Vacuum Tunneling in an Electroweak Model in Extra Dimensions With an External Flux

Abstract

With the standard system for an SU(2) Higgs field in M_4 X S_1, the top and bottom component of the Higgs spinor have exactly the same coeficients for the quadratic and quadric terms. This makes the vacuum degenerate and thus there are no tunneling effects to zeroth order in radiative corrections of the vector gauge felds in the standard model with this extra dimensional geometry. However, if we include an external magnetic flux that permeates our manifold, then the top component of the Higgs spinor will have an additional term in its coefficient due to this theoretical flux with the usual charge assignments for the standard model. This extra term gives rise to two nondegenerate vacuum states for the resulting potential. We will compute the tunneling probability per unit time per unit volume between these vacuum states for the zero modes of our system as well as investigate the masses for the fields of the model using the false vacuum.

Full text

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arXiv:0904.2082v1 [hep-th] 14 Apr 2009
UCI-TR-2009-02
Vacuum Tunneling in an Electroweak Model in Extra Dimensions
With an External Flux
Aaron Roy∗
Department of Physics and Astronomy,
University of California, Irvine, California 92697-4575
Abstract
With the standard system for an SU(2) Higgs field in M4⊗S1, the top and bottom component
of the Higgs spinor have exactly the same coefficients for the quadratic and quadric terms. This
makes the vacuum degenerate and thus there are no tunneling effects to zeroth order in radiative
corrections of the vector gauge fields in the standard model with this extra dimensional geometry.
However, if we include an external magnetic flux that permeates our manifold, then the top com-
ponent of the Higgs spinor will have an additional term in its coefficient due to this theoretical flux
with the usual charge assignments for the standard model. This extra term gives rise to two non-
degenerate vacuum states for the resulting potential. We will compute the tunneling probability
per unit time per unit volume between these vacuum states for the zero modes of our system as
well as investigate the masses for the fields of the model using the false vacuum.
∗Electronic address:roya@uci.edu
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I. INTRODUCTION
In this paper we will discuss the tunneling between vacuum states in M4⊗S1[1,2,3] that
are no longer degenerate with the inclusion of an external magnetic flux. We will be following
the formalism of [4] as well as adopting the same notation, even though our problem is more
involved mathematically from that example. The Higgs potential for this paper will get an
additional contribution to the upper component of the Higgs spinor that will involve the flux
[5]. With the usual charge assignments, only the top component of the spinor is affected by
this flux since it is charged, the bottom component is unaffected since it is neutral. This
will create a local minimum in our theory as well as the usual global minimum which will
be denoted as the true vacuum.
We will compute the tunneling probability per unit time per unit volume from the false
vacuum to the true vacuum. This tunneling probability will be a function of the flux for
our model. In the limit as the flux goes to zero, the tunneling probability will go to zero
as expected, for then the vacuum becomes degenerate. Obviously there is still tunneling
between the degenerate vacuums but this is tunneling in a different context than we are
analyzing in this paper. A WKB approximative method [4] will be used to determine the
predominate portion of the tunneling probability, which involves a decaying exponential.
There is also a multiplicative constant that is a parameter [4] to the tunneling probability
formula that we will not be concerning ourselves with since it, to my knowledge, can only be
determined numerically and does not affect the overall behavior of the tunneling probability
for the numbers that will be used in this paper. In Sec II a summary of the general theory [5]
will be presented. In Sec. III the masses for the fields in the model using the false vacuum
will be determined. In Sec. IV we will discuss the Higgs potential for the tunneling and the
Lagrangian density for the Higgs field. We will also integrate out the extra coordinate by
forming the action for the model thus leaving an effective 4-D Lagrangian density in terms
of the field modes. In Sec. V the problem of calculating the probability of tunneling for the
zero mode (standard model Higgs field) between the false vacuum and the true vacuum will
be analyzed. In Sec. VI the general result for the tunneling probability will be presented as
well as a discussion of appropriate limits and then a short calculation using the upper limit
for the ratio of flux to compactification size from [5].
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II. SUMMARY OF THE THEORY
Originally from [5],
L = (DAϕ)†(DAϕ) −1
where A = 0,1,2,3,5 and as usual DA= ∂A+ igWA+i
2Tr(FABFAB) −1
4fABfAB+ µ2ϕ†ϕ −λ
2(ϕ†ϕ)2
(1)
2g′BA. The metric used was
gAB=











0 if A ?= B,
if A = B = 1,2,3,5,
−1
1 if A = B = 0
(2)
where y is the extra coordinate on the circle of radius R for our manifold. Then we added
an external flux permeating the manifold. This flux only affects the charged fields in the
model with an additional phase eiQby/Rwhere Q is the charge of the field and b =
e
¯ hc× flux
(Gaussian units). We will be using (¯ h = c = 1). Given the usual charge assignments for the

01
standard model we found φ −→
flux
eiby/R0

φ or
φ −→



fluxBφ
(3)
where B =

eiby/R0
01

. Similarly WA=
√2eiby/RW+
A
−1
1
2W3
√2W−
A
1
√2W+
2W3

A
1
A−1
e−iby/R0
A

−→

flux


1
2W3
A
1
1
√2e−iby/RW−
A
2W3
A

=

eiby/R0
WA−→
01
WA
fluxBWAB†.
01
or
(4)
For Tr(FABFAB) = Tr(FµνFµν) + 2Tr(Fµ5Fµ5) it was found that
Tr(FµνFµν) −→
fluxTr(FµνFµν) and
Tr(Fµ5Fµ5) −→
flux
1
2F3
µ5F3µ5− (∂µW−
5− ∂yW−
µ+ib
RW−
µ)(∂µW+
5− ∂yWµ+−ib
RWµ+)
+ (cubicandquadricterms) (5)
where as usual W+
A=
1
√2(W1
A− iW2
A) and W−
A=
1
√2(W1
A+ iW2
A). Then with the field
definitions,
˜W−
µ= W−
µ− Λ∂µW−
µ− β∂µW+
5
(6)
˜ W+
µ= W+
5,
(7)
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we finally had
Tr(Fµ5Fµ5) −→
flux
1
2F3
µ5F3µ5− (∂y˜ W−
µ−ib
R
˜ W−
µ)(∂y˜ Wµ++ib
R
˜ Wµ+)
+ (cubicandquadricterms)(8)
where Λ = (∂y−ib
to solve the degrees of freedom problem that arises from the charged W’s picking up a mass
R)−1and β = (∂y+ib
R)−1. The reason that these fields were redefined was
from the flux before any Higgs mechanism [5].
Now for the Higgs particle we had
(DAφ)†(DAφ) = φ†[← −
∂A+ igWA+i
2g′BA][− →
∂A− igWA−i
2g′BA]φ −→
flux
φ†B†[← −
∂A+ igBWAB†+i
2g′BA][− →
2g′Bµ][− →
2g′B5][− →
∂A− igBWAB†−i
∂µ− igWµ−i
∂y+ B†(∂yB) − igW5−i
2g′BA]Bφ
= φ†[← −
∂µ+ igWµ+i
2g′Bµ]φ
− φ†[← −
∂y+ (∂yB†)B + igW5+i
2g′B5]φ
or in terms of (6) and (7),
(DAφ)†(DAφ) −→
fluxφ†[← −
−φ†[← −
∂µ+ ig˜ Wµ+ ig∂µT +i
2g′Bµ][− →
2g′B5][− →
∂µ− ig˜ Wµ− ig∂µT −i
∂y+ B†(∂yB) − igW5−i
2g′Bµ]φ
∂y+ (∂yB†)B + igW5+i
2g′B5]φ (9)
where
˜Wµ =


1
2W3
√2˜ W−
µ
1
√2˜ W+
2W3


µ
1
µ −1
µ

and T =
R2 0
0 0
1
√2


0
βW+
5
ΛW−
5
0

. Combining the term
−φ†(∂yB†)(∂yB)φ = −φ†
b2

φ in equation (9) with the terms µ2φ†φ −λ
2(φ†φ)2, we
defined the potential,
V (φ†φ) = −φ†

µ2−b2
R2
0
0
µ2

φ +λ
2(φ†φ)2.
(10)
Minimizing the potential in equation (10), the true vacuum is ?φ?0=


0
?
µ2
λ

for the global
minimum and the false vacuum is ?φ?localminimum=



?
µ2−b2
λ
R2
0


. For more details of the
general theory please see [5].
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III. THE FALSE VACUUM
We now want to look at the masses for the particles in this theory using the false vacuum.

0
for the masses,
?
Denoting the false vacuum as
1
√2
v

where v =
g2(µ2−b2
?
µ2−b2
λ
R2
, then from (8) and (9) we find
mW=
R2)
4λ
+b2
R2
(11)
for˜ W+−
µ
and
mZ=
?
(g2+ g′2)(µ2−b2
4λ
R2)
(12)
for ZAwhere ZA=
1
√
g2+g′2(gW3
A−gBA). Note that the definitions for ZAand AAdiffer from the standard
model because of the polarization of the false vacuum. Then the mass for the remaining
A+ g′BA). The photon is massless were
AA=
1
√
g2+g′2(g′W3
Higgs field (it will be in the top component now with the false vacuum [5]) is
mh=
?
4µ2+b2
R2.
(13)
and the mass for W+−
5
is
mW5=g
2
?
µ2−b2
λ
R2
.
(14)
Also
mW
mZ
=
?
g2(µ2−b2
(g2+ g
R2) + 4λb2
′2)(µ2−b2
R2
R2).
(15)
With
−1
2F3
µ5F3µ5=
1
2(g2+ g′2)(g∂µZ5+g′∂µA5−g∂yZµ−g′∂yAµ)(g∂µZ5+g′∂µA5−g∂yZµ−g′∂yAµ)
+ (cubicandquadricterms)
and
−1
2fµ5fµ5=
1
2(g2+ g′2)(g∂µA5−g′∂µZ5+g′∂yZµ−g∂yAµ)(g∂µA5−g′∂µZ5+g′∂yZµ−g∂yAµ)
along with (9), we have for the modes
mWn=
?
g2(µ2−b2
4λ
R2)
+(n + b)2
R2
(16)
5