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NORDITA-2010-14
On the cancellation of 4-derivative terms in
the Volkov-Akulov action
A. A. Zheltukhin a,b,c∗
a Kharkov Institute of Physics and Technology,
1, Akademicheskaya St., Kharkov, 61108, Ukraine
b Fysikum, AlbaNova, Stockholm University,
106 91, Stockholm, Sweden
c NORDITA,
Roslagstullsbacken 23, 106 91 Stockholm, Sweden
Abstract
Recently Kuzenko and McCarty observed the cancellation of 4-
derivative terms in the D = 4 N = 1 Volkov-Akulov supersymmetric
action for the fermionic Nambu-Goldstone field. Here is presented a
simple algebraic proof of the cancellation based on using the Majorana
bispinors and Fiertz identities. The cancellation shows a difference
between the Volkov-Akulov action and the effective superfield action
recently studied by Komargodski and Seiberg and containing one 4-
derivative term. We find out that the cancellation effect takes place
in coupling of the Nambu-Goldstone fermion with the Dirac field.
1 Introduction
A general approach to the construction of the phenomenological Lagrangians
for the Nambu-Goldstone bosons associated with arbitrary group G, sponta-
neously broken to its subgroup H , was studied in the known papers [1],[2].
∗e-mail: aaz@physto.se
1

The Volkov’s approach [2] uses the powerful Cartan’s formalism of the ex-
terior differential ω-forms resulting in the invariant phenomenological La-
grangians of the interacting NG bosons
L = 12Sp(G
−1dG)k(G−1dG)k, G = KH, (1)
where the differential 1-forms G−1dG = H−1(K−1dK)H + H−1dH repre-
sent the vielbein (G−1dG)k, and the connection (G−1dG)h associated with
the vacuum symmetry subgroup H . The generalization of the NG boson
conception to the fermions with spin 1/2 associated with the spontaneous
breaking of supersymmetry was proposed by Volkov in [3] and their action
was consructed in [4].
The idea of the fermionic Nambu-Goldstone particles attracts much atten-
tion and was discussed in many papers. In the recent paper by Komargodski
and Seiberg [6] a new superfield formalism for finding the low-energy La-
grangian of the fermionic NG field was proposed, and its connection with the
Volkov-Akulov Lagrangian [4] was considered 1. The connection stimulates
some questions and further studies in this direction. Our interest in particular
is motivated by the Kuzenko and McCarty paper [5], where they observed
the complete cancellation among 4-derivative terms in the D = 4 N = 1
Volkov-Akulov supersymmetric action 2. This cancellation shows a differ-
ence between the VA [4] and KS [6] actions and gives rise to the question
about the minimal superfield action without 4-derivative and higher deriva-
tive terms. Another question is whether such a cancellation takes place in
the NG fermion couplings with other fields.
Here we present an independent proof of the cancellation effect [5], based
on using the Majorana bispinor representation of the D = 4 N = 1 fermionic
NG field and the corresponding Fiertz identities. We also find out that
the cancellation effect occurs in interactions of the NG fermion with other
fields. As a result, the 4-derivative and higher terms, associated with the
fermionic NG field, are absent in the minimal interaction Lagrangian. We
show that the maximal numbers of the NG fermions and their derivatives in
the minimal Lagrangian of interactions with the Dirac fields equal six and
three respectively. The 4-derivative term cancellation puts the question on
the constuction of the minimal superfield Lagrangians of the NG fermion
interactions equivalent to the Volkov-Akulov Lagrangian with couplings [4].
1Paolo Di Vecchia attracted my attention to Ref. [6]
2Sergei Kuzenko kindly informed me about Ref.[5].
2

In sections 2, 3, 4 we draw attention to the supersymmetry algebra in
the Weyl and Majorana representations, the Volkov-Akulov action and its
generalizations including the higher derivative terms. In section 5 we present
a new proof of the cancellations of 4-derivative terms in the Volkov-Akulov
action. In section 6 we find out that the cancellation effect takes place in the
NG fermion couplings with the Dirac and other fields.
2 Supersymmetry and superalgebra
The focus here is on the case of D = 4, N = 1 supersymmetry which trans-
formations are given by
θ′α = θα + ξα, θ¯′α˙ = θ¯α˙ + ξ¯α˙, x
′
αα˙ = xαα˙ +
i
2(θαξ¯α˙ − ξαθ¯α˙) (2)
in the Weyl spinor representation with xαα˙ = xmσmαα˙ 3. The Pauli matrices
σi and the identity matrix σ0 form a basic set σm = (σ0, σi) in the space of
all SL(2C) matrices. The Lorentz covariant description uses the second set
of the Pauli matrices with the upper indices σ˜m := (σ˜0, σ˜i) := (σ0,−σi)
{σm, σ˜n} = −2ηmn, Spσmσ˜n = −2ηmn, σmαα˙σ˜β˙βm = −2δβαδβ˙α˙ , (3)
where ηmn = diag(−1, 1, 1, 1). The matrices σm and σ˜m are Lorentz invariant
similarly to the tensors ηmn and the spinor antisymmetric metric εαβ with
the components ε12 = ε21 = −1. The supersymmetry generators Qα and
their complex conjugate Q¯α˙ := −(Qα)∗ have the form
Qα = ∂∂θα
− i2 θ¯α˙
∂
∂xαα˙
, Q¯α˙ = ∂∂θ¯α˙
− i2θα
∂
∂xαα˙
(4)
and form the supersymmetry algebra
{Qα, Q¯α˙} = −i ∂∂xαα˙
= 12 σ˜
α˙α
m Pm, (5)
{Qα, Qβ} = {Q¯α˙, Q¯β˙} = [Qα, Pm] = [Q¯α˙, Pm] = 0
together with the translation generator Pm = i ∂∂xm .
3We use algebraic agreements accepted in [7].
3

The supersymmetry transformations (2) and superalgebra (5) are pre-
sented in equivalent bispinor form after transition to the Majorana spinors
δθ = ξ, δθ¯ = ξ¯, δxm = −
i
4(ξ¯γmθ), {Qa, Qb} =
1
2(γmC
−1)abPm, (6)
where θ¯ = θTC with the antisymmetric matrix of the charge conjugation C
Cab =
(
εαβ 0
0 εα˙β˙
)
, Qa =
∂
∂θ¯a −
i
4(γmθ)a
∂
∂xm
. (7)
The Majorana spinors and the Dirac γ-matrices are defined as in [8]
θa =
(
θα
θ¯α˙
)
, ξa =
(
ξα
ξ¯α˙
)
, γm =
(
0 σm
σ˜m 0
)
, {γm, γn} = −2ηmn. (8)
3 The Volkov-Akulov action
To construct the phenomenological Lagrangian of the Nambu-Goldstone fermions
the elegant formalism of the invariant Cartan ω-forms [2], unified with su-
persymmetry by Volkov, was used in [4]. The supersymmetry invariant dif-
ferential ω-forms in extended superspace with the Grassmannian coordinates
θIα have the form
ωIα = dθIα, ω¯α˙I = dθ¯α˙I , ωαα˙ = dxαα˙ −
i
2(dθ
I
αθ¯α˙I − θIαdθ¯α˙I), (9)
where I = 1, 2, ..., N is the index of the internal SU(N) symmetry.
In the Majorana representation these fermionic and bosonic 1-forms are
ω = dθ, ω¯ = dθ¯, ωm = dxm −
i
4(dθ¯γmθ). (10)
The ω-forms (9) were used in [4] as the building blocks for the construction
of supersymmetric actions for the interacting NG fermions. Posssible actions
for the fermionic NG fields are constructed in the form of the wedge prod-
ucts of the ω-forms (9), forming hyper-volumes imbedded in the extended
superspace. The action candidates have to be invariant under the Lorentz
and internal (unitary) symmetries. In the case of the 4D Minkowski space
the invariant action of the NG fermions must include the factorized volume
4

element d4x. This requirement restricts the structure of the admissible com-
binations of the ω-forms. If such a combination is given by a wedge product
of the ω-forms (9) and their differentials, it should have the general number
of the differentials equals four. The conditition is satisfied by the well known
invariant [4]
d4V = 14!εmnpqω
m ∧ ωn ∧ ωp ∧ ωq, (11)
where ∧ is the wedge product symbol, that gives the supersymmetric exten-
sion of the volume element d4x of the Minkowski space. The supersymmetric
volume (11) is invariant under the Lorentz and unitary groups. It does not
contain the spinorial one-forms ωIα and ω¯α˙I , but they appear, e.g. in the
following invariant products [4]
Ω(4) = ωIα ∧ ω¯β˙I ∧ σ˜β˙αm d ∧ ωm, Ω˜(4) = εαβωIα ∧ ωJβ ∧ ω¯α˙I ∧ ω¯β˙Jεα˙β˙ , (12)
where d ∧ ωm is the exterior differential of ωm. The additional important
symmetry of the invariants (11) and (12) is their independence on the choice
of the superspace coordinate realization. It means that the four dimensional
hypersurfaces, associated with (11-12), may be parametrized by various ways.
Because the Volkov’s idea was to identify the Grassmannian θ coordinates
with the fermionic NG fields, associated with the spontaneous breaking of
supersymmetry, they must be considered as functions of x. This requirement
means transition to the non-linear realization of supersymmetry.
It explains why the pullbacks of the 4-form d4V (11) or its generaliza-
tions (12) on the 4-dimensional Minkowski subspace were proposed in [4] to
generate supersymmetric actions for the fermionic NG fields. As a result of
the observations, the differential forms ωm (10) and d4V are presented as
ωm = (δnm −
i
4
∂θ¯
∂xn
γmθ)dxn = W nmdxn, d4V = detWd4x. (13)
The identification of θ with the fermionic NG field is achieved by the change:
ψ(x) = a−1/2θ(x), where a has sense of the interaction constant [a] = L4
that introduces a supersymmetry breaking scale. This constant restores the
correct dimension L−3/2 of the fermionic field ψ(x) and the transition to ψ
in (13) and d4V yields the original Volkov-Akulov action [4]
S = 1a
∫
detWd4x (14)
5

with the 4× 4 matrix W nm(ψ, ∂mψ) defined by the following relations
W nm = δnm + aT nm, T nm = −
i
4∂
nψ¯γmψ. (15)
An explicit form of the action S (14) follows from the definition of detW
detW = − 14!εn1n2n3n4ε
m1m2m3m4W n1m1W
n2
m2W
n3
m3W
n4
m4 , (16)
where we chose ε0123 = 1. Using (15) and (16) presents S (14) in the form
S =
∫
d4x[ 1a + T
m
m +
a
2(T
m
m T nn − T nmTmn ) + a2T (3) + a3T (4) ], (17)
where T (3) and T (4) code the interaction terms of the NG fermions that are
cubic and quartic in the fermion derivative ∂mψ. The first term in (17) pro-
vides a non-zero vacuum expectation value for the VA Lagrangian, confirming
that it describes the spontaneously broken supersymmetry. In supergravity
this term associates with the cosmological term [9], [10]. The second term
reproduces the free action for the massless NG fermion ψ(x)
S0 =
∫
d4xTmm = −
i
4
∫
d4x∂mψ¯γmψ, (18)
The terms T (3) and T (4) cubic and respectively quartic in the NG fermion
derivatives were presented in [4] in the form
T (3) = 13!
∑
p
(−)pTmm T nn T ll , T (4) =
1
4!
∑
p
(−)pTmm T nn T ll T kk , (19)
where the sum
∑
p corresponds to the sum in all permutations of subindices
in the products of the tensors Tmn . The terms (19) describe the vertexes with
six and eight NG fermions.
4 Higher derivative generalizations
of the Volkov-Akulov action
The ω-form formalism [4] yields a clear geometric way to extend the VA
action by the higher degree terms in the NG fermion derivatives. In general
6

case the combinations of the ω-forms (10), admissible for the higher order
invariant actions, have to be the homogenious functions of the degree four in
the differentials dx and dψ. The latter condition guarantees the factorization
of the volume element d4x in the action integral. To restrict the number of
these invariants the minimality condition for the degree of derivatives ∂mψ
in the general action
S =
∫
d4xL(ψ, ∂mψ) (20)
was proposed in [4]. The minimality condition takes into account only the
lowest degrees of the derivatives ∂mψ in the Lagrangian and corresponds to
the low energy limit. To count the degree of ∂mψ in different invariants
observed was that these derivatives appear from the differentials dψ in the
fundamental ω-forms. From this point of view there is an important differ-
ence among the spinor and vector 1-forms (9). The spinor one-form contain
one derivative ∂mψ, but the vector form (13) either do not contain the ψ
fields at all or contain one derivative ∂mψ accompanied by ψ. As a result,
the whole number of the derivatives ∂mψ with respect to the whole number
of the fermionic NG fields is lower in the vector differential one-forms than
in the spinor ones. The invariants including the exterior differential of the ω-
forms like Ω(4) in (12) have the higher degree in ∂mψ in comparison with the
product of ω-forms themselves. The same conclusion concerns the invariant
Ω˜(4) including only the spinor forms.
Thus, the demand of the minimality of the number of the derivatives
∂mψ in S (20) will be satisfied if the admissible invariants will contain only
the vector differential one-forms ωm. The exact realization of the minimality
condition by the VA action uniquely fixes its solving the problem of the
effective action construction in the low energy limit.
5 Cancellation of 4-derivative terms in the
Volkov-Akulov action
For the case of N = 1 supersymmetry the algebraic structure of the terms
T (3) and T (4) (19) was analysed in [5] using the Weyl spinor basis. It was
observed that the terms having the fourth degree in ∂mψ and collected in
T (4) completely cancel out.
Here we consider an alternative proof of the observation using the Majo-
7

rana bispinor representation. In correspondence to representation (16) the
term T (4) (19) may be written as
T (4) = − 14!εn1n2n3n4ε
m1m2m3m4T n1m1T
n2
m2T
n3
m3T
n4
m4 = (21)
− 14!(εn1n2n3n4ψ¯
,n1
a1 ψ¯
,n2
a2 ψ¯
,n3
a3 ψ¯
,n4
a4 )(ε
m1m2m3m4γa1b1m1 γ
a2b2
m2 γ
a3b3
m3 γ
a4b4
m4 )
(ψb1ψb2ψb3ψb4),
where γabm = (Cγm)ab is a symmetric matrix in the bispinor indices (a, b =
1, 2, 3, 4) and the condenced notation ψ¯,na := ∂nψ¯a is introduced. The product
ψb1ψb2ψb3ψb4 in (21) is a completely antisymmetric spin-tensor of the maximal
rank four since of the Grassmannian nature of the spinor components ψb.
Then we find that the product may be presented in the equivalent form as
ψb1ψb2ψb3ψb4 = −(C−1b1b2C
−1
b3b4 + C
−1
b1b3C
−1
b4b2 + C
−1
b1b4C
−1
b2b3)ψ1ψ2ψ3ψ4, (22)
where the antisymmetric matric C−1 is inverse of the charge conjugation ma-
trix C (6). The representation (22) collects all spinors ψ without derivatives
in the form of a scalar multiplier. The substitution of (22) in (21) transforms
it into the sum of products of the bilinear spinor covariants
T (4) = 34! ΦΞ, Ξ := ψ1ψ2ψ3ψ4, (23)
Φ := εn1n2n3n4εm1m2m3m4(ψ¯,n1Σm1m2ψ,n2)(ψ¯,n3Σm3m4ψ,n4),
where Σmn := 12 [γm, γn] are the Lorentz transformation generators.
Taking into account the well known property of Σmn
εm1m2m3m4Σm3m4 = −2γ5Σm1m2 , γ5 := γ0γ1γ2γ3 =
(
−i 0
0 i
)
, (24)
one can present the Lorentz invariant Φ (23) in the compact form
Φ = −2εn1n2n3n4(ψ¯,n1Σm1m2ψ,n2)(ψ¯,n3Σm1m2γ5ψ,n4). (25)
Using representation (25) we shall prove the vanishing of Φ. To this end let
us recall the known Fierz relation for the Grassmannian spinors χi
(χ¯1χ2)(χ¯3χ4) = −
1
4
16
∑
N=1
(χ¯1ΓAχ4)(χ¯3ΓAχ2), (26)
8

where the 16 Dirac matrices ΓA and their inverse ΓA = (ΓA)−1, defined as
ΓA := (1, γm, Σmn, γ5, γ5γm), (27)
ΓA := (ΓA)−1 = (1, −γm, −Σmn, −γ5, −γ5γm),
form the complete basis in the space of 4×4 matrices. As a result, we obtain
Φ = 12εn1n2n3n4
16
∑
A=1
(ψ¯,n1Σm1m2ΓAΣm1m2γ5ψ,n4)(ψ¯,n3ΓAψ,n2). (28)
The r.h.s of (28) includes the products of two bilinear covariants. The second
(right) of them (ψ¯,n3ΓAψ,n2) is either symmetric or antisymmetric under the
permutation n3 ↔ n2. Only the antisymmetric covariants generated by ΓA =
(−γr,−Σrs) give non-zero contribution to (28). The first (left) covariant in
(28), corresponding to the above choice of ΓA, includes either the matrix Lv
or Lt given by the expressions
Lv = Σm1m2γrΣm1m2γ5, Lt = Σm1m2ΣrsΣm1m2γ5. (29)
Using the representation of Σm1m2 in the form Σm1m2 = (ηm1m2 +γm1γm2) we
obtain the following relations
Σm1m2ΓAΣm1m2 = 4ΓA − γm1γm2ΓAγm2γm1 , (30)
γmγrγm = 2γr, γmΣrsγm = 0
which show that
Lv = 0, Lt = 4Σrsγ5. (31)
Using the results (31) permits to present (28) in the next form
Φ = −2εn1n2n3n4(ψ¯,n1Σrsγ5ψ,n4)(ψ¯,n3Σrsψ,n2). (32)
Taking into account the symmetry property (CΣrsγ5)ab = (CΣrsγ5)ba and
changing the summation indices n3 ↔ n1 one can present the expression
(32) in the form
Φ = 2εn1n2n3n4(ψ¯,n1Σrsψ,n2)(ψ¯,n3Σrsγ5ψ,n4). (33)
The matching (25) and (33) yields the expected result
Φ = −Φ ⇒ Φ = 0, T (4) = 0 (34)
9

which proves that the 4-derivative term T (4) (21) actually vanishes in agree-
ment with the observation [5].
Thus, the maximal number of derivatives present in the Volkov-Akulov
action reduces to three and the action takes the following form
S =
∫
d4x[ 1a + T
m
m +
a
2(T
m
m T nn − T nmTmn ) +
a2
3!
∑
p
(−)pTmm T nn T ll ] (35)
with the maximal number of NG fermions in the vertices equal six.
Matching the Lagrangian (35) and the Komargodski and Seiberg La-
grangian [6], having the form
LAV = −f 2 + i∂µG¯σ˜µG+
1
4f 2 G¯
2∂2G2 − 116f 6G
2G¯2∂2G2∂2G¯2, (36)
shows their difference, because of the presence of one 4-derivative term in-
cluding eight NG fermions in (36). It hints that the KS superfield Lagrangian
might not be minimal and then the question about the minimal superfield
action arises. The second question concerns a possibility of such a type can-
cellations in the NG fermion couplings with other fields.
6 Coupling of the fermionic Nambu-Goldstone
fields with the Dirac field
Here we show that the above discussed cancellation of the 4-derivative terms
also occurs in the NG fermion couplings with other fields. It is easy to see
by the application of the general Volkov’s method [2] in the construction of
the phenomenological Lagrangian, describing the NG particles interacting
with other fields. The extension of this method, aimed at including the
supersymmetric couplings, is based on joining of the differential dχ of a
given field χ, carrying arbitrary spinor and unitary indices, to the set of
the supersymmetric ω-forms [4]. Then the above described procedure for
the minimal VA action construction, using only the ω-forms (9), may be
applied to the enlarged set of these supersymmetric one-forms. The only
restriction on the admissible χ-terms is the demand of their invariance under
the Lorentz and the internal symmetry groups. The effective actions must be
the homogenious functions of the degree four in the differentials dx, dψ and
dχ, and generally it has to restict the number of the derivatives ∂mψ to be
10

less than four. Then the considered cancellations are not relevant. However,
if dχ is absent in the couplings then the 4-derivative cancellation may take
place and will reduce the derivative ∂mψ number in the coresponding vertices.
An instructive example of the described possibility gives the N = 1 min-
imal supersymmetric coupling of the fermionic NG particle with the massive
Dirac field χ in the low energy limit [4]
S =
∫
[ i2εmnpq(χ¯γ
mdχ− dχ¯γmχ) ∧ ωn ∧ ωp ∧ ωq + (37)
mχ¯χεmnpqωm ∧ ωn ∧ ωp ∧ ωq ].
The kinetic term of the Dirac field in (37) includes the differential dχ and
the cancellation is absent here. The maximal number of the NG fermions at
this term nNGf equals six and the maximal number nNGd of their derivatives
equals three, just as in the case of the VA Lagrangian (35) after 4-derivative
cancellation. The mass term in (37) does not include dχ and respectively it
includes the supervolume form d4V (11), because of the minimality condition.
Then the cancellation effect does work and results in the same maximal
numbers nNGf = 6 and nNGd = 3 as in the kinetic term. To present (37) in
the standard notations [4] we substitute the ω-forms (13) in (37) and obtain
S =
∫
d4x[Rmm + a(RmmT nn − Rmn T nm) +
a2
2
∑
p
(−)pRmmT nn T ll + (38)
a3
3!
∑
p
(−)pRmmT nn T ll T kk +mχ¯χ detW ],
where Rmn := i2(χ¯γm∂nχ − ∂nχ¯γmχ) is the kinetic term for χ. Using the
expression for detW from (35), the mass term in (38) is presented as
mχ¯χ detW = mχ¯χ + amχ¯χ[Tmm +
a
2(T
m
m T nn − T nmTmn ) + (39)
a2
3!
∑
p
(−)pTmm T nn T ll ],
where T nm = − i4∂nψ¯γmψ in accordance with the definition (15).
The mass term (39) contains the maximal number of the NG fermions
nNGf = 6 and respectively the derivative number nNGd = 3, as a consequence
of the cancellation of 4-derivative terms. These maximal numbers nNGf = 6
11

and nNGd = 3, characterizing the structure of the interaction action (37), are
the same as for the VA action (35). The considered example shows that the
cancellation effect takes place in the supersymmetric couplings containing
the supervolume (11).
Thus, we obtain that an enough condition for the 4-derivative cancellation
in the couplings of the fermionic NG particles is the presence of d4V (11)
there. The observation puts the question on the restoration of the minimal
superfield action with couplings.
7 Discussions
Here we presented an independent algebraic proof of the cancellation of 4-
derivative terms in the D = 4 N = 1 VA action using the Majorana bispinor
representation and the Fiertz identities. The Majorana representation may
simplify investigation of such a type cancellations in the case of extended
supersymmetries and/or of the higher dimensional spaces. We observed that
the cancellation results in the difference between the Komargodski-Seiberg
superfield [6] and Volkov-Akulov [4] actions. The difference gives rise to
the question about restoration of the minimal superfield action without 4-
derivative terms which is equivalent to the VA action.
The second question arising from the cancellation concerns its presence
in the NG fermion interactions with other fields. We found out that the
cancellation occurs in the coupling of the fermionic NG field with massive
Dirac fields. It yields the maximal number of the NG fermions nNGf and their
derivatives nNGd in the interaction Lagrangian equal six and respectively
three. The maximal numbers nNGf = 6 and nNGd = 3 are the same as in the
VA action describing the NG fermion interactions between themselves. The
observation puts the question on the restoration of the minimal superfield
Lagrangian of interactions equivalent to the VA Lagrangian with couplings.
Taking into account the freedom in the NG field redefiniton and/or in the
change of the superfield constraint [6] may turn out to be enough to prove
the equivalence of the KS and VA actions.
Acknowledgments
I would like to thank Paolo Di Vecchia and Fawad Hassan for the in-
teresting discussions and critical remarks and Sergei Kuzenko for the letter
concerning the paper [5]. I am grateful to the Department of Physics of
Stockholm University and Nordic Institute for Theoretical Physics Nordita
12

for kind hospitality. This research was supported in part by Nordita.
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