Relativistic three-fermion wave equations

PHYSICS LETTERS A

PhysicsLettersA 158 (1991) 361—366 North-Holland

Relativistic three-fermion wave equations W.C. Berseth and J.W. Darewych Physics Department, York University, Downsview, Toronto, Ontario, Canada M3J 1P3 Received 3 May 1991; revised manuscript received 17 July 1991; accepted for publication 19 July 1991 Communicated by J.P. Vigier

Coupled integral eigenvalue equations are derived variationally, within the Hamiltonian formalism of QED, for a relativistic two-fermion—one-antifermion bound system, each of arbitrary mass and charge. The ansatz employed is sensitive to all terms of the Hamiltonian. These equations are uncoupledapproximately and their applications are discussed.

Bound state problems in quantum field theory have conventionally been treated within the Bethe—Salpeter formalism (see, for example ch. 10 ofref. [1] and ref. [21) where bound states are identified as poles in Green functions. This method is in practice perturbative and is very difficult to implement for systems of three or more particles [3]. Recently, the variational principle, within the Hamiltonian formalism of QED, has been applied [4] to derive a tractable relativistic wave equation for a two-fermion—one-antifermion bound system. It is pointed out there that while solutions expanded in a series of a will contain order a4 relativistic kinetic energy corrections, corrections to the interaction energy will contain Coulomb contributions only, as transverse effects were not included in the derivation ofthe equation. This is due to the limited nature ofthe ansatz that was used, which was insensitive to the transverse part of the interaction Hamiltonian which is linear in the photon field. Here we modify the ansatz and derive two coupled relativistic two-fermion.-one-antifermion bound state equations that include effects of transverse photon exchange. This modification is analogous to that of the two-body case, which in the perturbative regime gave the same order a4 physics as covariant perturbation theory [5,6]. In addition, we consider here the generalized case of particles of arbitrary mass and charge. The equations are uncoupled approximately and particle masses renormalized. The standard QED Hamiltonian density extended to include three different fermions of mass m 1 and charge q1, in the Coulomb gauge, is ~ [~(x)(

~ (1)

+~~$d3y1)3~03??) +1{A2(x)+[VxA(x)]2},

where p3(x)= ~ q,wt (x)wi(x). Because we use the Coulomb gauge in the Hamiltonian formalism, these expressions and others do not appear to be manifestly covariant. The electromagnetic and fermionic field operators, A“(x) and yi1(x), satisfy the usual commutation and anticommutation rules with different fermionic field operators anticommuting. We follow the notational conventions of Bjorken and Drell [7] with the Fourier decomposition of the fermion fields given by 3p ~ (2)

~J

,

d

0375-9601/9l/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

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where here and henceforth we use the notation U1Q, s) = ~/m1/w,(p) u,(p, s), etc., where u~Q,,s) is the usual free fermion spinor of type 1 and w,(p) = ,,Jp2 + m The b1Q,, s) and dt (p, s) are fermion annihilation and antifermion creation operators, respectively, of mass m1, charge q1, momentum p and spin index s, and satisfy the anticommutation relations 3(p—p’). (3) ~.

[b,(p,s),b~(p’,s~)J÷=[d1(p,s),dj~(p’,s’)]÷=ö,,ö55.ô

We identify the m

0, in eq. (1) as the bare masses of the Lagrangian while the m, of eq. (2) are, at this stage, adjustable parameters which will be identified as the physical particle masses. To describe a system of two fermions plus one antifermion, all ofarbitrary mass and charge, we use the ansatz ~

(4)

where

3p l;l~l~ >= ~ Jd3pt d 2f(p1,p2,s1,s2,s3)bj,si)b~(p2,s2)d~(p3,s3)I0>

(5)

SI S2S3

and 1-1-’~ abcT = ~ S152S3a

$

3p~d3p~d3p~g(p~

s

d

2, s3,A)b~(p~, si)b

,s2)d~(p~,s3)a~(p’4,A)I0>, (6)

where a ÷(p, A) is a creation operator for a (free) photon of momentum, and polarization index A, and +P2 ~ = 0, p~ +p~+p~+p’~= 0 in the rest frame ofthe three-body system. The subscripts a, b, care chosen to be 1, 2 or 3 depending on the system of interest. For systems containing a fermion and antifermion of the same type (e.g. ee~) we would have to supplement the ansatz (4) with terms like Il; y> + Il; 1~1~1~1~‘y> in order to include virtual annihilation contributions fully [6]. However we shall not include them at the present time. Applying, as in ref. [41,the variational principle

Pi

~<3I:H3—E:I3>=0

(7)

(note that we normal order the Hamiltonian), yields the following coupled integral eigenvalue equations for the coefficients, FQ,1,p2, s1, S2, s3)=f(p1,p2, s1, s2, S3)~öabf(P2,P1,S2,Sl,S3)

(8)

,

(9) and eigenvalue E: (wa(qi )+w~(q2)+w~(q3)+ (moa—ma)ma + (m0 — —

~

f ~(

1 SIS3

j

m~)m,+ ~

_E)F(qi, q2, a~,a2, a3)

U(q1,a1)U~(pj,s1)V~(—p1—q2,s3)V~(q3,a3) 2 F(p1, q2, S1, a2, 53) Pi —q1

Pi 1, q~q~

U~(q2,a2)Ub(jiI,sI)V~(—p1—qI,s3)V~(q3,a3)

+q~q~

2

F(q1,p1, a1,s1,s3)

p~—q2 —q,~qb

362

U(qI,al)Ua(fll,s1)U~(q2,a2)U~(—q3—p1,s3) 2 F(,p1, —q3—p1,s1,s3,a3) p~—q1

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ôacqaqc

~

PHYSICS LETTERS A

16 September 1991

1q212

a

U~(q

2 2,a2)V~(q3,a3)V~(—pj ~ 1q11

+

“

“

~

(

d~p1~q~

2)3/2~j

~

~IqI

—psi

,,J21q2—p11

~

q2 p~a1 a2 s1 A))

~J2Iq3—p1I

—qC

(10)

and (Wa(PI)+Wb(P2)+Wc(P3)+

1P4

~ (moa—ma)ma oa(Pi) + (mob—mb)mb Wb(p2) + (mo~—m~)m~ w~(p3) _E)

XG(p1,p2,p3,s1, s2, s3,2) = (21

)3~~J( U~(j2,s2)Ub(P2

Iq —P3 +~3

GQ~+p3—q,p2,q,a1,s2,a3,A)

2

—q, a1)V~(q,a3)V~(p3,s3) G(p1 ,P2

iq—p3i~

+P3

—

q, q, S~,a1, a3, A)

2 U(ji,si)Ua(q, a1)U~Q,2,s2)U(fl1 +P2 —q, a3) G(p Iq—p11

—q~qb

ôacqaqc

—

1+j,2—q,q,p3,a3,a~,s3,A)

U~ (pr, s~)V~(p3,53) V~(q, a3)U~(p1+p3—q,a1) G(,p1 +p3—q,p2, q, a1, 52, a3, A) Ipt+P312

a

~ öb~qbq~

+~(2)32II

1P2

+~3

12

~(qau~(Pi,si)c~.E~p4,2)ua(_p2 Cl

—p3,a1)F(—p2 —p3,p2,a~,s2,s3)

\

+qbU~(P2, s2)~.e(y4,A)Ub(—pi

J33,

a1)F(y~,—P1 —P3, S1, Ci,

—q~V~ (—~~ —P2, C1 )~•(j~,A) V~(P3,s3)F(~~ ,P2,

Si,

52,

53)

a~))~

(11)

where q3=—(q1+q2) and p4=—Q1+p2+p3). We note that if particles a and bare identical in eq. (4) the correct particle interchange symmetry results naturally in eqs. (8)—(11).

The coupled equations (10) and (11) are difficult to solve. We therefore follow the approximate decoupling 4. To lowest order in scheme usedfrom previously [5,6], which, in the perturbative regime, is justifiable to order a q, we have, eq. (11),

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G(j.,~,p

2,p3,Si, ~2, S3, 2)

1

~ ~(qau:cPi,SiEP4,2ua(_P2_P3,ci)F(_P2_P3,P2,ai,s2,S3) .J(2~2Ip4’ ~ +qbUb(p2,S2)~.E(fl4,A)Ub(—p1—p3,aI)Ft’flI,—pI—p3,si,ai,s3) —q~V~(—p~ —P2, a~)~(P4,2)V~(p3,s3)F(P1,p2,s~, 52, as)),

(12)

giving for eq. (10)

[wa(qi)+wb(q2)+wc(q3) —E]F(q~,q2, a1, a2, a3) 2 = (2E)3~*’ThE q~q~ /

ac

K~(q1,q2,p1,a~,s3,s~,a3) 1q21

~

Ip~—qiI2 +qbqc(

K~(q

2

IP~—q21 ~

Pt

(

I~i 2 2,q~,p~,a2,s3,s~,a3)

ôbc

C 1 S~ s3)

2

~ —q21

K~(q

—q~qb

1,q2,p1,a1,a2,s1,s3) p~—q~ 2

K~,(q1,q2,p1,a1,a2,s1,s3) —

1q1 —P~2

)F(pi, —q3 —p1,S~,S3,a3)]~

(13)

where K~c(qi,q2,pi,ai,s3,si,a3)U~(qi,ai)Ua(fl1,s1)V~(—pi—q2,s3)Vc(q3,a3),

(14)

K~(q~,q2,p1,a~,a2,s~,s —p1,s3) 3)\—TT+(q1,aI)U~(P1,sI)U~(q2,a2)U~(_q3 ‘—‘a

(15)

~

,

q2,pi, a~,S3,s1, a3) = ~ U~(q~, a1)~e(q~Pi, A)UaQ’i, Si) V~( Pi

q2,

53)

~-(q1

—pi,

A)V~(q3,a3)

,

K~(q~,q2,p~,a~,a2,S1,S3)

(17)

= ~ U~(q~,a1)~-(q1 Pi, A)Ua(pi,Si) U~(q2,a2)~~(q~—Pt, A)U~,(—q3

K~(q~,q2,p1, ai,S3,Si,a

3)\_IT+ ‘-‘a

(16)

PI,53)

,

(q~,a~) Va(q3,a3)V~(—p1—q2,s3)U~(p~,s1) .

(17) (18)

To obtain the expected kinetic plus rest energy on the left hand side of eq. (13), with the parameters ma, m~, and m~identified as the physical particle masses, these parameters were chosen to make terms of the form (for particle a) 2wa(P moa—m a

q~m a 3~ a —p1,A)U~(Pi,s3)U~(j1,s3)~(qi —pi,A)U~(qj,ai (2~)3S~AJd ~i U~(q1,s~)~(q1 21q1—p1I 1)

vanish, and similarly for particles b and c, where ë’ (q, A) = (0, 364

E(q,

A)) for A = 1, 2. These are just the mass

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renormalization contributions arising from transverse photon exchange. The Coulomb mass renormalization contribution is suppressed here by the normal orderingprocedure. Note that because we use the Coulomb gauge in the Hamiltonian formalism, this and other expressions do not appear to be manifestly covariant. The same 1a’)’>, with mass renormalization condition results if one considers the single particle ansatz, I~> = I l~ > + I the variational principle, eq. (7), and demands the correct relativistic one-particle energy E 1 =co~(q).Ofcourse this mass renormalization could equally be achieved in the conventional way, by using ma = moa etc. throughout, but adding suitable mass counter terms to the Hamiltonian. In eq. (13) the kernels Kc, KC and KT, Kr arise from Coulomb and transverse photon exchange, respectively, while the KA are the Coulomb virtual annihilation contributions. Setting the transverse kernels, KT and KT’, to zero and all masses and charges equal in eq. (13) we obtain the equation derived previously for pure Coulomb interactions [41.Note that the Coulomb and transverse kernels Kc and KT can be combined into the form K(q~,q2,

0j53, S1a3)KC_KT

2 =

~ ~U~—(q~,a1)~(q1 —p1,A)P~,(P~,s1)V~,(—p~ —q2,s3)~(q1—p1,A)V~(q3,a3)

(20)

5=0

and similarly for Kc~_KT~, where ë’(q, A)= (1, 0) for 2=0 and as defined previously for 2=1, 2, and ~= I for 2=0 and ~=—l for 2=1,2. 2/2m, U~(q, s)U(P, a)=ö In the nonrelativistic limit we have w(p)=m+p 50, U~(q, s)V(p, a)= U~(q,s)a.e(q—p, 2)U(p, a)=0, etc., and lettingF(q1, q2, a~,a2, a3)=Ø(qj, q2)x(a1, 02, a3) in eq. (13) and factorising out the spin wave function reduces eq. (13) to ~,

+ 11~ + (2~)3Jd3PI(_ q1’

112ø(P1~~2)_ q2....p112ø(~1~P1)+

where e = E— ma mb — m~.This is just the momentum space Schrödinger equation for three particles of arbitrary mass and charge interacting via mutual Coulomb potentials. For particles ofequal mass and charge this is identical to the nonrelativistic-limit equation obtained previously pure interactions. 4~t) limit of[4] eq.for (13), as Coulomb demonstrated for the analSince eq. (21) is also the weak coupling (small a~= q.q~/ ogous two-particle equation [5,6] then solutions of (13), if expanded in a series of a, will be identical to those of (21) to lowest order (i.e. a2) in the energy eigenvalue. Order a4 terms will contain proper relativistic kinetic energy corrections [4] and the corrections to the interaction energy will contain both Coulomb and transverse contributions. We note that eq. (13) can be applied to a number of relativistic three-body systems, including e~ee, ~ s~te, H, H~ (in the approximation that protons are fundamental fermions), etc., and the respective charge conjugated systems. Also, setting the mass of the antifermion to infinity will give the equations for two fermions in a static external Coulomb potential, e.g., helium or a high-Z ion. Approximate solutions of eq. (13) for particular J’~states are in progress and will be discussed at a later date. —

We thank the Natural Sciences and Engineering Research Council of Canada for financial support.

References [1) C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980).

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[21W. Johnson, P. Mohr and J. Sucher, eds., Relativistic quantum

electrodynamics and weak interaction effectsin atoms, A.I.P. Conf.

Proc. Series No. 189 (American Institute of Physics, New York, 1989).

[31M.H. Mittleman, Phys. Rev. A 39

(1989) 1. [4) J.W. Darewych, Phys. Lett. A 147 (1990) 403. 5] J.W. Darewych and M. Horbatsch,J. Phys. B 22 (1989) 973; 23 (1990) 337. [6] W. Dykshoorn and R. Koniuk, Phys. Rev. A 41(1990) 60. [7] J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw-Hill, New York, 1965).

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