lnt in th~s limit:
(al+a 2 for J = 0 , < ~ i n t > N N ~- /tSI~-S 2
for
T = 1, J = I, T = 0
(2.15)
For (2.15) to be repulsive or zero, one has then the conditions al+a 2
>=0,
Sl+S 2__>0,
whmh do not conflict with (2.13) 3. N u c l e o n - A n t i n u c l e o n B o u n d S t a t e s In this section we try to answer the question whether an interaction which brads most strongly the psendoscalar, T = 1, nucleon-antlnucleon state can also have the type of spin and isobaric spin dependence, defined through condition (2 13), which could lead to the observed type of s-wave plon-nucleon attraction For this purpose we conmder the bound state solutions of a covariant
PION-NUCLEON ATTRACTION AT SHORT DISTANCES
19
two-body wave equation of the Bethe-Salpeter-Nambu type 1 ) ( ~1+ M
, v 2 , ~% 2 -w~
M ' ~Jt x
~ 1, x2) =
--3
fd 3dx4z((xlx x3x4) e(x3,
x4)
(3 1)
where the x carry both four-vector and spmor suffixes and rz acts to the right on antinucleon spmors (the suffix T denotes the transpose). Tins approach becomes rather doubtful under strong-couphng conditions, when the binding energies become of order of the two-particle rest mass, since, as w l l be seen below, some of the solutions have anomalous properties of the Klein-paradox type Since it would require a many-pair treatment to overcome these difficulties we can only regard the present work as gavmg some sort of qualitative indication as to the binding properties of the basic lnteractmn, and do not attempt any quantitative calculation of the bound state properties. To relate the kernel :Yd of (3 1) to the basic interaction introduced in sect. 2, eq. (3.1) can be derived in a two-body approximation as an equation of motion for the Feynman amplitude
~(xx, Xe) =.
(3 2)
For this purpose the wave equation for the ~ is deduced using (--~lnt) of (2 1) as a Lagrangian density, and the free field commutation relations are used (Makl 3)) However, the resulting two-body equation chverges on account of the contact character of ~¢t~=t, and to avoid this we Introduce a form factor which, in order to be able to solve explicitly for the bound states, we assume to be of separable form *. Introducing centre of mass and relative momenta through P = (P~+P2),
q=
1 -~(Pl--P2)
(3.3)
the two-body equation (3 1) now takes the form, in momentum representation, 1 1 [~'~,(~Pt,+qt,) - - , M ] [r T 2(½Pt,--qt,)+,M] T ( P , q)
= --2~ f dq'H(q)H(q')O~(P, q'), (3.4) where H (q) is a cut-off function, and since ~ is fully antlsymmetnzed through (2.2), (2 5), the spmor product can equally be represented as 0~v = 0~p~a ~8~ = --~p~* ~vp~.
(3.5)
We now define an external spmor wave function
A~B(P ) = f dq H(q)~p(P, q)
(3.6)
* Some care must be employed m the use of a separable form factor in order to preserve crossing symmetry I n t h e b o u n d s t a t e c a s e , t h e a n t m y m m e t r y r e q u i r e m e n t , < 0 [ N ( ~ v ~ ) ] z t ) = - - <0[N(-vmd)[~t ) is s t i l l s a t m f m d m t h e s e p a r a b l e e q u a t m n (3 4)
20
S
D O N I A C H AND E
YAMADA
which satisfies a matrix eigenvalue equation
A =/XeA,
(3.7)
where X is a spinor operator defined by HZ(q)
X = i f dq ()'~(½p~,+ql,)_,M)()'T,(½pt_q~,)+iM ) and the q, integration is taken around a Feynman contour Then if depends only on q2, X must generally have the covariant form X :
1 1 (Pv)'~ T 2 )T(P 2 )--~()'t,)'~* 1 1 T 2)I(P2) --~(P~,)'~,) 1 1 T2 1 T + [~(P~)'a)(P~)'~ )--,MPt,()'~,--)'~,2)+Ma]S(p2),
(3 8) H(q)
(3.9)
where T, I, S are real functions of Pz defined through
S
H2(q) × { 1
I~5~,~+ T P~,P,/P* =*
dq ((½P+q)*+M*) ((½P--q)2+M*)"
(3 10)
For a particular elgenvalue, (3.7) has the character of a Kemmer equation with non degenerate mass operator since ~1 {r. 1a - - y .aT 2 ~ j has the properties of the Kemmer-Duffin fla as most easily can be seen b y a charge conlugation of the equation with respect to the antiparticle 2, when ~t7~--7~ 1, 1 w2,) becomes of the well-known form-~W~, )'i,}" However, we fred it more convenient to use a direct product representation for the two-particle spinors, based on the set of 16 ),-matrices This has the advantage of being manifestly Lorentz covariant in the nucleon-antinucleon case: ( 0 1 N ( ~ q p ) l a ) transforms in the same w a y as a one-particle spinor operator. With this definition the mult~phcation rule becomes ()'~A)~p ----- ()'a)~,A,a;
()'w~,ZA)~ a = A~,()'a), a
(3.11)
and in the rest frame, P~ = (0, z/z), where /z is the bound state rest mass, (3 7) splits into a set of 2 × 2 submatrIces whose elgenfunctlons are linear combinations of the pairs t (1,)',),
(~',, r.,),
(rs,, rs,,),
()'5, rs,)
(3.12)
which transform respectively as scalar, vector, pseudovector and pseudoscalar states The interaction operator, d~aT~, is diagonal in this representation and in t I n a general f r a m e , w i t h c e n t r e of m a s s m o m e n t u m - v e c t o r Pl*, t h e m g e n f u n e t l o n s (3 12) u n d e r g o a t r a n s f o r m a t i o n of t h e t y p e
C~s, ~5~*) ~ (Ys,
*~'6Y~,PI,/~/-1:'*)
T h e five coefflcmnts t h e r e b y i n t r o d u c e d f o r m (in th~s e x a m p l e ) a n e l g e n v e c t o r for t h e fivedHnenslonal s u b m a t r l x d e c o m p o s i t i o n of t h e K e m m e r e q u a t i o n
PION-NUCLEON ATTRACTION AT SHORT DISTANCES
21
order to get some idea of the eigensolutions of (3 7) it is convement to consider cases where 0 projects onto only one part of a pair in (3.12). Eq (3.7) then reduces to a linear equation for this particular state, and the elgenvalue # satisfies the relation
/,
(3 13)
where < X ) a is the diagonal element of X of eq. (3.9) with respect to the state m question. < X ) -1 thus plays the role of an internal kinetic energy ior the bound state. The dependence of
--
(3 14)
~2+q~
?-o
A=2M
Fig 2 Sketch of t h e d e p e n d e n c e of dtagonal m a t r i x elements ( X ) A = ¼ Tr {~,Ax~ A} on t h e rest mass # Here,
=/'
is/
Iy,(S')
(v/ '
(],,,,(V')
(AI '
I~5~,(A' ) '
Iv. (P/ I:ys,(p' )
In tins simple case X can be evaluated analytically b y the Feynman parametnzation techmque. It may now be seen that many of the solutions have an anomalous character: the Internal kinetic energy may increase with decreasing binding energy (increasing rest mass) of the pair, or even become negative. This is the behaviour we attribute to the Klein paradox But since the state most easily bound, namely that with pseudoscalar wave furmtlon component ~'5, has a normal behavlour, it seems reasonable to draw quahtative conclusions from its properties We thus conclude that st is fairly easy to choose an interaction operator ~, which, provided st has non-zero projection onto the 7s
22
S DONIACH AND E YAMADA
wave function, will lead to a pseudoscalar state as the most easily bound state, in agreement with the conclusions of Fermi and Yang. However, we must now discuss the much stronger condition that the lowest bound state should have T = l, since no isobaric spin zero pion has been observed with comparable mass.
In the direct product representation the isobaric spin elgenfunctions are represented b y the three v, for triplet states and b y the unit m a t n x for the singlet state, so that the operator v 1 • v 2 has elgenvalues q-3, --1, for isobaric spin 0, 1, nucleon-antlnucleon states. Thus the effective nucleon-antmucleon interaction operator becomes [½(d÷S#) d~= ( ½ ( 3 d _ S a )
for for
T = 1, T=0,
(3.15)
where d , 6 a are defined in (2.5). A possible reason for reduced binding in the T = 0 state could therefore arise out of the change in sign of the symmetric part of the interaction. On evaluation one finds (~¢)e ---- ¼Tr(75d75 ) = --a2--4a3;
(d~)p = sx--4s ~.
(3.16)
Since the mixing in of the 764 component in (3.9) is proportional to (it~M) we neglect this for the T = 1 state and find as a condition for binding to be possible ( ~ ) p + ( 5 ~ ' ) p > 0.
(3 17)
A similar approximation for the T = 0 solution suggests a condition 3(d)p--(SP)p < (~¢)p+(~)p
(3.18)
for the T = 0 coupling to be weaker than that in the T = 1 state Combining these results in one condition, using (3.16), we have Sl--4s ~ > la2q-4a31 ~_ 0.
(3 19)
Thus it lS possible, b y suitable choice of parameters, e.g. s 2 < O together with In21 < Is~l etc., to satisfy both condition (3.19) for T = 1 lowest state, and (2 13) for T ---- ½ plon-nucleon attraction. We therefore conclude that, within the hmitatlons of the model, these two conchtions are not in chrect contradmtion with each other. E. Yamada wishes to thank Professor Frohhch for his hospitality at the Department of Theoretical Physics. References 1) 2) 3) 4) 5)
S F E d w a r d s a n d P T M a t t h e w s , Phil M a g 4 2 (1957) 1 E F e r m i a n d C N Y a n g , P h y s R e v 76 ( 1 9 4 9 ) 1 7 3 9 Z Makl, P r o g T h e o r P h y s 16 ( 1 9 5 6 ) 6 6 7 Y A t a k a , P r o g T h e o r P h y s 22 (1959) 321 A P a l s a n d R J o s t , P h y s R e v 82 (1952) 840
7.C
]
Nuclear Phvs,cs 18 (1960) 14--22, ~ ) North-Holland PubhsMng Co, Amsterdam Not to be reproduced by photoprmt or nncro'fllm without written permission from the pubhsher
PION-NUCLEON
ATTRACTION S DONIACH and E
AT SHORT DISTANCES YAMADA t
Department o/ Theoret,cal Physics, Umvers,ty o/ L,verpool Received 10 March 1960
A b s t r a c t : I t is suggested t h a t a t h e o r y which t r e a t s the ~ - m e s o n as a strongly b o u n d nucleona n t m u c l e o n s t a t e could promde a n explanation for t h e observed : s o b a n c - s p m dependence of t h e meson-nucleon s-wave scattering length The consistency of this proposal is tested using a simple model
1. D i s c u s s i o n of a M o d e l a n d C o n c l u s i o n s The observed attraction in the s-wave, T = ], : plon-nucleon scattermg phase shift has so far proved rather hard to reconcile wlth the usual one or two pion approximataons of the Yukawa type meson theory (Edwards and Matthews x)) In this paper we propose a quahtative explanation of this effect on the basis of a strong coupling model for the nucleon-antlnucleon system. It is supposed that a nucleon and antinucleon interact so strongly at short distances that they form a bound state - - the plon. Such an idea was first proposed b y Fermi and Yang *) and has since been taken up b y a number of other people, (See, e.g. the paper b y Maki s) and references given there). But at this stage the idea does not necessarily go beyond the usual Yukawa theory, the nucleon-antinucleon scattering matrix must certainly contain a pole corresponding to the pion state, Independently of any model for their interaction Our proposal is, however, that the bas:c nucleon-antlnucleon mteraction should not be treated as having purely pseudoscalar properties, but must involve various types of exchange of lsobanc spin, ordinary spm and p a n t y This observatxon has two main consequences: (I) the nucleon-antlnucleon system m a y have bound states other than the isobaric triplet (T ---- 1) pseudoscalar pion state (though they need not necessarily be stable ones), (fi) in addition to the usual Yukawa mechamsm for p:on-nucleon anteraction, whach m the present model proceeds via annihilation and creation of nucleon-antinucleon pairs, the basic :nteraction now provades a further mechanism for plon nucleon interaction whlch proceeds via the direct mteractlon of a constituent nucleon or antinucleon in the paon with the external nucleon (fig 1) It as this direct process which, we propose, may lead to a short range pion-nucleon attraction of the observed type *t t On leave of absence from Nagoya Umversaty, Nagoya, J a p a n tt We find t h a t S A t a k a *) has independently p u t forward a similar proposal 14
PION-NUCLEON ATTRACTION AT SHORT DISTANCES
15
This hypothesis appears plausible on order of magnitude grounds on account of the considerable strength required of an interaction which can bind the pion: for this purpose it must have total energy (well depth) of order at least 2M, where M is the nucleon mass; and if we suppose its range to be about 1/M (in umts of length), the Born approximation s-wave scattering length for the direct pion-nucleon scattering process will be of order 2/~2M(1/M) 3 m 0.1/~-1, where is the plon rest-mass, winch is comparable with the observed scattering lengths. However, for matrix elements of the type of fig. lb) to exist for s-waves, the basic interaction must contain scalar or vector components since no spin or parity IS to be exchanged at this vertex Furthermore the observed scattering is strongly dependent on the relative isobaric spin states of meson and nucleon.
a) Yukawo type
~ Fig
b) direct
ptonstate ~ b a s ~ nu~-nuck~nInteract~n 1
Pzon-nucleon mteractlon
mechamsms.
Can an interaction winch leads to strongest binding for a pseudoscalar, T = l, plon satisfy these conditxons~ To answer tins question some sort of a model is required for the basic interaction. In sect. 2 we consider the general form of an Interaction of contact (6-functlon) type between identical fermions. On account of the exclusion principle tins interaction must contain mixtures of scalar, vector, e t c , components, and is now determined by 5 real parameters as opposed to 10 for a non-antisymmetrized interaction. With this degree of freedom still remaanlng it should not be difficult to satisfy the above conditions, and we proceed to estimate in more detail what range of parameters is hkely to lead to the observed type of plon-nucleon attraction, using a Born approximation argument In sect 3 we consider the bound-state problem in a two-body covarlant approximation winch is subject to the usual strong-couphng difficulties. For tins equation the pseudoscalar state appears the most easily bound, but the condition t h a t the lowest state should have T ---- 1 Imposes a stronger restriction on the form of the basic interaction. It turns out that the latter restriction does not contradict the condition that the direct interaction mechanism can lead to the observed type of plon-nucleon attraction, so that we have shown our main hypothesis to be consistent, within the hmltatlons of our approach.
1{~
S
DONIACH
AND
•
YAMADA
2. P r o p e r t i e s of t h e D i r e c t I n t e r a c t i o n
We start this section b y considering the general form of a contact interaction between identical fermions Consider matrix elements of the form
= l¢
m
(x),
(2.1)
where / is a coupling constant and the ~v~ are nucleon wave functions, spmor suffix ~, with spinor summation imphed ~ava is a two-particle spmor and isobaric s p a operator which can generally be written in the form d~ = ¼(3@31 • 32)~g+1(1-- 31 • 32)If
(2 2)
where d and 5f are linear combinations of the Lorentz-invanant 7-matrix products
S = IXP,
V
=
7"plOp2,
T = (½~[7'j,, 7.])I(½,[7/,, 7'.]) 2,
A ---- (~7'57~,)I(z757'/,) 2,
P = 751752
(2.3)
B y the exclusion principle the expression (2.1) must be antisymmetrlc under interchange of the wave functions ~v or ~. Therefore, since ¼(3+31 • 3z), ¼(1--31" 32) are isobaric triplet and singlet projection operators, off and ~9~ should be purely antlsymmetrlc and symmetric, respectively, against particle interchange. From the Fierz identities one finds five independent hnear combinations of the 7' products, three antlsymmetric, two symmetric, satisfying the above requirements.
off1=
I(S+P--T),
,5~1 = - ~ ( S + P + ½ T ) ,
off2 ---- ¼ ( V - - A ) ,
off3 = I ( A - - T + 2 P ) ,
,9", = I ( V + A - - § T - - 4 P ) .
(2 4)
Hence a general contact interactmn between identical fermlons must be of the form gaven b y eqs. (2 1), (2.2), where
off = aloffl+a,offg.+a3off3;
~9~ = S l , ~ l - - ~ - S 2 ~ a 2
(2 5)
and a and s are arbitrary real coefficients Using this contact model for the basic nucleon-antmucleon interaction we now consider m somewhat more detail the dependence on isobanc spin, etc., of the direct plon-nucleon interaction mechanism proposed in sect. 1 We confine ourselves to a consideration of the sign of the isobaric spin dependent part of the Born approximation direct scattering matrix element (fig lb), since it so happens that this can be evaluated without any detailed knowledge of the plon wave function It is hoped that tins calculation will indicate what sort of basic interaction Is needed to gave the observed type of isobaric spin dependence for the plon-nucleon attraction, since, apart from radiative corrections, which we assume not too important in the low energy hmlt, this sign would still be
PION-NUCLEON
ATTRACTION
AT S H O R T
DISTANCES
17
preserved in an exact treatment of the static source s-wave scattering problem, as in the Wentzel pair theory Of course the usual Yukawa process (fig la) also contributes to the s-wave phase shift. The magnitude of this contribution is rather dependent on the pion-nucleon coupling type, pseudoscalar or pseudovector, which, in the present model is deduced from the details of the pion wave function and the form of the basic interaction. In the pseudovector limit, the s-wave contribution is extremely small, while in the pseudoscalar case the predicted scattering is purely repulsive 1), so that we do not consider this contribution in further detail. In the Born approximation the S-matnx element for the direct pion nucleon scattering process reads, in the forward scattering hmit,
S~ = 2,/(V~,p~,af d4x(vr(k)[eA(x)y,p(x)[z~(k) )(q[(,~,(x)y,~(x)[q),
(2.6)
where In(k)) represents a bound pion state, four-momentum k, and Iq) a free nucleon state, momentum q The interaction is as defined above. Since the nucleon number is conserved at the direct interaction vertex, the transfer of spin, isobaric spin and parity from meson to nucleon must have a boson symmetry character Therefore, by a generahzed Furry's theorm (Pals and Jost 5)) the only components of ~ B ~ to contribute to this process are S,
P,
A,
v 1. T2V,
1"1. v2T.
(27)
For s-waves, only the forward scattering limit is of interest, so that only the scalar and vector terms in eq (2.7) will contribute to this process, and of these the scalar part is independent of the isobaric spin, leaving the vector part to carry the lsobarm spin dependence"
SB(~) = 2z/l(a2--s2) f d'x(n(k)lg(x)vyt, lg(k)) " (q(~)(x)ryaW(z)lq)
(2 8)
The form of the Isobaric spin dependence of (2 8) follows from lnvarlance in isobaric spin space: since [z~(k)) is a T ---- 1 state, (~[r]n) must be proportional to matrix elements of the isobaric spin 1 vector operator ¢o This can conveniently be represented by introducing an effective plon field 6~ (x), in terms of which (2 8) is represented as a first order S-matrix element of the effective symmetrlzed Interaction
~ e r r - Fo~p I ¢ . ( x ), ( 1 ~¢B(x)) j ~(x)vT~v/(x) •
(2 9)
It remains to determine the relation between the effective couphng constant F and the constants in eq. (2.8). Fortunately the magnitude of the matrix element (2 8) can be darectly determined from current conservation since for normalized plon states (~r[, f d% (~vay,~2)Ivr) = 1.
(2.10)
18
S DONIACH AND ]~ YAMADA
Thus, the required relation is 1 ~ 7 ~___ 2~t~--1/½(~2__$2).
(2.11)
In the static limit (2.9) gives rise to Born apprommation scattering lengths proportional to F#,X/+I --
for for
T:~ T =
I
(2.12)
m the two-plon nucleon isobaric spin states, Thus, in order tohave more attraction in the T = { than T = ~- states we require a ~ - - s z > 0.
(2.13)
Hence, the right sort of isobaric spin dependence can result, provided that the basic interaction has a non-zero vector component whose sign is determined through (2.13) Before going on to consider the bound state problem, it is of interest to test the effect of the condition (2.13) on the sign of the Born approxlmatmn matrix element for nucleon-nucleon scattering. It seems reasonable that this matrix element should be either repulsive or zero, since, on account of its strength, any attractive effects of the interaction would tend to de-saturate the internucleon force In the static nucleon-nucleon limit one has
(S,
V,
T,
A,
P)---> ( P I z, I l I 2, a l . a ~, a l . a ~, 0)
(214)
and since the interaction (21), (2.2) is antlsymmetrised through (24) only the J = o, T = 1 and J -- 1, T = 0 two-nucleon states will yield non zero matrix elements of 3~lnt in th~s limit:
(al+a 2 for J = 0 , < ~ i n t > N N ~- /tSI~-S 2
for
T = 1, J = I, T = 0
(2.15)
For (2.15) to be repulsive or zero, one has then the conditions al+a 2
>=0,
Sl+S 2__>0,
whmh do not conflict with (2.13) 3. N u c l e o n - A n t i n u c l e o n B o u n d S t a t e s In this section we try to answer the question whether an interaction which brads most strongly the psendoscalar, T = 1, nucleon-antlnucleon state can also have the type of spin and isobaric spin dependence, defined through condition (2 13), which could lead to the observed type of s-wave plon-nucleon attraction For this purpose we conmder the bound state solutions of a covariant
PION-NUCLEON ATTRACTION AT SHORT DISTANCES
19
two-body wave equation of the Bethe-Salpeter-Nambu type 1 ) ( ~1+ M
, v 2 , ~% 2 -w~
M ' ~Jt x
~ 1, x2) =
--3
fd 3dx4z((xlx x3x4) e(x3,
x4)
(3 1)
where the x carry both four-vector and spmor suffixes and rz acts to the right on antinucleon spmors (the suffix T denotes the transpose). Tins approach becomes rather doubtful under strong-couphng conditions, when the binding energies become of order of the two-particle rest mass, since, as w l l be seen below, some of the solutions have anomalous properties of the Klein-paradox type Since it would require a many-pair treatment to overcome these difficulties we can only regard the present work as gavmg some sort of qualitative indication as to the binding properties of the basic lnteractmn, and do not attempt any quantitative calculation of the bound state properties. To relate the kernel :Yd of (3 1) to the basic interaction introduced in sect. 2, eq. (3.1) can be derived in a two-body approximation as an equation of motion for the Feynman amplitude
~(xx, Xe) =
(3 2)
For this purpose the wave equation for the ~ is deduced using (--~lnt) of (2 1) as a Lagrangian density, and the free field commutation relations are used (Makl 3)) However, the resulting two-body equation chverges on account of the contact character of ~¢t~=t, and to avoid this we Introduce a form factor which, in order to be able to solve explicitly for the bound states, we assume to be of separable form *. Introducing centre of mass and relative momenta through P = (P~+P2),
q=
1 -~(Pl--P2)
(3.3)
the two-body equation (3 1) now takes the form, in momentum representation, 1 1 [~'~,(~Pt,+qt,) - - , M ] [r T 2(½Pt,--qt,)+,M] T ( P , q)
= --2~ f dq'H(q)H(q')O~(P, q'), (3.4) where H (q) is a cut-off function, and since ~ is fully antlsymmetnzed through (2.2), (2 5), the spmor product can equally be represented as 0~v = 0~p~a ~8~ = --~p~* ~vp~.
(3.5)
We now define an external spmor wave function
A~B(P ) = f dq H(q)~p(P, q)
(3.6)
* Some care must be employed m the use of a separable form factor in order to preserve crossing symmetry I n t h e b o u n d s t a t e c a s e , t h e a n t m y m m e t r y r e q u i r e m e n t , < 0 [ N ( ~ v ~ ) ] z t ) = - - <0[N(-vmd)[~t ) is s t i l l s a t m f m d m t h e s e p a r a b l e e q u a t m n (3 4)
20
S
D O N I A C H AND E
YAMADA
which satisfies a matrix eigenvalue equation
A =/XeA,
(3.7)
where X is a spinor operator defined by HZ(q)
X = i f dq ()'~(½p~,+ql,)_,M)()'T,(½pt_q~,)+iM ) and the q, integration is taken around a Feynman contour Then if depends only on q2, X must generally have the covariant form X :
1 1 (Pv)'~ T 2 )T(P 2 )--~()'t,)'~* 1 1 T 2)I(P2) --~(P~,)'~,) 1 1 T2 1 T + [~(P~)'a)(P~)'~ )--,MPt,()'~,--)'~,2)+Ma]S(p2),
(3 8) H(q)
(3.9)
where T, I, S are real functions of Pz defined through
S
H2(q) × { 1
I~5~,~+ T P~,P,/P* =*
dq ((½P+q)*+M*) ((½P--q)2+M*)"
(3 10)
For a particular elgenvalue, (3.7) has the character of a Kemmer equation with non degenerate mass operator since ~1 {r. 1a - - y .aT 2 ~ j has the properties of the Kemmer-Duffin fla as most easily can be seen b y a charge conlugation of the equation with respect to the antiparticle 2, when ~t7~--7~ 1, 1 w2,) becomes of the well-known form-~W~, )'i,}" However, we fred it more convenient to use a direct product representation for the two-particle spinors, based on the set of 16 ),-matrices This has the advantage of being manifestly Lorentz covariant in the nucleon-antinucleon case: ( 0 1 N ( ~ q p ) l a ) transforms in the same w a y as a one-particle spinor operator. With this definition the mult~phcation rule becomes ()'~A)~p ----- ()'a)~,A,a;
()'w~,ZA)~ a = A~,()'a), a
(3.11)
and in the rest frame, P~ = (0, z/z), where /z is the bound state rest mass, (3 7) splits into a set of 2 × 2 submatrIces whose elgenfunctlons are linear combinations of the pairs t (1,)',),
(~',, r.,),
(rs,, rs,,),
()'5, rs,)
(3.12)
which transform respectively as scalar, vector, pseudovector and pseudoscalar states The interaction operator, d~aT~, is diagonal in this representation and in t I n a general f r a m e , w i t h c e n t r e of m a s s m o m e n t u m - v e c t o r Pl*, t h e m g e n f u n e t l o n s (3 12) u n d e r g o a t r a n s f o r m a t i o n of t h e t y p e
C~s, ~5~*) ~ (Ys,
*~'6Y~,PI,/~/-1:'*)
T h e five coefflcmnts t h e r e b y i n t r o d u c e d f o r m (in th~s e x a m p l e ) a n e l g e n v e c t o r for t h e fivedHnenslonal s u b m a t r l x d e c o m p o s i t i o n of t h e K e m m e r e q u a t i o n
PION-NUCLEON ATTRACTION AT SHORT DISTANCES
21
order to get some idea of the eigensolutions of (3 7) it is convement to consider cases where 0 projects onto only one part of a pair in (3.12). Eq (3.7) then reduces to a linear equation for this particular state, and the elgenvalue # satisfies the relation
/,
(3 13)
where < X ) a is the diagonal element of X of eq. (3.9) with respect to the state m question. < X ) -1 thus plays the role of an internal kinetic energy ior the bound state. The dependence of
--
(3 14)
~2+q~
?-o
A=2M
Fig 2 Sketch of t h e d e p e n d e n c e of dtagonal m a t r i x elements ( X ) A = ¼ Tr {~,Ax~ A} on t h e rest mass # Here,
=/'
is/
Iy,(S')
(v/ '
(],,,,(V')
(AI '
I~5~,(A' ) '
Iv. (P/ I:ys,(p' )
In tins simple case X can be evaluated analytically b y the Feynman parametnzation techmque. It may now be seen that many of the solutions have an anomalous character: the Internal kinetic energy may increase with decreasing binding energy (increasing rest mass) of the pair, or even become negative. This is the behaviour we attribute to the Klein paradox But since the state most easily bound, namely that with pseudoscalar wave furmtlon component ~'5, has a normal behavlour, it seems reasonable to draw quahtative conclusions from its properties We thus conclude that st is fairly easy to choose an interaction operator ~, which, provided st has non-zero projection onto the 7s
22
S DONIACH AND E YAMADA
wave function, will lead to a pseudoscalar state as the most easily bound state, in agreement with the conclusions of Fermi and Yang. However, we must now discuss the much stronger condition that the lowest bound state should have T = l, since no isobaric spin zero pion has been observed with comparable mass.
In the direct product representation the isobaric spin elgenfunctions are represented b y the three v, for triplet states and b y the unit m a t n x for the singlet state, so that the operator v 1 • v 2 has elgenvalues q-3, --1, for isobaric spin 0, 1, nucleon-antlnucleon states. Thus the effective nucleon-antmucleon interaction operator becomes [½(d÷S#) d~= ( ½ ( 3 d _ S a )
for for
T = 1, T=0,
(3.15)
where d , 6 a are defined in (2.5). A possible reason for reduced binding in the T = 0 state could therefore arise out of the change in sign of the symmetric part of the interaction. On evaluation one finds (~¢)e ---- ¼Tr(75d75 ) = --a2--4a3;
(d~)p = sx--4s ~.
(3.16)
Since the mixing in of the 764 component in (3.9) is proportional to (it~M) we neglect this for the T = 1 state and find as a condition for binding to be possible ( ~ ) p + ( 5 ~ ' ) p > 0.
(3 17)
A similar approximation for the T = 0 solution suggests a condition 3(d)p--(SP)p < (~¢)p+(~)p
(3.18)
for the T = 0 coupling to be weaker than that in the T = 1 state Combining these results in one condition, using (3.16), we have Sl--4s ~ > la2q-4a31 ~_ 0.
(3 19)
Thus it lS possible, b y suitable choice of parameters, e.g. s 2 < O together with In21 < Is~l etc., to satisfy both condition (3.19) for T = 1 lowest state, and (2 13) for T ---- ½ plon-nucleon attraction. We therefore conclude that, within the hmitatlons of the model, these two conchtions are not in chrect contradmtion with each other. E. Yamada wishes to thank Professor Frohhch for his hospitality at the Department of Theoretical Physics. References 1) 2) 3) 4) 5)
S F E d w a r d s a n d P T M a t t h e w s , Phil M a g 4 2 (1957) 1 E F e r m i a n d C N Y a n g , P h y s R e v 76 ( 1 9 4 9 ) 1 7 3 9 Z Makl, P r o g T h e o r P h y s 16 ( 1 9 5 6 ) 6 6 7 Y A t a k a , P r o g T h e o r P h y s 22 (1959) 321 A P a l s a n d R J o s t , P h y s R e v 82 (1952) 840