A chiral theory on a lattice
G. PREPARATA and SHE-SHENG XUE(*)
INFN, Sezione di Milano - Via Celoria 16, Milano, Italy
(ricevuto il 5 Agosto 1996; approvato il 16 Ottobre 1996)
Summary. — We present the details of a theoretical analysis of a SUL( 2 )7UR( 1 ) chiral theory with multifermion couplings on a lattice. The existence of a possible scaling region in the phase space of multifermion couplings necessary for defining the continuum limit of chiral fermions is advocated. In this region no spontaneous symmetry breaking occurs; the «spectator» fermion cR(x) is a free mode and thus decoupled; doublers decouple as massive Dirac fermions consistently with the
SUL( 2 )7UR( 1 ) chiral symmetry, whereas the normal mode of ciL(x) is plausibly expected to be chiral in the continuum limit. Such conclusions are not in agreement with the general belief of the definite failure of theories of this kind.
PACS 11.15.Ha – Lattice gauge theory. PACS 11.30.Rd – Chiral symmetries.
PACS 11.30.Qc – Spontaneous and radiative symmetry breaking.
1. – Introduction
Since the «no-go» theorem [1] of Nielsen and Ninomiya was demonstrated in 1981 the problem of chiral fermion «doubling» and «vector-like» structure on a lattice still exists, if one insists on preserving chiral symmetry. One of the ideas to get around this «no-go» theorem was proposed by Eichten and Preskill (EP) [2] ten years ago. The crucial points of this idea can be briefly described as follows. Multifermion couplings are introduced such that, in the phase space of strong couplings, Weyl states comprising three elementary Weyl fermions (three-fermion states) are bound. Then, these three-fermion states pair up with elementary Weyl fermions to become massive Dirac fermions. Such Dirac fermions can be massive without violating chiral symmetry, due to the appropriate quantum numbers and the chirality carried by the three-fermion states. The binding threshold of such three-fermion states depend on the elementary
(*) E-mail address: xueHmilano.infn.it
Weyl modes residing in different Brillouin regions. If one assumes that the sponta-neous symmetry breaking of the Nambu-Jona Lasinio (NJL) type does not occur and such binding thresholds separate the weak coupling from the strong-coupling symmetric phase, there are two possibilities to realize the continuum limit of chiral fermions. One is the crossing over the threshold for the three-fermion state of chiral fermions, another is of a wedge between two thresholds, where the three-fermion state of chiral fermions has not been formed, provided all doublers sitting in various edges of the Brillouin zone have been bound to be massive Dirac fermions and decouple.
To probe this idea, EP proposed a model [2] of multifermion couplings with SU( 5 ) and SO( 10 ) chiral symmetries and suggested the regions in phase space where it is possible to define the continuum limit of chiral fermions. However, the same model of multifermion couplings with SO( 10 ) chiral symmetry was studied in ref. [3], where it was pointed out that such models of multifermion couplings fail to give chiral fermions in the continuum limit. The reasons for such negative result are that in an NJL spontaneous symmetry breaking phase separating the strong-coupling from the weak-coupling symmetric phase, the right-handed Weyl states do not completely disassociate from the left-handed chiral fermions and the phase structure of such a model of multifermion couplings is similar to that of the Smit-Swift (Wilson-Yukawa) model [4], which has been very carefully studied and shown to fail. It is thus a general belief [5] that the constructions [2, 6, 13] of chiral fermions on the lattice with multifermion couplings must fail to give chiral fermions on the basis of the argument that multifermion and Yukawa couplings should be in the same universality class. In fact, this opinion is well founded only if one considers multifermion and Yukawa couplings only for continuous field theories, in the sense that the two theories should have the same spectrum and relevant operators at the ultraviolet fixed point [7]. However, we know of no proof of a one-to-one correspondence between the phase spaces of multifermion couplings and Yukawa couplings on a lattice, where they exhibit exactly the same spectra and relevant operators not only for chiral fermions but also for doublers. Even for Yukawa couplings, models with different symmetries could be in different universality classes [8]. Generally speaking, multifermion couplings possess more symmetries than Yukawa couplings in a lattice theory. All symmetries of the Standard Model can possibly be preserved by multifermion couplings, but not the Wilson-Yukawa couplings on a lattice.
Thus we should not be surprised that a particular model of multifermion couplings does not work. This does not mean that EP’s idea is definitely wrong, to prove this there must be another generalized «no-go» theorem on interacting theories [9] for a whole range of coupling strength; we will come back to this point in sect. 6. Actually, Nielsen and Ninomiya gave an interesting comment on EP’s idea based on their intuition of anomalies [10]. We also note that the phase space of multifermion coupling models in the EP type [2, 3] has not been completely explored. We thus believe that further considerations of models of chiral fermions on the lattice with multifermion couplings and careful studies of the spectrum in each phase of theories so constructed are necessary.
In sect. 2 we present a model of chiral fermions with multifermion couplings on the lattice and discuss the cR(x) shift-symmetry and its related Ward identity. The
analyses of the weak-coupling and the strong-coupling phases are given in sect. 3 and 4. The thresholds and wedges suggested by EP are qualitatively determined and discussed in sect. 5. The existence of a scaling region for the continuum limit of lattice chiral fermions is discussed in sect. 6.
Let us consider the following action of chiral fermions with the SUL( 2 )7UR( 1 )
global chiral symmetry of the lattice:
.
`
/
`
´
S 4Sf1 S11 S2, Sf4 1 2 a!
x!
m(
c –i L(x) gmDijmc j L(x) 1c–R(x) gm¯mcR(x))
, S14 g1!
x c –i L(x) Q cR(x) c–R(x) Q ciL(x) , S24 g2!
xc –i L(x) Q [DcR(x) ][Dc–R(x) ] Q ciL(x) . (1)In eq. (1) Sfis the naive lattice action of chiral fermions, a is the lattice spacing and the
SUL( 2 ) chiral symmetry is actually local and can easily be gauged by setting
!
m gmD m 4!
m(
Um(x) dx , x 1m2 U † m(x) dx , x 2m)
, Um(x) SUL( 2 ) , (2)but we will impose Um(x) 41 so that the SUL( 2 ) is a global symmetry. S1and S2are two
extra multifermion couplings, where
.
/
´
DcR(x) f!
m [cR(x 1m)1cR(x 2m)22cR(x) ] , Dc–R(x) f!
m [c – R(x 1m)1c–R(x 2m)22c–R(x) ] . (3) In the action (1) ciL(i 41, 2) is a SUL( 2 ) gauged doublet, cRis a SUL( 2 ) singlet and
both are two-component Weyl fermions; cRis treated as a «spectator» fermion. ciLand
cR fields are dimensionful [a21 O 2]. The first multifermion coupling S1 in eq. (1) is a
dimension-6 operator relevant both for doublers p 4pA1pA and for normal modes p 4 pA of ci
L and cR fields. Note that all momenta are scaled to be dimensionless, the
physical momentum ( pA) of normal modes and the momentum p 4pA1pA of doublers
are
pAC0 , p 4pA1pA,
(4)
where pA runs over fifteen lattice momenta pAc 0 . The second multifermion coupling S2 in eq. (1) is a dimension-10 operator relevant only for doublers, but irrelevant for
normal modes of ci
L and cR. The multifermion couplings g1 and g2 have dimension
[a22]. The action S
1is similar to the mass term in lattice QCD and the second term is
similar to the Wilson term. They are quadrilinear in order to preserve the chiral gauge symmetries.
The action (1) has an exact local SU( 2 ) chiral gauge symmetry, the symmetry possessed by the continuum theory (the target theory). The global flavour symmetry UL( 1 )7UR( 1 ) is not explicitly broken in eq. (1). When g14 0 , action (1) possesses a cR
shift-symmetry [11], i.e. the action is invariant under the transformation
c – R(x) Kc – R(x) 1e–, cR(x) KcR(x) 1e , (5)
where e is independent of space-time.
consider the generating functional W(h , J) and the partition functional Z(h , J) of the theory, W(h , J) 42 ln Z(h, J) , (6)
.
`
/
`
´
Z(h , J) 4 f expk
2S 1 x (c–i LhiL1 h–iLciL1 c–RhR1 h–RcR1 AmJm)l
, f 4[ dci LdcRdAm] , (7)where Am(x) refers to the SUL( 2 ) gauge field (2) defined on the lattice. Then we define
the generating functional of one-particle irreducible vertices
(
the effective action G(cR8i, c 8R, A 8m))
as the Legendre transform of W(h , J),G(c8Li, c8 , A 8R m) 4W(h, J)2
x(c–L8ihiL1 h–iLc8Li1 c–8RhR1 h–Rc 8R1 A 8mJm) ,
(8)
with the relations
A 8m(x) 4 aAm(x)b 42 dW dJm(x) , c8Li(x) 4 aciL(x)b 42 dW dh–i L(x) , c–8Li(x) 4 ac –i L(x)b 4 dW dhi L(x) , (9) c8 (x) 4 acR R(x)b 42 dW dh–R(x) , c–8 (x) 4 acR – R(x)b 4 dW dhR(x) , (10)
in which the fermionic derivatives are left-derivatives, and
Jm(x) 42 dG dA 8m(x) , hiL(x) 42 dG dc–8Li(x) , h–iL(x) 4 dG dcL8i(x) , (11) hR(x) 42 dG dc–R8 (x) , h–R(x) 4 dG dcR8 (x) , (12) In eqs. (9), (10), aQ Q Qb indicates aQ Q Qb 4 1 Z
f (Q Q Q) expy
2S 1 x (c–i LhiL1 h–iLciL1 c–RhR1 h–RcR1 AmJm)z
, (13)which is an expectation value with respect to the partition functional Z(h , J). Making the parameter e space-time dependent, and varying the generating functional (6) according to the transformation rules (5) for arbitrary e(x) c 0 , we arrive
(14) e–(x)
o
1 2 agm¯mc
R(x) 1g1c–iL(x) Q cR(x) ciL(x) 1
1g2D
(
c–iL(x) Q DcR(x) ciL(x))
1 hR(x)p
4 0 .Together with (12), the Ward identity in terms of the primed fields corresponding to the cR shift-symmetry of the action (1) is given by
(15) 1 2 agm¯ m c 8R(x) 1g1ac –i L(x) Q cR(x) ciL(x)b 1 1g2
a
Dc –i L(x) Q DcR(x) ciL(x))b
2 dG dc–8R(x) 4 0 . Based on this Ward identity, one can get all one-particle irreducible vertices containing at least one external cR.The two-point function is obtained as
x exp [2ipx] d ( 2 ) G dc 8R(x) dc–8R( 0 ) 4 i agmsin (p ma) , (16)which shows that cR does not receive a wave-function renormalization.
3. – The weak-coupling region
Our goal is to seek a possible regime where an undoubled SUL( 2 ) chirally gauged
fermion structure is exhibited in the continuum limit in the phase space (g1, g2, g),
where «g» is the gauge coupling, regarded as a truly small perturbation g K0 at the scale of the continuum limit we consider. Thus, we impose g 40 and Um(x) 41 in
eq. (2). In the weak-coupling limit g1b1 and g2G 1 (area 1 in fig. 1), the action (1)
defines an SUL( 2 )7UR( 1 ) chiral continuum theory with a doubled and weakly
interacting fermion spectrum that is not the continuum theory we seek.
Let us consider the phase of a spontaneous symmetry breaking in the weak-coupling g1, g2limit. Based on the analysis of the large-Nf(Nfis an extra fermion
index, e.g., color, Nc) weak-coupling expansion, we show that the multifermion
couplings in the action (1) undergo Nambu-Jona Lasinio (NJL) spontaneous chiral-symmetry breaking [12]. In this symmetry-breaking phase, indicated 2 in fig. 1, the SUL( 2 )7UR( 1 ) chiral symmetry is violated by
1 2S i (p) 4g1
d4x exp [2ipx]ac –i L( 0 ) Q cR(x)b0c 0 . (17)Assuming that the symmetry breaking takes place in the direction 1 in the 2-dimensional space of the SUL( 2 ) chiral symmetry
(
S1( p) c 0 , S2( p) 40)
, one finds afermion spectrum that contains a doubled Weyl fermion c2
L(x) and an undoubled Dirac
fermion made up of the Weyl fermions c1
Fig. 1. – The phase diagram for the theory (1) in the g1-g2 plane.
fermions can be written as
S21 b1 (p) 4 i a
!
m gmsin p mZ 2(p) PL1 i a!
m gmsin p mP R1 S1(p) , (18) S21 b2 (p) 4 i a!
m gmsin p mZ 2(p) PL. (19)The SUL( 2 )7UR( 1 ) chiral symmetry is broken to UL( 1 )7U(1) with three Goldstone
modes and a massive Higgs mode that are given in the report [13]. The fermion self-energy function Si( p)
(
eq. (17))
for i 41 is given by 1 2S 1 (p) 4
d4 x exp [2ipx] d ( 2 ) G dc8L1(x) dc–8R( 0 ) . (20)Based on the Ward identity (15) originating from the cR shift-symmetry, one can
obtain an identity for the self-energy function Si( p)
(
eq. (19))
. Performing a functionalderivative of eq. (15) with respect to c 8i
L( 0 ) and then putting the external sources h and
J equal to zero, we obtain 1 2 S i (p) 4g1ac –i L( 0 ) Q cR( 0 )b01 2 g2w(p)ac –i L( 0 ) Q DcR( 0 )b0, (21)
where the well-known Wilson factor [14] is
w(p) 4
!
m
(
1 2cos (pma))
, 2 w(p) 4D(p) 4 d4x exp[2ipx] D(x) . (22)
Fig. 2. – The effective four-point interacting vertex.
derivatives of the Ward identity (15) with respect to c–8Li( 0 ), c 8Li(y) and c 8R(x) and
obtains (23)
xyz exp [2iyq2ixp2izp 8] d ( 4 ) G dc8Li( 0 ) dc–L8i(y) dc 8R(z) dc–8R(x) 4 4 g11 4 g2wg
p 1 q 2h
wg
p 81 q 2h
, where p 1qO2 and p81qO2 are the momenta of the cR(x) field; p 2qO2 and p82qO2are the momenta of ci
L(x) field (q is the momentum transfer as shown in fig. 2). These
two identities eqs. (21), (23) show us to two consequences of the cR shift-symmetry
when g14 0 ; i) the normal modes of ciL and cR are massless
Si
( 0 ) 40 ,
(
O(a))
; (24)ii) the normal modes of cR(x) and ciL(x) are free
(
O(a2))
from the four-fermioninteraction, only the doublers of cR(x) and ciL(x) have a non-vanishing four-fermion
interacting. We will come back to these two points in sect. 5 and 6.
Owing to the four-fermion interaction vertex (27), the fermion self-energy function S1( p) in eqs. (17) and (20) obeys the NJL gap-equation in the large-Nf weak-coupling
expansion as shown in fig. 3
S1(p) 44
q
S1(q)
den (q)
(
gA11 4 gA2w(p) w(q))
, (25)where
q f 2p p d4q ( 2 p)4 , den (q) f!
r sin 2 qr1(
S1(q) a)
2, gA1f g1Nfa2, gA2f g2Nfa2.We adopt the parametrization [3]
S1(p) 4S1
( 0 ) 1gA2v1w(p) , S1( 0 ) 4rv1,
(26)
where r depends only on the couplings gA1, gA2, aand v1plays a role as the v.e.v. violating
SUL( 2 )7UR( 1 ) chiral symmetry. We can solve the gap-equation (25) by using this
parametrization. For v1
4 O( 1 O a), one obtains (see appendix B) r 4 gA1gA2I1 1 2gA1I0 , r 4 1 24gA2I2 4 I1 , (27)
where the function In(v1), (n 40, 1, 2), are defined as
In(v1) 44
q wn(q)!
r sin 2 qr1(
S1(q) a)
2 . (28)eq. (26) leads to a crucial result:
gA14 0 , r 40 and S1( 0 ) 40 ,
(29)
which is due to eq. (21), resulting from the Ward identity (15). This means that on the line g14 0 , the normal modes ( p 4 pAC 0 ) of c1L amd cR are massless while their 15
doublers ( p 4pA1pA) acquire chiral-variant masses S1(p) 4gA2v1w(p)
(30)
through the multifermion coupling g2 only. In this case (g14 0 ), the gap-equation is
then given by eq. (27) for r 40,
1 24gA2I2(v1) 40 , i.e. 1 416gA2
q w2(q)
!
r sin 2q r1(
gA2v1w(q) a)
2 . (31)The Wilson factor w2(q) contained in integral (31) indicates that only doublers
contribute to the NJL gap-equation. As v1
K 0 , eq. (27) gives a critical line gA1c(g2c):
gAc 14 1 24gA2cI2( 0 ) 4 gAc 2I12( 0 ) 1I0( 0 ) 24gA2cI0( 0 ) I2( 0 ) , (32)
gA1c4 0 .4 , gA2c4 0 ; gA1c4 0 , gA2c4 0 .0055 ,
(33)
as indicated in fig. 1. These critical values are sufficiently small even for Nf4 1 .
This broken phase cannot be a candidate for a real chiral gauge theory (e.g., the Standard Model) for i) c2
L is doubled
(
see eq. (19))
; ii) the spontaneous symmetrybreakdown on the SUL( 2 ) chiral symmetry is caused by the hard breaking Wilson
term [14] (18) (a dimension-5 operator), which must contribute to the intermediate gauge-boson masses through the perturbative gauge interaction and the disposal of Goldstone modes. The intermediate gauge boson masses thus turn out to be O( 1 Oa). This is, of course, phenomenologically unacceptable.
4. – The strong-coupling region
We now turn to the strong-coupling region, where g1(g2) are sufficiently larger than
a certain critical value g1c(g2c) (area 3 in fig. 1). We can show that the ciLand cR in (1)
form the three-fermion bound states
Ci R4 1 2 a(c – RQ ciL) cR; cnL4 1 2 a(c –i LQ cR) ciL. (34)
These three-fermion states are Weyl fermions pair up with (c–R) and (c–iL), respectively,
to become massive, neutral Cn and charged Cic Dirac modes,
Cic4 (ciL, CiR) , Cn4 (CnL, cR) .
(35)
These three-fermion states (34) carry the appropriate quantum numbers of the chiral group that accommodates ci
Land cR. CiRis SUL( 2 )-covariant and UR( 1 ) invariant. CnL
is SUL( 2 )-invariant and UR( 1 )-covariant. Thus, the spectrum of the massive composite
Dirac fermions Cic and Cn is vector-like, consistent with the SUL( 2 )7UR( 1 ) chiral
symmetry.
On the basis of the 1PI vertex functions eq. (16), we can determine the inverse propagator of the neutral composite Dirac fermion Cn(x) to be
Sn21(p) 4 i a
!
m gmsin p mZ 2n(p) PL1 i a!
m gmsin p mP R1 M(p) , (36)where the unknown Z2n( p) and M(p) are mass and a wave-function renormalization for
CnL(x) field.
The propagator of the charged composite Dirac fermion (35) is
(37) aCic( 0 ) C – c j(x)b 04 4 aciL( 0 ) c– j L(x)b01 aCiR( 0 ) c–L j (x)b01 acLi ( 0 ) C –j R(x)b01 aCRi ( 0 ) C – R j (x)b0,
which we compute in the strong-coupling expansion. In order to understand the three-fermion bound states, we henceforth focus our attention on the region (g1c1 ,
g24 0 ). We make a rescaling of the fermion fields,
ci
L(x) K (g1)1 O4ciL(x) ; cR(x) K (g1)1 O4cR(x) ,
and rewrite the action (1) and the partition function in terms of the new fermion fields Sf(x) 4 1 2 ag11 O2
!
m(
c –i L(x) gm¯mciL(x) 1c–R(x) gm¯mcR(x))
, (39) S1(x) 4c–iL(x) Q cR(x) c–R(x) Q ciL(x) . (40)When the coupling g1K Q , the kinetic terms Sf(x) can be dropped. With the S2(x)
appearing in eq. (40), the integral of exp [2S2(x) ] is given
(
see eq. (80) with D(x) 41and g2K g1 in appendix A
)
byZ 4Pxia
[ dc–aR(x) dcaR(x) ][ dc–Lia(x) dciaL(x) ] exp [2S1(x) ] 424 N,(41)
where «N» is the number of lattice sites. Equation (41) shows that the strong-coupling limit is non-trivial.
In order to calculate eq. (37) we can now perform the strong-coupling expansion in powers of 1 Og1 about the strong-coupling limit. To all orders in this expansion, the
spectrum of the theory contains only massive states (37) even though the SUL( 2 )7UR( 1 ) chiral symmetry is exact. We define the following two-point functions
for the charged Dirac particle (37)
SLLij (x) f aciL( 0 ), c– j L(x)b , (42) SMLij (x) f aciL( 0 ), [c– j L(x) Q cR(x) ] c–R(x)b , (43) SMMij (x) f a[cR( 0 ) Q ciL( 0 ) ] cR( 0 ), [c j L(x) Q cR(x) ] cR(x)b . (44)
We compute them in the strong-coupling expansion in powers of O( 1 Og1). Using the
relations (A.15), (A.17) with D(x) 41 and g2K g1 in appendix A, in the lowest
non-trivial order, we obtain the following recursion relations:
SLLij (x) 4 1 g1
g
1 2 ah
3!
† m S ij ML(x 1m) gm, (45) SMLij (x) 4 d(x) dij 2 g1 1 1 g1g
1 2 ah
!
m † SLLij(x 1m) gm. (46) SMMij (x) 4 1 g1g
1 2 ah
!
† m gmg0SML ij† (x 1m) g0, (47)where for an arbitrary function f(x),
!
m † f (x) 4!
m(
f (x 1m)2f(x2m))
. ( 48 )SXij(p) 4
d4x exp [2ipx] S ij X(x) ,(49)
one gets three recursion relations in momentum space
SLLij (p) 4 1 g1
g
i 4 a3h
!
m sin p mSij ML(p) gm, (50) SMLij (p) 4 dij 2 g1 1 i g1a!
m sin p mSij LL(p) gm, (51) SMMij (p) 4 1 g1g
i ah
!
m sin p m gmg0S ij† ML(p) g0. (52)We solve the recursion relations (50), (51), (52) and obtain
SLLij (p) 4PL dij(i O2a)
!
m sin p m gm ( 1 Oa2)!
m sin 2 pm1 M12 PR, (53) 1 2 aS ij ML(p) 4PL dij( 1 O2) M(p) ( 1 Oa2)!
m sin 2p m1 M12 PL, (54)g
1 2 ah
2 SMMij (p) 4PR dij(i O2a)!
m sin p m gm ( 1 Oa2)!
m sin 2 pm1 M12 PL, (55)where the chiral-invariant mass is
M14 2 g1a .
(56)
The second two-point function in eq. (37) is given by
(57) 1 2 aa[c – R(x) Q c j L(x) ] cR(x), c–iL( 0 )b 4 1 2 ag0S †uj ML(x) g04 4 PR dij( 1 O2) M1 ( 1 Oa2)
!
m sin 2 pm1 M12 PR.We substitute eqs. (53)-(55), (57) into eq. (37), in the lowest non-trivial order of the strong-coupling expansion and obtain the massive propagator of the charged Dirac fermion Ci c, Scij(p) 4
d4x exp [2ipx]aCic( 0 ) C –j c(x)b 4dij (i Oa)!
m sin p mg m1 M1 ( 1 Oa2)!
m sin 2p m1 M12 . (58)be calculated as Sn(p) 4
d4x exp [2ipx]aCn( 0 ) C – n(x)b 4 (i Oa)!
m sin p mg m1 M1 ( 1 Oa2)!
m sin 2p m1 M12 , (59)which coincides, for M(p) 4g, a and Z2n( p) 41, with eq. (36) that is derived by using
the Ward identity (15) and the cR(x) shift-symmetry. Equations (58), (59) show that
the spectrum is vector-like and massive, consistent with the SUL( 2 )7UR( 1 ) chiral
symmetry. In this strong coupling symmetric phase, all fermion modes including the doublers and normal modes of ci
L(x) and cR(x) are bound to be three-fermion states
and then form massive Dirac fermion states. The spectrum of normal modes and doublers is massive and vector-like. This is certainly not what we desire.
5. – Different thresholds for the formation of three-fermion states
The three-fermion states (34) are composed of three elementary Weyl modes through the multifermion couplings S1(x) and S2(x) (1). As expected by Eichten and
Preskill [2], due to the fact that the multifermion coupling S2(x) gives different
contributions to the effective value of g1at large distance for the sixteen Weyl modes of
ci
Land cR in the action (1), these sixteen modes have different thresholds g1c(g2c) for
the formation three-fermion states. In fact, we can explicitly see this point by looking at the four-fermion 1PI vertex function (23), which is exactly obtained by the Ward identity (15) and the cR(x) shift-symmetry
G( 4 ) (p , p 8, q) 4g11 4 g2w
g
p 1 q 2h
wg
p 81 q 2h
, (60)where p 1qO2 and p81qO2 are the momenta of the cR(x) field; p 2qO2 and p82qO2
are the momenta of ci
L(x) field (q is the momentum transfer as shown in fig. 2). In the
case g14 0 the multifermion associated with the normal modes of cR(x) and ciL(x) is
very small
(
O(a2))
.These different thresholds g1c(g2c) can be qualitatively determined by the following
arguments. We consider a complex composite field,
Ai
4 c–RQ ciL,
(61)
its real and imaginary parts are four composite scalars (i 41, 2)
.
`
/
`
´
Ai 14 1 2(c –i LQ cR1 c–RQ ciL) , Ai 24 i 2(c –i LQ cR2 c–RQ ciL) . (62)These composite scalars and their propagators are determined by the two-point function of the theory,
Gij (x) 4 a Ai ( 0 ), A†j (x)b 4 ac–R( 0 ) Q ciL( 0 ), c–R(x) Q c j L(x)b . (63)
G( 4 )
(p , p 8, q) 4g1.
(64)
Adopting the strong-coupling expansion in powers of 1 Og1 (g1c1 ) and the relation
(A.20) with D(x) 41 and g2K g1 in appendix A, we obtain the following recursion
relation Gij (x) 4 d(x) dij g1 1 1 g1
g
1 2 ah
2!
6m Gij (x 1m) . (65)Going to momentum space, we have
Gij
(q) 4
d4x exp [2iqx] Gij(x) ,
(66)
where q is the momentum of the composite scalar Ai
4 c–RQ ciL. The recursion relation
(65) in momentum space is given by
Gij (q) 4 dij g1 1
g
1 2 a2h
1 g1!
6m cos qmGij(q) . (67)As a result, we find the propagators for these four massive composite scalar modes of Ai 4 c–RQ ciL, Gij (q) 44 dij ( 4 Oa2)
!
m sin 2 (qmO 2 ) 1 m2 ; (68) m2 4 4g
g12 2 a2h
, (69)which are degenerate owing to the exact SUL( 2 )7UR( 1 ) chiral symmetry. Thus,
m2
Ai
Ai†
(70)
effectively gives the mass term of the composite scalar field Ai in the effective
Lagrangian. We assume that the 1PI vertex Aj
A†j Ai A†iis positive and the energy of the ground state of the theory is bounded from below. Then, we can qualitatively discuss [3] the second-order phase transition (threshold) from the strong coupling symmetric phase to the weak-coupling NJL broken phase by examining the mass therm of these composite scalars (70). Spontaneous symmetry breaking SU( 2 ) KU(1) occurs, where m2
D 0 turns to m2E 0 . Equation (69) for m24 0 gives rise to the critical point:
gc
1a24 2 , g24 0 ,
(71)
(as indicated in fig. 1), where a phase transition takes place between the NJL symmetry-breaking phase and the strong-coupling symmetric phase.
As for the case g2c 0 , the second multifermion coupling in eq. (60) has to be taken
into account. Note that the thresholds gc
1(g2c) depend on the sixteen modes of ciL and
eq. (69) by the coupling (60) involving g2. As a result, the thresholds for binding
three-fermion states can be qualitatively determined by
m24 4
g
g11 4 g2wg
p 1 q 2h
wg
p 81 q 2h
2 2 a2h
4 0 . (72)Let us first consider the multifermion couplings of each mode «p» of the ci
Land cR,
namely, we set p 4p8, qb1 in the four-point vertex (60). We obtain m2 4 4
g
g11 4 g2w2(p) 2 2 a2h
. (73) Thus, m24 0 gives rise to the critical lines (thresholds):
g1ca24 2 , g24 0 ; g14 0 , a2g2c , b4 0 .008 ,
(74)
where the first binding threshold of the doubler p 4 (p, p, p, p) is located, and g1ca24 2 , g24 0 ; g14 0 , a2g2c , a4 0 .124 ,
(75)
where the last binding threshold of the doublers p 4 (p, 0, 0, 0) is located. In between (area 4 in fig. 1) there are the binding thresholds of the doublers p 4 (p, p, 0, 0) and p 4 (p, p, p, 0) in eq. (73), and the binding thresholds of the different doublers pcp8 in eq. (72). Above g2c , aall doublers are bounded. As for the normal modes ( pA) of ciLand
cR, when g1b1 , the multifermion coupling (60), G( 4 )4 g11 4 g2w2( pA), is no longer
strong enough to form the three-fermion states (c–i
LQ cR) ciL and (c–RQ ciL) cR unless
a2g
2K Q as indicated in fig. 1.
Thus, as expected in ref. [2], several wedges open up as g1, g2increase in the NJL
phase (area 5 in fig. 1), in between the critical lines along which three-fermion states of normal modes and doublers of c–i
L and cR, respectively, approach their thresholds. In
the initial part of the NJL phase (the deep NJL broken phase), the normal modes and doublers of the c–i
L and cR undergo the NJL phenomenon and contribute to eqs. (18),
(19), as discussed in sect. 2. As g1, g2 increase, all these modes, one after the other,
gradually disassociate from the NJL phenomenon and no longer contribute to eqs. (18), (19). Instead, they turn out to associate with the EP phenomenon and contribute to eqs. (58), (59). The first and last doublers of c–iL and cR making this transition are p 4
(p , p , p , p) (74) and p 4 (p, 0, 0, 0) (75), respectively. At the end of this sequence, the normal modes ( p 4pA) make this transition, due to the fact that they possess the different effective multifermion coupling (60). There are two possibilities that one might expect to arise in the continuum limit of chiral fermions defined either on one of these thresholds or within one of these wedges.
Had these thresholds separated the two symmetric phases (the strong-coupling and the weak-coupling symmetric phases), we would have found a threshold over which an EP phase transition takes place, namely, the massless normal modes of the ci
L and cR becomes massive, while the doublers of c–iL and cR acquire
chiral-invariant masses and decouple (58), (59). We would define a continuum theory of massless free chiral fermions [2] on such a threshold. However, this is not really the case [3] for when g24 0 , as has been seen in eq. (69), m2D 0 turns
to m2
E 0 at g14 g1c4 2 , indicating a phase transition between the strong-coupling
symmetric phase and the weak-coupling symmetric phases. Thus there is no EP phase transition over any one of those thresholds. This can be clearly seen (as
(74), (75) and the NJL phase transition (33).
Had one of these wedges contained a spectrum in which all doublers of c–i
Land cR
decouple as Dirac fermions by acquiring chiral-invariant masses (58), (59) and the normal modes of c–iL and cR remain massless and free, within that wedge we would
have obtained a continuous theory of massless free chiral fermions [2]. However, this seems not really the case for the reason that all these wedges from the deep NJL broken phase to the deep EP symmetric phase are continuously connected.
So far, we have almost no possibility to find a distinct threshold for the second-order phase transition to define the continuum limit for chiral fermions and a distinctly chiral-symmetric region in the phase space where a desired spectrum of chiral fermions exists. Nevertheless, a possible resolution of this undesirable situation could probably take place in a particular symmetric phase, i.e. in a segment in the phase diagram, where the doublers of c–i
L and cR have formed three-fermion states
(c–RQ ciL) cR and (c–iLQ cR) ciL via the EP phenomenon, while the normal modes of c–iL
and cR have neither formed such bound states yet nor are they associated with the
NJL-phenomenon. If this were the case, we might find a scaling region of the continuum limit of lattice chiral fermions.
6. – A scaling region of chiral fermions
The possible scaling region of the continuum limit of lattice chiral fermions can probably be found on the line where g14 0 and g2is in a certain segment A, indicated in
fig. 1. Assuming that above g2c , a all doublers are strongly bounded, above g2c , Q all
modes are strongly bounded and g2c, QDg2c, ac1, this peculiar segment A is given by
A 4 [g14 0 , g2c , aE g2E g2c , Q] .
(76)
In order to show that this segment is a possible candidate for the scaling region of the continuum limit of lattice chiral fermions, we must demonstrate that on it the following properties hold:
1) the normal mode of cR is free and decoupled;
2) no spontaneous chiral symmetry breaking occurs;
3) all doublers are strongly bounded and become massive Dirac fermions; 4) the normal modes of ci
L(x) and cR(x) are notbound as three-fermion state
(c–RQ ciL) cR, and an undoubled chiral mode of ciL(x) exists in the low-energy
spectrum.
We have already established the first property in sect, 2. In the case of g14 0 , the
norma mode of cR(x) is massless and does not receive any wave function
renormalization (16)
S21
RR4 igmpAm.
Furthermore in sect. 4, based on the Ward identity (15) associated with the cR(x)
shift-symmetry, we have determined the propagator of the neutral Dirac particle (36), up to a wave function renormalization for Cn
L(x) field.
As for the other properties, since g14 0 , g2c1 in this segment, we can make use of
imply such properties of the theory. Thus, we make a rescaling of the fermion fields,
ci
L(x) K (g2)1 O4ciL(x) ; cR(x) K (g2)1 O4cR(x) ,
(77)
and rewrite the action (1) and the partition function in terms of new fermion fields
Sf(x) 4 1 2 ag21 O2
!
m(
c –i L(x) gm¯mciL(x) 1c–R(x) gm¯mcR(x))
, (78) S2(x) 4c–iL(x) Q [DcR(x) ][Dc–R(x) ] Q ciL(x) . (79)When g2K Q , the kinetic terms Sf(x) can be dropped and we consider the
strong-coupling limit. With S2(x) given in eq. (79), the integral of exp [2S2(x) ] can be
computed (see the beginning of appendix A)
(80) Z 4Pxia
[ dc–aR(x) dcaR(x) ][ dc–iaL(x) dcLia(x) ] exp [2S2(x) ] 424 N(
det D2(x))
4,where the determinant is taken only over the space-time lattice and N is the number of lattice sites. For the non-zero eigenvalues of the operator D2(x), eq. (80) proves the
existence of a sensible strong-coupling limit. However, for the zero eigenvalue of the operator D2(x), this strong-coupling limit is trivial and non-analytic, and the
strong-coupling expansion in powers of 1 Og2 breaks down.
We now discuss property 2), i.e. that this segment is entirely symmetric. The vanishing of the 1PI self-energy function S( p) (17) indicates the absence of spontaneous symmetry breaking. In sect. 2 and 3, on the basis of both the Ward identity (21) of the cR shift-symmetry and the explicit computation in the large-Nf
weak-coupling expansion (29), we have shown that this 1PI self-energy function S( p) vanishes at zero momentum, provided g14 0 ,
S(p) 40 , p 40 . (81)
Obviously, we need to show that this 1PI self-energy function S( p) vanishes for p c 0 in this segment as well. For this purpose, we have to calculate the following two-point functions: SRLj (x) f acR( 0 ), c– j L(x)b , (82) SMRj (x) f acR( 0 ), [c– j L(x) Q cR(x) ] c–R(x)b . (83)
In appendix A, using the strong-coupling expansion in powers O( 1 Og2) above the
non-trivial strong-coupling limit (80) for the non-zero eigenvalues of D2(x), we obtain
the relations (A.15), (A.17). Armed with these relations, in the lowest non-trivial order, one gets the following recursion relations:
SRLj (x) 4 1 g2D2(x)
g
1 2 ah
3!
m † SMRj (x 1m) gm, (84) SMRj (x) 4 1 g2D2(x)g
1 2 ah
!
m † SRLj (x 1m) gm. (85)momentum space are SRLj (p) 4
d4x exp [2ipx] S j RL(x) SMRj (p) 4d4x exp [2ipx] S j MR(x) .For p c 0 and the non-zero eigenvalue D( p) 42w( p) c0 (22), one gets two recursion relations in momentum space,
SRLj (p) 4 1 4 g2w2(p)
g
i 4 a3h
!
m sin p mSj MR(p) gm, (86) SMRj (p) 4 1 gw2(p)g
i 4 ah
!
m sin p mSj RL(p) gm. (87)The solution to these recursion relations is
g
(
8 ag2w2(p))
21 1 a2!
m sin 2 pmh
S j RL(p) 40 . (88)For p c 0 , clearly we must have
SRLj (p) 40 , S j
MR(p) 40 .
(89)
Similarly, we can prove the vanishing of the following two-point functions:
a[c–i L( 0 ) Q cR( 0 ) ] ciL( 0 ), c– j L(x)b , a[c–i L( 0 ) Q cR( 0 ) ] ciL( 0 ), [c– j L(x) Q cR(x) ] c–R(x)b .
This demonstration can be straightforwardly generalized to show the vanishing of all n-point Green functions that are not SUL( 2 )7UR( 1 ) chirally symmetric. As a
consequence, the segment is entirely symmetric and no spontaneous symmetry breaking takes place.
Now we turn to the discussion of the property 3), that all doublers are decoupled as massive Dirac fermions consistently with chiral symmetry. Analogously to the case g1c1 , g24 0 in sect. 4, we need to compute the two-point functions (42)-(44) in the
propagator of the charged Dirac fermion (37). Performing a strong-coupling expansion in powers of 1 Og2 above the non-trivial strong-coupling limit (80) for non-zero
eigenvalues of D2(x), we compute these two-point functions. Through the relations
(A.15), (A.17) and (A.18) that are obtained in appendix A, in the lowest non-trivial order, we obtain the following recursion relations:
SLLij (x) 4 1 g2D2(x)
g
1 2 ah
3!
m † SMLij (x 1m) gm, (90) SMLij (x) 4 d(x) dij 2 g2D2(x) 1 1 g2D2(x)g
1 2 ah
!
m † SLLij (x 1m) gm. (91)SMMij (x) 4 1 g2D2(x)
g
1 2 ah
!
m † gmg0S ij† ML(x 1m) g0. (92)In the non-trivial strong-coupling limit (80) of p c 0 and with D( p) 42w2( p) c 0 , for
the Fourier transforms one gets three recursion relations
SLLij (p) 4 1 4 g2w2(p)
g
i 4 a3h
!
m sin p mSij ML(p) gm, (93) SMLij (p) 4 dij 8 g2w2(p) 1 i 4 g2w2(p) a!
m sin p mSij LL(p) gm. (94) SMMij (p) 4 1 4 g2w2(p)g
i ah
!
m sin p mg mg0S ij† ML(p) g0. (95)We solve these recursion relations (93), (94), (95) and obtain
SLLij(p) 4PL dij(i O2a)
!
m sin p m gm ( 1 Oa2)!
m sin 2p m1 M22(p) PR, (96) 1 2 aS ij ML(p) 4PL dij( 1 O2) M2(p) ( 1 Oa2)!
m sin 2p m1 M22(p) PL, (97)g
1 2 ah
2 SMMij (p) 4PR dij(i O2a)!
m sin p m gm ( 1 Oa2)!
m sin 2 pm1 M22(p) PL, (98)where the chiral-invariant mass is given as
M2(p) 48g2aw2(p) , p c 0 .
(99)
The second two-point function in eq. (37) is given by
(100) a[c–R(x) Q c j L(x) ] cR(x), c–iL( 0 )b 4 4 1 2 ag0S †ij ML(x) g04 PR dij( 1 O2) M2(p) ( 1 Oa2)
!
m sin 2 pm1 M22(p) PR.We substitute eqs. (96)-(98), (100) into eq. (37) and obtain the propagator of the charged Dirac doublers Ci
c ( p c 0 ), Scij(p) 4
d4x exp [2ipx]aCic( 0 ) C – c j (x)b 4dij (i Oa)!
m sin p mg m1 M2(p) ( 1 Oa2)!
m sin 2p m1 M22(p) . (101) Since w2( p) c 0 and M2( p) c 0 all SUL( 2 ) charged doublers are decoupled as massive
Dirac fermions. The massive spectrum for these doublers turns out to be SU( 2 )-QCD vector-like, and in perfect agreement with the SUL( 2 )7UR( 1 ) chiral symmetry.
of the ci
L(x) and cR(x) are massless and chiral in this segment. It is most difficult to
hve a rigorous proof of this property for the time being, since for these normal modes
(
p 4pA40, D( pA) 40 in eq. (80))
, a sensible non-trivial strong-coupling limit does not exist and the strong-coupling expansion in power of 1 Og2fail to converge analytically.We are actually not able to compute the spectra (propagators) of these normal modes to see whether or not they are chiral in this segment. However we expect that in the limit g1K 0 , no EP-mechanism being available the long wave-length mode will remain
chiral. The argument on which we base our expectation is that the effective multifermion coupling
(
the 1PI four-point vertex function (60))
is strongly momentum dependent and becomes negligible for the long wave-length modes, the long wave-length modes receive no mass from such NJL-term, there remains only the momentum-dependent term to possibly modify their chiral nature. However when we let the momentum of the three-fermion state decrease toward zero, due to the vanishing of the coupling in the limit, we shall hit a momentum threshold where the three-fermion bound states disappear and with them their pairing with the Weyl state necessary to produce the Dirac-type fermionic states.Thus the spectrum of the theory in the segment is expected to be the following. There are 15 copies of SU( 2 )-QCD charged Dirac doublers ( p c 0 ) eq. (101) and 15 copies of SU( 2 ) neutral Dirac doublers ( p c 0 ) eq. (36). They are very massive and decoupled. Besides, the low-energy spectrum contains the two massless Weyl modes eqs. (18), (19) for g14 0 and p 4 pA,
SL21(pA)ij4 igmpAmZA2dijPL; SR21(pA) 4igmpAmPR,
(102)
which are in agreement with the SUL( 2 )7UR( 1 ) symmetry. Thus in this region, the
chiral continuum limit is very much like that of lattice QCD. We need to tune only one coupling g1K 0 in the neighbourhood of the segment A g2c , aE g2E g2c , Q. For g1K 0 ,
the cR shift-symmetry is slightly violated, the normal modes of ciL and cR couple
together to form the chiral symmetry breaking terms Si
( 0 ) c–i
LcR, which is a
dimension-3 renormalized operator and thus irrelevant at short distance. This scaling region is ultraviolet stable, where the multifermion coupling g1 is an effective
renormalized dimension-4 operator [15].
* * *
We thank M. Creutz, R. Shrock and M. Testa for discussions.
AP P E N D I X A
We keep in mind that ci
Lhas two components and cRhas one component (all times
a factor of 2 for spin degeneracy). In the strong-coupling limit, the kinetic terms Sf(x):
.
/
´
Sf(x) 4SfL(x) 1SfR(x) , SfL4 1 2 ag21 O2!
m c –i L(x) gm¯mciL(x) ; SfR(x) 4 1 2 ag21 O2!
m c – R(x) gm¯mcR(x) , (A.1)are dropped, the interacting action S2(x) (79) turns out to be bilinear in ciL(x) fermion
field at the same point «x». We first perform the integral of ciL(x) (i fixed) at the point
«x» and obtain (A.2) Di (x) f Pa
[ dc–iaLdcLia] exp[
2c–iL(x) Q [DcR(x) ][Dc–R(x) ] Q ciL(x) ] 4 4 detsu
Dc–1 R(x) Dc1R(x) Dc–2 R(x) Dc1R(x) Dc–1 R(x) Dc2R(x) Dc–2 R(x) Dc2R(x)v
4 2 PaDcaR(x) Dc–aR(x) ,where the determinant detsis taken over spinor space. Then, the partition function of
the one-site theory is given by the integral that is bilinear in fermionic variable ca
R(x),
(A.3) Zi
(x) 4Pa
[ dc–aR(x) dcaR(x) ] Di(x) 44 Pa
g
[ dc–aR(x) dcaR(x) ] 2 DcaR(x) Dc–aR(x)h
4 22dets(
D2(x))
4 22(
D2(x))
2.The total partition function in the strong-coupling limit is then obtained,
Z 4Pi , x
g
Pa[ dc–aR(x) dcaR(x) ] 2 DcaR(x) DcaR(x) Dc–aR(x)h
4 24 N(
detx D2(x))
4, (A.4)
where the determinant detx is taken only over the lattice space-time and N is the
number of lattice sites.
In order to obtain the recursion relations (45)-(47), (90)-(92) and (65) satisfied by two-point functions, we consider an integral of one field cjL(x) at the point «x» defined
as P1js(x) f 1 Zj(x)
x R xj L c–Ljs(x) exp [2Sf(x) 2S2 j (x) ] , (A.5)where the measure of the fermion fields cjL(x)
(
the SUL( 2 )-index j is fixed)
and cR(x)at the point «x» is given as
x R xj L f Pa[ dc–aR(x) dcRa(x) ][ dc–Lja(x) dcLja(x) ] . (A.6)To have a non-vanishing integral of cjL(x) at the point «x», we need a c
j L(x) field in the expansion of exp [2SL f (x) ], and obtain (A.7) P1 js (x) 4 4 1 Zj(x)
g
1 2 ag21 O2h
!
m † [c–jL(x 1m) gm]g x R xj L cLjg(x) c–jsL(x) exp [2SfR(x) 2S2 j (x) ] ,!
m † [c–jL(x 1m) gm]g4 [c– j L(x 1m) gm]g2 [c– j L(x 2m) gm]g. (A.8)Using eq. (A.2), we can first perform the integral over cj
L(x) in eq. (A.7) that is bilinear
in terms of cjL(x),
.
`
/
`
´
xj L cLjg(x) c–jsL(x) exp [2S2 j (x) ] 4u
D j (x) DcgR(x) Dc –s R(x)v
,u
Dj(x) Dcg R(x) Dc–sR(x)v
4 2 dg1ds1Dc2R(x) Dc–2R(x) 2dg2ds2Dc1R(x) Dc–1R(x) 1 1dg1ds2Dc2R(x) Dc–1R(x) 1dg2ds1Dc1R(x) Dc–2R(x) . (A.9) As a result, we have P1 js (x) 4 (A.10) 4g
1 2 ag21 O2h
!
m † [c–jL(x 1m) gm]g 1 Zj(x) x R exp [2SfR(x) ]u
Dj(x) DcgR(x) Dc–sR(x)v
.The remaining integral (A.10) of cR(x) at the «x» point is bilinear in terms of fermionic
variable ca
R(x), we need to have cR(x) and c–R(x) fields in expansion of exp [2SfR(x) ],
for the lowest order of O( 1 Og2), we obtain
P1js(x) 4 (A.11) 4
g
1 2 ag21 O2h
3!
m † [c–jL(x 1m) gm]g[gmcR(x 1m) ]a[c–R(x 1m) gm]bUbags j (x) , where Ujbags(x) 4 1 Zj(x)Pl [ dc–l R(x) dclR(x) ] cbR(x) c–aR(x)u
Dj(x) Dcg R(x) Dc–sR(x)v
. (A.12)Using eqs. (A.3), (A.9) and the following relation:
(A.13) 1 Zj(x) Pl
[ dc –l R(x) dclR(x) ][DcgR(x) Dc–sR(x) ] cbR(x) c–aR(x) 4 4 1 Zj(x) dbsdga D2(x) 2 2det s(
D 2(x))
4 dbsdga D2(x) ,we obtain Uj bags4 dbsdga D2(x) , (A.14) P1js(x) 4 1 D2(x)
g
1 2 ag21 O2h
3!
m † [c–jL(x 1m)QcR(x 1m) ][c–R(x 1m) gm]s. (A.15)In eq. (A.15) the reason for the three fields cR, c–R and ciL being at the same point
«x 1m» is due to the lowest non-trivial approximation.
We define the integral of three fields P3js(x) at the site «x»
P3is(x) f 1 Zj(x)
x R xj L c – L ja (x) ca R(x) c–sR(x) exp [2Sf(x) 2S2 j (x) ] . (A.16)Analogously to the reason for eqs. (A.10), (A.12) to get non-trivial results, we only need a cjL(x) field in the expansion of exp [2SfL(x) ], and considering eqs. (A.9), (A.14) we
obtain (A.17)
.
`
`
`
/
`
`
`
´
P3js(x) 4 1 Zj(x)g
1 2 ag21 O2h
Q Q!
m † [c–jL(x 1m) gm]g x R xj L cLjg(x) c–Lja(x) caR(x) c–sR(x) exp [2S2 j (x) ] 4 4g
1 2 ag21 O2h
!
m † [c–jL(x 1m) gm]gU j bsgb(x) , P3js(x) 4 1 D2(x)g
1 2 ag21 O2h
!
m † [c–jL(x 1m) gm]s.We turn to consider the integral of the four fermion fields at site «x»,
(A.18) P4ij , us(x) f 1 Zj(x)
x R xj L cuiL( 0 ) c–Lja(x) caR(x) c–sR(x) exp [Sf(x) 2S j 2(x) ] C C 1 Zj(x) x R xj L ciuL( 0 ) c–Lja(x) caR(x) c–sR(x) exp [2S2j(x) ] 4d(x) dijUasua(x) 4 dusd(x) dij D2(x) . We compute the following integral Pj2(x) of two fields c–L ja (x) ca R(x) at «x», P2j(x) f 1 Zj(x)
x R xj L c – L ja (x) ca R(x) exp [2Sf(x) 2S2 j (x) ] . (A.19)(A.20) P2j(x) 4 1 Zj(x)
g
1 2 ag21 O2h
Q Q!
m † [c–i L(x 1m) gm]g x R xj L cLjg(x) c–jaL(x) caR(x) exp [2SfR(x) 2S2 j (x) ] 4 4g
1 2 ag21 O2h
2!
6m [c–jL(x 1m) gm]g[gmcR(x 1m) ]sUasga(x) 4 4 1 D2(x)g
1 2 ag21 O2h
2!
6m[c –j L(x 1m) cR(x 1m) ] .Armed with these eqs. (A.15), (A.17), (A.18), (A.20), one can get the recursion relations satisfied by two-point functions in the main text of the article.
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