ar
X
iv
:h
ep
-th
/0
40
61
82
v2
2
7
Ju
l 2
00
4
SIT-LP-04/06
hep-th/0406182
June, 2004
More on the universality of the Volkov-Akulov action under
N = 1 nonlinear supersymmetry
Kazunari Shima ∗ and Motomu Tsuda †
Laboratory of Physics, Saitama Institute of Technology
Okabe-machi, Saitama 369-0293, Japan
Abstract
We discuss further the universality of the Volkov-Akulov (V-A) action of a
Nambu-Goldstone (N-G) fermion for the spontaneous breaking of supersym-
metry (SUSY). We show general relations between the standard V-A action
and nonlinear (NL) SUSY actions including apparently (pathological) higher
derivatives of the N-G fermion. Composite fields of the N-G fermions are
found, which transform homogeneously under NL SUSY transformations of
V-A. Consequently, we obtain NL SUSY invariant constraints which connect
our NL SUSY actions with the V-A action. The constraints are explicitly
solved and we show examples of the NL SUSY actions which are equivalent
to the V-A action.
PACS:12.60.Jv, 12.60.Rc /Keywords: supersymmetry, Nambu-Goldstone
fermion
∗e-mail: shima@sit.ac.jp
†e-mail: tsuda@sit.ac.jp

Nonlinear (NL) realization of supersymmetry (SUSY) and NL SUSY action given
by Volkov-Akulov (V-A) [1] are decribed in terms of a Nambu-Goldstone (N-G)
fermion [2] indicating the spontaneous SUSY breaking [3, 4]. The equivalence of
the V-A model of NL SUSY to various linear (L) supermultiplets [5, 6] was shown
by many authors [7]-[10]. In the relation between the NL and the L SUSY, basic
fields in the L supermultiplets are expressed as composites of the N-G fermion in
SUSY invariant way, and this fact gives deep insight towards the unification of
spacetime and matter from the viewpoint of compositeness of matter as discussed
in [11]. While, it is known that there exists a nontrivial NL SUSY higher derivative
action of the N-G fermion as is exemplified in [12]. In order to understand the
implications of the nontrivial higher derivative action of the N-G fermion, it is
useful to investigate the relation among NL SUSY actions i.e., the universality of
NL SUSY actions with the N-G fermion. Recently, this problem has been discussed
in [13] in the viewpoint of the braneworld scenario. And by heuristic arguments we
have discussed in [14] the relation between the standard V-A action and a NL SUSY
action including apparently a (Weyl) ghost field which originates from pathological
higher derivatives of the N-G fermion.
In this letter, by extending the arguments of [14] with respect to the universality
of NL SUSY actions to the cases where the order of derivatives of the N-G fermion
is higher than in [14], we discuss more general relations between the standard V-A
action and NL SUSY actions including apparently (pathological) higher derivatives
of the N-G fermion under N = 1 NL SUSY. By using the algorithmic procedure
given in [15], we find composite fields of the N-G fermions which transform homo-
geneously under NL SUSY transformations of V-A. Consequently, we obtain NL
SUSY invariant constraints which connect our NL SUSY actions with the standard
V-A action. The constraints are explicitly solved and we show examples of the NL
SUSY actions which are equivalent to the V-A action. We show that the arguments
of [14] with respect to the universality of NL SUSY actions are derived as a solution
of the constraints in the examples.
Let us begin with the brief review of the NL realization of SUSY by V-A [1].
In the N = 1 V-A model, the NL SUSY transformation law of a (Majorana) N-G
fermion ψ generated by a constant (Majorana) spinor parameter ζ is ‡,
δQψ =
1
κζ − iκ(ζ¯γ
aψ)∂aψ, (1)
where κ is a constant whose dimension is (mass)−2. The NL SUSY transformation
(1) satisfies a closed off-shell commutator algebra, [δQ(ζ1), δQ(ζ2)] = δP (v), where δP
‡In this letter Minkowski spacetime indices are denoted by a, b, ... = 0, 1, 2, 3, and we use the
Minkowski spacetime metric 12{γa, γb} = ηab = (+,−,−,−) and σab = i4 [γa, γb].
2

is a translation with a parameter va = 2iζ¯1γaζ2. Based on the invariant one-form
under Eq.(1), i.e., ωa = wabdxa = (δab − iκ2ψ¯γa∂bψ)dxa, the NL SUSY V-A action
SVA(ψ) is given by
SVA(ψ) = −
1
2κ2
∫
d4x |w|
= − 1
2κ2
∫
d4x
[
1 + taa +
1
2
(taatbb − tabtba)
−1
6
ǫabcdǫefgdtaetbf tcg −
1
4!
ǫabcdǫefghtaetbf tcgtdh
]
, (2)
where |w| = detwab and tab = −iκ2ψ¯γa∂bψ.
On the other hand, let us consider NL SUSY actions which include (patholog-
ical) higher derivatives of a N-G fermion in addition to the standard V-A action
as exemplified in [14]. Namely, we denote λ for the (Majorana) N-G fermion which
transforms into ψ in Eq.(2) through NL SUSY invariant constraints as will be shown
later, and we propose the actions S(λ) including apparently nontrivial terms with
(pathological) higher derivatives of λ,
S(λ) = SVA(λ) + [ higher derivative terms of λ ] (3)
which are invariant under the NL SUSY transformation of λ,
δQλ =
1
κζ − iκ(ζ¯γ
aλ)∂aλ. (4)
Note that the form of Eq.(4) is the same as Eq.(1).
In order to construct the NL SUSY invariant constraints between the N-G
fermions ψ and λ, we use the algorithmic procedure given by Ivanov [15] to pass to
a relevant NL SUSY theory from another one. Indeed, first we introduce the fields,
λ+
∑
n≥1
cn (iκ
1
2 )nγA∂Aλ, (5)
as the most general form of O(λ1) in terms of λ and its higher derivatives with the
arbitrary coefficients cn, where γA and ∂A are defined respectively as
γA =
n
∏
α=1
γaα = γa1γa2 · · · γan ,
∂A =
n
∏
α=1
∂aα = ∂a1∂a2 · · ·∂an (6)
3

with a1, a2, ..., an being Minkowski spacetime indices, and γA∂A means γA∂A = 6∂n.
Second we show explicitly the following finite transformations of the fields (5) for
the simplified case of cn = 1,
λ˜(ζ) =
(
1 + δζ +
1
2!
δ2ζ +
1
3!
δ3ζ +
1
4!
δ4ζ
)



λ+
∑
n≥1
(iκ 12 )nγA∂Aλ



= eδζ



λ+
∑
n≥1
(iκ 12 )nγA∂Aλ



(7)
which are generated by the NL SUSY transformations (4). Note that in Eq.(7) the
terms higher than δ4ζ vanish by means of ζn = 0 for n ≥ 5. By replacing the spinor
parameter ζ in Eq.(7) by −κψ, we finally define the composite fields,
λ˜(ψ) = λ˜0(ψ) + λ˜1(ψ), (8)
where λ˜0(ψ) and λ˜1(ψ) are the fields for the finite transformation of λ and its higher
derivatives, respectively, i.e.,
λ˜0(ψ) =
(
1 + δζ +
1
2!
δ2ζ +
1
3!
δ3ζ +
1
4!
δ4ζ
)
λ |ζ→−κψ, (9)
λ˜1(ψ) =
(
1 + δζ +
1
2!
δ2ζ +
1
3!
δ3ζ +
1
4!
δ4ζ
)
∑
n≥1
(iκ 12 )nγA∂Aλ |ζ→−κψ. (10)
The explicit form of λ˜0(ψ) becomes
λ˜0(ψ)
= λ− ψ + ηa(ψ) ∂aλ+ iκ2ηa(ψ) ψ¯γb∂aλ ∂bλ+
1
2
ηa(ψ)ηb(ψ) ∂a∂bλ
−κ4ηa(ψ) ψ¯γb∂aλ ψ¯γc∂bλ ∂cλ+
i
2
κ2ηa(ψ)ηb(ψ) ψ¯γc∂a∂bλ ∂cλ
+iκ2ηa(ψ)ηb(ψ) ψ¯γc∂aλ ∂b∂cλ+
1
6
ηa(ψ)ηb(ψ)ηc(ψ) ∂a∂b∂cλ
−iκ6ηa(ψ) ψ¯γb∂aλ ψ¯γc∂bλ ψ¯γd∂cλ ∂dλ− κ4ηa(ψ)ηb(ψ) ψ¯γc∂aλ ψ¯γd∂b∂cλ ∂dλ
−κ4ηa(ψ)ηb(ψ) ψ¯γc∂aλ ψ¯γd∂cλ ∂b∂dλ−
1
2
κ4ηa(ψ)ηb(ψ) ψ¯γc∂a∂bλ ψ¯γd∂cλ ∂dλ
−1
2
κ4ηa(ψ)ηb(ψ) ψ¯γc∂aλ ψ¯γd∂bλ ∂c∂dλ+
i
2
κ2ηa(ψ)ηb(ψ)ηc(ψ) ψ¯γd∂a∂bλ ∂c∂dλ
+
i
2
κ2ηa(ψ)ηb(ψ)ηc(ψ) ψ¯γd∂aλ ∂b∂c∂dλ+
i
6
κ2ηa(ψ)ηb(ψ)ηc(ψ) ψ¯γd∂a∂b∂cλ ∂dλ
+
1
24
ηa(ψ)ηb(ψ)ηc(ψ)ηd(ψ) ∂a∂b∂c∂dλ (11)
4

with ηa(ψ) = iκ2ψ¯γaλ. On the other hand, in order to derive the explicit form of
λ˜1(ψ), we use the Leipniz rule of derivatives for some products A1A2 · · ·Am,
∂A(A1A2 · · ·Am) =
n
∏
α=1
∂aα(A1A2 · · ·Am)
=
∑
k1+k2+···+km=n
n!
k1!k2! · · · km!
×
k1
∏
α=1
∂aαA1
k1+k2
∏
β=k1+1
∂aβA2 · · ·
n
∏
γ=k1+k2+···+km−1+1
∂aγAm, (12)
where kj ≥ 0 (j = 1, 2, ..., m) and the indices aα, aβ, ..., aγ are totally symmetrized.
According to Eq.(12), λ˜1(ψ) is given by
λ˜1(ψ)
=
∑
n≥1
inκn2 γA∂Aλ
+
∑
n≥1
in+1κn2 +2γA
∑
k1+k2=n
n!
k1!k2!
∏
α
ψ¯γb∂aαλ
∏
β
∂aβ∂bλ
−
∑
n≥1
inκn2 +4γA
∑
k1+k2+k3=n
n!
k1!k2!k3!
×


∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβ∂bλ
∏
γ
∂aγ∂cλ
+
1
2
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
∂aγ∂b∂cλ


−
∑
n≥1
in+1κn2 +6γA
∑
k1+···+k4=n
n!
k1! · · · k4!
×


∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβ∂bλ
∏
γ
ψ¯γd∂aγ∂cλ
∏
δ
∂aδ∂dλ
+
1
2
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγ∂b∂cλ
∏
δ
∂aδ∂dλ
+
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγ∂bλ
∏
δ
∂aδ∂c∂dλ
+
1
6
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγλ
∏
δ
∂aδ∂b∂c∂dλ


5

+
∑
n≥1
inκn2 +8γA
∑
k1+···+k5=n
n!
k1! · · · k5!
×



∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβ∂bλ
∏
γ
ψ¯γd∂aγ∂cλ
∏
δ
ψ¯γe∂aδ∂dλ
∏
ǫ
∂aǫ∂eλ
+
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγ∂bλ
∏
δ
ψ¯γe∂aδ∂c∂dλ
∏
ǫ
∂aǫ∂eλ
+
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγ∂bλ
∏
δ
ψ¯γe∂aδ∂dλ
∏
ǫ
∂aǫ∂c∂eλ
+
1
2


∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγ∂b∂cλ
∏
δ
ψ¯γe∂aδ∂dλ
∏
ǫ
∂aǫ∂eλ
+
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγ∂bλ
∏
δ
ψ¯γe∂aδ∂cλ
∏
ǫ
∂aǫ∂d∂eλ
+
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγλ
∏
δ
ψ¯γe∂aδ∂b∂cλ
∏
ǫ
∂aǫ∂d∂eλ
+
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγλ
∏
δ
ψ¯γe∂aδ∂bλ
∏
ǫ
∂aǫ∂c∂d∂eλ


+
1
6
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγλ
∏
δ
ψ¯γe∂aδ∂b∂c∂dλ
∏
ǫ
∂aǫ∂eλ
+
1
24
∏
α
ψ¯γb∂aαλ
∏
β
ψ¯γc∂aβλ
∏
γ
ψ¯γd∂aγλ
∏
δ
ψ¯γe∂aδλ
∏
ǫ
∂aǫ∂b∂c∂d∂eλ



. (13)
By straightforward calculations we can prove that in all orders the fields λ˜0(ψ) and
λ˜1(ψ) transform homogeneously under the NL SUSY transformations (1) and (4),
i.e.,
δζ λ˜0(ψ) = ξa∂aλ˜0(ψ),
δζ λ˜1(ψ) = ξa∂aλ˜1(ψ), (14)
so that δζ λ˜(ψ) = ξa∂aλ˜(ψ). Therefore, the constraints,
λ˜(ψ) = λ˜0(ψ) + λ˜1(ψ) = 0, (15)
are NL SUSY invariant and those explicitly give NL SUSY invariant relations which
connect our NL SUSY actions (3) including apparently (pathological) higher deriva-
tive terms of λ with the V-A action (2) described by ψ. §
§Note that if we consider the NL SUSY invariant constraint, λ˜0(ψ) = 0, then it gives the unique
solution ψ = λ by means of Eq.(11). This is just the case, [ higher derivative terms of λ ] = 0 in
Eq.(3), i.e., S(λ) = SVA(λ) = SVA(ψ).
6

Here we briefly discuss on a simple example of the NL SUSY invariant relations
derived from the constraints (15). Indeed, for the case of n = 1 in the above
discussion, i.e., for the case where the terms up to the first-order derivative of λ are
considered in the fields (5), the composite fields λ˜1(ψ) of Eq.(10) are
λ˜1(ψ) =
(
1 + δζ +
1
2!
δ2ζ +
1
3!
δ3ζ +
1
4!
δ4ζ
)
iκ 12 6∂λ |ζ→−κψ, (16)
and the constraints (15) have the following form at leading orders,
λ˜(ψ) = λ− ψ + iκ 12 6∂λ + iκ2 ψ¯γaλ ∂aλ
−κ 52 (ψ¯γa∂bλ γb∂aλ+ ψ¯γaλ ∂a6∂λ)
−κ4
(
ψ¯γaλ ψ¯γb∂aλ ∂bλ+
1
2
ψ¯γaλ ψ¯γbλ ∂a∂bλ
)
−iκ 92
(
ψ¯γa∂cλ ψ¯γb∂aλ γc∂bλ+ ψ¯γaλ ψ¯γb∂a∂cλ γc∂bλ
+ψ¯γaλ ψ¯γb∂aλ ∂b6∂λ + ψ¯γa∂cλ ψ¯γbλ γc∂a∂bλ
+
1
2
ψ¯γaλ ψ¯γbλ ∂a∂b6∂λ
)
+O(κ6)
= 0. (17)
Solving Eq.(17) with respect to ψ as a function of λ gives the NL SUSY invariant
relation,
ψ = λ+ iκ 12 6∂λ + κ 52 (λ¯γa6∂λ ∂aλ− λ¯γa∂bλ γb∂aλ)
+iκ3(6∂λ¯γa∂bλ γb∂aλ+ λ¯γa6∂λ ∂a6∂λ)
+iκ 92 (λ¯γa6∂λ λ¯γb∂aλ ∂bλ+ λ¯γa∂cλ λ¯γb∂aλ γc∂bλ) +O(κ5). (18)
Substituting this into the V-A action (2), we have the following NL SUSY action,
S(λ) = SVA(λ) +
∫
d4x
[
κ 12∂aλ¯∂aλ+
i
2
κ∂aλ¯γa2λ
]
+O(λ4) (19)
except for total derivative terms. It can be understood from the higher derivative
terms in O(λ2) of the action that Eq.(19) includes apparently a (Weyl) ghost field
in the higher derivative fermionic field theory (for example, see [16]). However,
those terms in O(λ2) do not alter the pole structure of the N-G fermion for the on-
shell amplitudes because Eq.(19) is equivalent to the standard V-A action SVA(ψ)
of Eq.(2) through the NL SUSY invariant relation (18). Eqs.(18) and (19) are just
the relations as discussed in [14] by heuristic arguments.
7

In the same way, for more general cases of n ≥ 2 where the terms up to the
second-order and the higher derivatives of λ are considered in the fields (5) and in
λ˜1(ψ) of Eq.(10), the constraints (15) can be solved unambiguously with respect to
ψ as a function of λ in all orders. By substituting the obtained general NL SUSY
invariant relations as, at O(λ1),
ψ = λ+
∑
n≥1
{(iκ 12 )n 6∂nλ+O(κn2 +2)}
= λ+ iκ 12 6∂λ− κ 6∂6∂λ + · · ·+
∑
n≥1
O(κn2 +2) (20)
into the V-A action (2), we have also the corresponding NL SUSY actions including
apparently (pathological) higher derivative terms of λ,
S(λ) = SVA(λ) +
∫
d4x


i
2
∑
n≥1
(iκ 12 )n{λ¯6∂n+1λ+ (−)n6∂nλ¯6∂λ}
+
i
2
∑
n≥1,m=1,...,n
(−)n(iκ 12 )n+m6∂nλ¯6∂m+1λ

+O(λ4)
= SVA(λ) +
∫
d4x
[
−1
2
κ 12 (λ¯2λ− ∂aλ¯γa6∂λ)
+
i
2
κ(∂aλ¯γa2λ− λ¯26∂λ − 2λ¯6∂λ)
−1
2
κ 32 (6∂λ¯26∂λ − 2λ¯2λ) + i
2
κ22λ¯26∂λ + · · ·
]
+O(λ4) (21)
which are equivalent to the standard V-A action. Our results, e.g., Eqs.(20) and
(21), are also valid for the arbitrary coefficients cn in Eq.(5).
To conclude, we have extended the arguments in [14] with respect to the univer-
sality of the NL SUSY actions with the N-G fermion to more general cases where ψ
in (2) is expanded in terms of λ in (3) and its higher derivatives at O(λ1) as Eq.(20).
In order to determine higher order terms of Eq.(20) in NL SUSY invariant way, ac-
cording to the algorithmic procedure given in [15], we have found the composite fields
λ˜(ψ) defined by the sum of Eqs.(9) and (10) (or explicitly, Eqs.(11) and (13)) which
transform homogeneously under the NL SUSY transformations (1) and (4). Conse-
quently, we have obtained the constraints (15) which connect our NL SUSY actions
(3) with the V-A action (2). In our previous work in [14], we have constructed a NL
SUSY invariant relation by heuristic arguments for the case where ψ is expanded
up to the first-order derivative of λ at O(λ1), i.e., ψ = λ + iκ 12 6 ∂λ + O(κ 52 ). We
have proved in this letter that the NL SUSY invariant relation is derived in Eq.(18)
as a solution of the constraint (17). We have also discussed in Eqs.(20) and (21)
8

that more general NL SUSY invariant relations and the NL SUSY actions including
apparently (pathological) higher derivatives of the N-G fermion can be obtained by
solving the constraints (15) with respect to ψ as a function of λ and by substituting
the solutions into the standard V-A action (2).
9

References
[1] D. V. Volkov and V. P. Akulov, JETP Lett. 16 (1972) 438; Phys. Lett. B46
(1973) 109.
[2] A. Salam and J. Strathdee, Phys. Lett. B49 (1974) 465.
[3] P. Fayet and J. Iliopoulos, Phys. Lett. B51 (1974) 461.
[4] L. O’Raifeartaigh, Nucl. Phys. B96 (1975) 331.
[5] J. Wess and B. Zumino, Phys. Lett. B49 (1974) 52; Nucl. Phys. B78 (1974)
1.
[6] P. Fayet, Nucl. Phys. B113 (1976) 135.
[7] E. A. Ivanov and A. A. Kapustnikov, Relation between linear and nonlinear
realizations of supersymmetry, JINR Dubna Report No. E2-10765, 1977, un-
published;
E. A. Ivanov and A. A. Kapustnikov, J. Phys. A11 (1978) 2375; J. Phys. G8
(1982) 167.
[8] M. Rocˇek, Phys. Rev. Lett. 41 (1978) 451.
[9] T. Uematsu and C.K. Zachos, Nucl. Phys. B201 (1982) 250.
[10] K. Shima, Y. Tanii and M. Tsuda, Phys. Lett. B525 (2002) 183; Phys. Lett.
B546 (2002) 162.
[11] K. Shima, Z. Phys. C18 (1983) 25; European Phys. J. C7 (1999) 341;
K. Shima, Phys. Lett. B501 (2001) 237;
K. Shima and M. Tsuda, Phys. Lett. B507 (2001) 260; Class. Quantum Grav.
19 (2002) 5101.
[12] S. Samuel and J. Wess, Nucl. Phys. B221 (1983) 153.
[13] T. Hatanaka and S. V. Ketov, Phys. Lett. B580 (2004) 265.
[14] K. Shima and M. Tsuda, Mod. Phys. Lett. A19 (2004) 1357.
[15] E. A. Ivanov and A.A. Kapustnikov, Theor. Math. Phys. 129 (2001) 1543.
[16] E. J. S. Villasen˜or, J. Phys. A35 (2002) 6169.
10