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arXiv: [hep-ph] 18 Apr 2007
Cosmological Symmetry Breaking, Pseudo-scale
invariance, Dark Energy and the Standard Model
Pankaj Jain and Subhadip Mitra
Physics Department, I.I.T. Kanpur, India 208016
Abstract: The energy density of the universe today may be dominated by
the vacuum energy of a slowly rolling scalar field. Making a quantum expansion
around such a time dependent solution is found to break fundamental
symmetries of quantum field theory. We call this mechanism cosmological
symmetry breaking and argue that it is different from the standard phenomenon
of spontaneous symmetry breaking. We illustrate this with a toy
scalar field theory, whose action displays a U(1) symmetry. We identify a
symmetry, called pseudo-scale invariance, which sets the cosmological constant
exactly equal to zero, both in classical and quantum theory. This
symmetry is also broken cosmologically and leads to a nonzero vacuum or
dark energy. The slow roll condition along with the observed value of dark
energy leads to a value of the background scalar field of the order of Planck
mass. We also consider a U(1) gauge symmetry model. Cosmological symmetry
breaking, in this case, leads to a non zero mass for the vector field. We
also show that a cosmologically broken pseudo-scale invariance can generate
a wide range of masses.
1 Introduction
The current cosmological observations [1, 2, 3, 4, 5] suggest that the energy
density of the universe gets a significant contribution from vacuum energy.
For a review see [6, 7]. This may be modelled by simply introducing a
cosmological constant [6, 7, 8, 9, 10, 11] or dynamically by a scalar field
slowly rolling towards the true minimum of the potential [12]. If we assume
the existence of such a field then it implies that in the current era its lowest
energy state is not the true vacuum state of the theory. In order to study
the spectrum of this theory one needs to make a quantum expansion around
a time dependent field. This has interesting implications for the physics of
such models, not explored so far in the literature. In particular since we
are expanding around a time dependent field and not the ground state, the
resulting physics need not display the symmetries of the original lagrangian.
Hence this may provide us with another method of breaking fundamental
symmetries. We point out that in general a theory displays the symmetries
of the action provided the ground state of the theory is symmetric under
the corresponding transformations. In the present case, however, the ground
state is irrelevant since the field never reaches this state. The time dependent
solution, around which we need to make the quantum expansion, need not
display the symmetries of the original action. Hence the theory may display
broken symmetries over a large period in the life time of the universe.
2 Symmetry Breaking for a Scalar Field
We consider a simple model of a complex scalar field with a global U(1)
L[(x)] = ∂μ∗(x)∂μ(x) − m2∗(x)(x) − λ(∗(x)(x))2. (1)
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This Lagrangian has global U(1) symmetry, i.e., it is invariant under (x) →
ei(x). So far we have neglected the effect of the background metric which
will also be included later. We assume that the potential is sufficiently gentle,
with mass parameter sufficiently small, such that the scalar field is slowly
rolling towards its true minimum. Let η(t) be the classical solution to the
equations of motion. Here we assume that this solution is independent of
space and is in general complex.
We split (x) into two parts: (x) = η(x) + φ(x). Here η(x) is the
classical solution of the equation of motion. Let us assume the classical
solution depends only on time but is in general complex, i.e, η(x) = η(t).
Hence (x) = η(t) + φ(x) and ∗(x) = η∗(t) + φ∗(x).
L[η, φ] = η˙∗ η˙ + η˙∗ ˙φ + η˙φ˙∗ + ∂μφ∗∂μφ − m2(φ∗φ + ηφ∗ + η∗φ + η∗η)
− λ{(φ∗φ)2 + (ηφ∗)2 + (φη∗)2 + (η∗η)2 + 2(ηφ∗ + η∗φ)φ∗φ
+ 4η∗ηφ∗φ + 2η∗η(η∗φ + ηφ∗)}. (2)
&From this we identify the classical Lagrangian.
LClassical[η(t)] = η˙∗ η˙ − m2η∗η − λ(η∗η)2. (3)
The classical field η satisfies the equation of motion,
¨η + η(m2 + 2λη∗η) = 0. (4)
If we assume that the quantum fluctuations die sufficiently fast at t = ±∞,
then we can write
Z d4x ˙ η(˙φ + φ˙∗) = −Z d4x ¨η(φ + φ∗) = Z d4x η(m2 + 2λη2)(φ + φ∗).
Hence we get,
L = LClassical[η(t)] + ∂μφ∗∂μφ − m2φ∗φ
− λ{(φ∗φ)2 + (ηφ∗)2 + (η∗φ)2 + 2φ∗φ(ηφ∗ + η∗φ) + 4η∗ηφ∗φ}. (5)
Let η = η0ei0 and φ(x) = (φ1 + iφ2)/√2. This gives
L = LClassical[η(t)] +
∂μφ1∂μφ1 +
∂μφ2∂μφ2 −
− φ22
) cos 2θ0 + 4φ1φ2 sin 2θ0 + 4(φ21
)2 + 4η0(φ21
)(φ1 cos θ0 + φ2 sin θ0)]. (6)
It is clear that the two modes do not have the same mass. For a complex
classical solution, i.e. with θ0 6= 0, the two modes are coupled. They can be
decoupled by the rotation in internal space


cos β −sin β
sin β cos β 


We assume adiabaticity and hence we ignore the time dependence of rotation
matrix in the kinetic energy term. The rotation angle β is found to be equal
to θ0 and the two mass eigenvalues are found to be m2+6λη2
0 and m2+2λη2
Hence the two modes pick up a different mass and the symmetry is broken.
We do not find a zero mode, in contrast to the case of spontaneous symmetry
breaking. We call this phenomenon Cosmological Symmetry Breaking.
3 Gravitational Background
We next consider the complex scalar field model in a background gravitational
field, which would be considered classically. The action may be written
S = Z d4x√−g hg∂∗(x)∂(x) − m2∗(x)(x) − λ(∗(x)(x))2i. (8)
We assume the FRW background metric with the Hubble parameter H(t)
and the expansion factor R(t). This model also displays the global U(1)
symmetry (x) → ei(x). We again write (x) = η(t)+φ(x), where η(t) is
a space independent solution to the classical equations of motion. The real
and imaginary parts η1 and η2 satisfy
∂ηi
for i = 1, 2. Here the potential V (η) = m2η∗η + λ(η∗η)2. The model again
displays symmetry breaking as long as η(t) is different from zero. This is
allowed as long as the conditions for slow roll is satisfied. The Hubble parameter
is determined by the entire matter content of the universe and here
we shall consider it as an independent function of t. We shall assume it to
be approximately constant as is the case for a vacuum dominated universe.
The slow roll condition is satisfied if the mass parameter m << 3H. We
may expand the potential around the classical solution and find that the
mass spectrum is same as found in the earlier section. We, therefore, find
a squared mass splitting 4λη(t)2, whose scale is determined by the Hubble
parameter. Since the splitting is determined by the value of the Hubble parameter,
its value in the current era is quite small. However the splitting
need not be small in the early universe. At that time it may lead to large
observable consequences.
Although the mass splitting is very small, the value of the field η(t) can
be large. This depends on our choice of the coupling λ. By choosing λ
sufficiently small we can make η(t) arbitrarily large and still maintain slow
roll conditions. The non-zero value of this field can lead to a wide range of
breakdown of symmetries, including Lorentz invariance. Lorentz invariance
is broken because the classical field only has time dependence. Furthermore
if gauge the U(1) invariance then the gauge symmetry will be broken. The
mass of the gauge field in this case depends on η(t) and can be quite large.
4 Dark Energy
So far we have considered a slowly rolling complex scalar field and shown
that it leads to breakdown of symmetries of the original lagrangian. We
next consider the possibility that the complex scalar field itself leads to dark
energy. We determine the range of allowed values for the background scalar
field in order that it leads to vacuum energy equal to the observed dark energy
density. For this purpose we consider the equation of motion in gravitational
background in terms of the two real fields η1 and η2. For orders of magnitude
estimate we may assume η1 &#8764; η2. The analysis is easily modified if this is not
the case and does not lead to any essential difference. For slow roll, the second
derivative term is negligible. We may consider two separate cases where either
mass term or the quartic coupling term in the potential dominates. If the
mass term dominates then the slow roll condition is satisfied if m2 << 9H2. If
the quartic coupling term dominates then we find the condition λ <> ρ1/2
V /H ≈ s 3
where MPL is the Planck mass. Here we have used the fact that the vacuum
energy density is almost equal to the critical energy density. Hence we find
that in our model the value of the slow roll scalar field has to be of the order
of the Planck mass or higher.
We point out that the value of the coupling constant λ turns out to be
extremely small in our model for dark energy. This by itself need not lead to
a fine tuning problem since we are free to choose a value for this parameter.
Indeed a parameter as small as this is expected due to the widely different
scales of Planck mass and the Hubble constant. The fine tuning problem
may arise if at higher orders we need to adjust this parameter to very high
accuracy. This may happen if it undergoes large quantum corrections. This
is an important check of the theory which we shall address in detail in a
future publication.
5 Gauge Theory
We next gauge the U(1) symmetry considered in the earlier sections. The
resulting Lagrangian, in the presence of background gravity, can be written
S = Z d4x√&#8722;g g(D(x))&#8727;D(x) &#8722;
ggFF &#8722; m2&#8727;(x)(x)
&#8722; λ(&#8727;(x)(x))2i (11)
where D = &#8706; &#8722; igA is the covariant derivative and F
the field strength
tensor of the U(1) gauge field A. This Lagrangian is invariant under (x) →
ei(x)(x). We can parameterize (x) by:
(x) = (η0 + ρ/√2) exp "i
+ . . .!ei0 .
Hence, for small oscillations ρ(x) and σ(x) are φ′
1(x) and φ′
2(x) respectively.
We define new fields,
′ = exp "&#8722;i
√2η0!# = η0 + ρ/√2,
Bμ = Aμ &#8722;
If we neglect the gravitational field, the Lagrangian becomes:
L = (Dμ(x))&#8727;(Dμ(x)) &#8722;
4FμFμ &#8722; m2(η0 + ρ/√2)2 &#8722; λ(η0 + ρ/√2)4.
We now follow the same procedure as before. We split the gauge field
into two parts - i) the classical gauge field, βμ(t) which depends only on time
and ii) Bμ(x), the quantum field i.e., Bμ(x) = βμ(t)+Bμ(x). So the classical
Lagrangian becomes:
LClassical = η˙2
0 + g2βμβμη2
0 &#8722; m2η2
0 &#8722; λη4
where fμ = &#8706;μβ &#8722; &#8706;βμ. The equations of motion are:
β0 = 0, (12)
¨ βi = &#βiη2
¨ η0 = &#8722;η0(g2~β2 + m2 + 2λη2
If we assume βμ = 0, then using these equations and dropping the total
derivative terms one can rewrite the Lagrangian as:
L = LClassical +
&#8706;μρ&#8706;μρ + g2B2
+ √2ρη0!
&#8722; (m2 + 6λη2
(ρ4 + 4√2ρ3η0) &#8722;
FμFμ (15)
where Fμ = &#8706;μB &#8722; &#8706;Bμ. The gauge field has acquired a mass, mB = gη0.
Hence the gauge invariance is broken in this theory. In the simplest case,
discussed in the last section, this mass will be of the order of Planck mass,
assuming gauge coupling of order unity and if we require that the scalar field
vacuum energy gives dominant contribution to dark energy. However if we
do not impose the condition that the vacuum energy associated with the field
 is equal to the observed vacuum energy, then the mass of the gauge field
is an independent parameter which can be fixed by a suitable choice of the
classical solution. In section 7 below we also provide another generalization
of the lagrangian so as to generate a different mass scale.
6 Pseudo-scale Invariance
We have so far introduced a new method of breaking symmetries of a field
theory. The procedure is found to naturally lead to dark energy in the form
of the vacuum energy of a slowly rolling scalar field. However so far we
not addressed the question of why the cosmological constant is so small.
The problem is ofcourse well known. Quantum field theory in general produces
cosmological constant many orders of magnitude larger than what is
observed. In the absence of any symmetry, which may demand absence of
cosmological term in the action, this is a serious problem in fundamental
physics. In this section we identify a symmetry which eliminates cosmological
constant both at classical and quantum level.
We first consider actions which are scale invariant. It is clear that at the
classical level scale invariance eliminates all dimensionful parameters from
the action, including a cosmological constant term. However in the quantum
theory, it is well known that scale invariance is anomalous and hence may
generate a cosmological constant. In very interesting papers Cheng [13] and
Cheng and Kao [14] have argued that scale transformations can be broken
into a general coordinate transformation and what is refered to as the pseudoscale
transformations. Under the pseudo-scale transformations
gμ → gμ/2
Aμ → Aμ. (16)
The matter part of the action is invariant under this transformation. The
gravitational action is not invariant but, as explained in Ref. [13, 14], it can
be easily generalized so that it is invariant. One simply replaces [14],
R → β&#8727;R (17)
where G is the gravitational constant, R the Ricci scalar and β a dimensionless
constant. Next assuming a slow role scalar field discussed in earlier
sections, with its value of order the Planck mass, we find that the resulting
action will have predictions identical to Einstein’s gravity, at leading order.
The cosmological constant term is not invariant under pseudo-scale invariance
and hence eliminated in the classical action. The theory now has exactly
zero cosmological constant as long as the symmetries of the action are not
The pseudo-scale invariance is, however, broken through our cosmological
symmetry breaking mechanism, discussed earlier. Hence this mechanism will
generate a nonzero cosmological constant, or dark energy. We can directly
borrow the results obtained in section 4 with the mass parameter m set to
zero. The theory discussed in section 4 now displays pseudo-scale invariance,
besides invariance under the U(1) transformations. Both of these symmetries
are broken cosmologically and the slow roll condition, along with the value
of the observed dark energy density sets the mass scale of the classical scalar
field of the order of Planck mass.
We next consider a regulated action, assuming dimensional regularization.
We first focus only on scalar fields. In this case we consider the scalar field
action in n dimensions,
S = Z dnx√&#8722;g hg&#8706;&#8727;(x)&#8706;(x) &#8722; λ(√&#8722;g)(n&#8722;4)/n(&#8727;(x)(x))2i. (18)
The action is invariant under the transformation
 → /a(n)
gμ → gμ/b(n) (19)
where b(n) = 4a(n)/(n &#8722; 2) and we may choose a(n) to be any function
of n. This generalizes the pseudo-scale transformations to n dimensions.
The action displays exact symmetry under pseudo-scale transformations in
n dimensions. However the action has general coordinate invariance only in
4 dimensions. In dimensions other than 4 the potential term violates general
coordinate invariance. Here we take the point of view that the fundamental
quantum theory may obey a more general transformation law rather than
general coordinate invariance. We are guided primarily by data and the
absence of general coordinate invariance in dimensions other than four will
give modified predictions only at very high energy scale of the order of Planck
mass. At this scale the theory is so far untested and we cannot rule out our
To summarize, we find that we can impose pseudo-scale invariance as an
exact symmetry. The symmetry is not anomalous. This symmetry prohibits
us to introduce a cosmological constant, both at the classical and quantum
level. Hence it may provide an explanation for why the cosmological constant
is so small.
7 Generating Masses
The basic problem of generating realistic masses of the observed particles,
however, still remains in our theory. Pseudo-scale invariance prohibits any
mass terms in the action. Hence the standard Higgs mechanism is not applicable
and all the Standard Model fields, for example, will remain massless.
The pseudo-scale invariance is ofcourse broken cosmologically and hence one
may expect that we may be able to break the standard model gauge symmetry
also by a slowly rolling scalar field. However we have to do this such
that the mass of the scalar field is sufficiently large and not ruled out experimentally.
In the construction so far, the mass of the scalar field has been
found to be very small. One possibility is that this scalar boson is eliminated
from the spectrum by gauging the pseudoscale invariance [13, 14]. In this
case the Higgs boson will be eliminated from the spectrum. An alternate
construction, which does not involve gauging the pseudo-scale invariance, is
described below.
We next construct a toy model such that, besides generating dark energy,
it also breaks another U(1) symmetry with a sufficiently large mass of the
scalar field. This is a toy model which can be generalized to construct an
acceptable Standard Model of particle physics. We consider a model with
two complex scalar fields  and
. Here  will be considered as a slowly
rolling field which gives rise to dark energy. We construct an action such
that it is invariant under the transformation  → ei and
well as the pseudo-scale transformations. Here we restrict ourselves to four
dimensions.
S = Z d4x√&#8722;g"g&#8706;&#8727;(x)&#8706;(x) + g&#8706; &#8727;(x)&#8706;
(x) &#8722; λ(&#8727;(x)(x))2
&#8722; λ1( &#8727;
&#8722; λ22
&#8727;)2#. (20)
Here λ1 is taken to be of order unity. Its precise value will be fixed by the mass
particle. The coupling λ2 << 1 and will be fixed by the magnitude
of the classical solution of the field
, which is eventually determined by the
scale of symmetry breaking of the U(1) group of transformation
We again expand these two fields as  = η(t) + φ and
= ζ(t) + ψ,
where η(t) and ζ(t) are the time dependent classical fields and φ and ψ are
the quantum fluctuations. The classical fields satisfy,
2)ηi + λ1λ22
2 &#8722; λ22
2)]ηi = 0, (21)
2 &#8722; λ22
2)]ζi = 0. (22)
We consider a slow roll solution such that all the second derivative terms
are negligible. We set ζi = (1 + δ)λ2ηi, where δ << 1. We can determine δ
perturbatively by solving the differential equations. As we have seen earlier,
slow roll condition requires that λ << H2/η2. Here we have an additional
term proportional to λ1 in the equation of motion for ηi, eq. 21. Substituting
ζi = (1 + δ)λ2ηi in eq. 22, we get an estimate of dηi/dt. Substituting this in
eq. 21 we find
We want to choose λ2 such that ζi is of order of theWeinberg Salam symmetry
breaking scale. It is clear that in this case δ << 1, which is required for the
self consistency of the perturbative solution.
We can now determine the contribution of the term proportional to λ1 to
the equation for η1. We find that it gives a contribution of order λ22
2)ηi, which is much smaller compared to leading order term λ(η2
Hence the term proportional to λ1 can be treated perturbatively. We also
find the λ1 term gives a correction of order (λ/λ1)λφ4 to the vacuum energy.
Since λ/λ1 << 1, this correction is negligible. We, therefore, find that the
new term in the lagrangian, proportional to λ1, can be ignored at the leading
order in the equation of motion for η1 and also gives negligible correction to
the vacuum energy density. Furthermore the complete solution in its presence
can be determined by treating this term perturbatively.
Finally we estimate the mass of the
particle. For this we expand the
potential in terms of the fields φ and ψ and collect terms which are second
order in these fields. We find
V = λ(&#8727;)2 + λ1( &#8727;
&#8722; λ22
&#8727;)2
+ 2η1η2φ1φ2 +
(η1φ1 + η2φ2)2 &#8722; δλ1λ42
+ λ1[(ζ1ψ1 + ζ2ψ2)2 + δλ22
(ζ1ψ1 + ζ2ψ2)(η1φ1 + η2φ2)] + . . . (24)
where we have only displayed the quadratic terms in the fluctuations. We
now need to diagonalize the mass matrix. The form of these terms suggests
that we define the fields
φ+ = φ1 cos θ0 + φ2 sin θ0
φ&#8722; = &#8722;φ1 sin θ0 + φ2 cos θ0
ψ+ = ψ1 cos θ0 + ψ2 sin θ0
ψ&#8722; = &#8722;ψ1 sin θ0 + ψ2 cos θ0 (25)
where cos θ0 = η1/qη2
2 and sin θ0 = η2/qη2
2. In terms of the rotated
fields the potential can be written as
λ + (1 &#8722; δ)λ1λ42
2 &#8722; δλ1λ42
+ λ1(1 + 3δ)λ22
+ δλ1λ22
(1 + δ)φ+ψ+)(η2
We find that the states φ+ and ψ+ mix with one another. The mass matrix
can be diagonalized and we find that the four states
+ = ψ+ cos θ1 &#8722; φ+ sin θ1
+ = ψ+ sin θ1 + φ+ cos θ1 (27)
ψ&#8722; and φ&#8722; with mass squared eigenvalues, 2λ1λ22
2), 3λ(η2
2), λ(η2
2) and λ(η2
2) respectively. The mixing angle θ1 << 1 is approximately
equal to λ2. We find one particle with relatively large mass of order
2). By adjusting λ2 we can choose this to be of order of 100
GeV and hence can model the Higgs particle. The remaining three particles
have very small masses. If we gauge one of the U(1) symmetry then one of
these particles will be eliminated from the spectrum and will instead give rise
to a massive gauge boson. The theory then predicts two very light weakly
coupled particles.
In this section we have considered a toy model which illustrates that we
can generate any mass scale by cosmological symmetry breaking in a theory
with pseudoscale invariance. The precise gauge group used U(1)×U(1) is not
essential for this purpose and the construction can be easily generalized to the
standard model. It is ofcourse important to check that quantum corrections
do not lead to acute fine tuning problems. We postpone this necessary check
to future research. We point out that an alternative to the construction in
this section is to simply gauge the pseudo-scale invariance [13, 14, 15]. This
eliminates the Higgs boson from the particle spectrum and instead predicts
a new vector boson with mass of the order of Planck mass.
8 Conclusions
We have shown that a slowly rolling solution to scalar field theories, leads
to breakdown of symmetries of the action. We call this phenomenon cosmological
symmetry breaking and show that it is intrinsically different from
spontaneous symmetry breaking. We argue that if we impose pseudo-scale
invariance on the action then it sets the cosmological constant to zero both
in the classical and the quantum theory. The pseudo-scale invariance is also
broken cosmologically, leading to a slowly varying cosmological constant. We
further show that cosmologically broken pseudo-scale invariance can lead to
a wide range of particle masses and it appears possible to impose this symmetry
on the full action of fundamental particle physics.
Acknowledgements: We thank S. D. Joglekar for useful discussions.
References
[1] P. M. Garnavich et. al., ApJ 509, 74 (1998).
[2] S. Perlmutter et al., ApJ 517, 565 (1999).
[3] J. L. Tonry et al., ApJ 594, 1 (2003).
[4] B. J. Barris et al., ApJ 602, 571 (2004).
[5] A. G. Riess et al, ApJ 607, 665 (2004).
[6] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003).
[7] T. Padmanabhan, Phys. Rep. 380, 235 (2003).
[8] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
[9] S. M. Caroll, W. H. Press and E. L. Turner, Ann. Rev. Astron. Astrophys.
30, 499 (1992).
[10] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 2000.
[11] J. R. Ellis, Phil. Trans. Roy. Soc. Lond. A 361, ).
[12] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80,
[13] H. Cheng, Phys. Rev. Lett. 61, ).
[14] H. Cheng and W. F. Kao, MIT preprint Print-88-); W. F.
Kao, Phys. Lett. A 154, 1 (1991).
[15] D. Hochberg and G. Plunien, Phys. Rev. D 43, ); W.R.
Wood and G. Papini, Phys. Rev. D 5, ); J. T. Wheeler,
J. Math. Phys. 39, 299 (1998); A. Feoli, W.R. Wood, G. Papini, J.
Math. Phys. 39, ); M. Pawlowski, Turk. J. Phys. 23, 895
(1999); H. Nishino and S. Rajpoot, hep-th/0403039; D. A. Demir,
Phys. Lett. B 584, 133 (2004); R. Foot, A. Kobakhidze and R. R.
Volkas, hep-ph/.
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