A quasi-relativistic equation for a fermion-boson system

Physics Letters B 293 (1992) 265-269 North-Holland

PHYSICS LETTERS B

A quasi-relativistic equation for a fermion-boson system C.Y. Cheung and S.P. Li Institute of Physics, Academia Sinica, Taipei 11529, Taiwan, R OC Received 1 July 1992

We propose a new quasi-relativistic two-body equation for a system consisting of a spin-½ fermion and a spinless bosom It involves coupling a Dirac equation to a Duffin-Kemmer-Petiau equation. A scheme for solving this two-body equation is presented.

In a recent paper [ 1 ], we studied the problem of solving a quasi-relativistic equation for a system of two Dirac particles. We showed that, for a rather general class of potentials, the problem could be reduced to that of solving a pair of coupled SchrSdinger-type equations. Our scheme permits transparent and general discussions of the mixing of various partial wave channels (including isospin) and the question of the Klein paradox. In the present work, we extend our study to systems consisting of a spin-½ fermion and a spinless boson. There has been relatively few studies of relativistic dynamics in fermion-boson systems in the literature, which is unfortunate since useful applications can be found in various baryon-meson or lepton-meson systems. A naive approach to the relativistic fermion-boson problem would be to couple the Klein-Gordon equation to the Dirac equation. However, one is immediately confronted with the problem that the Dirac equation is linear in the energy operator whereas the Klein-Gordon equation is quadratic. This difference in dimensions is also a source of difficulties in the Bethe-Salpeter approach for a fermion-boson system [ 2 ]. A possible solution to this impediment is to use a linear equation for the boson. The idea of linearizing the Klein-Gordon equation is in fact not new. It was first done in the 1930's [3-8], and the resulting equation is commonly referred to as the DuffinKemmer-Petiau (DKP) equation. The DKP equation is manifestly covariant and formally looks identical to the Dirac equation. The free DKP equation is given by (h = c = 1 )

(1)

(flup~'_rn )~u=O ,

where q/is the boson wave function, m is the mass, and the ffs are matrices which satisfy the following commutation relations: (2) There are 126 linearly independent elements for this algebra compared to only 16 in the Dirac case. The rank of the matrices turns out to be 16 and the corresponding representation can be decomposed into three irreducible ones, with dimensions one, five, and ten. The one-dimensional representation is trivial, while the five- and ten-dimensional representations describe scalar and vector particles respectively. Explicitly, the//-matrices in the five-dimensional irreducible representation are given by

ii) /000000) 0

po=

0 0 0 0 0 0 0 0 0

0 0 0 0 0

0

0

0

,

"

f l l = O 0 0 0 1 0 ,

~00 0 i

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

0 0

(3)

0 0

265

Volume 293, number 3,4

132=

°°i) (i ° °°i) 0 0 0 0 0 0 i

(i °

PHYSICS LETTERS B

0 0 0 0

0

B3=

0

0 0

0 0 0

0 0 i

(3 cont'd)

The potentials S' and V' give rise to nonlocal terms in the corresponding Klein-Gordon equation, even if they are local themselves. Hence if only local potentials are to be considered in the corresponding Klein-Gordon equation, one should only allow S and Vin eq. (7). The equation we propose to use for a fermion-boson system takes the form [Hf(p) + H b ( - - p ) +H~t ] ~ = E ~ ,

With these matrices, and writing the DKP wave function in the form

(9)

in the center of mass frame, where ~Vis the two-body wave function,

Hf(p)=yo(y'p+M)

(10)

and

01 ~=

29 October 1992

02 ,

(4)

Hb(p) = (flofl_flflo).p+ flom

( 11 )

3

one can readily verify from eq. ( 1 ) that O satisfies the usual Klein-Gordon equation for a free particle, and

O"= ±0~0.

(5)

m

Eq. (5) shows that although q/has five components, only one (¢~) is independent, as expected. In the Klein-Gordon equation, tile particle interacts with external fields via the familiar scalar (s) and vector ( vu) potentials: [ (pg --Yu)2+ (m+s)2]O=O,

(6)

where 0 is the one-dimensional Klein-Gordon wave function. In the DKP formulation, however, there are more possibilities:

[flU(pu-Vu-~V'u)+m+~S+S']~,=O

,

(7)

where (S, V) and (S', V') are two possible sets of scalar and vector potentials, and N is a projection operator which projects out the last (fifth) component of ~u. In the above equation we have left out two possible tensor potentials, due to the fact they give rise to noncausal effects [8]. When eq. (7) is converted back to an equivalent Klein-Gordon equation, one finds that one can recover eq. (6) if

S=2s+s2/m, V=v, 266

V'=0.

are the free fermion and boson hamiltonians respectively, Hint is the interaction hamiltonian, M (m) is the mass of the fermion (boson), and p is the momentum operator for the fermion in the center of mass frame. Due to the particular commutation relations, eq. (2), for the fl-matrices of the DKP equations, the bosonic hamiltonian is not obtained by simply multiplying eq. (1) with flo [see eq. (11 ) ]. Consequently, the introduction of two-body interaction terms in the proposed equation (9) is not as obvious as in the case of two Dirac particles [ 1 ]. With potentials S and V only, the one-body interaction hamiltonian in the DKP formulation is given by hint = V ° - V" (flo fl__ flflo ) _{.flo p s

+ i [0p l n ( m + ~ S ) ] (flpflO_gpO) i + 2(m+~S)

(8)

(12)

where

o~u~=Ou V~-O~V u .

(13)

For the two-body fermion-boson case, the source of the potentials in the above expression is the fermion. Therefore, in analogy with the two-fermion case, we make the following substitution in eq. (12):

Vu~),oYuV,

S'=0;

~°(flufl°flp-gU°flP)'

such that

S~yoS,

(14)

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PHYSICS LETTERS B

Hi.t = V - V~,°r • (/~o#_ #//0) +/~o ~ @ o

body wave function

+ i [0p I n ( m + NS7 °) ] (flpflo _gpO) i + 2 ( m + ~ S y O ) ~o(flufl°flP-gU°flP).

29 October 1992

~-Iljm, w e

have (i = 1, 2, 3 ) (19a)

Fi =fi( r)(gljm

(15)

and

Gi=gi(r)(9~,jm , This completes the specification of the proposed quasi-relativistic two-body equation (9) with scalar and vector interactions. In the following we present a scheme for solving the proposed fermion-boson equation. We first notice that the wave functions for the fermion and the boson are four- and five-component column matrices respectively. Therefore, analogous to the approach taken in ref. [ 1 ], we write the composite fermionboson wave function Tas a 4 × 5 matrix, with the column (row) index corresponding to the fermionic (bosonic) component. Thus, for instance, the free two-body wave function is given by

where (90,~is the angular momentum part of the wave function, with l=j+ ½ and I'=j-T- ½. The upper and lower angular wave functions, (90,, and (gtvm, are related by

a'p x

i-rn

1 exp(ip.r),

which is a property due to the Dirac equation. Substituting eq. ( 18 ) into eq. ( 17 ), we get a set of six differential equations for the unknown functions F and G:

(E-M-Vc)F1 = - (g.p)Gt - (rn+S)F3 + VZF2,

= (a-p) VG2 -VF~ +

where Z is the two-component Pauli spinor, E~ = (p2+M2)~/2, and E2= ~2+m2)~/2. For the sake of demonstration, we shall in the following consider the case with scalar (S) and Coulomb interactions ( Vu= Vcruo ) only. The equation to be solved is then given by

vvc m

= - (~.p)F~ - ( m - S ) G 3 + VZG2,

VVc

~p= iFl iG1

[(E+M- Vc)2-m(m-S) '

(18)

where the F ' s and G's are two-component functions of the relative coordinate r and Pauli spinors of the fermion. Parity conservation implies that there is no partial wave mixing, so that the orbital angular momentum (l) as well as the total angular momentum (3.) are good quantum numbers. Thus for the two-

(21d)

m

G3,

(21e) (21f)

After some algebra, we obtain two coupled first-order differential equations involving F3 and G3 only, viz.,

[(E-M-

G3

(21c)

( E + M - Vc) VG2

( E + M - Vc)G3 = (a-p)F2 - m G ~ . (17)

(21b)

(E+M-Vc)G~

and we shall assume that S and Vc are functions of the relative distance r only. In this relatively simple case, the wave function ~ c a n be written as VF2 VGz

F3,

Vc)F3 = (~'p)G2 - m F l ,

= ( a ' p ) VFz-VG~ +

E~P=(Hr(p) + l i b ( - - p ) + Vc + [3°~S7 °

i ) 2(m+-~$7o ) ~o([3u[3°]3P-gU°[3 p) ~P,

(E-M-

(21a)

Vc) VF2

(16)

+Mj

(20)

(glVm =O" r (gljrn ,

(E-M-i-m

(19b)

V c ) X - m ( m + S) ]F3

= 2 ( E - Vc)(a'p)G3 - (~'p Vc)G3,

(22a)

]G3

=2(E-Vc)(~r'p)F3-(a'p

Vc)F3.

(22b)

From eq. ( 19 ) and eq. (20), one can deduce the following relations: a ' p Gi( r ) =(90,. Dogi( r )

~'p F~(r) = (gZ'jmD 0 f ( r ) ,

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Volume 293, n u m b e r 3,4

PHYSICS LETTERS B

where Do= -i(d/dr+ko/r), with ko= - I and l+ 1 for j = l+ ½and j = l - ½respectively. With the help of these relations, the angular momentum components can be separated out in eqs. (22); the result is

[ ( E - m - Vc)2-m(m+ S) ]f3 = 2 ( E - Vc)

dVc

Dog3 -

-~r

g3,

(24a)

the upper (F3) and lower (G 3 ) components of a free fermion. Using eq. (27) in eq. (22a), we obtain [ ( E ' - Vc)2-p2-m(m+S)+O(l/M) If3 = 0 , (28) which is the usual Klein-Gordon equation with correction terms resulting from the recoil of the fermion. The correction terms up to order 1/M are 1

[ ( E + M - Vc)2-m(m+S) ]g3 dVc_ = 2 ( E - V c ) DrJ3- --~r J3 •

29 October 1992

O(1/M) = ~ [ - (E'- Vc)p 2 (24b)

+ ½(pZVc) + (pVc)"p].

(29)

Note that eqs. (24) can be converted into two second-order differential equations each involving f3 or g3 only. The other four components can be expressed in terms off3 and g3 through the following relations:

If instead the boson is heavy, i.e., m>> (M, S, Vc), we write p2 E~-m+ ~m +E'.... (30)

f~ =

E-M-m Vcf3+ ---~D°g3'

(25a)

Then from eq. (22b),

gl =

E + Mm- Vc g3 + ~ 1 Dlfl~

(25b)

o"p

A f2= m' g2-

1( E'-Vc G3 "~ ; 1+ --m

g3 • m

VCF3,

(25c) (25d)

Vc) 2. Substituting eq. (31) into eq. (22a) gives

p2

(26)

where E' is the system's energy after separating out the kinematic part of the fermion's energy. Then we have, from eq. (22b),

( (E'-M-

½S- V¢)y-p 2+ Y [tr-p( ½S- Vc) ] tr.p

+O(1/m))F3=O ,

(~'p Vc) F3 ,

1

O( l/m)= m[((tr'p Vc) +2 E ' - Vc tr.p(½S- Vc)) er.p ] Y - 2 ( E ' - Vc)p2 + ½(p2Vc) + (PVc)"P 2y

o.

z o.) (33)

(27)

which is the correct nonrelativistic relation between 268

(32)

where the correction terms are given by

+ ½y[p2+ ( E ' - M - Vc)2] 1 . G3 ~ ~ - - ~ ' p - ~

(31)

2my where y=-E'+M+½S-Vc and z--p2+(E'+ M -

Thus in this simple example, the problem is reduced to solving two coupled first-order differential equations, which is equivalent to two decoupled secondorder ones. For the sake of calculating nonrelativistic corrections, and also for checking purposes, it is useful to examine the nonrelativistic limits of eqs. (22), or equivalently eq. (24). In the case when the fermion is heavy, i.e., M>> (m, Vc, S), we approximate the total energy E as

E,-,M+ ~-~ +E'+ ....

1 z) 2my (°"p)V3

If one ignores the O ( 1 / m ) correction terms due to the recoil of the boson, it is easy to verify that eqs.

Volume 293, number 3,4

PHYSICS LETTERS B

(31), (32) are equivalent to the following Dirac equation with external scalar and Coulomb potentials:

(puYU + yo Vc - M -

½S)~u=0,

(34)

where the factor of ½ associated with S is due to the projection operator ~ in eq. (17). Finally we would like to point out an interesting observation, when only the Coulomb potential (Vc) is present. One notices that an important feature of the D K P equation is that it is manifestly covariant. If one does not insist on this requirement, then there exist other methods of linearizing the Klein-Gordon equation. A naive way to do so is by taking the square root of the Klein-Gordon equation: 2 O=EO.

HbO=~+m

(35)

In this case, the boson wave function remains onedimensional, so that the combined fermion-boson wave function is a four-component object. The twobody equation with Coulomb potential only is given by [Hr(p) +

px/~m 2 ] T = ( E - Vc) T .

(36)

Another way is to use the Feshbach-Villar formalism [9]. In this case, the boson wave function has two components, and the Klein-Gordon equation is rewritten as

(2--~ ('/73+il"2)p2+ m ' f 3 ) ¢ = E ¢ ,

29 October 1992

One can readily verify that both eq. (36) and eq. (38) give the same result [i.e. eqs. (22) ] as eq. (17) with S = 0 . The reason has to do with the fact that in all these three equations, the introduction of the Coulomb interaction via minimal substitution is unique. However, in eqs. (36) and (38), the momentum operator p appears quadratically in Hb, but linearly in Hf; consequently, there is no unique way to introduce the vector potential V, and different ways of introduction lead to different results. The same can also be said about the scalar potential S. In summary, using the DKP formalism for the boson, we have constructed a quasi-relativistic equation for a system consisting of a spin-½ fermion and a spin-0 boson. To our knowledge, this is the first of its kind in configuration space. The corresponding twobody wave function is a 4 × 5 matrix. We show that, with scalar and Coulomb potentials, the proposed equation can be reduced to a pair of coupled firstorder differential equations. We also show that the proposed two-body equation has the correct nonrelativistic limit when either one of the particles becomes heavy. This work was supported in part by the National Science Council of the Republic of China under grants NSC 81-0208-M-001-54 and NSC 81-0208-M-00152.

(37) References

where zi are the usual Pauli matrices. If one uses this form for the free boson hamiltonian, the composite fermion-boson wave function will be a 4 × 2 object. The two-body wave equation then takes the following form (with Vc only): +

1

= (E- Vc)T.

q_iz2)p2 + m z 3 ) ] T (38)

[ 1] C.Y. Cheung and S.P. Li, Phys. Len. B 234 (1990) 444. [2] See, e.g., D.A. Owen, Phys. Rev. D 42 (1990) 3534. [3] N. Kemmer, Proc. R. Soc. London A 173 (1939) 91. [4] R.J. Duffin, Phys. Rev. 54 (1938) 1114. [ 5 ] G. Petiau, Acad. R. Belg.CI. Sci. Mem. Collect. 8, 16 ( 1936) No. 2. [6] A. Klein, Phys. Rev. 82 (1951) 639. [7] E. Fischbach, M.M. Nieto and C.K. Scott, J. Math. Phys. 14 (1973) 1760. [8 ] R.F. Guertin and T.L. Wilson, Phys. Rev. D 15 (1977) 1518. [9] H. Feshbach and F. Villars, Rev. Mod. Phys. 30 (1958) 24.

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