Preprint typeset in JHEP style - HYPER VERSION

UT-KOMABA/10-9
December 2010

Reﬂection Positivity of N = 1 Wess-Zumino modelon the lattice with exact U(1)R symmetry

Yoshio Kikukawa
Institute of Physics, University of Tokyo, Tokyo, 153-8902, Japan
E-mail: kikukawa@hep1.c.u-tokyo.ac.jp

Kouta Usui
Department of physics, University of Tokyo, 113-0033, Japan
Institute for the Physics and Mathematics of the Universe (IPMU), the University ofTokyo, Chiba 277-8568, Japan
E-mail: kouta@hep-th.phys.s.u-tokyo.ac.jp

Abstract: By using overlap Majorana fermions, the N = 1 chiral multiple can be formulated so that the supersymmetry is manifest and the vacuum energy is cancelled in the free
limit, thanks to the bilinear nature of the free action. It is pointed out, however, that in thisformulation the reﬂection positivity is violated in the bosonic part of the action, although
it is satisﬁed in the fermionic part. It is found that the positivity of the spectral density of the bosonic two-point correlation function is ensured only for the spacial momenta
a|pk | . 1.72 (k = 1, 2, 3). It is then argued that in formulating N = 1 Wess-Zumino modelwith the overlap Majorana fermion, one may adopt a simpler nearest-neighbor bosonicaction, discarding the free limit manifest supersymmetry. The model still preserves thewould-be U(1)R symmetry and satisﬁes the reﬂection positivity.
Keywords: overlap fermion, reﬂection positivity, admissibility condition.

Contents
1. Introduction

1

2. N = 1 chiral multiple with overlap Majorana fermion

2

3. Violation of the reﬂection positivity in the bosonic part
3.1 Reﬂection positivity condition
3.2 Reﬂection positivity of the free overlap boson
3.3 Källén-Lehmann representation of the free overlap boson propagator
3.4 Estimation of the violation in the momentum space

3
3
5
8
9

4. Reﬂetion positivity of lattice Wess-Zumino model

10

5. Discussion

12

A. Spectral density ρ

12

1. Introduction
The chiral multiplet of N = 1 supersymmetry[1] can be formulated on the lattice so that thesupersymmetry is preserved and the vacuum energy is cancelled in the free limit, thanks
to the bilinear nature of the free action. By using overlap (Majorana) fermion [2, 3, 4]
for the fermionic component, species doublers[5, 6, 7] are successfully removed and U(1)Rsymmetry can be maintained at the same time[8, 9, 10]. With this chiral multiplet, one
may formulate lattice N = 1 Wess-Zumino model with exact U(1)R symmetry[11, 12, 13,
14, 15, 16]. A numerical study of this lattice N = 1 Wess-Zumino model has recently beenreported in [17].
The purpose of this short article is, however, to show that in this formulation of thechiral multiplet, the reﬂection positivity[18, 19, 20, 21, 22] is violated in the bosonic partof the action, although it is satisﬁed in the fermionic part, as shown recently in [23]. We
will also examine the spectral density of the bosonic two-point correlation function (cf.
[24]). It is found that the positivity of the spectral density is ensured only for the momenta
a|pk | . 1.72 (k = 1, 2, 3), and the mode with a negative density appears at the energy as
low as aE ≅ 0.69 for the momenta ap = (π, 0, 0), (0, π, 0), (0, 0, π).
We will then argue that in formulating the lattice N = 1 Wess-Zumino model with theoverlap (Majorana) fermion, one may adopt the simpler nearest-neighbor bosonic action,
discarding the free limit manifest supersymmetry. The model so constructed still preserves
the U(1)R symmetry and satisﬁes the reﬂection positivity.

–1–This paper is organized as follows. In section 2, we review brieﬂy the N = 1 chiralmultiple on the lattice formulated with overlap Majorana fermion. In section 3, we showthat the reﬂection positivity is not fulﬁlled in the bosonic part of the action. The spectral
density of the bosonic two-point correlation function is also examined. In section 4, weshow that it is possible to formulate lattice Wess-Zumino model which possesses both
the reﬂection positivity and the exact U(1)R symmetry, by adopting the simpler nearestneighbor bosonic action. Section 5 is devoted to discussion.

2. N = 1 chiral multiple with overlap Majorana fermion
The action of the free N = 1 chiral multiplet is given by
S0 = a4∑{ 1
x2

χT CD1 χ + φ∗ D12 φ + F ∗ F}
1+ χT CD2 χ + F D2 φ + F ∗ D2 φ∗ .
2

(2.1)

In this expression, we have used a decomposition of the overlap Dirac operator [2][3],D = D1 + D2 , where1
D1 = γµ (∂µ∗ + ∂µ )(A† A)−1/2 ,
2

}
1{
1
1 − (1 + a2 ∂µ∗ ∂µ )(A† A)−1/2 ,
a
2

(2.2)

}
1{
γµ (∂µ∗ + ∂µ ) − a∂µ∗ ∂µ .
2

(2.3)

D2 =

and
A = 1 − aDw ,Dw =Note that D1 and D2 have diﬀerent spin structures with respect to spinor space. In
particular, we have {γ5 , D1 } = 0 and [γ5 , D2 ] = 0. In terms of this decomposition, theGinsparg-Wilson relation γ5 D + Dγ5 = aDγ5 D [8] is expressed as
2D2 = a(−D12 + D22 ),(2.4)

and as a consequence, we have relations
1
1
1
γ5 (1 − aD)γ5 (1 − aD) = 1 − aD2 ,22
2

1
γ5 (1 − aD)γ5 D = D1 .
2

(2.5)

It is also understood that the 4 × 4 identity matrix in operators D12 and D2 is omitted
when these operators are acting on bosonic ﬁelds.
It is straightforward to see that the above free action S0 is invariant under “lattice
N = 1 supersymmetry”:√√δϵ χ = − 2P+ (D1 φ + F )ϵ − 2P− (D1 φ∗ + F ∗ )ϵ,√√δϵ φ = 2ϵT CP+ χ,δϵ φ∗ = 2ϵT CP− χ,√√δϵ F = 2ϵT CD1 P+ χ,δϵ F ∗ = 2ϵT CD1 P− χ,

–2–

(2.6)

where ϵ is a 4 component Grassmann parameter. We also note that the free action S0
possesses three types of U(1) symmetry [10]. The ﬁrst is a rather trivial one acting only
on bosonic ﬁelds and is deﬁned by the transformation:δα χ = 0,δα φ = iαφ,δα F = −iαF,(2.7)

where α is an inﬁnitesimal real parameter. The second one is nothing but the chiralsymmetry introduced by Lüscher,1δα χ = iαγ5 (1 − aD)χ,
2

(2.8)

Thirdly, somewhat surprisingly, the bosonic sector of S0 possesses a U(1) symmetry analogous to eq. (2.8):
1
1
δα φ = +iα{(1 − aD2 )φ − aF ∗ },2
2
1
1
δα F = +iα{(1 − aD2 )F − aD12 φ∗ }
(2.9)
2
2
due to the Ginsparg-Wilson relation. The lattice action S0 is not invariant under a uniform
rotation of the complex phase of bosonic ﬁelds, φ, F , due to the presence of terms F D2 φand F ∗ D2 φ∗ . The above provides a lattice counterpart of this uniform phase rotation ofbosonic ﬁelds under which the free action S0 is invariant. Using a linear combination of
the above three U(1) symmetries, it is possible to deﬁne the U(1)R symmetry [10] in theinteracting system.
1δα χ = +iαγ5 (1 − aD)χ,2
1
1
δα φ = −3iαφ + iα{(1 − aD2 )φ − aF ∗ },2
2
1
1
δα F = +3iαF + iα{(1 − aD2 )F − aD12 φ∗ }.
22(2.10)

3. Violation of the reﬂection positivity in the bosonic part
3.1 Reﬂection positivity condition
In this subsection, we will formulate the reﬂection positivity condition. It has been rigorously shown that the lattice theory satisfying the reﬂection positivity condition corresponds
to the quantum theory with unitary time evolution [18, 19, 20]. Here we consider the genericcase in which there are both a bosonic ﬁeld φ and a fermionic ﬁeld ψ. Let us assume that
S(φ, ψ, ψ̄) is the action of a lattice model1 and its partition function Z is given by the path
integration∫
Z = [DφDφ∗ ][DψDψ̄] e−S(φ,ψ,ψ̄) .
(3.1)
In the following, we write the bosonic ﬁeld argument of a function like S(φ) instead of S(φ, φ∗ ) for thenotational simplicity. This notation never means that S is an analytic function of φ.
1–3–

We set the lattice spacing a to be unity, and assume the ﬁnite volume hypercubic latticeΛ = {−L+1, −L+2, . . . , L−1, L}d ⊂ Zd . We impose the anti -periodic boundary conditionin the time direction for the fermionic ﬁeld ψ, while the periodic boundary condition forthe bosonic ﬁeld φ. In the spacial directions, periodic boundary conditions are imposed forboth ﬁelds.
To formulate the reﬂection positivity condition, we ﬁrst introduce the time reﬂection
operator θ as follows. For each site x = (t, x) ∈ Λ, we denote θx = (−t + 1, x). This is thetime reﬂection with respect to the t = 1/2 plane. We deﬁne the operation of θ for bosonicﬁelds as(θφ)(x) = φ(θx)(3.2)

and for functions of bosonic ﬁelds F(φ) as(θF)(φ) = F ∗ (θφ),(3.3)

where * means complex conjugation. For fermionic ﬁelds, the θ reﬂection is deﬁned as(θψ̄)(x) = γ0 ψ(θx),(3.4)(θψ)(x) = ψ̄(θx)γ0 .

(3.5)

We extend this θ operation to the whole ﬁeld algebra A. We deﬁne the ﬁeld algebraA, the algebra of observables, as the Grassmann algebra generated by the fermionic ﬁelds
with the coeﬃcients of the continuous functions of bosonic ﬁelds which are integrable withrespect to the bosonic Gaussian functional measure. For F, G ∈ A, the θ operation isdeﬁned by the relations
θ(FG) = θ(G)θ(F),(3.6)

θ(αF + βG) = α∗ θ(F) + β ∗ θ(G).

(3.7)

For instance, if F has the form ofF(φ, ψ, ψ̄) = f (φ)ψ̄a1 (x1 ) . . . ψ̄an (xn )ψb1 (y1 ) . . . ψbm (ym ),(3.8)

its θ reﬂection should beθ(F)(φ, ψ, ψ̄) = f ∗ (θφ)(ψ̄γ0 )bm (θym ) . . . (ψ̄γ0 )b1 (θy1 )(γ0 ψ)an (θxn ) . . . (γ0 ψ)a1 (θx1 ). (3.9)
Let Λ+ (resp. Λ− ) be the set of lattice sites with positive (resp. non-positive) timecomponents, and A± be the subalgebras of A, which depends only upon ﬁelds on Λ± . In
this notation, θ is a map from Λ± into Λ∓ and from A± into A∓ .
Reﬂection positivity condition is deﬁned through this θ map. For a lattice theory withthe expectation functional 〈·〉 deﬁned as
∫
1
〈F〉 =
[DφDφ∗ ][DψDψ̄] e−S(φ,ψ,ψ̄) F(φ, ψ, ψ̄),F ∈ A,(3.10)
Z
we say the theory is reﬂection positive with respect to θ if any function F+ ∈ A+ fulﬁlls
the inequality〈θ(F+ )F+ 〉 ≥ 0.

–4–

(3.11)

3.2 Reﬂection positivity of the free overlap boson
In this subsection, we investigate the reﬂection positivity of the bosonic sector of the freechiral multiplet (2.1). It will be shown in the following that the bosonic sector does notsatisfy the reﬂection positivity condition. After integrating out the auxiliary ﬁeld F , wehave the overlap boson system which is deﬁned through the lattice action on Λ
Sb (φ) =∑φ∗ (x)¤Λ φ(x),(3.12)x∈Λ

where we have deﬁned¤Λ (x, y) =∑¤(x + 2nL, y),(3.13)n∈Z4

D† D = ¤ · 1̂,(3.14)

with 1̂ being the unit spinor matrix. This ¤ given above is the bosonic overlap operator
on Zd and ¤Λ is that on Λ with periodic boundary conditions. The ﬁeld algebra A ofthis overlap boson system is deﬁned as the set of all continuous functions of bosonic ﬁeld
conﬁgulations φ = {φ(x)}x∈Λ , which are integrable with respect to the bosonic Gaussian
measure[Dφ][Dφ∗ ] e−Sb (φ) .

(3.15)The expectation of this theory is deﬁned by the bosonic path integration∫
1〈F 〉 =[Dφ][Dφ∗ ] e−Sb (φ) F(φ),
F ∈ A.
Z(3.16)The standard way of investigating the reﬂection positivity of lattice ﬁeld theory is toprove that the action can be written in the form of−Sb (φ) = B(φ) + θ(B)(φ) +∑θ(Cs )(φ)Cs (φ),

B, Cs ∈ A+ ,(3.17)swhere in the third term Cs are elements of A+ parametrized by some discrete parameter s[20]. To see that the equation (3.17) indeed implies the reﬂection positivity (3.11), we ﬁrst
note that for an arbitrary F+ ∈ A+ ,∫〈θ(F+ )F+ 〉0 :==[Dφ][Dφ∗ ]θ(F+ )F+∫ ∏dφ(x)dφ∗ (x) F+ (φ) ·

x∈Λ+

∫

∏

dφ(x)dφ∗ (x) θ(F+ )(φ)

x∈Λ−

¯2
¯
¯
¯∫
∏
¯
¯
∗
¯
dφ(x)dφ (x) F+ (φ)¯¯ ≥ 0.
=¯¯¯ x∈Λ+–5–(3.18)

If the action is given in the form of (3.17), we obtain for all F ∈ A+ ,〈 −S〉e b θ(F+ )F+ 0〈〉
P= eB+θ(B)+ s θ(Cs )Cs θ(F+ )F+
0
〈
〉
∞
(
)n
∑
∑
1
θ(Cs )Cs θ(F+ )F+= θ(eB ) eB
n! s
n=0

=
=

∞
∑
1 ∑
n=0
∞
∑
n=0

n! s

···

∑〈

0

〉
θ(eB ) eB θ(Cs1 )Cs1 . . . θ(Csn )Csn θ(F+ )F+ 0

sn

1

)〉
)
∑〈 (
1 ∑
.
θ eB Cs1 . . . Csn F+ eB Cs1 . . . Csn F+
···
n! s
0
s

(3.19)n1This last expression is clearly positive from (3.18). This immediately implies the reﬂection
positivity because〈 −S〉
e b θ(F+ )F+ 0〈θ(F+ )F+ 〉 =≥ 0,
F+ ∈ A+ .
(3.20)〈e−Sb 〉0
We note that the third term in (3.17) may be given by an integration over a continuous
parameter s as∫ds θ(Cs )(φ)Cs (φ),
Cs ∈ A+ .
(3.21)
This type of the action appears in the case of overlap fermions. See Ref.[23] for detail.
Therefore, to prove the reﬂection positivity of the ‘overlap boson’ system reduces toﬁnd the decomposition of the action (3.12) into (3.17). We ﬁrst note that Sb can be written
as
∑
∑
Sb =
φ∗ (x)¤(x, y)φ(y) +φ∗ (x)¤(x, y)φ(y)x,y∈Λ++2∑x,y∈Λ−φ∗ (x)¤(x, y)φ(y),(3.22)x∈Λ+ ,y∈Λ−

where ¤(x, y) is the kernel of the operator ¤ on Λ. To establish the decomposition (3.17),
we should ﬁnd that (i) the second term is the θ reﬂection of the ﬁrst term, and that (ii)the last term is written in the form of∫− θ(Cs )Cs ds(3.23)
for some Cs ∈ A+ parametrized by some parameter s. Note that this second condition isequivalent to say that∫∑∗−
φ (x)¤(x, y)φ(y) = f (s) θ(Cs )(φ) Cs (φ) ds(3.24)x∈Λ+ ,y∈Λ−–6–for some non-negative function f (s). In this bosonic system, while (i) holds true, theproperty (ii) breaks down, as will be shown below.
To show this, we will derive the spectral representation of the kernel ¤(x, y). First,
the Fourier transformation ¤(p) is given by:∑
1 − µ (1 − cos pµ )¤(p) = 1 − ∑[]2∑
2µ sin pµ + 1 −µ (1 − cos pµ )=1+ √b(p) − cos p0a(p) − 2b(p) cos p0,(3.25)

wherea(p) = 1 +∑sin2 pj + b(p)2 ,(3.26)j∑(1 − cos pj ).
b(p) =(3.27)j

From this formula we obtain the following representation of the kernel ¤(x, y),∫¯
d3 p ip·(x−y)¯¤(x, y)¯e
I(x0 − y0 ),=(2π)3x0 ̸=y0(3.28)

where we have deﬁned∫
I(x0 − y0 ) =b(p) − cos p0
dp0 ip0 (x0 −y0 )√e.
2πa(p) − 2b(p) cos p0(3.29)

Furthermore, by Cauchy’s integration theorem, I(x) deﬁned above can be estimated as∫¯
b(p) − cos p0
dp0 ip0 x0¯√
I(x0 )¯e=2πx0 ̸=0a(p) − 2b(p) cos p0∫ ∞b(p) − cosh E
dE −|E|x0√e,(3.30)=
2b(p) cosh E − a(p)E1 πwhere E1 is the positive solution of
2b(p) cosh E1 − a(p) = 0.

(3.31)

Since the kernel of the operator on the ﬁnite lattice ¤Λ (x, y) is deﬁned as (3.13), it isstraightforward to derive the following spectral representation of ¤Λ ,¯
1 ∑ ip(x−y) ∑¯=e
I(x0 + 2n0 L − y0 )¤Λ (x, y)¯(2L)3 px0 >y0n0 ∈Z∫
1 ∑ ip(x−y) ∞ dE
1
=
e
×
3
−2EL
(2L) pE1 π 1 − e[]b(p) − cos p0
× e−E(x0 −y0 ) + e−E(2L−x0 +y0 ) √.
(3.32)
a(p) − 2b(p) cos p0–7–

The second term represents a ﬁnite volume eﬀect.
Now we can see the property (3.24) does not hold. From eq.(3.32), we obtain∑−φ∗ (x)¤Λ (x, y)φ(y)x∈Λ+ ,y∈Λ−=∫
1 ∑ ∞ dE1
cosh E − b(p)√×(2L)3 p E1 π 1 − e−2EL 2b(p) cosh E − a(p){}∑×e−E(x0 −y0 ) eip·(x−y) φ∗ (x)φ(y) + e−2EL e+E(x0 −y0 ) eip·(x−y) φ∗ (x)φ(y)x∈Λ+ ,y∈Λ−∫1cosh E − b(p)1 ∑ ∞ dE
√
=
×
3
−2EL
(2L) p E1 π 1 − e2b(p) cosh E − a(p)
{ (
) (
)
(
) (
) }
× θ CE,p (φ) CE,p (φ) + θ e−EL C−E,p (φ) e−EL C−E,p (φ) ,(3.33)

where we deﬁneCE,p(φ) =∑e−Ex0 eip·xφ∗ (x) ∈ A+ .

(3.34)x∈Λ+In this case, (E, p) plays a role of the parameter s in (3.24). For the condition (3.24)
to be satisﬁed, the coeﬃcient factor cosh E − b(p) should be non-negative for any (E, p)satisfying E1 ≤ E, but this is not the case. In fact, cosh E − b(p) can become both positive
and negative in general, which prevents us from proving the reﬂection positivity.
3.3 Källén-Lehmann representation of the free overlap boson propagator
Another way to see the violation of unitarity of the overlap boson system is to study
the Källén-Lehmann representation of the propagator, (¤)−1 , which should carry all theinformation of Hilbert space of state vectors and the spectrum of Hamiltonian [24]. It issuﬃcient to consider the inﬁnite volume propagator,¯1¯∆+ (x, y) = (x, y)¯¤x0 >y0∫4d p ip(x−y) 1.
(3.35)
=
e
(2π)4
¤(p)
It is possible to evaluate this propagator explicitly to obtain the spectral representation∫ ∞∫dE
d3 p −E(x0 −y0 ) ip(x−y)∆+ (x, y) =eeρ(E, p).
(3.36)π(2π)30The spectral density ρ(E, p) is given by√(cosh E − b(p)) 2b(p) cosh E − a(p)
ρ(E, p) = (isolated poles) +θ(E − E1 ).
cosh2 E − a(p) + b(p)2(3.37)In the second term, there appears the factor cosh E − b(p) which is not positive deﬁnite
and this violates the positivity of ρ(E, p).

–8–

3.4 Estimation of the violation in the momentum space
From the explicit form of the spectral density (3.37), we can ﬁnd where in the Brillouinzone the reﬂection positivity is violated. One notes that there is the region in the spacial
Brillouin zone where the spectral density ρ(E, p) can not become negative. Let us callthis region S. The region S is characterized by the condition that the negative value ofcosh E − b(p) be avoided. The necessary and suﬃcient condition on spacial momenta p to
avoid negative cosh E − b(p) is thatcosh E − b(p) ≥ 0,

∀E ≥ E1 ,(3.38)

which is equivalent to the conditioncosh E1 ≥ b(p)

1+

i.e.

∑

2
pk − (
k sin∑2∑k (1 − cos pk ))k (1 − cos pk )

2

≥ 0.

(3.39)

By using the identity sin2 pk +cos2 pk = 1 and completing the square with respect to cos pk ,the ‘safe’ region S is given by
{
}
d−1
∑
1
d
+
1
(cos pk − )2 ≤S = p ∈ [−π, π]d−1 :,(3.40)
24
k=1

where d is the spacetime dimension.
Now let us estimate the size of S to investigate whether we can ignore the violationof the reﬂection positivity or not. In the case of d = 4, there are three spacial momentum
components. First, we consider the case in which p1 = p2 = p3 =: p. In this direction, thesafe momentum region has the extent−1.72 ≤ p ≤ 1.72.

(3.41)

Second, we consider another direction p1 = p, p2 = p3 = 0. In this case, in the safe region
S, p is restricted by 2
−1.95 ≤ p ≤ 1.95.

(3.43)

These regions are a little bit lager than or the same as [−π/2, π/2]d−1 .
When the spacial momenta p does not belong to S, the spectral density ρ(E, p) has
to become negative on the energy interval E1 ≤ E < Ec , where E1 and Ec are determinedbycosh E1 =

a(p)
,
2b(p)

cosh Ec = b(p),(3.44)

2This restriction value of spacial momenta p is exactly the same as for the two dimensional case. In thecase of d = 2, the spacial momenta has only one component and S is given as
S = {−1.95 ≤ p ≤ 1.95}.

–9–(3.42)

since ρ(E, p) < 0 is equivalent to a(p)/2b(p) ≤ cosh E < b(p) when p ̸∈ S. We will
numerically estimate E1 and Ec , the lower and upper bound of the energy interval on
which the spectral density become negative. For instance, if d = 4, these energy values arecomputed as shown in the following table:p(π, π, π)(π, π, 0)(π, 0, 0)

b(p)
6
4
2

a(p)
37
17
5

a(p)/2b(p)
37/12
17/8
5/4

E1 = cosh−1 a(p)/2b(p)1.79 . . .
1.39 . . .
0.69 . . .

Ec = cosh−1 b(p)2.48 . . .
2.06 . . .
1.32 . . .

Whether these values are large enough or not should depend on the physics one wants tosee through the overlap boson.

4. Reﬂetion positivity of lattice Wess-Zumino model
To remedy the violation of the reﬂection positivity, one may adopt the simpler nearestneighbor action for the boson ﬁelds, φ and F as follows3 :S0′ = a4∑{x}
1
− χT CDχ + φ∗ (−∂µ∗ ∂µ )φ + F ∗ F .
2

(4.1)

This action still possesses three types of U(1) symmetry, Eq. (2.7), (2.8) andδα φ = +iαφ,δα F = +iαF,(4.2)

instead of Eq. (2.9).
In this formulation of the chiral multiplet, the action of the lattice N = 1 Wess-Zumino
model may be given as follows:
S = a4∑{ 1
1− χT CDχ + φ∗ (−∂µ∗ ∂µ )φ + F ∗ F + X T CX
2ax−g χ̃T CφP+ χ̃ − g ∗ χ̃T Cφ∗ P− χ̃ + gF φ2 + g ∗ F ∗ φ∗2

}
,

(4.3)

where X(x) is an auxiliary Majorana fermion ﬁeld and χ̃(x) = χ(x) + X(x). Then one
may deﬁne the U(1)R symmetry as follows:1δα χ = +iαγ5 (1 − aD)χ,2
δα φ = −2iαφ,δα F = +4iαF
3

(4.4)

Here, we have changed the sign convention of the fermionic action by introducing new Majorana ﬁeld
χ = iχ. Of course this does not change any physical results. It is simply because this convention has beenused in the proof of the reﬂection positivity for the overlap fermions in our previous work [23].
′– 10 –

The reﬂection positivity is now satisﬁed in this formulation of the Wess-Zumino model.
The θ-reﬂection is deﬁned for the bosonic ﬁelds χ, F in the same way as in the generic case(3.3),
θφ(x) = φ(θx)(4.5)

θF (x) = F (θx)(4.6)

and for the fermionic ﬁelds χ, X as in (3.4),(θχ̄)(x) = γ0 χ(θx),(θχ)(x) = χ̄(θx)γ0 ,(4.7)(θX̄)(x) = γ0 X(θx),(θX)(x) = X̄(θx)γ0 .

(4.8)

Note that this deﬁnition of θ reﬂection does not contradict to the Majorana conditionsχ̄ = χT C and X̄ = X T C. Our ﬁeld algebra A here is that of the polynomial algebra offermionic ﬁelds whose coeﬃcients are the well-behaved functions of the bosonic ﬁelds. We
extend θ operation to whole algebra A, by the relations (3.6) and (3.7).
To prove the reﬂection positivity of the Wess-Zumino model, it is suﬃcient to showthat the action (4.3) can be rewritten in the form of (3.17)∑−S = B + θ(B) +θ(Cs )Cs ,B, Cs ∈ A+ ,(4.9)
s[23]. Let us ﬁrst consider the free part of (4.3). The ﬁrst term in (4.3), the overlap Majorana
fermion, can be written in the form of (3.17) as is shown in ref [23]. On the other hand,the second term can be written in the form of (4.9), as is well-known. Furthermore, thethird and fourth terms in (4.3) is θ reﬂection of themselves and do not contain any ‘time
hopping terms’. Therefore, these terms can be written in the form of
}
∑{
1
a4
F ∗ F + X T CX
a
x∈ Λ
}
[ ∑ {
}]
∑ {
1
1
= a4
F ∗ F + X T CX + θ a4
F ∗ F + X T CX .
(4.10)
a
a
x∈Λ+

x∈Λ+

The rest of the terms in (4.3) are interaction terms,}∑{
Sint := a4−g χ̃T CφP+ χ̃ − g ∗ χ̃T Cφ∗ P− χ̃ + gF φ2 + g ∗ F ∗ φ∗2 ,(4.11)

x

which are all strictly local. They are equal to theta-reﬂection of themselves again, and do
not contain any nonlocal ‘time hopping’ terms either. This means that Sint can also be
written as
Sint = B + θ(B)
with
B=

}
∑{
−g χ̃T CφP+ χ̃ − g ∗ χ̃T Cφ∗ P− χ̃ + gF φ2 + g ∗ F ∗ φ∗2 ,(4.12)(4.13)x∈Λ+

which obviously belongs to A+ . Therefore, one concludes that this lattice Wess-Zumino
model satisﬁes the reﬂection positivity condition.

– 11 –

5. Discussion
Preserving R symmetry exactly is a useful way in formulating supersymmetric ﬁeld theories
on the lattice. This point has been emphasized by Elliot, Giedt and Moore [25] in theirformulation of four-dimensional N = 4 super Yang-Mills theory. The discrete R symmetry
in the two-dimensional N = 2 Wess-Zumino model [26] has played an important role in
the numerical study of the correspondence to N = 2 conformal ﬁeld theories[27].
In formulating the exact R symmetry on the lattice, however, there is a freedom inthe choice of the bosonic part of the action. When one can preserve some part of theextended supersymmetries in the theories with N ≥ 2 [26, 28], it seems useful to adopt
the bosonic actions to preserve the supersymmetries, although one should take into care apossible eﬀect of the violation of the reﬂection positivity. But, for the theories of N = 1, it
seems diﬃcult to preserve the supersymmetry in general[29]. The free limit supersymmetry
may still help in the convergence to the supersymmetric limit in the interacting models.
Otherwise, the reﬂection positivity condition may give a possible guideline to choose abosonic action.
It would be interesting to examine further the inter-relation among the reﬂection positivity, the vacuum energy cancellation(the exact supersymmetry) and the exact U(1)Rsymmetry of free chiral multiplet on the lattice. If one adopts the Majorana Wilson fermion
for the fermionic component of the chiral multiplet, one can show that the bosonic partof the supersymmetric action now fulﬁlls the reﬂection positivity condition. In this case,the U(1)R symmetry is not manifest. But, through the block spin transformation, it isrecovered in the ﬁxed point action [8]. In this course of the renormalization group transformatons, it seems possible to maintain the vacuum energy cancellation by adjusting theparameters in the block-spin kernels and the normalization factors. Then, if the reﬂection
positivity could also be maintained through the block-spin transformation, all the three
conditions could be fulﬁlls in the ﬁxed point approach[30, 31, 32, 33, 34].

Acknowledgements
K.U. would like to thank Tsutomu T. Yanagida for continuous encouragement. He also
thanks Kenji Maeda for valuable discussions. K.U. is supported by Global COE Program“the Physical Science Frontier”, MEXT, Japan. This work was supported by World Premier International Center Initiative (WPI Program), MEXT, Japan. Y.K. is supported inpart by Grant-in-Aid for Scientiﬁc Research No. 21540258, 21105503.

A. Spectral density ρ
In this appendix, we derive eq.(3.37). Deﬁne a function f (z) as
f (z) =

1
(p0 = z, p).
¤– 12 –(A.1)

It is straightforward to ﬁnd the explicit form√(√)a(p) − 2b(p) cos za(p) − 2b(p) cos z − (b(p) − cos z).
f (z) =
a(p) − b(p)2 − cos2 z(A.2)

f has a branch cut where inside the square root becomes negative,∑
1 + j sin2 pj + b(p)2a(p)a(p) − 2b(p) cos z < 0 ⇐⇒ cos z >
=
.
2b(p)
2b(p)

(A.3)

Since
1+

∑

2
2
j sin pj + b(p)

2b(p)

√
≥

1+

∑

sin2 pj ≥ 1,

(A.4)

j

Equation a(p) − 2b(p) cos z = 0 has two pure imaginary solutions ±iE1 (E1 > 0) :
(
)
a(p)
−1
E1 = cosh,
E1 > 0.
2b(p)f has isolated poles where the denominator vanishes,∑cos2 z = a(p) − b(p)2 = 1 +sin2 pj

(A.5)

(A.6)

j

To solve this equation, put z = x + iy, x, y ∈ R and it becomescos2 x cosh2 y + sin2 x sinh2 y − 2i cos x cosh y sin x sinh y = a(p) − b(p)2 ≥ 1.

(A.7)

Noting that the imaginary part must vanish, one ﬁnds that (A.7) is equivalent to{cos x = 0or

sinh2 y = a(p) − b(p)2{
sin x = 0

(A.8)

cosh2 y = a(p) − b(p)2 .

The solution of these equations are :z = x + iy = kπ ± iE0

1
or z = x + iy = ( + l)π ± iE22(k, l ∈ Z),(A.9)

where E0 and E2 are deﬁned as the unique positive solution of cosh E0 = a(p) − b(p)2 andsinh E2 = a(p) − b(p)2 respectively.
From these observations, the analytic structure of the function f can be drawn in thecomplex z-plane as in the ﬁgure below. The application of Cauchy’s integration theorem
to the contour drawn in the ﬁgure tells us that∫ π
∫ ∞dp0 ip0 (x0 −y0 )idEef (p0 ) = 2πi × (residue from poles) +(f+ (iE) − f− (iE)),
−π 2π
E1 2π
(A.10)
where in f+ one chooses the branch in which√−1 = i, in f− the branch in which– 13 –√−1 = −i.

It is not diﬃcult to see that only the pole z = ±iE1 has a non-zero residue and therefore
contributes the ﬁrst term of eq (A.10). The other poles are removable singularities. The
contributing residue coming from z = iE1 can be computed to give the result√√a(p) − 2b(p) cosh E1 ( a(p) − 2b(p) cosh E1 − (b(p) − cosh E1 ))Res (f, z = iE1 ) =χS (p)
4πi cosh E1 sinh E1(A.11)
where χS is a characteristic function of S deﬁned as{
1 (p ∈ S)
χS (p) =0 (p ̸∈ S).

(A.12)The second term in (A.10) is also easily calculated as∫ ∞idE(f+ (iE) − f− (iE)) =
E1 2π

∫ ∞
0

√
dE (cosh E − b(p)) 2b(p) cosh E − a(p)θ(E − E1 )π
cosh2 E − a(p) + b(p)2(A.13)

Substituting (A.11) and (A.13) into (A.10), we get the spectral density ρ√√a(p) − 2b(p) cosh E( a(p) − 2b(p) cosh E − (b(p) − cosh E))
πχS (p)δ(E − E1 )ρ(E, p) =2 cosh E sinh E√(cosh E − b(p)) 2b(p) cosh E − a(p)+θ(E − E1 ).
(A.14)
cosh2 E − a(p) + b(p)2

References[1] J. Wess and B. Zumino, Phys. Lett. B 49, 52 (1974).
[2] H. Neuberger, Phys. Lett. B 417, 141 (1998) .
[3] H. Neuberger, Phys. Lett. B 427, 353 (1998) .
[4] Y. Kikukawa and H. Neuberger, Nucl. Phys. B 513, 735 (1998) [arXiv:hep-lat/9707016].
[5] L. H. Karsten and J. Smit, Nucl. Phys. B 183, 103 (1981).
[6] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981) [Erratum-ibid. B 195, 541(1982)].
[7] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 193, 173 (1981).
[8] P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 (1982).
[9] M. Luscher, Phys. Lett. B 428, 342 (1998) .
[10] T. Aoyama and Y. Kikukawa, Phys. Rev. D 59, 054507 (1999) [arXiv:hep-lat/9803016].
[11] K. Fujikawa and M. Ishibashi, Nucl. Phys. B 622, 115 (2002) [arXiv:hep-th/0109156].
[12] K. Fujikawa and M. Ishibashi, Phys. Lett. B 528, 295 (2002) [arXiv:hep-lat/0112050].
[13] K. Fujikawa, Nucl. Phys. B 636, 80 (2002) [arXiv:hep-th/0205095].

– 14 –

[14] M. Bonini and A. Feo, JHEP 0409, 011 (2004) [arXiv:hep-lat/0402034].
[15] Y. Kikukawa and H. Suzuki, JHEP 0502, 012 (2005) [arXiv:hep-lat/0412042].
[16] M. Bonini and A. Feo, Phys. Rev. D 71, 114512 (2005) [arXiv:hep-lat/0504010].
[17] C. Chen, E. Dzienkowski and J. Giedt, arXiv:1005.3276 [hep-lat].
[18] K. Osterwalder and R. Schrader, Commun. Math. Phys. 31, 83 (1973).
[19] K. Osterwalder and R. Schrader, Commun. Math. Phys. 42, 281 (1975).
[20] K. Osterwalder and E. Seiler, Annals Phys. 110, 440 (1978).
[21] M. Luscher, Commun. Math. Phys. 54, 283 (1977).
[22] P. Menotti and A. Pelissetto, Nucl. Phys. Proc. Suppl. 4, 644 (1988).
[23] Y. Kikukawa and K. Usui, Phys. Rev. D 82, 114503 (2010) [arXiv:1005.3751 [hep-lat]].
[24] M. Luscher, arXiv:hep-th/0102028.
[25] J. W. Elliott, J. Giedt and G. D. Moore, Phys. Rev. D 78, 081701 (2008) [arXiv:0806.0013[hep-lat]].
[26] Y. Kikukawa and Y. Nakayama, Phys. Rev. D 66, 094508 (2002).
[27] H. Kawai and Y. Kikukawa, arXiv:1005.4671 [hep-lat].
[28] Y. Kikukawa and F. Sugino, Nucl. Phys. B 819, 76 (2009) [arXiv:0811.0916 [hep-lat]].
[29] M. Kato, M. Sakamoto and H. So, JHEP 0805, 057 (2008).
[30] P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B 427, 125 (1998)[arXiv:hep-lat/9801021].

Im
¾

6
¾
6

r

iE1
¾?
t iE

r

?

−π

6

π/2
r + iE2
r

0

−π/2

r

0

π/2
r

- -

π

Re

r

r

r

Figure 1: Integration contour of f . Only the z = iE0 pole has non-vanishing residue.

– 15 –

[31] P. Hasenfratz, Nucl. Phys. B 525, 401 (1998) [arXiv:hep-lat/9802007].
[32] T. L. Bell and K. G. Wilson, Phys. Rev. B11 (1975) 3431
[33] U. J. Wiese, Phys. Lett. B 315, 417 (1993) [arXiv:hep-lat/9306003].
[34] H. So and N. Ukita, Phys. Lett. B 457, 314 (1999) [arXiv:hep-lat/9812002].

– 16 –