We study the connection between N=2 supersymmetry and a topological bound in a two-Higgs doublet system having an SU(2)\times U(1)_Y\times U(1)_{Y^{\prime}} gauge group. We derive Bogomol'nyi equations from supersymmetry considerations showing that they hold provided certain conditions on the coupling constants, which are a consequence of the huge symmetry of the theory, are satisfied. Certain interesting limiting cases of our model are analysed

Edelstein, J D

Núñez, C

CERN Document Server

A Topological Bound for Electroweak Vortices from
Supersymmetry
Jose D. Edelstein and Carlos Nu~nez
Departamento de Fsica, Universidad Nacional de La Plata
C.C. 67, (1900) La Plata
Argentina
hep-th/9507102
Abstract
We study the connection between N = 2 supersymmetry and a topological bound in a two-Higgs-
doublet system having an SU(2)  U(1)Y  U(1)Y 0 gauge group. We derive Bogomol’nyi equations
from supersymmetry considerations showing that they hold provided certain conditions on the coupling
constants, which are a consequence of the huge symmetry of the theory, are satised.
Supersymmetric Grand Unied Theories (SUSY GUTs) have attracted much attention in connection
with the hierarchy problem in possible unied theories of strong and electroweak interactions [1, 2]. In view
of the requirement of electroweak symmetry breaking, these models necessitate an enrichment of the Higgs
sector [3], posing many interesting questions both from the classical and quantum point of view. In fact,
many authors have explored the existence of stable vortex solutions in a variety of multi-Higgs systems
[4, 5] which mimic the bosonic sector of SUSY GUTs, as it happens in the abelian Higgs model [6].
Vortices emerging as nite energy solutions of gauge theories can be usually shown to satisfy a topo-
logical bound for the energy, the so-called Bogomol’nyi bound [7]. Bogomol’nyi bounds were shown to
reflect the presence of an extended supersymmetric structure [8]-[11] - this requiring certain conditions on
coupling constants - where the central charge coincides with the topological charge. Being originated in
the supercharge algebra, the bound is expected to be exact quantum mechanically.
Since multi-Higgs models can be understood to be motivated by SUSY GUTs, Supersymmetry is a
natural framework to investigate Bogomol’nyi bounds. We shall study, then, the supersymmetric general-
ization of the SU(2)U(1)Y U(1)Y 0 model with two-Higgs rst introduced in Ref.[5]. The theory has the
same gauge group structure as that of supersymmetric extensions of the Weinberg-Salam Model that arise
as low energy limits of E6 based Grand Unied or superstring theories. In spite of being a simplied model
(in the sense that its Higgs structure is not so rich as that of Grand Unied theories), it can be seen as the
minimal extension of the Standard Model necessary for having Bogomol’nyi equations. We show that the
Bogomol’nyi bound of the model, as well as the Bogomol’nyi equations, are straight consequences of the
requirement of N = 2 supersymmetry imposed on the theory. We also show explicitely that a necessary
condition to achieve the N = 2 model implies certain relations between coupling constants that equal those
found in [5] for the existence of a Bogomol’nyi bound.
The SU(2) U(1)Y  U(1)Y 0 gauge theory in 2 + 1, introduced in Ref.[5], is described by the action
S =
Z
d3x
"
−
1
4
~W  ~W
 −
1
4
FF
 −
1
4
GG
 +
1
2
2X
q=1
jD(q) (q)j
2 − V ((1);(2))
#
(1)
where (1) and (2) are a couple of Higgs doublets under the SU(2) factor of the gauge group, A and B
are real scalar elds and ~W = W aa is a real scalar in the adjoint representation of SU(2). The specic
Consejo Nacional de Investigaciones Cientcas y Tecnicas.
1
form of the potential will be determined below. The strength elds can be written in terms of the gauge
elds A, B and ~W. The covariant derivative is dened as:
D(q) (q) =

@ +
i
2
gW a 
a +
i
2
(q)A +
i
2
(q)B

(q); q=1,2 (2)
where g is the SU(2) coupling constant while (q) and (q) represents the dierent couplings of (q) with
A and B. A minimal N = 1 supersymmetric extension of this model is given by an action which in
superspace reads:
SN=1 =
1
2
Z
d3xd2
h
ΩAΩA + ΩBΩB + Ω
a
~WΩ
a
~W
−DADA−DBDB −DW
a
DWa + 1A+ 2B
+
1
2
2X
q=1

(r(q)(q))
a(r(q)(q))
a + iy(q)
q
2
(q)
1 A+
q
2
(q)
2 B +
p
23W
aa

(q)
#
(3)
where
r(q)(q) =

D +
i
2
g[Γ ~W ; ]−
i
2
(q)[ΓA; ] +
i
2
(q)[ΓB; ]

(q): (4)
(q)  ((q);Ψ(q)) are a couple of complex doublet superelds, A  (A;A), B  (B;B) and W 
(W a; a~W )
a are real scalar superelds and ΓA  (A; A), ΓB  (B; B) and Γ ~W  Γ
a
~W
a = (W a ; 
a)a
are three spinor gauge superelds in the Wess-Zumino gauge. ΩA, ΩB and Ω
a
~W
, are the corresponding
supereld strengths. Concerning 
(q)
1 , 
(q)
2 , 3, 1 and 2, they are real constants whose signicance will
be clear below. Finally, D is the usual supercovariant derivative, D = @ + i 6@, while the γ-matrices are
represented by γ0 = 3, γ1 = i1 and γ2 = −i2. In the sake of simplicity, we shall consider congurations
with vanishing A, B and ~W 1. Then, the Higgs potential in (3) takes the form:
V ((1);(2)) =
 
2X
q=1
q

(q)
1 
y
(q)(q) −
1p
2
!2
+
 
2X
q=1
q

(q)
2 
y
(q)(q) −
2p
2
!2
+3
 
2X
q=1
y(q)
a(q)
!2
: (5)
In order to extend the supersymmetric invariance of the theory to N = 2, we consider transformations
with a complex parameter [9, 11]. We rst combine all the spinors into Dirac fermions as:
A  A − iA ; B  B − iB and 
a  a~W − i
a: (6)
The fermionic contribution to the (non-minimal part of the) interaction lagrangian can be written as
LFer;int =
2X
q=1
24(q) +
q
8
(q)
1
4
Ψ(q)A(q) +
(q) +
q
8
(q)
2
4
Ψ(q)B(q) +
g +
p
83
4
Ψ(q)
aa(q)
−
(q) −
q
8
(q)
1
4
Ψ(q) ~A(q) −
(q) −
q
8
(q)
2
4
Ψ(q) ~B(q) −
g −
p
83
4
Ψ(q) ~
aa(q)
35 ; (7)
where ~a, ~A and ~B are the charge conjugates of 
a, A and B respectively. Now, transformations
with complex parameter  are equivalent to transformations with a real parameter followed by a phase
transformation for fermions, fa;A;B;Ψ(q)g −! e
ifa;A;B;Ψ(q)g. Then, N = 2 supersymmetry
requires invariance under this fermion rotation. One can easily see from (7) that fermion phase rotation
invariance is achieved if and only if:
3 =
g2
8
; 
(q)
1 =
2(q)
8
and 
(q)
2 =
2(q)
8
: (8)
1The SU(2) U(1)Y  U(1)Y 0 theory with its full eld content is considered in Ref.[12].
2
These conditions, imposed by the requirement of extended supersymmetry, x the coupling constants
exactly as they appear in [5]. Thus, we have shown that the potential and the coupling constants of the
SU(2) U(1)Y U(1)Y 0 model are simply dictated by N = 2 supersymmetry. This result is analogous to
that recently found in the Abelian Higgs model [11].
We shall now analyse the N = 2 algebra of supercharges for our model. To construct the conserved
charges we follow the Noether method and obtain Q[]  Q+Q, with
Q = −
i
2
Z
d2x
(
yA
"
1
2
Fγ +
2X
q=1
(q)
2
y(q)(q) − 1
#
+ yB
"
1
2
Gγ +
2X
q=1
(q)
2
y(q)(q)
− 2] + 
ya
"
1
2
W aγ +
g
2
2X
q=1
y(q)
a(q)
#
− i
2X
q=1
Ψy(q)γ
D(q) (q)
)
(9)
Since we are interested in connecting the N = 2 supercharge algebra with Bogomol’nyi equations and
bound, we impose static congurations with A0 = B0 = W
a
0 = 0, and we restrict ourselves to the bosonic
sector of the theory after computing the algebra. We obtain, after some calculations
f Q;Qg = 2γ0P
0 + Z (10)
where
P 0 = E =
1
2
Z
d2x
"
1
2
(W aij)
2 +
1
2
(Fij)
2 +
1
2
G2ij +
2X
q=1
jD(q)i (q)j
2 + V ((1);(2))
#
(11)
while the central charge is given by:
Z = −
Z
d2x
"
1
2
ijFij
 
2X
q=1
(q)
2
y(q)(q) − 1
!
+
1
2
ijGij
 
2X
q=1
(q)
2
y(q)(q) − 2
!
+
g
4
ijW aij
2X
q=1
y(q)(q) + i
ij
2X
q=1
(D(q)i (q))(D
(q)
j (q))

#
=
1
2
I
Vidx
i; (12)
where Vi is given by
Vi =
 
1Aj + 2Bj + i
2X
q=1
y(q)D
(q)
j (q)
!
ij : (13)
Finite energy dictates the following asymptotic behaviour for the Higgs doublets [5]
(1)1 =
0p
2

0
exp in(1)’

; (2)1 =
0p
2

exp in(2)’
0

; (14)
where n(1) and n(2) are integers that sum up to the topological charge of the conguration m,
m  n(1) + n(2): (15)
Then, coming back to eq.(12) for the central charge, after Stokes’ theorem, we see that
Z =
I
(1Ai + 2Bi)dx
i = −420m (16)
that is, the central charge of the N = 2 algebra equals the topological charge of the conguration. It is
now easy to nd the Bogomol’nyi bound from the supersymmetry algebra (10). Indeed,
f Q;Qg =
Z
d2x
"
(a)y(a) + (A)
y(A) + (B)
y(B) +
2X
q=1
(Ψ(q))
y(Ψ(q))
#
 0: (17)
3
the lower bound being saturated if and only if a = A = B = Ψ(q) = 0. Non-trivial solutions to
these equations force us to choose a parameter with denite chirality, say +. Now, conditions
+
a = +A = +B = +Ψ(q) = 0 (18)
are nothing but the Bogomol’nyi equations of the theory:
1
2
ijFij +
2X
q=1
(q)
2
y(q)(q) − 1 = 0 ;
1
2
ijGij +
2X
q=1
(q)
2
y(q)(q) − 2 = 0 (19)
ijWij
a + g
2X
q=1
y(q)
a(q) = 0 and (D
(q)
i − iijD
(q)
j )(q) = 0: (20)
Note that, for this chiral parameter, eq.(17) implies the Bogomol’nyi bound of our model,
M  220m: (21)
Let us remark on the fact that eld congurations solving Bogomol’nyi equations break half of the
supersymmetries (those generated by −), a common feature in all models presenting Bogomol’nyi bounds
with supersymmetric extension [13]. Were we faced with an antichiral parameter, we would have obtained
antisoliton solutions with broken of the supersymmetry transformation generated by +.
The connection of our model with realistic supersymmetric extensions of the Standard model, and its
coupling with supergravity (the possible existence of string-like solutions in this last theory) remain open
problems. We hope to report on these issues in a forthcoming work.
References
[1] G.G.Ross, Grand Unied Theories, Frontiers in Physics Vol.60, The Benjamin-Cummings Publishing
Company (1984).
[2] Particle Data Book, Phys.Rev.D50(1994)1173.
[3] See for example P.Nath, R.Arnowitt and A.H.Chamseddine, Applied N = 1 Supergravity, ICTP Series
on Theoretical Physics, vol.1, World Scientic (1984) and References therein.
[4] M.A.Earnshaw and M.James, Phys.Rev.D48 (1993)5818; G.Dvali and G.Senjanovich, IC/93/63
(1993); L.Perivolaropoulos, Phys.Lett.B316 (1993)528; H.S.La, CPT-TAMU-1/93 (hep-th/9302220).
[5] G.Bimonte and G.Lozano, Phys.Lett.B326(1994)270.
[6] H.B. Nielsen and P. Olesen, Nucl.Phys.B61(1973)45.
[7] E.B.Bogomol’nyi, Sov. Jour. Nucl. Phys. 24 (1976) 449; H.de Vega and F.A. Schaposnik, Phys. Rev.
D14 (1976) 1160.
[8] E.Witten and D.Olive, Phys.Lett.78B(1978)97.
[9] C.Lee, K.Lee and E.J.Weinberg, Phys.Lett.B243(1990)105.
[10] Z.Hlousek and D.Spector, Nucl.Phys.B370(1992)143; B397(1993)173; Phys.Lett. B283(1992)75;
Mod.Phys.Lett.A7(1992)3403.
[11] J.D.Edelstein, C.Nu~nez and F.Schaposnik, Phys.Lett.B329(1994)39.
[12] J.D.Edelstein and C.Nu~nez, Phys.Rev. D55(1997)3811.
[13] C.M.Hull, Nucl.Phys.B239(1984)541.
4


A topological bound for electroweak vortices from supersymmetry

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