The Akulov-Volkov Lagrangian, Symmetry Currents and Spontaneously Broken Extended Supersymmetry

17 Pages·0.15 MB·English

by  T. E. Clark

The Akulov-Volkov Lagrangian, Symmetry Currents
and Spontaneously Broken Extended Supersymmetry
T.E. Clark∗ and S.T. Love†
Department of Physics, Purdue University, West Lafayette, IN 47907-1396
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Abstract
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n
a Ageneralization of theAkulov-Volkov effective Lagrangian governing theself
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6 interactions of the Nambu-Goldstone fermions associated with spontaneously
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broken extended supersymmetry as well as their coupling to matter is pre-
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5 sented and scrutinized. The resulting currents associated with R-symmetry,
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2 supersymmetry and space-time translations are constructed and seen to form
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a supermultiplet structure.
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PACS number(s): 11.30.Na, 11.30.Pb, 14.80.Ly
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Typeset using REVTEX
∗e-mail address: [email protected]
†e-mail address: [email protected]
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I. INTRODUCTION
Many proposed fundamental theories of nature require there to be supersymmetry
(SUSY) in extra dimensions. When reduced to four dimensions, this results in models
exhibiting an extended supersymmetry, N 2. Since the low energy standard model of
≥
elementary particle interactions has no supersymmetry, N = 0, the question of the nature
of the chain of supersymmetry beakdown from higher N to N = 0 is quite relevant. In addi-
tion, forthose models withN = 1 supersymmetry atthe electroweak scale, thebreakdown of
SUSY from a higher N > 1 to N = 1, either in stages (partial supersymmetry breaking), or
directly, is of paramount importance. Models of partial supersymmetry breaking have been
extensively studied and require exploiting one of two possible avenues of realization. Either
the supersymmetry current algebra requires the inclusion of certain central charges, [1]- [3],
or when embedded in a supergravity model, the gravitino gauge field contains negative norm
states [4]- [6]. In this paper, we present the construction of the Akulov-Volkov effective ac-
tion [7] describing the spontaneous breakdown of N-extended supersymmetry to N = 0 [8]
[9]. This model contains only the Nambu-Goldstone fermions necessary for it to describe the
spontaneous breakdown of the extended supersymmetry. Since the supersymmetry charges
are in the fundamental representation of an accompanying (unbroken) SU(N) symmetry,
R
their Akulov-Volkov realization depends on a single scale, the common Goldstino decay con-
stant. This fact is explicitly exhibited by exploiting this non-Abelian SU(N) symmetry of
R
the N-extended SUSY algebra which manifests itself through a rescaling invariance of the
action resulting in its single scale dependence. The Noether currents associated with the chi-
ral SU(N) symmetries, the N supersymmetries and the space-time translation symmetries
R
are constructed and the supersymmetry transformations of the currents are determined. In
particular the R-currents, supersymmetry currents and the energy-momentum tensor are
shown to form the components of a supercurrent [10]- [11]. This extends previous work [12]
for N = 1 supersymmetry. Finally, the couplings of the Goldstino fields to matter and gauge
fields are delineated once again generalizing the construction of the N = 1 case [13]- [14].
2
The N-extended supersymmetry Weylspinor charges QA andQ¯A, where A = 1,2,...,N,
α α˙
obey the supersymmetry algebra (in the absence of any central charges)
QAα,Q¯Bα˙ = 2δBAσαµα˙Pµ
n o
QAα,QBβ = 0 = Q¯Aα˙,Q¯Bβ˙
n o n o
Pµ,QAα = 0 = Pµ,Q¯Aα˙ (1)
h i h i
with Pµ the energy-momentum operator.
This algebra can be realized by means of the Akulov-Volkov nonlinear transformations
[7] of spontaneously broken N-extended supersymmetry. The N Weyl spinor Goldstino
fields λ′α(x) and their hermitean conjugate fields λ¯′A(x) are defined to transform under the
A α˙
nonlinear extended supersymmetry as
δQ(ξ,ξ¯)λ′α = f ξα +Λρ(ξ,ξ¯)∂ λ′α
A A A ρ A
δQ(ξ,ξ¯)λ¯′A = f ξ¯A +Λρ(ξ,ξ¯)∂ λ¯′A, (2)
α˙ A α˙ ρ α˙
where ξα, ξ¯A are Weyl spinor N-SUSY transformation parameters, f are (for the moment
A α˙ A
2
independent) constants with dimension mass and
1
Λρ(ξ,ξ¯) = λ′ σρξ¯ ξ σρλ¯′ . (3)
if A A − A A
A
h i
[Comment on notation: Throughout this paper, a summation convention is employed in
which all repeated indices in a single term are summed over. The only exceptions are the
first terms on the right hand side of Eq. (2) and Eq. (7). In addition, we shall often supress
the indices when they are summed over. Thus, for example, ξσµξ¯ ξασµ ξ¯Aα˙.]
≡ A αα˙
Further representing the space-time translations as
δP(a)λ′α = aρ∂ λ′α
A ρ A
δP(a)λ¯′A = aρ∂ λ¯′A, (4)
α˙ ρ α˙
it is readily established that the above extended SUSY algebra is indeed satisfied by the
associated variations.
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Using these Goldstino field extended SUSY transformations, it is straightforward to gen-
eralize the Akulov-Volkov N = 1 SUSY construction to form the extended SUSY invariant
action, Γ = d4x , where
AV AV
L
R
f2
= detA (5)
AV
L − 2
where f is a scale with dimension mass2. The Akulov-Volkov vierbein A ν is the 4 4 matrix
µ ×
defined as
i ↔
A ν = δ ν + λ′ ∂ σνλ¯′ . (6)
µ µ f2 A µ A
A (cid:18) (cid:19)
Expanding the determinant, a canonically normalized kinetic term for each of the Goldstino
fields is secured after the rescalings
f
λ′ = Aλ . (7)
A f A
Since all self interactions of the Goldstinos have the functional form (in a short hand nota-
tion) f2F(λ′A∂λ¯′A, ∂λ′Aλ¯′A) = f2F(λA∂λ¯A, ∂λAλ¯A), it follows that there is really only one scale
f2 f2 f2 f2
A A
in the action and the Akulov-Volkov Lagrangian can be written as
f2
= detA (8)
AV
L − 2
where
i ↔
A ν = δ ν + λ ∂ σνλ¯ . (9)
µ µ f2 A µ A
(cid:18) (cid:19)
Since this Akulov-Volkov action is the unique extended SUSY invariant structure containing
the lowest mass dimension Goldstino self interactions, it follows that the extended SUSY
algebra is completely broken at this single scale, √f, which can be identified as the common
Goldstino decay constant [8]. That is, one cannot sequentially break the various supersym-
metries at different scales and still realize the algebra of Eq. (1). This conclusion is in accord
with a heuristic argument [15] which follows from the form of the extended SUSY algebra,
Eq. (1). If onedemands that theHilbert space ofstates bepositive definite, thenthe algebra
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dictates that it cannot be that some of the supersymmetry charges annihilate the vacuum
while others do not. As was previously pointed out [1], however, this argument is purely
formal since for spontaneously broken symmetries, the associated symmetry charges really
do not exist. In our approach, we reach an identical conclusion using very concrete effective
Lagrangian techniques which explicitly encapsulate the relevant dynamics. The result is
certainly a very strong constraint and is very different from the situation often encountered
involving multiple global symmetries which can generally be broken at different scales. Thus
we encounter yet another example [16] of how supersymmetry very tightly constrains the
allowed dynamics of a model.
Further note that in terms of the rescaled (unprimed) fields, the extended nonlinear
SUSY transformations take the form
δQ(ξ,ξ¯)λα = fξα +Λρ(ξ,ξ¯)∂ λα
A A ρ A
δQ(ξ,ξ¯)λ¯A = fξ¯A +Λρ(ξ,ξ¯)∂ λ¯A, (10)
α˙ α˙ ρ α˙
with
1
Λρ(ξ,ξ¯) = λ σρξ¯ ξ σλ¯ . (11)
A A A A
if −
h i
Clearly, the supersymmetry algebra
δQ(ξ,ξ¯),δQ(η,η¯) λα = 2iδP(ξ ση¯B η σξ¯B)λα
A − B − B A
h i
δQ(ξ,ξ¯),δQ(η,η¯) λ¯A = 2iδP(ξ ση¯B η σξ¯B)λ¯A (12)
α˙ − B − B α˙
h i
continues to be satisfied. Here the space-time variations, δP(a), of the unprimed fields are
of the same form as for the primed fields (cf. Eq. (4)).
The Akulov-Volkov Lagrangian is invariant while the Goldstino SUSY transformations
are covariant under a global SU(N) symmetry which has the properties of an R symmetry.
Under this SU(N) symmetry, the Goldstino fields transform as
R
δR(ω)λα = iλαω (Ta)B
A − B a A
δR(ω)λ¯A = iω (Ta)A λ¯B, (13)
α˙ a B α˙
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where ω , a = 1,2,...,N2 1, are the SU(N) transformation parameters and Ta denote
a R
−
the fundamental representation matrices of SU(N). It follows that
δR(ω),δQ(ξ,ξ¯) λα = δQ(ξ ,ξ¯ )λα
A R R A
h i
δR(ω),δQ(ξ,ξ¯) λ¯A = δQ(ξ ,ξ¯ )λ¯A, (14)
α˙ R R α˙
h i
which is precisely what is required of an R-symmetry. Here
ξα iξαω (Ta)B
RA ≡ B a A
ξ¯A iω (Ta)A ξ¯B. (15)
Rα˙ ≡ − a B α˙
This symmetry isalsoreflected intheextended SUSYalgebra, Eq. (1), which remains in-
variant under an SU(N) rotation of the supersymmetry charges. The QA and Q¯ transform
A
as the N and N¯ fundamental representations of this SU(N) symmetry, so that
R
Ra,QB = i(Ta)B QC
C
h i
Ra,Q¯ = iQ¯ (Ta)C , (16)
B − C B
h i
while the energy-momentum operator is R invariant:
[Ra,Pµ] = 0. (17)
It should be noted that Eq. (16) requires the Akulov-Volkov realization of the SUSY trans-
formations to form fundamental representations of SU(N) which, along with Eq. (13),
R
implies that the Goldstino decay constants must all be the same.
Combining these various results, it follows that the Akulov-Volkov Lagrangian [8] [9] of
Eq. (8) is invariant under SU(N) transformations while transforming as a total divergence
R
under SUSY and space-time translations:
δR(ω) = 0
AV
L
δQ(ξ,ξ¯) = ∂ Λρ(ξ,ξ¯)
AV ρ AV
L L
h i
δP(a) = ∂ [aρ ]. (18)
AV ρ AV
L L
Hence the action constructed from this Lagrangian is invariant under these transformations.
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II. SYMMETRY CURRENTS
Noether’s theorem can be used to construct the conserved currents associated with the
abovesymmetriesoftheaction. Firstintroducethelocalvariationbymeansofthefunctional
differential operator
δ δ
δ(x) = ζα(x) +ζ¯Aα˙(x) , (19)
A δλα(x) δλ¯Aα˙(x)
A
which allows for space-time and field dependent Weyl spinor variation parameters, ζ(x) and
ζ¯(x). When applied to the Akulov-Volkov Lagrangian, Eq. (8), this yields
f
δ(y) (x) = detA(x)(A−1) µ(x)∂x δ4(x y)Λν(ζ(x),ζ¯(x))
LAV 2 ν µ −
h i
iδ4(x y) detA(x)(A−1) µ(x) ζ (x)σν∂ λ¯A(x) ∂ λ (x)σνζ¯A(x) . (20)
− − ν A µ − µ A
h (cid:16) (cid:17)i
Further defining the variation operator,
4
δ = d xδ(x) , (21)
Z
it follows that
i ↔ ↔
δ (x) = detA(x)(A−1) µ(x) ζ (x) ∂ σνλ¯A(x)+λ (x) ∂ σνζ¯A(x) , (22)
LAV −2 ν A µ A µ
(cid:20) (cid:21)
while the local variation of the Akulov-Volkov action, Γ = d4x , is secured as
AV AV
L
R
f
δ(x)Γ = δ (x) ∂ detA(x)(A−1) µ(x)Λν(ζ(x),ζ¯(x)) (23)
AV AV µ ν
L − "2 #
which constitutes Noether’s theorem.
To extract the various currents and their conservation laws, one allows the local trans-
formation parameters ζα and ζ¯A to assume the various forms of the associated Goldstino
A α˙
transformation laws. For example, taking the explicit form to be that of a space-time trans-
lation of the Goldstino field
ζα(x) = aρ∂ λα(x)
A ρ A
ζ¯A(x) = aρ∂ λ¯A(x) (24)
α˙ ρ α˙
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and substituting into Noether’s theorem, Eq. (23), leads to the conserved Noether energy-
momentum tensor
f2
Tµ(x) = (detA(x))(A−1) µ(x). (25)
ν − 2 ν
and its conservation law
a ∂ Tµν(x) = δP(x;a)Γ . (26)
ν µ AV
with
δ δ
δP(x;a) = a [∂µλα(x) +∂µλ¯Aα˙(x) ]. (27)
µ A δλα(x) δλ¯Aα˙(x)
A
Similarly, the parameters can be chosen as
ζα(x) = iλBα(x)ω (Ta)B
A − a A
ζ¯Aα˙(x) = iω (Ta)A λ¯Bα˙(x) (28)
a B
whichcorrespondtotheformofaGoldstinoSU(N) transformation. Thistimesubstitution
R
into Noether’s theorem produces the conserved R current
2
Rµ(ω) ω Raµ(x) = Tµ(x) λ (x)ω (Ta)A σνλ¯B(x) . (29)
≡ a f2 ν A a B
h i
and its conservation law
∂ Rµ(ω) = δR(x;ω)Γ , (30)
µ AV
where the Ward identity functional differential operator describing R transformations is
given by
δ δ
δR(x;ω) = ω i(Ta)C λα(x) +i(Ta)B λ¯Cα˙(x) . (31)
a"− B C δλαB(x) C δλ¯Bα˙(x)#
Finally, the parameters can be chosen to have the form of the Goldstino nonlinear SUSY
transformations
ζα(x) = fξα +Λρ(ξ,ξ¯)∂ λα(x)
A A ρ A
8
ζ¯Aα˙(x) = fξ¯Aα˙ +Λρ(ξ,ξ¯)∂ λ¯Aα˙(x), (32)
ρ
with the ξ and ξ¯ appearing on the right hand side being the usual constant spinor SUSY
transformation parameters. So doing, the supersymmetry currents Qµ and Q¯µ are ob-
Aα Aα˙
tained from Noether’s theorem as
Qµ(ξ,ξ¯) = 2Tµ(x)Λν(ξ,ξ¯), (33)
ν
where Qµ(ξ,ξ¯) = ξαQµ (x)+Q¯µ (x)ξ¯α˙A. The current conservation equation takes the form
A Aα Aα˙
∂ Qµ(ξ,ξ¯) = δQ(x;ξ,ξ¯)Γ , (34)
µ AV
with
δ
δQ(x;ξ,ξ¯) = fξα +Λρ(ξ,ξ¯)∂ λα(x)
A ρ A δλα(x)
A
(cid:16) (cid:17)
δ
+ fξ¯Aα˙ +Λρ(ξ,ξ¯)∂ λ¯Aα˙(x) . (35)
ρ δλ¯Aα˙(x)
(cid:16) (cid:17)
The conserved currents in the effective theory are related through their SUSY transfor-
mation properties. The SUSY transformation, δQ(ξ,ξ¯) = d4xδQ(x;ξ,ξ¯), of the R current
R
produces the supersymmetry current plus a term that is a Belinfante improvement term for
the supersymmetry currents as well as additional Euler-Lagrange equation terms:
δQ(ξ,ξ¯)Rµ(ω) = Qµ(ξ ,ξ¯ )+∂ qρµ(ω,ξ,ξ¯)+Λµ(ξ,ξ¯)∂ Rρ(ω). (36)
R R ρ ρ
−
Here the ρµ antisymmetric improvement terms are defined as
qρµ(ω,ξ,ξ¯) = Λρ(ξ,ξ¯)Rµ(ω) Λµ(ξ,ξ¯)Rρ(ω). (37)
−
Notethaton-shell, thedivergenceoftheRcurrentvanishesasaconsequence oftheR-current
conservation law and the field equations.
Analogously, the Noether SUSY currents transform into the Noether energy-momentum
tensor, its Belinfante improvements, other trivially conserved (without need of field equa-
tions) antisymmetric terms and Euler-Lagrange equation terms that enter through the di-
vergence of the supersymmetry currents which again vanish on-shell:
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δQ(η,η¯)Qµ(ξ,ξ¯) = 2i ξσ η¯ ησ ξ¯ Tµν +∂ Gρµ(η,η¯,ξ,ξ¯)+Λµ(η,η¯)∂ Qρ(ξ,ξ¯). (38)
ν ν ρ ρ
−
(cid:16) (cid:17)
The ρµ antisymmetric improvement terms Gρµ are defined as
Gρµ(η,η¯,ξ,ξ¯) = Λρ(η,η¯)Qµ(ξ,ξ¯) Λµ(η,η¯)Qρ(ξ,ξ¯). (39)
−
Finally the SUSY variation of the energy-momentum tensor does not lead to another
conserved current but only to a trivially conserved antisymmetric term and the on-shell
vanishing Euler-Lagrange equation terms so that
δQ(ξ,ξ¯)Tµν = ∂ Λρ(ξ,ξ¯)Tµν Λµ(ξ,ξ¯)Tρν +Λµ(ξ,ξ¯)∂ Tρν. (40)
ρ ρ
−
h i
Hence, the currents Rµ, Qµ, and Tµν form the component currents of a supercurrent. Given
the R current, the SUSY currents and the energy-momentum tensor canbe defined by SUSY
variations of Rµ(ω) through the above transformation equations. Due do the R invariance of
the N extended SUSY algebra, all of the lower component R and SUSY currents transform
into the energy-momentum tensor. The form of the currents, their conservation laws and
multipletstructureexhibitanentirelysimilarstructuretothatfoundfortheN = 1nonlinear
SUSY theory [12].
III. MATTER AND GAUGE FIELDS
The N extended supersymmetry algebra can also be nonlinearly realized on matter (non-
Goldstino, non-gauge) fields, generically denoted by φi, where i can represent any Lorentz
or internal symmetry labels, using the standard realization [13], [14], [17]
δQ(ξ,ξ¯)φi = Λρ(ξ,ξ¯)∂ φi. (41)
ρ
The matter fields also carry a representation of the chiral SU(N) symmetry so that
R
δR(ω)φi = iω (Ra)i φj, (42)
a j
with (Ra)i constituting a set of SU(N) representation matrices. It is straightforward to
j
check that the extended SUSY algebra is obeyed by these transformations:
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