On the spectrum of excitations in some strong coupling gauge models

Volume 220, number 4 PHYSICS LETTERS B 13 April 1989

ON THE SPECTRUM OF EXCITATIONS IN SOME STRONG COUPLING GAUGE MODELS

V.P. GUSYNIN, V.A. KUSHNIR Institute for Theoretical Physics, 252 130 Kiev 130, USSR

and

V.A. MIRANSKY l Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan

Received 12 December 1988; revised manuscript received 28 January 1989

The spectrum of spinless excitations in (3 + 1 )-dimensional ladder QED with additional four-fermion interaction and in (2 + 1 )- dimensional QED is considered.

Recently, there has been considerable interest in strong coupling gauge theories. On the one hand, non- asymptotically free (NAF) gauge theories with the bare coupling c~ t°l in the supercritical (c~ ~°) > a c ~ 1 ) phase can lead to a new class of four-dimensional continuum field theories with non-trivial S-matrix [ 1-4 ] (for a recent discussion of this possibility see ref. [ 5 ] ). On the other hand, both NAF and asymptotically free (AF) gauge theories in the dynamical regime with a "walking" near-critical [a (q2) - c~. << 1 ] coupling can find their application in technicolour dynamics [6-9] (for a recent discussion see ref. [10]).

Also, there are similarities [ 11-14] between the non-perturbative dynamics in four-dimensional strong coupling QED and that in (2+ 1)-dimensional QED with four-component spinors [ 15 ]. The last model may be important [ 14 ] for some attempts to understand theoretically the recently discovered high temperature superconductors.

In the large majority of papers dealing with strong coupling gauge theories the approximate (usually ladder) Schwinger-Dyson (SD) equation for the fer-

Permanent address: Institute for Theoretical Physics, 252 130 Kiev 130, USSR.

mion propagator is studied. From the Ward identi- ties for the chiral currents, it follows that such a method is equivalent to looking for solutions of the approximate Bethe-Salpeter equations for the corre- sponding massless Goldstone bosons. The aim of the present paper is to study the spectrum of spinless excitations in such models, i.e., we want to describe not only massless but also massive spinless ferrnion-an- tifermion bound states there. The emphasis will be on the study of four-dimensional ladder QED with additional four-fermion interaction. This model has recently attracted considerable interest both from the point of view of the existence of non-trivial continuum NAF theories and that of its application to the technicolour dynamics [ 3,16-19 ]. We shall also consider the spectrum of the radial excitations of Goldstone bosons in (2 + 1 )-dimensional QED.

The main results of the present paper are the following.

We show that in the massless ladder QED with additional chiral-invariant four-fermion interaction,

AL= ½G(°) [ (~ / / ) 2__ (~e~IS ~/,/)2 ], ( 1 )

the form of the spectrum ofspinless excitations in the non-perturbative phase with chiral symmetry break- down essentially depends on the values of the gauge

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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coupling c~ ~ and the d imensionless four-fermion coupling g - G ~°)N~A2/4z~2 (Nf is the number of fer-

mion flavours and A is the ul traviolet cu tof f ) . In a certain range of these parameters there are no spinless exci tat ions but in others there are an infini te number of them. We give a s imple in terpre ta t ion o f this phenomenon. Also, we show that in this model the lightest scalar bound state is always massive.

In the case of ( 2 + 1 ) -d imensional QED we show that in the approx imat ion used by Appelquis t et al. [ 11,15 ] there are an infinite number of radial exci- tat ions in the channels with the quan tum numbers of Golds tone bosons.

The direct study of the ladder Bethe-Salpeter (BS) equat ions for massive f e rmion-an t i f e rmion bound states in gauge theories is a very compl ica ted prob- lem (for a general discussion see ref. [2] ). They can be analysed only in some special cases [ 20 ]. To over- come this obstacle, we shall use a trick suggested in ref. [ 21 ]. The essence of the trick is the following.

In the case of dynamical chiral symmetry breaking there are tachyonic fe rmion-ant i fe rmion bound states in the symmetr ic (uns table) phase. The appearance of the fermion dynamical mass leads to the stable phase in which some tachyons become goldstonions, while others acquire a real ( M 2 > 0 ) mass. The sta- bi l izing role of a fermion mass is connected with the general fact that, for a given dynamics of interact ion, the square of the mass of a composi te state M 2 increases monotonica l ly with increasing mass of its const i tuents (see fig. l ) . Therefore, at some critical value m~, a tachyon bound state ( M 2 < 0 ) becomes a massless (Go lds tone ) one. At m > m,, it will become a massive state with M 2 > O.

Let us define the critical value m~. '~ of the fermion mass for a bound state l i ) as the value of the mass at which this state would be massless. It is evident from the preceding discussion that the dynamical fermion mass coincides with the largest value of the critical masses in . . . . . . , y , - .... . , _..,, >~m~ ') for all i ( in general case i is a mul t i index) . Indeed, in this case, because of the fact that M 2 increases with increasing fermion mass, all bound states have non-negative M 2. Otherwise, if m,~ was equal to a non-largest crit ical mass m~ j''~ < m~/''~ , some bound states (in part icular , Iio) ) would remain tachyonic (see fig. 1 ) ~ . The critical values m~ ~} with i¢ io correspond to bound states which become massive ones ( M 2 > 0 ) in the

M2~

I(a)

M2~

(b)

Fig. 1. An illustration of the behaviour of,,l/2 as a function of the fermion mass in for two bound states Ii ° ) and Ij ° ) in the case without (a) and with (b) the level crossing.

(,o) Therefore the spec- stable phase with m o y n = m c •

t rum of m~ ~ for massless BS equat ions determines the number o f massive bound states in different channels. These equat ions are considerably s impler than BS ones for massive bound states.

Let us apply this approach to the ladder (3 + 1 )-

~t We emphasize that this conclusion is general and it remains to be valid also in the case of level crossing (see fig. 1 ).

636


dimensional QED with additional four-fermion interaction (1). In this model the BS equation for massless pseudoscalar composites takes the following form in the euclidean region ( 2 - 3a~°~/4n):

.12

Cps(k 2) = j dq" q2 1 q2+(mpS)2

0

× [2K(k 2, q2) +A -2g]Co~(q2 ). (2)

Here the kernel K(k 2, q2)=O(k2_q2)/k2 + 0 (q 2 _ k 2 ) / q 2 and the amputa ted BS wavefunction for massless pseudoscalars is

T',~ (k) = i752"Cp~ (k 2 )

= G -~ (k) f dax exp( ikx)

× (OIT~ ' (½x)~( - ½x)l~")G - ' (k),

where a = 0, 1 ..... N ~- - 1 and the fermion propagator G(k)=i[A(k2)~C-B(k2)] - ' . We note that eq. (2) is written in the Landau gauge (the special role of this gauge in the ladder approximat ion is discussed, for example, in refi [2 ] ) . In this approximat ion the function A (k 2 ) = 1 and eq. (2) coincides with the li- nearized version of the ladder SD equation for the function B (k2).

Eq. (2) can be solved in the standard way used ear- lier in pure QED [2,22]. One finds that the solution of this equation is expressed through the hypergeometric function:

C.,,,(k2)=cF(½+u, ½ - v, 2; -k2/(m~S)Z), (3)

where u = ½ ( 1 - 42) 1/2 and c is a normalizat ion constant. One also finds that the solution satisfies the condition

[(l+g/2)k2C'o~(k2)+Co~(k2)]l,2=~,2=O. (4)

The analytical expression for rn~ S can be found at m~VA << 1, i.e. in the region near the critical line in the plane (2, g) , where, by definition, the ratio m{?VA equals zero. The critical line for this model has been recently obtained in refs. [16,17]. It is g = ~ [ 1 + ( 1 - 4 2 ) ' / 2 1 2 at 2<2c=-~ and 2=2~ at g~< ~ (see fig. 2 ).

At mP~/A << 1, using the asymptot ic expansion for the hypergeometric function in eq. (4) , one finds the following spectrum for the critical fermion mass I n P S :

1

9

i i

1 .4

o o !

7~ Fig. 2. The critical line in the (2, g) plane is depicted by the solid line. The shaded region bound by the critical line is the perturbative phase where the chiral symmetry is unbroken.

(1) 0<2<2 , ,=¼:

r(1 + 2 v)r2(~ - 1.,) g - (½ + u)2"~ ' /4" rnOc S=A F ( l _ 2 ~ ) F 2 ( 3 + u ) g _ ( ½ _ u ) 2 j

(5)

in the near-critical [ g - ( ½ + z,) 2 << 1 ] region; (2) 2=£=¼,g>I :

1 m~'S-~0"5 A exp ( 2(g-- ~ ) ) (6)

a t g - ¼ << 1; (3) 2 > 2 c = ~ :

I "~ (a) 2> ~, g > z - y - , where y - ½ ( 4 2 - 1 )~/2:

rn~ pS~°t-~0.5Aexp - 2 ( g _ ¼ + 7 2 ) , (7)

- , n = l , 2 . . . . . ( 8 )

a t 2 - ¼ << 1 and g - (¼ - 7 2 ) << l; (b) 2> ¼,g< ~-72 :

mps =4A exp ( ~ z ~ ) - , n = 1, 2 , . . . , ( 9 )

at 2 - ¼ << 1. The interpretation of the critical values m pS con-

sidered above leads us to the following conclusion about the spectrum of pseudoscalar composites in the

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non-per turba t ive phase of this model. In the region where 2 ..~ 2,. = a, there are N ? massless Golds tone bosons and there are no radial exci tat ions for them ~2. In the region with the supercri t ical gauge coupling 2 > 2~ = ¼ there are an infinite number of pseudoscalar excitations. Moreover, in the region with 2 > ~ and g > J - 7 2 , there is an addi t ional pseudoscalar com- posite in compar ison with pure QED (it is clear that in this region the dynamical mass mdy n coincides with the larger value from two ones, m~ °~ ~o) and m~ "s ~) [ see eqs. (7) and (8) ] ).

This spectrum admi ts the following simple inter- pretat ion. In the region with 2 ~< 2c = ¼, only the short- range four-fermion interact ion is responsible for the format ion of f e rmion-an t i f e rmion bound states. But in the region with 2 > 2c = ~ supercri t ical long-range Coulomb-l ike forces lead to the appearance of infinite number of addi t ional pseudoscalar composi tes [see eq. ( 8 ) ] ~3. When the four-fermion coupling constant g decreases and becomes smaller than ¼ - 72, one pseudoscalar composi te disappears . In this region Coulomb-like forces alone are responsible for the ibrmat ion of the composites.

Although this picture is ob ta ined only in the near- : r i t ica l region, we believe that the character of the spectrum remains, in principle, the same in the whole non-per turba t ive phase.

The analysis of the spectrum of scalar ( J P = 0 +) bound states can be realized in the same manner . The amputa ted BS wavefunction for massless scalars takes the form

T~'(k) =i2"[ C~(k 2) +~D(k 2 ) 1. (10)

Then, from the BS equat ion one finds that D (k 2 ) = 0 and the function C~(k:) satisfies the equat ion

. 12

~dq2q2 ( 1 2(mS) 2 "~ C~(ke)= J \ q2+(mg)2 [q2+(mg):]2j 0

× [2K(k 2, q2) +A -2g] C~ (q2). ( 11 )

~-" Th relation (7) cannot be directly applied at 2=0 ( v= ½ ) (the Nambu-Jona-Lasino model ). However, the conclusion about the absence of the radial excitations remains, of course, to be valid in this case too.

~3 In pure (g=0) ladder QED one can explicitly show that the appearance of a photon mass results in cutting out the levels m~ ~ ¢" with large n, i.e. in a finite number of the radial excitations [2,20].

The solution of this equat ion is the function

C s ( k 2 ) = c ( 1 + k2 ) , /2+, ,

X F ( I + / t + v, 1 + / z - v , 2 ; - k 2 / ( m S c ) 2) (12)

[ # = 1 ( 1 - 82) J/2 ] satisfying the condi t ion

[(l+g/2)k2C's(k2)+C~(k2)llk:=A~=O. (13)

Then, repeat ing the preceding analysis, one finds that the form of the spectrum of scalars is s imilar to that of pseudoscalars but the rat io of the largest values of the fermion critical mass for these two channels is

m ~ s

m s

= \ F 2 ( 3 + v ) F ( l - l z - v ) F ( l + I t - v ) 1 - 2 v J

(14)

F rom this one finds that in the near-cri t ical region the rat io m ~ S / m ~ 3 J/2 when u- . ½ (g--, 1 ) and

rn~°---~ --*exp[ 1 + R e T(1 + ½i) - T( 3 )] m~

_~ (3 .6 )1 /2=2

when u ~ 0 [here T ( x ) = d In F ( x ) / d x ] . At inter- p s S media te values o f v, 0 < u< ½, the rat io mc /mc is in

p s S the interval 3 ~/2 < m c / m c < 2. p s S Thus, the rat io m c / m ~ is always larger than one

and therefore all scalars are massive in the supercri t- ical phase ~4

This fact admi ts the following simple interpreta- t ion. Compar ing eqs. (2) and ( 11 ) we see that there is the addi t ional term - 2 ( m ~ ) 2 / [ g 2 + (mS)2] 2 in

the large parentheses on the r ight-hand side o f the equat ion for scalars. The minus sign in this term im- plies an addi t ional repulsion in the scalar channel as compared to the pseudoscalar one. This repulsion provides a non-zero mass for scalars in the non-per- turbat ive phase with the chiral condensate.

Let us apply this approach to (2 + 1 ) -d imensional

~4 In the present approximation all spinless bound states are stable. But it is clear that pseudoscalar radial excitations and scalar composites will become unstable if the approximation is improved.

638


Q E D with f o u r - c o m p o n e n t spinors. In this case the

ini t ial " c h i r a l " U (2N~,) s y m m e t r y o f the lagrangian

is spontaneous ly broken to U (Nf) × U (Nt-) [ 15 ]. The

BS wave func t i ons o f 2N { G o l d s t o n e bosons take the

fo rm

tP~( k ) =i732"C3(k2),

~P'~(k) =iv52"Cs(k2). (15)

O n e can show that in the a p p r o x i m a t i o n used by

Appe lqu i s t et al. [ 1 1,15] bo th func t ions C3(k 2) and

C s ( k 2) satisfy the same equa t ion which co inc ides

with the SD one for the f e rmion mass func t ion

B (k 2) ~5. The spec t rum of the cr i t ical f e rmion mass

takes the fo l lowing fo rm at (32Dz2Nr) - 1 << 1 [ 1 1 ]:

( - 2 ~ n ) m~, ''1 ~ a exp (32 / l r2Nf - 1 ) 1 / - ' ,

n = 1, 2,... , ( 16 )

where o~ = ~e-Nr. F r o m the d iscuss ion above we im-

media te ly conc lude that there is an inf in i te n u m b e r

o f radial exc i ta t ions In; 3 ) and In; 5 ) (n>~2) in the

model . This in turn impl ies that long-range forces

must be present there.

In conclus ion , we wou ld like to e m p h a s i z e that the

present m e t h o d is not l imi t ed by cons ide r ing the

spec t rum o f spinless b o u n d states. It wou ld be inter-

esting, for example , to apply it to the s tudy of the

spectra o f vec to r and tensor compos i t e s in gauge

theories .

One o f us (V .A .M. ) thanks the J apan Socie ty for

the P r o m o t i o n o f Science for f inancia l suppor t . He

also thanks K. Y a m a w a k i for useful discussions.

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On the spectrum of excitations in some strong coupling gauge models