Wilson fermions in a two-dimensional model

Nuclear Physics B212 (1983) 461-500 © North-Holland Publishing Company

W I L S O N F E R M I O N S IN A TWO-DIMENSIONAL MODEL F. GUI~RIN

Physique Theorique, Universite de Nice*, Nice, France Received 14 June 1982

Wilson's action for fermions on a lattice is compared to the continuum action in a model obtained from the chiral Gross-Neveu model by performing a chiral transformation. The local definition of the axial current leads to two anomalies unrelated by the constraint of Lorentz invariance. In the large-N limit, the mass counterterm of the action is determined; this term is unnecessary in the Osterwalder-Seiler regularization. An expansion in the fermion propagator and in the axial current coupling may be formulated and summed to all orders for large N.

I. Introduction

Lattice gauge theories [1] provide a tool for the study of non-perturbative phenomena. One hope is to extract physical insight by comparing the numerical results of the Monte Carlo method [2] with the results of analytical methods, even approximate, such as the strong coupfing expansion [1, 3], the mean field approach [3, 4] or the variational approach [5]. The extension of the numerical simulation to theories with fermions [6] encounters the problem of the choice among the lattice actions with fermions. The naive discretisation of the continuum Dirac action is chiral invariant but produces too many fermions: the degeneracy is 2 d in d dimensions. A lattice fermion theory without species doubling either breaks explicit continuous chiral symmetry or does not lead to the expected continuum limit in weak coupling perturbation theory [7]. Chiral symmetry is supposed to be an approximate symmetry of the strong interactions. In the continuum limit of QCD, the flavour axial singlet symmetry is explicitly broken due to the triangle anomaly [8]; however, the flavour non-singlet axial symmetry should be realized in a spontaneously broken way. It is now established [9, 10] that the spontaneous breaking of chiral invariance is a general feature of the strong coupling limit of lattice gauge theories, in the approximation where internal fermion closed loops are neglected. It has also been found in numerical simulations [11]. One effect of the inclusion of fermion loops should be to introduce the chiral singlet anomaly. * Equipe de Recherche Associ~e au C.N.R.S. Postal address: Physique Th+orique, I.M.S.P. Universit~ de Nice, Parc Valrose 06034 Nice Cedex, France. 461

462

F. Gu~rin / Wilsonfermions

We review the several ways of avoiding the multiplicity of fermions that have been proposed. The SLAC fermions [12] have manifest chiral invariance, but further specification is needed [13] to reproduce the expected continuum limit in weak coupling perturbation theory. Wilson's modification [1] to the naive fermion action is to split the mass of the fermions by giving to (2 d - 1) fermions a mass of the order of the cut off; it leads to the expected weak coupling continuum limit [14, 7], but a zero physical mass is not natural to the theory: a mass counterterm has to be carefully adjusted to obtain that result [15]. Osterwalder and Seiler [16] have proposed a related regularisation scheme where the mass-splitting term violates parity in addition to chiral symmetry. Susskind's method [17, 18] is to make use of the set of discrete transformations relating the 2 d fermions and to project on the eigenstates of the 2 d/2 commuting elements of that set. The degeneracy is reduced to 2 d/2 and the action has a discrete chiral invariance which prevents the occurrence of a mass counterterm if one starts with a zero bare mass. Banks and Casher [19] have proposed a regularisation for the fermion determinant based on the second-order Dirac equation which gives the expected weak coupling continuum limit; however, one parameter has to be tuned to obtain a zero renormalized fermion mass. Two-dimensional models provide a testing ground for the properties of these fermionic actions [12, l l]. In addition, the large-N limit of these models possess simple analytical solutions and a comparison can be made of the results of the continuum and lattice regularisations. In this paper we shall consider the chiral Gross-Neveu (GN) model [20]. This two-dimensional model possess a continuous chiral symmetry which cannot be broken. The complete solution of the model through the Bethe ansatz [21] shows that the fermion particle which spontaneously acquires a mass, does not carry any chirality; the latter is carried by a massless particle. The lattice regularisations which do not have explicit chiral invariance cannot be used for the action of this model. To circumvent this difficulty we turn to an equivalent action. In sect. 2 we shall perform a chiral transformation on the fermion field of the chiral G N action; the resulting action will contain a fermion field unaffected by the global chiral symmetry and two types of four-fermion interaction, a scalar one with coupling strength g2 and an axial-axial one whose strength ~r/N is determined by the anomaly. The large-N limit of this action will show the following features: the fermion acquires a mass and one fermion-antifermion bound state exists at threshold in the scalar channel; the axial density fluctuates about zero and no particle pole is generated in that channel. We then consider a lattice regularization of the model. Our choices will be (i) the Wilson (W) regularization and the related one by Osterwalder and Seller (OS); (ii) the local version of the axial current rather than the non-local one recently proposed [ 18]. This lattice regularization has been shown to be equivalent to the Pauli-Villars one for the vacuum polarization tensor and the triangle anomaly [7]. In sect. 3 it will be

F. Gu~rin / Wilsonfermions

463

pointed out that these regularizations differ for the fermion loop with two axial currents: in two-dimensions the axial-axial anomaly will be found to depend on one parameter of the lattice regularization and will not obey the constraint imposed by Lorentz invariance on the two anomalies. As a byproduct, one feature of Wilson's regularization will appear: the 2 d - 1 large mass fermion poles are present in the amplitudes and the limit of the lattice spacing a ~ 0 has to be taken carefully in order for these poles not to affect the physical results. This problem is avoided in the OS regularization where the physical mass does not interfere with the degeneracy regularization. In sect. 4 we shall study the large-N limit of the lattice regularization for the transformed chiral GN model obtained in sect. 2. The dynamical breaking of the discrete chiral invariance of the model is correctly reproduced for a specific value of the mass counterterm of the W action. The dependence of this counterterm on the cut-off is determined. No such counterterm is needed in the OS regularization. The continuum results are reproduced in the axial-axial channel. In sect. 5, to leading order in 1/N a gauge-invariant expansion in the fermion propagator and in the axial coupling will be formulated for the W and OS regularizations. This expansion will then be summed to all orders and the large-N results of sect. 4 will be recovered. 2. Anomaly and chiral transformation of the fermion fields 2,1. RESULT OF AN INFINITESIMAL CHIRAL TRANSFORMATION

In an euclidean, two dimensional space-time we consider the lagrangian ~ ( x ) = - ~ ( x ) ( ~ + iV+ il#'y5 + m)t~(x),

(2.1)

where V~and W~ are external vector and axial fields, y~ and 3'5 are the Pauli matrices (~,~,y,= ~,~- ie~y5 ). The generating functional for the one-fermion loop amplitudes is

An infinitesimal chiral transformation of the fermion variables results in ~p(x) ~ eiY'8*(~)q~( x ) ,

~m~k ~ ~m(1 + 2iysSq~(x)) ~b.

(2.3)

The anomalies a(x) of the axial current should be generated in that transformation,

F. Gukrin / Wilsonfermions

464 i.e. one should have

8e (x).= 8 ~ ( x ) ( Oi,( ~i'r, Y5q J ) - 2mfiysqJ + a ( x ) ) .

(2.4)

The presence of the anomaly in (2.4) has recently aroused interest. Kerler [22] has shown how its appearance may be connected to the necessary breaking of continuous chiral invariance by the part of the lagrangian describing the gauge-invariant regularization [8]. Alternatively, Fujikawa [23] starts from the ill-defined lagrangian (2.1) and regulates the path-integral measure: under the infinitesimal chiral transformation a jacobian is generated, D+ D q 7 ~

D+DqTexp(f

d2xSdp(x)a(x)).

Eq. (2.4) wiU be our starting point. 2.2. ANOMALIES IN TWO DIMENSIONS The axial current-vector current loop is finite in two dimensions (due to a trace anomaly); however, by the standard argument [8] the divergence of the axial current can be written as a difference of two ill-defined linear divergent integrals plus a mass term. If one uses the Pauli-Villars regularization, i.e. subtraction of a heavy mass M, the linear divergent integrals cancel but a new term is generated:

qjdWV= ieo~qp ( i(q2, m 2) _ i(q2, M2)), with m2

1

I( q2, m2 ) = fo dx-~ mZ + q 2 x ( 1 - x ) '

(2.5)

so that in the limit M --+ oo

') .

(

q~Idw v = ieo"q p I( q 2, m 2 ) --~ This calculation leads to the divergence equation i

O~( fA~'ts q, ) = 2mf/i3's qJ + -~ %, OoV, ( x ) , and this anomaly appears in (2.4):

i al( x ) = - -~eo~OpV~( x ) .

(2.6)

F. Gudrin / Wilson fermions

465

Another anomaly exists in the axial-axial current loop. The same argument leads to

q.I~.WW__ - - G ( Z ( q 2 , m Z ) - z ( q a , M2)) and in (1.4) to 1

a 2 (x) = ~ O~W~(x).

(2.7)

These two anomalies are related since in two dimensions Y~Y5 = iGoYo, so that an axial current may be treated as an effective vector current Vpeff--~ ie~,om. .

(2.8)

For later comparison with lattice regularizations we give the two-current loops in the Pauli-Villars regularization, which obeys (2.8):

IVV=(g,.-q.qJq2)F(qZ,

ma),

Iwg=i(%,~-%,oqpqJq2)F(q2, m2), l~,~w = _ q~,q~/q2F( q2, rn2 ) ,

(2.9)

with

F(q2, m2)=I(q2, m2)

1 9I"

and I(q 2, m e) defined in (2.5). It is worth noting that all three amplitudes vanish as q2 ~ 0, and that

ilL(q2)= lf~(q2)= i1~(q2).

(2.10)

Are these anomalies unique or may they depend on the regularization? Another regularization may differ from the Pauli-Villars one by terms of dimension two:

(x) = Evv (x) + aV.V~ + flie~.V~W. + yW~W~, with

£pv ( x ) = - ~ ( ~9+ iV+ iWy 5 + m)~p - ~( ~ + iV+ iWy 5 + M )4~,

where the 4~ field obeys Bose statistics; a = fl = 0 by gauge invariance but the last

466

F. Guerin /

Wilson f e r m i o n s

term can be eliminated only if one imposes on the lagrangian the additional invariance under the transformation

V~(x)~ V~(x)+q,(x),

W~(x)--+ W~(x)+ie~,ovo(x),

(2.11)

so that only the combination (V~ + ie,~,W,) may appear in e and a(x), We shall encounter lattice regularizations which do not have the invariance (2.11), for which ~, will differ from zero. This extra term will generate in eqs. (2.2) and (2.4) an extra anomaly in the axial-axial loop, and we write instead of (2.7)

a'a(x) = I O,,W,, + 270,,W,, = b o,,W,,. ,,l"r

q'r

(2.12)

2.3. THE RESULT OF A FINITE CHIRAL TRANSFORM

Roskies and Schaposnik [24] notice that the term arising from the anomalies in an infinitesimal chiral transformation can be written in (2.4):

a(x)SeO(x) = 8eO(x) l o,,(-ie,,~,V~ + bW,,),

a(x)~,c,(x) =

- L ~W~(x)(-i~v~

+

'7"1"

bW~),

since from (2.3) O,,(Sq,(x))= 8W,,(x) is the change in the axial current generated by the chiral transformation, so that

a(x ) 8 ~ ( x ) = 1 ( 6W~ie~,V~ - ½bS(W,W,)). In a finite chiral transformation, the terms resulting from the succession of infinitesimal transformations just add up for the second part of the anomaly, leading to

~,az( x )aeO( x ) = - 2-~ ( W,,WS ~ - W,,W,,i~tia~) .

(2.13)

Roskies and Shaposnik give the Schwinger model as an example. The initial lagrangian is

e= -~/ ( O + ielV)q~-¼r~,,,r~,,, with F~, = O~,V,,- 0,V~. A chiral transformation will lead to ,l,( x ) = e',,°~x)~,( x ),

e'(x)= ~,(-#){-

e2

' -2~ - Vy~, ~F~.F~.-

F. Gu&in/ Wilsonfermions if

O(x)

467

is chosen so that

eV~ = ieouOoO(x ), in a regularization which obeys the invariance (2.11). £' describes a massive vector field and a massless fermion field.

2.4. THE TRANSFORMED CHIRAL GROSS-NEVEU MODEL The chiral Gross-Neveu model [20] possesses a continuous chiral symmetry that cannot be broken. We shall perform a chiral transformation of the initial lagrangian and obtain another lagrangian where the conservation of the chiral current will be a triviality. The lagrangian of the chiral Gross-Neveu model is

~= _~O~p + ½gZ((ftp)2

(fy54,)2),

(2.14)

where ~p describes a set of N fermion fields. We introduce auxiliary fields o ( x ) and It(x) and alter the lagrangian density

ff,--*£-

2--" 2 ), 1 ((o+g2¢~p)2+(~r+g,p,3,5~p) 2g 2

then change to the auxiliary fields

O(x)

02 _{_ ~ 2 .~. p2,

and

O(x)

defined by

o + iys rr = pe ivS°.

The vacuum to vacuum amplitude is

ew=SPDoDODq, D+exp[-fd2x(+O~+p+eiv'oq~+-~g2) ]. (2.15, This lagrangian is invariant under the transformation

~p(x)--+e'V'°~(x),

O(x)--,O(x)-2a.

We introduce a source for the associated conserved current:

e'(a)=-

(

~(O+i%y,V,

pi

Nb

)

+ 0 e ' V , o ) ~ p + ~ g Z + 2---~-%a, .

F. Gu~rin / Wilsonfermions

468

E'(a) is invariant under the transformation ~ ( x ) --* em~°(x)q,( x ) , O(x)~O(x)-260(x),

,,~(x) -~ ~ . ( x ) - o~(~O(x)). The NbaZ/2~r term is needed to cancel the term arising from the anomaly of the N fermion loops, as discussed in subsect. 2.1. We now perform the finite chiral transformation

+ (x) = e-,,,o(~)/~¢ (~). E'(a) is transformed to

e,,.ep= t )

T t . + i , %t _ O . O / 2 , T . )V s + p , r )W v/k#

2gz

Nb2~r(a._O.0/2)2, (2.16)

since, according to (2.13), the contributions of the anomaly sum up to

Nb 2 ~r

((

0)2)2

- 0.-~

%

- a.

~"(a) is invariant under the transformation

¢-~¢, a.0(x) -~ a A ( x ) - 20.80(x), ~.(x) -~ ~.(x) - 0.~0(x). The ~ field does not carry any chirality and the conservation of the chiral current of the initial lagrangian has turned into the equation of constraint of the auxiliary field. Integration over the auxiliary field O(x) in g"(a = 0) produces a four-fermion interaction

~r 8.8~ _ 7r ( 0.0~) 2Nb~biT,75qJ---~iT~Ts~l/=~-N--[~qdT.~ g . ~ - 0----5- ~T,~P,

(2.17)

where the equality holds in a regularization which obeys the invariance (2.11). The same lagrangian E"(a = 0) has been obtained by the same method by Duerksen [25]. However, we shall disagree with his conclusions in the large-N limit.

F. Guerin/ Wilsonfermions

469

It is convenient to substitute to O(x) the auxiliary field W

(x) = - a , ° ( x ) , ,L

with the auxiliary condition

The transformed lagrangian ~"(a = 0) (2.16) reads

(

2g 2

~

2.5. THE LARGE-N LIMIT OF THE TRANSFORMED CHIRAL GN MODEL The vacuum to vacuum amplitude is

eW=fDCDf~DpDW.exp-fd2x

~(O+ilJ/ys+p)~b+~g2

+~--~W~W~

,

(2.18) with the auxiliary condition

e~O,W~(x)=O.

(2.19)

A gauge-invariant regularization is assumed for the fermion coupling to the axial field; b/~r is the axial-axial anomaly, one should have b = 1 by Lorentz invariance. The lagrangian possess the discrete chiral symmetry 4~~ 3'5~P,

P ~ -P

The integration over the fermion fields of the action of (2.18) leads to an effective action

l(o,W~)=N[Trln(O+iW,,+p)-f d'x( ~PZ+ __b . W~W r r _ ~_In p) Tr is a trace over space and Dirac indices. The leading contribution for large N comes from the stationary points. Restricting to constant fields p(x) = #, W~(x) = W~, they obey f

d2p tr

1

Wrs)+ -~ f bw~_j~

d2p tr

iT~T5 i(p+W-/s)+ff

~ 09

(2.20)

=0.

(2.21)

F. Gu~rin / Wilsonfermions

470

Different regularizations will be used for the two kinds of fermion loops. In the p sector, the divergence can be absorbed in the bare coupling constant and the model will display asymptotic freedom; we restrict the integral to p2 < A2. In the W~ sector, the Lorentz-invariant, gauge-invariant Pauli-Villars regularization is used, i.e. one sets b = 1 in the lagrangian of eq. (2.18) and adds a term at=

+

+ m),,

where ¢ describes N fields of large mass M, obeying Bose statistics. From eqs. (2.20), (2.21) the stationary points corresponding to the minimum of the potential obey

=0,

__fa

d2p

2

(2r:)2 p2 + t52

0.

(2.22)

Expanding the action I(p, W~) around p(x) = ~, W~(x) = 0, the quadratic terms are identified with the # and W~ propagators. The p sector is identical to the large-N limit of the Gross-Neveu model [20]. The W~ propagator involves the axial-axial fermion loop encountered in eq. (2.9)

D~,(qZ)=N[~_

~ + 7q~'q.{ ~ i ~ q " 2 ,,if)

1)] ,

(2.23)

with i(q2, if2) defined by eq. (2.5)

q~qv

(2.24)

The auxiliary condition (2.19) on the W~ field says that the second term does not contribute to the propagator since

. g~-

q,q, 1 q2 =-Se~pqoevoqo. q-

The anomaly disappears and one is left with a W~ propagator which has a constant sign and no singularity for q2 > _ 4ff2. The result at q2 = 0 may be simply stated in terms of the four-fermion interaction. The leading order contribution to the four-point function is the sum of graphs with fermion loops as illustrated in fig. 1, with the four-fermion interaction given by (2.17). At zero momentum, the fermion loop vanishes as a consequence of Lorentz invariance [see eq. (2.9)] and one is left with the contact interaction, the scattering amplitude is finite. This conclusion disagrees with Duerksen's conclusion [25] of a massless particle in that sector.

F. Gu~rin / Wilsonfermions

471

Fig. 1. Leading order, fermion four-point function.

To conclude, in the large-N limit the model's features are: The fermion has acquired a mass ~. In the ~ channel the theory is asymptotically free and identical to the Gross-Neveu model. The p propagator shows a fermionantifermion bound state at threshold. The axial field W~ = -O~O(x)/2 has small fluctuations around W~= 0. The W~ propagator shows no sign of a particle in the qTi]'~'ts~Pchannel. 3. A lattice regularization for the fermion loop involving two currents There are several ways of putting fermions on a lattice, the so-called naive, SLAC, Wilson, Susskind, Casher-Banks fermions. Our choice will be motivated by the model's properties in the large-N limit: (i) the continuum limit should be obtained for weak coupling of the gauge fields; (ii) the axial current should be associated with a chiral transformation which exponentiates simply; (iii) an expansion in the fermion propagator should be allowed. In order to satisfy (ii) we choose the local version of the axial current and discard the non-local one, recently proposed by Sharatchandra, Thun and Weisz [18]. Our choice will go to the Wilson regularization [1] and to its related one proposed by Osterwalder and Seiler [16]. Although similar, they will have different analytical properties. For these choices, it will be shown that the axial-axial anomaly depends on the parameter r of the regularization and does not satisfy the constraint of Lorentz invariance in the limit of the lattice spacing a ~ 0. The other parts of the amplitude will have the correct continuum limit. For this regularization the triangle anomaly and the vacuum polarization have been thoroughly studied by Karsten and Smit [7] in four dimensions. Our study of the corresponding amplitudes will be an illustration of their results in the very simple two-dimensional case where analytical results are transparent. A new feature will be encountered in the axial-axial current fermion loop. We first consider the naive lattice fermion, then study the Wilson (W) and Osterwalder-Seiler (OS) regularizations. In two dimensions, their common features are (i) The propagator carries four fermions, in momentum space they correspond to the four regions of the Brillouin zone Px =Py = 0; Px = O, p y = ++~r/a; p y = O, p x = + ~ r / a ; Px = + , r / a , p y = --k ~ ' / a .

(ii) All fermions have the same vector charge, two of them have an axial charge + 1, while the two fermions with one component of p = ~ r / a have an axial charge --l.

F. Gu~rin / Wilsonfermions

472

(iii) The W and OS regularizations give three fermions a mass of the order of which goes to infinity as a ~ 0.

3. I. T H E N A I V E L A T T I C E R E G U L A R I Z A T I O N

AND THE FERMION

1/a,

LOOP

The lattice version of eq. (2.1) is [1,7, 14] ~N = -- '~

1

-~a [ f/(x)y~V~,,(x + a~,) - ~ ( x + a~,)y,U~+qJ(x)] - m~ (x)qJ(x),

(3.1) where a t is a vector along the/~ direction with length a, and U~= exp(ia (V~(x + 1 % ) + .rsl,V~(x + ½a~))).

(3.2)

Eq. (3.2) is appropriate to the weak coupling of the gauge fields; the line integral of the fields has been approximated [14] fxx + aj,

dxV~(x)=aV~(x + ½a~) + O ( a 3 ) .

In dealing with anomalies, one has to keep track of terms of order a in the lagrangian since a loop may be of order 1/a. For m = 0, EN is invariant under the following transformation:

,; (x) --,

(x),

aV~(x + ½a~,) ~ aV~(x + ½at)+ a ( x + aN)- A ( x ) , aW~,(x+½a~,)--*aW~,(x+½a~,)+O(x+a~)-O(x).

(3.3)

The Feynman rules obtained from eq. (3.1) are the following [7]. The range of momenta is restricted to - ~r/a < p~ < + ~r/a. The fermion propagator is

S -'(p)

=

E i'/,1 sin a e ,

+ m.

l

There are vertices with multicurrent lines. The first vertex functions involving the currents are illustrated in fig. 2. The ~qJV~ and ~pW~ vertex functions are

Fv,(p, p') = i,6,cos½a(p +p')~,

Fw~(p, p') = iy~,yscosla(p +p')~,

F. Guerin / Wilsonfermions t~

473

~

p'

p

p'

Fig. 2. Fermion-currentvertices. and the vertices with two currents are

Fv~v. ( p, p') = Fw~w. ( p, p') = - g~,~aiy~,sin ~-a( p + p')~,, Fv.w. ( p, p') = - g~,~aiy~,yssin ½a ( p + p')~. Fig. 3 shows the diagrams contributing to the amplitudes involving two currents I vv, 1 wv, I ww. Diagram (b) is a q-independent counterterm. Gauge invariance reads ' ~aq~ , _- 0 = /~w Vsin ½aq~. I ~v zsm

(3.4)

A direct check of eq. (3.4) makes use of the property that the periodicity of the integrand allows shifts in the internal momentum. Amplitudes at zero momentum. The integrand in the vector-vector current amplitude is a total derivative, since inspection of the lagrangian (3.1) shows that a derivative with respect to V~ is equivalent to a derivative with respect to the fermion momentum

vz ( q = O ) = 1~.

; Op~ap~ 0 0

trln(iysinpa+ma)=O

where

fp=

a

(3.5)

2"

b

Fig. 3. One-fermionloop contributionto the amplitudesinvolvingtwo currents.

474

F. Gu~rin / Wilson fermions

Also

)

sin ap~ = 0 , I;WV( q = O) = fp ie~'~c°s(ap~') O~-~ a-~u-(-~

(3.6)

D N ( p ) = ~_~sin2apt + (ma)2.

(3.7)

with

l

However, the axial-axial amplitude is non-zero:

I~WW( q = O) = I f v( q = O) - g.. f a 2 4( ma )2 c°s2ap, , ~p D ~ ( p ) so that in the limit a ~ 0 ww

IL

(q = o) =

4

The four fermions carried by the propagator contribute equally to that result. The axial-axial amplitude does not satisfy one of the constraints of Lorentz invariance [see eq. (2.9)]. One way to state the problem is that no part of the counterterm changes sign when one goes from the I vv to the I ww amplitude, in contrast to the Pauli-Villars regularization. Momentum dependence of the amplitudes. We examine the constraint (2.10) of Lerentz invariance IVY (q) = I~.~ ww (q) = fp a2

N(p,q) Du(P+½q)Du(p-½q)

'

(3.8)

with D u given by (3.7) and N ( p, q ) = 4 cos apxCOSapy (sin ½aqx sin ½aqyCOSapxcos apy - ( p ~ ½q )).

(3.9)

In the limit a ~ 0 the integrand is peaked towards the fermion poles

ww (aqx)(aqy) 4 ( 1 - ½ ) + IVVv(q) = I;.~ (q)a~o (ma) z

O((q)4).

(3.10)

This is the expected result (2.9) except for a factor of 4 coming from four types of fermions. The same results holds for the/~ = u terms to the same order. However, the wv I~=~(q) amplitude vanishes identically: two of the ferrnions contribute positively to

F. Guerin /

Wilson fermions

475

the amplitude, two negatively; in equation (3.9) the factor cos aPxCOSapy is replaced by cosZap~.

3.2. THE WILSON AND OSTERWALDER-SEILERREGULARIZATIONS AND THE FERMION LOOP These regularizations will split the masses of the four fermions and three fermions will have heavy masses of the order of 1/a, playing the role of the heavy mass of the Pauli Villars regularization. The most dramatic change will be in the I wv amplitude which will have the correct continuum limit. The lack of Lorentz invariance of the I ww anomaly will persist. Wilson [1] adds a term to the action

lw=EeN(x)+L,

(3.11)

x

1} 11= Y" ~-~-d[ ~p ( x )rei"V.(x+~./2)t~ ( x + a.) + ~b ( x + au)re- i~V.(x+~./2)~b( x ) X,~

- 2 ~ (x)rq~(x)],

(3.12)

where r is an arbitrary parameter and EN is given by (3.1). 1(1) breaks chiral invariance, under the gauge and chiral transformation specified by (3.3):

Ir(1)'-'~ E ~---d[~P(x)rei~V"eiVs(°(x)+°(x+~"))~P( x

+at,)

x,p,

+~p (x + a~)re-i"V, eiV'(°(x)+°(x+a,))qJ(x) - 2 f (x)reiV52°(x)tp(x)]. (3.13) Osterwalder and Seiler replace r by riy 5 in (3.12); the extra term now also breaks parity. We shall consider successively three versions for the I r term: (i) the axial gauge field is not coupled to the r terms as in (3.12) and the vertices involving axial currents are unchanged with respect to the naive action; (ii) the axial field couples to the r terms so that either all these vertices are modified, or only those with an even number of axial lines are changed. Going from action (3.1) to action (3.11) the modifications to the Feynman rules are: the fermion propagator is S - l ( p ) = E i , / l l sin /

ap, + ra E (1 - cos ap,) + m ; l

F. Gu&in / Wilsonfermions

476

the vertices with one and two vector current lines are [7]

l"v~( p, p') = iT~cos ½a ( p + p')~ + r sin ½a ( p + p ' ) . , rv, v. ( p, p') = - a g ~ (iT'~sin ½a( p + p')~ - rcos ½a( p + p ' ) , ) . The extra terms in the vertices guarantee the gauge invariance of the amplitudes. The Ward identity reads

0

0__ s - ' (

Fv,,...v,( P, P) = O p t , " " Opt,.

P )"

Replacing r by ri'y5 gives the rules for the OS regularization. The main effect of the extra terms is to modify the denominator of the integrand in the fermion loop. Eq. (3.7) is replaced by

Dw(p, m)= E sin2apt+ (aM(p) +am) 2,

(3.14)

l

D o s ( P , m) = Esin2apt + ( a M ( p))2 + (am)2,

(3.15)

1

with

a M ( p ) = Z r ( 1 - cos ap,).

(3.16)

l

The W propagator carries one fermion with mass m 1 = m from the region px = py = O, two fermions with mass m 2 = m + 2 r / a from the regions p~ = 0, py = ~r/a and Px = ~r/a, py = 0, and one fermion with mass m + 4 r / a . The OS propagator carries fermions whose mass are respectively m , ( m 2 -4- 4 r 2 / a 2 ) 1/2, ( m 2 + 16r2/a2) 1/2. Amplitudes at zero momentum. By the same argument as for the naive fermion action

I~W~V( q = O) = 0 = IVV( q = 0). For the amplitude I w w (q = 0), the vertices are unchanged and the consequence of the modification of the propagator appears in full. The W and OS propagators will lead to different analytical properties in the limit a ~ 0. Fig. 3 shows the diagrams contributing to I ww; one obtains

I~W(~ w) ( q = O) = - g ~ 2 fpa 2 c°sZapx D 2 ( P ) '[ D ( "p ) " - 2sinZapy),

I/,~w(b%=

sin2aPx

_

g~,~2~p f a 2 D(p)

(3.17)

(3.18)

F. Guerin / Wilsonfermions

477

where D ( p ) is either D w ( p, m) given by (3.14) or Dos(P, m) given by (3.15). Both contributions have the same sign and one does not obtain I ~ W ( q = 0 ) = 0 as required by Lorentz invariance. It is interesting to examine the value of this amplitude in the limit a ---,0. We discuss first the OS regularization. The pseudoscalar terms that would violate parity, vanish by periodicity of the integrand. In eqs. (3.17), (3.18), the fermion of mass m 1 = m is always the closest pole to the integration domain. In the limit a ~ 0, amplitude (a) has, as a function of r, a narrow variation between its r = 0 value and its value for large r" 4(1-¼1r)>~-l(a~W(q=O)>~l(1-

3-~ln(2v/2r)) 2r 2

The numerator of the integrand in (3.17) can be written as a sum of four terms which contribute to the result as follows: (rna) 2 brings an r independent term 1/2,r, ( a M ( p ) ) 2 contributes the remaining part and the contribution of the other terms decreases as 1 / r 2 since cos2apx (sin2ap~ - sin2apy ) = - ½(sin2ap:, - s i n a p y ) 2 2 + X , where X are odd terms in the exchange x o y . from its r = 0 value towards zero:

Similarly, amplitude (b) decreases

For the W regularization two new features are encountered: (i) The closest pole to the integration region can be, at will, any of the fermion poles of the naive version. The amplitudes (3.17) and (3.18) possess x - - , - x symmetry, where x = am + 2r. As x -* 0, D w ( p ) vanishes at p~ = 0, py = + 7r/a and at py = O, Px = + ~ r / a (these fermions have vector charge + 1 and axial charge - 1); as ma--* O, D w ( p ) vanishes at px = 0 = p y , as ma + 4 r - * O, D w ( p ) vanishes at Px = +~r/a = p y (these fermions have vector and axial charge + 1). The limit a ~ 0 may be taken in various ways, keeping fixed either m I = m, or m 2 = m + 2 r / a , or m 4 = rn + 4 r / a . As an illustration, ita)WW(q = 0) and I(~ w are drawn as a function of m a in fig. 4. (ii) The interference in D w ( p ) between the mass term and the M ( p ) term brings in some amplitudes an infinite slope as a --* 0. In fig. 5, I(aW)w (q = 0) is drawn as a function of m a for the OS and W regularizations. The interference term in the numerator of the integrand brings a contribution whose slope behaves as - I n m la as m l a ~ O.

F. Guerin / Wilson fermions

478

_l ww

i •4

o

! -4

b

i

-2

I

I

0

1

t

ma

Fig. 4. The contribution of diagrams (a) and (b) of fig. 3 to the amplitude with two axial currents, at zero m o m e n t u m , as a function of ma, for the Wilson regularization exhibiting the presence of the four fermions; r = 1.

,.i w w

...4

.,4

o

i

I

I

me

Fig. 5. The contribution of diagrams (a) and (b) of fig. 3 at zero m o m e n t u m as a function of ma for the OS regularization (dashed line) and the W regularization (solid line); r = 1.

F. Gubin

/

Wilson fermions

419

To conclude, both regularizations give the same results as a --, 0 if one keeps m I or m4 fixed in the Wilson regularization. The limit is smoother for the OS case. The axial-axial current amplitude at zero momentum has in this limit an r-dependent value. Momentum dependence of the amplitudes. In the OS case, as a --* 0, m fixed, the denominator vanishes only for p, =p,, = 0. Extra terms appear in the numerator of the integrand of eq. (3.8); however, they do not contribute asp + 0. As compared to (3.10), one now obtains the correct relation IVV p*Y

In

W case,

,IWW,IWV p-=v

=

-+)+o((;)4).

m

m,a or

qpqy/mi,

J+(l

or mga

and I$:

to zero,

and I,“=: as q&m:,

-2q,q,/&

qpqy/m&

as q,q,/mf, illustrating

the

multiplicity and quantum numbers of the fermions. The limits are reached smoothly, in a way similar to amplitude ICbj ww of fig. 4. Anomalies. The anomalies may be extracted in a simple way from the OS regularization where the mass term does not interfere with the degeneracy regularization and the amplitude for the ( ys - ys y,) loop takes a simple-minded form. The total amplitude with two axial currents can be written in the limit a + 0

with IPyCTntgiven by (2.9) and

1”“,~,1(~“(

q = 0) + I,K”.

The anomaly for the lattice regularization continuum anomaly

is obtained

by adding Iww to the

From the above discussion of the various terms contributing separate the r-dependent terms:

with Dos( p) given (3.15) and uM(p) parameter; the bounds are 3 b -- T”--(r)<

to Iww one can

by (3.16). The integral only depends on the r

- --& 2mr

ln2JZr.

F. Gu~rin / Wilsonfermions

480

The vector-axial amplitude obeys in the limit a - , 0 wv

I~vlatt

wv

I~vcoll t

The corresponding anomaly is obtained by setting m = 0 in the contraction q~IWV(q)cont . In the lattice case, contracting with q~, setting m = 0, then taking the limit a ---,0 with q fixed, results in q,I~wv (q = 0) with I ~ v (q = 0) given by eq. (3.6) where Dos (p; m = 0) is substituted to DN(p). A careful integration around p = 0 gives the correct anomaly [7, 18]. To conclude, for this choice of regularization and of axial current one can reproduce the continuum results for the loop with two currents ~ith one exception: the axial-axial anomaly has the wrong sign and depends on the parameter r of the regularization; it is given by eq. (3.19). 3.3. OTHER CHOICES FOR THE LOCAL AXIAL CURRENT

Instead of the action L (1) of (3.12), one may try to couple the r-dependent term to the W~ field in the following way: 1

I(2'(V,W) = E ~a[~P (x)rexp(ia(V~ + -/5I,V~) (x +½a.))~p(x + a¢) x, lx

+~p (x + a~)rexp(-ia(V~- ~,sW~)(x + ½a~))~p(x) - 2~ (x)r~p(x)].

(3.20)

Such a term does not have the correct continuum limit as a ---, 0, and one has to add another term in the action, tentatively / ( 3 ) = _ E ~ P ( x ) r ( eiaz'SW~(x)- l ) ~ p ( x ) . X,,tt

The Feynman rules are modified correspondingly: to the qTffW~vertex one associates

Fw~( p, p ' ) = i),~yscos½a ( p + p')~ +/),sr (1 - cos l a ( p + p')~). Such a vertex amounts to including in the axial current vertex a term which is precisely, up to a factor of a, the chiral symmetry breaking term which appears in the divergence of the axial current, eq. (3.13). In the IWW(q=O) amplitude the modifications contribute an additional isotropic term of the (~'5 - Ys) type

81~=, aZr z (1 w w ( q - O) = 6I~WW(q = O) = _

cosapx)(1 -

cosapy)

(3.21)

481

F. Gu~rin / Wilsonfermions

The limit a ~ 0 amounts to setting D ( p , m = 0) in the integrand of that integral. The finite result one obtains worsens the problem IWW(q = O) ~ O. Another possible coupling of the r-dependent term to the W~ field is #4)= ½(/r(Z)(v, W) + Ir(2)(V,- W ) ) ,

(3.22)

with I}2)(V, W) given by (3.20). This term has the correct continuum limit, the r-dependent term couples only to an even number of W~ fields. Under the gauge and chiral transformations specified by (3.3) /r (4) "+ E ~ 1a [~p(x)r(eiv,2e(X)ei~(v,+r,w. ) + ei.1,20(x+~%)eia(V_v,w,))~p(x x,~

-{- at,)

+ ~ (X + a,)r(eirsZe(x)e ia(v~- rsw,)+ eirs:e(x+aDeia{V,+r,w~)) ~ ( x ) - 4 ~ (x )reir52e
Fw, ( p, p') = ig~g5 cos ½a ( p + p') ~, Fw, w, ( p, p') = - ag,~( iy~sin ½a( p + P')u -- rcos ½a( p + p ' ) , ) . As compared to the results of subsect. 3.2, only the counterterm of the axial-axial amplitude is affected. Eq. (3.18) has to be replaced by

ww _ _ gu,2 fpa 2 sin2apx - raM( p )cos aPx

In the limit a ~ 0, upper and lower bounds can be given to this integral. Indeed

r ( p , r 2 - 1)~
j

~

" 2 -~py)(1 a . 2 -~py)) a + r2(sin2apx\2 + san 2a2px+ s,n

so that in the limit a + 0

1-

K

'/] < --I{bWW <

rr r 2 + 1

where K is the elliptic integral [26]. As a function of r, -I(bW)W starts from its r = 0 value 1, changes sign for r = 1.5 and behaves for large r as (1 - 27r- qn2f2-r). For

482

F. Gudrin / Wilsonfermions

r= 1, - I(b) w w _-- 1 -- 4/3f3-. The cancellation of I(aww ) (q = 0) given by (3.17) occurs for r = 2.5. To conclude, restoration of Lorentz invariance is achieved for this amplitude at a particular value of r. The anomaly is

b(

1 r) = 2~r

I

[ a 2 2sin2apxsin2apy + ( a M ( p ) ) 2 - raM( p ) D( p, m = O) Jp D2(p,m=O)

Apart from the problem of the anomaly, this two-dimensional example has shown that the Wilson amplitudes keep track of the four fermions carried by the propagator and any of the fermions can be made to dominate in the limit a ~ O. The same fermion always dominates the Osterwalder-Seiler amplitudes. This distinctive feature will appear again in the next section when we study a gap equation.

4. Lattice regularization for the transformed chiral GN: the iarge-N results We write down the large-N results of the lattice regularization of the model defined in subsect. 2.4 from the chiral ON model. The value of the mass counterterm needed in the Wilson regularization will be obtained. The fermion field interacts with two auxiliary fields a scalar and an axial ones. The lattice version of the action (2.18) is, for the Wilson regularization I

=

I N + It,

IN= -- ~ [ ~

~----a(~(x)7~,exp(i'ysaW~( x + ½a~,))~/(x+ G )

(x +

_ +o(x)~/(x)¢(x)+ _°2(z) + 2g 2 I~(') =

+

+ (x)) 1

E2-g~(

Nb r)W~(x)W~(x)[

~a(~(x)rqJ(x+a,)+~/(x+a~)rqJ(x

- 2~/ ( x )r~b( x ) ) - 3 r n ~ ( x )tk( x ) ] .

l

(4.1)

)

(4.2)

F. Guerin / Wilsonfermions

483

In sect. 3 was discussed the choice for the axial current, the dependence of the anomaly b(r) on the regularization and an alternative choice for Ir,

X ~ ( x + ag)r(eiv'aW~ + e-'r~aw*)t~(x) -- 4{ (x)r~(x))

-- t~m{ (x)t~(x)] . (4.3)

Ir°) and I~ 2) will give identical results for the model to leading order in 1/N, 3m is a needed counterterm to be discussed later on. One goes to the regularization of Osterwalder-Seiler by the substitution r ---,ri'/5 and 3m = O. Integrating over the fermion fields gives an effective action

+ W~W~(x) 21r ] + T r l n

p(x)+6m

1 iaP X +~da~(e ,y~,exp(iysaW~(x+½a~,))

2a E (eiae" +

e-

- 2

,

(4.4)

/L

where P~ is the translation operator. Tr is a trace over space and Dirac indices. The leading contribution for large-N comes from the stationary points. Restricting to constant fields p(x) = if, W~(x) = W~, these are solutions of

OI (p,W,)p=~,w, =

Op(x) 31

(p,

=

=0,

(4.5)

=0.

(4.6)

Eq. (4.6) has W~ = 0 as a solution. The discussion of the solutions of eq. (4.5) differs for the OS and W regularization.

484

F. Guerin / Wilsonfermions

4.1. THE OS REGULARIZATION AND THE LARGE-N RESULTS

The potential of the p field can be read from

I(p, W~) in

eq. (4.4) for the case

W~=0:

with Dos(P, p) given by (3.15) and fp by (3.5). The potential possess the p ~ - p symmetry and the stationary points obey ~ = 0 or

1 £a2

2

= 0.

(4.8)

Dos( P, 0) The minimum of the potential corresponds to ~. Keeping t5 and r fixed as a ~ 0 brings only one singularity close to the integration region, the integrand is peaked towards Px = Py = 0 and eq. (4.8) is 1

X=

1

- -- In I~a I + O( tSa ),

(4.9)

which is the result (2.22) for the dynamically generated mass in an asymptotically free theory. Expanding the action I(p, W~) of eq. (4.4) around p(x) = if, W~ = 0, the quadratic terms are identified with the inverse of the p and W~ propagators. In momentum space 4(pa)2 = N1 . D p - I ( q = 0) = __d2V p= = N 2fJap dp 2 D 2 s ( p , ~) a--,0 ~r

Dp- ~(q) - Do- l ( q = 0) is a convergent integral; in the limit a --, 0 only one fermion runs around the loop, for qa << 1 one recovers the continuum result. For the W~ propagator, one finds

D~'(q)=N( g~b(r)cr I~WW(q))' where IWW(q) is the fermion loop with two axial currents, with fermion mass m = 15, fully discussed for that regularization in subsect. 3.2: -rr D~|(q)=N ( g~ b(r)

I~ww(q = O)- (I~ww( q ) - w1 ~w

(q=0)

)) .

The anomaly b(r) cancels in the first term and the second term is a converging integral; for qa << 1, the result (2.23) of Pauli-Villars regularization is recovered.

F. Gu~rin / Wilsonfermions

485

The choice of i~2) defined in (4.3) instead of I~ ° given by (4.2) would only affect the anomaly (see subsect. 3.3), the W~ propagator would be unchanged. To conclude, this lattice regularization gives back all the features of the large-N limit of the model obtained with the Pauli-Villars regularization.

4.2. THE WILSON REGULARIZATION AND THE LARGE-N RESULTS

The effective potential for the 0 field is read from eq. (4.4) by setting ~ -- O:

V(p)=N

~-

InDw(p,o+Sm ) ,

(4.10)

Dw( P, p + 8m) = • sin2ap, + (a(p + 6m) + aM(p)) 2, 1

aM(p) = Y'.r(1 - cos apt),

(4.11)

/

and the stationary points of the potential obey

P X

fpa22 p + S m + M ( p ) Dw( p, p + 8m) = O.

(4.12)

The second term on the right-hand side of (4.10) possess x -o - x symmetry where

x = a(o + 8m)+ 2r; indeed a change of x into - x can be compensated by the changepp -o¢r -Po of the integration variables. As already discussed in subsect. 3.2 a singularity comes close to the integration domain whenever one of the following quantities vanishes

am, = a ( p + S m ) ,

am2=a(p+Srn)+2r,

am4=a(p+Sm)+4r.

If one chooses a 6m + 2r = 0, the full potential is symmetric around p = 0 and the potential possess a minimum P, which is the dynamically generated fermion mass. However, as a -o 0, keeping t~ fixed, two fermion poles come close to the integration domain, the poles Px = 0, py = + ~r/a and py = O, Px = +-1fla. As a consequence eq. (4.12) reads in the limit a --* 0 --P+ p 2 1 n ~ - ~ + p f ( r ) = O X

(Etcos apt)2 f(r)=

fpa2r2 (zlsin2apz + ra(Ztcosapt)2)2 .

F. Gu~rin / Wilson fermions

486

The relation between the coupling constant and the cut-off is wrong by a factor of two, due to the contribution of the two fermion poles. Another choice for the counterterm is am+M(p) D-w-((-l~,p+am) = 0 ,

(4.13)

with D w and M ( p ) defined in (4.11). The discussion is more conveniently made in terms of the physical mass m = P + am.

(4.14)

Fig. 6 shows a a m ( m a , r) as a function of ma for r = 1; a a m + 2r is an odd function of rna + 2r; aarn = ma when one fermion pole touches the integration domain, in particular a a m ~ 0 as ma ~ O. The behaviour of the effective potential may be studied as a function of m, in eq. (4.10) with V ( p = m - am(m)); the location of the extrema obey

OV

o = - ~ d---~ =

X

a2

dV[l

Dw(p,m)

d(a,,,) ]

:'

=

-~ -

a2Dw(p,m

) (m-am).

Fig. 7 shows the X dependence of extrema of the potential and shows for comparison the analogous result (4.8) of the OS regularization. The effective potential is drawn in fig. 8 as a function of ma and compared to the corresponding potential of the OS regularization. The symmetry point of the potential is am + 2r = 0 in the W case and am = 0 in the OS case. The OS potential is, for pa < 2, identical to the potential of a

6m a

-4 I

!

frl8

I

-

--'7

Fig. 6. T h e m a s s c o u n t e r t e r m 8rn as a f u n c t i o n o f ma, w h e r e m is the p h y s i c a l m a s s ; r = 1.

F, Gu6rin /

Wilson fermions

487

.2 l

-1

-2

0

!

I

1

2

ma

Fig. 7. The location of the m i n i m u m of the potential versus ~,- 1 in the OS case (dashed line) and the W case (solid line); r = 1.

V., 2 -2 !

-I

i

~0

1

2

t

I

arTl

Fig. 8. The effective potential for A- i = 0.4 as a function of ma in the OS case (dashed line) and the W case (solid line); r = 1. The symmetry point of the potential is am = 0 in the OS case and am + 2 = 0 in the W case.

F. Gu~rin / Wilson fermions

488

continuum momentum cut-off,

Vcont(P)=No2( 1

1 (ln(Pa)2)) 41r 1 4~r

2X

"

The minima of the Wilson potential obey

a2Dw(p,m ) = 0 .

7~

Keeping ~ fixed as a ---, 0, only one fermion pole touches the integration domain and one obtains in this limit 1

1

~- + -- In A

ma = O,

aSm fp 2aM(p) ~, =g(r)= a2Dw(P,m=O), p=m-Sm,

g ( r = 1) = 4/3¢~-.

(4.15)

The fermion mass of the GN model is generated by the self-interaction of the fermion field. In the large-N limit, this self-interaction is just the one-fermion loop. For the large momenta flowing around the loop, the Wilson degeneracy regularization is an effective mass which has to be compensated by the counterterm (4.15). Eq. (4.13) may be written

fp(~m + M(p))tr(~Sw( p, rn)75Sw( p, m)) = O,

(4.16)

where Sw ( p ) is the Wilson propagator •

1

.

Swl(p) = Y',rt, aSmapt+ M(p) + m. 1

The W~ propagator involves a fermion loop with fermion mass m and leads to a result identical to the OS case.

5. A strong coupling expansion for the transformed chiral GN model To leading order in 1/N an expansion in the fermion propagator and in the axial current coupling will be formulated for that model. This expansion will then be summed to all orders; one will recover the large-N results of sect. 4. For clarity we shall proceed in three steps of increasing complexity. We shall formulate a strong coupling expansion successively for three regularizations of the

F. Guerin / Wilsonfermions

489

fermion propagator: the continuum momentum cut-off, the naive lattice one, and the W-OS regularizations. The expansion in the OS case will differ from the Wilson expansion; the problem of the mass counterterm will show up again in the W case. 5.1. THE C O N T I N U U M MOMENTUM CUT OFF

We want to show how the expansion in the fermion propagator decouples from the expansion in the ferrnion-fermion-axial field vertex and reproduces the strong coupling expansion [27] of the G N model which we shall briefly recall. For that purpose we consider a continuum momentum cut off for the fermion propagator without worrying about gauge invariance. The generating functional for Green functions involving fermions and axial fields is

Z(n,~I,J.)= foDoDW~Dq~D~exp['+f dZx(i-l++fn+J~W~)J,

(

p2)

- 2 . qTOqJ+ O~b + W~iy~,ys~p + ~~vb w . w ~ + -2g

I = f d2x -

(5.1)

The fermion propagator and the fermion-fermion-axial field vertex are extracted via

o 0~' o OJ. a )foDoDW~D+DC~ Z(n,~,J.)=K On' xexp

Nb

If

d2x - 2g 2

(5.2) with K

07' 0~1'OJ,

v.,(x)~d(x-Y)v., (Y) -fd2xo-~(x) f-~(x'i3'.Y,~-~(x)] •

The bare fermion propagator is

Go(X--Y) = --O~A(X-y) = --0[ J

d2p~eipxo(A2-p2). (2,~) ~

We formally divide space into cells of volume a 2 = 8A(0) - 1

~___

4~.//A2.

(5.3)

F. Gu~rin / Wilsonfermions

490

Performing the integration over the fermion and axial fields gives

Z ( ~ , ~, J.) = K

(,,,)

(s

O~' 0~1' OJ. eW°°7")exp

d x--f~J.J~

)

,

(5.4)

where

eW°(~'~)=fpDpexp[-Nfd2x(-~X-Sa(O)lnp2--~p)]. In the large-N limit W0(O1) can be evaluated to leading order in 1 / N using Laplace's method. The stationary points obey P X

28a(0) ~1 + =0, p NO2

(5.5)

which has two solutions at each space-time point. We break the symmetry by hand by picking the solution

p+= + (2hSA(O)) V2 + 0 ~

,

and the zeroth-order potential may be expanded as a power series in ~*I/N. It determines the vertices of the strong-coupling graphs

WO(~n)N~

~ -

Vo(~,) = N

Vo(~n)=-NE

f d2xVo(Wn),

- ~(o)1~ A -

~,7 Np+

2N 1 { ~/r/(x) ]k k!t

N

; v2,,

]

'

(5.6) (5.7)

k~l

with

v2, = ( ¢ ~ ~A(o)3/2) -k ~a (o) c,,

(5.8)

where ck is a number. Expanding the operator K(O/O~, 0/0~, O/OJ~) in eq. (5.4) gives an expansion in the fermion propagator and the fermion-fermion axial field vertex. The diagrams

F. Gukrin /

Wilson fermions

491

rules are: (i) with each fermion line carrying momentum p, associate a propagator G o ( p ) = _ i[;O( A 2 _ p 2 ) ;

(ii) With each axial field line associate a propagator gj,~rr/Nb; (iii) With each vertex consisting of k incoming and k outgoing fermion lines associate v z k / N k - l; (iv) With each vertex consisting of one axial line, one incoming and one outgoing fermion line associate -iy~,ys. These fermions lines must be stuck to one leg of the fermion vertices defined in (iii). The other rules are identical to the weak-coupling Feynman rules [27]. In the large-N limit the strong coupling expansion of the p field is uncoupled to the weak coupling expansion of the IV, field, and can be summed to all orders in the following way. One carries out a vertex renormalization and a propagator renormalization. The renormalization of the two-point vertex is achieved by inserting loops as shown in fig. 9:

V2R =

E

I l)2(k + 1) k~.~

(5.9)

k=0

In the one fermion loop a(v2R ) there appears the fully renormalized fermion propagator obtained by inserting the renormalized two-point vertex on a bare propagator as illustrated in fig. 9:

a(VZR)=

_Nf,d2P2tr Go(p) J(2

)

1-v2RGo(p)

(5.10) '

where the trace is over Dirac indices.

q2R---- ~

Fig. 9. The renormalized two-point vertex in terms of the bare vertices and the loop; the renormalized fermion propagator.

F. Gu~rin/ Wilsonfermions

492

Fig. lO. The one-ferrnionloop contribution to the propagator of the axial field (continuum momentum cut off). One notices that the renormalized vertex as given by eq. (5.9) is d

v2R =

v0(

)

~-~(v2.) -

1

(5.11)

since V0 is given by eq. (5.7) or by eq. (5.6); P+(a) satisfies eq. (5.5) with ~1~/replaced by a. This is just the gap equation for the fermion mass m = p+(a): m

26A(0 )

X

m

1 2 d~p Go(p) m~ f (2~r) 2 Tr 1 - (1~re)Go(P) = 0 ,

(5.12)

which reduces to the gap equation (2.22) of the model in the large-N limit. In the W~ sector, the renormalization of the W~ propagator is obtained by inserting fermion loops on a bare propagator: 7r D~(q)=

1

Ub 1 - ( ~ r / N b ) t ( q ) '

(5.13)

where the one-fermion loop contribution is shown in fig. 10:

Ioo(q)=-Nl~v.pTr

.] (27/.)2 z

i70"y5 1 +

n=[

(vzRG0(P))" i'Yo75 1 + Z(vzRGo(p')) n=l

Ioo(q ) = - N ( -dzp T r i 7 Y5 1 1 j (2qr)Z o V2r~_Go(p)iTdrSv~_Go(p,) •

,

(5.14)

Eqs. (5.13) and (5.14) give the W~ propagator (2.23) of the large-N limit of the model. This is only a formal result since our choice of G0(p) breaks gauge invariance, an essential ingredient for the definition of the anomaly. A Pauli Villars regularization is unsuited to a strong coupling expansion. The tadpole diagram of fig. 11 vanishes to all orders in the expansion. 5.2. THE NAIVE LATTICE FERMION PROPAGATOR IN THE STRONG COUPLING EXPANSION The theory is now defined on a square lattice of spacing a. To the action of eq. (5.1) one has to substitute the action (3.1) of the naive lattice fermion regularization

F. Guerin/ Wilsonfermions

493

Fig. 11. A tadpole term.

and one choice for the axial current. In equation (5.2) the fermion propagator is extrated via

( 0 0

0 ) = e x p ( - ~x ~1a ~[ ~0 (X)y~ei~saO/OJ~(x+½a~) ~._O(x+a~) O~/'

K O~l'O~l'OJ~

O (x+a~) -i'saa/oJ, ix+ka,) O-~-(x)]). On y,e O~ To leading order in 1 / N only the two first fermion-fermion axial field vertices will be needed:

eiVsaO/OJu= I + iy~a

a + ½(ia)2 02 OJu(x + ½a~) OJ~(x + ½a~)cgJ~(x + ½a~)

+ •...

The modification to the diagrams rules of subsect. 5.1 only concerns the bare fermion propagator which is

G0(p)

=

-

ai

y . ylsin aPl. /

and the fermion-fermion axial field vertex. (iv) With each vertex consisting of one axial line, one incoming fermion line carrying momentum p and one outgoing fermion line carrying momentum p' associate

-iy~,yscos½a( p + p')~,. (v) With each vertex consisting of two axial lines and two fermion lines associate

gj,fia3,~,sin½a( p + p'),. The substitutions are 8A(0)~ 1/a z and fd2p/(2~r) 2---, fp defined by (3.5). The summation of the strong coupling expansion in the p field sector is identical to the

494

F. Gu~rin /

Wilsonfermions

__(3--(3Fig. 12. A gauge-invariant contribution of order v~ to the propagator of a vector gauge field.

continuum momentum cut off case. The four fermion species bring a wrong factor of 4 in the gap equation [27]. If we consider the axial field propagator the two types of diagrams of fig. 3 contribute to the one-fermion loop correction Ipo(q) of eq. (5.14): (5.15)

Ioo( q) = Ioo(~( q) + Ioo~b), with

(v2ROo,p'n) Xi3~o75cos½a(p +P')o 1 +

(v2RGo( p

,

rt=l

I p ° ( b ) = - N f p gp°triT°asinapp'v2R(l+

n=l y'~ (v2RGo(p))n)"

Eqs. (5.13) and (5.15) give the axial field propagator of the naive lattice fermion regularization. It is worth noticing that a similar expansion for a vector gauge field would be gauge invariant to each order in 1/~. As an example the sum of the four diagrams of fig. 12 would satisfy the lattice Ward identity

I~oV( q ) sin ½aqp = O, where the derivation makes use of the periodicity of the integrand. 5.3. THE WILSON REGULARIZATION AND THE STRONG COUPLING EXPANSION

The W and OS regularizations add an extra mass term or pseudomass term to the action; it will modify the expansion in the sector of the 0 field. The expansion in the W~ sector will be gauge invariant in a way similar to the naive lattice fermion regularization.

F. Gu~rin /

Wilson fermions

495

The Wilson regularization adds a term 81 to the naive fermion action. The axial field may or may not couple to the extra term as in actions (3.12) or (3.22). Both expansions are similar. We choose the action (3.12)

1 8 I = ~x ~ a Y ~ ( ~ ( x ) r ~ ( x + a ~ ' ) + ~ ( x + a ~ ) r ~ p ( x )

-2~ (x)r~(x))-Sm~

(x)~(x)] .

(5.16)

The easiest way to define the strong coupling expansion is to split this term into a static term ~ x ) ( S m + 2r/a)Lk(x) to be added to the term p ( x ) ~ ( x ) + ( x ) , and to include the hopping term into the bare propagator [1] aw(?)

a1 (iytsin apt

= -E

rcos apt ) .

1

The zeroth-order potential analogous to eq. (5.6) is

22,

a z l n p + + 8rn +

N p++Sm+2r/a

'

where the stationary points p + obey

p 2`

2 a2

1 + ~__~_~ 1 =0, p ++ 8m + 2r/a N (p+ + 8m + 2 r / a ) z

(5.17)

or

p±=

a

with =

adm + 2r

(5.18)

The P ~ - P symmetry is broken to zeroth order in the strong coupling expansion if /~ * 0. The vertices of the strong coupling graphs are defined as in eq. (5.7), the c k numbers of eq. (5.8) now depend on the /~ parameter. The diagrams rules are identical to the rules of subsect. 5.2 with the substitution of G w ( p ) to Go(p). The two-point vertex is renormalized V2R = _ _~d V ( a )

aa

= 1 ~="(V2R) p ++ 8m + 2r/a "

F. Gu~rin/ Wilsonfermions

496

The inverse of the renormalized fermion propagator VzRI - G w ( p ) gives m = p++ 6 m , as the fermion mass in the limit a ~ 0. Eq. (5.17) becomes the gap equation O+

2 1 p++ 8m + 2 r / a a 2

fp v Z T r G w ; ~ P 'J

1 - v2RGw(P) '

(5.19)

which is eq. (4.12). Up to now, the mass counterterm 3rn has been left unspecified. In the strong coupling expansion, the gap equation (5.19) appears as a self-consistent equation for the renormalized two-point vertex. Indeed eq. (5.9) reads a

(C,(l~)+c2(l~)M+c3(#)M2+...)

1

v2R [ a 2 T r a 2 G 2 ( p ) ( 1 2 2 a Jp ~" + v 2 R a w ( p ) +

I)2R- 2¢2~ M=

'

... )

with/z defined by (5.18). A criterion has to be found to determine 3m order by order in the strong coupling expansion. The choice ~ - 0 would lead the expansion towards the wrong fermion poles, as discussed in subsect. 42. Another choice for 3m is the constraint (4.16) which reads in terms of the variables of this strong coupling expansion

192R f p ( # - _ _1

~_,rcosapl Tr3'5 1 - v 2 R G w ( P )

ix= --~r22 V2R (1-- ( - ~ )2½(3r2 -

D2R ~'5 1 - v 2 r G w ( P )

=0,

i)(3r2--2)+ ...).

At zeroth order in the strong coupling expansion/~ = 0 and the p --* - p symmetry is unbroken. The expansion in the W~ sector is similar to the naive lattice fermion action of subsect. 5.2 and presents no new feature. The tadpole terms of fig. 10 vanish by periodicity of the integrand.

5.4, THE OSTERWALDER-SEILERREGULARIZATIONAND THE STRONG COUPLING EXPANSION The extra term in the action is obtained from eq. (5.16) by substituting riy 5 to r and setting 3m = 0. We split this term into a static part ( 2 r / a ) f ( x ) i 3 , s q ~ ( x ) to be

F. Guerin / Wilsonfermions

497

added to O ( x ) f ( x ) ~ p ( x ) and a hopping part, so that the expansion is in terms of a1 (i3,tsin aPl _ riyscos apt) .

Oos(p) = -E l

The zeroth-order potential analogous to eq. (5.6) is Vo(£171,~liy, n ) = N

~

In 02++4

--~

1

1

p++2iT, r/a~l ;

the stationary points obey O+

2p+

1

(5.20)

N ~ (o+ + 2 i y s r / a ) 2 ~ l = O .

a 2 p2+ 4r2/a 2

The O --* - P symmetry is unbroken at zeroth order in the strong coupling expansion p+=+~/2~-4r2 -

+O(~/

a

~i75 ~/)

N'

N

"

The vertices of the strong coupling graphs are more numerous: '

k=0 z=0 k!l! \ N

-

-

V2k'2l'

Vo( ~71, ~ltYs~ ) = - ~171v2 - ~tiys ow 2 + . . . ,

with a

(

v2 = V~.o = ~

2r2) 1/2 ' 1 - --U

/-

w2 = Vo,2 = - a ~ .

The renormalized fermion propagator is obtained as in fig. 9 by inserting the two-point vertex (VzR + w2Ri3,5) on a bare propagator. In the renormalization of the two-point vertices V2R and wzR, as shown in fig. 9, two kinds of loops appear

,

(o,

)k(o2~ ( v 2 . , w 2 . ) ), ,

v2.= E E ~.l,.v2~k+,~2,-g(v2..w2.) k=O /=0

with a,(v2R, WzR) = - N

fp

1 TrGos(P ) 1 - (I)2R --k i]fsw2R)Gos(P )

a 2 ( v2r~ , w2R) = - N f p T r i~5 Gos ( p )

'

1 - (vzR + iysw2R)Gos(P ) "

(5.21)

F. Gu~rin / Wilsonfermions

498

An equation similar to eq. (5.21) defines WzR. This strong coupling expansion may be summed v2R

=

W2R ~---

aVo l

I ~,=~,(v2..w2.)

aVo (al, a2 )

= - 2r/a ~,=~,(v2,, w2,) p2+(a,,Ctz)+4rZ/a 2'

0erE

=

,

P2+(a,,ct2)+4r2/a2

where p+ satisfies the gap equation (5.20) with ~ replaced by aj and fliys,i replaced by a 2. Since 1 D2R "-[- iYsW2R = p + ( 0 t l ' 0/2) "-~i752r/a '

the gap equation (5.20) becomes p+

1 1 a2 tr P++ 2iysr/a

fp

1 tr ( P + + 2iysr/a) 2

1 × G ° s ( P ) 1 - (v2R + i~,sW2p.)Gos(P ) = O,

and leads back to the gap equation (4.8) for this regularization. The W~ sector presents no new feature. The parity-violating terms vanish to each order in the strong coupling expansion, by periodicity of the integrand. All tadpole terms vanish. No mass counterterm appears, at the cost of more complexity in the strong coupling expansion.

6. Conclusion

The conclusion of our study of this two-dimensional model are: At one-loop approximation the Osterwalder-Seiler regularization presents no parity-violating term and possess good analytical properties. The limit a ~ 0 is always reached in a smooth way and reproduces the continuum results, except for one anomaly. The strong coupling expansion, although more complex than for other regularizations, may be summed to all orders to leading order in 1/N. In the case of the Wilson regularization the interference between the mass term and the degeneracy regularization allows the high mass fermions to have a sizeable effect on the behaviour of the amplitudes. Also a mass counterterm is needed; an explicit expression has been given for that term that might generalize to other cases. Both regularizations give identical results in the limit a --, 0 and a comparison of their results for a *= 0 might prove useful.

F. Guerin / Wilson fermions

499

T h e lattice r e g u l a r i z a t i o n is n o t e q u i v a l e n t to the P a u l i - V i l l a r s r e g u l a r i z a t i o n for the f e r m i o n l o o p i n v o l v i n g two axial c u r r e n t s , if o n e c h o o s e s the local v e r s i o n of the axial c u r r e n t . I n t w o d i m e n s i o n s it shows u p i n o n e a n o m a l y ; i n h i g h e r d i m e n s i o n s the effect s h o u l d b e a b s o r b e d i n c o u n t e r t e r m s [23]. T h e a u t h o r is very i n d e b t e d to C. Becchi w h o s e help was c r u c i a l for the f o r m u l a t i o n of the t r a n s f o r m e d chiral G . N . m o d e l . D i s c u s s i o n s w i t h R. K e n w a y a n d B. P e t e r s s o n are g r a t e f u l l y a c k n o w l e d g e d .

References [1] K.G. Wilson, Phys. Rev. DI0 (1974) 2445; in New phenomena in subnuclear physics, ed. A. Zichichi (Plenum Press, N.Y., 1977) (Erice, 75) [2] M. Creutz, Phys. Rev. Lett. 43 (1979) 553; Phys. Rev. D21 (1980) 2308; Phys. Rev. Lett. 45 (1980) 313 [3] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. DI0 (1974) 3376; D11 (1975) 2098, 2104; D19 (1979) 2514; J.B. Kogut, Rev. Mod. Phys. 51 (1979) 659; J.M. Drouffe, K.J.M. Moriarty, Phys. Lett. 108B (1982) 333 [4] J. Greensite and B. Lautrup, Phys. Lett. 104B (1981) 41; P. Cvitanovic, J. Greensite and B. Lautrup, Phys. Lett. 105B (1981) 197; J.M. Drouffe, Phys. Lett. 105B (1981) 46 [5] D. Boyanosky, R. Deza, L. Masperi, Phys. Rev. D22 (1980) 3034; J. Caxdy and H. Hamber, Nucl. Phys. BIT0[FS1] (1980) 79; L. Masperi and C. Omero, preprint IC/81/172; D. Horn and M. Weinstein, Phys. Rev. D25 (1982) 3331 [6] F. Fucito, E. Marinari, G. Parisi and C. Rebhi, Nucl. Phys. BI80[FS2] (1981) 369; E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B190[FS3] (1981) 734; D.J. Scalapino and R.L. Sugar, Phys. Rev. Lett. 46 (1981) 519; D. Weingarten and D. Petcher, Phys. Lett. 99B (1981) 333; H. Hamber, Phys. Rev. D24 (1981) 951 [7] L.H. Karsten and J. Smit, Nucl. Phys. B183 (1981) 103 [8] S.L. Adler, 1970 Brandeis University Summer Inst. in Theoretical physics, ed. S. Deser, M. Grisaru and H. Pendleton (M.I.T. Press, Cambridge, Mass.) [9] B. Svetisky, S.D. Drell, H.R. Quinn and M. Weinstein, Phys. Rev. D22 (1980) 490, 1190; J. Smit, Nucl. Phys. B175 (1980) 307; J. Greensite and J. Primack, Nucl. Phys. BI80[FS2] (1981) 170; J.M. Blairon, R. Brout, F. Englert and J. Greensite, Nucl. Phys. B180[FS2] (1981)439; H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B190[FS3] (1981) 504; N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100 [10] N. Kawamoto and K. Shigemoto, Phys. Lett. l14B (1982) 42; V. Alessandrini, V. Hakim, A. Krzywicki, Nucl. Phys. B205[FS5] (1982) 253; H. Kluberg-Stern, A. Morel and B. Petersson, Phys. Lett. 114B (1982) 152; A. Hasenfratz, Z. Kunst, P. Hasenfratz and C. Lang, Phys. Lett. 117B (1982) 81 [11] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; H. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. 108B (1982) 314; D. Weingarten, Phys. Lett. 109B (1982) 57 [12] S. Drell, M. Weinstein and S. Yankielowicz, Phys. Rev. DI4 (1976) 1627 [13] L.H. Karsten and J. Smit, Nucl. Phys. B144 (1978) 536; Phys. Rev. 85B (1979) 100; J.M. Rabin, Phys. Rev. D24 (1981) 3218

500

F. Gu&in / Wilsonfermions

[14] H.S. Sharatchandra, Phys. Rev. D18 (1978) 2042 [15] J. Shigemitsu, Phys. Rev. D18 (1978) 1709 [16] K. Osterwalder and E. Seiler, Ann. of Phys. 110 (1978) 440; W. Kerler, Phys. Lett. 100B (1981) 267 [17] L. Susskind, Phys. Rev. DI6 (1977) 3031 [18] H.S. Sharatchandra, H.J. Thun and P. Weisz, Nucl. Phys. B192 (1981) 205 [19] T. Banks, A. Casher, Nucl. Phys. B169 (1980) 103 [20] D.J. Gross and A. Neveu, Phys. Rev. DI0 (1974) 3235 [21] N. Andrei and J.H. Lowenstein, Phys. Rev. Lett. 43 (1979) 1698 [22] W. Keder, Phys. Rev. D23 (1981) 2384 [23] K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; Phys. Rev. D21 (1980) 2848 [24] R. Roskies and F. Schaposnik, Phys. Rev. D23 (1981) 558; R.E. Garnboa Saravi, F. Schaposnik and J. Solomin, Nucl. Phys. B185 (1981) 239 [25] G. Duerksen, Phys. Lett. 103B (1981) 200 [26] I.S. Gradshteyn and I.M. Ryznik, Table of integrals, series, and products ed. A. Jeffrey (Academic Press, N.Y., 1965) p. 904 [27] F. Guerin and R.D. Kenway, Nucl. Phys. B176 (1980) 168