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Variational calculation of the bound-state wavefunction in strongly coupled QED
Volume 176, n u m b e r 1,2
PHYSICS LETTERS B
21 August 1986
V A R I A T I O N A L C A L C U L A T I O N OF THE B O U N D - S T A T E W A V E F U N C T I O N IN S T R O N G L Y C O U P L E D Q E D R o m a n K O N I U K and Jurij W. D A R E W Y C H Department of Physics, York University, Toronto, Ontario, Canada M3J I P3
Received 27 May 1986 The variational method is used to calculate the fermion-antifermion binding energy and wavefunction in strongly coupled QED. The calculation is carried out in the Coulomb gauge using an ansatz that is insensitive to the transverse photon degrees of freedom. The results are of interest as they bear on the highly relativistic bound systems in QCD.
Essentially all work on two-body b o u n d states in the q u a n t u m field theory has been carried out in the framework of covariant perturbation theory in the Bethe-Salpeter formalism [1]. On the other hand, the variational method [2-4], although under-utilized, appears to be well suited for investigating the bound-state spectrum for all values of the coupling constants. In this note we will study the f e r m i o n - a n t i f e r m i o n binding energy and wavefunction in Q E D for large values of a (fine structure constant). The relativistic rather than quantum-field theoretic effects will be emphasized. In radiation gauge the Q E D hamiltonian is
H= fd3x
½[E2(x)+nZ(x)]},
~+(x)[a'((1/i)V-eA)+flm]~p(x)+
(1)
where E 2 = E t2 + E 2E,
Et= - A ,
B = V × A ,
e2 f ~t(x)~(x)+*(Yl+(Y) E 2 = -4--~ d 3Y Ix-y[
Setting e = 0 for the moment, we sandwich the hamiltonian between a variational trial vacuum state which is just the free field vacuum with variational mass parameters A for the p h o t o n and ~2 for the fermions. The divergent result is 2 I , ( A ) -- A210(A) -- 411(I2 ) + 4(rn -- g2) $210(12),
(2)
where I~(m)=f -(2~r) d~p (,~2)u 3 2~0 '
w2= (p2+m:).
This vacuum energy is minimized at the values A = 0 and I2 = m, namely at the bare mass values. N o w for e ~ 0 we see that the p h o t o n will continue to be massless as the e~b+ot "A~k term is linear in A and gives no contribution given our free field (gaussian) ansatz. There is, however, an infinite fermion mass renormalization due to the C o u l o m b self energy. Since we are only interested in energy differences here, and identical infinite shifts will appear for any (multi) particle state we consider, we can simply normal-order the theory in what follows and c o m p u t e the f e r m i o n - a n t i f e r m i o n bound-state energy directly. 195
Volume 176, number 1,2
PHYSICS LETTERSB
21 August 1986
Our ansatz for the bound state is the Fock-space state
o(p)x, ob+(p, s)d+(-p, a) 10),
[e+e ) = f E d 3 p
(3)
So
i.e. a coherent superposition of electron-positron states of momentum p and - p respectively and where X,,o is the total spin wavefunction. We emphasize that this limited ansatz is insensitive to the interaction term of the hamiltonian that is linear in A and so only explores the relativistic kinematics and Coulomb dynamics. Sandwiching the hamiitonian between this state we find that the total energy, expressed as a functional of o(p) is 2fd3q M[o(p)]
02(q)Eq
=
f d3q 02(q)
=f d3q d3k
o(k)4w m 2 -ut(q, Iq-~[ -2-EqEk
- (2~) - 3 s'o'EEX~.oX' ~oo(q)so
s')u(k,
s)t;t(-k,
o)t)(-q, o')
f d3q o2(q) (4) The first term is the relativistic kinetic energy of two particles and the second the Coulomb energy. We now vary o ( p ) and obtain the integral equation o ( k ) [ 2 E k - M] d3q ~ _/(2~)3 G,
. 4re
Ex,,o,X,oO(q)so
~Iq-kl
m2
EqEkut(q" s')u(k, s)vt(-k,
o)v(-q,
o')=0.
(5)
Somewhat unexpected there is a spin-dependent effect even though our limited ansatz is not sensitive to the q,la-A~b term in the hamiltonian. One would have to include admixtures of the one-photon, the fermion-antifermion-photon and in fact the two-fermion-two-antifermion-photon Fock-space states to incorporate the effects of dynamical one-photon exchange consistently. Evaluating the spin sums we obtain
$t J=l
(1/v/2)(1"~ + $1 ") S J,
(A + iB.f A2-BZ+2B 2,
J=0
( 1 / v / 2 ) ( 1 " $ - ST)
A2+B 2,
(6)
(A - i B = )
where
A= [(Eq+ m)(E,, + m)+p'q]/2m(im+ E )Im+
B=p ×q/2m¢(m+ Et,)(m+ Eq).
We note in passing the nonrelativistic limit of eq. (5) is the SchfiSdinger equation k2
°(k)~.t~
_
d3q
(2~.) 3 Ik
4__~ o(q) = ( M - 2rn)o(k) ql 2
for two particles bound in a Coulomb potential with binding energy 2m - M. 196
(7)
Volume 176, number 1,2
PHYSICS LETTERS B
21 August 1986
M /.O
m
7-
I
I
I
I
1
.4
,8
/2
Z6
2.0
II 2,4
\ 2.8
CZ
Fig. 1. Plot of the total two-particle energy versus c~. The solid curves show the relativistic (singlet and triplet) and the dashed curve shows the nonrelativistic values. One could now proceed to solve the fully relativistic integral equation (5) numerically, however, since one knows the solution o(k) exactly in the nonrelativistic limit (small c~), we have instead variationally optimized the b o u n d state energy M [ o ( p ) ] using the hydrogenic trial form and varied the Bohr radius parameter. This a p p r o a c h has been used in previous work [5] where three figure agreement was obtained with respect to the exact energy eigenvalues. The results are presented in fig. 1. The most surprising feature of this graph is the rather small departures of the eigenvalues from their nonrelativistic values. Up to c~ = 1 the three curves are indistinguishable! The spin singlet curve falls through zero at ~ --- 2.3 where presumably the true v a c u u m is a state populated by f e r m i o n - a n t i - f e r m i o n pairs. The singlet-triplet splitting becomes appreciable for c~ > 1.5. N o t e that none of the pathologies of the Dirac equation, for example imaginary eigenvalues, appear at large c~. A comparison between the relativistic and nonrelativistic m o m e n t u m wavefunctions shows that for a < 1 there is a modest difference between the wavefunctions but for larger values the difference is striking. F o r example at c~ = 2 there is a factor of two increase in the m o m e n t u m expectation value ( p ) . We feel that the main conclusion to be drawn from this work is that even for highly relativistic b o u n d systems the relativistic corrections to the nonrelativistic energy eigenvalues are rather small. It is, therefore, perhaps not surprising that the tightly b o u n d ground state mesons are well described by the nonrelativistic quark model. To apply this formalism to the meson problem one is really forced to a more sophisticated ansatz so that the effects of dynamical boson exchange and thus all the important spin-dependent effects can be included. Of course for light mesons the potential is not coulombic but is believed to be a combination of linear plus Coulomb. Short of solving Q C D such a potential can only be inserted in a heuristic manner. Such an a p p r o a c h to quark-model calculations would at the very least treat the relativistic effects in a rigorous manner. The authors have benefited immensely from conversations with T. Barnes and M. Horbatsch. The Natural Sciences and Engineering Council of C a n a d a is acknowledged for its financial assistance. 197
Volume 176, number 1,2
PHYSICS LETTERS B
References [1] [2] [3] [4] [5]
H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Plenum, New York, 1977). L.1. Schiff, Phys. Rev. 130 (1963) 458. T. Barnes and G.I. Ghandour, Phys. Rev. D 22 (1980) 924. P.M. Stevenson, Phys. Rev. D 32 (1985) 1389. J.W. Darewych, M. Horbatsch and R. Koniuk, Phys. Rev. Lett. 54 (1985) 2188; Phys. Rev. D 33 (1986) 2316.
198
21 August 1986
PHYSICS LETTERS B
21 August 1986
V A R I A T I O N A L C A L C U L A T I O N OF THE B O U N D - S T A T E W A V E F U N C T I O N IN S T R O N G L Y C O U P L E D Q E D R o m a n K O N I U K and Jurij W. D A R E W Y C H Department of Physics, York University, Toronto, Ontario, Canada M3J I P3
Received 27 May 1986 The variational method is used to calculate the fermion-antifermion binding energy and wavefunction in strongly coupled QED. The calculation is carried out in the Coulomb gauge using an ansatz that is insensitive to the transverse photon degrees of freedom. The results are of interest as they bear on the highly relativistic bound systems in QCD.
Essentially all work on two-body b o u n d states in the q u a n t u m field theory has been carried out in the framework of covariant perturbation theory in the Bethe-Salpeter formalism [1]. On the other hand, the variational method [2-4], although under-utilized, appears to be well suited for investigating the bound-state spectrum for all values of the coupling constants. In this note we will study the f e r m i o n - a n t i f e r m i o n binding energy and wavefunction in Q E D for large values of a (fine structure constant). The relativistic rather than quantum-field theoretic effects will be emphasized. In radiation gauge the Q E D hamiltonian is
H= fd3x
½[E2(x)+nZ(x)]},
~+(x)[a'((1/i)V-eA)+flm]~p(x)+
(1)
where E 2 = E t2 + E 2E,
Et= - A ,
B = V × A ,
e2 f ~t(x)~(x)+*(Yl+(Y) E 2 = -4--~ d 3Y Ix-y[
Setting e = 0 for the moment, we sandwich the hamiltonian between a variational trial vacuum state which is just the free field vacuum with variational mass parameters A for the p h o t o n and ~2 for the fermions. The divergent result is 2 I , ( A ) -- A210(A) -- 411(I2 ) + 4(rn -- g2) $210(12),
(2)
where I~(m)=f -(2~r) d~p (,~2)u 3 2~0 '
w2= (p2+m:).
This vacuum energy is minimized at the values A = 0 and I2 = m, namely at the bare mass values. N o w for e ~ 0 we see that the p h o t o n will continue to be massless as the e~b+ot "A~k term is linear in A and gives no contribution given our free field (gaussian) ansatz. There is, however, an infinite fermion mass renormalization due to the C o u l o m b self energy. Since we are only interested in energy differences here, and identical infinite shifts will appear for any (multi) particle state we consider, we can simply normal-order the theory in what follows and c o m p u t e the f e r m i o n - a n t i f e r m i o n bound-state energy directly. 195
Volume 176, number 1,2
PHYSICS LETTERSB
21 August 1986
Our ansatz for the bound state is the Fock-space state
o(p)x, ob+(p, s)d+(-p, a) 10),
[e+e ) = f E d 3 p
(3)
So
i.e. a coherent superposition of electron-positron states of momentum p and - p respectively and where X,,o is the total spin wavefunction. We emphasize that this limited ansatz is insensitive to the interaction term of the hamiltonian that is linear in A and so only explores the relativistic kinematics and Coulomb dynamics. Sandwiching the hamiitonian between this state we find that the total energy, expressed as a functional of o(p) is 2fd3q M[o(p)]
02(q)Eq
=
f d3q 02(q)
=f d3q d3k
o(k)4w m 2 -ut(q, Iq-~[ -2-EqEk
- (2~) - 3 s'o'EEX~.oX' ~oo(q)so
s')u(k,
s)t;t(-k,
o)t)(-q, o')
f d3q o2(q) (4) The first term is the relativistic kinetic energy of two particles and the second the Coulomb energy. We now vary o ( p ) and obtain the integral equation o ( k ) [ 2 E k - M] d3q ~ _/(2~)3 G,
. 4re
Ex,,o,X,oO(q)so
~Iq-kl
m2
EqEkut(q" s')u(k, s)vt(-k,
o)v(-q,
o')=0.
(5)
Somewhat unexpected there is a spin-dependent effect even though our limited ansatz is not sensitive to the q,la-A~b term in the hamiltonian. One would have to include admixtures of the one-photon, the fermion-antifermion-photon and in fact the two-fermion-two-antifermion-photon Fock-space states to incorporate the effects of dynamical one-photon exchange consistently. Evaluating the spin sums we obtain
$t J=l
(1/v/2)(1"~ + $1 ") S J,
(A + iB.f A2-BZ+2B 2,
J=0
( 1 / v / 2 ) ( 1 " $ - ST)
A2+B 2,
(6)
(A - i B = )
where
A= [(Eq+ m)(E,, + m)+p'q]/2m(im+ E )Im+
B=p ×q/2m¢(m+ Et,)(m+ Eq).
We note in passing the nonrelativistic limit of eq. (5) is the SchfiSdinger equation k2
°(k)~.t~
_
d3q
(2~.) 3 Ik
4__~ o(q) = ( M - 2rn)o(k) ql 2
for two particles bound in a Coulomb potential with binding energy 2m - M. 196
(7)
Volume 176, number 1,2
PHYSICS LETTERS B
21 August 1986
M /.O
m
7-
I
I
I
I
1
.4
,8
/2
Z6
2.0
II 2,4
\ 2.8
CZ
Fig. 1. Plot of the total two-particle energy versus c~. The solid curves show the relativistic (singlet and triplet) and the dashed curve shows the nonrelativistic values. One could now proceed to solve the fully relativistic integral equation (5) numerically, however, since one knows the solution o(k) exactly in the nonrelativistic limit (small c~), we have instead variationally optimized the b o u n d state energy M [ o ( p ) ] using the hydrogenic trial form and varied the Bohr radius parameter. This a p p r o a c h has been used in previous work [5] where three figure agreement was obtained with respect to the exact energy eigenvalues. The results are presented in fig. 1. The most surprising feature of this graph is the rather small departures of the eigenvalues from their nonrelativistic values. Up to c~ = 1 the three curves are indistinguishable! The spin singlet curve falls through zero at ~ --- 2.3 where presumably the true v a c u u m is a state populated by f e r m i o n - a n t i - f e r m i o n pairs. The singlet-triplet splitting becomes appreciable for c~ > 1.5. N o t e that none of the pathologies of the Dirac equation, for example imaginary eigenvalues, appear at large c~. A comparison between the relativistic and nonrelativistic m o m e n t u m wavefunctions shows that for a < 1 there is a modest difference between the wavefunctions but for larger values the difference is striking. F o r example at c~ = 2 there is a factor of two increase in the m o m e n t u m expectation value ( p ) . We feel that the main conclusion to be drawn from this work is that even for highly relativistic b o u n d systems the relativistic corrections to the nonrelativistic energy eigenvalues are rather small. It is, therefore, perhaps not surprising that the tightly b o u n d ground state mesons are well described by the nonrelativistic quark model. To apply this formalism to the meson problem one is really forced to a more sophisticated ansatz so that the effects of dynamical boson exchange and thus all the important spin-dependent effects can be included. Of course for light mesons the potential is not coulombic but is believed to be a combination of linear plus Coulomb. Short of solving Q C D such a potential can only be inserted in a heuristic manner. Such an a p p r o a c h to quark-model calculations would at the very least treat the relativistic effects in a rigorous manner. The authors have benefited immensely from conversations with T. Barnes and M. Horbatsch. The Natural Sciences and Engineering Council of C a n a d a is acknowledged for its financial assistance. 197
Volume 176, number 1,2
PHYSICS LETTERS B
References [1] [2] [3] [4] [5]
H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Plenum, New York, 1977). L.1. Schiff, Phys. Rev. 130 (1963) 458. T. Barnes and G.I. Ghandour, Phys. Rev. D 22 (1980) 924. P.M. Stevenson, Phys. Rev. D 32 (1985) 1389. J.W. Darewych, M. Horbatsch and R. Koniuk, Phys. Rev. Lett. 54 (1985) 2188; Phys. Rev. D 33 (1986) 2316.
198
21 August 1986