Lattice supersymmetry in one dimension
with two supercharges
Alessandra Feo
Universitá di Torino and INFN Torino
Sestri Levante 2008 Convegno Informale di Fisica Teorica
Sestri Levante, June 4-6, 2008
Based on:
Arianos, D’Adda, A.F., Kawamoto and Saito, arXiv:0806.0686 [hep-lat];
Arianos, D’Adda, Kawamoto and Saito, arXiv:0710.0487 [hep-lat]
Introduction
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Lattice formulation of Super Yang-Mills theory: Problems
• The major obstacle in formulating a supersymmetric theory on the lattice
arises from the fact that the supersymmetry algebra, which is an extension of the Poincaré algebra, is explicitly broken on the lattice [Dondi and
Nicolai 1977] .
In particular, the Super Poincaré algebra is given by the anti-commutator
of a supercharge Qα and its conjugate Qβ yields the generator of infinitesimal translations Pµ. Schematically,
µ
Pµ
Qα, Q†β = 2σαβ
On the lattice there are no infinitesimal translations and therefore the
supersymmetry algebra must be broken.
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Ordinary Poincaré algebra is also broken by the lattice but the
hypercubic crystal symmetry forbids relevant operators which could
spoil the Poincaré symmetry in the continuum limit →
The Poincaré invariance is achieved automatically in the continuum limit without fine tuning since operators that violate Poincaré
invariance are all irrelevant.
However, in the case of the super Poincaré algebra, the lattice crystal
group is not enough to guarantee the absence of supersymmetry violating
operators.
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Failure of the Leibniz rule
On the lattice the Leibniz rule does not hold anymore. [Fujikawa, hepth/0205095]
1
(f (x + a)g(x + a) − f (x)g(x)) =
a
1
1
= (f (x + a) − f (x))g(x) + f (x)(g(x + a) − g(x))
a
a
1
1
+a (f (x + a) − f (x)) (g(x + a) − g(x))
a
a
= (∇f (x))g(x) + f (x)(∇g(x)) + a(∇f (x))(∇g(x))
the breaking of supersymmetry is of order O(a).
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• If the supersymmetric theory contains scalar mass terms they break supersymmetry. Since these operators are relevant fine tuning is needed in
order to cancel their contributions.
• A naive regularization of fermions results in the doubling problem
[Nielsen and Ninomiya, 1981] → wrong number of fermions and violation of
the balance between bosons and fermions
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Solutions
• Without exact lattice supersymmetry one might hope to construct
non-supersymmetric lattice theories with a supersymmetric continuum limit.
This is the case of the Wilson fermion approach for the N = 1 supersymmetric Yang-Mills theory in 4d where the only operator that violates
supersymmetry is a fermion mass term.
By tunning the fermion mass to the supersymmetric limit one recovers
supersymmetry in the continuum limit [Curci and Veneziano, 1987; I. Montvay, hep-lat/0112007, hep-lat/9510042; A.F., hep-lat/0305020; Taniguchi and Suzuki;
Maru and Nishimura]
In the case of theories with extended supersymmetries the fine tuning of
coupling constant is neither feasible nor theoretically practical.
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– Alternatively, using domain wall fermions [Kaplan and Schmaltz hep-lat/0002030;
Ginsparg and Wilson, 1982] or overlap fermions [Huet, Narayanan, Neuberger,
hep-th/9602176] this fine tunning is not required [Fleming, Kogut and
Vranas; Itoh, Kato and Sawanaka; Harada and Pinsky]
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• A number of approaches that allow to realize part of the supersymmetry
as an exact symmetry on the lattice in certain class of theories with
an extended supersymmetry have been proposed in recent years [Kaplan,
Cohen, Katz and Unsal; Sugino; Catterall; Damgaard and Matsura; Giedt; Onogi,
Ohta and Takimi] Two approaches: Orbifold methods. SYM case. Twisted
formulations.
This exact lattice supersymmetry is expected to play a key role
to restore continuum supersymmetry without (or with less) fine
tuning of the parameters of the action.
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• A more ambitious approach, aiming to preserve exactly all supersymmetries in some extended supersymmetric models (including N = 2 WessZumino, N = 2 SYM and N = 4 SYM), was also proposed [D’Adda,
Kanamori, Kawamoto and Nagata, hep-lat/0406029; hep-lat/0409092; hep-lat/0507029;
arXiv:0707.3533; arXiv:0709.0722]
Here the key ingredient is the use of Dirac-Kähler fermions to overcome
the doubling problem and most important the introduction of a mild
non commutativity to preserve the Leibniz rule.
Susy transformations on the lattice can be defined without ambiguity
with the aid of the Modified Leibniz rule.
– Recent claims of inconsistency of the formalism [Bruckmann and de Kok,
hep-lat/0603003] will be discussed in this talk and shown to be baseless.
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– Other examples of lattice exact supersymmetry for a simpler model
as Wess-Zumino (kinetic part) Golterman and Petcher, Bietenholz; Catterall,
Karamov and Gregory; Fujikawa and Ishibashi; Fujikawa; Wipf; Giedt and Poppitz;
Kato, Sakamoto and So; Beccaria, Campostrini and A.F.]
– For the N = 1 Wess-Zumino model in 4 dimensions an exact lattice formulation have been achieved using the Ginsparg-Wilson formulation (also checked the Ward-Takahashi identity: exact at fixed
lattice spacing and in the continuum limit at one loop order →The
renormalization wave function of all fields coincide) [Bonini and A.F.,
hep-lat/0402034, hep-lat/0504010]
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Recent reviews
1. J. Giedt, hep-lat/0701006, (Plenary talk at Lattice 2006), “ Advances
and applications of lattice supersymmetry”.
2. S. Catterall, hep-lat/0509136, (Plenary talk at Lattice 2005), “DiracKahler fermions and exact lattice supersymmetry”.
3. A. F., hep-lat/0410012, (Review for MPLA), “ Predictions and recent
results in SUSY on the lattice’
4. D. B. Kaplan, hep-lat/0309099, (Plenary talk at Lattice 2003), “Recent
developments in lattice supersymmetry”.
5. A. F., hep-lat/0210015, (Plenary talk at Lattice 2002), “Supersymmetry
on the lattice”
6. I. Montvay, hep-lat/0112007, (Review), “Supersymmetric Yang-Mills theory on the lattice”
7. I. Montvay, hep-lat/9709080, (Plenary talk at Lattice 1997), “SUSY on
the lattice”
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Plan of the Talk
Example of exact lattice supersymmetry for all supersymmetries
Lattice formulation of N = 2 susy model in one dimension [Arianos, D’Adda,
Kawamoto and Saito, arXiv:0710.0487]
• Modified Leibniz rule
• Modified Ward-Takahashi identities [Arianos, D’Adda, A.F., Kawamoto and
Saito, arXiv:0806.0686]
Discussion of some recent claims of inconsistency.
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The N = 2 Supersymmetric Model in 1d: Continuum
The model we are going to discuss is the supersymmetric quantum mechanics
with two supersymmetry charges that have the following algebra
Q21 = Q22 = Px ,
{Q1, Q2 } = 0,
Px =
[Px, Q1 ] = [Px, Q2] = 0
∂
.
∂x
A superspace representation of the algebra may be given in terms of two
Grassmann odd, real coordinates θ1 and θ2 , namely:
Q1 =
∂
∂
+ θ1 ,
∂θ1
∂x
Q2 =
∂
∂
+ θ2 .
∂θ2
∂x
The field content of the theory is described by a hermitian superfield Φ(x, θ1, θ2 ):
Φ(x, θ1 , θ2 ) = ϕ(x) + θ1 ψ1(x) + θ2 ψ2(x) + θ1 θ2 F (x),
where ψ1 and ψ2 are Majorana fermions.
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The supersymmetry transformations of the superfield Φ are given by
δj Φ = [ηj Qj , Φ]
j = 1, 2
where ηi are the Grassmann odd parameters of the transformation. In terms
of the component fields the susy transformations reads,
δj ϕ = iηj ψj
δj ψk = iδj,k ηj ∂x ϕ + ǫjk ηj F
δj D = −ǫjk ηj ∂xψk
Supersymmetry transformations of superfields products obey ordinary Leibniz
rule
δi(Φ1 Φ2 ) = (δiΦ1 )Φ2 + Φ1 (δiΦ2 )
To construct the action we need to introduce the superderivatives,
Dj =
∂
∂
− θj .
∂θj
∂x
which anticommute with the supersymmetry charges Qj and satisfy the algebra
∂
Dj2 = −
{D1, D2 } = 0
∂x
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The supersymmetric action can then be defined in terms of the superfield Φ
as
Z
1
dxdθ1 dθ2
D1ΦD2 Φ + iV (Φ)
2
where the superpotential is defined as
V (Φ) =
1
1
mΦ2 + gΦ4
2
4
the action written in terms of the component fields is
Z
1
2
2
S =
dx{ (∂x ϕ) − F − ψ1∂x ψ1 − ψ2∂x ψ2
2
−m(iψ1 ψ2 + F ϕ) − g(3iϕ2ψ1ψ2 + F ϕ3 )}
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Matrix Representation and Modified Leibniz Rule
Consider a one dimensional lattice with N sites and periodic boundary conditions. Let a be the lattice spacing. The N sites of the lattice will be labeled
by a coordinate x = ra. L is the lattice size, L = aN .
And a scalar field ϕ on the lattice defined by a set on N numbers ϕr (r =
1, 2, · · · , N ), ϕ(x) = ϕ(ra) ≡ ϕr .
The N numbers ϕr can be regarded as the
matrix ϕ:
ϕ1 0
0 0
0 ϕ2 0 0
0 ϕ3 0
ϕ=0
...
...
...
...
0
0
0 0
eigenvalues of an N × N diagonal
···
···
···
...
···
0
0
0
...
ϕN
whose rows and columns are in one to one correspondence with the sites of
the lattice. Derivatives are replaced on the lattice by finite differences
(∂+ ϕ)r = (ϕr − ϕr−1 ) .
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In matrix notation finite differences may be represented using the
trices ∆+ and ∆− :
0 0 0 ···
0 1 0 0 ··· 0
1 0 0 · · ·
0 0 1 0 · · · 0
0 1 0 · · ·
0 0 0 1 · · · 0
−1
=
,
∆
=
∆
∆+ = .. .. .. .. . .
0 0 1 · · ·
−
.
+
. ..
. . . .
... ... ... . . .
0 0 0 0 · · · 1
0 0 0 ···
1 0 0 0 ··· 0
shift ma0
0
0
0
...
1
1
0
0
0
...
0
namely, in components
(∆+ )rs = δr,s−1 ,
(∆− )rs = δr,s+1 .
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In the continuum the derivative ∂ϕ is just the commutator [∂, ϕ]; on the lattice
however the commutator [∆+ , ϕ] is not diagonal. In order to write a diagonal
matrix we have to defined it as
ϕ1 − ϕN
0
0
(∂+ ϕ) = ∆− [∆+ , ϕ] =
...
0
0
0
0 ···
ϕ2 − ϕ1
0
0 ···
0
ϕ3 − ϕ2 0 · · ·
...
...
... . . .
0
0
0 ···
0
0
0
...
ϕN − ϕN −1
The factor ∆− in front of the commutator is responsible for the violation of
the Leibniz rule, in fact we have:
(∂+ ϕψ) = (∂+ ϕ)ψ + ∆− ϕ∆+ (∂+ ψ)
The factor ∆− ϕ∆+ is a shifted field equivalent to (∆− ϕ∆+ ) = ϕr−1 .
The modified Leibniz rule reflects the fact that translational symmetry on the
lattice is a discrete, and not continuous symmetry.
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Consider an action given as the trace of a product of fields ϕi,
S = Tr ϕ1 ϕ2 · · · ϕr
The trace corresponds to the sum over all lattice sites, and translational
invariance can be simply expressed as the invariance of the action under
ϕi → ∆− ϕi ∆+ = ϕi + δϕi
where δϕi = −(∂+ ϕ).
The variation of the Lagrangian can be cast in the form:
δ (ϕ1 ϕ2 · · · ϕr ) = (δϕ1 )ϕ2 · · · ϕr−1 ϕr + (ϕ1 + δϕ1 )(δϕ2)ϕ3 · · · ϕr + · · ·
+(ϕ1 + δϕ1 )(ϕ2 + δϕ2) · · · (ϕr−1 + δϕr−1 )(δϕr )
which is again the modified Leibniz rule.
The linear terms in δϕi give the ordinary Leibniz rule typical of the continuum
limit.
The situation is different in susy because supersymmetry charges are non
diagonal and hence supersymmetry transformations of a product of superfields
obey a modified Leibniz rule on the lattice.
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If δα is the supersymmetry variation generated by the SUSY charge Qα and and
Φi(θ, x) (i = 1, 2) are superfields, the shift aα associated to Qα will determine
the modified Leibniz rule:
δα (Φ1(θ, x)Φ2 (θ, x)) = (δαΦ1(θ, x)) Φ2 (θ, x) + Φ1(θ, x + aα) (δαΦ2 (θ, x))
However, these modified Leibniz rule can not be derived (in this present
formulation) from a field transformation as ϕi → ∆− ϕi ∆+ = ϕi + δϕi.
The conseguences:
The action will be invariant under the susy trasformations given above using
the modified Leibniz rule for a product of fields.
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Matrix Representation of a Grassmann Algebra
In order to define the N = 2 supersymmetric quantum mechanics on a one
dimensional lattice it is convenient to introduce a matrix representation for
the two Grassmann variables θ1 and θ2
θ1 ≡ σ+ ⊗ 1 ⊗ ∆+ ,
∂
≡ σ− ⊗ 1 ⊗ ∆− ,
∂θ1
θ2 ≡ σ3 ⊗ σ+ ⊗ ∆− ,
∂
≡ σ3 ⊗ σ− ⊗ ∆+ ;
∂θ2
or explicitly
0
0
θ1 ≡
0
0
0 ∆+
0
0
0
0
0
0
0
0
∂
0
0
≡
∆−
0
∂θ1
0 ∆−
0
∆+
0
0
0
0
0
0
0
0
0
0
0 ∆−
0 0
θ2 ≡
0 0
0 0
0
∂
∆
≡ +
0
∂θ2
0
where the entries are matrices N × N matrices.
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0
0
0
0
0 −∆−
0
0
0
0
0
0
0
0
0 −∆+
0
0
0
0
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Fields and Superfields
The next ingredient we need in order to construct a supersymmetric lattice
theory is a matrix representation of the fields. As usual we deal with bosonic
fields, fermionic fields and superfields, defined as follows,
• Bosonic field: a field which commutes with all θ’s and
ϕ
0
0
0
0 ∆+ ϕ∆−
ϕ̂ ≡
0
0
∆− ϕ∆+
0
0
0
∂
’s.
∂θ
ϕ ≡ N × N matrix
θiϕ̂ = ϕ̂θi
∂
∂
ϕ̂ = ϕ̂
∂θi
∂θi
0
0
0
ϕ
• Fermionic field: a field which anticommutes with all θ’s and
ψ
0
0
0
0
0
0 −∆+ψ∆−
ψ̂ ≡
0
0
−∆−ψ∆+ 0
0
0
0
ψ
∂
’s.
∂θ
ψ ≡ N × N fermionic matrix
θi ψ̂ = −ψ̂θi
∂
∂
ψ̂ = −ψ̂
∂θi
∂θi
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• Superfield: a field which commutes with all θ’s but not with
a standard expansion in powers of θ’s:
∂
’s.
∂θ
It has
Φ = ϕ̂ + θ1 ψ̂1 + θ2 ψ̂2 + θ1θ2 F̂ .
In our matrix representation it can be written as
ϕ −ψ2∆− −ψ1∆+
−F
0
∆+ ψ1
0 ∆+ϕ∆−
.
Φ=
0
0
∆− ϕ∆+ −∆− ψ2
0
0
0
ϕ
The building blocks of the model are the matrices without hat ϕ, ψ1, ψ2, F .
To describe a one dimensional lattice of size N we need to apply to Φ an
orbifold condition. The conditions over the components fields requires that,
ϕ and F are diagonal matrices while ψ1 = ∆− ψ̃1 and ψ2 = ∆+ ψ̃2 .
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Supercharges and susy transformations
The two supercharges of the N = 2 supersymmetric quantum mechanics are
given in the continuum theory by
Qi =
∂
∂
+ θi
∂θi
∂t
As we assigned to θi a shift operator corresponding to one lattice unit, the
time derivative ∂t will be associated on the lattice to the two units shift
operator ∆2± . The correspondence between continuum and lattice operators
is
∂t → N ∆2±
The supercharges on the lattice are defined without ambiguity as
!
0
0
N∆
0
∂
ˆ 2− =
Q1 =
+ N θ1 ∆
∂θ1
Q2 =
∂
ˆ 2+ =
+ N θ2 ∆
∂θ2
0
∆−
0
0
∆+
0
0
0
0
∆−
N ∆+
0
0
0
Sestri Levante 2008, Convegno Informale di Fisica Teorica
0
0
0
−
0
0
0
−∆+
N ∆−
0
0
,
0
0
−N ∆+
0
!
.
24
Q1 and Q2 satisfy the algebra of supersymmetric quantum mechanics written
in Majorana representation
ˆ 2− ,
Q21 = N ∆
{Q1 , Q2 } = 0 ,
ˆ 2+ ,
Q22 = N ∆
[Q1, 2, ∆±] = 0 .
Supersymmetry transformations are naively obtained by taking the commutator of Q1 and Q2 with Φ. However, for consistency we want the supersymmetry variations of Φ to commute with Ω, just like Φ, and also to commute
with all the θ’s. This is obtained by defining
ˆ + [Q1, Φ] ,
δ1Φ = η̂1 ∆
ˆ − [Q2, Φ] ,
δ2Φ = η̂2∆
where
1 0
0
0 −1 0
η̂1 = η1
0 0 −1
0 0
0
0
0
,
0
1
1 0
0
0 −1 0
η̂2 = η2
0 0 −1
0 0
0
0
0
0
1
and η1, η2 are odd Grassmann parameters:
ηi ψj = −ψj ηi
∀ i, j = 1, 2 .
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The susy variation of a product of two superfields follows a modified Leibniz
rule. For instance, we consider the variation under Q1 :
δ1(Φ1 Φ2) = η̂1∆+ [Q1, Φ1 Φ2 ] = (δ1Φ1 )Φ2 + (∆+ Φ1∆− )δ1Φ2
where ∆+ Φ1 ∆− is a shifted superfield. Similar expresion can be obtain for
variations under Q2 .
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By doing explicit matrix computations we obtain, in terms of component fields
δ1ϕ = η1 ∆+ ψ1 ,
δ1ψ1 = −η1 N ∆+ [∆2− , ϕ] ,
δ1F = η1 N ∆+ [∆2−, ψ2 ] ,
δ1ψ2 = −η1 ∆+ F .
and
δ2 ϕ = η2 ∆− ψ2 ,
δ2ψ1 = η2 ∆− F ,
δ2F = −η2N ∆− [∆2+ , ψ1 ] ,
δ2ψ2 = −η2N ∆− [∆2+ , ϕ] .
These susy transformations close the algebra.
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The Inconsistency Claimed
[Bruckmann and de Kok, hep-lat/0603003; Bruckmann, Catterall and de Kok, hep-lat/0611001]
claim an inconsistency in the lattice formulation:
Consider the action written in terms of the component fields is
Z
1
2
2
S =
dx{ (∂x ϕ) − F − ψ1∂x ψ1 − ψ2∂x ψ2
2
−m(iψ1 ψ2 + F ϕ) − g(3iϕ2ψ1ψ2 + F ϕ3 )}
When one applies the susy transformation to a product of fields, as for example:
δs(F (x)ϕ(x)) = (δsF (x))ϕ(x) + F (x + a)(δsϕ(x))
the result is not equal to the one when is applied to
δs(ϕ(x)F (x)) = (δsϕ(x))F (x) + ϕ(x + a)δsF (x)
and it might give wrong results.
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The action
Here our aim is to construct a matrix (lattice) action which reproduces action
of the continuum limit. Introduce the covariant derivatives D1 and D2 ,
∂
∂
ˆ 2− ,
ˆ 2+ .
D2 =
− N θ1 ∆
− N θ2∆
∂θ1
∂θ2
which anticommutes with all the supercharges and also
ˆ 2+ .
ˆ 2− ,
D22 = −N ∆
D12 = −N ∆
D1 =
(1)
A suitable candidate for our matrix action is the following
∂
∂
1
S = Tr
,
, [D1, Φ][D2, Φ] + iF (Φ)
∂θ2
∂θ1 2
This action is invariant under the supersymmetry transformations.
The action written in terms of the component fields is
Skin ∝ Tr
− N ψ1[∆2+ , ψ1] − F 2 − N 2[∆2−, ϕ][∆2+ , ϕ] + N [∆2−, ψ2]ψ2
i
+ m(−ϕF − F ϕ − ψ2ψ1 + ψ1ψ2)
2
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The kinetic term of the action written in components as
Skin = Tr − N ψ1∆2+ ψ1 + N ψ1ψ1∆2+ − F 2 − N 2∆2− ϕ∆2+ ϕ + N 2 ∆2− ϕϕ∆2+ + N 2ϕ2
− N 2ϕ∆2− ϕ∆2+ + N ∆2− ψ2ψ2 − N ψ2∆2− ψ2
i
+ m(−ϕF − F ϕ − ψ2ψ1 + ψ1ψ2)
2
where the order of the factors in each term has been preserved. On the lattice
reads (g = 0),
Skin = a
X 1 x
a2
− ϕ(x)ϕ(x + 2a) − ϕ(x)ϕ(x − 2a) + 2ϕ2(x)
− F 2 (x)
1
1
+ ψ1(x) ψ1(x − a) − ψ1(x + a) + ψ2(x) ψ2(x + a) − ψ2(x − a)
a
a
i
+ m(−ϕ(x)F (x) − F (x)ϕ(x) − ψ2(x + a)ψ1(x) + ψ1(x − a)ψ2(x))
2
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• This action has the correct continuum limit and satisfy the requirement
of reflection positivity [Osterwalder, Schrader, 1973,1975; Lüscher, 1977] which
means that is bounded, symmetric and positive operator. The positivity
is essential for the existence of a self-adjoint Hamiltonian.
• Once the reflection positivity is taking into account the point is that, the
objections raised by [Bruckmann and de Kok, hep-lat/0603003] do not have any
meaning since, although the Modified Leibniz rule (MLR) may be apply
differently in some terms when the order is interchanged, in the case of
a physical theory, there is only one way to write an action that satisfy
the reflection positivity.
• An important point is that only the action above is invariant under the
MLR.
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31
Using the Modified Leibniz rule (MLR) we can show that the lattice
action is invariant under the SUSY transformations, if one uses the
modified Leibniz rule for product of fields:
δ1(A(x)B(x)) = (δ1A(x))B(x) + A(x + a)(δ1B(x))
M LR
S = 0.
→ δ1,2
While, using the “normal” Leibniz rule (LR) we can show that the
action is not invariant under the SUSY transformations due to the
fact that the action does not satisfy the invariance
ϕA → ϕA + δϕA .
LR
S 6= 0 .
→ δ1,2
SUSY breaking
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The Ward-Takahashi identities
The question is to understand whether this exact lattice supersymmetry is
preserved at the quantum level.
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33
Definition of the source term and Ward identities
Let us define the source term needed to construct the Ward identities.
Z
SJ =
dθ1 dθ2 T r(JΦΦ) ,
Z
=
dθ1 dθ2 T r Jˆ0(ϕ̂ + θ1ψ̂1 + θ2 ψ̂2 + θ1 θ2F̂ )
Jˆ1 (θ1ϕ̂ + θ12ψ̂1 + θ1 θ2 ψ̂2 + θ12 θ2F̂ )
Jˆ2 (θ2ϕ̂ + θ2θ1 ψ̂1 + θ22 ψ̂2 + θ2θ1 θ2 F̂ )
where
Jˆ12 (θ1 θ2ϕ̂ + θ1θ2 θ1 ψ̂1 + θ1θ22 ψ̂2 + θ1 θ2θ1 θ2 F̂ )
= T r Jˆ0 + Jˆ1 ψ̂2 − Jˆ2ψ̂1 + Jˆ12 ϕ̂ ,
Φ = ϕ̂ + θ1 ψ̂1 + θ2ψ̂2 + θ1 θ2 F̂
and
JΦ = Jˆ0 + θ1Jˆ1 + θ2Jˆ2 + θ1θ2 Jˆ12 ,
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34
where
Jˆ0 =
J0
0
0
0 !
0 ∆+ J0∆−
0
0
,
0
0
∆− J0∆+ 0
0
0
0
J0
Jˆ12 =
J12
0
0
0 ∆+ J12 ∆−
0
0
0
∆− J12∆+
0
0
0
0 !
0
,
0
J12
Jˆi =
Ji
0
0
0 −∆+ Ji∆−
0
0
0
−∆−Ji∆+
0
0
0
0 !
0
,
0
Ji
where
J0 = J˜0I ,
J12 = J˜12 I ,
J1 = J˜1 ∆− ,
Sestri Levante 2008, Convegno Informale di Fisica Teorica
J2 = J˜2∆+ .
35
The source term is
SJ = T r
JJ
0
0
0
0
0
∆+ JJ∆−
0
0
∆− JJ∆+
0
0
0 !
0
,
0
JJ
where
JJ ≡ (J˜0 F + J˜1 ∆− ψ2 − J˜2∆+ ψ1 + J˜12ϕ) .
The generating functional is defined as
Z=
where
Z
DΦ exp(−Ssusy + SJ )
SJ = JϕA ϕA .
Let us make a change of variables
ϕA → ϕA + δϕA
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36
where the measure is invariant:
DϕA = D(ϕA + δϕA ))
and
Z
Z=
DϕA exp − Ssusy (ϕA + δϕA ) + SJ (ϕA + δϕA)
Z
=
DϕA exp(−Ssusy (ϕA) + SJ (ϕA)) exp(−δSsusy + JϕA δϕA )
where we defined
δSsusy ≡ Ssusy (δϕA )
then,
DϕA exp(−Ssusy + SJ )(1 − δSsusy + JϕA δϕA) −→
Z
Z = Z + DϕA exp(−Ssusy + SJ )(−δSsusy + JϕA δϕA )
Z=
that means
Zδ ≡
Z
Z
DϕA exp(−S + SJ )(−δS + δSJ ) = 0 ,
Sestri Levante 2008, Convegno Informale di Fisica Teorica
δSJ = JϕA δϕA
37
A two point Ward identity is obtained by taking the derivative on two sources
and then taking to zero all of them. For example,
δ2
δ J˜12(r)δ J˜2 (s)
which gives a two point Ward identity are:
Z
Dϕ exp(−S) − ∆+ ψ1(s)δ1ϕ(r) − ϕ(r)δ1 (∆+ ψ1(s))]
1
+ [∆+ ψ1(s)ϕ(r) + ϕ(r)∆+ ψ1(s)](δ1(LR)S) = 0 ,
2
or equivalently,
< −∆+ ψ1(s)δ1ϕ(r) > − < ϕ(r)δ1 (∆+ ψ1(s)) >
1
+ < [∆+ ψ1(s)ϕ(r) + ϕ(r)∆+ ψ1(s)](δ1(LR)S) >= 0 ,
2
Although these Ward identities are satisfied on the lattice (to tree level), they
do not represent a symmetry of the action because here δ1,2S is different from
zero.
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38
Two point Modified Ward Identities
What we propose are Modified Ward identities that mimics the modified
M LR S = 0.
Leibniz rule and uses the fact that δ1,2
< δ1M LR (∆+ ψ1(s)ϕ(r) + ϕ(r)∆+ ψ1(s)) >= 0 ,
more explicitly,
< δ1(∆+ ψ1 (s))ϕ(r) > + < (∆+ ∆+ ψ1(s)∆−)δ1ϕ(r) >
+ < δ1ϕ(r)∆+ ψ1(s) > + < (∆+ ϕ(r)∆− )δ1ψ1 (s) >= 0 .
Let us use (s → x, r → y) and ψ1 = ∆− ψ̃1
=< δ1(∆+ ψ1 (x))ϕ(y) > + < (∆2+ ψ1(x)∆− )δ1ϕ(y) >
+ < δ1ϕ(y)∆+ ψ1(x) > + < (∆+ ϕ(y)∆− )δ1 ψ1(x) >=
= − < (∆+ η1 N ∆+ [∆2−, ϕ(x)])ϕ(y) > + < (∆2+ ψ1(x)∆− )(η1∆+ ψ1(y)) >
+ < (η1 ∆+ ψ1(y))(∆+ ψ1(x)) > − < (∆+ ϕ(y)∆− )(∆+ η1 N ∆+ [∆2− , ϕ(x)]) >
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39
= η1
− N < ϕ(x)ϕ(y) > +N < ϕ(x + 2a)ϕ(y) > − < ψ̃1 (x + a)ψ̃1(y) >
+ < ψ̃1(y)ψ̃1(x) > −N < ϕ(y + a)ϕ(x) > +N < ϕ(y + a)ϕ(x + 2a) >
and in Fourier transform, we have
Z Z
1
dk dt
= η1 −
< ϕ̃(k)ϕ̃(t) > (1 − exp(2ika)) exp(ikx) exp(ity)
a
(2π) (2π)
Z Z
dp dq
<˜ψ1(p)˜ψ1(q) > exp(ipa) exp(ipx) exp(iqy)
−
Z Z (2π) (2π)
dp dq
+
<˜ψ1(p)˜ψ1(q) > exp(ipy) exp(iqx)
(2π) (2π)
Z Z
1
dk dt
−
< ϕ̃(k)ϕ̃(t) > exp(ika)(1 − exp(2ita)) exp(iky) exp(itx)
a
(2π) (2π)
substituting the propagators
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40
= η1
−
Z
Z
1
dt
exp(−ita)(exp(ita) − exp(−ita)) exp(−it(x − y))
4
2
(2π) a2 sin (ta)
2i
dq − a sin(qa)
exp(−iqa) exp(−iq(x − y))
4
2
(2π) a2 sin (qa)
2i
dq − a sin(qa)
exp(iq(x − y))
+
4
2
(2π) a2 sin (qa)
Z
dt
1
(exp(−ita) − exp(ita)) exp(−it(x − y)) = 0
−
(2π) a42 sin2(ta)
Z
due to,
(exp(ita) − exp(−ita)) = 2i sin(ta) .
All the other modified Ward-Takahashi identities that are different from zero
in the kinetic term of the action (g = 0, m = 0) are satisfied at fixed lattice
spacing (to tree level).
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41
The mass term
The mass term (for g = 0) in the action
X 1 Skin = a
− ϕ(x)ϕ(x + 2a) − ϕ(x)ϕ(x − 2a) + 2ϕ2(x) − F 2 (x)
2
a
x
1
1
+ ψ1(x) ψ1(x − a) − ψ1(x + a) + ψ2(x) ψ2(x + a) − ψ2(x − a)
a
a
i
+ m(−ϕ(x)F (x) − F (x)ϕ(x) − ψ2(x + a)ψ1(x) + ψ1(x − a)ψ2(x))
2
is also invartiant under the modifield Leibniz rule, i.e.,
M LR
Sint = 0
δ1,2
and also the two points Ward-Takahashi identities are satisfied, for example,
< δ1M LR (F (x)(∆+ ψ1(y)) + (∆+ ψ1(y))F (x)) >=
=< δ1F (x)(∆+ ψ1(y)) + (∆+ F (x)∆− )δ1(∆+ ψ1(y)) >
+ < δ1(∆+ ψ1(y))F (x) + (∆+ ∆+ ψ1(y)∆− )δ1 F (x) >= 0
Sestri Levante 2008, Convegno Informale di Fisica Teorica
42
Interaction term
Up to now we show that the two points Ward-Takahashi identities are satisfied
to tree level. Let us now check whether it holds also at one loop (g order).
The superpotential is given by
V (Φ) =
1
1
mΦ2 + gΦ4
2
4
and gives a symmetric piece:
1
g(ϕ(x)3 F (x) − ϕ(x)2 F (x)ϕ(x) − ϕ(x)F (x)ϕ(x)2 − F (x)ϕ(x)3
4
− ϕ(x)2 ψ̃2(x + a)ψ̃1(x) + ϕ(x)2 ψ̃1(x − a)ψ̃2(x) − ψ̃2(x + a)ψ̃1(x)ϕ(x)2
ψ̃1(x − a)ψ̃2(x)ϕ(x)2 − ϕ(x)ψ̃2 (x + a)ϕ(x + a)ψ̃1(x)
− ϕ(x)ψ̃2(x + a)ψ1(x)ϕ(x) − ψ̃2(x + a)ϕ(x + a)2ψ̃1(x)
− ψ̃2(x + a)ϕ(x + a)ψ̃1(x)ϕ(x) + ϕ(x)ψ̃1 (x − a)ϕ(x − a)ψ̃2(x)
+ ϕ(x)ψ̃1(x − a)ψ̃2(x)ϕ(x) + ψ̃1(x − a)ϕ(x − a)2ψ̃2(x)
ψ̃1(x − a)ϕ(x − a)ψ̃2(x)ϕ(x))
(1)
Sint
=
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A two point Ward identity is
< δ1M LR (F (x)(∆+ ψ1 (y)) + (∆+ ψ1(y))F (x)) >= 0
and can be written to order g as
< (ψ̃2(x) − ψ2(x + 2a))ψ̃1(y) >(1) − < F (x + a)(ϕ(y) − ϕ(y + 2a)) >(1)
− < (ϕ(y) − ϕ(y + 2a))F (x) >(1) − < ψ̃1(y + a)(ψ̃2(x) − ψ2(x + 2a)) >(1)
= η1
...work in progress...
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44
Conclusions
A consistent formulation of a fully supersymmetric theory on the lattice has
been a long standing challenge.
We have shown, in the simple one dimensional example of N = 2 supersymmetric quantum mechanics, that supersymmetry transformations on the
lattice can be defined without any ambiguity with the aid of the modified
Leibniz rule if the superfield formalism is consistently used.
The other problem we approached is the derivation of the Ward identities,
At least in the case of the theory without interaction (but including the mass
term) exact modified Ward identities hold, that reflect the modified Leibniz
rule of the original symmetry.
We expect that the situation of higher dimensional models will be similar.
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