Confinement and chiral condensates in 2d QED with massive N-flavor fermions

ELSEVIER

16 May l996

Physics Letters B 375 (1996) 273-284

PHYSICS LEl-fERS B

Confinement and chiral condensates in 2d QED with massive N-flavor fermions

Ram6n Rodriguez ’ , Yutaka Hosotani 2 School of Physics and Astronomy, Universi~ of Minnesota Minneapolis, MN 55455, USA

Received 8 February 1996 Editor: M. Dine

Abstract

We evaluate Polyakov loops and string tension in two-dimensional QED with both massless and massive N-flavor fermions

at zero and finite temperature. External charges, or external electric fields, induce phases in fermion masses and shift the value of the vacuum angle parameter 8, which in turn alters the chiral condensate. In particular, in the presence of two sources of opposite charges, q and -4, the shift in 0 is 24 q/e) independent of N. The string tension has a cusp singularity

at 0 = f7~ for N 2 2 and is proportional to v?“~+‘) at T = 0.

Two-dimensional QED, the Schwinger model, with massive N-flavor fermions resembles four-dimensional

QCD in various aspects, including confinement, chiral condensates, and 6 vacua [ 1,ll I. Much progress has been made recently in evaluating chiral condensates and string tension in the massive theory [ 12,161. In this paper we shall show that the three phenomena, confinement, chiral condensates, and 0 vacua, are intimately related

to each other. In particular, the string tension in the confining potential is determined by the 8 dependence of

chiral condensates ( $@ ) . The behavior of the model is distinctively different, depending on whether N = 1 (one-flavor) or N 2 2

(multi-flavor), and on whether fermions are massless or massive. The massless (m = 0) theory is exactly

solvable. (Fe), $0 for N = 1, but (.$?Ifi), = 0 for N 2 2 [ 17,181. In either case the string tension between

two external sources of opposite charge vanishes [4,8,12].

In the massive (m # 0) theory ((CHIC, jB is proportional to ~x(~--I)I(~+~) COS~~~(~+‘)( e/N) at T = 0 where

8 = 0 - 2?r[ (0 + or) /2~] [ 5,131. For N > 2 the dependence on m is non-analytic. It also has a cusp singularity

at 0 = i-n-. A perturbation theory in fermion masses is not valid at low temperature. The confinement phenomenon can be explored in various ways. One way is to evaluate the Polyakov

loop Py(x> = exp{iqJ!dTAg(7,.x)} at finite temperature T = pm' [19,23]. F(T) = -Tln(P,(x)) or

-Tin (f’q(x>tpq(~> ) measures the increase in free energy in the presence of an external charge q or a pair of

’ E-mail: [email protected]. 2 E-mail: [email protected].

0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved

PfI SO370-2693 (96) 00240-7

274 R. Rodriguez. Y. Hosotani/Physics Letters B 375 (1996) 273-284

charges q and -4. In particular, the latter is written as UIX - yI for large IX - yI where u is identified with the string tension. This method has the advantage of giving the temperature dependence directly.

Alternatively, one may determine the ground state to evaluate the change in the energy density (at T = 0) when a pair of sources of charge q and -q is placed [4,24,25]. This method has the advantage of showing how external charges affect the 6 parameter and chiral condensates.

We employ both methods in a unified manner. Years ago, Coleman, Jackiw, and Susskind showed the confinement of fractional charges in the N = 1 theory adopting the latter method [4]. Recently Hansson,

Nielsen and Zahed applied the functional integration method to evaluate the Polyakov loop correlation function

[ 121. The argument has been generalized to finite temperature by Grignani et al. [ 151. Ellis et al. [ 81 and Gross et al. [ 141 have presented the mechanism of confinement in terms of soliton solutions in the bosonized

form. All of these arguments are given in the one-flavor (N = 1) case and rely on the validity of a perturbation

theory in a fermion mass. Recently, chiral condensates with arbitrary fermion masses m, vacuum angle 8 and temperature T have been

evaluated in the N-flavor model [ 131. The problem was reduced to solving a quantum mechanical system of N

degrees of freedom. It was shown that the m -+ 0 and T + 0 limits do not commute for N > 2. In particular,

the m-dependence of physical quantities is singular at T = 0. We analyse the model

c= -t FwF”” + eIV,(y’(iJ, - eA,)}& + L,,,, , Ll=l

N

C mnss = - I{ m, ,“u M, + @. M; > (m, 2 0))

Gl

defined on a circle with a circumference L [ 24,351. The model defined at finite temperature [ 36,461, on a torus

or sphere [ 47,521, or on a lattice or light-cone [ 53,621, has been also extensively discussed in the literature. We impose boundary conditions AP( t, x + L) = A+( t, X) and 1,9~ (t, x + L) = -@a (t, x). On a circle the only

physical degree of freedom associated with gauge fields is the Wilson line phase Ow (t) [ 241,

L

e i@hv(r) _ - exp ie {S

dxAl(t,x)

0

’ . (2)

In Matusbara’s formalism the finite temperature field theory is defined by imposing periodic or anti-periodic

boundary conditions in the imaginary time (7) axis on bosons or fermions, respectively. Mathematically the model at finite temperature T = p-’ is obtained from the model defined on a circle by Wick rotation and

replacement L + p, it ---f x and x 4 7. Furthermore, the Polyakov loop of a charge q in the finite temperature

theory corresponds to the Wilson line phase,

Py(x) =exp{iqfdiAo(r.i)} oexp{iSBw(t)}.

0

(3)

We bosonize fermions in the Coulomb gauge [5,17,24,13]. Take yp = (~1, ic2) and write #i = ((crz, @! ). In the interaction picture defined by free massless fermions,

G$(~,~) = _& cf efi{&+2T’:(t*x)/L} : ,*ifi’&(tJ) : e2riP: j phys) = I phys) , (4)

R. Rodriguez, Y. Hosotani/Physics Letters B 375 (1996) 273-284

where CT = exp{ir Cil,‘(pt + pb)} and Cf = exp{hrbl (P: - ~5) ). Here

215

[4$,&l =iFb,

c#Y* (I, x) = F(4m) --li2 {c’i,n e-2Tin(rfx)lL + h.c.} , [c$,~, c!&] = Pb&, . (5) Il=l

The : : in (4) indicates normal ordering with respect to (c,, CA). In physical states p$. takes an integer

eigenvalue.

Conjugatepairsare{p~,,q,}={~~~~+~~~,qa++~},{~~,4~}={p~-p~,~~qa+-q~~},{~,~w},a~~

{&,4,, = P+ + 45). The H amiltonian in the Schrodinger picture becomes

Hi,, = HO + H# + Hmass + (constant) ,

(6)

In the mass term

where Nfi [. . .] indicates that the operator inside [ ] is normal-ordered with respect to a mass ,u. In general a mass-eigenstate field xa with a mass ,u, is related to c& by an orthogonal transformation xa = U,,&. In (7) we have

(8)

As [p,,, Ht,,t] = 0, we may restrict ourselves to states with j& = 0. H maSS gives a rather complicated coupling between the zero and &, (xa) modes, whose effects are non-

perturbative for N > 2. As in previous papers [ 131, the vacuum wave function is written in the form

2TN-1

I@“,,tW= s

~&,f(CPa;&ff) 1@0(4%+&;W (9)

0 0=l

where


{

N-l

[email protected]( po,,; 0) ) = (2~) -Ni’2 C exp in8 + i C rap, l@r+r'*...,n+rN-'Tn) ) ,

{n.r.} ll=l ?

We have generalized the expression to incorporate the phases 6,‘s in the above for the vacuum is good for mrr < e.

The eigenvalue equation (HO + H,,,,) I@,,,( 0) ) = El@,,,( 0) ) reads

where

mass parameter in ( 1). The ansatz

(11)

andE=NEL/2m(N- 1). For the xn fields, the vacuum is defined with respect to their physical mass ,uu,‘s which needs to be determined

self-consistently from the wave function in (9). In the symmetric case m, = m, one has ,u:! = . . . = ,UN, ,uy =

,u’+pi and B,= B(,xIL)'I~B(~~L) (N-')IN E i?. The potential is reduced to

NmL

Ko= (N- I)97 ,-dWjj.

Further, & is determined by

- -r/NpL

(cow)p

(13)

(14)

where the f-average is given by (g(p) )f = j’[ dp] g(p) If( p) 12. We have made use of the fact that ( ei’+‘<< >f is independent of a. Eqs. ( 11) and ( 14) are solved simultaneously. Evaluation of these equations was given in I 131. An important point in the following discussion is that f( 9; &.r) or a,,,( 0) is determined solely by m, L (or T), and f&f.

R. Rodriguez, Y. Hosotani/Physics Letters B 375 (1996) 273-284 211

Now let us evaluate the Polyakov loop at finite T, or equivalently (e ik@vf(‘) )@ on a circle. The parameter

k corresponds to q/e where q is the charge of an external source. Since the expectation value is r-translation invariant, it is sufficient to evaluate at t = 0. Making use of (9), one immediately finds

N-l

( @( 0’; cp’) leikew I@( 0; cp) ) = &.( 0 - 6’ - 2rk) n &,, a=l

c,o‘, - p: - T) e--?rkZ@L/4N. (15)

It follows that

( 4, jB.r = ( eikew )e,L=r-~

=

{

0 for k $ an integer,

e-kZnp/4NT s

[drpl f(~;&n)*f (qO + 2$;&) fork= an integer.

The vanishing of the Polyakov loop for a fractional k is due to the invariance under

(16)

large gauge transforma-

tions. The Hamiltonian (6) is invariant under &+I -+ Ow + 27r and pa + pa - 1. In other words,

(17)

which implies the vanishing of ( eikew )@ for a non-integer k.

In the N = 1 (one-flavor) case, there are no +Q degrees of freedom. Eq. ( 16) reduces to ( & )e,r = e-k2Vfi/4T,

which agrees with the result of Grignani et al. The factor k*r,u/4 is understood as the self-energy of the source

[151. In the N > 2 (multi-flavor) case the overlap integral for the f( 40; &a) factor becomes relevant. In the

massless (m = 0) case, however, f( 4p) is constant as the potential VN( pp; 8,~) in ( 11) vanishes. Hence

( Pkr )H,r = e-k2rfi/4NT in the massless theory.

When m $ 0, the overlap integral needs to be evaluated numerically. In two limits, namely T/p <

(ml& N’(Nf’) and T/p > 1, analytic expressions are obtained. It is instructive to examine the free en-

ergy F,(T) = -T In ( P, )B,T for k = I. At sufficiently low T = L-‘, KO > 1 in ( 13) so that f(q) has a sharp peak at the minimum of the potential,

e min = 2 (Dt, N ’

f( cp) is approximately given, up to a normalization constant, by

(18)

f = exp N-l -Ko COS -

2N (19)

Hence the overlap integral gives an additional damping factor in (16). In the opposite limit T/p > 1, KO < 1

so that f N constant. Hence we find, for an integer k = q/e,

k2r,u -

F,(T) = 4N I- l+(N- 1) for T << mN/(N+l)~l/(N+~)

k*n-,u (20)

4N for T >> ,u .

The free energy is finite. It does not diverge even at T = 0. An integer external charge is screened.

278 R. Rodriguez, Y. Hosotani/Physics Letters B 37.5 (1996) 273-284

N=3 m/p =O.Ol I _ +*++++**++++*++*+*++++ 0.25 -

a + e.** t 00 0 08 - + t 02 -

e

*0 l .?....eeOIIO..*.......**

F/cl I + Overlap

integral * 0 < p, 0,s N=3

: t

m/u =O.Ol

.

Fig. I (a) The Polyakov loop ( P ) and the overlap integral in ( 16) are plotted as functions of T/p for k = 1 (q = e), IV = 3. m/p = 0.0 I

and H = 0. (b) The free energy F/p = -(T/p) In ( P ) and the chiral condensate ( q@ )/CL arc plotted with the same parameter values.

In Fig. 1 we have depicted the temperature dependence of the Polyakov Ioop, free energy, and chiral

condensate in the N = 3 case with rn/,u = 0.01 and &rr = 0. Notice that all these quantities show cross-over transitions, but at different temperatures.

The vanishing of the Polyakov loop for a fractional charge q = ke does not necessarily imply the confinement as it follows from the gauge invariance. To obtain information on the confinement or the string tension, we

evaluate the Polyakov loop correlator,

G,(X) =(p,(.~)~p,(O)),,, w Gy~~e(t)=(T[e~ik~w~r~eik~w~O~l)B~L. (21)

Without loss of generality we suppose that x > 0 and t > 0 (x ++ it). Note that Gq( t) is gauge invariant.

In the N = I mode1 the correlator G,(t) was first evaluated by Hetrick and Hosotani [ 241. For a genera1 9%

c,(x) has been evaluated by Hansson et al. [ 121 and by Grignani et al. [ 151. Consider

G(t; k, 1) = (T[ e-ik@W(t) efi&V(0) ] )B.L (22)

in the massless N-flavor model, which we denote by G(O) (t; k, 1). In this case the zero modes (Ow, qu) and

oscillatory modes decouple so that the model is exactly solvable. The Heisenberg operator OW( t) can be expressed in terms of Schrodinger operators,

N

@w(t) = Owcospt + Q.r& YPwsin~t+$~~o(cospt-l).

a=1

Making use of (23), we find

e-‘k@hv(t) e+“@~(0) = exp iklr,uL ikr,uL

2N slnpt+F sin@(kcospr-1)

2ikr x exp - N(cospt - 1) cpo - i(kcospt - I)&

ikrpL - - sin@Pw

N

(23)

(24)

In taking the vacuum expectation value of (24), we encounter the factor &[ B - 8’ + 277( k - 1)] as in ( 15). It follows that


for k - I$ an integer,

_ ?!!$ (kg + 12 _ 2kje-‘Pili) for k - I= an integer. (25)

This implies that the increase in the free energy in the presence of a pair of charges 4 and -q is

Fyi’(T)(O) = -Tin ( Pq(x)+Pq(0) ). = g (s)’ (1 - e--I.Lixi). (26)

In the massless theory external charges are completely shielded and the string tension vanishes, as shown by

various authors [4,12,14,15]. If fermions are massive, the situation qualitatively changes.

analysed for which a perturbation theory in fermion masses is

and S = 0 (therefore 8,~ = e), we find

C”)(t; k, k) = G(t; k, k) - G”‘(t; k, k) M

In the literature only the N = 1 case has been

valid. Restricting ourselves to N = 1 with k = 1

=-iJ { ds ( T[ e-ik@h(l)’ ei”W(o)’ fZi,t( S)‘] ) - G'O'( t; k, k) ( Hi”,(s)’ )} 3

-m L

fMs)’ = - mB(pL) L

J { & ,+(S)‘N P[ ei6&&x)‘] + h.c. . 1

0

(27)

Here the superscript I indicates the interaction picture defined by a massless fermion. To O(m) the C#J field part

of Hint does not contribute.

In the second term in the expression of G”), (Hi,t( s) ) = mile -TlfiLcosf9. In evaluating the first term we

need, in addition to (23),

(28)

A useful identity is

(e *iq(s)’ e-ik@w(t)’ eikOw(0)’ )B = G”‘( t; k, k) eF i@ pr/~L e+vrk( 1 -e’p’)e-‘“’

(29)

Without loss of generality we take t > 0. The integral over s in (27) splits into three parts: lz,, &, and

St”. It is easy to check that the first integral is the same as the third integral after a change of variables, and

each of them vanishes. The manipulation is justified with the hypothesis of adiabatic switching of interactions implicit in the derivation of Gell-Mann-Low relations.

The second integral gives the sole contribution to G”),

G”‘( t; k, k) = iw~Be-“‘~~G’~)( t; k, k)

x jds{ ,-i+ 2*ike-i~k(r-““-“l+e-‘~~) -1 +(B+-O,k+-k)

> (30)

0

The integral is expressed in terms of Bessel functions. The correction to the free energy is, after making a Wick rotation it = x > 0.

FP~~‘(-Q)“’ = -TG”’ Y I

G’O’ = -2l~lmTBe-“f“~ Jo(2rkz) cos(8 - 2rk) - cos6’

280 R. Rodriguez, Y. Hosofani/Physics Levers B 375 (1996) 273-284

loo +/4x1 n=, -c( e -r(B-27rk) + (-1 jne+i(B-27rk)

> fwn - z”)J,,(27Az) ,

1

where z = e -~/‘I/‘. For ,u]x~ > 1, z K 1, so that

Here (T is a “string tension”. Since (F$)o = -2Te-1TipB(p/T) UN@, we find

(31)

(32)

In other words, the major effect of a pair of external sources of charges q and -q is to shift the 0 parameter

in the region bounded by the sources by an amount 2r( q/e), which changes the chiral condensate [4,12 1. A

linear potential results because of this. We shall show below that this is true even for N 2 2. The expression

(33) is valid at arbitrary temperature. The string tension (T can be either positive or negative, depending on the values of 8 and q/e. This implies

that the 8 $0 vacuum is unstable against pair creation of sufficiently small fractional charges.

The perturbation theory in fermion masses cannot be employed in the N 2 2 case as physical quantities are

not analytic in m at T = 0 [ 5,131. The perturbation theory can be applied only in the high temperature regime.

There is a better way to explore the problem. We place external charges on a circle and solve the Hamiltonian

as was done in the N = 1 case in [ 241 and [25]. In the presence of external charges f&, = -Aopext, Gauss’s law implies

a,&xt(X) = PedX) . (34)

Let us restrict ourselves to static sources pext(x) where Jo’ dx pext = 0. Then EeXt(x) = Ei,9’ - Art(x)‘. Here

E”’ is constant and Aext(x) ext 0 = - Jt dy G( x - y ) p,,.(y). In particular, for a pair of sources located at n = 0

and at x = d,

text = q{&(x) - ~L(X - d)}, Eexttx) = E:,9’ + Em(x)(‘),

forO<x<d,

E (‘) ext = -AT’(x)’ = (35)

ford<x<L.

Note that lo” dx E,,t(x)c’) = 0.

Suppose that m, = m << ,x. The total charge density is &,, = C,“=, @A&, + Pext, and the Coulomb energy

becomes

L L

ffCoulomb = - 1 .I’

dxdy_&&,x)Gtx - Y)_&&,Y) = J dx ;(,ux, - E;;,))2. (36)

0 0

Here xi = N-Ii2 Cy=, 4,. In view of (36) we write

R. Rodriguez, Y. Hosotani/Physics Letters B 375 (1996) 273-284

The total Hamiltonian is now

H i? = Ho + &I + H, + Hmss , L

H,,= (,&‘)2+(~~?-E:;~)2 0 L

281

(371

(38)

HO and Mass are given by (6). When m < ,u, ~1 N p. In H,,,,,,

Np [ e’++XI ] = ,;&h? Ng [ e’@,tl ] . (391

In other words, the net effect of a pair of external sources is to give x-dependent fermion mass phases

8, = ,/&$?xt’( x) in ( I). Suppose that CL-’ < d << L. Sufficiently away from the source,

N $ E&r)(‘) = for p”-’ < x -C d - ,c’ ,

(40) _- - for d + p-l < x < L - p-’ .

Finding the exact form of the ground state wave function of (38) is rather involved. Instead, we content ourselves with finding an approximate wave function, noticing that 8,” is almost constant between the two

sources. The entire circle is divided into two regions, the inside region 0 < x < d and the outside region d < x < L

(,u”-’ << d < L). For the evaluation of local physical quantities in each region, one can approximately write

the ground state as a direct product of ground states in the two regions: \W,) N (ly)i” 8 /P),,U,. In the absence

of sources, jPC) N l@)i” @ le),,,. In the presence of sources,

IT,) N 10 + 68; Sin)in @ 10 + 80; Put),“, . (41)

In addition to the effect of gff an overall shift 68 in the 8 value results as s”,” is x-dependent. After all, there is only one 0 parameter globally.

To determine 68, we utilize the fact that local physical quantities in the infinite volume limit L -+ cc must

reproduce results in the Minkowski spacetime. In particular, physics in the outside region d+p-’ < x < L-,u”-’ (L + a) must be essentially the same as physics in the absence of sources. In other words, 10 + 68; Put),,, N

(8; w = O),,,. As shown above, physics depends on 8 through the combination &.ff = 0 - C,“=, 8,. This determines

60 Npt= -2!!!d, = e L

Hence, the net effect is summarized by

IpJ - l&ff)in ~3 le),,, , eeff = 8 - T. (43)


Consequently, the change in the energy due to the external sources is, to O(d/L),

AE= Nmd(i&$>e,.f, - iB$),) -

so that the string tension is

(44)

The result generalizes to finite temperature, as was seen above in the N = 1 case. Note that the parameter B is not completely equivalent to the electric field E. Indeed, in the absence of

sources, ( E). = ( ePw - Al, )e = 0. External sources, or external electric fields, induce effective fermion mass

phases s’,” in the Hamiltonian, which in turn changes the effective &rr through the chiral anomaly. We also remark that the Coulomb energy (36) is 0[ (d/L)‘]. The linear potential results from the change in the chiral

condensate.

The chiral condensate ( qt+b )H at arbitrary temperature T = L-’ has been evaluated in Ref. [ 131. With given

m, the dependence of 0 on charge 9 is essentially the 6’ dependence of (p$)o. At T = 0 it has a CUSP at

8,fr = 71 (mod 2~). More explicitly,

a;* = +*I$ (zu’ ;)‘“‘““‘{ ( cos ,)2N’(N+” _ ( cos ,)2N’(N+‘J}. (46)

Notice the singular dependence of g on m as well. A mass perturbation theory cannot be employed at low

temperature for N > 2. In the high temperature limit,

(47)

for N 2 3. There appears no B dependence to this order. Hence the string tension g is at most O(m3) in this

regime.

For N = 2, the expressions for (q$ jH in (47) is multiplied by a factor 2cos2 i@. Therefore,

(48)

For a general value of T/p, o- must be evaluated numerically. In Fig. 2 we have displayed the q/e dependence

of ~/J.L’ at T/p = 0.003, 0.01, 0.03, and 0.1 with m/p = 0.01 and 0 = 0 in the N = 3 case. One can see how a cusp behavior develops at q/e = 0.5 (0,~ = n) as the temperature goes down. At higher temperature the magnitude of the string tension rapidly diminishes. For instance, u/p2 = 1.8 x 10m7 at T/p = 1 and q/e = 0.5.

In this paper we have shown that the confinement of fractional charges in the massive N-flavor Schwinger model results from the effective change in the 8 parameter which alters chiral condensates. In the multi-flavor case (N > 2) the string tension at zero temperature has a singular dependence on fermion masses and the

a-parameter.

This work was supported in part by the US Department of Energy under contracts DE-FGO2-87ER-40328 (R.R.) and by DE-AC02-83ER-40105 (Y.H.). Y.H. would like to thank Jim Hetrick and Satoshi Iso for useful communications.


Fig. 2. String tension a//~,* in (45) is depicted as a function of q/e at various temperature in the N = 3 case with m/p = 0.01 and @ = 0

A cusp develops at q/e = 0.5 in the T -+ 0 limit.

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Confinement and chiral condensates in 2d QED with massive N-flavor fermions

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Transcript of Confinement and chiral condensates in 2d QED with massive N-flavor fermions