Parity doubling at the critical point in tightly bound systems

Volume 229, number 1,2

PHYSICS LETTERS B

5 October 1989

PARITY D O U B L I N G AT T H E C R I T I C A L P O I N T I N T I G H T L Y B O U N D S Y S T E M S William D Y K S H O O R N , R o m a n K O N I U K

Departmentof Physics, YorkUniversity,Toronto,Ontario,Canada and R a m o n MUI~IOZ-TAPIA

Departamentde FisicaTe6rica,E-08193Bellaterra(Barcelona),Spain Received 11 April 1989; revised manuscript received 12 July 1989

Integral equations are developed for bound states in QED. These equations are then solved numerically and the critical values, i.e. the values of the coupling constants at which the bound-state mass vanishes, are obtained. We find that at the critical values the pseudoscalar and scalar as well as the vector and pseudovector states are degenerate.

Essentially all work on two-body states in 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 [2,3] although under-utilized, appears to be well suited to the investigation o f the bound-state spectrum for all values o f the coupling constants. In this note we will derive integral equations for bound states within a limited Fock-space approximation using the variational principle [ 4 ] in massive quantum electrodynamics ( Q E D ) . In radiation gauge the Q E D hamiltonian is

H = f dax{~t{at • [( 1/i)V-eA] +flm}~ +½ ( E E + B 2 ) } ,

(1)

where

E2=E2+E 2, E t = - , ' ] ,

le+e - )

=~ d3p ~ F(p,t~,t~)b*(p,a)d*(-p,~)}O>

which is a coherent superposition of electron and positron states of m o m e n t u m p and - p respectively. The function F(p, tr, ~) is obtained by constructing a bi-local operator of the form ~(x)F~u(y)f( Ix-Yl ), and integrating over all x and y. The matrix F is Ys, Y, YsYor 1 depending on whether we wish to construct a pseudoscalar, vector, pseudovector or scalar state. The resulting function F(p, tr, ~) will then b e f ( p ) O(p, tr)l-'v(-p, ~) where f(p) is to be determined variationally. The hamiltonian is now sandwiched between our ansatz (eq. ( 2 ) ) and the quantity corresponding to the bound state mass

B=V×A,

M[f(p ) ] = e2 f

E2 = ~

d3y

gt*(x)gt(x)qP(y)~(y)

(e+e- IHle+e-) (e+e-le+e -)

(3)

Ix-yl

Our ansatz for the bound state is the Fock-space state

132

(2)

is now constructed and variationally optimized by functionally differentiating with respect t o f ( p ) and setting the result to zero. For the states considered here the angular integration can be performed analvt0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Volume 229, number 1,2

PHYSICS LETTERS B

ically a n d this leads to one-variable integral equations for the f u n c t i o n f ( p ) :

Mf(p) = 2 E p f ( p ) -

~

K(p, q)f(q) dq.

(Ep+Eq) 2 ,

K~(p, q) = -2-

EpEq

2pq ~Eq'

m

(5)

\\

(Ep + Eo)2(EpEq - m z) log P+q pqEpEq p-q

2m 2

- -

IP+q[

2

(4)

In the interest o f brevity we present the K(p, q) o f the pseudoscalar a n d scalar states only as examples:

Kp~(p, q)--

5 October 1989

C 0.0 (6)

We will present the kernels K(p, q) for all other states as well as resuRs on scalar e l e c t r o d y n a m i c s ( S Q E D ) elsewhere [ 5 ]. We note that H a r d e k o p f a n d Sucher [ 6 ] have obt a i n e d the pseudoscalar equation in the context o f a t w o - b o d y Dirac-type equation a u g m e n t e d by positive energy projection operators. It is encouraging that identical equations arise from very different formalisms. These integral equations were then solved numerically b o t h by the basis-expansion m e t h o d a n d by opt i m i z e d discretization o f the integral a n d solving the resulting eigenvalue problem. The results are presented in fig. 1, where we have plotted the b o u n d state energy d i v i d e d by m versus a . All four states follow the nonrelativistic curves for small a . We r e m i n d the reader o f the nonrelativistic correspondence: (jpc, 2S+~Lj), (0 - + , ISo), (0 ++, 3po) , ( 1 - - , 381), (l ++, 3PI). All four states exhibit d r a m a t i c t u r n a r o u n d behaviour near the critical value o f a¢, where the b o u n d state mass vanishes. In fact we find a degeneracy o f the 0 - + a n d 0 + + states as well as a degeneracy o f the 1 - - and 1 + + at the critical values. Note in both cases that this is a degeneracy o f what were nonrelativistically the 1S a n d 2P states! The states have t r a d e d partners; a nonrelativistic degeneracy has been traded for an ultra-relativistic one. This degeneracy can be u n d e r s t o o d in the following way. I f one puts m = 0 in the integral equations, the corresponding integral equations b e c o m e identical. This is expected as massless Q E D is chirally in-

I 0.5

1 1.0

I 1.5

I

2:0

2.5

Fig. 1. Two-particle bound-state mass versus a. The quantum numbers of the curves are indicated 0- +, 1- -, 0 + +, and 1+ +. The dashed curves show the nonrelativistic dependence 2-Or2/ 4n 2. variant. An explicit realization o f this s y m m e t r y would be parity doubling in the spectrum. States with the same q u a n t u m numbers but differing parity should be degenerate. It can be argued that ac is a dimensionless p a r a m e t e r and since m is the only dimensionful p a r a m e t e r o f the massive theory, otc m u s t be i n d e p e n d e n t o f m. O f course ot~ is i n d e p e n d e n t o f m in the massive theory, as is the entire curve, a n d it appears the l i m i t to m--,0 is smooth. We have explicitly solved the massless theory [ 5,6 ] ~' and have found that 4

ac-

n/2 +2/rr 1.81

pseudoscalar, scalar 0 - + , 0 ++ ,

8/,/3 ~2.20

vector, p s e u d o v e c t o r 1 - % 1 ++ .

(7)

O u r best numerical values are a ¢ = 1.826

pseudoscalar,

a c = 1.828

scalar,

ac = 2 . 2 2 6

vector,

a¢ = 2.221

pseudovector.

(8)

~1 The pseudoscalar massless result was obtained previously in ref. [ 7 ]. The value of a~ in this case was found by an asymptotic analysis of the massive theory in ref. [6]. 133

Volume 229, number 1,2

PHYSICS LETTERS B

W i t h each increase in accuracy (as the n u m b e r o f basis states and mesh points is increased) we a p p r o a c h the analytic massless theory values a n d therefore strongly believe we are seeing a massive theory behave as a massless one at strong coupling. The lack o f convergence near the critical value is the result o f large cancellations between the kinetic and potential energies which just precisely balance when the wavefunction goes over to the massless form f(p) =p-2. This is a state which is extremely localized in configuration space, sitting deep in the potential well at the origin. The energies involved are so large that a small fermion mass becomes irrelevant. Although expanding the Fock-space would lead to quantitative changes, such as some m o v e m e n t in the critical coupling [ 6 ], we would find it very surprising if the massive theory which j o i n s so smoothly to the massless theory at c% would cease to do so with an i m p r o v e m e n t in the calculation. Thus we expect parity doubling to not be an artifact o f the approximation. Finally, our critical coupling seems to be very much related to the "critical coupling" ( a c = n/3 ) discovered in the l a d d e r a p p r o x i m a t i o n to the B e t h e Salpeter equation (for the pseudoscalar bound-state) in massless Q E D [8]. At ~x= ac there is a "collapse o f the wavefunction" where the massless bound-state goes over to a 1 / r form. We, however, are claiming that the massive theory also goes over to this form at a=ac. (The numerical difference is due to the fact that our approach is equivalent to s u m m i n g a l a d d e r o f Coul o m b photons. ) The critical coupling in the massless theory is o f some interest as recent investigations o f quenched Q E D [ 9 ] have associated it with an ultraviolet-stable fixed point where ( i f a p p r o a c h e d from the strong coupling side) a nontrivial q u a n t u m field theory, with

134

5 October 1989

spontaneous chiral s y m m e t r y breaking and confinem e n t can be defined. The authors wish to thank J.W. Darewych a n d M a r k o Horbatsch for n u m e r o u s stimulating discussions. Two o f us (R.K. and W . D . ) would like to thank R o l f Tarrach a n d the D e p a r t m e n t d ' E s t r u c t u r a i Constituents de la M a t e r i a at the U n i v e r s i t a t de Barcelona for their gracious hospitality at a time when this work was begun. We would like to acknowledge the financial assistance o f the Natural Sciences and Engineering Research Council o f C a n a d a a n d CAICYT, Spain.

References [ 1] H. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Plenum, New York, 1977). [2] L.I. Schiff, Phys. Rcv. 130 (1963) 458; T. Barnes and G.I. Ghandour, Phys. Rev. D 22 (1980) 924; P.M. Stevenson, Phys. Rev. D 32 (1985) 1389. [3] L. Polley and D.E.L. Poninger eds., Proc. Workshop on Variational calculations in quantum field theory (Wangerooge, FRG, 1987). [4 ] J. Finger, D. Horn and J.E. Mandula, Phys. Rev. D 20 ( 1979 ) 3253. [5]W. Dykshoorn, R, Mufioz-Tapia and R. Koniuk, York University preprint, in preparation. [6] G. Hardekopfand J. Sucher, Phys. Rev. A 30 (1984) 703; A 31 (1985) 2020. [ 7 ] J.W. Darewych and M. Horbatsch, York University preprint (1988). [8l C.N. Leung, S.T. Love and W.A. Bardeen, Nucl. Phys. B 273 (1986) 649; V.A. Miransky, Nuovo Cimento 90A (1985) 149; V.P. Gusynin and V.A. Miransky, Phys. Lett. B 191 (1987) 141; R. Fukuda and T. Kugo, Nucl. Phys. B 117 (1976) 250. [9] J. Kogut, E. Dagotto and A. Koci6, Phys. Rev. Lett. 61 (1989) 1001; B. Holdom, Phys. Rev. Len. 62 (1989) 997.